This is essentially the "calculus vs. statistics" debate that has been going on for some time now. I'm definitely on the side of making statistics mandatory, and then making pre-calc/calc electives for motivated students who want to enter STEM fields. Primary reason for this is that I believe a base knowledge of statistics is so important for everyone in modern society, and it directly goes to better being able to evaluate the constant flood of information we are now all subjected to.
I agree in principle that statistics education should be given to all students. The problem I have with the proposal is that I have very little confidence that teachers are prepared to provide a rigorous and thorough statistical education.
For an example, the replication crisis in the social sciences. These studies, containing flawed statistics, are carried out by social scientists with PhDs who have been required to take courses in statistics designed for their discipline.
If they can’t get the statistics right, how can we expect it from teachers with far less education?
> For an example, the replication crisis in the social sciences. These studies, containing flawed statistics, are carried out by social scientists with PhDs who have been required to take courses in statistics designed for their discipline.
> If they can’t get the statistics right, how can we expect it from teachers with far less education?
This isn’t really a statistics education issue, much like Enron wasn’t an accounting education issue.
Social scientists abuse statistics because the incentives are all aligned with abusing statistics. To get a good job you have to publish lots of papers. To publish you have to have statistically significant results. To get into top journals you need surprising results. You see other people in your field playing fast and loose with their data and getting rewarded for it. Why not exclude that one problematic subject from your data analysis? Without them the p-value (from a regression on a carefully chosen eight of your eleven measured variables) drops to .038, and you can publish...
It’s not that nobody thought to teach the scientists about corrections for multiple comparisons or the dangers of picking observations to exclude as outliers based on what gives you the result you want. They learned all those things. But it’s so much harder to get results when you’re not willing to play games with the data, and they need results. The people who do try to play by the rules wash out, either by choice after getting fed up with the fraud they see all around them or by failing to publish N papers in top journals.
(Social) scientists don't necessarily know statistics even though they have had some courses. It's very common e.g. not to understand what p-value actually means. It's disturbingly common that non-linearities or heteroscedasticity are totally ignored in regression. Assumptions of statistical tests are typically not understood.
It's largely a cargo cult. For example p-values are reported as t(N) = x, p < threshold but very few understand why and keep on doing it and even demanding it from others.
(The why is because the p-values had to be read from tables before computers. It makes no sense nowadays.)
this is not the only reason 'why'. Another persistent reason is the function of the alpha value in typical null hypothesis testing leading to the (misguided) idea that a p value of .04 is functionally equivalent to a p value of 2.3e-10 since both are below threshold.
So i would argue that this is more damning - essentially a misunderstanding of probability and what p values tell us at a fundamental level.
Typically you see p < 0.01 etc even with 0.05 alpha. A lot of stats software gives only those inexact values.
But yes, the interpretation of p-values and confidence levels are wildly misunderstood. p > alpha is often taken as "evidence of absence" of an effect, which is just wrong. Or when for some quantity p1 < alpha and other p2 > alpha, it's often intepreted that the quantities differ.
Most study outcomes have a political cause connected to them, even if that wasn't the author's intent. Those movements want these study outcomes to give their cause legitimacy and the membership of such movements often have a large amount of overlap with those employed in social science academia. It's little different than petrol companies that either fund or amplify studies with outcomes that benefit their business.
While there are definitely still papers being published as a result of p-hacking (intentionally or not), psychology has been undergoing a renaissance in this area for a few years now. See the "Reproducibility Project: Psychology": https://osf.io/ezcuj/wiki/home/
In short, there is a movement to validate past results, starting with those most influential in the field. A lot of progress has been made in that area.
Because all social sciences are not "roughly comparable to Enron." Hacker News is getting a bit ridiculous with the utter disdain shown for sciences and academia. The problem identified by the parent of this post is a problem of incentives that are misaligned with what we as a larger society want out of basic research. But every energy company had the same incentives as Enron. Not every energy company published fraudulent financial statements.
Poor replicability of social sciences is multifactorial. To some extent, what they're studying is a moving target that is not all that amenable to scientific methods, which tend to assume static reductionist laws dictating system behavior. The dynamics of how fundamental forces dictate everything from gravity to covalent molecular bonding don't change from culture to culture as well as over time as trends. When you're studying human behaviors and preferences, I'm sure some of it is governed by more or less immutable eternal laws, but some of it is semi-random diffusion of learned trends and what is true today of one group of people may not be true of any other group or even the same group at some later point in time.
Some of it is statistical illiteracy and not understanding the limitations of the techniques you're applying.
Some of it is outright fraud.
There is also interplay between those two because outright fraud is facilitated by peer reviewers not having the statistical maturity to be able to detect it.
> Hacker News is getting a bit ridiculous with the utter disdain shown for sciences and academia.
Just to be clear, nobody actually said the social sciences are roughly equivalent to Enron. I used Enron as a hyperbolic example to emphasize that there are reasons for poor behavior other than lack of education. I was not implying that academic fraud is at the same level as Enron fraud, and after rereading what I wrote I’m comfortable with how I worded it.
> But every energy company had the same incentives as Enron. Not every energy company published fraudulent financial statements.
Accountants and corporate executives have substantial disincentives against publishing fraudulent financial statements, like going to jail. Academics mostly do not have similar disincentives against abusing statistics. There have been a few high profile embarrassments, but for the most part even people who are widely known to have p-hacked their way to dozens of questionable publications are still sitting comfortably in their tenured professorships.
Because if we did we’d have to admit just how little real information we know in so many fields. We want to think that (as a culture) we’ve got it all just about figured out.
My anecdotal experience in a psychology faculty makes me think there is an issue of statistics mastery in the social sciences. There's also an issue of corruption like p-hacking, but so far the statistics courses I've had were poor and the teachers didn't really understand what they were teaching.
Yes, but this is not necessarily (only) because of the teacher quality. Statistical testing is quite complicated and very easy to do wrong (violate assumptions). On average scientists aren't very technically or mathematically apt, so in the few short courses things have to be taught in a "cookbook style".
Short term individual and small group behavior in artificial conditions can be somewhat reliably measured. How well and to what situations these genralize to is a harder question.
Larger groups and timescales are (in practice) outside the domain of science (in the most rigorous form) and must be studied with other methods and epistemological criteria. This also leads to major abuse of statistics as the inference has to be done with unrealistic assumptions and unwieldly models.
The "brand" of science is so strong that many fields, especially economics, want to appropriate it even though they don't and can't do science in the strict definition.
> If they can’t get the statistics right, how can we expect it from teachers with far less education?
Because it’s their job—and if the statistics curriculum is mandatory, then the teachers might spend inservice days developing statistics curriculum, going to workshops to learn how to teach statistics, etc. Teachers will develop statistics curriculums and share them with each other. With NSF grants, you can fund teacher outreach programs, put statistics exhibits in science museums, etc.
A lot of teachers will struggle to teach various subjects. That’s why we have support networks in place, to help develop curriculums and provide training for teachers. I know that the support network has a lot of problems—but many teachers would struggle to teach, say, biology, history, or algebra, too, without support.
I think what's far more likely is that the curriculum-writers will see it as an opportunity to dumb down the math. They'll make it into a very basic course on study design and data collection, with a handful of descriptive statistics thrown in so the students have something to do with their scientific calculator's STAT mode.
Will they teach discrete and continuous probability distributions? The binomial, Poisson, and normal distributions? Dependent and independent random variables? Bayes' Theorem? Measures of statistical significance and hypothesis testing? Chi-squared and student-t distributions? Confidence intervals and p-values? Maximum likelihood estimation?
Highly doubtful, since many of those topics are built on top of university-level calculus.
People who have some understanding of study design and data collection would be in a much better spot to understand and interpret day-to-day news / “information flood” than those who have done a lot of calculus-based probability. You can go all the way through rigorous measure-theoretical probability and come away with almost nothing useful for interpreting a study.
Most problems I see with moderns statistics aren’t of the form “ohhh, they fooled you by using a subtly wrong statistical metric to ascribe significance” but “the way the data was gathered/interpreted is fundamentally wrong and made to mislead”
Some of the problems come from statistics itself. “All models are wrong, but some are damn hard to interpret.” Let’s make that a corollary to Box’s statement about the utility of models. Why does so much statistical analysis take place without even an expectation of human comprehension? It’s just magical sigils for most papers.
Many midlevel statistical practitioners suffer from a holier than thou complex, where a “correct” approach to statistical analysis might buy a little more precision at the expense of a lot of comprehension.
Box plots or Bar charts with error bars, using randomized data collection. That’s like 90% of the interpretive value right there. Statistics is a UI for math and it could use improvement if we expect so much from it.
See Brett Victor’s “Kill Math” for more context on why we should expect more from our mathematical interfaces.
http://worrydream.com/KillMath/
> "Math" consists of assigning meaning to a set of symbols, blindly shuffling around these symbols according to arcane rules, and then interpreting a meaning from the shuffled result. The process is not unlike casting lots.
@dr_dshiv Edit: Sorry about the bad quote, I should have included the prefix "When most people speak of Math ...". And the negative comment about the blog post.
Now I've read / skimmed all of it and it was interesting, I hope the new methods he wants to use for teaching maths will work fine. (I'm sceptical, but still seems like worth to give it a try.)
I think his project title does his project a disservice: "Kill maths"? That sounds silly to me.
And how it starts -- I got annoyed and stopped reading (until I went back two days later).
Another more positive project title maybe could be "Maths for everyone" or "A new approach for teaching maths"?
In my experience they (try to) teach those somewhat. Not in a calculus-type rigor, or even very mathematically, but conceptually yes.
The problem is that those things are not "immediately" needed, so students don't learn them or immediately forget them if they do. What students "immediately need" is to run some test in some application and check if some value passes some magical threshold.
These students then become researchers and these researchers become professors.
>I think what's far more likely is that the curriculum-writers will see it as an opportunity to dumb down the math.
I'd go as far as to argue that has already happened. The reason math doesn't really teach problem solving and instead opts for working through things with formulas etc is precisely because that's a dumbed down version of math.
Not all topics are equally easy to teach and assess in a deep way, with limited time and resources.
Statistics is IMO a lot like security. Unless it's at a very high level, just follow a basic check list and don't do anything creative. Calculus is more like algorithms - you can get to deep and creative levels at an earlier stage.
My first real stats course was a 300-level engineering stats class that kicked my ass pretty well.
The class had a calc 3 prereq I found the computation generally the easiest part. Truly grasping the topic takes patience and work but it's pretty rewarding. That said, it must be difficult to find genuinely insightful instructors who can make the material remotely interesting because good god it's necessary.
I think Americans do calc 1 which is the basic derivatives and integration you typically learn in high school, then calc 2 which is more like a college calculus course, then calc 3 is multi variable calculus.
I don't recall if multivariable was pushed in at the end of my differential and integral calculus course or at the beginning of my differential equations course. It's possible it was also somehow tucked into my linear algebra course (though, I doubt it).
In any case, we did cover multivariable as a pretty straightforward extension of single-variable calculus, without making it a separate course. Do I likely have some huge blindspot as a result of not spending a full course on multivariable calc?
(All of my formal education was in the U.S., for what that's worth. Though, it was an accelerated magnet program teaching middle school students algebra and trigonometry and covering geometry, calculus, linear algebra, and differential equations in high school.)
Did you cover Jacobian matrices and how to to calculate and classify local extremes in a multivariate function? Do you remember saddle points? If not you did miss multivariate.
The main thing you lose then is you don't know how to apply calculus on non linear coordinates like spherical coordinates and so on. It is useful for data analysis if your data is easier to work with after a non linear transformation, but if you don't work with that sort of thing then probably not very useful.
Then you did multivariate calculus in the linear algebra course. It isn't that strange to do it that way since the hard parts of multivariate has more to do with linear algebra than calculus.
Thinking back to my own statistics at university… how do you even teach statistics without a lot of other math first?
Analysing data is similar. I'm all for it, it's clearly useful on the days when my main working tools are slack and powerpoint, but it's not clear to me how teach that without math. One of my math textbooks even had the word "analysis" in its title. ("Calculus and analytic geometry" perhaps?)
You can absolutely teach basic statistical literacy without too much other maths and certainly without any higher maths. That’s how it’s done in schools in other countries - for example here is the statistics component of the maths GCSE for one of the exam boards in the UK (the others will be similar) https://www.aqa.org.uk/subjects/mathematics/gcse/mathematics...
GCSE starts after the third year of secondary school in the UK so most people will have completed their GCSE in the year they turn 16. Then 16-18 year olds wanting to study STEM subjects go on to do maths A-level, which includes more topics that in the US would be considered precalc as well as some calculus, more probability and statistics and some mechanics. Here’s the A-level syllabus from the same board. As you can see, for stats it includes distributions and hypothesis testing, so much more depth https://www.aqa.org.uk/subjects/mathematics/as-and-a-level/m...
A level maths is not a necessity for all STEM courses. My daughter is doing AS maths (half way between GCSE and A level, heavily weighted to stats), alongside A levels in Biology and Chemistry (and one non science A level)
I'd argue that very little - or even no - math is required to teach the concepts.
The ideas of standard deviation and confidence intervals can be taught visually, for instance. You needn't be able to calculate them to understand what they mean when they're presented.
The extent of my statistics education in high school was "mean, median, mode". That was it, and I exhausted every math course that was offered at my school by the time I entered 11th grade.
You can teach basic statistics without a lot of math background.
If your class of 12 year old physics students has just timed a block of wood sliding down a sloped plank at different angles, and plotted an X/Y chart of their results, you can just have them put a best fit line through the points by eye.
No need for matrices or differentiation or X-transpose-X-inverse-X-transpose-y - just bang a line through the data by eye.
My wife is an academic and is repeatedly raising the issue of the replication crisis in medical fields too. This is an issue across all of academia and isn't just an issue with flawed statistics but pressures to publish, a lack of replication in the peer review process, a lack of replication by subsequent work building on that original research, using more tools like Machine Learning, etc...
Metascience is a growing field at many top universities because of this issue and the belief that modern science may be very flawed right now.
So we ram language constructs into people’s brains, which we started doing centuries ago to preserve knowledge.
Yet it comes with none of the warnings by the long dead mathematicians who initially built out statistical tooling[1]. Statistical tools are intentionally crude to help communicate the complexity of stats and yet we build society on crude leaky abstraction. Not out the obvious day to day right in front of our faces.
Modern science isn’t flawed. Society is. Boomers and GenX are proper gold star for nothings who lucked into living in the only country that could manufacture anything after WW2. They dismantled the New Deal that propped them up and made us their serfs.
They didn’t fight the war or build the economy. They have no muscle memory for doing anything “real”. They went to college (much less rigorous in the 50-70s), memorized the cliff notes and recited the catechisms. Neither generation struggled materially as any generation before them (yes yes distributions, ranges, gradients of truth; relative to material conditions before 1950s).
New generation comes along with no awareness of that time and doesn’t feel much obligation to status quo. GenX and Boomers are big mad about it, never mind they walked away from religious life. This time the future will stick to our script.
Society wants less to do with their hyper industrialized life due to war time manufacturing habits they had rammed down their throats by a much more imperialist society of decades gone.
We keep letting people who cheered on conquest of other countries to stay ahead of them keep running things. Everyone is too apathetic to tell grandpa it’s time for hospice. We let elders implicitly engage in ageism against youth.
Everyone is acting shocked the old warlords are looking for idiot soldiers to serve their fiefdoms? All they knew for their formative years was war time …hustle. Embedded deep.
Stop with the meta bullshit. Day to day life is just this. The abstract mental models are not helping. They’re distractions apathetic people escape into to avoid reality. Boomers down to apathetic centrists who love their material privilege despite the environmental toll… this culture is a joke
Yes, it is. A lot of the issues found in modern science with replicable studies comes down to the publish-or-peril approach. If you have academics on temporary postdocs having to publish X papers to get an extension, find a new postdoc, or maybe get a professorship then you're going to have issues. Add to this the lack of incentives to replicate papers under this stress and people build up on top of them rather than validating them first. Especially when the studies are expensive/time-consuming with MRI machines etc.
> Stop with the meta bullshit
The impact bad research has both financially and in society is huge. For example the issue recently with Alzheimer's where loads of work was built up on a 2006 seminal study that wasn't replicable (because of academic fraud). Finding incentives to catch bad science early is important.
I have no idea what the rest of your message is about.
We humans are soooo good at fixing systemic issues. Surely we’ll eradicate this decades old issue at great cost, plus the same old costs, and new costs; it’ll be just as tidy and a big win as your tidy little post puts it. It’s so simple it needs but a paragraph to explain, afterall. Really, no greed or other perverted incentives will creep up as the process moves on and government loses interest again.
The rest of my post was to suggest where the flawed incentives come from; educationally outdated meat suits and apathetic voting public who think they’re off the hook to society. Elder politicians enable such perverse incentives because they care about fiat currency flow, not science. Ignore it and dig into vacuous meta theory because the public is kowtowed by threats by the seniles
It’s amazing to me how many think choices today are guaranteed to matter tomorrow. You have no idea if what you say is possible given the state changes that occur constantly reshaping global society
In the end you’re peddling high minded BS
We can’t get the world to agree on climate change. Surely we’ll keep all hackneyed science from propagating, certainly we’ll keep the costs from ballooning to serve any perverse incentives that pop up as the public lacks any real command of the political system and it’s pork spending
Sure, sure. Musk will have a full colony on Mars first
To be honest, I'm not that concerned with high schoolers needing "a rigorous and thorough statistical education."
I'm more concerned with the populace at-large being able to understand and apply fundamental statistical concepts. For example, another commenter mentioned Bayes Theorem, and how it's a very powerful idea and not that difficult to grok. Related, predictive value positive and negative, and how they are calculated from (but also very different from) sensitivity and specificity, are extremely valuable concepts to understand.
To what you point out, I think the concept of "p-hacking" is super important to understand, but I'd be less concerned about a student needing to hand-calculate the steps to run a t-test (that's what a college-level class is for). That said, I decided to look up T-test on Wikipedia while writing this comment, and I found this interesting tidbit. Great example of the applicability of statistics, and something that I think would peak the interest of many high-schoolers:
> Gosset had been hired owing to Claude Guinness's policy of recruiting the best graduates from Oxford and Cambridge to apply biochemistry and statistics to Guinness's industrial processes.[13] Gosset devised the t-test as an economical way to monitor the quality of stout.
> If they can’t get the statistics right, how can we expect it from teachers with far less education?
I’m unconvinced that the incentives pull them towards “better use proper statistical methods” or penalize them when they don’t.
This is, IMO, not insufficient education; if it’s not, the premise that math teachers given incentives to get it right couldn’t accomplish it because they have less education than a PhD sociologist is flawed.
Teachers not being skilled enough for a high school subject is not a problem in 2023.
By the time the curriculum gets reformed, it'll be like 2025. By which time students can ask GPT5 for any statistics question.
Statistics is so unbelievably broadly useful at low levels, compared to calculus. Understanding
1. Selection bias
2. Normal distribution + standard deviation
3. Central limit theorom
Is a massive help in modern society. You don't even need any math equations, just understanding them on a rough conceptual level would help.
If we decide that statistics are more necessary in earlier education there is nothing stopping us from building a system with more teacher training for stats education.
I dreaded statistics in high school. When I started out with probability I was excited and thought this stuff was useful.
But when I got the about two-tailed tests and p-values, it was all just so opaque. I didn't feel it was that useful, plus the whole argument about rejecting or not rejecting the null hypothesis felt too philosophical and contrived. And why 95% confidence?
I hated statistics throughout undergrad because of those philosophical contortions, which seemed very arbitrary to me.
It wasn't until grad school when I discovered the statistical learning side of things (PCA/PLS) that statistics became exciting to me -- because suddenly statistics became useful and able to predict things.
I'm still convinced the 80% of people don't need to know p-values and null hypotheses. Some might say, but what! These are used all over the social sciences! Umm no.
Let the people who need to learn that learn, and actually teach statistical learning (linear regression, logistic regression, Bayesian statistics) to the rest of us in school.
Same, with the extra comment that every other math course I took had some way of verifying your result. You do your work and the result can be plugged into the original question to calculate some other element and see if it matches, or the value can be checked visually, or the equation will cancel out completely, or ... Not with the statistics as taught at the time. Your result is 0.48 - did you use the right test? was it supposed to be two tailed? did you calculate properly? did you choose the right distribution? does that method work with this distribution? Who knows. It's basically a memory exercise.
Couldn't agree more. I waited until the junior year of my undergrad to take stats. That was much, much too late.
Everyone should know Bayes' theorem. It's not that hard to grasp, but it's a wildly powerful idea. Even if you aren't plugging-and-chugging, just use it to dissect logical fallacies.
That is basic probability, not statistics. Basic probability is taught before high school in many parts of the world and is very easy to teach, there is no reason to put a lot of focus on this in high school.
You have the kids write down probability graphs from events, that takes a few lessons. Then you do some basics about confidence intervals and sample sizes. Basically every kid learns this at some point, just that they forget and now they think it is missing. Kids forget about what they learned about statistics since it doesn't seem relevant to them.
If we allowed schools to also operate as casinos, students would remember probability better. There would be a whole host of other side effects, sure, but maybe a well-educated populace is worth it.
There's an episode of The Wire, in which Presbo finds kids in his class betting on dice; so he collects the dice from all the board-games in the school store, and starts running classes on probability. The kids are totally engaged, and they start winning money from street-corner dice games.
I vaguely recall doing probability in middle school, but it was never framed in formal terms. The lessons never really transcended "let's roll some dice and draw some graphs." The college course helped me project those ideas into other domains. You can use probability theory to dissect rhetoric, for example. Never thought about that.
You learn about conditional probabilities and how to calculate around those. They teach Bayes theorem but doesn't use that name or the formula, since the formula is too hard for kids to read, but you are given the intuitive explanation for how conditional events are related to each other and you write those out in event trees where it is easy to see. I clearly remember getting taught the exact equivalent to Bayes theorem in middle school, they just didn't call it that.
If anyone who learned that sees Bayes theorem later it is intuitively obvious, so there is really no need to bring it up, and there is no reason to take a course to understand Bayes theorem later as an adult. Those things are easy thanks to what you learned in middle school.
Yeah Bayes' Theorem and confidence intervals, population vs sample means, and margins of error are all things any college graduate should understand. When you realize you can bake correlations/associations to imply whatever you want, you start taking any percentages people toss around a lot less seriously.
I agree with you that statistics is incredibly valuable skill, but teaching it is not that different from teaching calculus. Besides notions of set theory, you need a pretty sound understanding of the concept of integration. For that you need to still understand real numbers, functions, graphing, and possibly even limits [1].
If you are teacher, you learn pretty quickly that every time you invoke magic or "trust me it works this way" you lose students. Students are clever and have self-respect and they don't like being lied to, or when information is hidden from them. You have to present a coherent picture, and I think teaching statistics without calculus is incredibly difficult to keep coherent.
[1] I am not sure if it's possible to argue without limits that the normal distribution extends to infinity, yet has a finite area under the curve.
If you wanted anything other than hand waving, you can't even define a normal distribution (because you can't define `exp`) without limits. `exp` is a transcendental function.
You don't have to lie to students to define the exponential function itself. They are familiar with pi, so it is not difficult to introduce another irrational number, and define f(x) = e^{-x^2}. There is no lying here.
The need for limits and such arises when you try to differentiate the exp function. But we don't need differentiation for basic statistics. Just integration.
How do you introduce e? Just say it's 2.718...? The usual definitions of e involve a limit, and to my knowledge there's no simple geometric definition like there is for pi. Likewise I don't know of a definition of exp that doesn't involve a limit, and there's no simple geometric one that I know of like there is for sin/cos.
(There is a geometric definition of exp that I know of, but it's that it turns a vector into an integral curve, so not so useful without calculus or limits)
You also need limits to be able to talk about the central limit theorem/when a normal even ought to be used. Otherwise you get confused people thinking everything is normally distributed by default.
Remember that my larger point is that statistics can't be taught without teaching most of calculus, so just teach calculus first anyway. But, if we hypothetically tried to get away with the minimum...
> The usual definitions of e involve a limit.... I don't know of a definition of exp that doesn't involve a limit ...
Yes, I agree with you that if you want to define e as interesting itself, you need to use limits. Similarly, the way exp(x) was introduced to me in high school was as a function whose derivative was equal to itself (i.e. as an interesting function) - which also requires limits but I think my teacher/curriculum just handwaved away that part.
But in our hypothetical curriculum, I am indeed proposing that we just say that e = 2.718..., since we are not interested in e, but in its usage for defining continuous probability distributions. Then to compute something like e^2 you just plug it into the calculator (like you do sin/cos) and it will give the answer. But again, we will have to put in effort to argue that something like e^(5/4) or e^pi is a computable real number.
> You also need limits to be able to talk about the central limit theorem
Indeed, but I think rigorous usage of the central limit theorem is quite beyond high school mathematics.
I was agreeing with/augmenting your larger point: I don't see how you can do any justice to the subject at all without calculus, just like I don't see the point in teaching a bunch of solutions to memorize to particular setups and calling it physics.
Even Bayes' theorem is, IMO, most obvious in the continuous setting where you can interpret it in terms of relative areas, which gives a nice, easy picture. Making big tables and trees obscures the basic geometry.
One of the thing that strikes me with the BLM and DEI movement is that the vast majority of people seems to be incapable of thinking in term of distributions, and rather think in term of stereotypes, which leads to what is in my opinion, the very definition of racism and sexism, ie attributing to an individual some attributes derived rightly or wrongly from some metrics (average or percentile) of a distribution purely on the basis of skin colour or gender. Instead of treating an individual as an individual which could be anywhere on those pretty wide distributions.
And I observe that even from friends that are reasonably well educated but have done little to no statistics.
This lack of basic stats skills does lead to bad public policies.
Sorry for my potential lack of understanding to what you are saying. But people don't think in terms in stereotypes because they don't look into distribution. You can still look at a distribution of people and form a stereotype because for most people and most cases you cannot form good enough sample to represent any significant statistics to draw any conclusions in everyday life interactions.
And in the case of BLM. I am not deeply involved with following what the development of the movement but I think at it core (at least when it was formed) is because statistically speaking, you are more likely to be killed by police id you are black. If you look at US prison distribution per capita and race, you can draw a similar conclusion. Stereotypes are a cause not an effect. You get some of that in part because some people are racist and have racial stereotypes.
And it is great that you are saying everyone should be treated as a unit itself. But we all know what this is not the case when you have a significant portion of the people think of people from their lens of stereotypes., Black peoole, muslims, Asians, Jews..etc and this have real consequences on the life of this people.
Good luck telling an TSA officer that you should treat me as individual and don't pick me "randomly" because I have have the wrong color to not be suspicious.
For the BLM, the stats actually support that you are slightly less likely to be killed by the police if you are black. But there are vastly more interactions of black people with the police, and those are aligned with crimes committed, including crimes which stats are unlikely to be affected by policing practice (eg murders). Now we can debate why the black population commits more crime, but it’s not a racist policing discussion.
But it’s even more mundane things. People are focusing on extreme percentiles of distributions, like % of engineers at google, or board members of major companies, ie looking at 0.001%-ish percentiles, where even minor differences in distributions may have a dramatic effect. And this leads to people reacting to this with “so you are saying that women engineers at google are less capable” (read many times on HN during the google memo controversy) which is absolutely not what those differences in distribution mean. They just mean you may see more of one group and less of another, but if the recruitment process is fair, all the people who passed the threshold are equally capable. And then you have to compare that to the distribution of people who actually apply, which has its own biases.
> Now we can debate why the black population commits more crime, but it’s not a racist policing discussion.
Criminal statistics are a record of who is arrested and prosecuted, not a record of who commits the most crime. If a black criminal is more likely to be arrested than a white criminal, then the statistics will reflect that.
Yes, it's more reliable to consider number of crimes reported and number of victims. I think you'll see why there are more police in black communities if you consider those statistics.
What confuses me is that this is now painted as racist and somehow Republican. I clearly remember that back in the 90s it was the Democrats, at the urging of black leaders, who supported harsher sentences and funding for more police (most prominently the Clinton crime bill [1] and three strikes laws in Washington and California [2]).
And we can see that black people still want the police to focus on their communities, as they elect leaders like Eric Adams (over the objections of white progressives), and a Gallup poll in 2020 found that 81% of black people want police to spend same amount of or more time in their area.[3]
That naturally leads to more interactions with police and more incidents when terrible things happen. Policing is always a trade off between the harms that police cause (through mistakes, misconduct, and misunderstandings) and the harms they prevent.
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1: "the largest crime bill in the history of the United States... provided for 100,000 new police officers, $9.7 billion in funding for prisons". "Then-Senator Joe Biden of Delaware drafted the Senate version of the legislation".
2: "The first true "three-strikes" law was passed in 1993, when Washington voters approved Initiative 593. California passed its own in 1994, when their voters passed Proposition 184[16] by an overwhelming majority, with 72% in favor and 28% against."
Which is why I tend to look at homicides as it will be less biased. Every homicide will be accounted for (materially). And if anything you would expect less efforts to be made by the police to identify the author in poorer black areas, which would lead to an under representation. But from memory I think close to 50% of the homicide in the US have black authors (and predominantly black victims too which is consistent with your point). Which is broadly in line with their representation in police shootings.
> For the BLM, the stats actually support that you are slightly less likely to be killed by the police if you are black
I think you meant more likely not "less". And even in thay case, this [1] is statistics up to before covid that shows that it is actually 3 times ratio between black and white risk of being killed by a police. This does not constitute "slightly".
> But there are vastly more interactions of black people with the police, and those are aligned with crimes committed, including crimes which stats are unlikely to be affected by policing practice (eg murders).
That would be true of any race or group of people. They have much more interactions with police other than those which will result in death.
> People are focusing on extreme percentiles of distributions, like % of engineers at google, or board members of major companies, ie looking at 0.001%-ish percentiles, where even minor differences in distributions may have a dramatic effect
Yes, and that's actually how personal stereotypes works. Your experience with people is very limited to form an opinion about race, nationality...etc. It is not even extreme percentile like top or bottom but it is a small percentage that people use to build their opinions.
> but if the recruitment process is fair, all the people who passed the threshold are equally capable
If the process involve human judgment or evaluation, then biases start to be a huge factor that prevents establishing a clear threshold. Unless you have something like a standard exam that will be taken without human interactions. You will always be working under the assumption that the process is not 100% fair. That is actually hard problem to solve especially at big companies. I don't know a solution that can give the best outcome and I don't envy people who have to do it.
But this is a good example of the problem with statistics and the public. Fryer’s analysis was highly touted, but not a single person I talked to could explain it to me or say why his results differed from other analysis.
I think the link you provided argues implicitly that looking at shootings as percentage of police interactions is wrong because policing practices may be biased (ie black population may be more policed than other segments of the population).
What I am saying is that you find a similar black over representation in crimes that are less likely to be overpoliced (homicides which stats should be fairly reliable, ie every incident accounted irrespective of the race of the author, and which if anything would be less policed in poor neighbourhoods where gang violence is more common and less effort is made to identify the author).
I don't think anyone is arguing black overrepresentation in crime. Rather is arrests to shooting ratio a good measure of racial bias. As Fryer himself notes, shootings are very different than almost anything else police do -- its a life changing event. In his own research he notes that excessive force does show racial bias, but it's not viewed as a life changing event for the cop.
In fairness to Fryer (and yourself), we may not have the tools to determine this definitively either way today. And really my more important point is that as a society we should approach with caution statistical claims where we don't have a good understanding of the methods and the pros/cons of the methods. After becoming familiar with Fryer's methods in this study I'm probably leaning toward his conclusions being wrong -- but I wouldn't wager large sums of money on my leaning.
They're both. Some are a cause, others like for example "black people can't swim" are an effect of racist laws. Changing how you treat a group creates its own set of stereotypes. Then you can also get a multilayered stereotype pile from things like cake walk.
And that's an axiomatic error, not statistical. They hold it as an axiom that they should think in terms of percentage instead of individual outcomes. Statistics can't fix it, because you can't question axioms with mathematics alone.
AP Statistics has existed since 1996 and my high school (and many others) offered it as a path for students in addition to the calculus path. That is, you could head down the statistics path or the calculus path (or both) depending on your interests.
I finished the AP Calc path my junior year, but was told colleges would still expect to see a 4th year, so I signed up for AP Stat.
It was mind-numbing how easy it was by comparison. We got to the end of the semester and I realized we'd only just reached some of the principles from the first week of calculus. I don't remember why he was so proud to point out that we could now calculate derivatives (it was a long time ago), but I was thoroughly unimpressed.
By the second semester, I had my college acceptance letter, so I didn't bother with the other half.
The computation in calculus was the hard part... the intuition made complete sense almost all of the time. In stats it seemed the other way around. Interesting that you seem to feel otherwise, but different strokes.
I feel like it depends on the teacher at the end of the day.
At my relatively average Bay Area public high school, we offered AP Stats, AP Calc AB, AP Calc BC, and AP Physics E&M. This was the kind of HS where 10% of the student base finished BC by 10th grade and began taking Multi and Linear Algebra from our local community college or Cal.
Our AP Stats teacher integrated a lot of Calc and E&M concepts into the curriculum and stressed Probability theory quite heavily, the same way our Calc and Physics teachers dug deeper into Numerics/Interpolation than the AP Board demanded.
That said, the AP Stats test was an absolute joke.
I would like to see the basics linear algebra taught in high-school. Lack of knowledge of linear algebra is extremely insidious. If someone who doesn't know calculus encounters a calculus problem, they recognize it as something they can't solve. If someone unfamiliar with linear algebra comes across a linear algebra problem, they see it as an extremely difficult algebra or trig problem. I've personally seen programmers apply trigonometry to problems that only require addition and subtraction to solve.
I've never understood the "calculus vs statistics" argument. Calculus is an unquestionable prerequisite for any meaningful statistics work.
If you don't understand the concept of "integrating a function" they how can you possibly make sense of virtually any part of statistics? 90% of practical statistics can be boiled down to understanding the basic algebra of normally distributed random variables and then doing basic calculus on the result.
Statistics without calculus is the worse kind of statistics where students are taught to blindly throw tests at a problem without having a clue as to why they are doing this. Statistics without understanding is worse that no statistics at all.
I don't think it's necessarily 1 class of statistics vs 1 class of calculus, but the fact that your typical calculus ladder is 4 semesters worth, most of which being about symbolic solutions. Handling integration and derivation of simple polynomials is fine, but the majority of the computational part of calc 2 is just as detached from modern calculus as asking people to multiply five digit numbers in your head.
The useful parts of calculus are typically the easiest for students, and could be taught in a semester. Instead, your typical calculus track is 2 whole years.
#1. split into three. Next two are other classes. Some universities combine diff eq and linear algebra in one semester. Some universities probably have classes that cover the three semesters in two. I think most prepared students arrive with AP calc BC and just start on the last semester (multivariable..)
>> Martin said it’s also important to remember that vocational training is not the only purpose of math education.
This is true, and not just for high school but for college. Neither are out to teach specific vocational skills - rather they provide the underlying tools that students need to learn their vocation later on.
Sure we could stream kids from grade 1 for their "allocated profession", teaching them only with regard to their eventual job, but there's a reason we don't go that pretty dystopian route.
With that in mind we can argue that one curriculum is better than another for a specific path or other. But really that's a fruitless, and pointless argument. Future STEM students need Calculus, future programmers need Logic, future artists need Color Theory.
If I had to argue for a curriculum change I'd lobby for things like budgeting and basic accounting - but that's just me and everyone will have a different opinion.
I agree - as long as it's not an excuse to dumb things down statistics should be taught before Calculus. It's a quirk of history, I think, that we have it the way it is. Statistics is still going to be important to understand for 99% of STEM majors, so I don't think there's much downside to flipping the order.
The probability theory, statistics and data analysis you need for typical everyday life is already taught in middle school. A statistics course is overkill, the problem here isn't that people weren't taught those things it is that they don't remember it afterwards.
You can read the standards here, everyone learns this before high school:
No, I would argue that what is taught in middle school is the "basic of the basic", e.g. stuff like average vs. median, etc.
I'm more talking about the high school version of statistics, and the link you gave gives good info: https://www.thecorestandards.org/Math/Content/HSS/introducti.... That curriculum is more what I'm referring to, but as far as I know is not often taught to HS students in the US.
It isn't just median vs mean basics, see for example:
"For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be."
This is proper statistical understanding, since it includes how sampling is made and confidence intervals. They just don't use the names of theorems, they just teach the understanding for estimating values etc. I don't see the value in drilling math equations for calculating those values and forcing the kids to remember lots of names.
Making (pre)calc elective would have a lot of good outcomes, but it would also drag the decision point for whether someone wants to pursue a STEM career down to earlier in their life - you'd have to commit to going on the "STEM track" or the "non-STEM track" before starting High School.
For people who know they don't want to go on the STEM track by then already, the change is a net positive. However, it might also mean STEM loses a few more people from underrepresented groups as a result, who would have done well in STEM but are now locked out of that track earlier.
Pre-calc is a weird course. It includes functions, exponentials and logs which are fairly key concepts in STEM, along with a bunch of fiddly trigonometry that just becomes superfluous once you introduce complex numbers. You could teach the first part of precalc and then do whatever part of trig+complex numbers alongside calc.
this is no longer the debate. the debate is whether kids should have to take math classes at all in order to graduate. And based on what i'm seeing, the answer the schools came up with is - No.
I don't know if I'd want Linear Algebra proofs - they tend to veer into results over vector fields rather than the more useful "solve Ax=b" and "here is how AI works" parts (i.e we should teach real-world computational linear algebra rather than the pure parts of field).
That said, I think proofs are an important thing we should teach in high school - its more a flavor of "real math", and logical thinking involved is a better transferrable skill than the symbol manipulation and bag-of-tricks-you-never-really-use-in-the-real-world you get from high school calculus courses. I'd just focus it on something easier to grasp, like basic number system construction.
The foundations of statistics are very much based on calculus, though. You can't really have one without the other. Even study/experiment design is at its root an optimization problem.
The foundations of statistics are very much based upon Galton observing:
"the middlemost estimate expresses the vox populi, every other estimate being condemned as too low or too high by a majority of the voters"
at a livestock weight guessing contest and conceiving of a measure to quantify normal variation: the standard deviation without the use of calculus.
To be sure this evolved to expressions that included an integral sign or two .. but the foundations were founded with no more than a Sigma and some division.
There's a stronger case for those than care to make it that statistics is more dependant on linear algebra, if one takes the view that statistics is about finding fewer lower dimensional fair representations of many higher dimensional values.
Calculus really should be thought of as dependent on linear algebra (the derivative of a function is the best linear function that approximates it; it just happens that in 1D the only linear functions are multiplication by a single number). Linear algebra seems like it would be far more useful than algebra 2 (things like conic sections?), trigonometry, and "pre-calc".
It's possible that you could do something like algebra (middle school)->geometry (maybe introduce the notion of a group here and focus more on symmetry and not so much on figuring out the missing angle/length in a complicated diagram)->linear algebra->probability/statistics. Concurrent with linear algebra, have kids learn calculus in physics class. After physics 1, they can do chemistry and/or e&m. Do vector calculus in e&m. Basically trim all the useless stuff out of high school and add the first couple semesters of college instead. Offer analysis as an elective after linear algebra/physics 1, and put a proper account of the n-D derivative and things like the Newton–Raphson method there.
Obviously that's a STEM bound curriculum, but at least in my school growing up, you only needed algebra 1 and geometry to graduate, which the honors kids did in middle school. So I assume any curriculum more advanced than that is for STEM bound kids.
Reforming the overall approach would probably do more for numeracy than tweaking the end stage curriculum.
As it is now, you have ~3 groups of students in most K-12 math classes. A group that is bored because they have mastered the concepts being presented, a group that is benefitting from the concepts being presented and a group that doesn't have the framework to benefit from the concepts being presented.
There are of course lots of teachers that will be doing what they can to address the gaps, but it needs to be systematic.
I would much rather see a one semester course on formal logic and then a basic probability course than throwing students into the misery of spreadsheets and proprietary stats software.
Eh. I think calculus should be taught theoretically. It’s honestly a very simple concept and helpful to have intuition for. Just don’t waste time on the implementation. Folks aren’t going to remember the chain rule years later.
Chain rule is a bad example. If you cannot remember the chain rule, you do not understand the theory at all. And if you do understand what a derivative is, the chain rule is trivial.
Right, the chain rule is foundational. If you don't understand the chain rule, then related rates and lots of your real-world engineering applications of calculus would seem to require memorizing a crazy number of specific solution templates.
Though, if typical calculus courses don't spend the time to ensure a good understanding of the chain rule, that would explain why so many students seem to think calculus is a gigantic sea of random rules.
I’ve read a couple of texts by Paul Lockhart. His opinion (which I support) is that what is taught in high school as math isn’t math, as in what a mathematician does. As an example: Pythagoras proved the relation between the lengths of the sides of a right triangle. The journey towards finding the answer to this is math. Or even exploring why this relation even exists based on his findings is math. The answer to the question is expressed as a2+b2=c2. What we’re asking students to do is just to take that formula and apply it hundreds of times without going into the actual math behind it. This, he says, isn’t math at all. The whole “math” step is skipped.
He compares it to if for example we were teaching music in school without ever touching a musical instrument, and just learning how to read and write notes. Or teaching art without ever touching a pencil or brush.
The risk is that students think they’re bad at math, while in fact they’re bad at mindlessly memorizing and applying formulas. The same way we could totally miss out having students never discovering their artistic talents if art class was taught without ever actually doing any art.
His text that explains this most clearly is “A Mathematician’s Lament”. It’s a great read and not very long.
This is very similar to what I was coming to post. There is very little mathematics taught in k-12 in the US. There are almost no proofs taught, there is a lack of mathematical thinking taught. My classes personally were done as a series of rote memorization steps to perform before putting it into the calculator. The mathematical logic, proof, rigor, and how to create a model is what is needed and that is missing. The specific courses taught don't matter as much if they aren't taught correctly.
> The mathematical logic, proof, rigor, and how to create a model is what is needed and that is missing.
This is interesting to me. My experience was the opposite. So much time in my honors math in high school was focused on proofs and deriving them, that developing and intuitive understanding of the material and how to apply the math to solve problems was secondary.
This may or may not be comparable to what is referred to as high school math in the general sense. I don't know how your curriculum worked, but presumably the students who are taking honors math had already demonstrated a capable, if not above average faculty for the minimum level of rote memorization necessary to simply get the basic arithmetic and algebraic symbol manipulation right most of the time.
This would leave class time and homework time available to explore the matrix in which these manipulations are grounded, since "how" they work is taken to be table stakes.
But the vast majority of students don't have the extra bandwidth available, due to competing pressures in other areas of life, or simply insufficient short-term memory capacity, to get past "how to" with enough time left over for "but why" even if they have a natural curiosity and interest in the question.
Beside irony told this is that the more one understands why, the easier how it becomes to memorize. But finding this balance seems to be a very unique mixture for students who are not extraordinarily capable in the ladder, there is a huge advantage to being able to build a working toolchain of mechanical spellcasting to reliably obtain repeatable, testable results so that you can be productive when postulating why the spells work at all.
Essentially it helps to be capable.of mimicking an ersatz computer when you need to develop human intuition fpr computation. but computers are getting so good at thinking like a human that humans who best think like computers are going to be less prized than they have been in, say, the 30s, 40s, 50s and so on, when the premises that are still baked into our current pedagogical sensibilities were formed.
I suppose the commonality is just that primary/secondary math education is often bad, but doing proofs without intuition seems to be completely missing the point. A decent proof should mostly be the intuition. Once you've learned the basic techniques, you just say "by induction X", "evidently X", etc. to wave away routine calculations, and mostly stick to the "meat" of the argument, with explicit calculations for technical steps only if something is tricky.
Rather than just being able to solve a problem, people need to be able to create the questions that beg the answers.
Physics class helped me understand differential calculus because I was able to create problems that I could then solve with tools learned in math.
There are also some mathematical tools that or often quite simple but tremendously help with solving equations. Extending a formula with neutral or inverse elements for example. This is the same in almost all fields in math, but it is rarely though as a set of tools.
That said, if studies like PISA are to be believed, the problems are much more fundamental.
"The risk is that students think they’re bad at math, while in fact they’re bad at mindlessly memorizing and applying formulas. The same way we could totally miss out having students never discovering their artistic talents if art class was taught without ever actually doing any art."
This is more or less what I've found as an adult who is trying to improve their math skills. Like 90% of what I learned in school was never really applied, I couldn't tell you why I learned them, just that I was told they would be essential for my later life if I chose certain paths.
I'm not an educator or a pundit and won't express any personal opinion on what the purpose of primary math education should be, but as far as I can tell, the purpose as it stands in both K-12 and college undergraduate education if you're not a math major is not to teach math. It's to teach analytic and computational techniques necessary for research and practice in business, finance, economics, sciences, and engineering. They're not trying to get you to understand what a continuous function is or why it matters. They're trying to teach you just enough about how to compute compound interest that getting an amortization schedule out of a spreadsheet won't be incomprehensible magic.
That said, as Temporal said, even this isn't necessarily done well. It's entirely possible that teaching students to derive the techniques from first principles based on actually understanding math might lead to better outcomes. But it would also be quite difficult to do and I'm not sure existing K-12 teachers are even qualified to do it. To Lockhart's credit, he walked the walk and actually taught children at a primary school. How many similarly qualified people are willing to do that?
Society thinks of math as technical and functional, rather than beautiful.
The math used in jobs are also to “get things done” rather than for the sake of their beauty.
A very tiny fraction of population will ever be pure mathematicians. Most just need to know enough to “get things done”.
Considering how much society throws money at STEM, rather than art and music, it makes sense society wants a return in their investment in terms of training practical people.
> Society thinks of math as technical and functional, rather than beautiful.
The math people suck at is the technical and functional part, just as much as the beautiful parts. I'd frame the issue like this: there's the journey of arriving at the result, there's wielding the result, and then there's applying it to a very particular problem.
Schools are teaching mostly the last part - applying a^2+b^2=c^2 to a set of contrived problems that are meant to exercise your ability in arithmetic, variable substitution, and simple symbolic transformation. They do not teach people how to wield the formula - as in, how can you realize on your own that any particular real-life challenge calls for application of that tool (or of mathematics in general). That's the middle part that's directly valuable, in the immediate term. That's the difference between a carpenter, and a person who figured out how to use a hammer to drive a nail into a wall.
The beauty and journey part? That's study of history and art of inventing new types of hammers and other impactors, and perhaps inventing your own. Even further upstream of the "knowing how to use it to build something" part, but neither is taught in schools.
I tend to not like A Mathematician’s Lament for being so lofty, but there's a more straightforward reason to care about "pure" math: students complain that math "doesn't make sense" or is a bunch of random difficult to remember rules, which is absurd. More than any other field, math has logical reasons for everything.
You can figure out pretty much anything you'll encounter in school by just thinking about it; you don't need to have any external knowledge from having done some experiment or being told some fact. On the flip side, if you do use physical intuition, you'll usually go down the right path (for anything pre-university, or even most undergraduate level material).
Regardless of how beautiful music is, teaching how to read sheet music without actually listening to music is stupid because you're not teaching what the symbols even mean. Of course no one will understand what you're talking about.
> A very tiny fraction of population will ever be pure mathematicians. Most just need to know enough to “get things done”.
I agree with this, but I want to point out that the few "pure mathematicians" are not the only ones who can enjoy the beauty of math, similar to how the few classical musicians (or classical music experts) aren't the only ones who can enjoy classical music.
In fact, looking at the success of various YouTube math channels, or Science channels, etc, there are clearly orders of magnitude more people interested in material on Science and Math than there are people who actively engage in these fields professionally. (This is true of probably every field, btw.)
> The risk is that students think they’re bad at math, while in fact they’re bad at mindlessly memorizing and applying formulas.
I'm skeptical that the kids who struggle with memorization will suddenly be able to write proofs with ease. I'd argue proof-writing is much harder to wrap one's head around than being forced to remember when to apply a2+b2=c2.
I think this is a false dichotomy of what it means to be "smart" or "good at math/studies".
Someone can jbe bad at arithmetic, but good at writing proofs. Hell, plenty of mathematicians say they are bad at arithmetic!
More importantly in the context of education, someone can be completely unmotivated by arithmetic, but be more motivated by something else, leading to different amounts that they apply themselves, leading to different outcomes.
proofs are actually what made me get interested in math, proofs allowed for such a range of self expression and creativity that i was hooked even though they usually required more thought. Compare this to the studying the myriad of integration techniques which i found very boring. My wife on the other hand seemed to enjoy rote mechanical memorization. I guess the fundamental question is, long after most students graduate high school or college and forget most of the math they learned, what are they supposed to keep from the many years of mathematical education? If you ask me, it is the ability to reason that they should keep and proof based math teaches you exactly that.
I agree it should be an option for the children who are capable, but as I mentioned in another thread, I don't think the median child is capable... or they may be capable after far more effort from the institution than is acceptable.
I'd imagine you'd have to start kids writing extremely basic baby proofs in elementary school, and then continue to hammer it in every single year until they graduate high school to get basic competency in proofwriting. That's a long time commitment for little visible gain.
A lot of people talk about adult mathematical/statistical illiteracy and how it leads to major problems with cognitive biases, poor decision making, etc. However I haven't seen anything that explicitly ties having had a class in X leading to that person applying X more frequently in their life.
The breakdown as I imagine it is:
1. The top 10-20% of students implement X vaguely into their life
2. The bottom 80-90% merely take the class to pass and forget about X and fail to apply it to their life. They are still prone to cognitive biases because they haven't extrapolated the effectiveness of what they've learned to the real world.
I think generally, at least in America, a general culture of anti-intellectualism prevents people from actively caring about what they learn in school and applying it to the real world. It's not a mere matter of education, it's about making education meaningful to people over the alternative race to the bottom.
Related, there's no evidence that people behave more ethically after taking a course in ethics. But the education system has one hammer, teaching courses about subjects, so every problem looks like a nail. They want their students to behave ethically so for school systems there can be only one solution; they teach ethics courses.
It's likely that the true ethical education children receive is observing the behavior of people in their family and community from a very young age to learn what is or isn't acceptable. Schools can't do much about that, and if anything most modern schools are harmful in this regard because they generally award petty cheating and excuse spinning. And so too do schools teach the wrong approach to critical thinking and cognitive biases. Schools reward students for conforming to what their teachers say and give students trouble when they think on their own. This is particularly true during the youngest grades. Some high-concept courses about critical thinking in highschool won't undo the damage; by that time students already learned to either conform with authority or become equally unthinking reflexively contrarian rebels.
There's confusion over the correlation of education vs ability to make good decisions -- I don't think the causal relationship between learning maths/logic and better decision making is established at all. It could be just that better "general intelligence" enables one to get good grades and also make better decisions, instead of the education having a meaningful impact.
And then there's an oversell of how useful some concepts are. I think it's fair to say that for me personally, I've had more success integrating into my life what I learned about constitutional law than linear algebra. Before this generative AI thing I don't think I've ever used applied any knowledge of linear algebra.
There's a trend of people over-estimating the usefulness of their subject. Mathematicians tend to think everyone else needs to know advanced math. Historians think everyone should learn history. Tech people here think everyone should be more technologically literate. Judges think everyone is supposed to know the law ("ignorantia juris non excusat").
In the end the body of knowledge out there is just too vast, life is too short, and it's actually a good thing that people learn different things, even at expense of being "illiterate" at some subjects. I think the "80-90%" who never integrated the stuff they were taught at school is evidence that they should have the option to learn something else instead of being forced to sit in classes that they don't feel like taking.
While in general I don't condone anti-intellectualism in the sense of being proud of being ignorant, I think to some extent it is a reaction against
forms of out-of-date intellectual-elitism, especially the kind that considers people who took a classical education as superior than those who have not. For example, knowing how to build a house is definitely more useful than knowing the cause of the fall of the Roman Empire, but the house builder is presumably not looking down upon the history major for lack of house-building knowledge. But somehow there is (or at least was) a snobbishness among the educated class that did view the house builder as less "sophisticated". (Of course we all know history majors can end up worse off financially than blue collar workers now, but these days it's the STEM people who still somewhat maintain this elitist attitude and clinging onto century-old math curricula.)
And some people find it hard to accept that the classical education isn't as useful as they claim to be.
> A lot of people talk about adult mathematical/statistical illiteracy and how it leads to major problems with cognitive biases, poor decision making, etc.
Yeah, I find that difficult to imagine too. The assumption behind that seems to be that with the proper statistical knowledge, people are able to understand ... scientific articles? Because I don't really know where else you could find a relation between cognitive bias and knowledge of statistics.
But first, articles have the statistics done. Second, knowing statistics isn't going to make you understand the article, nor spot the errors. Third, most articles still rely on poor statistics, because many of the authors and reviewers still think that e.g. null-hypothesis testing is just fine. As a corollary, most articles are wrong, and should not be used for decision making.
Man oh man does this bring back past trauma. I graduated high school in 2006 and despite me being in one of the top half rated high schools in my state and having a consistent B average, I was not prepared at all for when I entered my engineering school (NJIT). A lot of my math was lacking fundamentals and instead was just memorization and 'plug and chug' style learning. That does not fly at university especially an engineering school. Schools (at least my school) really expected you to learn fundamentals as their tests would not just be a variation of the problems taught in the lectures but a unique twist on the concept itself. The school tested me on my skills before entering and put me in a 'remedial' course called Pre-Calculus with a Calculus Perspective. They even wrote a custom textbook to really fine tune getting prepped for what was to come. (Calc 1, 2, 3, Diff-Eq). I wasn't enough because the new thought process requires a great deal of effort to internalize in the timeframe given.
I was barely treading water my first semester and ended up failing math and physics and got put on probation. I eventually graduated and am doing quote ok in life career wise but it was an extremely painful ordeal. Not being prepared and really ready to go remains to this day one of the biggest regrets of my life. I really feel like I let down the people who made my education possible (parents, teachers, schools).
Sometimes I wonder why is there such a dumbing down of mathematics in the US? The US is powerful and at the top of their game because of science and technology. Why try your best to kill your golden goose?
Similar story here. What I learned in math in high school was an absolute joke. It in no way prepared me for physics and math in college. In addition, most college majors do not need it, they need an understanding of statistics, and a much better understanding of 'business math' and algebra. I think dividing kids into two or more tracks does not hurt them, because they can always take extra course work in college if they ultimately decide they want to pursue STEM. Where it hurts them are the standardized tests, this seems to be the issue.
> instead was just memorization and 'plug and chug' style learning.
Kind of surprised to see that, my experience with college calc was it was so heavy on memorization, that was my struggle.
i ended up failing out of school too, lol. i was clicking around khan academy recently through their calc stuff and was surprised i struggled so much with it, 15-ish years ago.
Same experience 17 yrs earlier and not in the US. Went from breezing through high school calculus and physics, to completely floundering and flunking out the first year of university. The gap was enormous - probably another 3 years of learning at the high school pace.
I noticed this when I went to RPI (I graduated HS in 2007). My high school wasn't ranked at all (or at least no one cared where it was ranked) and AP courses weren't offered until after I graduated. Based on reports from my younger sister, the AP curriculum they did add later seems to have restricted what was taught due to the scheduling rigidity rather than improving it. (Students could take the AP tests on their own though.)
I found that I was far more ready than many of my fellow students for math in college (though this could also be due to aptitude given my handle, even the physics and chemistry in my high school went further than AP curriculum). I attribute this to MathCounts (sadly just a 7th/8th grade thing) and the accelerated math track a friend and I were allowed to take. This track ended up getting us calculus concurrently with physics which made a lot of things there make way more sense and gave examples of real use of calculus outside of the math classroom. I also ended up taking statistics which I found more interesting/useful than calculus at the time (possibly because my intuitions weren't as applicable).
This always degenerates into a calc vs stats debate when really the problem is the vast majority of the calc curriculum is a waste of time. Even if you become a professional engineer, you're not going to be solving lots of complicated integrals by hand.
If the curriculum started with 'imagine the student has access to a computer' you could easily cover not just stats and calc but a whole lot of discrete math etc. as well, just from the time saved not teaching everyone the huge 'bag of tricks' necessary to solve a few equations by hand (in the real-world most things don't have closed-form solutions so the tricks are utterly pointless).
Note the problem goes far beyond high-school curricula. I studied a subject on queuing theory at one point and realised that everyone in the class was less equipped to deal with any real-world queuing problem than any of my CS friends who could code. Simulating the queue allows you to solve most things, whereas only an extremely narrow class of problems could be solved by hand. The only reason the maths curriculum is the way it is is because it's barely changed since computers were invented.
The teachers who grew up without computers wouldn't change the curriculum because they wouldn't see the point.
Even those who grew up with computers might not have realized you could solve problems easily with computers instead of the older methods. (I believe most people don't know they could plug a formula into Wolfram Alpha and get its integral for free.) Even if they knew, some of them might take the dogmatic view that the extra effort in learning so called "fundamentals" would build character of sorts.
So the few people who actually believe that the curriculum should be changed have to wait until they get older, rise to positions of authority, and then they maybe they could propose changing some things around. So you get at least a generation or two of students learning things the old way, because society moves slower than technology does.
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While we're on the topic -- I've always thought that the philosophy department took a long break during the mid-20th century. The logic department teaches Gödel, maybe passingly mentions Turing's work, but while everyone is busy trying to correct other people's interpretation of Gödel, they seem to be oblivious to almost 100 years of advances in our understanding of computation and programming language theory (the language of logic is analogous to programming languages). It seems somewhat similar for physics too -- interpretation of quantum physics seems to be a niche thing.
I'm not saying there's nobody who knows these stuff, but it just seems like the teaching of philosophy has stagnated for almost a century, given the weight put on learning the classics vs the newer discoveries in the past 100 years.
The nice thing is that we have the Internet these days, so at least people who are interested can find what the material they want to learn.
Even in the STEM world I don't think statistics is taught all that well.
I studied Engineering a STEM subject, this was sometime ago (2000's) and at that time we were required to take a "statistics for engineers" course in the second year of the degree. The subject was mostly useless (it pretty much boiled down to "plug this data into Excel and this is how you interpret the ANOVA output").
What I really wish they focused on was regression analysis and especially non-linear regression, logistic regression etc. I had to learn all about it in the workplace after graduation.
Even the statistics knowledge we were taught was incomplete - for example I did not learn about Cross-validation at university.
talking to more recent graduates I don't think the situation has improved all that much in the intervening years.
The "STEM World" is a large one and "XXX for Engineers" courses in undergraduate university are long on material to rote memorise and short on depth required to understand and expand.
I took engineering myself, the Physics, Maths, Chemistry 110 "for Engineers" variations were shallow compared to the Math 100 etc core courses for those who wanted to study Physics, Mathematics, specialise in Chemistry, etc.
Mathematics, medical, and biology students who take statistics for epidemiology and other sensitive applications get a much better grounding in the pitfalls and meaningful operations of low dimensional summaries of high dimensional data.
I teach in a high school that is very humanities-focused (and the students explicitly want that, they chose it). I realized statistics was not taught, and I introduced a module on it for my classes. I thought it was insane to do three chapters on trigonometry, including solving useless equations with goniometric functions, and doing zero stats and probability. These students will likely (and statistically) choose uni careers such as medicine, psychology, law, politics -- the ones who choose engineering are very rare. Statistics is THE ONE field of math that they need.
I want to know what world this author is from. High School math in public schools has been dumbed down repeatedly over the years.
Calculus in High School? Increasingly rare to non-existent.
What we need is the opposite - a return to traditional curriculums, and stop non-sense like a 5 credit course in graphic novels (a thing in our district school).
It would be perhaps more enlightening to see the percentage of students that are in a school that offers it rather than the percentage of schools. I don't see that figure in this table, but it does show 4.7% of high school students enrolled in calculus.
I think it would actually make sense as, say, an art elective or something.
My school required six credits of some kind of art (music, visual arts, performing arts), and I can't think of any great reason this would be less worthy than any other kind of niche art class.
The curriculum consists of reviewing the history of comic books, reading a lot of comic strips, and watching several graphic novel driven movies such as Superman and Watchmen.
It is a consumption course. Kids don’t actually learn how to write graphic novels. That might actually might be worthwhile.
The curriculum consists of reviewing the history of literature, reading a lot of literature, and reading several novel literature such as Great Gastby and Catcher in the Rye.
It is a consumption course. Kids don’t actually learn how to write literature. That might actually might be worthwhile.
That doesn't sound normal. Ever highschool English class I ever heard of was an equal split writing and literature class. You read things and were expected to write as well. Not until university did I ever have a literature class where I wasn't expected to produce creative output of my own.
And no one said they don't write in the class. THe original post was complaining that they do not learn to write graphic novels
Which is normal. We don't tend to teach high school students how to write entire novels. We aren't really doing creative work at the high school level either: Most papers are research papers or talking about themes of novels the entire class read. Hardly creative stuff.
There is no reason a highschool English class shouldn't have students writing short stories or short 'graphic novels'. If they're just reading and not exercising their own creative processes then the class is a farce.
English classes aren't there for creativity - that's a secondary option.
English classes exist to help you process what others are communicating and to facilitate your communication with others. Most of this won't be creative endeavors, and the focus usually shys away from this stuff as kids get older. This is especially true with "advanced" and "AP" classes, which tend to model themselves after a more scholarly look at literature. There isn't much room for creativity there.
This doesn't make things a farce, but merely a different take than what you think you'd want for a fun class. IIRC, they offered a creative writing elective when I was in school. Anyone actually interested would take it, but in modern times, I'd guess students are more likely to get tips from places like NaNoWriMo - and all it takes if for a teacher to mention that once or twice for interested students to do it. For fun.
Kids don't learn to write novels in literature class either - you've missed the point of such classes.
The point is to read and be exposed to different sorts of literature. This is much easier to do this with material that students are actually interested in instead of 150-year-old romance novels. And as a bonus, the students are a bit more likely to understand the symbolism and references than they would in an old book - the older literature is honestly more suited for a class that hooks both history and literature in the same course.
Literature classes are consumption courses, too, and lead a few to develop a lifelong reading (and consumption) habit, btw. We definitely watched movies based on books: Part of my high school Lit class was doing a comparison between A Brave New World and Demolition Man. (We also had a 12-page senior paper, mine was on Kurt Vonnegut. But that's beside the point.)
And perhaps part of the issue is that you haven't really read a lot of graphic novels so you misunderstand the depth that they can have, especially if you get into more obscure titles.
What highschool "literature" class isn't heavy with writing exercises? I've never heard of an English class where you just read (except in college, where you're expected to also be taking dedicated writing classes.)
Why do you assume that the class mentioned doesn't include writing?
I'd assume they were writing about the graphic novels. But I also assume they were discussing the themes in class and we took multiple choice tests about things we read. It wasn't always writing. In current times, this would especially be the case if the school system had an initiative to cut down on the sheer workload of homework. (Which I approve of, most teens don't need hours of homework on top of school)
What you don't learn to write are actual freaking novels, just like they aren't going to teach you how to illustrate and write a graphic novel in most high school classes. You write essays, research papers, and things like that. Not entire novels.
I responded to a (your) comment claiming that highschool English classes are consumption-mode literature classes.
> Kids don't learn to write novels in literature class either - you've missed the point of such classes.
> Literature classes are consumption courses, too, and lead a few to develop a lifelong reading (and consumption) habit, btw.
Kids don't write full novels in those classes because the timeframes involved couldn't possibly permit it, but nevertheless half the time in those courses is dedicated to getting students to write short stories, poetry, (not merely essays!) At the highschool level the courses are about writing as much if not more than about consuming literature.
You don't get to pick and choose which taxes you agree with. Part of living in society is paying taxes, part of which won't directly benefit you at all.
That doesn't mean that you personally get to decide. Living in a democracy doesn't mean you'll have any fair say in things - or a say at all (see: felons and immigrants) and it does mean that things are going to happen that you think is ridiculous.
If we let things pass because (sometimes misinformed) person thought it was ridiculous, there wouldn't be a government at home and I, being female, might not have any rights whatsoever.
Everybody gets their say in the decision. It's called democracy. Somebody spouting off their opinion saying the process should be changed is part of the process; they not only get to do that, they're encouraged to.
They definitely do not get their say. If groups of folks cannot vote in any election (in the US, this is the case: Immigrants here get to vote in local elections but no such rights generally exist in the US. Felons cannot vote. Children and teens cannot vote).
Representatives in the US tend not to listen to those that cannot vote and generally dismiss a lot of folks that Do.
Democracy is a name, sure, but flawed democracies are more common than unflawed ones. Those flaws generally mean that some folks' voice is supressed. The US was a "democracy" even while it enslaved millions of folks that it did not allow to vote. Once these folks became "free", many still could not vote. IT was a democracy then, too.
The child's parents also pay taxes so surely they should have a voice.
And it seems like if the child is creative and has zero plans to go into STEM then he shouldn't have his time wasted. Which is a win-win for everyone since disengaged students often disrupt the teacher and other students.
I would only teach applied math in high school, not the Applied Math curricula from most universities, but a vertical that covers things that can be made sense without depth, just breadth. I'd leave most abstractions and numeric solving for later in life or STEM students. I just feel I did so much equation solving for nothing in my student life! I remember too many times the teacher would assign the equation solving exercises 1 - 10 as homework, and only problems 11 and 12, leaving problems 13 to 20 out. Why? Well, problems are hard to teach, to grasp and to solve!
Most math is actually very easy if kept easy, even integrals only have a few rules. Don't make a student do 700 variations of integral solving. Find a practical use for it (ie find the satellite velocity and position), introduce the subject with the problem "Today we're going to find a velocity of a satellite, there's a thing called integral that helps us do that, this is why and how..." and do that problem 20 times over and over. A variety of real-world problems have very simple math to solve, like stress and strain in materials engineering, center of gravity in physics, etc. Problem-solving is the worst of all math for most kids - so do repetitive problems and teach kids to extract information from problems into formulas and equations. And, yes, do plenty of statistics which is rich in real-world use too. Having a problem-solving, math applying mind and probabilistic view of the world are the skills to have at any field!
I teach Communication at college but I was a math minor in undergrad. I hear my students complain about math class all the time, how they hate math, and it isn't rare to hear students say they have taken the same math class multiple times. I've glanced at the math book and it is basic stuff. I don't know how but we need to get kids to not hate math before we can do anything else.
I think its because of how its taught. one thing is its often for a lot of people just learning for the sake of learning once your doing anything like algebra or harder.
The worst part about these discussions is that no one actually remembers what they learn in high school unless they build upon it further in college (or it’s something they’re more likely to encounter it in day to day life, such as history, geography, literature, etc).
The real purpose of education, in high school but in general until your junior year of college in the American system at least, is to learn how to learn.
And advanced math involves the most unique and mind challenging ways of learning in a high school education. It involves memory, application, pattern recognition, etc but it’s about the only part of the high school curriculum that encourages abstract logical thinking. Literature does this too, but it doesn’t require it, and it doesn’t require the rigor in abstract thinking that’s needed in advanced math. In an argumentative essay you could hide flawed thinking with wordplay and any ways the logical accuracy of the argument only forms a small part of the grade but with advanced math there is no place to hide flawed logical thinking…there being a single correct answer, and the fact that math is a language that is designed first and foremost to eliminate ambiguity, means it’s hard to get away with sloppy thinking.
Advanced math that isn’t necessarily useful in your college career is absolutely essential to teach high quality thinking.
A statistics course, on the other hand, will almost certainly devolve into the practical and the useful, so much like high school physics will not teach thinking but simply knowing how to apply the right formula at the right time. That’s a useful skill but one that’s taught all over the current curriculum.
To the extent algebra and calculus are not useful outside of STEM degrees or in “real life” is indeed what makes them useful since it allows them to remain essentially the only abstract thinking course high school students will be exposed to.
> Advanced math that isn’t necessarily useful in your college career is absolutely essential to teach high quality thinking.
You've put into words something I wish I could tell everybody. The impact of achieving a level of rigorous logical thinking through math classes goes far beyond "Will I use it in my job," it makes a fundamental difference to the way you see the world and think through every decision in life.
What's so unfair is that, in my experience, whether a given student in average circumstances successfully attains this in either high school or college is, essentially, a crap shoot depending on the combination of teacher, student motivation and support, and any number of other environmental factors.
> calculus are not useful outside of STEM degrees or in “real life” is indeed what makes them useful since it allows them to remain essentially the only abstract thinking course high school students will be exposed to.
Nah, I would rather argue that discrete math is more valuable to students in both abstract thinking and in real life. In basic calculus, mostly you are just remembering formulas and equations like d/dx(e^x)=e^x. Not even touching the edge of abstract thinking.
That equation is extremely important though, it tells you that the rate of change for exponential functions is proportional to the current value. So in reverse, anything where the rate of change is proportional to the value is an exponential function and now you instantly know how fast those will grow.
So by knowing that we do get a lot of intuition for so many systems and processes. That is the power of math, understanding one thing improves your understanding for so many different otherwise unrelated systems.
So, play this for me.. Which specific profession I should be in for this to matter to me? Not that I should be curious about fundamentals, but something I can apply in a meaningful way?
Finance, economics, ecology, virology, electrical engineering, mechanical engineering, chemical engineering, software engineering, pretty much any field with a technical component (i.e one that uses numbers).
You could reasonably just look at 2^n for software, though in general usually continuous math is simpler then discrete math IMO. Doubling vs. e^x is kind of an exception to the rule, and if you do any software involving signal processing or simulations, you'll want to understand the continuous version.
Everyone has use for understanding and detecting exponential processes. And the most important and easiest to identify part of exponential processes is the derivative, their rate of change is proportional to their value.
It is like understanding basic dice outcomes, everyone should know that since it is so basic to understanding events that happens in the world, political discussions and advertisements and products.
> advanced math there is no place to hide flawed logical thinking
If that's what you think, your math isn't advanced enough :P (disclosure: neither is mine)
These days mathematical proofs tend to be dozens of pages long and require multiple experts days to check and validate.
I'm pretty sure programming fits your descriptions of nowhere to hide flawed thinking though.
> To the extent algebra and calculus are not useful outside of STEM degrees or in “real life” is indeed what makes them useful since it allows them to remain essentially the only abstract thinking course high school students will be exposed to.
Sure, but if "useless" things are desired, why don't we teach them comparative history of Hobbit society instead?
> so much like high school physics will not teach thinking but simply knowing how to apply the right formula at the right time...
Unfortunately, that seems to apply to a lot of high school math, including a lot of calculus classes. And, to shoot into my own ranks, college math and especially statistics classes are far from immune to this as well.
I took both ap statistics and ap calculus, got a 4 on statistics and a 5 in calculus, and in my opinion the calculus was way more helpful in college. But at the same time, I didn't feel like I was unprepared for college math.
And I got a degree in an engineering field at Georgia Tech, so I don't think the classes were just easy or anything.
So in my opinion people have to take a look at the standardized test scores, and not just 'this teacher chose to give them an A'.
Yes. MIT came out and pushed back against all of the anti-standardized testing narrative to show the data. At least for math, those standard scores are incredibly reliable indicators of success.
Interestingly, Caltech extended their standardized testing moratorium for another 3 years[1] out to 2025. Wishful thinking on my part, but I wonder if the two admissions departments would ever get together and compare notes, maybe even in a public forum.
Caltech at least will continue studying the performance of their cohorts through the moratorium extension (including a graduating class). I find it fascinating that two of the supposedly most rigorous academic American STEM undergraduate programs arrived at completely different conclusions from their internal studies (thus far).
One line of the article stood out for me: "College admissions officers value calculus, almost as a proxy for intelligence." I think what lack of Calculus signals is more lack of facility with Mathematics generally.
When I speak with students who have weak (or missing) Calculus backgrounds, I usually find that they are uncomfortable with something much more basic: they simply do not understand what it means to use a symbol for a quantity. Not understanding that, they don't see how manipulating symbols (e.g. subtracting something from both sides of an equation to move a term to the other side) makes any sense. It's as though they had a mental block on the day when a teacher said "let x be the unknown". They usually bluffed their way through that class, and the next and the next, as they proceeded through middle school. But they never really got their heads around the ideas. And this, not intelligence, is the problem with their later success in STEM fields.
Helping such students is a real challenge. It's a matter of establishing a connection and a trust that will let you probe back into their past until you find the place where the problem arose. This is like psychotherapy. It takes one-to-one work and it takes a long time to build trust, before the probing can begin. None of this is practical in a traditional college teaching framework, and that is why college admissions offers key on Calculus.
As for the discussion of Statistics, I agree that this is more important for general students. STEM students need to add Calculus as well.
> (e.g. subtracting something from both sides of an equation to move a term to the other side)
You mention that as if it was a trivial operation, but it only works because subtracting a constant is total and injective. In general if a=b then f(a)=f(b) [the substitution property for equality] for any total function f - but the converse is not true in general.
US-centered response, assuming the article is about US education.
The article is off track vs what I observed with my kids math (typical middle of the country small public hs). Nobody gets to do trig until they've done boat loads of stats.
The thing I saw missing was the kind of top notch teacher that can inspire and intrigue the students. This can only be addressed by paying teachers much more. That's not going to happen. Our school only had one "real" math teacher, and he retired last year.
Paying more goes hand and hand with testing teacher (or holding them responsible for students grades), and generally that's not very well received by teacher unions.
The example I always drop is my ap calculus teacher whose students had an average test score of a 4.7, but the ap physics teacher didn't have a single student get a 5. They never shared their average for obvious reasons, but I'd assume it was closer to the national average of 2.3 based on what I heard.
Clearly the school had students capable of putting in the effort, if the teacher was capable of teaching the material.
> But Martin said that non-STEM students didn’t really need to learn trigonometric functions, which are used in satellite navigation or mechanical engineering.
For me, trig has been the least useful math. I never once used it in college (2 years as a CS major, then an Econ major) or in the working world (tax lawyer, startup founder). I'm sure there are some jobs where it is irreplaceable, but this might be a minority of STEM jobs. Do biologists need to know trig?
I once TA'd a Calculus for Business course; there were two fundamental differences between that and the regular course: 1) the problems and exam material were much easier and 2) it lacked any mention of trigonometric functions.
When I realized the second part, I went to the instructor and asked: "Without the trig functions, how are they supposed to model and analyze oscillatory phenomena in business settings?".
The instructor turned and gave me the biggest smirk I had ever seen.
Nah, they slap a few linear variables on it, throw that in a model that adds lag/time dependent variables, and they're ready to tell you the pendulum is starting to swing in the other direction.
I hated math in middle school and high school. This is despite loving science, and enjoying math all the way through Algebra 1. However, when I started taking Algebra 2 and beyond, it seemed like we shifted away from why things work to solving ever larger polynomials and other tedious tasks, getting points taken off along the way for a flipped sign or digit or moving a decimal point, even when the entire rest of the calculation was correct. I ended up doing a degree in computer engineering, and every class that involved algorithm design or programming I excelled at, while every "hard math class" (for example, statistics, electromagnetism, etc.) I barely passed.
I didn't fall back in love with math until I took discrete mathematics and signal processing in the same semester, and then later a class on algorithms and formal models of computation. They showed me that math was actually beautiful and fascinating, an art form unto itself. These three classes showed me how amazing math can be; I remember particularly that the day we learned about FFT's my brain felt like it had ascended into another dimension. They truly felt magical. Same thing with learning about recurrence relations and finally "getting" dynamic programming.
Now I love math, and I am planning on taking this holiday break to finally crack open the Principia Mathematica and trying to really understand it (a personal goal of mine for several years). I can't wait to share the joy of mathematics with my kids when they're old enough. I just hope that they don't get bogged down in the mechanical, number crunching part like I did and manage to continue to see the beauty of the underlying ideas.
Maybe history, social studies, government, or economics classes in highschool should use a spreadsheet to show basic data analysis and statistics (mean, median, mode, stdev).
It would be good if there was a more basic understanding of how an "average" can hide a lot of inequality. I don't really see getting rid of angles and algebra to cover something "outside of STEM".
This is where taking a more holistic approach to curriculum planning would pay off. If the weeks after you were taught these (or any other) concepts in math class they start showing up in your science and geography classes, you'd probably remember them better and have a better understanding of how to use them.
How about we start with high school students being prepared for math that concerns their adult life. Like how debt works, compound interest, savings, budgeting, etc?
Many schools do teach this. It’s not required but some schools do offer it as an elective. However, as practical as it may be, many teenagers just aren’t interested in learning about personal finance.
At a more philosophical level, should schools even teach personal finance? The math itself is very simple and many adults can learn it on their own, without guidance from any instructor. All they need is a couple of good blogs and YouTube videos.
The classes schools teach and the curriculums they offer are limited. There’s only so much time in the day to teach a wide variety of topics.
So, should educators spend time teaching subjects students could otherwise learn themselves? Without an instructor, assignment feedback, etc, would people still pick up something like physics or calculus on their own?
My personal opinion is that K-12 and college should focus on complex topics like algebra, chemistry, and physics, while simpler courses like personal finance could be taught by parents or by a students self. Happy to hear the debate around this though, it’s always fairly interesting.
A lot of kids parents can’t teach this or don’t know how or if they even need to. Yes give complex topics for those that can. But have a basement level of basic personal finance. Don’t care if not interested because it will be relevant. Make it mandatory.
Learn yourself or parents teach it - great! Test out and go to advanced. For everyone else we require finance literacy.
Reminds me of that joke that crops up in April, about how grateful we are for all those hours in our high school math classes this parallelogram season.
I don't understand the statistics-vs-calculus debate at all. Probability theory is at the heart of statistics, and how can a student understand statements like 'the integral of a probability density function must equal one over its possible range' without taking some calculus?
The problem with calculus as taught today in most schools is that most of the mathematical tricks for integrating functions that they hammer the students with are essentially useless in real-world problems so you might as well just start with numerical methods of approximation right at the beginning, once basic concepts like 'the integral of a function corresponds to the area under the curve of that function' are grasped. I know many math purists are dedicated to the pencil and paper approach, but it's not all that useful for non-math majors who will certainly be building or using computational models almost exclusively.
Another prerequisite for statistics should be linear algebra, because without a grasp of vector representations of data and matrix operations on that data, a lot of statistical concepts, e.g. multivariate statistics, won't be very understandable.
Fundamentally, trying to use statistics without this deeper understanding of the mathematics underlying much of it can lead to all kinds of issues, like choosing the wrong statistical method for a given problem.
I think a "statistics" class could still help a lot of people without ever touching continuous distributions (which as you mention are more naturally built off calculus) or multivariate distributions (which as you mention are more naturally built off linear algebra). Simple stuff like basic conditional probability, Bayes' rule, confidence intervals, basic hypothesis testing, and general "statistical thinking" don't really require calculus or linear algebra. Just getting people to a point where they understand some of the really basic tools seems pretty valuable, and pretty far from the current state of things.
A mandatory nitpick: you cannot do conditional probability without multivariate distributions, but you are right, you do not need linear algebra for a basic introduction. Even continuous distribution can be introduced, you just start with discrete cases, introduce areas in a histogram, and then approximate the histogram by a continuous curve. You can even get to the Riemann integral that way if you want to. And nobody actually calculates normal probabilities by integrating the Gaussian function, it's kind of hard to do.
I’ve always felt that the best capstone high school class is neither stats nor calc, but finite math, which ends up being a mix of some linear algebra, combinatorics and probability with a dash of discrete mathematics. This is stuff that has a lot more applicability in business and social science and is under-taught (one example was one that came up when I was getting my MS/teaching credential was my fiancée’s step-father wanted to be able to estimate the length of a roll of sandpaper given the diameter and thickness of the sandpaper. The engineers at the company came up with a correct answer by writing a program to add up the lengths of each layer of sandpaper, I verified their answer with a bit of pencil-and-paper algebra impressing my future in-laws. Someone who had taken a finite math class would¹ be able to do the same.
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1. Although perhaps I’m being optimistic since I’ve neither taken nor taught finite mathematics, I’ve only paged through a textbook I saw in a bookstore once.
How much time roughly do you spend in math classes in school in the US?
I'm finding it a bit baffling reading these comments ('calculus vs statistics' etc.) from my UK perspective. We had I think roughly^ 3x1h maths lessons each week, and it didn't seem to be a problem to do both, as well as whatever else. Nor did we wait until secondary school to start stats (calculus sure, for the usual colloquial definition of it).
Or are we talking specifically about some arbitrarily advanced level of each that somehow everyone's on the same page on without describing?
I have a kid starting 1st grade and I have just been exposed to common core math for the first time. At a glance I do not understand how it prepares for a future in mathematics or related fields. Interested in hearing from others who have been through common core math first hand or raised a child through the program
Also the way i learned to divide in my head is completely backwards and, though it works for me, when i tried to explain it to my wife i realized how completely backwards and inefficient it is, so maybe common core helps with these and other hard-to-internalize concepts?
Well, one thing is that I can’t help my kids with their homework. I simply don’t understand the way they do the math these days. They get the same answers, but go through entirely different steps. I remember one parent teacher conference where my kid was in attendance (4th grade). The teacher was showing my kid how to do a problem. I was able to get the answer in my head, but she was using all these boxes and things to get the answer and I could not follow along. I still have no idea how she got the right answer doing that. It seemed incredibly complicated.
The point of most of the common core math changes were to teach numeracy rather than rote arithmetic.
Remember how some teachers said "you won't just have a calculator on you at all times" and it turned out that we all carry around what would have been supercomputers for the time? Early math education realized that too, and shifted the focus to quickly estimating solutions through understanding the relationships of their numbers so you can go "hey, something's not here" when you work with your calculator, rather than just accepting whatever garbage out came from your garbage in.
It is complicated, but the point isn't to get the right answer per se, it's to give kids a geometric intuition of numerical relationships by going through the whole rigamarole.
This isn't the first time such changes have happened. https://en.wikipedia.org/wiki/New_Math, target of the famous Tom Lehrer song by the same name. The idea even seems exactly the same as common core:
>In that unfamiliar context, students couldn't just mindlessly follow an algorithm, but had to think why the place value of the "hundreds" digit in base seven is 49. Keeping track of non-decimal notation also explains the need to distinguish numbers (values) from the numerals that represent them
I do not have a child or direct experience with common core.
Personally I suspect issues lie more in teachers themselves not being taught correctly than the actual concept behind common core though.
From what i can gather it is trying to introduce the concept of factoring earlier which in my opinion makes sense since that is really useful in Algebra and above.
That means the implementation of common core is also upon teachers with decade(s) of experience being told to teach differently.
Regardless, if your child has a nack for maths that is a great advantage over the other children and worth investing in since so few people bother to learn it at all.
Additionally it leads to better reasoning and logic in other academic and professional settings as well. Which means a better chance at a higher income and ability to sustain and secure their livelihood and thus afford having their own children possibly. Which means a better chance for continuation of a little bit of yourself.
Primary school common core math is focused on teaching numeracy and conceptual understanding. Students who want a future in mathematics or related fields will take more formal rigorous math in secondary school, often via a "compaction" program that resembles older math programs that result in completion of Calculus by the 11th or 12th grade.
Once your kids get into higher grades, you may end up seeing them be taught the "tricks" that you may have come up with yourself as a math person in previous years. These solutions leverage the students ability to have a functional mental model of how the math works, even if they haven't done as much rote memorization. In a modern world with calculators everywhere, it's better to teach students to be able to identify incorrect calculations quickly over repetitive practice on large, complex sums and products. (Times tables up to 12x12 are still ubiquitous and emphasized, as having those down helps a lot when doing other math.)
Repetitive practice is important - it's about getting exposure to formal reasoning in a "toy" context. You can't get "conceptual understanding" in math without learning what formality and rigor is for.
My kids are now college age, but when they were in K-12, I looked up common core, since the standards are published in several states. Honestly it looked quite a lot like the math that I learned in school, give or take. I took exactly the same school math subjects.
The one noticeable difference was that my kids did virtually no proofs. When I was in school, my district used a "new math" curriculum that introduced sets in first grade, and included derivations and proofs. (I'd call a derivation a lightweight proof). High school geometry was almost 100% proofs, and it was a class that a lot of students remembered as their favorite.
My kids: No proofs. They solved lots of problems where they were expected to inspect a problem, choose an algorithm, then crunch through it to an answer.
Proofs were what made math come alive for me. I could crunch numbers in science class, or on a computer, as I learned programming in 11th grade. I ended up majoring in math, and then added a physics major as well.
Math is an extremely confusing and fraught topic for parents, because we all viscerally know it's important for some reason, but nobody can really put their finger on why. Some parents treat it as a form of obedience training, or expect that it will magically confer special thinking skills. We know math is a sorting hat for getting into vaunted STEM programs in college.
Very few people use their school math after they finish school. In school, it's treated as a tournament, to reach "levels" and get good "scores." Many of the brightest kids are repelled by this. I would have been.
The college math topics are the same as they've been for 50 years. A number of years ago, in between jobs, I taught an introductory math course at a Big Ten university, and the students didn't touch a computer for the entire course. My office didn't have a network connection.
If it were up to me, I'd add a lot more computation and data work to K-12 math. And I'd bring back proofs. I envision a balance of four quadrants, not in any particular sequence, but perhaps in a cycle:
1. Arithmetic, which is symbol manipulation up through calculus
2. Computation, which is using computers to solve problems
3. Working with data
4. "Theory" which I associate with proofs and abstract topics
My advice as a parent is, first, be prepared for them to follow their own interests. This might include not being interested in math. Be prepared to help them deal with the competition, and to not let it cause them to lose interest. Next, treat it as something interesting and fun at home, separate from grinding through problems.
I’d just add to this excellent comment that, when I was in school a few decades ago, we also did essentially no proofs. Geometry was presented as rote memorization (as was pre-calc). I was utterly unprepared for college physics and differential equations (the mathy freshman CS courses) as a result - I struggled to figure out how to build on equations to find new solutions, and struggled with understanding the logical rules of things like Scheme.
I wish we had had a more proof-based math education in high school. But I’m also unconvinced that the quality of math teachers in this country is up to that task (though maybe I just had a string of really mediocre ones, punctuated with a couple exceptions).
> Also the way i learned to divide in my head is completely backwards and, though it works for me, when i tried to explain it to my wife i realized how completely backwards and inefficient it is, so maybe common core helps with these and other hard-to-internalize concepts?
What way? The classic way to learn is to be proficient at multiplication up to 10x10 and then solve for the reverse.
i think you're exactly right -- it is trying to take some of the mental tricks that "good at math" students would intuit for themselves, and teach them explicitly.
this is good overall, although of course it's possible to ruin anything via bad teaching.
Basically what we’ve found is that students need a better understanding of statistics, visualizations, etc.
But just because these are mathematical concepts why do they have to replace math in the curriculum? Logically, shouldn’t they replace the least valuable parts of the curriculum, irrespective of whether that’s math or not?
So the question really shouldn’t be statistics vs calculus (or advanced algebra) but rather statistics vs whatever is considered the weakest part of the curriculum across the board, not just limited to the field statistics belongs to.
One of my freshman classes was game theory. A prerequisite was trigonometry, and I only had pre-Calc. The best part about it was I was doing averages and which great which number is greater in my senior high school class after taking pre-Calc as a Sophomore. Of course I complained about this to both schools but that wasn't really their problem. I suppose they just didn't care.
Old fart here: In high school, math was broad spectrum. Calculus, Algebra, Geometry etc. Too early for specialized topics like Probability and Statistics. Those were first-year university material, where you start getting steered toward the actual math skills you'll need for your curriculum. What's wrong with that?
Funny that most humanities degree holders that I am aware of do not agree that humanties events can be expressed in numbers. They can't comprehend anything rather than over simplified stories. What I mean by degree holders, I mean Phd degree holders, professors.
What being said in the article reflects misunderstandings of math from the arts and humanities community. What they need is a general understanding of discrete math rather than specifically statistic. Calculus is valuable to STEM because of the smooth and continuous natural of macroscopic world. Contrast that cultural/social/econ events are discrete. If you want to draw a beautiful smooth line which fit so badly, that do not tell the whole story. However we see people drawing line cut through a circle in a graph in most humanities papers and they claim there is a positive relationship whatsoever.
And here is how flawed the survey is.
>>> Survey target: Majors that require calculus were excluded.
>>> Many high school math topics were unimportant to college professors. For example, most professors said they wanted students to understand functions, particularly linear and exponential
Linear functions are already taught in pre-calculus. You want calculus but you don't want calculus.
>>> the ability for patterns and relationships and make generalizations
This is unrelated to math. More related to IQ. Seems that students in the non STEM realm generally lack the ability to find patterns as professors witnessed. People should take more concerns on this phenomenon.
> This is unrelated to math. More related to IQ. Seems that students in the non STEM realm generally lack the ability to find patterns as professors witnessed. People should take more concerns on this phenomenon.
Yep, this holds true for programming too. You either have the mental capability to turn a problem into a set of instructions that can be coded (and then grind through the syntax and language specifics) or you don't. I'm not that young anymore, so my first formal programming course was in college (at the time when pretty much every stem student had a computer at home, some even a laptop for college), but it was just like this... with zero correlation to math grades, some would understand the concept and break the "idea" down to basics programmable steps, and some would get stuck at the "design" phase.
Also, this was basic programming... eg. fibonacci sequence, so they could explain recursion after that. Basically, some would understand "ok, I have to start at zero and go on, but i don't need to remember just the last two numbers, so two variables, previous and previousprevious, and a loop,..." (and then debug the for loop since it starts at zero, and <= was used for n or whatever), and some would get stuck in a mental loop, if they have a "n", how do they get the previous two numbers, if they need previous numbers for those too.
So this is way before the level of patterns and generalizations, where you need the subject matter knowledge to even know, if you should fit the curve to a straight line or some higher function.
I have a friend who has been a high school math teacher for 15 years. When she started, she asked me and the rest of our friends to make note of any times we use algebra or more advanced high school math in our daily lives.
Both times it has happened, I've been sure to let her know.
Once, the Pythagorean theorem helped me figure out how long the sides of square table decorations needed to be for my sister's wedding if they were to be inset in round tables with 5-foot diameter.
The other time, I was able to prove, conclusively, that I could not possibly have been speeding before another driver caused an accident, but none of the police officers could follow the math, and since I wasn't a "certified accident reconstruction expert," they wouldn't believe me anyway. (Hiring one would have been more than the repair cost.)
It didn't go over well when I told the supervising officer that just because he couldn't do the math, it didn't mean it couldn't be done, even by some dork who had a liberal arts degree but had, in fact, studied calculus in high school.
I had the accident record pulled from the vehicular equivalent of my car's black box that said how fast I was going at impact. The distances involved made it such that I could not have braked to that speed from a higher speed than the speed limit in the required space.
One thing that stood out in university was learning the algebra version to so many basic math formulas in any field, whether it might be physics, math, biology, chemistry, or most anything.
As for your use case, sometimes, some people are hired to not understand things.
Data point, (or rant) about high school (Abitur) situation in germany: it is the same or worse.
I was quite good in math in school, yet in the beginning did not understand anything math related in university. Which was no surprise, because the math in school for me was basically memorizing algorithms to solve known problems of style X. But not really understanding it. And then in the final years in school, while in theory doing higher math, the main focus shifted on how to use a graphical calculator to do the math for us.
But in university I suddenly had to understand what I was doing, to be able to use it. Oh my. So it was just really hard grind, to recover all that I did not learn before (that grind was highly beneficial, though). In theory the universities knew about the problem and they complained about it to us - but as far as I know, nothing has changed since then. So Math will remain a weeding factor for lot's of students, who otherwise might have been great engineers, but maybe had the bad luck, of having bad math teachers on top of the bad curriculum.
It is also a really bad sign, that math is so universally hated among students. Because math done right, is just thinking done right - and I think the world might benefit from more of it.
Honestly, the bigger issue here is that good math teachers in the US are just in short supply. Sometimes I wonder if the way we teach algebra here is flawed, like something pedagogically or conceptually, and good teachers break through that to give students what they need to prepare their students.
1. Can you teach statistics without doing calculus first?
It's quite common. We had such a course which was typically taken by social science majors who needed some statistics in their fields. (I never taught it.) The prereq was just a little algebra. A typical text was Mario Triola's "Elementary Statistics" (see here for the table of contents: https://www.amazon.com/Elementary-Statistics-13th-Mario-Trio...). The students used Minitab as part of the course. But we also had prob & stat courses which used calculus.
1. (a) How can you define the exponential function (or the number e) without calculus (limits)?
When I taught college algebra (= review of high school algebra, around two courses below Calc 1) I'd say something like: "There's a number called e which is approximately 2.7118281828459045 ... - it's kind of like pi, an infinite decimal that never repeats. We've seen there are exponential functions like 2^x, and we have an important exponential function called e^x. You're probably wondering why we're using such a weird number in an exponential function. You'll see how it comes up if you go on to take calculus." No one much complained about any of that.
Even a typical calc course skips a lot of the theoretical background - you don't see the background unless you take a course in real analysis. In teaching college math, you're always "starting in the middle", assuming a lot, and in basic courses you're omitting a great deal of the rigor.
(Also, in calculus it's common to define logs first using a definite integral, then discuss inverse functions, then define e^x as the inverse of ln x [so in particular e is just e^1]. You justify the notation by showing ln x and e^x have the properties you "expect" logs and exponentials to have [e.g. ln (a b) = ln a + ln b].)
2. Why not teach statistics/linear algebra/logic in high school?
High school curricula are often strongly determined by state testing, and by the needs of students who are going on to college to have the math that colleges expect. I took courses in math logic and matrix theory in high school, but they were electives; the "big" course for me from the point of view of college admissions was BC calc. I certainly got no college credit for the logic or matrix theory courses.
So e.g. if you tried to do linear algebra or logic in high school it would come at the expense of topics that a state is testing - then your students do poorly (even though they might know some pretty good math) and you and your school get in trouble.
There is more of a movement to teach stat and data analysis in high school and college - things are changing.
3. Why not teach statistics/linear algebra/logic before/in place of calculus in college?
College math departments are heavily service departments. The bulk of the credit hours are in courses taken by majors from other departments. Those departments (science, engineering, business, and so on) tell math departments what their majors need, and math department have to ensure that the math courses suit. The other departments in turn are often required to have majors take certain courses with certain topics by program accrediting agencies. So there can be a lot of inertia there.
