While well-intentioned, we believe that many of the changes proposed by the CMF are deeply misguided and will disproportionately harm under-resourced students. Adopting them would result in a student population that is less prepared to succeed in STEM and other 4-year quantitative degrees in college. The CMF states that 'many students, parents, and teachers encourage acceleration beginning in grade eight (or sooner) because of mistaken beliefs that Calculus is an important high school goal.'
The updated CMF looks better, but I just don't see how an educator who knows math or how to teach math could come to such a conclusion (that Calculus should not be a goal). If it is well-intentioned, what was the intention... to dumb down math in high school? Perhaps we need to educate those who are coming up with the math frameworks in math and science, or to get people who care on the California Department of Education?
> If it is well-intentioned, what was the intention... to dumb down math in high school?
Bingo. It's well intentioned, but the intentions aren't to ensure that America can keep up with a rising China.
It's shocking to me that people in California aren't more worried about this. About 15 years ago, I was talking to an engineer at Juniper/Cisco. We were joking about how Huawei had copied one of their router designs down to the silk screened assembly instructions (in English!) on the PCBs. Fast forward to today, Huawei is making fully custom equipment down to state of the art switch and router chips, and Chinese companies are white boxing lower end products made by American brands.
There's a big bet out there that the U.S. can survive on software and social media alone. I would think the success of Tik Tok would have blown even that rationalization out of the water.
On the general point of U.S. math education: my cousin who lives in a nice California suburb was complaining that the math education her early high school student is receiving is several grade levels behind what she got--in Bangladesh. My mom, who also went to school in Bangladesh (in the 1960s!) was deeply unhappy about the math education in our affluent Virginia suburb, until I got into a top STEM magnet high school. My own kids go to an expensive private school, but are still getting math tutoring on the side. Math is just a shockingly low priority for Americans.
In the 1960s, it was the USSR. In the 1980s it was Japan. Now it's China.
I'm not trying to suggest that the US is fine and we shouldn't fix anything, but if you look at the world by comparing test scores and grade levels in mathematics, you're going to come to some very warped perceptions about what is important. I'm speaking as someone passionate about STEM education, who got a B.S. in mathematics.
The whole situation is warped. The USA accounts for 4% of the world population, and 40% of the top 100 universities in the world. That's fucking weird. I don't have an explanation for it. I'm just saying that the different signals we use for evaluating how good our education system is functioning are giving us radically different pieces of feedback, and our understanding needs to be correspondingly sophisticated.
There are all these narratives about how China is going to eat our lunch (like Japan in the 1980s, or the USSR in the 1960s) and while I don't feel comfortable betting on long-term US hegemony, and while I do think we should put more work into our mathematics education, I do think that looking at the world through high-school mathematics test scores is going to give you anxiety more than it's going to give you an accurate picture of what are problems really are.
To take another statistic into account, there are actually many STEM graduates in the US. What do we do with this information? How do we change our policies? It's unclear.
We lost entire industries of good paying jobs in consumer electronics and appliances to Japan, Taiwan, and Korea. It wasn’t the total eclipse people predicted—in part because our population control evangelists helped derail Japan’s development—but our shift to software and IP didn’t replace the kinds of jobs we lost.
Finance, insurance, social, and professional services will keep our GDP numbers afloat long past the point where the real economy has been surpassed. A better comparison is Britain. Once the real economy moves, most of the ancillary services do too. Banking and insurance keeps Britain going, but it’s actually kinda poor compared to America. (The US is to the UK as the UK is to Hungary or Poland.)
> To take another statistic into account, there are actually many STEM graduates in the US.
No there aren't, at least if you only count homegrown STEM graduates. USA can be a tech powerhouse thanks to brain drain, not thanks to its STEM education. USA is one of the worst in OECD.
> The USA accounts for 4% of the world population, and 40% of the top 100 universities in the world. That's fucking weird. I don't have an explanation for it. I'm just saying that the different signals we use for evaluating how good our education system is functioning are giving us radically different pieces of feedback, and our understanding needs to be correspondingly sophisticated.
What would our STEM university departments look like if the next generation of Indian and Chinese kids with good K-12 math educations decided opportunities were good enough at home that they didn’t need to leave their families behind and immigrate to America?
For those of who were around in that era and watched as the US auto industry and consumer electronics industry were devastated as their Japanese counterparts ran rings around them, Japan did eat America's lunch and with considerable gusto. Sony products were everywhere, as were Toyotas and Hondas.
Japan doesn't support your case. (Neither does China really.)
Isn't Toyota making cars in the U.S. now? Aren't iPhones proudly "designed in California"? The old factory jobs were doomed by technical change either way, so picking up the slack and looking for newly-opening opportunities (including in consumer tech) was an entirely sensible response on the U.S. side. But that requires fixing the dysfunctional K-12 public education system.
> The whole situation is warped. The USA accounts for 4% of the world population, and 40% of the top 100 universities in the world. That's fucking weird.
No it isn’t. The best researchers want to go to the best universities and the universities are ranked on research performance. Its a feedback loop.
If we look at countries beyond the US and China... what are they surviving on?
Should math be a higher priority in the US? Should working hours in the US be the same as in China? Should the academic pressure on kids be as high in the US as in China?
The US is much smaller population-wise, would we actually need to try five times harder than China?
Is it not enough to compare today's high school overachievers with those of 20 years ago, all still fighting for the same universities but with all similarly-inflated resumes? Do we actually need to push them even further?
Do we instead want to be more like the European countries that currently put themselves under less pressure than the US?
I don't doubt that the people crafting these proposals care. I think they truly believe they are doing the right thing. I personally think it's just increasingly popular, mistaken moral beliefs that inform these types of proposals. Some of the underlying beliefs:
1. Blank slate - All humans are of equal ability
2. Any observable differences between humans are merely the result of social factors
3. Any observable differences in outcomes between groups of humans are the result of oppression from the majority group
4. If you observe differences at your org/institution, it's your moral duty to create policies which disfavor groups of humans performing better and to favor groups of humans performing worse, as those performance differences are due to oppression.
If these beliefs undergird your worldview, and your social groups/information environment reinforce and reward these beliefs, it is of no surprise that we'll see a lot of people soberly propose the types of policies we see here. I can empathize that they really do think they are fighting the good fight, and are doing the right thing for society.
I don't think that those beliefs are a workable explanation here.
These proposals come from committees and groups of people, and it's just not realistic to write off the entire group of people behind these proposals as having some uniform set of beliefs like that, especially when they give other rationales for the proposals!
The current school system makes decisions in middle school (8th grade and earlier) which determine whether or not each particular student will be able to take calculus in high school. This is, simply put, insane.
Because it's obviously insane, when you introduce questions of race and class into the mix, then it's easy to apply pressure to the department of education to come up with a proposal that changes things. And then you end up with bad proposals... why? Because these proposals are produced by poorly-shepherded committees full of government employees under political pressure, and it's much easier to come up with a bad proposal that responds to political pressure than it is to come up with a good proposal.
There's just no need to try and explain that this proposal is bad because the people who made it have bad beliefs. I'd characterize this as fundamental attribution error here... "the committees made a bad proposal because of wrong beliefs" versus "the committees made bad proposals because it's easier to respond to political pressure than to write a good proposal".
If you talk to these people though. They DO have bad beliefs. Let's not even talk about politics. Their goal is to get a kid a degree,(especially in groups that dont normally get degrees) and many of them believe just having the degree to get past the job resume hurdles is good and helping people.
They ignore the fact that the degree is supposed to be a proof of "This person has X skillsets at minimum". It no longer is. There is basically no job in the United States that cares if you have a high school diploma or GED other than government jobs. They are repeating this process with College right now. The College Diploma is now the new High School Diploma, and pretty soon jobs are gonna want Masters or Doctorates for entry level.
This is a bad belief, they don't (or won't) understand that they are ruining and devaluing a thing by their actions and beliefs.
You can't get a college degree without passing the College Algebra weedout course (let alone Calculus, which is required for the bulk of STEM courses), and you can only realistically pass College Algebra by getting a lot of rigorous math in K-12. Lowering the bar is doing every student a disservice, and the most vulnerable students will be the hardest hit.
Yes pretty much every college requires "college algebra". But some schools have "college algebra for stem majors" vs "college algebra for non stem majors". Guess which one is easier, has lots of bonus credit and extra curricular stuff to earn extra credit. (You attended the college showing of "vagely related math movie?! Here's 10 points on your final!") And also grades on a curve. Also you only need 69.5 (and sometimes just a D!) to graduate.
There's also other cheats/hacks. Like lots of state schools will let you transfer from a community college with credit for your "core courses", and some of those have questionable standards. There's also the fact that college algebra usually has some kind of test out or online option. There was a whole sub industry of "pay you to take the online test for me" at colleges for stuff like college algebra. Some of those courses did have some kind "you have to take 1 test in person so we know its you" rule. But they didnt check super heavily that you were actually that person other than a cursory examination of your drivers license name matched who was supposed to take the test. (and oh boy let me tell you about how covid and masks interacted with all of that)
Dont get me started on the "Statistics for Sociology" that was different than actual "Statistics" (but fulfilled the Stats requirement for the degree)
This is also ignoring that taking College Algebra to begin with IN COLLEGE. Was a major sign you were not a Stem Major. Stem Majors took that in high school and were taking at minimum precalculus. (and even that was viewed as the slow lane, you should be talking Calc 1 as a red blooded STEM freshman)
An acquaintance asked me if she should take "Calculus for Artists" after I suggested she take a calculus course. I laughed, and said that such a course should be named "Pretend to Learn Calculus". She should take a real calculus course, which she did, and did well in it.
If you're in college, stick to the real math classes, not the "math for losers who are forced to take a math class". You'll be with other students who want to learn math, and you'll have a prof that wants to teach math (the loser math course has a prof who doesn't want to be there, either). It'll be a much more pleasant experience.
Hey, if I was running a college, I'd have the two track math system, too. That way the students who want to learn won't be bothered by the ones who don't.
And yet people don’t feel the same way about “Calculus for Engineers” despite it also being a ridiculously dumbed down course full of rote memorization of formulas, “procedures,” and “strategies” with the bare minimum of hand-wavy theory. Compared to the courses for maths students it’s just as much pretending to learn calculus.
dae STEM has got to be one of the most exhausting memes on the internet. Hell, I’m 3/4 of the letters and even I find this aspect of our culture insufferable.
Knowing the theorems behind calculus is difficult. But they don’t magically make you able to solve limits, integrals, etc. So having different courses that focus on different parts is smart.
E.g., there can be a course on compilers that teaches automata theory. There can also be a course that just jumps in and codes a compiler. Both courses teach different, but valuable skills. Now, having a course “Compiler for Artists” where you’re taught such useful gems as “a compiler translates between human-level languages and machine code,” not so much.
I don't think that's fair. I can't say anything about Calculus for Artists, but Calculus for engineers won't be dumbed down. It will be focused more on the practice of using calculus to solve real-world problems, while a 'regular' Calculus course will be focused more on the development of theory, proofs, etc.
I've seen the real world result of dumbed down math for engineers - applying the wrong formula for the task, inability to adapt the formula to the task, and poking around in the dark because they were terrified of math. "Walter, can you go help him out" is what I'd hear.
So, yeah, in the real world, it doesn't work out so good.
I eventually learned to appreciate that Caltech never taught how to use formulas, but instead taught where the formulas came from. I recall a class on jet engines (really, an awesome class!), where the prof spent the entire lecture deriving the formula for a jet engine's performance. It was breathtaking. I knew where every term came from. I finally understood how the damned thing worked. All those other handwavy explanations mystified me.
If I was just handed the formula, it wouldn't have meant much of anything to me.
The theorems they prove in mathematician's calculus are useless for engineers, and they take time away from learning useful tricks for doing integrals, which if not that critical for a world with numerical methods, are crucial for solving problems on future tests.
> The specific theorems students prove are not really the point.
Right, that's why they should skip the calculus and start students directly on Real analysis. Then they might actually have some use for what they prove in the class.
I prefer to know where the formulas come from, to know when the formulas apply, when they don't apply, what their limits are, and how to adapt them to a specific problem.
> pretty much every college requires "college algebra".
Caltech didn't. In fact, they expect you to know calculus before entering. I didn't, and that nearly capsized my college career before it left the dock.
They now do have a remedial kind of class, "Math 1d", which is to be taken concurrently with Math 1a. I was placed into it after not trying very hard on my placement test, but they let me drop it when I told them I didn't need it.
Sad to hear that. It suggests their admissions process has succumbed. When I attended, it was very rare for the admissions people to admit someone who couldn't do the work. I only knew one who couldn't.
Yeah, heaven forbid 17 year olds decide they want to take on a career in STEM later in High School. They must be real dummies. REAL engineers decide to be engineers as HS Freshmen and never waver from, struggle on, or doubt the One True Path.
Colleges should only admit those who pledge themselves to the Guild of Software Engineering at 13, so we can keep the noble profession of building CRUD apps in JavaScript free of riffraff.
Grandparent was talking specifically about Caltech. I don’t see how anyone can go to Caltech without knowing calculus, possibly even linear algebra. This school has tougher admission standards than MIT.
Frankly, a major thing that Caltech admissions looked for is a very strong interest in engineering and science. It's gotta be in your bones, or you ain't going to thrive there. With their limited resources (it's small) it's just not pragmatic to accept those with a more casual interest. And really, why would you want to go there if you weren't? After all, 9 out of 10 classes were 90% math, math, more math, and math.
There should exist places designed to provide maximum learning opportunity to those who come in already knowing a fair bit. It is a loss to society if such places start disappearing, or perhaps a sign of such loss if the reason is less demand.
This does not in any way imply that those should be the only kind of places to exist.
This does not in any way imply that there's anything wrong with people who started to recently to have relevant background.
I could have put this comment nearly anywhere in this thread. As a German I'm really surprised by this. Calculus was part of regular math school education and just expected knowledge for the mandatory math classes during my CS undergraduate studies. I did graduate at one of the universities that do use the first years to weed out students though and dunno if less competitive ones are more lenient.
We also do split students as early as after the 4th year (end of elementary school) in three different school systems with about 50% of students attending the highest university track tier. The other two tiers might not have calculus or only in a limited form, but also don't qualify for university without further education. Thus setting students on different paths based on ability/performance early on seems very natural to me. Students always can up/downgrade schools based on performance at any year later on.
I’m a math tutor at a run-of-the-mill state university and I can tell you for certain that the average student in college algebra and especially in pre-calculus has to work hard to pass. Both classes have a substantial amount of easy credit like mandatory lab time, credit for filling out notes, etc. with tutors there on the clock and available, but in both classes student struggle mightily with the material and spend a ton of time in the lab beyond the requirement.
For cheating we proctor all of the tests, which are worth by far the highest portion of the grade. I’m sure people have cheated on the proctored tests, but there’s not rampant purchasing of passing grades.
This is specifically to rebut your implication that students are cheating or being carried by the grading structure, not necessarily to discount your position in the overall discussion.
I also worked at a run of the mill state university as a teaching and CS/IT assistant.
It was well known there was a side hustle for some of that. And certain proctors would at times be told to drop it if they pressed. It was notoriously an issue with the non english speaking population. They would show up with an id that was clearly not them, proctors would be told to just let them take the test, same student would still be at classes months later. Though to be fair foreign students and schools chasing that money is probably its own kettle of fish.
One of my favorite courses was an into to geology course I took on a lark. Turned out everyone (including the professor) called it "rocks for jocks". It was incredibly easy, and years after being out of college I don't recall much of anything I learned there that I didn't already know from high school or earlier, but I do remember the professor at least made the material engaging.
I do expect that such "workarounds" will always exist. I just don't think they're more realistic than just getting some good-enough math fundamentals in K-12, without mucking about with "data science" silliness. (Data literacy is of course appropriate to Science class, and the letter even mentions that.)
How the heck can you call college algebra a “weedout course”?! I was a math major and skipped over it to start college in Calc 2, but I saw my friends tutoring those intro classes and saw the work my professor-friends put into trying to make basic algebra accessible and get them over the line. They really did everything they could. Many students still struggled though because public high school utterly fails most people and/or those peoples brains just weren’t wired for math.
That's probably true. But there's just not much that you can do to teach years of junior high and high school math in a single college class, to students who never got the chance to seriously engage with that material because their school replaced actual math courses with this new "data science" thing (or for countless other reasons which would fully apply even today). "Weedout course" might not be the intention, but it's definitely the real-world outcome.
Given the situation on the ground you can comment that it'd be great to stop de-valuing college degrees but I think it's pretty clear that ship has sailed. A lot of jobs do unnecessarily demand college degrees, even going so far as to accept irrelevant degrees... anytime you see "Our ideal candidate has a BS from an accredited university" you can be confident that an arbitrary and discouraging requirement is being placed on the post. Now, in reality, a lot of employers don't actually care about those "requirements" but young people often don't realize that, and the ones that do can't be certain their uncertainty won't be invalidated by the time they secure a diploma.
I don't have a specific comment on the policy under discussion, I'm not a californian and I'm not familiar with the specifics - but telling people "You'll be fine without a degree" isn't going to go over well and, honestly, is asking the recipient to accept a large risk regarding their future while you, an employed person, have already passed that hurdle. American colleges definitely have issues with over enrollment but even if wanting that state is a "bad belief", it's certainly an accurate belief.
Employers only demand unnecessary college degrees because it's a quick and easy way to filter down a huge stack of applications. If there were fewer college grads available in the labor market then employers will just remove that filter and hire based on other criteria. It's not like they're going to leave those jobs open.
Does the why materially change what new market entrants experience? Right now a good chunk of High School students assume a college education will be necessary to get any sort of good job, unless we can reverse that conception[1] college will remain an assumed necessity for most folks.
I don't disagree at all that it's mostly BS, but, much like the stock market, it is functioning primarily on the social valuation - and currently degrees are seen as close to a necessity to enter the job market.
1. The only thing I've seen on this front is some advertising for trades work which can pay equal or better to non-SF software engineers.
While I agree that the proposals are bad, I don't blame "committees full of government employees". One of the lead proponents/authors is distinguished Stanford Professor Jo Boaler. It's interesting that a lot of the arguments made in favor of the changes are done in the name of equity, but Boaler herself has been put on the wrong side of racial equity, threatening to call the cops on a Black Berkeley CS professor. This article [0] is gossipy, but it's both interesting and relevant how "Nice White People" can hurt the minority groups they are supposedly trying to help. Hurt by taking away opportunities to take calculus, and hurt by threatening legal action against one of the few minority CS professors at a leading research institution.
Oh, I missed the context: https://stanfordreview.org/boaler-professor/. I didn’t realize Boaler had a history of doing that (though I guess I’m not surprised). Prof. Nelson sounds like a champion.
Boaler taught sixth and seventh grade math before becoming a professor of education. She might know how to teach pre-algebra concepts, but I do not believe she understands which high school math is most useful.
A lot of students in the US learn physics without calculus, which is basically useless. It's just a set of magic formulae to plug and chug. We should not go further in that direction but instead go in the other direction.
> One of the lead proponents/authors is distinguished Stanford Professor Jo Boaler..
I have an issue with the parent's gossipy put-down of Professor Boaler.
I think it's important to emphasize that genuine intellectuals who have put serious thought into this proposal support it. I would prefer more serious engagement with it, even if on HN the majority disagree.
That is just an overall terrible outlook for the UN, how can this be the common denominator for nations? It seems that it becomes the personal outlet for some people.
Not really specific to the UN. These sorts of international NGOs all have the same problems. The WHO is worse. The whole idea of global governance is really only attractive to one particular type of person, and they aren't accountable to anything really, so of course you end up with this sort of groupthink.
There are no two people in the world with the same beliefs - nobody is suggesting that.
There is however a very broad movement of people who believe that unequal outcomes are a manifestation of racism and so they act accordingly in their roles in government etc..
Many of these people are in the civil service and so this will influence their view.
It would be 'insane' to ignore this movement, it's one of the most powerful social forces in the US right now.
I certainly agree racism gets overemphasized as the sole cause of disparities compared to other factors like culture, economic status, and selective immigration (I don’t see how racism explains, for example, why the children of Nigerian immigrants outperform those of Mexican immigrants, or why Asians outperform whites).
But even conceding that racism is the dominant factor in performance differences, the framework proposed here is plain ridiculous. In essence, what they’re advocating is that because disadvantaged groups underperform in advanced math, advanced math should be scrapped altogether in public schools. The effect of this will be to handicap disadvantaged groups even further, as they’re less likely to attend private school or have access to private tutoring, and thus will be locked out of advanced math entirely while advantages groups continue to receive advanced math instruction in private schools.
I know progressives love to portray things as a dichotomy between “fascism” and “equity” but that leaves out the position that the majority of people in minority groups actually support—color blindness: https://www.pewresearch.org/fact-tank/2019/02/25/most-americ...
Well Mr/Ms Drewwwww - you have very graciously made point for me.
Since I have no such 'scientific racism' views and claims of 'proto-fascism' are ridiculous at face value ...
... that a common person would take some rather mundane comment somewhere to mean those things, implies that they've been radicalized in some way in the manner that I've described.
"Racists is everywhere, under my bed, in the jingle for that product, in our schools! Math is racist! We Need Action Now!"
'Anti Racism' is a reasonable concept at face value, but the issue has obviously created groups of wayward ideologues in large swaths in the US, who are more likely than not to be involved in the civil service, particularly in education.
Since 'improving education' is a perennial issue of contention anyhow - if we throw 'math is racist' into the fire, along with legions (at least a critical mass) of fervent supporters - and then finally allow the hollow politicians and media to misrepresent and aggrandize all of it in bad faith (votes, money, attention, power etc.) ... then we have yet another toxic cocktail of public malaise and dysfunction distracting us from dealing with the core issues.
You're saying it's one of the "most powerful social forces in the US right now"... is there any particular reasoning here? To be honest, it sounds like an extremist take on the nature vs nurture argument which has been playing out for millennia and I don't see a reason to believe that somehow the extreme version nurture side of the argument has become dominant here. If anything, it's easy to remember the pendulum swinging back and forth in the past decades, and the pendulum doesn't seem to swing that far in either direction.
> Many of these people are in the civil service and so this will influence their view.
It influences the political pressures placed on people in the civil service more than anything else. People in the civil service are in the civil service for a long time, typically. Often they are in the civil service for decades. On the other hand, elected officials and political appointees rotate in and out much more quickly.
I know people in the civil service, they have a much more long-term and level-headed view of things, and my impression is that they "weather the storm" of changing political pressures. At least, the ones who survive in the job do. Political appointees and elected officials are much more mercurial. Appointees know that their position is (somewhat) an extension of their own political power to begin with.
Christians are fond if it: "You have an innate, direct relationship with God, in his eyes you are equal to the The King" etc..
But we've only had 'governance' in the broad sense for 100 years.
And we've never really tried to apply such principles into education until the 1960s.
Now we have actually made incredible progress on social issues, we have our 'wars' in Social Media with Holy Anti Racism Fanatics trying to do their best - because 'racism is bad' - which of course it is - and 'systematic racism exists' - which of course it does - but the 'kernel of truth' of these issues drives people into ideological fervour as though it's some giant overwhelming issue, when really it's not. Racism is still pernicious, but it's not fundamental.
And FYI don't think it's all rubbish.
For example - 'Math' is heavily based on prerequisites. If you 'fall back' in Grade 4, you may never be able to 'catch up'. While that's true in general, it's not as acute as in math.
Poor kids might be far more likely to 'fall of the bandwagon' and a lot of poverty might be due to systematic racism, and so the 'Hard Requirements' for certain things may not be ideal.
You could have 'summer school' or 'after school' or 'accelerated catch up' programs.
Those would be 'reasonable' solutions in my view - and FYI these are mostly issues of poverty, not race, they are couched as racial issues because that's what fires people up.
Edit: and yes, I agree 'most people in the civil service' are level-headed. Most people actually are. But some groups have outsized voices, amplified by 'allies' elsewhere.
The 'Anti Racism Agenda' is a 'fundamental pillar', like a religion, of 25% of the US population, and they are pretty active about it. And the actions of the most extreme 5% end up really upsetting other people. Much like a very powerful fool using made up constitutional manoeuvres to try to take over the government would upset a lot of people as well.
Good intentions, surely, but that doesn't make them right.
>> Racism is still pernicious, but it's not fundamental.
How did you reach this conclusion in the face of 300+ years of verifiable, well-documented federal, state and local racist legislation and policies? If it’s pernicious now then it’s always been pernicious since generations have persevered irregardless of these policies… but that doesn’t make sense to me.
Institutional racism really does not exist, and in general, people really are not racist. Obviously, subtleties exist, and there is work to do - but by and large, people are treated fairly in 2020, moreover, there are numerous 'affirmative action' programs in place to try to make up for the issues.
>>and 'systematic racism exists' - which of course it does
Systematic racism only exists today to positively discriminate against people belonging to so-called disadvantaged races, at the expense of those belonging to so-called privileged races. That's why White and Asian kids are discriminated against in college admissions.
Oh yeah? Then why am I stuck paying a higher property tax rate than my white neighbors who inherited their house from their parents that purchased during an era of segregation?
First of all, are we talking about “systematic” or “systemic” or “structural” racism. These terms are not interchangeable. Put that aside for a second…
I assume your example is talking about Prop 58. [1] Is Prop 58 racist? Of course not, the CA Constitution prohibits racist and sexist laws and the courts are very good at enforcing that prohibition.
Prop 58 means a lower property tax bill can be inherited by the children with the property. By your reasoning, isn’t allowing property inheritance at all racist?
To me this is a meaningless injection of “equity” into a discussion on equality. Racism is not a word that means “inequity” - as in - everyone not consuming, producing, contributing, and owning the exact same quanta. That of course is a bizarre fiction.
We are talking about a tax policy which generationally preserves a tax break. It had a meaningful basis in law and policy which does not discriminate based on race, but it certainly provides a significant tax benefit to generational land owners.
Now again, that’s a drop in the bucket compared to the Federal estate tax lifetime exclusion amount, which provides orders of magnitudes greater tax benefits to preserving generational wealth, so Prop 58 is just a bit player in this scheme.
If the argument is that a tax policy which is advantageous to anyone with generational wealth is “systemically racist” then I think the argument has gone off the rails.
There are important justifiable public policy goals to ensuring a certainly level of generational wealth. There is no and never will be an important justifiable public policy goal of being “racist”.
If someone wants to argue for wealth confiscation, redistribution, and equity (e.g. some form of communism) then by all means give it a shot. But I don’t abide calling people racists if they disagree with that as a policy goal. If the argument is “merely” that there should be no tax benefit which preserves generational wealth, you actually immediately end up in exactly the same place.
The answer is of course more likely along the lines of “we do both”. We have policies that provide for protecting and maintaining a certainly level of generational wealth. And we also have policies which provide a massive welfare system for people without any generational wealth.
The original commenter obviously meant "systemic" racism, and you're not discussing in good faith if you're trying to misinterpret this. Ditto when you digress about calling people racists when no one in this thread has done that.
As to estate tax exemptions, they aren't nearly as egregious. There can certainly be an argument for reparations of ill-gotten gains to level the playing field, but it's at least possible for me to accumulate wealth myself and pass it on to my children. In contrast, it's simply impossible for me to acquire and pass on the same property tax rate that my white neighbors have.
I guess to me the words matter, and calling it “systemic racism” means that somewhere in there, there should be identifiable racism and not merely structural inequity, aka property ownership and capitalism.
I think saying you can’t go back in time and capitalize on an investment 60 years ago that also comes with an accrued property tax advantage is true, but has nothing to do with race. Anyone can start accruing their own tax advantaged status under the exact same Proposition, and no one can ever try to take it away from them on the basis of a protected trait, right?
I guess what I’m asking is, is there anything specific about this tax policy that accrues benefits to a historical capital investment which makes it “racist” over any other tax policy that accrues benefits based on historical capital investment?
There are so many race neutral policies that provide accrued benefits like this. Even social security benefits accrue based on historical wages. But also things like Roth IRAs which provide compounded tax free growth. You can’t contribute the same dollars today to a Roth IRA and get the same tax advantages as someone who contributed the same amount but over the last 60 years, the benefit can only accrue over time, just like the CA property tax benefit can only accrue over time.
Not everyone pays the same tax rate on their income or their property, nor are they provided the same level of government benefits. In many cases that tax rate or benefit level will be based on historical personal or family wealth. Often times it works out to be a progressive tax, sometimes it’s regressive depending on so many policy goals that are often in conflict with each other.
Simply put, any of your neighbors who have owned their (family) property for several decades have accumulated a several thousand dollar a year tax benefit for being a long-time owner. You too can accrue a several thousand dollar a year tax benefit the longer that you hold your property. You don’t get to buy in and get that benefit on Day 1, and neither did they.
The difference in your experience from that of your neighbor is not a case of racial discrimination. Even if this afflicted more black people in general, a racial disparity in the impact of a policy is not evidence of it being systemically racist, so would not validate the claim that the US is systemically racist toward black people.
This can be trivially demonstrated: ten times more men are arrested and incarcerated than women, yet no one would claim this suggests the US criminal justice system is systemically sexist toward men.
Twitter's algorithm promotes whatever tweet will make people angry, because angry people spend more time on Twitter. I'm on Twitter, but I've been on Twitter for a while now, I've selected who I follow, and my feed just doesn't get that kind of noise in it at any significant level. Most of what I see on Twitter is people arguing about Elden Ring or showing off how they can access YouTube from a Mac SE or something.
"The news" is dominated by organizations which are trying to maximize their social media engagement metrics so they can get more money from advertisers, so they're subject to the same forces that drive Twitter.
Neither of these sources reliably give you a picture of national affairs. Right now the best I can do is get a picture of local affairs by talking with people that I happen to encounter because I live near them.
It sounds like you and OP differ only in that he thinks the people on the committee have these beliefs, while you’re implying that they’re afraid of “political pressure” from people who hold these beliefs. Which I suspect is correct.
I was thinking less that they are "afraid" of political pressure, and more that responding to political pressure is one of the things that these committees do, by design.
Political pressure is a manifestation of the population's problems / beliefs / perceptions. It's wrong to bow completely to political pressure, but it's also wrong to ignore it.
I agree with you somewhat, but I think there's an important distinction to be made.
If a political decision-making process concludes "men account for more road traffic deaths than women, but we've decided that gender can't be taken into account when pricing car insurance" that's fine; the facts have remained the facts, and a political decision has been made by a political process.
But if the same process concludes "men and women cause the same number of road traffic deaths" the process has gone off the rails despite the fact the outcome is the same because in one situation the facts have been acknowledged and the decision to act contrary to them made knowingly; while in the second situation that isn't the case.
> The current school system makes decisions in middle school (8th grade and earlier) which determine whether or not each particular student will be able to take calculus in high school. This is, simply put, insane.
It's not clear that this is the case. The "current approach" is to offer Algebra I in middle school, which ought to leave plenty of time for students who want to shape up in math and be prepared for HS calculus to do so. Push advanced math later in the curriculum, and you just expose the students to even higher-stakes dilemmas. Lowering standards is no solution, since you'll just end up with lower education quality for all students, that will make it even harder for them to catch up to reasonable levels. This is the broad background of OP's letter.
> These proposals come from committees and groups of people, and it's just not realistic to write off the entire group of people behind these proposals as having some uniform set of beliefs like that, especially when they give other rationales for the proposals!
You don't have to write off entire groups of people - just the few at the top. They got there by sucking up to those who were already at the top. Everyone else just has to keep quiet if they want to keep their job. Worse, they're expected to express visible support for their "superiors" and their ideas if they want to keep their jobs.
I don’t necessarily agree with the proposal in california, but do all students need to take calculus in high school? What about solid coverage of algebra 2 and pre calc before higher education. Community college is a great place to take a calc class (or even a pre calc class) affordably. Source: took pre calc, calc and stat in community college after being signalled to that I was terrible at math throughout high school.
> No one is suggesting that all students take calculus in high school.
It’s very commonly taught in 10th grade in France, Germany, Singapore and Taiwan (where I used to teach). It’s not universal by any means but as far as I can tell, the idea that calculus should be delayed until university is a nearly uniquely American idea.
"Very commonly" in STEM-focused prep schools (i.e. the "academic" part of the tracked education system that's common outside the US). Which leaves you with very roughly the same percentages as the U.S. approach where Calculus is an elective course.
> roughly the same percentages as the U.S. approach where Calculus is an elective course
I can’t speak with certainty about other places but this directly contradicts my experience living most of the past 20 years in Taiwan (and speaking with friends who taught mathematics in the public school system).
Many, many people not on a STEM track learn calculus in high school.
My wife did the fine art/liberal art at a good mainland Chinese high school and didn’t really learn calculus, and definitely not later on in the fine art university she went to. I get the feeling that most Chinese students don’t learn calculus, but the ones that do are still slot (since China has a huge student population).
> but do all students need to take calculus in high school?
Imho, 50% or more of students don’t even need to take algebra.
For that matter, that same number pretty much don’t even need to go to high school (although some sort of voluntary coop with practical skills training might be better than current high school or nothing).
HS is glorified baby sitting for most students. It sometimes is useful for people who want to go to “competitive” schools or enter stem fields in slightly less competitive schools.
As for your solution, learning math starting at algebra or pre-calc in a CC is fine if that is what the person needs for their chosen career path. I think the general idea is that the lack of calculus in high schools prunes the career path tree fairly aggressively in some relatively well-paid fields (esp. STEM).
I feel like in this case fundamental attribution error would go the other way, no? The explanation you offer is that there aren't circumstantial factors dominating the decision (the beliefs on this particular issue), but a fundamental flaw in how committees work. To be clear, I agree that this is an inevitable result of the decision-making structure, I've just only ever seen fundamental attribution error referring to mistakes in the other direction.
Ah, now I get it! I was thinking of the committee as the entity, not the people on it. Then the question is "why did the committee make a bad call?" where "the topic in question coincidentally misaligned with the views of the members" is the specific cause and "committees always make bad calls" is the general cause. But looking at it from the perspective of the people makes it clear what you were going for
> 4. If you observe differences at your org/institution, it's your moral duty to create policies which disfavor groups of humans performing better and to favor groups of humans performing worse, as those performance differences are due to oppression.
I don’t know anyone who seems to disadvantage high-achieving individuals. Only people who wish to raise the performance of as many as possible to the same level. And yes, I see oppression everywhere, including education.
It would be a net loss to society to deliberately reduce the performance of anyone.
I never understood why people try to make “blank slate” into a binary thing.
Can people not have varying degrees of physical and neural plasticity? Perhaps some people are more like blank slates and can adapt more readily than others? Maybe plasticity changes with age?
There's almost no opponent of the blank slate theory who thinks societal factors never matter. There's only one side that takes a hard-line dogmatic view on this issue and it's the blank slatists.
Of course they do. Even if someone actually believed in some 'hard' type of ideology, they might not act that way.
It's just a 'general set of principles', intuitions, pop culture ideals that lead a large group of people to assumptive believe that 'unequal outcomes are driven mostly by systematic racism' and that's that.
Ergo 'the world is deeply racist' and 'there is racism around every corner' including in your math textbooks and teaching.
Ad nauseum.
Some of it is actually correct.
Some of it is reasonable from an intellectual perspective, but it's hard to take anything from it.
It's hard for me to square your claim that there's a dominant belief that all humans are blank slates of equal ability with the sheer volume of messaging I see in both government-sponsored and private media about embracing differences, follow your own goals, find your talent, etc.
I see a lot more stuff that would lead a kid to believe "it's ok that I'm not good in math" rather than "I could be good in math if I wanted to be."
Frankly, I think this is actually worse educationally than what you suggest.
We need to find more ways to reward effort instead of pre-existing ability (regardless of how that pre-existing ability is gained... the kid whose parents got him ahead of the curve through high school math and then bombs out after taking university-level Calculus is similarly harmed by the current system as the one who's shunted away from ever being challenged).
You are leaving out who is involved and what commercial interest will be benefitted from these policies. It is likely those commercial interests are the ones sponsoring and pushing these by finding sympathetic folks.
The important thing to note here is- if you reduce the bar in high schools, a lot more students will end up in college - more money will spent, more loans will be written out etc.
I don't understand how these people could consistently ignore facts. Case in point, I could earn way more than Scott Aaronson or had way more social privilege than him, but you'd think I'm crazy if I claim that I can be as good at maths or quantum computing as Aaronson.
That kind of arguments goes in favour of the CMS. If you assume that only a handful of geniuses can do maths then as a society it makes little sense to allocate resources toward something completely inaccessible to the masses. Designing the education system for Scott Aaronson to the detriment of everybody else would be a mistake socially and economically. That is how these people think, not some nonsense blank slate theory.
In reality it's not quantum computing that we're talking about, it's high school calculus and algebra. You don't have to be a hardcore blank slate proponent to believe that most people can learn it. And that is what these people don't believe.
It's important to consider the goals of this committee. They propose this reform because they oppose the blank slate theory. The current structure really isn't appropriate to most people. Because it relies on a wrong form of the blank slate theory. They offer the wrong solution in my opinion, because they end up going too far the other way.
They're working with a bizarro Blank Slate theory according to which every student should simply be learning their math by themselves, and if they fail it's their own darn fault, or perhaps society's fault, or anyone else's fault, but certainly not the teacher's fault. Because the teachers all have Education degrees, and that's what they were told in Ed School. So don't anyone dare "demean" their job by suggesting that they have actual work to do in properly educating their students.
> Because the teachers all have Education degrees, and that's what they were told in Ed School. So don't anyone dare "demean" their job by suggesting that they have actual work to do in properly educating their students.
You have been to a remarkably bad college of education, if that was your experience. Which is saying something, because most of them aren't great, but for very different reasons.
It actually is possible that geniuses are rare but still contribute a significant fraction to the society. (Which means allocating resources to them will be the socially optimal choice.)
Not that taking calculus takes a genius, as evidenced by more sensible countries.
it’s not about being better than him. it’s about the theoretical possibility of being better + the virtue signaling that comes with the theoretical possibility
Not only does the earlier version of the framework explicitly reject this view, it cited specific empirical studies that the broad approach targeted (which I gather had not changed in the revisions which is why the complaints remain despite some revision to details) was better for people across the ability spectrum.
Similar points apply to each of your bad-faith assumptions about the underlying beliefs.
> Not only does the earlier version of the framework explicitly reject this view
Can you share that explicit rejection of the idea that there are not innate differences in ability, in the CMF? I have not seen it myself, thank you ahead of time.
To share what I've read, and colors my views a bit, is the following, from 'Chapter 1: Mathematics for All' of the Second Field Review [1]:
> An aim of this framework is to respond to the structural barriers put in the place of mathematics success: equity influences all aspects of this document. Some overarching principles that guide work towards equity in mathematics include the following:
> All students deserve powerful mathematics; high-level mathematics achievement is not dependent on rare natural gifts, but rather can be cultivated (Leslie et al., 2015; Boaler, 2019 a, b; Ellenberg, 2014).
...
> All students, regardless of background, language of origin, differences, or foundational knowledge are capable and deserving of depth of understanding and engagement in rich mathematics tasks.
> Hard work and persistence is more important for success in mathematics than natural ability. Actually, I would give this advice to anyone working in any field, but it’s especially important in mathematics and physics where the traditional view was that natural ability was the primary factor in success.
—Maria Klawe, Mathematician, Harvey Mudd President (in Williams, 2018)
> Fixed notions about student ability have led to considerable inequities in mathematics education.
Note that my pointing to this doesn't mean I inherently disagree that hard work/education can't help improve outcomes. I show the above citations to show that the CMF is not explicitly denying the Blank slate theory, which is what you are suggesting. If anything, they go out of the way to view ideas of innate ability negatively. I'm happy to also look at the references that you alluded to but did not cite.
[1] https://www.cde.ca.gov/ci/ma/cf/ (it's in the bottom section, the format is .docx, so don't want to directly link to it as that format can sometimes be cause for concern on random links shared on the internet)
As for 3, "average observable differences" in mental ability can largely be explained by socio economic factors. Case in point, East Asian IQs stagnated behind US IQ averages 50 years ago, but are significantly ahead now. I dont think they underwent a general transformation in 50 years time.
4. Is again a strawman. Affirmative action is used for a limited number of historically discriminated populations in very limited contexts.
If you are interested in an actual debate don't invent a caricature of the opposing position.
Admittedly I am a little confused by your comment. You made the above statement, and then follow up with this statement:
> As for 3, "average observable differences" in mental ability can largely be explained by socio economic factors
If we are to assume you are being earnest here in your first statement in that you believe there ARE innate ability differences between humans and social factors don’t explain everything, can you explain how much innate ability differences are a factor in intelligence differences (using it since you brought it up)? And how do you determine it? What methodology are you using for that, and if you aren’t basing it on anything, what methodology should we use?
If you are going to dodge the question and suggest you were talking about averages between groups (for which the question does still apply) and that between groups are a special case, then please share how much of a factor innate ability differences are in intelligence within groups? And how you determined that?
Before I start, you really need to watch your tone and not pretend that you are a hunter on a hunt about to kill a nasty prey. My responses are clear, comprehensible and from your reply it is clear that you understood all of my points perfectly well, yet chose to use phrases like
1. If we are to assume you are being earnest
2. If you are going to dodge the question
etc.
I am a statistician and all of your questions have banal answers.
For individual differences you can plot the bell curve and the standard deviation gives you the individual variances.
I was clearly talking about population averages in 3.
> can you explain how much innate ability differences are a factor in intelligence differences (using it since you brought it up)? And how do you determine it?
AFAIK remember, heritable IQ/ height is estimated to be a bigger factor at an individual level. However, there is no existing causal methodology to determine it. Instead we have correlational regression models that make a lot of strong assumptions that are practical. There are various papers but it is clear that IQ is highly heritable like human height.
Also, I am going to replace "innate" ability with "inherited" ability because there is no methodology that can tell you how much of Roger Federer's tennis ability is "innate" and how much is "coached".
> talking about averages between groups
Your wording is ambiguous. But assuming you are talking about intergroup differences. When you sum up IQ or height within a group, the observation from regression models is that individual hereditary differences cancel out, and intergroup height and iq differences are easily explained by socio economic factors in a correlational regression model and heritable height/iq is an insignificant factor. Once again- this is not a causal model. Only correlational.
And this is just common sense. We know from life experiences that different people have differing IQs and heights from an early age. We are also aware of how important it is to have a school system - instead of just letting the kids figure things out on their own. The environment matters because we do have to work to nurture the inherent talents of a kid, otherwise that potential will be lost.
> please share how much of a factor innate ability differences are in intelligence within groups?
I expect the per sample standard deviation to be the same across different ethnic groups. For inherited "ability" i expect the total variance explained by "inheritance" factor to be the same across ethnic groups. I doubt if such an analysis has been conducted, because I don't see the value in this kind of subgroup analysis. Do you have any references to such studies? Have you seen data that contradicts this?
The assumptions above should result in ensuring educational resources are available for people who are struggling. They should also show people in which careers calculus is useful, to motivate people. Making calculus mandatory for everyone might also help on the margin, though this bears empirical investigation.
This is laziness of the moral clowns who know the easiest way to remove a difference in success is to eliminate the successful. As they don’t care about anything but the theatrical performance, they naturally take this route of least resistance.
Do you disagree with the following re-formulations:
Some observable differences are due to social factors.
Some observable differences by certain groups are the result of past actions by other groups.
You should favor policies to correct for the result of past harms.
One cannot reasonably claim that no groups in the US are still disadvantaged today due to actions taken on a centuries-long timescale. It seems willfully unfair to stick your fingers in your ears and just say "I'm not actively discriminatory, so there's no need to try to mitigate things, everything is peachy."
> Some observable differences are due to social factors.
> Some observable differences by certain groups are the result of past actions by other groups.
Agreed with this as well.
> You should favor policies to correct for the result of past harms.
Mostly disagree. I'm wholly convinced there's never going to be even remote consensus about ancestral oppression and victimhood. I think that if you want to correct harms, you should focus on the current social issues. Not only do I think that has much less shitty side effects (such as exacerbating group identity), but I also think it will be more effective at reaching those that have been mistreated in the past - both directly and their ancestors. The US in particular has, let's say, a decent amount of low-hanging fruit in terms of social issues to address (homelessness, lack of preventative health care, incarceration for victimless crimes etc etc).
> there's no need to try to mitigate things, everything is peachy
This is not the only option. You can recognize that things are fucked but have a vastly different opinion on how to improve.
I've sometimes wondered if these anti maths things, could partly be psyops by a foreign nation state that wants to long term damage STEM in the US.
I really don't think it is. And if it was, it'd be only partly, since apparently lots of this is coming from people inside the US.
But what if there're foreign state controlled FB groups that try to manipulate people in the US, spreading these anti maths ideas? I'd be surprised but not super surprised.
Still, if you were Putin or Xi, why would you not want to damage STEM in the US? And even if you didn't start these ideas in the first place, why not try to amplify them, once they already exist
It hugely is, and everyone in the most peaceful, free, and prosperous time is being 100x tricked, duped, and screwed out of realizing what they already have.
There are lots of people who advance agendas (whether in the office or in the government) that they think are best for their careers. Whether they believe the agenda is right or wrong is secondary.
No, even Ayn has never claimed to be a psychic, able to penetrate the deepest thoughts of school board members to distill their entire world view down to a bulleted list.
1. This is kind of a founding principle of the country, so...
2. This follows from #1, if we rule out nature then it must be nurture. Also, you must not be a parent if you'll accept "some kids are just dumb" as an excuse
3,4. Replace "oppression" with "competition". I think it might sound better to you. But the conclusion is the same.
You want to prevent winners from accumulating an advantage, eliminating all others (which sounds vaguely genocidal in this context), then you have to handicap winners and support the others. And the fact that the wealth distribution in America is so uneven certainly suggests that the initial premise is true (i.e, winning allows you to accumulate and compound advantages with repeated victories).
1. This is kind of a founding principle of the country, so...
Is it? Saying "all mean are created equal" has a lot of interpretations, of which many valid ones do not include "all men are born of equal ability in every regard." Given that some people are born to grow up to be 5'4 and weigh 100lbs and others are born to grow up to 6'7 and 320lbs, it should be clear that not everyone is "equal" at least in terms of their physical abilities. I'm pretty sure the Founders were aware of this, making it highly unlikely that their version of "created equal" meant "exactly equal in all terms of ability."
Regardless of interpretation, I'm pretty sure deciding whether a kid is fit for the elite/intellectual track or the physical labor track at the tender age of 10 based on whether they can do some tests is not in the spirit of "all men are created equal"
And I would agree. But your earlier comment seemed to imply a much more absolute stance. That's what I would disagree with.
Could you point out where anyone (besides someone like you arguing for these reforms) has said kids have to pick their course at 10 based on some tests? As far as I can tell, your side of the argument made this up.
It is ironic as you are sayings forcing kids down one path is bad, yet your solutions is to force kids down one path. The current system does not make you choose your path based on a some test at 10.
a founding principle is not everyone is of equal ability. The founding principle equality of opportunity. The government won't hold you back because you are a (peasant|lower caste|other arbitrary decision). i.e. Everyone gets to go to school, but not everyone learns the same (qualitatively or quantitatively).
'Equal opportunity' is not a 'founding principle'.
Just that they are 'equal' i.e. before God, or before the Law.
That one man is not from some superior lineage, that makes him a superior being.
I would be the founding fathers would have no problem if one man decided to 'discriminate' among others for some arbitrary reason - even if they were landholding men of high status etc..
When I went to public school in a privileged neighborhood, the "honors" track was opt-in for the parents and students. I don't think you have to go straight from "the government shouldn't decide whether you are fit for college" (which I agree with) to "there should only be one curriculum for everyone."
The government wouldn’t prevent you from going to college, it just wouldn’t voluntarily pay for you to go if you didn’t test well. You could pay your own way, seek external scholarships, etc.
As of now we effectively underwrite anyone who wants to go, often at the expense of the student racking up debt for a useless degree and later the taxpayer who will inevitably have to subsidize them.
1. So the country is founded on a lie or you misread that founding principle.
Those compound advantages you're talking about are good things that we want people to have because they help them do more good for society. You seem to want to handicap them in the name of fairness. Where does that thinking end when you realize that the founding assumption (1. above) is false? Disfiguring beautiful people to prevent them accumulating the compounding advantages that come with beauty? Brain damaging intelligent people?
The founding fathers almost all believed slavery was wrong but they were born into a society where it was a core institution and it wasn't obvious how exactly to reform it.
Entitlement principles are why the West was the 1st place in the world to abolish slavery and then went about and abolished it throughout the world.
People can find extreme examples that “disprove” this but they’re generally wrong. Most things people do aren’t that hard and people have the ability to learn to do them — they either choose a different path, have fewer choices, or just don’t care.
And yes, before you ask, this includes computer programming.
This is so obviously false that I'm always amazed there are people who actually believe this.
In _every_ activity I've ever participated in where I can observe many people's performance and progression - including powerlifting, bodybuilding, various ball sports, mathematics, chess, theoretical CS, software engineering, etc - it is transparently obvious that people's natural abilities vary dramatically.
Although it's not the most common scenario (training and experience do matter), I have seen many situations where someone with, say, 6 months' haphazard and lazy experience will absolutely crush the performance of someone with 3 years of serious and dedicated training.
I'm not sure what "generally" means here. In professional sports, my failure of not getting into any major league will be due to your lack of talent. In higher-level math, my failure of passing any exam will be due to your lack of talent. In chemistry, my failure of not being able to consistently reach precision of under 0.1% is due to my lack of talent (and trust me, I really tried and followed all kinds of instructions to the greatest details, or so I thought). In mechanical engineering, my failure of not being able to piece out a 3D model from a 2D schematics is due to my lack of talent. In medicine, my failure of not being able to memorize thousands of latin terms for all the bones and organs is due to my lack of talent. In biochemistry, my failure of not being able to internalize the energy cycle in human body is due to my lack of talent. But on the other hand, you didn't even use a computer until switching your major to CS when you were 20 yet you became the best student in every single class in a prestigious university. That's your talent. You studied world history until you were 30 years old, yet you switched to physics and somehow got Fields medal, that's your talent.
Not all failures are due to lack of talent, for sure. A blank statement like
" ascribing failure or lack of progress to lack of talent is a mistake " in the context of our discussion is nonetheless a mistake as well.
I don't know. Have you observed multiple kids in the same family? Same parents. Same "privilege". Same pressure. Same education down to the same teachers, tutors, books, and parent temperament. And you know what, some of the kids simply beat their siblings who can be years older, in STEM or writing or reading or leadership without even trying. Have you observed your classmates? Some are driven, ambitious, self-disciplined, had access to all the education they needed, and got perfect grades before grade 8 or whatever. Then just one day, he simply couldn't understand maths or physics or chemistry or computer science, and they simply got left behind and couldn't even study STEM in college because they couldn't pass the placement test. In the meantime, their classmates, less privileged, didn't really understand everything taught in elementary school, didn't have nor need tutoring or even challenging text books, simply became the best students at anything STEM in high school and in college, and again, without much trying.
Or, do you really think every 2-year old kid can teach their neighbor kids maths and then explained what groups are when he was 7-year old like Terence Tao did? I tried my own kids. Needless to say, I failed, miserably.
Me & my sister grew up, like you say, in the same family, same parents, same pressure. We largely went to the same schools, even. She probably acceled further, academically, than I did. (She probably went to the more prestigious college, her GPA/SAT/etc. were better, she earned a doctorate, while I got a BS…)
Without pointing to a vague notion of "we're different people", I think there's a few key things that were different, despite everything that was the same:
* She's the second child, I'm the first: there were some things in my education that my mother literally said "we are fixing that for her". (And I should note that I don't resent this: my mother was clearly doing the best she could with the information she had — and because she loved us. But she had more information during Round 2.)
* Education is a finite resource: in my home state, whether I got into a decent school (i.e., a magnet school) was dependent on the literal roll of a dice. (Literally literally. I.e., list of names goes in, gets shuffled, top n go to good school & gets educated, bottom m talent gets wasted.) In the worst case I was 5th? 6th? from the bottom of a several hundred person long wait list. She got in. She got a year in the magnet system that I didn't (I got entry a little over a year later). That missing year was an enormous detriment to my education and growth; it was such a clear detriment my parents were contemplating whether they could afford a Catholic private school (we're not Catholic) or simple home-schooling. Had they had the gift of clairvoyance, I think they would have the moment I was denied.
* Almost certainly the gift of a computer got me interested in CS. She didn't get one, and her interests are different. (She's still STEM, likely due to our parents.)
To digress a bit: good education will be a finite resource as we have finite number of good educators and good schools. I don't think it's possible for everyone to access good education, especially given that we have different definition for "good education". Saying everyone should go to MIT (or any scarce education resource) is like saying living in beach property is human right. Maybe so, but it'll be a different topic.
Books are a finite resource, but not really limited for any practical purposes at least in the US. Used books are inexpensive, libraries are readily available, most things out of copyright are available online, etc.
Education is following as similar course. Things like MITs open courseware, edX, etc. are making it increasingly easy to get the educational content from top teachers regardless of how limited these teachers are.
(Having access to an education is not the same as actually getting a degree, and getting a degree isn't always the same as getting an education.)
But there has probably never been a time in history where more people had free or inexpensive access to the top educational content in the world.
Education content is definitely ample now, including text books and references. we even have great communities to get answers to our questions. Unfortunately the bottleneck of education just switched to access to good teachers. A good teacher inspires students, identifies exactly why each student has difficulty understanding something, explains intuitions behind the most difficult concepts, designs highly tailored homework, leads engaging seminars, and keeps students in their discomfort zone. As in STEM field in general, lab staff, equipments, chemical agents, lab materials are generally scarce resources too.
I understand what you are saying, but I would argue that lack of access to teacher is less of a bottleneck than drive, desire, and motivation. A motivated individual is going to have no trouble finding what they need to learn and places to ask questions for things they don't understand.
I see where you are coming from on access to labs, chemicals, and equipment. But someone who has fully availed themselves of everything they can learn from free/inexpensive online classes, books, forums, emailing people, etc. is headed on a path where they have a high probability of getting access to those types of things once that is the only thing blocking their continued education.
I don’t disagree with you. I just think “drive, desire, and motivation” is part of one’s talent. The progressive policies will not hurt the best students because they students will find their resources anyway. It is the middle, the vast majority like me, who would get hurt. They would think that they got good education, and then realize that their understanding of maths is so shitty that they can’t even pass city college’s dead simple placement test. Oh, I didn’t make this up, either. NYT reported this miserable experience of a straight A student, and I was shocked to read it.
Not suggesting that, and it's wrong anyways: the point here is that we, as a society, want people to have a basic floor of an education by the time they reach adulthood. Not necessarily MIT, but not nothing either, and the track I was on for a bit in my childhood was closer to "nothing" than it was to "decent", and it was certainly way off course if I were to shoot for MIT.
For the students that never left that track… I cannot imagine that school did much, if anything, to produce a citizen with a high-school education.
Problem being an extremely significant portion of that blank slate is written before the child even enters school, let alone makes it to middle and high school, and even during those times it is impossible for a school to make up for a horrible home life. We single out and spread stories about people who raised themselves up because it's not the norm. The vast majority of people whose parents don't talk to them as infants, don't read to them as toddlers, don't listen to them as children, and don't keep from hitting them as teenagers will radically underperform both as students and as adults. Mucking about with the education of students who haven't been sabotaged by their parents is governmental thumb-twiddling.
>Most things people do aren’t that hard and people have the ability to learn to do them
That's true, but it's the small minority of tasks that require real intelligence that often matter the most. A regular person can probably be trained to 95% the skill of an anesthesiologist just by following instructions, but then they would kill the patient during edge cases.
The same thing applies to programming too. I had an internship at a regular company, and now work at a FAANG. The developers at the regular company are likely better at programming than me, especially when it came to regular tasks, but some of their technical decisions just didn't seem to make any sense.
> A regular person can probably be trained to 95% the skill of an anesthesiologist just by following instructions, but then they would kill the patient during edge cases.
You’re making the mistake of assuming anesthesiologists are something other than regular people with training.
Anyone who makes it through medical school is definitely smart by some standard, especially if they end up in a highly competitive specialty like anesthesiology. Does all that intelligence translate to talent for being an anesthesiologist? Probably not, but I'm sure a lot of it does.
I think this is where we’ll really disagree. Completing med school is a function of opportunity, determination, and to a much lesser degree intelligence. Doctors are just people and people have roughly the same innate abilities.
> Most things people do aren’t that hard and people have the ability to learn to do them
Have you considered that maybe you're just unusually talented? I know most things people do are very hard for me, and the few things I do are very easy. Learning history is an uphill slog even though I love it, but math doesn't warrant effort. For a while I just figured people that failed math were lazy and people that passed history were geniuses, but it turns out people have different amounts of natural talent. It's not the start, it's the slope
As narcissistic as I am, I don’t think I’m “talented.” I’m just like everybody else — can get pretty decent at a lot of things. But also just like everybody else I like to pretend I’m talented at the things I put work into. And I like to pretend that people who are good at things I haven’t put work into are talented in a way that I’m not.
I mean, by hours at this point I've definitely put more time in to math than any other subject, but it was easy from the start! Maybe I'm enough of a fuckup that it intensifies differences in ability which would otherwise be too small to notice - like math took close to 0 effort to do and want to do, so anything else is unbearable. Doesn't square with just how much effort I put into history though.
As narcissistic as I am, I don’t think I’m “talented.”
Huh. To me it seems more narcissistic for someone to say that they have no special talent and that their success is entirely due to their superior work ethic and years of study and sacrifice.
I’m just like everybody else — can get pretty decent at a lot of things.
That's a very different claim than "innate talent doesn't exist".
yes and no. the old nature vs nurture + not all people are genetically gifted and a small generic advantage can mean a huge difference in capabilities.
I don't think it has to be related to any point you put in here. I think when STEM people comment on math education, they easily forget K-12 math education is for all students, not future college STEM students.
Lots of controversies in math education between STEM professors (especially mathematicians) and K-12 math educators/researchers are rooted in this. In the community of math and science education, we educators/researchers always focus on average students who will grow into future citizens, not STEM workers. This is really a different mindset to STEM professors.
Middle and high schools start auditioning and training students into “advanced” art, preparing them to go onto competitions and onto serious careers in programs differentiated from the casual art classes.
Learning calculus won’t be enough as a professional in STEM either — but AP Calculus is the equivalent of audition-only advanced art classes. (Which exist all over.)
Do most public schools really have the capacity to train students for serious art careers? At least in dance and music, at the highest levels everything is purely student driven (school dance and music is typically not super competitive for the serious artist) and presumably visual art is as well. Unless you go to an arts high school, but there aren't that many of those
I went to two high schools, one very poor and one moderately wealthy.
Both had audition programs in art for “advanced” classes (both music and drawing), where students were matched with more serious training and where their bands went to competitions and drawings were entered in regional shows.
I think you’re confusing “good, on track for professional” with “absolute top tier” — many students from regular schools go on to, eg, be animators at a studio or choir directors.
The same split exists in math:
- advanced classes you have to place into exist at most schools, eg AP calculus
- but to be the “absolute top tier”, you’re talking about STEM schools and private mentoring programs
You need the first for engineers, scientists, etc — even if they’re not going to be Terry Tao.
That's fine, but it's the reason we have tracks. Future STEM workers can go into the advanced track and everyone else can go into their own track.
Germany separates people into vocational school and something closer to what we'd consider high school in the US by 9-10th grade. If you embrace the idea that some people are simply less suited for intensive math - whether it be because of work ethic, inherited IQ, lack of interest, etc. - and give them a path towards jobs that better fit their skillset, I think you'd see a lot less people drowning in college debt because they got a degree in sociology when they got weeded out of Calculus 101.
Totally agree. That's why there is a CTE (career and technical education) movement in the US now. Perkins V is the strong push in this regard. https://cte.ed.gov/legislation/perkins-v
So then let's have a few separate paths instead of only one curriculum. I believe that there is a lot of value to a basic algebra + personal finance + probability math track that helps a future plumber understand everything they need for their career.
Depriving college-bound students of calculus, though, is a bad move. A lot of philosophy also involves calculus-related arguments (ie if we cut up space in really small pieces, what do we get?), so it has applications outside the STEM fields.
Is calculus an important highschool goal? I feel like I may have benefited more if that time was spent on statistical literacy than calculus. I encounter stats very often in my adult life, calculus style problems are rare and I don't remember the formulas offhand, so end up just looking up what I need.
It's true that to someone familiar with collegiate mathematics, it doesn't feel too important to make that the goal - sure, why not statistics (except perhaps that a thorough understanding of statistics requires some calculus), or why not discrete mathematics, or number theory, or linear algebra, or set theory...there are lots of topics! Mathematics really is a tree with many branches, and you're correct that the high school track towards calculus just develops one trunk with a couple stunted growths, which is definitely unfortunate.
Unfortunately, I think it comes down to resource constraints: When I attended a relatively wealthy and large suburban school district that offered more courses than most other school system in the state, there were only 18 other students who took AP Calculus BC our senior years (and one anomaly who took it his junior year). There were a couple classes for Calc AB, mostly seniors and a few juniors. That special 18-student course was already pushing the limit on the minimum class size, a couple years prior they hadn't had enough students and didn't offer it at all.
If you'd split the curriculum into discrete math and statistics as well, there wouldn't be enough resources to support those branches. To take a chainsaw to the analogy, you wouldn't have the straight but sturdy tree trunk we have now, you'd have a stump or maybe a shrub.
The point isn't that every single student should take calculus. The goal is to make it so different students can choose the path that's best for them. California's proposed changes make it much tougher for students to take calculus in 12th grade, let alone earlier.
Stats requires calculus; you can't even define a probability distribution without some pretty advanced notions of real analysis. Discrete math, linear algebra etc. are viable alternatives.
Stats requires calculus; you can't even define a probability distribution without some pretty advanced notions of real analysis.
Only in the same sense in which operating a car requires advanced knowledge of mechanical engineering, electrical engineering, aerodynamics, fluid dynamics, thermodynamics, etc. IOW, in no sense related to the everyday, ordinary practice of operating a car.
Sure, statistics requires calculus... and real analysis, and probability, and measure theory, and FSM knows what else - IF you're doing statistics research or trying to break new ground, or do really advanced things. But all of this is light years away from the level of statistics knowledge needed by Joe Q. Public to better understand (and not be misled by) the "statistics" frequently thrown out in news articles, government reports, etc.
Please, for the love of FSM, can we stop this HN "thing" of assuming that every mention of any mathematical topic implies that the goal of the user/learner is to do original research in the field?
Is this really true, in practice? For example, given its importance in every day life, I think everyone should understand test sensitivity and specificity, and how these relate to but are quite different from predictive value positive and predictive value negative. All of those topics can be understood with basic algebra. Similarly, I recall my introduction to biostatistics class I took at an Ivy League institution, and I don't really recall using calculus much in any of it.
At some point, someone was saying precisely this about literacy. And it was as true then as your comment is now. You only need some range of function around the baseline education to operate in society. But if you want society to progress, then everyone needs to learn to read and write.
Bad analogy. Being able to understand statistics at an everyday usage level is the equivalent of being able to read and write. Understanding calculus needed for statistics is the equivalent of needing to understand Latin or comparative linguistics. Noble and useful pursuits for sure, but you can get far with an educated society that just knows how to read and write even if they're iffy on the Latin roots of it all.
This is quite clearly false. Most people on the planet have zero to little understanding of statistics. They get by just fine. The field of Calculus is actually fairly simple compared to stats. By and large calculus is the study of the slope of a line, and the area under the curve.
Statistics is more complicated philosophically and computationally, as you do in fact need calculus to do statistics, since a fundamental tool of statistics is evaluating the area under a variety of distribution curves.
You've misunderstood my post. I'm not arguing that needing to understand statistics is the same as needing to be able to read and write; I was responding to a post that was arguing that teaching stats without the underpinnings of calculus is like a society that is barely literate.
> Statistics is more complicated philosophically and computationally, as you do in fact need calculus to do statistics
This is just fundamentally false, as there are many statistical concepts and introductory statistics courses (given at some of the most renowned universities in the world) that can be taught without an underpinning of calculus. It's like arguing that you can't teach a programming class until someone understands digital logic design.
Again, philosophically and computationally, statistics is more complicated.
And again, you do need calculus to do statistics. Maybe you're saying: "If someone does the calculus part for you, you can use this one equation to solve this one question."
That's akin to calling a spelling test literature. You are confusing practice with knowledge. Anyone can bang on a drum. That does not make them a drummer.
At some point someone was saying the same thing about Greek and cursive, too, though, so being merely more than the baseline isn't sufficient evidence that it should be included in everyone's education.
Not at a high school level. AP Stats I believe does not require calculus as a prerequisite.
Discrete distributions can be defined, even infinite discrete distributions (as sequences and series are taught in pre-calc). Continuous distributions can't be formally defined, but a lot of intuition can be given with hand-waving e.g. the area under this curve. Probably most of the class is spent with counting and probability-type problems, but plenty of actual statistics can be done without calculus - we can learn about distributions, what a statistic is, expectation, sampling, counting, probability... the list goes on.
But that's like saying everybody who learns to program needs to know electrical engineering so they understand how CPUs work. Just as most programmers don't need that to be good programmers, knowing useful statistics doesn't mean deriving things from first principles but rather knowing what statistical test to apply to analyze data and how to interpret the results. This doesn't need calculus.
I took AP Stats in high school. I took a calculus based probability course at MIT. The former was extremely important to me and I learned a ton. The latter was interesting, but mostly unnecessary. A z-score lookup table is more than enough to teach the concept of a normal cdf without actually being able to derive it yourself.
As a professional data scientist I've never needed to use calculus unless you consider graphical reasoning on distribution diagrams to be calculus.
Most people who deal with data/statistics in their working lives do not need to know real analysis, do not utilize calculus to draw (correct) conclusions, and do not use linear algebra either. They learned all these things once and forgot them a long time ago, because they did not need them. They utilize statistics an order of a magnitude more often than they do calculus.
They are not statisticians, but people who need to deal with data as part of their job.
As someone who took both a stats and newtonian physics course before taking a calc course, I wish I hadn't. It was a waste of time. They can't explain why you use the formulas they do, they have to just say "trust us, and be able to regurgitate it on the exam". For me, learning means developing an intuition, which means resolving/building new facts from other facts I have already accepted. Being handed seemingly random formulas to memorize goes directly against this. Yeah, I can use my car without knowing how every piece inside works, but the moment something goes wrong, I don't know what to do. I would never say I am car-literate, and someone who hasn't taken calc cannot be stats-literate.
But how would you "motivate" the formulas without knowing where they came from? Why is this the formula we use and not something else?
To go back to the car analogy, I know why I need an engine, you might even say I know how to use the engine, but if the engine dies or I want to use the engine for some other purpose, I'm not equipped to do anything.
I don't have "literacy" with engines, I have rote memorization of a series of steps. I don't have enough information to know why the steps are what they are, nor could I know under what conditions the steps should change or what they should change to.
Sometimes they'll need formulas they don't understand (can't motivate) for doing work things?
Then what? Are they going to incorrectly use anther one, a wrong one, because they understand it better?
(Searching not where one lost the key, but where the light is stronger)
Hmm, I wonder if you studied stats? It sounds as if you studied lots of math.
I did long ago, mostly forgotten, and, hmm, I don't think there's a single real life useful probability function one can understand, without knowing the underlying calculus? Except for understanding in the sense of trusting the teacher.
You can't even define arithmetic on natural numbers without some pretty advanced notions of logic and set theory. But you can definitely get a lot of value from arithmetic without those definitions.
Colleges have varying levels of stats classes for different majors, and not all of them require rigorous understandings of calculus. The ideas and principles presented in those classes are still important for people to learn, even if they are unfamiliar with, or haven't mastered, calculus. It's possible to teach those principles in high school, as well.
There’s a diminishing return here. Single variable differential and integral calc is a sweet spot. Without that, there’s a ton of memorizing seemingly unrelated facts, but with it, you can learn a few principles that lead to a huge amount of practical stuff.
i’d argue that you can’t properly define probability without notions in measure theory, which is obviously far too advanced for a high school student. i’m not an educator, but some middle ground needs to be struck. i think it’s clear to many that the quality of education in american colleges far exceeds the quality of education in the average middle or high school. that’s the issue, imo
You only need measure theory when working with something that is not easily replaceable by R^n, Z^n, or finite sets to meaningfully define integration, otherwise (in)finite sums and Riemann integration get you very far.
I am a bit rusty on my advanced probability theory, but IIRC the only thing that required* measures was defining conditional probabilities and expected values on zero-probability events.
Of course redoing that class without Lebesgue integration sounds excruciatingly painful.
* Not just to make proofs nicer and theorems more powerful
I'm certainly very happy that I learned calculus in high school. I then got a 5 on the AP calculus test and tested in to third quarter calculus in college. It was important as an engineer to understand those concepts. I don't use them too often and I forgot a lot of it, but I really enjoyed it and I think it would be unfortunate if other students did not have that opportunity. I agree stats would be great too!
As a software engineer, I wish I had learned stats instead of calculus. Some exposure would have been great, but the high school & university requirements were way off target wrt its usefulness in computer science. It was a painful process of learning, failing, and re-taking calculus, squeaking by, only to never use it again. I was a straight-A student otherwise.
I get that. As a robotics engineer, some cursory understanding of integrals and derivatives is useful.
But what I really mean is that, as a person, I just really enjoyed calculus. I found I was very good at it, and that experience helped me understand why some people choose to focus their career on pure mathematics. I am happy I took calculus not as a means of training for the workforce, but because I found it enriching on its own. And I never would have taken all that time if it wasn't offered to me as a class in public school that counted as credits towards graduation.
I think if the policy was "calculus isn't an important goal, we should actually teach stats" the reaction would be different.
I give the edge to calculus because it allows students to go right into physics and be able to graduate college with an engineering degree in four years (saving them time and money), but any challenging quantitative material would be good for their development.
The big picture goal is to show them there is this big world of problems that can be approached with specialized knowledge and get them familiar with what it takes to gain that knowledge.
> any challenging quantitative material would be good for their development.
I think this is key. What exactly they study may not be quite as important as whether or not they are actually getting an opportunity to do some form of challenging mathematics.
The abstract concepts of calculus are useful and will shape the way you think about and go about solving problems, even if you don't explicitly employ an integral or derivative. Rates, sums, areas, volumes, etc.
Learn the nuts and bolts in highschool, use the intuition for the rest of your life.
Seriously this. The average American doesn't grasp how basic graphs work. This simple idea of trends and how different functions imply graph differences is a powerful basic thought model. "Is this a linear or exponential curve curve problem?"
The abstract concepts are useful, but in practice most effort is spent on applying rote symbol manipulation rules to questions specifically designed such that applying the most obvious rule at each step will reach the solution. The idea of a tree search in symbol manipulation is never taught, so if you try solving non-trivial real-world problems you will likely manipulate yourself into a dead end.
Highschool calculus should be taught with computer algebra software. That's what you'll use in real life as soon as you find an even slightly difficult calculus problem. There's not enough time to teach both the symbol manipulation rules and the intuition.
I would've done better in college - and probably have a better understanding of calculus today - if I hadn't tested out of University-level Calc I due to AP credit. I didn't really know what I was doing in the high-school course.
But I was a slacker and that experience doesn't necessarily transfer.
And yes, I'd generally favor stats over calculus as an additional HS class; however, I am hesitant about discouraging the opportunity to take either.
The quality of mathematical arguments presented in AP Calculus are significantly higher than other "standard" high school courses. So to the extent that it is a nationwide program that promotes some careful learning, it is a big plus.
I think statistical literacy is also important, but more than anything I think students benefit from learning how to think about hard(er) problems. If they learn that in statistics, great.
Generally I would say easy courses is the real problem, not content.
However, many many students enter college engineering programs with 1-2 semesters of calculus, so not having it could be a competitive disadvantage - presumably to your understanding of those first year classes.
I had calculus before college, got a 5 on the AP calculus exam, and it did not seem to have prepared me in any way for college-level calculus. I always thought that had been a total waste of one of my senior year class periods.
I took calculus in High School, it was a dual credit course with a state university. I got credit at the university for the first semester of calculus (which was taught over the entire year in High School, so at a slower pace, but also allowed the High School class to spend more time on review in the first month or so, compared to college courses which basically dive right in). We took the same exams as the university course.
I felt I was prepared for 2nd semester calculus when I took that as my first math course in college.
Actually, yes. The rules for manipulating 'expressions' with 'variables' in school algebra describe what are called "free objects" over some set of "generators". The whole setting generalizes pretty well.
Calculus is important, because in high school you only superficially know the breadth of the field you're interested in, where you may fit, and what you need to get there.
I unwittingly had a "low math" high school education, with pre-calculus only introduced spring of my last year. Freshman Engineering was a shock, with Calc, Physics, and Electrical Engineering using concepts I'd never heard of.
There's nothing worse than thinking you're accepted, prepared, and ready, and finding you're totally wrong.
Lets say you need to find the probability of something happening 10% of the time to 40% of the time, you need to perform definite integration of the curve ( lets say normal curve ) from 0.1 to 0.4 on the x axis multiplied by the normal curve function. This is one of the easiest examples I could remember from my undergrad. We could solve these problems with ease at undergraduate level because we grinded hard during our high school. And also these type of problems were just a subset of the huge variety of problems presented during our undergraduate. But lets say they started teaching calculus only during Undergrad it would have become a tremendous task just to first learn about calculus then start with applying it on other subjects. I am all in for teaching calculus during high school.
> Lets say you need to find the probability of something happening 10% of the time to 40% of the time, you need to perform definite integration of the curve ( lets say normal curve ) from 0.1 to 0.4 on the x axis multiplied by the normal curve function.
No one does this in the real world. Not even professional statisticians who know calculus. In the old days they used table lookups. Today they use software.
You need to understand the concept of areas under curves. Calculus is just a means to compute the area.
There's a lot of competing / strange interests in school systems that can have well intended but BIZARRE outcomes.
My wife works in early childhood education. At one point it was recognized that the early childhood department should be more involved in helping students with learning disabilities as soon as possible. There was lots of outreach to parents to get them into free classes and education, and most importantly screening so they could get free services if they qualified / needed them.
However, it was noticed at some point by some very vocal parents that some students with specific backgrounds were refereed to these services more than others. These services were provided in and out of school, the kids weren't moved to another school or anything like that, but despite all their efforts... The result was deemed to be some sort of bias, or outright racism.
Therefore it was made very clear that they could not disproportionately "single out" students of some backgrounds for these services, that are free, to help them learn.
1. Math is a differentiating subject for getting into those competitive colleges, departments, and professions. In the meantime, the progressives simply refuse to believe that some people are just better at studying math. The logical choice, then, is to dumb down math to "level the play ground". It's the same unspoken reason why so many people pushed the magnet schools to use lottery to pick students (I actually think lottery with threshold can be a good solution, but that's another subject).
2. Progressive math educators have been advocating self discovery and that everyone can learn math in their own pace for years. What educators need to do, per the progressive argument, is to protect the fragile passion and creativity of the kids. Jo Boaler even argued that kids should discover all maths by their own. Naturally, we have to dumb down math courses, otherwise we would inevitably hurt the confidence and passion of some kids. As progressives always said: no kid should be left behind and some people got better at math only because they were socially privileged. I disagree with the progressive view of math education based on my personal experience, as so many classmates of mine simply were not interested in STEM, and maths in particular. I'm not sure why we don't accept that most people will hit a wall sooner or later when learning maths. To some it is arithmetic, to some it is calculus, to some it is abstract algebra, etc and etc. To me I definitely lost my drive when taking courses like model logic, and I certainly do not have interest or talent to get good at things like functional analysis or topology or algebraic geometry, but I make peace with it. I really don't understand why the progressives are hell bent on insisting that everyone can learn maths equally.
Goodharts law is in effect. They have a target they are trying to hit and are aiming a different way towards it.
They are optimizing towards "High School Graduates" and "College Graduates". And if they need to destroy the value of being any kind of graduate to get there. So be it.
The intention was to dumb down mathematics. If you lower the bar sufficiently you can get everyone over it. Then we'll all be equal, which is the goal of this curriculum.
It's almost a case of inmates running the asylum --but in this case, it's not even the inmates but their caregivers who in their maternalistic view seem to think they know what's best for the "inmates" and are guiding them to the path to "hell" --hell being a reduced education in an increasingly competitive labor market.
I think you’re looking at it the wrong way. The job that the administrator is hired for is “make Black and White test scores identical.” With only the leverage of the school, and no broader socioeconomic levers, the only way to make this happen is to reduce all assessment to 1+1=? (Multiple choice)
I’d rather linear algebra and discrete math be the goal. Calculus is greatly overrated. I mean, sure you should take it, but IMO linear algebra is considerably more useful in the real world and most people never take it. Knowing how to integrate and differentiate in continuous space isn’t nearly as useful as learning how to count in discrete space. Most people operate in a discrete world.
My guess is the intention is to be able to say that the framework benefits disadvantaged people. But like almost any policy like this all it does at best is pull down people at the top, at worst pulls down everyone making the situation worse for everyone.
You can have separation of education by ability, and progress, or you can have equality, and everyone being pulled down to the same low level. And suffering for everyone. You can’t have both. Take it from someone who has direct experience with communism, which is the same mentality that drives this.
> You can have separation of education by ability, and progress, or you can have equality, and everyone being pulled down to the same low level.
This seems a little off. What we're talking about (and what it seems like you're defending) is directing more resources towards the most gifted. It's fine to believe that, but it's an argument to give the most to those who have the most. Nobody is pulling anyone down, and communists are as happy to grant power and resources to those with aptitude and connections as capitalists are.
edit: with the constant attacks on teachers, it might be more realistic to stop aiming for calculus in high school. Any kid who manages it within a gutted public system would have gotten there anyway, no matter what situation they found themselves in. They can download calculus books and calculus lectures now; with the internet a feral education is within everyone's reach.
As the parent of a child who is gifted at math this is wrong on so many levels, I’ll just state one. My kid only has so much time he can genuinely focus on “school work” in a day, why should he be forced to spend “school” time on things wildly beneath his level and then come home and spend his own time on additional “school” type work?
How is it directing more resources to allow students to take courses at their level? It's not like you have to pay high school math teachers a higher salary to teach calculus. Your typical public high school in California has 1,000+ students. With that many students it's not going to be hard to find 20-30 students to register for a calculus class. It's not like you're running a special private class just for a few gifted students.
Every student needs and should be entitled to an education that pushes their abilities. Without it, they won't learn to put in the effort required to be successful even at a level that doesn't make use of their enhanced abilities.
This might mean that there are additional costs for 'gifted' students. But that's no different than there being additional costs for mentally less advanced students: no matter what the students are, providing an education fit to their needs has costs... and a failure to provide it creates inequality when the most privileged families move their children (or relocate entirely) to places that will provide a more suitable education.
Being able to integrate and differentiate is less important than understanding that P(B|A) does not, in general, equal P(A|B), and you can learn that latter fact without learning how a probability distribution is defined. Wikipedia has a proof of Bayes' Theorem without calculus:
Yes, it is ultimately founded on calculus. But we don't teach numbers to kindergarteners starting from Peano's axioms, and we don't need to gate the essentials of conditional probability behind calculus. In fact, doing so is positively harmful to society at large.
I could see replacing Calculus with Statistics as a goal for High School. It seems more broadly applicable, and both are offered at the AP level. But that doesn't seem to be this.
The perception of math by the public has led to a cognitive predisposition that math (especially calculus) is beyond the ordinary person. I wish pop culture would transform this.
What does "calculus" here mean? I'm not American, no idea what's included in that word and what's outside, in this context. Does it mean limits, derivatives, integration (Newton), maybe even some high level talk about ODEs for the "very best" schools? Anything more, anything less?
Also, in case anyone is also wondering, 8th grade means 13-14 years old.
Wait, is this high school or college? That looks like 3 different courses you're listing there - I've never heard of a high school offering more than a single year of calculus.
Some schools which allow students to take calculus 1 and 2 before senior year also offer multivariate calculus (3), differential equations, and linear algebra courses to round out the fourth year of math. This is especially prevalent when students can take geometry in 8th grade which leads to Algebra II, Pre-Cal, Calculus, Advanced Math Electives as the four year progression.
My public high school in WA also had this. "AP Calculus" covered Calc I and II, and once you had completed that there was a special elective called "Advanced Calculus" that covered Calc III (and prepared you for the AP Calc BC exam). In my graduating class about 30/430 ended up in this Advanced Calculus class senior year having finished AP Calculus as juniors, but many more who completed "regular" AP Calculus as seniors also ended up in STEM fields.
I took Advanced Calc and AP Stats senior year, which in retrospect was a mistake :P
> limits, derivatives, integration (Newton), maybe even some high level talk about ODEs
My son is taking high school BC Calculus (one step above "AP" calculus) this year. It includes limits, derivatives, integration (including integration by parts and partial fraction decomposition), ordinary differential equations, infinite series and taylor/mcluarin series.
> My son is taking high school BC Calculus (one step above "AP" calculus) this year.
BC Calculus is the advanced version of AP calculus, not "one step above". Ideally people are moved into AB or BC Calculus based on skill level, but only take one AP course.
At my US school around the year 2000, precalc was one option for seniors, which primarily focused on limits and derivatives. The more advanced pace AP calc course also went into integrals. Beyond that I don't really recall, but by that point you also had gone through courses focusing on basic geometry, basic algebra (using variables, factoring), and a course dedicated to trigonometry (mostly memorizing the rules around figuring out angles).
There were some other courses that had math involvement, but were more business oriented (finance / accounting type stuff) and I don't recall if they counted towards core math credit requirements.
Calculus without analysis, so the mechanical rules and recipes of real analysis of well behaved scalar functions that an engineering course might use (and that were used by the developers of Newtonian mechanics in the pre-modern era), limits on intervals, Riemann integration, etc.
I don't think calculus should be a goal to be honest. Or at least not as taught. Calculus could be greatly condensed to a shorter theoretical view. The ideas of understanding differentiation and integration are great. Memorizing the rules of doing it is pretty painful and likely won't stick. But that's the bulk of classroom time, homework, and testing.
Maybe for most of them, however i found calculus totally useless for my CS degree (the only time i recall it mentioned was defining big-oh notation). Otoh i liked calculus so still time well spent.
Of course, that's not counting "mathamatical maturity" which is super important or if you're doing some specific thing that needs calculus (hello machine learning.)
You can teach these in university, it's not a problem. Calculus doesn't need to be taught in high school to everyone but it should be available and it should be the goal state in terms of curriculum pace for everyone so that you should have no problem taking it by the time you are 17 or 18 (which is what we're talking about).
Anything else propagates back to a regressive dumbing down in an earlier year, from an already dumbed down curriculum by international standards.
I'd guess the vast majority of software development jobs are like "gluing one API layer to another" and "writing simple-to-complex CRUD apps". Neither calculus or discrete mathematics really helps if your goal is to simply make a computer read data from database X and display it in webform Y.
I found all of the math required by my undergrad degree to be totally useless in real life programming. Whether you need any math at all will highly depend on the application domain you get in to. The most complex math I needed as a code monkey was vector arithmetic (3D graphics) and trigonometry (ocean and aero mapping navigation).
There is a move to get basic care into the hands of nurse practitioners and probably similarly most programmers shouldn't bother studying anything. If there is enough consumer demand to fill experience-based jobs then that's the market reality but it doesn't mean doctors who are meant to invent new cures don't need biochemistry or engineers who are meant to design new solutions don't need mathematics.
Most programmers will never design new solutions and are objectively terrible at their job.
> Most programmers will never design new solutions and are objectively terrible at their job.
I don’t think you know anything about professional programming if you think that makes them “objectively terrible at their job”. Sometimes businesses just need a website.
We're talking about doing CS as a degree in college, not coding in a job. You don't even need to study CS to get a coding job.
As an aside, those secretarial coding jobs will all go away within 50 years. They are only needed in the transition period where machines still depend on humans to talk to each other.
I found it incredibly useful for learning all sorts of probability theory despite hating calculus. And I really think to be a well rounded CS graduate you need some background in stats/ML nowadays. So many of our systems have some element of ML-based recommendation and it's important that a new grad can meaningfully engage with those systems in research and in industry.
I don't disagree, but it's an unfortunately circular argument. "Calculus is the peak of HS math because college STEM programs expect you to have calculus because it's the peak of HS math." I think probability and statistics is almost certainly more useful in the real world, but it will never become the baseline while everyone is fixated on calculus.
Many high schools offer mathematics through Calculus. You can typically get to that course at the regular high school pace if you were able to take Algebra 1 in 8th grade. If you were ahead of that pace, then you are typically left with few options outside of taking college courses.
SF CA resident, parent of two school age children chiming in. The direction with math seems pretty dismal, in that as of right now, everyone is singly tracked together for math through freshman year of high school. This results in children who have higher aptitudes[1] to not be well served by schools. The majority of people I know who have the means opt out of the public school system, which probably makes the problem worse generally but solves a pain point for them.
[1] - I take it as a fact that different people have different talent levels for different things, but not everyone agrees with that, and disagreement on this point is a big driver (but not the only driver) in the "everyone gets exactly the same" approach that is trending now.
I don't know if it's still there in the revision, but in chapter one of the earlier draft of the California framework it said, in a prominent place "we reject ideas of natural gifts and talents"
Edit: in the new version it has been changed to "high-level mathematics achievement is not dependent on rare natural gifts, but rather can be cultivated"
> "high-level mathematics achievement is not dependent on rare natural gifts, but rather can be cultivated"
I mean, I would hope this is true.
I'm not "naturally" gifted at mathematics, but like reading, writing, and other things, I can learn them in school and got quite good at them.
Public education is like mass transit. Not everyone gets their own Lamborghini. Most have to take the bus. Its goal should be providing the best general education it can for all people and making as many people as possible productive.
If you looked at society 500 years ago you could assume that only certain people were smart enough to read and write.
It really depends on how you define "high-level". Yeah the attitude that some people "just can't do math" is not good, I don't disagree with you there. But that's not the same as acknowledging that some people may pick up math more quickly.
Holding kids back is really the opposite of cultivating mathematical achievement. To use the reading analogy, do you think a kid that can read at 4th grade level should be forced to only read 1st grade books anyway because that's what their age is? I'm not sure what that accomplishes.
It'd be one thing to make a resource allocation argument but that's not even what this is. This curriculum is clearly a philosophical statement and personally I don't get it.
> But that's not the same as acknowledging that some people may pick up math more quickly.
Certainly some people are better at things than others - but so what? I know plenty of people that excelled in math in grade school and struggled in high school, also many more that excelled in high school and struggled in college. Some, like myself, struggled in grade school and excelled in high school. The difference was motivation.
> Holding kids back is really the opposite of cultivating mathematical achievement. To use the reading analogy, do you think a kid that can read at 4th grade level should be forced to only read 1st grade books anyway because that's what their age is? I'm not sure what that accomplishes.
And how many brilliant kids moved just a bit too fast and lost interest? The thing is you only view things one way. You forget that a fast ramp-up in difficulty can turn away many students who could've turned out to be brilliant scientists and engineers.
> It'd be one thing to make a resource allocation argument but that's not even what this is. This curriculum is clearly a philosophical statement and personally I don't get it.
I don't agree that it is a philosophical statement, having read it, it seems pretty straightforward. The alarmism about the woke mob is overstated.
As mentioned by OP:
> in chapter one of the earlier draft of the California framework it said, in a prominent place "we reject ideas of natural gifts and talents"
The people that wrote said publicly available draft are still involved with this plan, and have not personally nor explicitly backed down from the statement. It's a good sign that some moderation has been introduced to the text, but it seems clear to me there are still some pretty extreme beliefs amongst those leading this thing.
Whether the "woke alarmism" is over the top or not, I don't think it should be controversial to say that a philosophical statement is at the root of this plan. I wouldn't be surprised if the Equitable Math folks would agree with that assessment even.
I also don't think that criticism of a specific model of leveled courses should be used to dismiss all leveled courses. You talk about kids that are rushed ahead or that perform differently at different points in their math "career" - which could certainly be problematic if levels are rigid throughout the educational timeline and leave little choice to students.
Yes sometimes it is implemented that way. But it is not impossible nor even particularly impractical to implement a more flexible levels system that would mitigate those concerns. There are schools that have done this well, California school system was not one of them. This proposal is throwing out the baby with the bathwater.
I don't see a problem in that statement. The opposite would mean that you are born with ability and can't grow or improve it, which is absolutely false. The idea behind the statement is basically: people do not have to be naturally gifted to be good at math.
> I also don't think that criticism of a specific model of leveled courses should be used to dismiss all leveled courses. You talk about kids that are rushed ahead or that perform differently at different points in their math "career" - which could certainly be problematic if levels are rigid throughout the educational timeline and leave little choice to students.
But arguing for a gifted track IS rigid - it basically says "you must decide now if you're good at math, or not." That is deeply flawed. You can't add flexibility - you need algebra and geometry to do calculus. If you decide or become motivated too late (even if naturally gifted!), you have no recourse, as the "gifted track" starts before you can even know. The current system and system you advocate for could be removing huge numbers of potential STEM graduates from the mix.
> Yes sometimes it is implemented that way. But it is not impossible nor even particularly impractical to implement a more flexible levels system that would mitigate those concerns. There are schools that have done this well, California school system was not one of them. This proposal is throwing out the baby with the bathwater.
Not really - as is said in the framework, most foreign nations that outperform the US have a standard curriculum. So the problem isn't flexibility.
"The framework builds on the strategies used in a number of high-achieving jurisdictions (e.g., Estonia, Finland, Japan, and Korea) that pursue an integrated curriculum—connecting the domains of mathematics with one another as students collaborate in using data to solve real-world problems. These countries pursue a common curriculum in elementary and middle school, supporting more students in reaching higher level mathematics. The framework illustrates how this integrated approach with many different kinds of supports can be used to expand the number of students excelling in mathematics and heading for science, technology, engineering, and mathematics (STEM) careers."
If you define "natural talent" in such an unconventional way sure I guess. But the existence of innate potential does not imply that abilities cannot be improved. Derek Jeter has natural talent. Derek Jeter worked his ass off to cultivate those skills. There is nothing at all mutually exclusive about these things.
Schools spend a lot of time reviewing things over the course of the academic year, including things from prior years - I disagree that it is not possible for students to move tracks with a well thought out curriculum plan. But regardless the proposed curriculum eliminates material that would be covered in upper track courses, and explicitly states it does not aim to have students prepared to take calculus during high school. This is like putting all students on the lower track, which is a hell of a solution to the problem of students getting stuck on the lower track.
Japan's model is much more like putting every student in the high track, it is not comparable to what is being proposed here. Japanese high schoolers are able to take intro analysis in 11th grade, here is a translated textbook that would be extremely rare to see a US 11th grader cover the same material (even with standard tracks you'd be lucky to cover it all in 12th):
https://bookstore.ams.org/mawrld-11
I'm not familiar with math education in those other countries mentioned, but I imagine it is more similar to Japan than the US. Hilariously there have been some recent pushes in Japan to be more like the US education system as far as flexibility is concerned, in order to take some of the pressure off of students.
Actual math is irrelevant to this whole argument, just as actual literacy was irrelevant ~500 years ago. This isn't about math ability, it's about the fact that STEM is an obstacle to woke bureaucrats' projection of power.
I'm not sure if I buy the woke-alarmism that is rife on HN. Reading the framework, I don't see any of this "wokism."
It seems like any other guided/misguided attempt to improve the school system that has already taken feedback into account.
This entire HN thread is filled with "back in my days" and "wokism" but no one actually is disputing the content of the framework itself with actual evidence.
As a student who participated in Math Olympiads throughout middle and high school, having to be on the same track as everyone was downright painful. This type of thing really shouldn't exist.
That would drive my wife up the wall. Our school district in the Pittsburgh suburbs has 5 math tracks from grades 4-12. They just added linear algebra because so many kids were maxing out the available math curriculum.
My understanding is that in the US ‘linear algebra’ is used for both the thing that involves manipulating grids of numbers in various ways (so the basis is implicit), the thing that is a bi like algebra but for matrices and vectors, the thing you have in physics where linear maps have specific geometric meanings (so you care about being mostly basis-agnostic, and you care about how the objects change when you change basis), and the thing which is abstract algebra for vector spaces and so on.
When I was in school in the U.K. we did the first and second things, including eg multiplying matrices, eigenstuff, diagonalising them, inverting small matrices, some determinant/cross product stuff, and we maybe did the thing where you solve a first order linear ODE system by converting to matrix exponentiation, though I don’t quite remember. I think we just called it vectors and matrices.
There was some useful stuff there. The problem is that it was at a course so close to the leaves of the ‘x allowed to depend on material from y’ tree that we didn’t get to apply that much (related example: we had to waste a bunch of time on silly equations in physics because they couldn’t depend on us knowing about the y’ = kx ODE)
At university we did some courses in vectors and matrices / vector calculus that went down the practical route towards physics things and useful tools, and we had a course called ‘linear algebra’ that covered the abstract algebra side of things, where everything was lemmas/theorems/proofs beginning with e.g. suppose e1, e2, …, en is a basis for a vector space V over F, …. However it is certainly possible that the US terminology (linear algebra for everything) was more common outside of the courses I took.
I think I can respond to your footnote. I went to a Catholic school that split us up into separate tracks for maths specifically in grades 4-8. I was in the upper level math class for a year before they moved me. I had a teacher who celebrated and encouraged bullies, slapped children with a ruler, and threw a chalkboard eraser at me from the front of the classroom because I appeared to be falling asleep. When my grades fell the knee jerk reaction was that I was wrongly assigned to this class and it was expected to have below some magical threshold of attrition. The ramifications for me were that my old friend group would no longer interact with me the way they used to, I was immediately bored in our lower maths class, and I was now a "dumb" kid.
It wasn't until I'd dropped out of college and taught myself math, because of the interviews in this industry, that I learned to enjoy math again. My point is that you're really fucking with the social firmware of kids when you do that. Also, reading between the lines of my life, not being in that upper level math class clearly had no impact on the latter parts of my life.
First, I'm sorry you had a bad experience with that teacher and that it caused you problems for many years down the line.
I'm trying to figure out how this relates. It sounds like you had a bad experience with tracking, and there is a fundamental issue with tracking where some amount of people will have bad experiences, but ultimately you were able to achieve your potential anyway, so why make the experience bad with tracking if people will eventually get to the right level over time - is that a fair interpretation of your comment?
My lack of success in that math program was due to a "bad experience" (I chuckled a little at that phrasing). The fallout of which had social implications, made me entirely disinterested (if not hostile?) to maths, and followed me for years. Point being, separating our classes had little to nil positive impact (for me) in the long run, and the decision of which class I was in failed to address the foundational problems with the class. The latter bit being the most important; in my case it was an abusive teacher with an anger problem, but there's also a good chance that nobody actually sucks at math but that we teach it from a perspective that only a subset of the population will ever relate to or meaningfully learn from. I guess what I'd like to see is some reproducible results showing that giving gifted kids their own course program and separating them from their cohorts actually has a long term benefit when compared to keeping them all together.
Edit:
To make what I'm saying a bit more clear: There's a ton of people in these threads assuming that not everybody is born with math skills (or variants of that attitude). That is exactly what these teachers thought and how different my life would've been had I agreed.
"not everybody is born with math skills" isn't the argument though. The argument is "not everybody has the same math skills". It's a multidimensional and continuous thing, but it's not exactly feasible to teach that way, so some approximation is involved.
It sounds like that was implemented very poorly in your case. You yourself said "I was immediately bored in our lower maths class". I don't see how the solution to this is to make everyone take the lower math class. Certainly there is some middle ground between the California plan and what you experienced.
Same, I had a really hard time parsing that comment. My read is that they had a terrible teacher who had the power to label them as “average” instead of “above average”.
It’s hard to see that as undermining tracking as a concept, since the problem is actually the absence of oversight and termination of bad teachers. No one teacher should have the power to doom a student, but a school needs to be able to recognize and cultivate talent.
Eh I don't think things working out in the long term necessarily invalidates the criticism, we have no idea how OP would have turned out if in a different situation. I'm sure most people that should have been in fast track math but didn't have an option for it still figured shit out long term, but that doesn't mean they couldn't have benefited from better math education.
It sounds like the implementation of levels was especially bad in the OP's case though, so I agree it isn't really a good anecdote to argue against leveling entirely like California is doing. It's throwing out the baby with the bathwater.
In the year 2081, the 211th, 212th, and 213th amendments to the Constitution dictate that all Americans are fully equal and not allowed to be smarter, better-looking, or more physically able than anyone else
It doesn’t bother you sometimes that some people starve while others hoard wealth that rivals the gdp of entire countries?
People going hungry, especially children, is a travesty.
What is your solution to it? We already have food stamps, welfare, fully subsidized healthcare, child tax credits, social security disability, free school breakfasts and lunches, Section 8 Housing, SNAP, WIC, CHIP, Medicaid, churches, charities, social security death benefit and so much more. People are still hungry.
Should we do more? The lesson of Africa says we shouldn’t. Decades of food aid to Africa did little more than make the continent dependent on food aid and drive all the farmers out of business because they couldn’t compete against free.
And I haven’t even brought up lthat there simply aren’t enough billionaires to tax to make even a tiny difference in any of this. Even if you took literally their entire net worth, converted it to cash (which is literally impossible) and spent it on the poor it wouldn’t make any difference.
THe child tax credit cut child hunger by 26%. We let it end. Giving people food isn't an especially hard problem in the US, yet we're not able to do it.
Increasing the eligibility and amount of the credit is helpful, but doling it out in repeated checks that you then have to go back and report on your 1040 is annoying. How many news stories are run about people who depend on their annual tax refund for living expenses? Reducing that refund by paying the money out earlier and celebrating it as a free handout is awfully manipulative.
Not really negatively manipulative IMO, I've worked with many people who don't really understand taxes and they think that a refund is some kind of free money "bonus" and often use it to make frivolous purchases and blow the whole refund at once. Most of these people don't really plan or budget on how to use their refund because they have no idea how much it's going to be, it's just treated as kind of a random winfall.
Doling that out in smaller increments and making the purpose specific (child credit, instead of tax refund) seems like a good nudge into better spending habits... especially when the amount is known ahead of time.
they think that a refund is some kind of free money "bonus"
For many people that get the child tax credit, a bonus is exactly what it is.
The word you should have put in quotes is “refund”. My sister, for example, used to pay about $1,500 in tax throughout the year yet get a $7,500 “refund”.
Isn’t that something? Pay x and get 5x as a “refund”.
in my opinion, as long as people are starving we aren’t doing enough… if billionaires aren’t enough tax the millionaires, if the millionaires aren't enough then tax me too… I can’t look at our existing failures and say “good enough, it’s just too hard”
Nobody should care about disparity, people should care about maximizing benefit for every American. I'm not saying it is easy to evaluate this, but it is obviously all we should care about. The existence of gazillionares is fine so long as individual wellbeing in this system is higher relative to other potential systems.
Firstly, evidentially, people do care about disparity: increasing disparity seems to adversely affect people physiologically, independently from wealth. That is, having someone else be significantly richer than you, independently of your own health, seems to create stress effects in a population. By the metrics of national health and well-being, it seems like wealth disparity is a bad thing in its own right.
Secondly, the problem with billionaires is not that they simply have so much money, it's that one's ability to influence society, including making and breaking the rules of society, is intrinsically tied to money. For example, consider the recent news about Bezos buying a new boat, and having to take down and rebuild a bridge to get it to sea. On the one hand, it doesn't really matter to me how much he spends on that boat - it's his money, and he can use it as he likes. However, the people of Rotterdam were promised that the bridge would remain put, yet Bezos' money (presumably via the shipyard that organised this) was enough to override the democratic process, presumably alongside adding a significant inconvenience to the people living there.
Or consider the recent trend of billionaires buying media companies. On the one hand, it's kind of irrelevant how they want to invest their money, but on the other hand, these media companies afford significant impact on the views and perspectives seen by society. If we really want to claim that we live in a democracy, it seems dangerous to also accept that one person can, essentially on a whim, buy one of the largest social media platforms with only vague hints as to what he plans to do with it. That sort of power is absolutely not something that you (I assume) or I personally can wield, yet it could well have a significant impact in shaping public opinion.
As long as money can be roughly equated to power, then wealth disparity will remain a very important thing to be concerned about.
It’s honestly a fucking nightmare to live here, especially with kids. I’m a crazy “leftist” where I grew up but SF/CA is bonkers (e.g. re: [1] — are you fucking kidding me? Have you not met a single real fucking living human being? My family has a “curse” we ascribe to all the clumsy folks of our blood and I assure you there are more than can be reasonably blamed on chance).
(personally I don't understand what your saying / what the point is, but I'm also tired. I'm not GP btw. -- in what ways is it a nightmare for example? What's a real human and a not real human? "the clumsy folks of our blood" -- I don't understand, how are they clumsy for example)
Something is rotten in the state of academia when looking at the evidence there are mathematicians, scientists and social scientists standing bravely in favour of a high bar and a quality education program in mathematics.
On the other side of the argument you have people from the Department of Education who specialize in Mathematics Education who seem happy to lower the bar as far as possible in the name of equality.
When I was in University the Department of Education was the most woke department on campus, except for perhaps the Department of Gender Studies. We are now seeing policies that favour wokeness ahead of the best interests of the students affected by the policies.
> On the other side of the argument you have people from the Department of Education who specialize in Mathematics Education who seem happy to lower the bar as far as possible in the name of equality.
They should rename the Department of Education the Department of Equity. By pushing frameworks like this, they show that they are less interested in education and more interested in achieving equity. They'll even privilege equity at the expense of actual education.
The worst thing about the CMF effort is that it would only deepen the disparities between rich and poor. Public education is often the only shot poor kids have to gain knowledge and skills that might propel them into STEM fields.
Do we need to mail copies of Stand and Deliver to the entire California school board? Or am I the only one that recalls that movie... based on something that actually happened... in California.
Well, why do you think the elites are unanimously supporting the recent equity initiatives? Because in reality they penalize the potential contenders from the rank-and-file class, while the top ladder plays by their own playback.
I don't think that's the motivation. Elites don't mind if some bright poor kids get a good education-- all the better: more advanced education means that there is a larger hiring pool to fill complex jobs at the elite's companies.
When you're highly privileged it's extremely easy to imagine that most things are positive sum and not zero sum competition because when you're wrong and your cooperation was actually a disadvantage it's still no big deal for you.
Instead, I think it's just cheap feel good measures, virtue signaling, and not wanting to take the costs/risks of bucking a fad. If you don't know whats good or bad, well cheer for the popular change and pat yourself on the back. If you do know that the initiative is bad, speaking against it may get you called a racist-- better to say nothing and relocate to where your kids will get a good education.
Someone sufficiently wealthy never has to worry about a known bad policy resulting in a poor education for their kids-- they can always apply money to the problem, one way or another.
Sorry for being out of the loop and asking a (possibly) dumb question: What is the stated reason that the CMF suggested these changes?
Is it about budgets? Is it because some people might think these classes "aren't that important?". The open letter seems to suggest that it's about closing gaps between privileged and less privileged - is that it? Honest question - I'm not trying to stir the pot.
I believe the gist of the argument is that when you split students into 'normal' and 'advanced' classes at a young age, the students who are not put into the advanced classes will believe they are just naturally not good at math and will give up on trying to get better because they will think they just "don't have a math brain". Here is a short blurb about the idea:
> The framework would not forbid districts from accelerating students in middle school. It does, however, recommend that middle-school students all take the same sequence of “integrated” math classes that blend concepts from arithmetic, algebra and other subjects with the goal of cultivating a foundation and comfort level with numbers.
> On top of that, the framework recommends that schools postpone offering students Algebra 1 until 9th grade or later, when it says more students are likely to be able to master the material.
> “When kids struggle, they immediately say ‘I don’t have a math brain,’” Boaler said. “That changes how the brain operates.”
I am sympathetic to the idea that we don't want to send the message that some kids are just bad at math, but it does seem to be a bit of throwing the baby out with the bathwater by holding back the other kids who are doing well. Even if you keep the advanced kids in the same class, the kids are are struggling are going to be well aware that some of the kids are getting it really quickly.
I was in the "middle" tier math program in high school. But around sophomore year I wanted to get more into science/engineering but you can't switch tiers or catch up to those ahead of you, no matter if you're making extra effort and doing well. It was frustrating.
In my case I got a letter about summer school at a local university. So I pre-calced over summer school to get moved into calculus in high school. It honestly changed my path. I get having tiers, but once placed into one its hard to move. If I wasn't self motivated, and had the opportunity to try I would be in a different place.
My school allowed changing tiers, and I am very grateful that it did. I had a bad year in my early teens with some mental health stuff, and spent my first year in high school with kids who needed much more time and practice to get a handle on concepts than I did. If I had been forced to stay in those tracks, my life would have taken a drastically different course, as I didn't really need to work to learn. Getting bumped into a higher tier challenged me, and that challenge is what prepared me for college.
Had I gone into college without that work ethic, I almost certainly would have failed out early.
I read it as an unspoken "within the school system." It seems reasonable to expect schools to include a path for changing tiers if they put such a system in place, rather than leaving it up to students to find a workaround.
My school supported me in taking trig as an independent study over the summer (with a textbook and slides from one of the teachers plus a few meetings as needed.) This let me take AP Calc senior year; otherwise I would have missed that opportunity due to being placed in the wrong math class freshman year.
If he was only able to get on this track due to interventions from outside the school system which many students could not afford or otherwise could not access, then that's a failure of the public school system.
Yes, but these alternatives are much harder to find for students who already aren’t the best at navigating the school system. Requiring students to figure all of this out is going to reduce the number of students who benefit by quite a bit.
My understanding is that CA law requires public schools to accept any class that would receive credit by the UCs. They don't always, and I believe some bay area schools have been (successfully) sued over their non-compliance.
I know it's considered good form on HN to beat around the bush a bit when it comes to sensitive culture war topics, but I think it's worth pointing for the non-Americans here that the only reason this discussion is even happening is because of the demographics which are observed after the "split" occurs.
Black and Latino students are overrepresented in underperforming math classes, while White and Asian students are overrepresented in the high-performing math class. That's literally the only reason there's any controversy, and if said disparity didn't exist, or if the races were reversed, then we wouldn't be having this discussion whatsoever.
If you approached athletics with the same strategy, you'd end up with a similarly wonky outcome. Consider that Asian students have always been highly underrepresented in high school football.
> “When kids struggle, they immediately say ‘I don’t have a football body,’” Boaler said. “That changes how the body operates.”
Is the equitable conclusion to change the rules of football so Asian students perform better? Surely not, right?
Yep, tracking students into systems like high/low early on makes it very hard to ever escape that track, as they're a sort of self-perpetuating system. That has downstream effects for one's entire life. It's a crude method of personalizing education within the context of factory education.
Downsides are that kids develop at different times, have different educational needs, have home life issues that can temporarily derail progress, etc and if those happen around the time kids are getting tracked, they may not reach their full potential.
A good education system would offer students a way to rise up whenever they're ready to rise up, let them learn at their pace, focus on mastery, build upon knowledge gained rather than schedule followed, etc. There's a lot of edtech out there that incorporate these concepts but school models struggle to integrate it into the (literally) old school way they operate. It's quite difficult to reorient school around these new concepts at scale, it has to be done school-by-school, leader-by-leader, school board by school board.
Agree its complex, as it may be the 'best of the worst' option for certain contexts. Anything involving balancing equity/access/etc is like that.
From what I've seen there are three major problems with edtech that gives a personalized education:
1) A lot of K-8 education is babysitting. If you let kids do their own thing they'll just watch YouTube and play Roblox instead. Most kids are not _that_ self motivated at this point in life. It's hard for teachers to manage a classroom if everyone is working on different things.
2) Staring at a computer screen is not a great learning experience. A classroom is an interactive, social experience with active feedback. It's hard to socialize when the kid next to you is not working on the same activity or problems you are.
3) Personalizing education diminishes the importance of teachers in the classroom, which teachers unions obviously oppose. Teachers can't teach if every kid is learning something different, and online education strongly promotes winner-take-all dynamics where the best teacher and content can scale up infinitely and dominate.
Out of all these I think 2) and 3) are the hardest problems to solve and whoever solves them is going to meaningfully advance education. But I'm not very convinced by the startups I'm seeing in this space that anyone has solved it yet.
Yeah we’re early on in really nailing the formula. Butts in seats staring at screens doing single player activities isn’t a particularly compelling education environment nor one children are accustomed to biologically. We need more embodied, social, psychologically safe, and intrinsically motivating learning environments, and I don’t think the enabling technologies and designs have yet emerged to fully satisfy these needs.
That said some of these programs have solid learning science foundations and good outcomes. Teachers roles necessarily change to ‘guide on the side’ and motivator, there’s a lot more there to go into but basically it’ll take time.
Another non-obvious problem is that you can get mis-tracked too low even on the highest track. It happened to me. There was an assessment test on entering middle school for how many classes up you got shifted, I went into the highest bucket with 4 other kids. Last year of middle school we had to get bussed into the local high school for math education and back for everything else.
I was not seriously challenged and felt like math classes wasted my time.
> “When kids struggle, they immediately say ‘I don’t have a math brain,’” Boaler said. “That changes how the brain operates.”
This really jumped out at me.
I didn't read any context, but students CAN and SHOULD learn to struggle. Productively. Without thinking they are failing.
Imagine you thought everything should come easily? That's not my experience in the world.
The fact that students (are reported to) shut down when faced with difficulty is a failing of the educational system and something that should be worked against.
> I didn't read any context, but students CAN and SHOULD learn to struggle. Productively. Without thinking they are failing.
Unproductive struggling with math is the natural consequence of substandard math education, such as is encouraged by the unscientific and arguably insane notion (which is however common throughout the Education field) that all students can simply be expected to "learn their math by themselves", and therefore have no need for actual, focused and direct teaching of that subject.
Sounds like more woke nonsense. Sounds nice and easy to a layman from a super high level but not practical or put through any kind of rigorous rational thought.
You are right that this is a lot of nonsense. Specifically, 'you aren't good at math/math is hard' is the nonsense meme that gets hammered into student heads so frequently during school that most of them actually start to believe it.
It's not some kind of novel woke nonsense, though, it's how math instruction on this continent has been happening over the past X decades.
'you aren't good at math/math is hard' may be nonsense, no doubt.
'you can be good at math/math is easy' may be an equal nonsense.
This seems to be a symmetrical situation to me. You can absolutely underrate or overrate a person's abilities to do X. I don't see how one is preferable to the other. Both are pretty destructive when taken to their extreme logical conclusions. For example, from the relative underrepresentation of blacks in advanced math classes, you can draw a conclusion that math as a science is inherently racist/white supremacist. Such sentiments can be sometimes seen in discussions and I consider them dangerous, toxic nonsense.
>You are right that this is a lot of nonsense. Specifically, 'you aren't good at math/math is hard' is the nonsense meme that gets hammered into student heads so frequently during school that most of them actually start to believe it.
No, the nonsense is the idea that we are all cut out for math. That's the fundamental underpinning behind the "wokies" push for equity, a silent conflation of equality of opportunity with equality of outcome based on the totally untrue premise that we are all equally capable given identical environments.
The only possible resolution to this goal, given the obvious uneven distribution of innate human ability, is the handicapping of those who are capable, because there fundamentally is no way to boost those at the bottom to match the middle and top.
And I don't think people understand how dangerously pervasive this mindset has become, as it is also the foundation for diversity and inclusion in the workplace, the equally misguided idea that given equal opportunity all demographics would see equal representation in a true meritocracy.
> No, the nonsense is the idea that we are all cut out for math.
I think the nonsense is making a decision about who is and isn't cut-out for math at such a young age, and keeping them hemmed into that path for the duration of their education. That's not merely recognizing the top, middle, and bottom - it's creating it.
I see that as a worthy thing to try to avoid. I also think we should strive to avoid falsely concluding that all persons are equally capable.
But every decision is one that creates tradeoffs. I don't know what should be done. I'm an observer on this topic, and I think there's a lot of hubris in this thread from others oh so certain they know what's best.
Perhaps a simple solution is worth a try: publicly praise/acknowledge those who excel, while also teaching that it's okay to not be at that level [yet]. Encourage peer mentorship, so that the more advanced ones can help someone who struggles. For the outliers who are absolutely stuck in the "I don't care" mindset, apply additional resources to find alternate ways to make the material matter to that individual (practical examples, scenarios, hands-on application, etc.). Ask other students who are interested what real world uses they can think of for the material/topic/equation/concept. If something works, consider implementing that method for the entire class earlier on for the next class.
This is where the goalposts generally get shifted toward teacher resources and/or pay. That's fine to discuss as well, but likely not a significant factor for the above suggestions.
What's the longest we can go without streaming and still meet reasonable targets? The people designing this curriculum seem to say they can't get rid of streaming without dramatically lowering the bar.
This means an informed discussion needs to be had about the costs of lowering the bar against the costs of early streaming. I think people are rather strongly against lowering the bar to the point of effectively removing calculus from high school based on the general reaction in this thread.
> The people designing this curriculum seem to say they can't get rid of streaming without dramatically lowering the bar.
If over twelve years of math instruction you can't figure out how to teach the average child algebra, trigonometry, logarithms, and the very basics of calculus, I would advise the educators to look into why their peers in other countries are managing to accomplish these feats.
But, of course, it's easier to just throw your hands up into the air, and just bifurcate people at Grade 7 into 'good math' and 'bad math' tracks.
>If over twelve years of math instruction you can't figure out how to teach the average child algebra, trigonometry, logarithms, and the very basics of calculus, I would advise the educators to look into why their peers in other countries are managing to accomplish these feats
Their peers in other countries are working with culturally and genetically different populations. Intelligence is 70%+ heritable, you do the math, as taboo as it may be. Then add in the difference between a culture that prizes academic achievement versus one that is ambivalent or worse, prioritizes sports or music over education, and you have more than enough to explain the divergence between nations, as well as demographic groups in the US.
Parents and peers/communities. If the US is any indication, teachers are incapable of instilling appreciation for learning once scholastic achievement is branded "uncool".
You underestimate the impact that schooling has on culture. As a school-age child, you spend more time being socialized and educated by your teachers, than by your parents.
> If over twelve years of math instruction you can't figure out how to teach the average child algebra, trigonometry, logarithms, and the very basics of calculus, I would advise the educators to look into why their peers in other countries are managing to accomplish these feats.
You think they're doing it with "Common Core" and "ethnic" rainforest math, let alone this new "data science" insanity? You couldn't be more mistaken on that. Take a look at the popular Russian and Singapore Math. Not even the smallest trace of the failing "progressive education" thinking, just a lot of solid, high-quality, direct, rigorous, focused teaching.
Can't we agree that both extremes are wrong? While I agree that it is wrong to assume there is no such thing as innate human ability, and it is wrong to assume everyone can achieve equally, you seem to be arguing the opposite; that there is nothing that can be done to improve achievement for those who are struggling.
This simply isn't true. There are things that can be done to improve the outcome for students, and we should continue to work to try to improve the success of all students. This doesn't mean that you expect everyone to achieve equally, just that you can help people achieve more than they would have without the help.
I also find this argument a bit paradoxical; if you truly believe that innate ability is the only determining factor for how well students do, then why do you worry about handicapping those who are capable? It shouldn't matter if we force them into classes they are too advanced for, since how we educate them doesn't matter and only natural talent matters.
It seems that you believe schooling does affect achievement, since you want to make sure we aren't holding back the high achievers, yet you are saying at the same time we shouldn't worry about how we educate the low achievers because they are stuck where they are no matter what. You can't argue that it matters for high achievers but not for low achievers, that doesn't make any sense.
> There are things that can be done to improve the outcome for students, and we should continue to work to try to improve the success of all students.
How would you suggest we do this?
Without a dramatic reinvention of our education system, you have to fill a room with N students and 1 teacher. If you want that teacher to be maximally effective at "improving outcomes" - how do we do that?
The proposal here is to group the kids strictly by age. Every kid in grade X gets the same math class. This will inevitably lead to the math class being irrelevant to some portion of the class. Some kids will be so far behind the teacher may as well be speaking a foreign language, and some kids will be bored out of their mind because the material is moving too slow.
By being a little more intelligent in choosing our groups of N student, we can maximize the relevance of what the teacher is teaching and therefore better improve the success of all students.
I don’t know how we do it, I am not an education expert. I am simply saying we should keep trying new ways to try to help lower performing student improve until we find one that works. We shouldn’t just give up and write them off as being unable to improve.
I don't understand if you're saying that every kid is equally good at math. Or, similarly, that every kid that the same capacity for it or ability to pickup math concepts.
Because it seems to me that if you have experience with any sampling of children where N>1, you'll see that's simply not true.
> I don't understand if you're saying that every kid is equally good at math.
I'm not, there are always extreme outliers and exceptions, but I do believe that the vast majority of children can meet the incredibly low bar for mathematics education that is considered normal in North American schools.
I also believe that teaching them to be afraid of math, (and having their teachers be afraid of math) is a major contributing factor for why so many of them struggle so much to meet that bar.
I would agree with this. The standards aren't super high -- from my POV as someone who always excelled in math. But it's clear (to me, at least) that even the "incredibly low bar" is actually quite challenging, at every grade level, for very many students.
Speaking of teachers... my own grade-school math development, decades ago, was stunted by the fact that my teacher didn't know anything about linear algebra. I asked her for help deciphering my "Amiga 3-D Graphics Programming" book, and she concluded that the vector and matrix notation must be a bunch of typos. Arrgh!
This is a big one. I was in sixth grade when my science teacher told me that the boiling point of water was 132F, because she thought you added 32 to convert from Celsius to Fahrenheit.
This problem runs all the way down, from teachers colleges to the kinds of people who apply to be K-12 teachers. That fearing math is okay and normal is pervasive in the culture and it’s not clear to me you can even do anything about it other than implement gating math credentials for teachers that would exclude a huge fraction of teaching school graduates.
This is a huge part of the issue I feel. I know way too many elementary school teachers who are afraid of math themselves and struggle to understand it. Is it any wonder the kids they teach don't? It causes big problems when they get to me for mathematics in high school.
Like I said, sounds good to a layman in general terms (just how you explained it). But the actual implementation is half-baked, short-sighted, and favors a weak/easy solution rather than something more well thought out and complex.
Ok, so what would your approach be to address the issue of huge groups of kids underperforming what they are actually capable of?
I feel too often the people who play the 'woke nonsense' card think that we should just allow the current failings to continue, and any work to help struggling groups is wrong.
Wouldn't cutting out high level math courses make even more kids underperform below what they are capable of?
The cited issue was that higher level math courses were making other students feel like they weren't cut out for math. So it seems more like the issue is a mindset one. They shouldn't be looking at better performing kids and think "I can never do that". We should be instilling a better growth mindset to these kids, so they understand that they can overcome their inabilities.
The "woke" solution of removing high level courses actually achieves the opposite. It reinforces the idea that such a level is inachievable for some people so it should be cut out for all people.
The problem is that these higher level courses aren't 'extras', they are table stakes for getting an education.
If you think a high-achieving student won't get a good education in an curriculum where they are 'dragged down' by the low-level course... Why on earth do you think that a non-high-achieving student isn't going to get 'dragged down' by being pigeonholed into the low-level course?
If your goal is to just write those people off as lost causes, then sure, by all means, bifurcate the coursework. But then the criticism of this approach starts to sound rather on point.
> If you think a high-achieving student won't get a good education in an curriculum where they are 'dragged down' by the low-level course... Why on earth do you think that a non-high-achieving student isn't going to get 'dragged down' by being pigeonholed into the low-level course?
Why do you assume it is a zero sum game? We can have high level math courses and improve the quality of lower level ones (if you think they are problem). Or if you are saying that students shouldn't be forced into lower levels just because they aren't getting good grades, then yeah sure, let people join high level courses based on passion and not achievement. I think that's an entirely separate debate though.
> If you think a high-achieving student won't get a good education in an curriculum where they are 'dragged down' by the low-level course... Why on earth do you think that a non-high-achieving student isn't going to get 'dragged down' by being pigeonholed into the low-level course?
If you can run a < 5 minute mile, you are not going to benefit from jogging at a pace set by the slowest pace. If you're that slowest kid, being forced to jog will be hugely beneficial.
If you think the bar is too low for the non-advanced classes, that's fine. You should be advocating for more rigour and mathematics across the board, not less. The stratification of classes is orthogonal to math education not being rigorous enough in general.
I don't know how it works in CA, but where I grew up the regular math classes were perfectly good math classes. But if you excelled in math, and wanted to focus on it, you could take the honors and AP level classes. Most of the kids in the regular math wanted to instead focus their time and energy on AP history, or literature. I found the system to work quite well. Nobody was "pigeonholed" and everyone got the fundamental education in all subjects that they needed.
If you can't run a five-minute mile, and are therefore put into a PE class where you never run, do you think you'll ever get into a shape where you can?
> If you can't run a five-minute mile, and are therefore put into a PE class where you never run, do you think you'll ever get into a shape where you can?
People all over this thread are making the assumption that the non-advanced math class is equivalent to no math class at all. That doesn't make any sense to me, and does not match my experience of regular, honors, and AP classes in high school.
Accepting your premise, the issue in your case is that the PE class needs to run more. Pulling all the talented athletes into that shitty class without changing the curriculum at all will strictly cause harm.
> what would your approach be to address the issue of huge groups of kids underperforming what they are actually capable of?
Huh? Are you suggesting that removing upper level maths classes helps kids achieve their potential?
If the current system is untenable, then I would force all students to have one "tutor" period. Everyone has to take a tutor period, so the social stigmatization you're worried about it not a factor. This way, kids gets extra help in their "worst" subject (decided by some combination of grades / introspection/ parental involvement).
This way, the kids who need more help in math can get it, without pulling down the kids who belong in more advanced classes.
> Huh? Are you suggesting that removing upper level maths classes helps kids achieve their potential?
That is the purpose, but I agree that I don’t think it will work. I am saying we need to keep trying, and not dismiss any attempt to fix the issue as woke nonsense.
I didn’t think our understanding of the brain was that advanced yet. AFAIK we run some experiments and observe results, but we can’t explain why those results were observed.
Which is useful and awesome from a learning perspective, but extremely worrying we use it to craft public policy.
> we don't want to send the message that some kids are just bad at math
But some kids are just bad at math. Some kids are bad at sports, music, dance, etc. Some kids are good at some things and kids are good at different things.
Yes, some kids are bad at math... but they could be better than they are.
Let's use your example of sports, for example. Yes, no matter how much I train and practice running, I will never be as fast as Usain Bolt... but I sure will be faster than if I didn't practice at all.
You're making a straw man that non-advanced math class == no math class.
The lower level math class is exactly where you belong to train up your math ability. You wouldn't expect to practice running with the same routine that marathon runners use.
Why wouldn't you just let everyone take the same class and the same exams, but let the kids who have interest do extra work? Want to do calculus a year early? Here's the book, here's the exercises, why would I stop you?
What you suggest is already the case. The book and exercises are "here" for you to do, nobody to stop you. It's called the internet. The calculus police doesn't come get you if you're reading a calc book in 10th grade. It's just that you don't have any way of getting instruction or school credit - so you are very unlikely to be successful, and not very likely to have your university credit you as having mastered the material without taking a college class.
I think the teacher sometimes says and explains things to the whole class, and if what the teacher says, or homework s/he has prepared, applies only to 5 out of 25 kids, the others can get bored and unmotivated, sometimes even start disturbing the others.
And sometimes the kids talk with each other about the maths problems. But that won't work so well if they're in different books.
Simpler for both the teacher and students, if those in a class are at roughly the same level.
But I do think that what you say, would work in a group where everyone is pretty good at maths, enjoy it so much so "yes I got a new book" they think (when they get the next book). And dive into it, no matter if there's a teacher there or not.
Because that's inefficient. A class geared towards the bottom 50% of kids is going to waste a lot of time for the top 50% of kids. Simply giving the gifted kids more homework just wastes more time, and hardly encourages them to reach their potential.
From my understanding, they revisit this framework every 8 years. California is doing poorly in 8th grade math scores, so I think they want to make changes to improve that.
It seems like such a bonehead solution to the problem. Of course if you’re doing poorly in math scores, you can make math easier in the hope to increase scores.
It’s sad that the state is proposing these changes. I remember in school there were kids who argued “algebra is stupid, who needs it, why waste time” and there were one or two sympathetic teachers who would respond “well, I rarely have to use algebra to balance my checkbook” or something silly. It seems like those kids have grown up, gained power, and are literally pushing the argument that this math isn’t important.
I think the argument (though not necessarily one I agree with) is a spin on what you said:
The current system pushes 50% of the kids into calculus and 50% into 'I hate math.' Of the 50% that go into calculus, 50% go into STEM.
That leads to (hyperbole) 25% A's / 25% B's / 50% F's.
The intent of the new rule is to maybe be more like 25% A's / 10% B's / 50% C's / 15% F's.
The key questions are 1) Is that actually better (I certainly think bringing up the floor is a good idea, but at what cost)? 2) Is this policy even going to get us there?
I like your example breakdown, but my understanding that by removing calculus as an option it’s lowering the ceiling so we’d end up with: 0% A, 25% B, 25% C, 25% D, 25% F.
I guess there is a societal discussion to be had if we should trade off losing As to reduce Fs.
But I think that first there’s not a dichotomy between approaches so the only way to get fewer Fs is to reduce the number of As. I think there are ways to improve education that doesn’t remove the opportunity to excel. And framing it as the California proposal or failing math is unfair.
I think what surprises me so is that they continue to propose these solutions that seem to reduce the overall math capabilities being produced.
I think you should do some primary research. This threads article argues that calculus is impossible, but the actual reality is [0]:
The draft Mathematics Framework includes calculus in the possible high school pathways, and also suggests ways to enable more students to get access to calculus. It notes that many high schools currently organize their coursework in a manner that requires eighth grade acceleration in order to reach calculus or other advanced mathematics courses by senior year. While some students succeed with this approach, acceleration has proved a problematic option for other students who could reach higher level math courses but would benefit from a stronger foundation in middle school mathematics.
There is nothing in this about actually removing calculus for high achievers, and this there’s an argument against it, it’s what I laid out: “is it better to push more kids into calculus at all costs or should we focus on raising the floor?”
It’s honestly a very similar argument to the (popular on HN) argument of “not everyone needs college”
And to be quite frank, this isn’t coming from nowhere, this is modeled on the success of other western countries with higher math scores across the board.
The person quoted (https://en.wikipedia.org/wiki/Jo_Boaler) is a "Nomellini-Olivier Professor of Mathematics Education at the Stanford Graduate School of Education" who "won the award for best PhD in education from the British Educational Research Association"
I'm not saying I agree with the proposed California Framework. As a formerly gifted maths student, I hate it. But let's not dismiss the rigorous work of an academic who is attempting to improve education for a public education body for a state with 40 million people as "bonehead" (or "woke nonsense" as another commenter did).
90% of the reasons why kids say "<subject> is stupid, who needs it" it's because they are not enjoying it or struggling with it and using this as a defense mechanism. Noone who is doing WELL at a school subject dismisses it as useless.
So maybe, just maybe, it's worth evaluating the education process to make it easier to teach kids to give them the foundations that then the more gifted ones can invest and build on top of, and everyone comes out with baseline math competency.
I find this kind of shallow dismissal of a tenured Stanford professor's work based on their thesis unproductive.
Engaging with their current, relevant work would be more appropriate.
This is exactly what the GP is saying - many of us don't like the conclusions, but just blowing off a whole body of work in a sentence is pretty arrogant.
There’s tenured Stanford professors in many subjects, such as theology. I’m sure their work is impressive within the context of the field. But that doesn’t mean it’s rigorous or has real world application. PhD publications are supposed to be a serious contributions to the field. This particular work won a major award.
Stop it with the accusations of “arrogance” and naked credentialism. Any of the millions of people with an undergraduate STEM degree (mine is in aerospace engineering) learns enough about the scientific method to distinguish “rigorous” work from non-rigorous work. It’s actually kind of an important thing they try to teach.
Scientists and engineers who don’t call out non-rigorous work that claims the mantle of “expertise” are shirking their moral obligations and helping to erode the credibility of science as a larger discipline.
Arrogance is skimming a PhD thesis in a field wildly different from your own and deciding you are more competent in your ability to evaluate its "rigor" than the dozens or people in that field who evaluated it that had a different conclusion.
It's not an original arrogance - it's pretty common amongst the STEM educated to basically consider all humanities, philosophy, social sciences to be not "serious" fields.
Just because it's common, doesn't make it not arrogant.
Individual humans are not free body diagrams. And societies even more so.
The study of people and groups of people and the way to understand them, engage them, and influence them (which is what "education" is) is not going to be familiar to an aerospace engineer. That doesn't mean it isn't built on the work and research of thousands of people over centuries.
> Arrogance is skimming a PhD thesis in a field wildly different from your own and deciding you are more competent in your ability to evaluate its "rigor" than the dozens or people in that field who evaluated it that had a different conclusion.
What a flimsy and transparent effort to insulate non-rigorous fields from criticism.
Ironically, one of the things I do for a living these days, as an attorney, is attack the credibility of credentialed experts: https://en.wikipedia.org/wiki/Daubert_standard. There is a whole process where attorneys and judges without specific expertise in a field evaluate the reliability of assertions by expert by reference to accepted scientific principles. Because that’s a thing you can do! You don’t need to be an expert in a field to know whether they applied reliable methodologies.
More importantly, people will do it. If you tell them to trust stuff that doesn’t seem trustworthy based on what they learned about the scientific method in high school, then it erodes your credibility and the credibility of all authorities. There’s a lot of work out there that’s not rigorous, and we have a moral obligation to call it out: https://en.wikipedia.org/wiki/Cargo_cult_science
> It's not an original arrogance - it's pretty common amongst the STEM educated to basically consider all humanities, philosophy, social sciences to be not "serious" fields.
Theology, film, German literature, European history, etc., are all “serious fields.” They’re the product of work by thousands of people over decades or centuries. That doesn’t mean they’re rigorous. Their ability to provide truth value is fundamentally limited by the nature of the discipline.
And yes, maybe engineers have a big head about it. I suspect that’s because they don’t have to brow beat anyone into recognizing the seriousness of their field or the authority of the expertise it generates. You can go up to a random yokel in a Bangladeshi village and he’ll be impressed at your STEM degree. I’ll posit to you that there is a basis for that.
In methodology, the field of education is akin to something like the field of business. It’s based on observation and case studies. It’s not wholly without value, but holding it up as a rigorous discipline that commands deference is disingenuous.
Yes but what is your perspective on the study of individual personal behaviour and societies.
Are you saying that all the "serious but not rigorous" fields that do so are A) all doing it wrong, and that you know a better way, B) That there is a better way and noone has tried, Or C) that it is impossible and we shouldn't bother trying.
Because if it's A or B, you can see why someone would call you arrogant. And if it's C, is that not defeatist? Seems like trying to understand our fellow humans and the societies we create is a task worth pursuing?
(C). Human behavior and societies are generally not amenable to rigorous analysis, and therefore aren't the proper subject of academic fields that purport to generate truth value and produce expertise worthy of deference.
That is not to say we can't seek to understand these things. To the contrary, humans have myriad ways to understand and respond to these things, including religion, tradition, culture, convention, etc. Academic inquiry that has the veneer of rigor--through credentials, peer review, conferences, etc.--but isn't actually rigorous improperly displaces those other ways of knowing, and cedes social power to a narrow class of elites.
The education and socialization of children, for example, is one of the fundamental pillars of society. People around the world have their own ideas of how best to socialize their kids. Unless you can bring real science to the table--unless you can create reproducible results that speak for themselves--you shouldn't be so arrogant as to act like your work can substitute for those other ways of knowing.
Not the OP, but I would weigh in with D) it's worth trying, and we should learn what we can, but the phenomena being studied are too complex and/or subjective to yield the kinds of rigorous results that we expect from the harder sciences, and the opinions of experts are therefore necessarily more infused with personal opinion and bias.
I think it is incumbent on all fields to be exceptionally clear about which of their results are rigorous, and which are outside the realm of rigorous knowledge. For example, as someone with expertise in CS, I can give you very reliable information about computability theory, and pretty reliable information about what is technically feasible to implement using current technology, but my opinion on fair moderation policies for social media are not really buttressed by my CS expertise. I wish public health officials (as just one example) were similarly clear about what they have expertise about (how disease spreads and what interventions will have what outcomes) and what is their personal opinion (cost/benefit analysis of different options, who should get vaccines first, etc).
I love that response because I think it gets to the point and the problem.
Q1: "What is the worst-case runtime complexity of quick sort" is a question that has a rigorous answer that can be proven with fundamental scientific concepts.
Q2: "What is the programming language/framework we should use for this next project" is one that absolutely cannot.
Any answer to Q2 will invariably involve some personal opinion and bias, AND YET serious professionals with decades of experience - ones perfectly capable of answering Q1 would also be expected to weigh in on Q2, incorporating their knowledge, context, the skills of the rest of the team, the existing architecture, the requirements, etc, etc.
It would be absolutely arrogant to push back against the qualitative analysis of a senior software developer making a recommendation to Q2 with "That's just your own opinion and bias" - ESPECIALLY by someone outside the field who doesn't understand all the considerations being weighed.
I sympathize with your desire for health officials, but my theory is that everything about epidemiology, public health, disease spread, etc are all mostly Q2-category questions. You still need a PhD and rigorous understanding of all the Q1 questions but shaping public policy in a pandemic is all Q2.
Education is much the same way. Education of mathematics at scale is less about mathematics and more about people.
I think your Q1 vs Q2 is a good start, but I would add:
Q3: "Should the US regulate Bitcoin?"
CS expertise will give a person much greater understanding about the details of how Bitcoin works. But any layperson can learn enough about how Bitcoin works to form an opinion on the question of whether the US should regulate it. I would never tell a non-CS person that they don't get to weigh in on this issue because they are not in the field. Would you?
Ultimately what we are talking about here is policy, which affects everybody, and value judgments, which everyone is entitled to make.
OK, let's bring it back full circle: The context was maths education - not "Should the US teach math?" (to which every layperson can have an opinion), but "How should the US (California) teach math to high schoolers?"
That's as complex of a question as "How should the US regulate Bitcoin (if it were to)". And while it's true that laypeople do have lots and lots of opinions on financial regulations, if there's one thing that HackerNews can agree on is that most lay opinions on financial regulations are not rigorous or serious, and come with unexpected side effects and consequences. (Whether it's taxing billionaires on stock grants or restricting crypto).
So in that sense I do still this requires people with expertise. And gifted math nerds (which I were) that benefited from a dramatically accellerated program have as much knowledge about the subject as a stereotypical redneck saying "We don't need math out on the tobacco farm".
Yes, many policies have unexpected consequences. The question is whether the experts are doing better at predicting them. The most sweeping educational policy in my lifetime, No Child Left Behind, passed with large bipartisan support, only to be abandoned 15 years later for failing to achieve its goals. I cannot find any expert opinion from the time that accurately anticipated these unintended consequences. If experts are much better than the general public at predicting the effects of policy, I would expect to see education experts on the record predicting the ultimate effects of the policy.
> And gifted math nerds (which I were) that benefited from a dramatically accellerated program have as much knowledge about the subject as a stereotypical redneck saying "We don't need math out on the tobacco farm".
I think the "nerds" would know enough to understand that they benefitted from the accelerated program and would have not achieved as much if it were taken away.
A Professor of Mathematics Education is a role that fits into a woke section of academia and generally publishes woke forms of advocate research. Many people can study mathematics education without studying much mathematics at all.
When the Mathematicians and Scientists are screaming that the policy is nonsense, I'm not convinced by an advocacy researcher saying it's rigorous work.
Can you make specific arguments instead of devolving to shorthand dogwhistles that are completely up to the interpretation of the reader.
The term 'woke' has no meaning, depending on the context it's anything between "We should shame all white people for the crimes of their ancestors" to "We should make a movie with a female lead".
By using it, you leave it ambiguous as to where on the spectrum you fall.
The specific argument is that the Mathematicians and Scientists are screaming that the policy is nonsense. The other specific argument is that a researcher in mathematics education doesn't need to take a lot of mathematics. The inference is that we should trust the former set of people more even on matters of mathematics education.
My (anecdotal but common) presumption is that the disparity often comes from a cultural / behavioral divide, which is increasingly blurry along racial lines but still distinct enough to recognize usually. By behavioral I mean things that are disruptive to standard teaching, like, during a lecture or presentation, students are spending time on phones or chatting with each other or listening to music or drawing or walking around. General "I don't care, I'm going to do my own thing" behavior. Sometimes the behavior is based in a cultural or subcultural expectation or standard. What methods or authorities do teachers have to enact behavioral change in such cases?
In most cases, none at all. If any attempt isn't derided as racist (or other -ist/-phobic accusations) it's viewed as authoritarian/inhumane. Decades of legal precedent and risk aversion have caught up to education, perhaps rightly so. I don't think there's any chance in hell of going back to paddling and such, so we need to come up with newer ways to enact behavioral change. In order to do this, I think we have to stop being afraid of slights against culture. Unity of purpose, of many one. Diversity is not a strength if there is no unifying principle among the diversity.
> "Since achieving a solid foundation in mathematics is more important for long-term success than rushing through courses with a superficial understanding, it would be desirable to consider how students who do not accelerate in eighth grade can reach higher level courses, potentially including Calculus, by twelfth grade. One possibility could involve reducing the repetition of content in high school, so that students do not need four courses before Calculus. Algebra 2 repeats a significant amount of the content of Algebra 1 and Pre-calculus repeats content from Algebra 2. While recognizing that some repetition of content has value, further analysis should be conducted to evaluate how high school course pathways may be redesigned to create a more streamlined three-year pathway to pre-calculus / calculus or statistics or data science, allowing students to take three years of middle school foundations and still reach advanced mathematics courses."
At face value, that suggests that the root problem is that students reaching middle school Algebra 1 aren't ready and need more remedial math instruction. As an Electrical Engineering professor, I can definitely attest to the fact that students reaching higher level classes with a precarious foundation are rarely as successful as those whose foundation is more solid. I suspect Scott would also agree that barely passing calculus in high school is not an adequate preparation for a career in data science. As a parent of a kindergartner and a second grader, I can also see that there is opportunity to push more math further down, but even at that age there are kids who have a huge variability in how they view their math.
With regard to resources, I thought this statement in the CMF was particularly insightful:
> “While early tracking of students into low-level courses has been problematic, there is evidence that thoughtful grouping of students to ensure they receive high-quality instruction geared to their needs at a moment in time can be helpful. This includes students who need to fill in gaps in their prior learning and high-achieving students who are ready to be more intensely challenged. It is also true that teaching heterogeneous classes requires greater skill for differentiating supports than teaching in classes where the range of performance may be narrower, and should be accompanied by high-quality professional development to enable success.”
Update - having re-read the first post about this, it seems that the issue of resources is exactly the problem. I think that opponents of the CMF would prefer to see More Resources put into careful, thorough elementary/middle school math so that middle schoolers would thrive in 8th grade Algebra. And in their view, the CMF simply lowers the bar, masking the need for more resources.
Want to know the one resource that schools can never give students? Parental involvement. It also happens to be the #1 variable in success for students.
If parents aren't checking their children's grades on a daily basis, asking what they're studying, staying in frequent communication with teachers and the school, there is nothing that is going to replace that.
My oldest son was on a robotics team during high school in which we were a minority. All the other parents, their kids were studying calculus by the end of high school. I asked every one of them how their kid was able to do that and each one of them shrugged and said, "nothing".
To them, "nothing" was their child spending 2 hours per day at Kumon after school and a few hours on the weekend in addition to their constant checking of their work and insistence on academic excellence.
You also have to be careful about "well performing" school districts for a similar reason. I live in one such district and most of the parents I know have a regular, dedicated tutor for each of their kids.
The teachers expect a lot because of this. Good luck if you can't afford such help.
When my kid was in kindergarten, half the class had an outside tutor. I learned this the next year, when a parent offered to share her tutor with us.
This concept is strange to me; when I was growing up, a "tutor" was someone who helped kids who were behind. Nowadays the word seems to also refer to an outside coach who helps kids get ahead.
The lack of resources for public education, coupled with the every increasing level of competency that students are required to internalize over time drives many of the fights in education.
I can understand how an electrical engineering professor like yourself is rightly concerned about your incoming students having less calculus proficiency. On the other hand, there many (possibly far more numerous than EE) education and career paths that would benefit from better general numeracy but not necessary calculus.
The terrible thing here is that these goals should be set against one another due to resource limitations. The blame for that lies with the broader societal inequities and their reflection on the educational funding system.
I'm a product of public schools and a state college. Due to my experiences I will never vote to give teachers one more dollar in funding until major changes occur.
I had multiple high school teachers that were checked out. Would watch movies and do worksheets. We learned nothing in their class other than how bad they thought students were now days. Yet they were never fired. Some lasted another 10 years before retiring.
I also had multiple professors that would start terms by saying if you have conservative views and you express them in class I will fail you. Had another professor joke about how she found out a student was part of a club she didn't like so she failed him. Often times these were not GE classes but CS specific. Politics have nothing to do with a class on OOP or AI. That is not teaching IMO. I have a ton of friends that feel the same way. Until this issue is actually recognized and addressed I do not believe you see anything change.
What exactly do you think are the facts regarding the "lack of resources" in education and "the educational funding system?" Are you aware that the U.S. spends among the most on primary education (adjusted for purchasing power) of any developed country: https://www.statista.com/statistics/238733/expenditure-on-ed...
Or that 50% of total K-12 school funding comes from federal and state sources, which is directed mainly at lower-income districts? And that, when including those funds, only a few states (ironically blue ones) have more than a 5% gap between rich and poor districts: https://edtrust.org/wp-content/uploads/2018/02/Gaps-in-State.... Meanwhile about 20 states (including many red ones like Utah and Georgia) direct over 5% or more funding to poor school districts than rich ones?
> Are you aware that the U.S. spends among the most on primary education
Yes, and I'm also aware that in the US we expect the public school system to be a primary treatment center for the disadvantages and traumas associated with poverty and unequal opportunity.
It's more expensive (and harder) to educate hungry, ignored, and traumatized children than it is to educate children who are well take care of. They need more support staff, more psychologists, more free lunch programs, after-school care that their parents can never afford. And it's hard to attract good teachers to teach academics in those circumstances.
The major problems with primary education in the US are largely faced disadvantaged communities, not prosperous ones. If we decide to remove educational funding from disadvantaged communities, we have to be prepared to either put it into those communities in other ways, otherwise we'll face even greater problems in schools.
None of that is to say that there doesn't need to be more accountability in school systems - there is a lot that needs to be done to re-examine how education is delivered. But just looking at the situation by comparing $ spent on primary education between countries is oversimplifying things.
> Yes, and I'm also aware that in the US we expect the public school system to be a primary treatment center for the disadvantages and traumas associated with poverty and unequal opportunity.
That’s true in virtually every country, because the school system is always the government’s primary point of contact with poor families.
> It's more expensive (and harder) to educate hungry, ignored, and traumatized children than it is to educate children who are well take care of. They need more support staff, more psychologists, more free lunch programs, after-school care that their parents can never afford.
Agreed on school lunch. Large school districts already offer after school and summer programs.
Disagree on more staff and psychologists. That’s the kind of waste that detracts from money for instruction. Countries like Japan and Singapore went from being desperately poor (at a level unimaginable even to an inner city American) to developed in a couple of generations in the 20th century. I’m pretty sure they didn’t (and still don’t) have a bunch of school psychologists on staff.
> Countries like Japan and Singapore went from being desperately poor (at a level unimaginable even to an inner city American) to developed in a couple of generations in the 20th century
Those countries did so by making absolutely massive infrastructure and human development investments in their poor populations during their industrializing phase, very similar to what the US did for its previously poor white population after the 2nd world war.
The people who face the greatest educational obstacles in the US today are disproportionately people who were also largely excluded from the huge post-WW2 investments and ensuing economic miracle, and have since faced the economic brunt of de-industrialization.
> I’m pretty sure they didn’t (and still don’t) have a bunch of school psychologists on staff.
They have far less crime and trauma to deal with, are not awash with weapons, and had very strong communal support systems. They were/are also quite authoritarian. Those are very different societies with very different circumstances. You can't directly compare the situation in post war Japan and Singapore with inner city America. By dint of our own history, issues of education and equity actually quite a bit harder here.
> Those countries did so by making absolutely massive infrastructure and human development investments in their poor populations during their industrializing phase, very similar to what the US did for its previously poor white population after the 2nd world war.
Japan and Singapore aren’t Nordic social welfare states. They invested heavily in development but not particularly targeted at the poor.
> The people who face the greatest educational obstacles in the US today are disproportionately people who were also largely excluded from the huge post-WW2 investments and ensuing economic miracle, and have since faced the economic brunt of de-industrialization.
Only if you pretend that white people in Appalachia are the same group as white people in Massachusetts.
> They have far less crime and trauma to deal with, are not awash with weapons, and had very strong communal support systems. They were/are also quite authoritarian. Those are very different societies with very different circumstances.
That indicates that America’s problem is culture, not the availability of school psychologists.
> Japan and Singapore aren’t Nordic social welfare states. They invested heavily in development but not particularly targeted at the poor.
Nor are they comparable to American inner cities.
My own anecdote about Singapore is an old friend who grew up very poor in Singapore (themself a child of impoverished rural immigrants laborers from India), but whose family received subsidized housing, transportation, and most of all, stability and security. If their family had arrived in Singapore as slaves, their outcome might have differed.
> Only if you pretend that white people in Appalachia are the same group as white people in Massachusetts.
Who is pretending that? Not me. I agree that they also have born the brunt of disinvestment and de-industrialization, and we are seeing the effects of that in the opioid and methamphetamine epidemics in those areas. It's interesting that we don't usually call it a culture problem with them though, like we do with black inner city communities facing similar challenges.
> That indicates that America’s problem is culture, not the availability of school psychologists.
It indicates a problem of economics and disinvestment. The psychologists are only there to manage the impacts of that disinvestment on society, not solve the original problem of lack of economic opportunity.
I'm also happy to reduce the number of school psychologists when teachers no longer have to deal with traumatized children disrupting and endangering their classes. Until then someone has to manage those issues, and as you stated, that's what we expect public schools to do.
It's saying that it is impossible to get the bulk of students (with the teachers we have) to complete the standard mathematics curriculum by the end of high school. This has always been true, in all countries, hence "streaming". But now they're saying let's do away with the advanced stream, therefore students can't complete the last part of the US mathematics curriculum (which is called "Calculus"). Rather than justify that move in terms of cost or fairness, we're going to say "because Calculus isn't important now".
This is obviously completely bogus. If their assertion that Leibnitz-style calculus isn't important now, they could replace it with Linear Algebra, Number Theory, or some other "important now" subject.
Add to that, the fact that in the US the names of high school mathematics classes are by convention. "Geometry" isn't all geometry, for example. And "Calculus" isn't all calculus. The classes are really : Math 1, Math 2, Math 3, Math 4, AP Math.
Ultimately the solution is better STEM foundation at the very very early ages. Universal pre school would probably be very useful. Empirically [edit: n = 1 or 2], it seems that drilling basic arithmetic and then multiplication tables early in pre K and earlier elementary will give students a better intuitive math foundation to do algebra very well. That would enable everyone to go into more advanced classes at the same time (earlier) rather than these policies which want everyone to go in at the same time (later).
As a math PhD with dyscalcula, I'm very skeptical. I was nearly held back as a child because of poor arithmetic performance, and really only started to be above average when we started on algebra. Poor arithmetic isn't that uncommon among the mathematicians I know.
On a similar note, I have a friend who majored in math at Harvard. He once told me that he came into Harvard being arrogant because in high school he was always at the top of his class in math. He enrolls in his first college level math course thinking he's got this, but he soon realizes that "higher math", which is largely proof-based, is a completely different subject than what he learned in high school. A month in he bombs the first exam. He went to the professor, who is originally from Italy, and explained his situation and how he was a star in high school. He responds in a thick Italian accent "that was not math, that was computation. In this course I teach math".
The math you typically learn in high school is very important, but I wish that we did a better job of explaining to high school students that what they are learning is completely different from what "real mathematicians" study (although I do think that computation is quite important in engineering, for example).
For me it was the opposite, i was a pretty average math student in high school and fell in love with proof based math in college. Proof based math requires much more creativity and thus is much for fun for me at-least, i wish i was taught proof based math in HS.
One of the major problems is that most of the high school teachers are themselves not very familiar with math, and are more educated in the very different problem of corralling children.
To be fair to your friend, 90 years ago they probably could've had a productive career in applied math as an expert computer. There was a point when calc tricks could get you a PhD, but those days are gone.
Did that dyscalculia prevent you from learning and attaining familiarity with the standard algorithms? That's the sensible goal of "drill and kill" in early grades, not doing routine arithmetic with high amounts of significant digits.
I wish we did probability with equal gusto as kids. I very occasionally multiply 7 and 6 in my head, but have to reason with probabilities and statistics all day long.
Agree. I think getting really good at addition subtraction and multiplication and then division/fractions is the way to go though. Have really strong math fundamentals and then learn algebra and then everything else is far easier to pick up.
Fair point. My first introduction to stats was at second year in uni though, far too late. By that time I had already done a lot of calculus, which while important, hasn’t exactly been critical in my daily life.
Arithmetics is absolutely the most important topic to learn, since it is the basis for all other quantitative reasoning.
For example, it is really important to understand that 1 / 3 chance is the same thing as 3 / 9 chance. It is obvious to you now since you have done so much arithmetic's, but to someone who never properly learned it they wouldn't be able to properly compare those two and could think that one is a very different number than the other. Without basic understanding about quantities all other quantitative skills become worthless.
>"drilling basic arithmetic and then multiplication tables"
I get the sense that such rote methods are no longer encouraged and a lot of the "new math" in Common Core is aimed at approximation and reckoning so that students won't rely on memorization.
> Empirically, it seems that drilling basic arithmetic and then multiplication tables early in pre K and earlier elementary will give students a better intuitive math foundation to do algebra very well.
This aspect of the Common Core was about recognizing deficits in conceptual understanding resulting from rote methods of drilling arithmetic.
The empirical evidence is the opposite of OP's assertion, but the end point of giving students a better intuitive foundation for higher level math is indeed the goal!
Signed,
An elementary school math teacher who has studied the 60 years of math reform in America, internationally, and worked very hard to ensure all students have a foundation to succeed in higher level mathematics
If you want to teach people methods to solve equations, limits, integrals etc. speed with basic algebraic operations is necessary.
Facility with those methods is then necessary to be able to adequately follow important proofs and gain understanding of more advanced concepts.
I don't know how you would teach people important results in their fields (physics, computer science etc., I'm not talking about actual mathematicians) without those skills.
TL;DR: algebra != arithmetic AKA real math doesn't use numbers
I'm only good at _arithmetic_ because of making Warhammer 40k armies (true story bro).
I'm good at _algebra_ because I was taught well, on top of a knack.
Speed with basic algebraic operations was very helpful in many places but speed with arithmetic operations has only been helpful in board games.
I don't think anyone here would disagree with your point about algebra, but I think a lot of people such as myself would disagree that pre-K memorization of arithmetic helps with algebra later.
But strong arithmetic fundamentals are absolutely necessary for strong algebra. I've watched kids struggle with basic algebra because when they don't instantly recognize that 7x8=56, they also don't recognize that 7zx8z=56z(edit: squared).
Edit: thanks to the reply; HN ate my superscript 2. Apparently it doesn't like the unicode multiplication x, either: × ² ?
Hmm. Alright, I can see solid arithmetic being good for introducing the concept of a variable...
...but the only time I ever saw significant numbers when actually doing math was in toy problems that deliberately chose weird, big coefficients, where the arithmetic part was by far the least significant.
Performing a bunch of calculations for tabletop wargaming is basically the same as learning multiplication tables and solving related problem sets. It should help every time that for example you have to simplify a polynomial involving fractions and similar operations.
As I stated in another post, I don't know when it is neurologically ideal to learn arithmetics, it seems something that would be important to study carefully (personally I learnt before grade school, when I was 3-4 years old, but I didn't learn to read until I was 6, something that is often taught earlier).
More and more earlier and earlier doesn't comport well with child development and can backfire by making students feel incompetent. We need better teachers, probably by paying more so more talented people join the profession.
Other countries do a lot of "pre-algebra" in the later grades of primary education, when the kids are quite ready for it; "drill and kill" rote methods are generally focused on in very early grades, since they help build familiarity with the sort of rigorous, algorithmic thinking that's required for good math proficiency. This is what Russian Math, French Math, Singapore Math, etc. are built on, and the approach has stood the test of time indeed. The fuzzy "Common Core" approach pushes abstract content way too early, and ends up confusing kids as a result.
I wholeheartedly endorse Singapore Math, Russian Math, Canadian Math (this is a thing, check out John Mighton's fantastic JUMP program). ANYTHING but US math.
I think we could pay more, but only if we could actually get better teachers. So many teachers are bad and not worth even what they get paid (personal experience from a nominally quite good school district). My concern with paying more before figuring out how to weed out bad ones is that it would just be a waste.
Anecdote: my then 5 year old and I would "practice counting by different numbers" on the walk to school. By the end of kindergarten, she could count by everything up to 12s. In 1st grade, we started reversing it and asking how many 4s in 48 and the like, and by the start of second grade, we were firmly in adding and subtracting fractions with different denominators (though, on paper at this point, no longer mental math).
She had (has?) a solid grasp on numeracy. I recall asking her why, around 7th grade, "0.999..." is equal to 1. I was prepared to show some fancy algebra and she one upped me when she said "well, 1/9 is 0.111... so 9/9 is one and 0.999...".
She never liked math though. She spurned calculus.
What's the point of teaching kids to memorize something that they can't apply? When I was a kid, schools taught multiplication tables in 2nd grade, when most kids are 7 years old. The difference in cognition between a 4 year old and 7 year old is insane.
I'd be surprised if there were any countries where multiplication was formally taught to pre-K students as part of the standard curriculum, but i'd love to be proven wrong.
I don't know if there are countries. I believe that if there actually was a unified accelerated math framework that was really emphasized starting age 4/5 then kids would be absolutely fantastic at math.
> What's the point ... ?
Paraphrasing what I said a comment above, you drill addition and subtraction until everyone is good at it, then you drill multiplication, then you do basic division, then you start introducing basic one variable algebra with "move plus to the other side to get minus" etc. The application is using algebra for word problems; formalism can come later.
In context I meant get really good at addition/subtraction starting pre K and then multiplication once +/- is mastered.
Though empirically, I don't know about age 4 but kindergarten is definitely not too young for learning up to 12*12. And once you figure out multiplication and eventually mental division, it's not too big of a leap to have one variable algebra with "move a plus to the other side to become minus" etc. The formalism can come later but it's fantastic to have some exposure to moving numbers and symbols around from an early age.
My sixth grader and first grader score in the 90+ %ile in mathematics and didn't come close to learning multiplication up to 12 in kindergarten. In fact, the topic isn't even covered until second grade at the earliest.
I think establishing a foundation of addition and subtraction takes far longer for children to master than you're considering, especially since there is evidence that children of this age appear to intuitively view numbers logarithmically rather than linearly [0].
I suppose you could take advantage of this by somehow prioritizing multiplication and division over addition and subtraction, but I think there's too much value in comprehension of linear numbers and addition/subtraction since that is the lion's share of interactions they will experience at that age.
On the other hand, if you're merely talking about abstracting multiplication and division into patterns, then I wholeheartedly agree with you, and there is evidence supporting this [1]. Although pattern identification is already part of kindergarten/1st grade curriculum here.
Ultimately, IMO the most important aspect of education in general is covered in the open letter linked to the OP:
> There cannot be a “one size fits all” approach to K-12 mathematical education.
My children have thrived with their current math curriculum, and I know some of their classmates have struggled in contrast. One size does not fit all in education, nor in many aspects of life.
Out of curiosity, is that a hunch or are you aware of any schools teaching multiplication tables even in Kindergarten? This used to be done in second grade when I was a kid.
I'm not aware of any schools teaching multiplication tables in kindergarten, but I did memorize the 9x9 table when I was in kindergarten because my older siblings' Big Kids Notebooks all had the times table on the back and it formed a rhyme/ditty in the local language. After it was explained to me that multiplication was repeated addition, that made perfect sense.
But don't ask me about division, my siblings/parents/whoever tried to explain it as "the opposite of multiplication", which was complete nonsensical gibberish and I didn't learn division until years later.
When I was in kindergarten, I used to do math booklets at home with my mom for fun. I learned basic multiplication sometime around then. 13 years later I majored in engineering.
So I'm not saying it can't be done by any 5 year olds, but it seems young to teach this to the majority of 5 year olds.
There is little evidence universal preschool would reduce academic variance later on. On cognitive measures (though not necessarily social/emotional), randomized trials of such programs tend to show fade-out (no difference between control and treatment groups) within several years.
1. Make it possible for HS students who are interested in Calculus to take the course under the instruction from a college profess on the high school campus. That way it would be set up for students to get college credit for the class and they would not need to travel to a college campus or deal with the AP exam system.
2. Make it possible for HS students who are not interested (Yet) or at all to graduate with out taking the class. Lots of student are ready for Calculus until college anyway. No need to force them on a single path IMO.
Probably the optics that will ensue - Our public schools seem intent on making sure that to ensure everybody is as achieved as everyone else, but rather than improving education for the bottom percentile, they'll simply remove the advanced stuff for the top percentile.
When we focus narrowly on what brick and mortar Public HS should and should not be teaching in regards to math curium we sideline all the pathways for learning available outside of this bureaucratic model.
Folks in most places in the United States a least can check out a Chromebook from a local library and use their free internet to access this information.
I don't think it's reasonable to expect some kid to sit through their algebra class, and then go home and take a self-directed calculus class at home. Certainly only the kids who are hyper-math-ers are going to do that in the first place, and we miss out on the kids who would have taken calc had it simply been offered.
Highschoolers fork over swaths of their day to the public education system already. Let's try to not waste any more of their time.
Yeah this is a huge part of it. It's one thing to say "we don't have the resources for high achievers", but it's a million times worse to force them through a shit curriculum because we don't want to acknowledge our school system is largely glorified baby sitting.
Especially for kids from poorer backgrounds it is not easy to just sit on Khan Academy after school. If they can't teach leveled classes they should let kids test out of certain classes, and spend time in a semi-supervised study hall with those free resources instead.
But it's not about resources here, or at least that isn't the main argument they're giving. The mere existence of students that can test out of math is apparently detrimental to others. This seems like over the top sheltering to me.
My high school did exactly that in the 2000s. The local Community College teacher drove to the high school and taught a class. You got high school credit and college credit if you payed the $50 community College enrollment. No ap test required.
Students could also take additional ge's at college outside of school hours, and I entered University with about 80 credits
I'd be interested in trying out the following idea: How about we don't mandate mathematics and science courses in high school, and especially in early college? The benefits of this are two-fold. The people who want to pursue other interests are freer to do so. And, more importantly to me as a mathematics graduate student, the people who are interested in math and science can go much much faster, and attain a higher level earlier on. As far as I can tell, most other countries with successful mathematics cultures operate more like this. In my math PhD program, the foreign students almost universally arrived better prepared (more knowledgeable and skilled) than the Americans. I believe the gen-ed requirements and general lack of specialization in American education may have a role in this.
> As far as I can tell, most other countries with successful mathematics cultures operate more like this.
Which country do you mean? As far as I'm aware, it's the opposite: countries like China and South Korea have math as a required subject on their college entrance exams for all students. The non-STEM students may have fewer requirements, but I'm fairly sure they've still learned more than the average American high school student.
I was really thinking about the lack of 1-2 years of general education classes before a major can be declared, at university, with that particular remark. I know this to be the case in several European countries, and India; in India you choose a specialization at the "11th-12th" grade level I believe. I did assume that to be the case in China, though it looks like Chinese universities may actually spend a lot of time on gen-ed classes. I'll have to ask my Chinese friends about their experience. I think you declare a major in the first year of university in South Korea, but I could be mistaken. If anybody wants to shed light on this by sharing when the major/specialization is declared in their country, please do.
I do know there are countries which allow specialization at the high school level (India, Italy, Romania I believe).
I found this interesting list, which is more complete than I can be on this. I don't think it proves that countries with great mathematical cultures always allow specialization in high school, but it does have some examples of this.
Most colleges I'm aware of these days have a completely softball core math/science requirement. "Rocks for jocks" and that kind of thing. Whether it's a waste of time or not, I don't think it really impedes the serious math learners.
I think you are right that we need more flexibility on the high school level though, at least for juniors/seniors that have safely passed the earlier requirements. I like that we have a chance to dabble in various subjects in college here before committing to something, but it would be a lot more efficient if some of that could be done in late high school.
"Whether it's a waste of time or not, I don't think it really impedes the serious math learners."
I thought about this for a bit. Suppose we kept the number of teachers (or courses) fixed, but eliminated the gen-ed math requirement. You would end up with a better teacher : student ratio, and I believe this would benefit the serious math learners. And so by not doing this, I believe we may be implicitly impeding them.
My guess is there are lots of ways wasted resources (and I do believe any class which could be perceived as "rocks for jocks"-type is wasted resources, and may actually be a damaging use of resources - I say this as a TA for the pre-major courses*, where the students are constantly and overwhelmingly negative about the whole thing) are implicitly impeding the serious students.
(And, as a positive note, I think the students not characterized as "serious students" for the above discussion could have a chance at meaningful and valuable contribution elsewhere, if we weren't putting so much effort into wasting their time.)
Edit: Changed "lower level courses" to "pre-major courses"
Just my opinion but... is Calculus an important high school goal? I took AP Calc in high school, got a 4 on the exam. I did Electrical Engineering in college and took college level math through differential equations. And yet... a) I've never used calculus once in my STEM career, b) looking back I realize I never really understood calculus back when I was in high school and college.
I came to that realization a decade after college when I was digesting 3Blue1Brown's series on Calculus for fun and had it finally click. Before then I was basically a Chinese Room that was able to solve calculus problems via pattern matching (i.e. "oh, this problem fits the shape of these rules, etc.") without really understanding how calculus works.
> without really understanding how calculus works.
I think this is the key. No, calculus isn't incredibly applicable in day to day life. But understanding how it works builds critical thinking skills that absolutely are. Math should be about critical thinking, not memorizing. Common core seems to be moving us in the right direction.
It's kind of funny how both sides of the political aisle have extremists who find ways to argue that being dummer is better. They mirror each other in so many ways and sometimes the two extremes kind of wrap back around to the other side and have the same goals.
No, the two sides of this debate aren't mirroring their arguments.
One side (Equitable Math) is engaged in a discussion of white supremacy in Californian math pedagogy. The other side (Moses Charikar, Scott Aaronson, et al) is arguing against a weakening of math standards.
I don't think he's talking about this debate, just that the Equitable Math extremists have some common ground with e.g. southern conservatives in how they view educational achievements.
Parents aren't going to stay in school districts that prioritize "equitable outcomes" above all else. This is likely the reason why real estate values for towns with 95% asians/whites are exploding YOY.
In the last couple of months, two readings stay with me on challenging the notion that math/science are things that only “certain kinds of men” do (more a gendered stance in the u.s. than eastern europe / asia).
The first is The Dawn of Everything, by David Graeber. I’m left with the notion that what we now call science and mathematics emerged from tinkering, persistent experimentation done mostly by women, and that what we have now disciplined into mathematics emerges from the systematic study and production of pattern (basket making, the organization of communal structures), and Graeber seems to argue against the hierarchical gender divides when viewed across the broad stretch of human history.
I think that the larger point both are making is that disciplines don’t have to be the way they are constructed now.
My only “political”’statement would be the hope that states (particularly the u.s.) would invest as deeply in mathematics education at the primary and secondary level, for all of its communities, at the level of investment in big science and big military projects.
I also believe that maths and STEM is something that all people do. Well, maybe not all, but at least not just a few geniuses. For that matter, I believe the progressives prescribed the wrong solution. The ordinary people, those who are are not the top X% in schools, need the push from their teachers, need the challenges in course work to push themselves to truly learn, and need a certain repetitiveness to truly grasp the fundamental concepts and skills.
How do I know that? I was one of such students. I made it because my teachers pushed me. They used carefully designed homework to show me that I didn't really have the deep understanding of math as I thought. They used homework to show me that my understanding of Newton physics was inadequate. They used midterms to show me that I didn't really intuitively grasp algorithm analysis (I didn't know how to rigorously and succinctly argue that two pivots and one pivot are really the same in quicksort). I grew to enjoy STEM and CS precisely because I was lucky to have tough yet encouraging schools. Dumbing down courses would have taken away the push I needed, and I would likely have a lot more failures in my life.
Efforts like this are well meaning, but only treat symptoms. The major impetus for watered down curricula is the very rational fear that large swaths of the student population will fail if expected to perform at the prior standards of rigor. Schools are not prepared to hold back massive numbers of kids, drop out proclivity for students held back rises, and teachers will be poorly evaluated by virtue of their students' inability above and beyond reasonable expectations of what they can learn in a single year given prior failure to build a proper foundation.
How we got to this point is perhaps a lot more complicated and politically fraught, but it has to be dealt with. Administrators and state education leadership are often simply responding to the incentives and avoiding dire outcomes suggested by the data. They have to craft palatable excuses for it, and it's ultimately a waste of time to engage those excuses on face.
This is a natural consequence of the American myth/narrative that everyone is suited for academic high school and university. Other countries simply sort people into different schools much earlier, and spend the high school years teaching meaningful vocational skills to those not on the academic track, rather than wasting everyone's time.
I'm not sure it's merely that. Most vocational track high schools types (except the hotel / restaurant ones that are well designed) here in Italy are honestly bad, not really helping students reach their full potential, but the content of the courses isn't on-its-face farcical like that "data science" course.
There are clearly decision makers detached from reality involved here.
Unfortunately, tracking is awful for the low-track students because guess what, no self-respecting teacher wants to be stuck teaching underperformers. So the students are trapped in that situation, being taught by terrible teachers who don't actually care for their educational achievement, and unable to improve. You see those outcomes across the board, including in the celebrated German system of academic vs. vocational school tracks. Yes, there are ways to cross through to the highest tracks, but very few students can avail themselves of those practically.
> “…no self-respecting teacher wants to be stuck teaching underperformers.”
If you look at people that are not academically inclined and call them under performers then maybe that’s part of the issue. I think everyone has strengths and weaknesses, and if someone is more inclined to do vocational training rather than the standard track I think their strengths and interests should be developed / encouraged. An educator teaching someone that is actually interested in what they are being taught sounds like a more rewarding experience than lecturing a class where maybe 5 of 40 (not an actual stat) students are actually engaged.
That’s a good point. I am not aware of a “good” way to make that kind of judgement, especially considering the lack of resources available to those that would be making such a judgement (k-12 educators at public schools)
> If you look at people that are not academically inclined and call them under performers then maybe that’s part of the issue.
It's not me doing that, it's education schools telling prospective teachers that their students will just be "learning their math by themselves", and the teachers can simply be facilitators. It must be a comfy job teaching math class to little Carl Friedrich Gauss and the like, but what about the remaining 99% of students - who will need actual teaching?
Yes, but if you do sort students by academic achievement, you will send a smaller fraction of blacks and Hispanics to academic high schools than whites, and a smaller fraction of whites than Asians, since there are differences in academic achievement by race. I say so be it, but currently many people assume that disparate outcomes prove racism.
If it's based on ability, it should not matter what percentage of advanced students are white, black, asian, or other. All that matters is that if you're good enough, you get in.
It's possible this is a good thing. Though it's also possible sorting too early determines someone's life path before they are mature to understand their talents and abilities. So I don't think early sorting is superior to the American model because at least the American model can uplift late bloomers.
A famous example of this that absolutely blew my mind is Uğur Şahin.
For those not familiar, he is the founder of BioNTech, the German researchers who developed Pfizer's SARS-COV2 mRNA vaccine.
Uğur moved from Turkey to Germany at age 4. At the end of primary school his teacher had assigned him to 'hauptschule'. It was only because of a neighbour's intervention that he was later put through 'Gymnasium' i.e. on track to study 'higher' studies.
The rest is history, after med school he did a doctorate in Imunotherapy, he founded a 18billion revenue biotech company and saved countless lives with what we now call the Pfizer mRNA covid vaccine.
Had that primary teacher's decision been held it really would have been a 'butterfly effect' of catastrophic proportions for Humanity.
You don't know the counterfactual, though: how many Americans' talent is wasted because they are forced to sit through watered-down school until age 18 before they can begin serious studies?
This is probably the best counterargument to raise in response to the "late-bloomer" anecdotes that many people raise.
For every unit of societal productivity created by a late bloomer that is saved by a common-stream system, I would personally argue that there is an order of magnitude more societal productivity lost by holding back the more typical high performers.
Late blooming intellectuals aren't the norm. Most highly intelligent people begin performing as such from a young age.
Yes, and it’s not just the high-performers who are being held back in American-style systems, but low-performers too, people who are either unsuited for or uninterested in academic work are forced to waste their time on something that doesn’t benefit them.
This one example just stuck with me when I read it because of how serindipitous his mRNA research turned out to be during an all out worldwide pandemic.
I've talked to some high school students about going to college despite high costs, and there was a general fear (statistically backed) that they would make less over their lifetimes without a degree and that not going to college would put them in a worse position to do basic things like buy a house as costs continue to rise.
That doesn't explain what has changed within American public education, if it's true what the parent commenter said: "large swaths of the student population will fail if expected to perform at the prior standards of rigor"
England mostly doesn't (though there's some local variation). Most secondary (11-16/18) schools are 'comprehensive', covering the full range of abilities. It is standard to set pupils by ability in most subjects, but uncommon to track/stream pupils by general ability across the whole curriculum.
We previously ran a split system, with exams splitting pupils at 11 into academic ('grammar') and non-academic ('secondary modern') schools. Many grammar schools were good; most sec mods were awful. The exams also famously had pretty poor predictive power for underlying academic ability as an adult, so pathways had to be developed to allow bright pupils to go to higher education despite being mis-sorted in the original exam - which somewhat undermines the point of the system. Comprehensivisation has never been nationally mandated, but nearly all areas have now done away with academic selection at 11. The remaining holdouts are mostly suburban Conservative areas where political power is in the hands of the well-heeled upper middle classes who are strongly in favour of grammars (and expect their own children to go there). Interestingly, Margaret Thatcher (as Education Secretary under Heath) was responsible for more grammar school closures than Labour was.
Vocational/academic choices are now being made at age 14 and 16 either within the secondary schools or by moving to local further education colleges (similar to US community colleges and trade schools). Vocational classes are low status. Britain no longer has a strong industrial sector to use such skills, and although both parties advocate for better vocational education essentially everyone who matters would be most unhappy if their darling children were diverted from a university track in the direction of trades. That's something they want other (poorer) people's children to do, not their own. We've never developed good vocational training for offices/services jobs, and although such courses do exist they're not taken particularly seriously by employers, despite several rounds of national reforms.
In Germany you can be sorted into a Hauptschule around age 10. At this point you will definitely not get a university education via any usual path. Increasingly even "the trades" are closed off to graduates and people expect a Realschule (HS/GED equivalent) degree for those.
I don't think university is appropriate for everyone and I dearly wish skilled trades had a higher position in society, and generally don't believe in the idea of "unskilled labor". But the German system is ultimately as cruel as the American one.
I don't think either system is great right now. From an American perspective, I really wish that there were still trade-type classes in high school, and those were more accepted as a viable career path at that stage of life for those who want it.
One path could be partnerships with local community/technical colleges. My high school participated in a program called "Middle College" where high school students took classes at a local community college (can't remember if it was all or just some) and that seemed to work well.
Deciding the future of your life in 4th grade (when students are evaluated in Bavaria, at least) is maybe not great. Was living in Munich for five years and I'm glad we left before our son got sorted.
There's also a strong racial/class component to who gets into Hauptschule/Realschule/Gymnasium, surprise surprise.
Here in M-V I had to go through "Orientierungsstufe"¹ which I partly blame for my learning difficulties and anger issues later in school, since in those two years we later had to find out that we missed 1-1,5 years of material and/or learning methods depending on the subject.
I mean... e.g. in Philosophy the teacher was absent for most of those two years resulting in us having to entertain ourselves for those "lessons". When we finally entered Gymnasium at 7th grade we received a culture shock when we had to learn the "Zauberlehrling" by Goethe in 1,5-2 weeks for recital in the first weeks.
Realschule felt more like Kindergarten from the treatment by teachers like their attitude while teaching and them not being interested in bullying unless it turned really physical, then protected the bullies when the bullied hit back.
What I'm saying is, as the slightly fat kid who also didn't get all the shiny new things from his parents I was bullied by a group of other students surrounding and harassing me almost every single day during breaks those incompetent "teachers" only had the "advice" to basically let the bullies tire themselves out from bullying but heaven forbid once trying to break out of the encirclement i tried to hit one of those bullying lowlives, the teachers descended like vultures isolating me in a room for the rest of the break questioning why I did that instead of "just ignoring the bullies".
Those bullies never received any kind of discipline/punishment.
Later in Gymnasium, I once had a very heated verbal altercation with a classmate within earshot of a teacher, we were taken aside, our parents called in for an evening sitdown that same week, there we resolved our differences with some guidance from a teacher and remain friends even now.
Both schools were only staffed by older teachers with the youngest being late-40s/early-50s and several teachers retiring during my stay at the Gymnasium, so probably little to no influence from education during newer eras of teaching.
Honestly I wish I could smack that idiot Brodkorb for all the stupid shit he did as education minister.
¹: no split after 4th grade, instead have everyone go through Realschule for 5th and 6th grade, only starting the split at grade 7.
Germany famously does. Children are given a test and depending on the score they go to Gymnasium (in perpetration for university) or one of two other options which trains them for either trades or labor.
I think the cause/effect is reversed. Society is damaged because parents cannot raise kids properly for a variety of reasons. You have kids who don't come to school, entire classrooms where 80% of the time is spent managing behavior, kids who receive zero parental support at home because the single parent is working two jobs, etc.
That problem is hard to fix, whereas the curriculum is soft and malleable. You also have an entire industry of education PHDs who have never taught class for any appreciable amount of time who have a neoliberal fetish for minor policy tweaks as the path to heaven.
Teachers have the same problems as police, random societal functions have fallen to them by default because there's no alternative. They're surrogate parents, social workers, mental health counselors, etc. and there's barely any time to do any teaching afterwards.
I think America is going to have to take a hard look at its math education if it’s serious about re-industrializing. Where are all the extra engineers going to come from?
I think you are touching on theory I have that Americans are becoming increasingly resentful that technical skills are becoming more necessary for middle class living. This is a huge driver of the pervading sense of precariousness. For some reason there’s a huge math phobia in this country
These things go in cycles. A lot of people don't realize that Scopes actually lost the Scopes Monkey Trial, and that the tide didn't turn overwhelmingly to rationalism in public education until the 1958 National Defense Education Act (which was motivated by the idea that there was a risk of the nation falling behind scientifically). The tide will turn towards rationalism and rigor again, eventually.
Homer Hickman came to my mind reading your comment. His Sputnik moment catalyzing the journey from coal mining town to NASA is a metaphor for that ideal. However, I’d have a hard time imagining America in 2022 has anything in common with 1959’s West Virginia.
The unstated irony is that there’s a large overlap in the anti-immigration and re-industrialization crowd.
I think an interesting part of 20th century American industrial/scientific history is that many of the prominent figures were European immigrants (many Jewish) or children of immigrants.
Maybe there is something exceptional about the environment itself. But the talented and privileged native citizens rarely aspire to be an innovator; they dream of being a leader or strategic thinker or mover of capital.
That’s why I think hard math is so marginalized even in elite circles (“I’ve never been good at math”). Technical work is essentially blue collar to the upper middle class. Many of my peers are in this group, and the only time they were ever interested in math or programming, was due to its possibility as a conduit to a more prestigious position.
This attitude is directly ingrained in undergrad institutions. They focus on general knowledge to serve as a justification to skip over front line work to become a leader (military officer, factory manager, investment banker).
There are excellent American technicians no doubt but most of them don’t fit the typical WASP mold or are predisposed to obsessing over systemic topics (which describes myself, being ADHD, although I can only strive to be excellent).
And the upper class? They have never aspired to much of anything really other than hedonism and protecting their position of status. At least in other countries, the upper class ideal is a renaissance man.
Edit: some of these assertions are sweeping and maybe a little mean. But I do think the thesis is directionally correct. America will need to change the culture around education to succeed in the 21st century. Immigration has and will continue to be a boon; but we have to accept the possibility that America could become a less desirable immigration target
I have to imagine these countries are going to begin limiting or restricting this type of emigration as a nation's people is its biggest asset. Brain draining the world without reciprocity is the greatest foreign policy we've had.
Right. “Unfortunately” doesn’t mean we should not continue to rely on immigration for high tech. It means it’s unfortunate we’re not providing the a similar pipeline of people domestically.
Is the curriculum being watered down? My reading on this the last time it came up on HN is that the sequencing of math topics is being changed, which results in classes not having the familiar names like Algebra 2, Geometry, AB Calculus, etc. That doesn’t mean the concepts will not be covered by the end.
I remember the huge blowups over “Common Core” years ago, which included new ways of teaching math concepts. I got to see some of it in action during COVID as I sat in on elementary Zoom school with my kid. I have to say I was impressed; they used techniques I did not recognize, but they seemed to work well.
As someone with a math background I found the blow ups over common core to be ridiculous. Focusing on parents unfamiliarity in place of any actual discussion of effectiveness.
If we are going to get anywhere with math education, it can’t be based on pandering to parent’s expectations.
> The major impetus for watered down curricula is the very rational fear that large swaths of the student population will fail if expected to perform at the prior standards of rigor.
I don't think the previous standards of rigor were ever so high for most students.
When I was growing up in the 80s and 90s in the suburban working class Midwest, the vast majority of the students in my high school didn't advance beyond algebra. This was in a well resourced school district.
But that was a time when it was felt that most people didn't even need to know anything past basic math to be employable. Times have changed of course, but the vast majority of educational paths, even STEM paths, don't require calculus.
However a great many do require basic data analysis abilities, so it's reasonable to emphasize those.
This shouldn't be done at the cost of offering calculus as an option for students who are prepared for and motivated to do it, though. Of course in the end this is about cost. Assuming the same resources, to teach a broader set of students data science will require reducing the availability teaching resources for something else.
I also recall that in the 1980s in some places there were programs that taught calculus at public community colleges for students who were on an accelerated academic path in high school. That is another option to consider.
> How we got to this point is perhaps a lot more complicated and politically fraught, but it has to be dealt with.
How many folks are really willing to do that? The problem is that actually discussing the perceived root causes of this issue will get you shunned out of polite society.
> the very rational fear that large swaths of the student population will fail if expected to perform at the prior standards of rigor.
Why is this a rational fear? It seems shocking to me that students can't perform at the same prior standards given their parents are more educated than the prior generation's parents.
Potential reason it's rational: The shoulders of those giants are higher than ever before. Climbing to those heights is correspondingly harder to do.
Speaking for myself, I had to worry very little about computers until late in high school (Senior year, specifically). There were no spreadsheet or word processing classes, and the typing classes were only for the girls. There was Algebra (up through Calculus as an optional course), but many others that my niece has that I didn't.
Hard to know. The public perception of public employment as a whole has gone considerably down since about the Reagan era. The pay is mostly not very good, and states and the feds have much more power over things that were traditionally local.
Rather than standing tall for universal high standards, education officials pass the buck to look good. The result will be an increasingly unproductive, stratified, and unequal society.
You can pass and pass and pass people, but eventually an employer is going to need someone to do the job, and they will make sure they get that someone. So there will be a hard standard sooner or later. The only choice is whether we give all students a chance to meet that standard.
Maybe I am missing something, but how would allowing some kids to take more advanced math cause other kids to fail? You could allow kids to graduate without taking the advanced math, but still let kids who want to/are ready take the more advanced classes.
Okay but what if American children were subjected to the standards of Chinese schools? You'd see 70% failure rates overnight at every school in the nation. Surely, in that case, we'd see policies like these pushed even by white people.
Likewise, the minority parents and schools simply see upper-class white schools in this country as we do the Chinese schools. I'm not saying that this is the best solution to the problem but I can at least understand where these parents are coming from.
We're not talking about abruptly changing standards, we're talking about holding kids to existing standards for which our society is already calibrated. I don't think your analogy applies (also, are Chinese schools really so much more rigorous than American schools, or is this assumed based on performance of Chinese immigrants?).
Yes, they really are. Look up example questions for the Zhongkao (national high school entrance exam) or Gaokao (national university entrance exam) and ask yourself how North American students at those respective stages would fare.
Declining math scores can be tied almost entirely tied to economic disparity (which unfortunately tracks with race pretty closely). Rich kids are on track, poor kids are far behind.
Unfortunately I don't expect these changes to impact that. It will likely take more rich kids out of public schools and further widen the gap.
> Rich kids are on track, poor kids are far behind
The achievement gap is growing. Part of this is explained by union dynamics [1]. Part by California's elites caring less about addressing the gap than talking about it.
> Altogether, this study provides some evidence that contract changes are associated with the educational opportunities of school districts’ diverse and economically disadvantaged students.
Essentially, if you want to prioritize your child's education above all else, get them into the richest school district you can manage.
Here's my experience as a parent of a 20yo who went through the MVLA school district in Mountain View.
It's a warning to any parents of younger children: Unless something has changed radically in the past 8 years, your child will be put into a math track in 6th grade: Separated into standard, accelerated and advanced classes. Which track you're in is determined by grades, standardized tests and teacher input after 5th grade.
This track determines which classes you can take in 7th and 8th. If you were in the advanced class, you will have finished Algebra 1 by the end of 8th grade. This allows the student to begin 9th grade taking Algebra 2, and then extending from there so that by their senior year they can take AP Calculus.
If you want little Suzy to be in more advanced classes, you better be prepared to be the most vocal Tiger Mom Karen you can imagine, because you'll have plenty of competition. As a result, almost no child moves between tracks. And in fact, in my opinion, the difference between normal and accelerated is so little, I'm pretty sure it's there just to give those children somewhere to go.
In other words, if your child doesn't demonstrate math skills as an 11yo, they will unlikely be able to take AP calculus 7 years later without doing something extra like taking summer classes, redoing an entire school year (an option a fellow parent I know took), or extraordinary effort like that.
Even if the MVLA education system isn't exactly the same now, or you live in a district that does something totally different, or even if you're in another state, I suspect this sort of thing is happening everywhere.
I personally was happy my son was in accelerated classes, right up until 9th grade when I realized how this circumscribed his future options for classes. In the end he would never have wanted to take AP Calculus, so it was fine. But I personally felt like I had fucked up as a parent because I simply wasn't paying attention. Planning out your kid's future math classes in detail in the 5th grade never crossed my mind, or if it had, I would have dismissed it as ludicrous overparenting! Had I known, I might have sent him to a math camp or something if I had realized how important the difference between a B+ and an A in math was at that moment. And he my have really gotten into mathematics as a subject. I really don't know.
So anyways, that's my experience. California is such a massive change from where I grew up in rural NH, I honestly can't imagine where to begin to fix a system with so many millions of children from such a varied socioeconomic and cultural backgrounds. I barely got my one kid through the system unscathed, and I live in one of the wealthiest districts in the country.
In New Jersey and Massachusetts at least, there is absolutely no move to water down the math curriculum in any way. On the contrary, the schools here compete on how many AP courses are offered.
It depends on the part of Massachusetts. The suburbs close to and beyond the I-95 have astoundingly good public schools. They're also expensive to buy into.
Boston proper and surrounding areas have begun to suffer from the same epidemic of anti-gifted mania that plagues California and is beginning to affect New York City, especially with regards to the exam schools (i.e. Boston Latin et al).
When I moved to MA, it blew my mind when I learned that what Florida considers an 'advanced' or even 'specialized' education is literally the baseline education in Boston and the suburbs. The best schools in Miami Dade County would be median in Middlesex.
New Jersey also has an excellent practices of magnet schools at the county level. So that even if you are in a poorer school district you can go study and apply for a magnet school at the county that is better and more focused on high academic schools. These schools focus on basically college level education for their area in health care, biochem, technology, arts, etc. with many AP courses. The normal public schools are also good in NJ with a solid baseline, but the magnet schools open up access to what would otherwise be private specialized schools to all income levels.
One thing I notice in NJ is that although there are plenty of private schools and academy's, you still see a lot of people in multi-million dollar homes paying 60k a year in property taxes sending their kids to the public high school. Even the most progressive mother wouldn't do that if the schools weren't at least up to snuff at offering the curriculum that can get their kids into "good" universites.
Magnet schools benefit the individual student but they are a bit problematic for many schools because they pick off the highest performers and whoever is willing to do the work to get admitted / physically drive to the magnet.
And then the home school quality goes down and that hurts everyone else who isn't among the small group at the magnet.
If you take the position that calculus is the concept of limit and all its consequences, then things like exp() and log() are calculus and it's hard to get anything done in CS without those. In this view, saying that quicksort is O(n log n) is a statement of calculus.
If you say that calculus is derivatives and integrals, then I'd say that calculus is not that important in a digital world, and that discrete math is much more useful. However, discrete math is harder than calculus, but you can use calculus as an approximation to the discrete answer (i.e., compute the integral if you don't know how to compute a sum, or use a derivative to approximate a difference). Ironically, this is the opposite of the old attitude that the continuous answer was the true one and the discrete answer was a poor man's approximation to the true one.
Also you don't need calculus to "do" ML (even deep learning research!)
I got to the point of writing my own toy neural network from scratch, seeing backpropegation, figuring that I'd have to use the chain rule myself on my forward pass, understanding what "automatic differentiation" was and why it's important, and decided "screw that I'm not putting myself through this hell again" and decided to look into https://en.wikipedia.org/wiki/Derivative-free_optimization
You can literally just find your own hetrodox subset of scholars in your field who are like "calculus? pfft!"
I've never understood the cult of calculus. I think it comes from mistaking the importance of its discovery for the importance of teaching it. It was a huge unlock for science, but is in no way a huge unlock for most people's lives.
On the other hand, I feel psychology and stats are the biggest missing pieces in K12 education. We need to build greater awareness of human needs and fallibilities, and awareness of how to make decisions in uncertain environments by understanding probabilities. Both are about developing a nuanced perspective on life and making better, more sober decisions, and they build a great deal of empathy to boot.
Finally, psych & stats are inherently relatable - everyone deals with people and has to make decisions. So much of the K12 experience isn't relatable, which is why students often hate school.
As I recall from when this came up a few years ago, the "cult of calculus" was because in the post-war era 'the end-users of mathematics studies [were] mostly in the physical sciences and engineering; and they expected manipulative skill in calculus.' - https://en.wikipedia.org/wiki/New_Math .
One way to see how curriculum has changed over the last 70 years is in Sheldon Glashow's autobiography. He graduated from the Bronx High School of Science in 1950. Quoting https://www.nobelprize.org/prizes/physics/1979/glashow/biogr... , "High-school mathematics then terminated with solid geometry."
Calculus is critically essential for learning many later math fields, and many important topics. Mechanical Engineering, Electrical Engineering, Physics, Chemistry, Civil Engineering, Aerospace, etc. There is a lot of very critical fields to modern society that requires knowledge of calculus. You can't build a modern bridge without calculus.
I don’t dismiss its value in the broader education system, nor for certain industries and jobs. But specifically in the context of K12 requirements & expectations in the process of applying for college, it’s hardly foundational knowledge for most of life’s paths.
Ergo, it’s best to a) not to have college expectations be built around most/everyone having it before college and punishing those who don’t, b) focus on teaching it where it’s needed (eg when in college for those majors), and of course c) if a kid knows their path involves it earlier, make it available to learn when they want to.
“[current curriculum] creates students who see mathematics as an exercise in memorized procedures that match different problem types. Yet, university professors and employers want graduates with critical thinking and reasoning skills.”
Is there a summary of the CMF that removes all the bureaucrateese? I didn't get very far through that molasses. I had to scrape it off with my trusty strigil.
I don't live in CA and this isn't my circus, but I have some things to say about math education. From the statement:
> We write to emphasize that for students to be prepared for STEM and other quantitative majors in 4-year colleges, [...], learning the Algebra II curriculum [...] in high school is essential.
Problems with math are one of the most common reasons why students encounter difficulties in STEM education and careers. The most common problem is difficulty with high-school level algebra.
I agree, fundamentally, with the relevant premise of the CA effort here (and agree with Aaronson's criticism of its implementation). That premise is that you shouldn't have to be on an accelerated track in middle school in order to take calculus in high school. And yet... the fact is, we get a lot of adults in college or graduate school pursuing STEM degrees, who have shaky foundations in high-school algebra.
Just looking at the "typical" math track in US high schools it does seem a bit arbitrary. Algebra I, Geometry, Algebra II, Pre-Calculus, Calculus--this is the most common math track I see, with accelerated students starting Geometry in 9th grade, Calculus in 12th grade.
The thing is... individual performance is highly variable in math classes, and to make sure that everyone gets good foundations in mathematics, we see high-school mathematics curricula that repeat the core algebra concepts in different classes. This repetition and focus on fundamentals is why the division between classes seems so arbitrary--what is presented as a sequence of classes is really more of a unified curriculum spread across multiple years. When you combine these two factors (variable performance, repetition in the curriculum), you end up with a population of high-school students who develop good foundations in algebra early on and are bored by the repetition, and a population of students who really benefit from the time spent mastering algebra, and it's hard to serve both.
I think we can figure out a way to let high-school students take AP calculus in 12th grade without expecting them to take Algebra I in 8th grade, and we don't need to push everyone into calculus faster in order to do it. And yet, my experience with high-school education in the US has left me very cynical about it. Letting students progress through the high-school math curriculum at the right rate requires a kind of "personal touch" that seems to only happen to individual students when their parents are involved, but not pushy. It's rare. The school system would rather do the easy thing (everybody moves in lockstep to the next class in the sequence), and parents are largely either uninvolved or overinvolved.
(This is more or less what the article says, I'm agreeing with the article.)
Considering the fact that the vast majority of students aren't going to go onto 4 year STEM degrees, it doesn’t make sense to track all students towards that goal.
I feel as though there is too much focus on giving everyone more or less the same type of mathematical education in high school. This is probably due to limited resources (ie teacher availability and class sizes), but ideally there would be room for a more varied approach wherein students don’t need to have every year build on the next if the aren’t STEM tracked. Too many students fall behind and never are able to recover. Math class just becomes dead time, and those that do make it to college end up retaking the same subjects over again.
> Considering the fact that the vast majority of students aren't going to go onto 4 year STEM degrees, it doesn’t make sense to track all students towards that goal.
It sounds like we agree 100% on that point.
I'm mostly thinking about the students who are going into STEM degrees later in life, who will (hopefully) come from varied backgrounds in high school and middle school. If you decide in high-school that you're interested in STEM, then it makes sense to develop solid foundations in algebra during high-school. Just like it doesn't make sense for all people to take math like a STEM major, it doesn't make sense to fast-track all future STEM majors to take calculus in high-school, and it doesn't make sense to make decisions in middle school that lock students out of high-school calculus.
The thing that confounds this is that people overvalue high-school calculus as the ticket to a STEM degree, when (like the article says) many people would be better served by developing stronger foundations in algebra. And public schools are generally not good at educating students at their own rate & level.
The tension here is that educational achievement is a proxy measurement for IQ and in capitalist societies IQ is directly correlated with income. The cultural ideal here in the US is that hard work, effort, and grit is what is necessary for success. It is supposed to be purely egalitarian; you get what you give. However whether you are born with a high IQ is purely random, and even worse IQ is heritable. This is creating a technocracy which is at odds with the egalitarian ideal. This along with the expanding wealth gap is causing the schism you see today.
I mean they are completely different measures. They both contribute to success but can be tested independantly.
Also assuming that "capacity and willingness to do hard work" is a personality trait, that is also largely random. So no matter which way you slice it, you gotta get lucky on either the intelligence or industriousness axes (or both) to be successful.
Who you are born as is random, but who will have children with higher IQ is not. Work ethic is less nature and more nurture than IQ, or at least feels that way intuitively.
High IQ by itself doesn't get rewarded very much without hard work, effort, and grit. If you have all of that, yes you'll be rewarded more than others who lack one or more of those qualities. And you should be.
I have worked in education for 16 years and just wanted to add my perspective. Here in Illinois over the last few years the state has shifted its goals to "equitable outcomes". This in and of itself is responsible for much of the lowering of academic standards since it is a flawed (but perhaps well intention-ed) goal.
In a excellent school district in a suburb of Chicago a district goal was adopted to reach equitable outcomes in higher Math. In a nutshell since black students scored statistically lower on AP Calculus this was seen as a failure in the school district. Despite increasing the number of black students able to pass AP Calculus the school district looked to cancel offering the course since Asian and affluent white students still scored statistically higher.
The idea that all races, genders, or whatever categorization you can come up with must have the same equitable outcome is a flawed goal. Education used to be about taking a student where they are and showing improved learning outcomes.
Equality used to be about striving for equal opportunity. The shift to conflating equality with equalized outcomes simply doesn't work.
I would suggest that in fact it hasn't; it was just as virulent over the past 50 years as it was over the last 3, but you just noticed it more in the last 3 years.
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This is a real laptop class kind of statement, to borrow terminology from one of the more prominent anti-"woke" investors.
Living in California is expensive as hell, especially for younger folks with no inheritance, and the state operates at varying levels of dysfunction because it's got 40 million people in it.
While the original post borders on a cliche at this point, the criticism seems to be correct in this case. Equalising math outcomes by destroying math education for the gifted is akin to levelling wealth by destroying all the houses.
