I graduated with a mechanical engineering degree, meaning I took 3 semesters worth of Calc and 3 more semesters worth of differential equations. So I'm not one to hate on math or calculus specifically. But I think high schools should focus more on statistics.
Statistics is relevant to more fields than calc. It matters a lot more in business and in all the jobs where you use calc you will also have to use statistics. It's also useful for citizens to understand statistics better - and educating citizens is truly one of the purposes of public education.
I would love to see statistics taught more universally. But there's statistics and statistics.
I took a fantastic one-off class as an undergraduate called, IIRC, "Statistics for Physicists". The class taught us Bayes' Law, what tests were, what significance was, and how one might go about deriving and using simple tests. Everyone came out of the class with a very clear understanding of what statistics meant, and no one who took that class would take a statement like "this drug had a significant effect on such-and-such" without an appropriate grain of salt (specifically, how big was the effect). The class didn't include a giant grab-bag of tests.
One of my friends took a regular introductory stats class the following quarter. At one point, he asked me if I could help him do a such-and-such test (Student's t, maybe, but I don't remember which one). I asked him what he was trying to do with the test, and he didn't know.
I think it would be enormously valuable to teach most students what statistics means, what it can do, and what it can't do. I'm not sure how valuable it would be to teach everyone how to evaluate a big pile of magic statistical functions.
This is an area where I think the usual proper math approach of abstract patterns and theorems and proofs is all wrong for the general public (the ones in a general public school).
They should teach it from the perspective of what problems you are trying to solve, what your options are, and how those options work in practice.
The "in practice" part should be lots of simulations with something like a ChromeBook, so people could really see the ideas working. Create a (simulated) bucket with 70% red balls and 30% blue, then pull samples of various sizes and see how well the sample means match what you already know is the "truth" (70/30). You'll soon have everybody seeing that bigger samples aren't always better, but they tend to be, and they'll see how much better they get as a function of size. Make charts of the improvement. Talk about how good the estimate really NEEDS to be.
Then try some tests where the students DON'T know the "truth" about the population and have to use samples to estimate it. How far off do they think they are. How confident about various intervals, etc.
Then start explaining some of the math that lead to formulas that let you calculate these estimations, sample sizes, confidence intervals, etc.
Have estimation games: you get points for getting an estimate within some delta of the population mean, but you have to pay points to buy your samples. So, how big a sample does your team want to buy?
Then continue with other basic prob/stat ideas such as bayesian cancer tests (does this mean you have cancer or not?) and courtroom dramas (does this mean he's probably guilty or not?), commonly misunderstood situations (ex: Simpson's Paradox), how Gaussian estimators begin to fail when things aren't independent, and all sorts of other introductions to quantitative thinking about real life scenarios.
I was a very math oriented student at a high school with a heavy bias towards liberal arts ... So when it came to picking senior year electives, I was delighted to enroll in AP statistics.
The classes turned out to be tutorials on the built-in TI-83 stats functions, with step-by-step instructions on how to answer the various types of questions we would encounter on the test. I forgot everything and gained virtually no statistical intuition.
Thankfully I had a similar experience to you ("Probability and Statistics for Engineers") with a fantastic professor who researched computer networks. His lectures were sprinkled with tangents tying the current topic back to his own field / research, which to me made the biggest impact. Contextualizing a taught subject markedly improves the course experience.
There are good stats classes and bad stats classes, just like there are good calculus classes and bad ones. When I was a student, we spend a full term learning integration techniques, with almost no class time devoted do learning what an integral actually is. The fundamental theorem of algebra was glossed over in a way that makes me think the lecturer didn't really understand it. Differential equations was a catalog of recipes, with almost no modeling and certainly no qualitative analysis. I got an A+ in the course but learned almost nothing from it.
Fortunately, I had a great professor in Calc III who more than made up for this.
I second that. Stats is so much more applicable, and it is also the area where common sense will fail you the most. Simple things like birthday paradoxes, Gambler's fallacy, Monty Hall problem, and so on are all contrary to common sense yet not taught in schools.
And you need stats all the time. Every time someone says crime has never been worse, you need to know how to decide if you believe it. When someone says flying is safer than driving, you need to know how to compare. A lot of stats thinking is also not formulas, but qualitative things like "Am I comparing apples to oranges".
Calculus by contrast is pretty common sensical. Acceleration is the derivative of velocity, you find the local extremes by finding the zeros of the derivative, the integral of a net shaped field is the rim... that all makes sense with some pictures.
I also haven't come across it much in finance, apart from derivatives work, where funnily enough you can't do with high school calculus, but you need stochastic calculus. Which contains some quite statistics related concepts.
I suspect calculus is less commonsensical to most people than you think :-) That said, I've argued the parent's point for years. Basic stats/probability/economic biases are really fundamental stuff and don't require a lot of complicated math if you don't want them to.
There are some basic calculus concepts that are useful. Area under a curve. Basic derivatives (for finding maxima/minima, etc.). But the truth is that I've rarely used calculus outside an academic setting and I use at least basic statistics/probability thinking all the time.
I think the reason calc is preferred, to prob/stats is because the intuition is much more visual than the latter. Furthermore, students up until that point spend the majority of their time studying various functional forms (conic fxns, exps/logs etc.)
Functional analysis seems like an obvious next step.
Though stats/prob are perhaps more 'useful' in non-academic life. Imparting the notion of what probability actually is and how Bayes' rule fits into it requires a very gifted teacher with a fundamental grasp of the real mathematics and on top of that the ability to communicate it and link it to previously learned material aka algebra/ pre-calc. You can make it visual, but it would seem to require defining the concepts of sets, sampling, distributions etc.
The issue imo is that teachers tend not to teach the intuitions of math, rather they focus on the mechanics of it. As such, most students turn into machines calculating things as opposed to viewing math as a way to reason.
In the UK high school math is really a preparation for a hard science/engineering degree, and there's a lot more calc than stats in hard science/engineering.
That's why calc is central. I wouldn't be surprised if it's the same in the US.
This means most people have no idea what the math they learn is used for, because they never reach a level where they have to use it.
So they leave school with the idea that math is annoying bullshit that they don't understand and can't use.
I think it would be better to stream math by ability much earlier. Make everyday math compulsory for everyone - interest rates and loans, basic stats, currency rates, areas/volumes, simple trig, maybe some slightly more advanced finance, perhaps touch on some calc ideas.
Only teach calc and scientific math to those with the talent and interest for it. Give everyone else a taste so they understand a little, but don't try and get them ready for a career they'll never go into.
Having taken hard science in university/ during my PhD, it's hard for me to say that calc presents itself more than stats. Perhaps in engineering but only in the early years of uni, physics same thing- year two is an exposition to quantum mechanics... In chemistry and biology I would say stats/prob definitely predominates. In chemistry calc begins to appears in physical chem.
Why most people don't understand the math they learned and what it is used for, I would argue has much more to do with how they are not taught to interpret what they see around them mathematically- granted that takes years of practice, but we do not care to teach this to children.
Now streamlining math is all well and good, but we may loose some of the greatest mathematical minds by doing so. Many famous mathematicians (Euler, Lagrange for example) were slated to be someone else- professionally that is. And it took other great mathematicians to notice them.
>The issue imo is that teachers tend not to teach the intuitions of math, rather they focus on the mechanics of it.
Yes. I agree generally--often even at relatively advanced levels.
The stats course I took in engineering grad school was so heavy on the math that I'm not sure I appreciated what the math was in service of. By contrast when I took a less math-heavy stats course later I much better appreciated the underlying principles.
But yeah, it's very easy to turn any of these math-related topics into: memorize the formula, plug the numbers, come up with a result.
Consider 3 doors: A, B, C. If you pick the right one you win the prize. Say you pick door A. Monty Hall opens B or C and reveals that there's nothing behind. Now he asks if you want to stay with A or switch to the closed door. What should you do to maximize your chances of winning?
Right before any of the doors is opened we have no idea what to expect, so there are 3 equaly likely possibilities:
1. prize is behind A 2. prize is behind B 3. prize is behind C.
In case (1) you win by staying with your choice of A. In case (2) Monty Hall must open C (by the rules), so you'll win if you switch to B and in (3) the host will open B(by the rules) and you should switch to C.
In 2 cases out of 3 (2/3) you win by switching and in only one case out of 3 (1/3) you win by staying with your first choice. Since 2/3 > 1/3, you should switch.
This problem is easy if you know the defintions of sample space, event and the Equally Likely Probability Formula.
Yes, indeed it did for me, as well, but then I'm not trusting my intuition that failed me for the three door version, and I'm desperately trying to prove it wrong, because I'm biased by my first commitment to an answer.
I think that commitment bias or what it's called is also an explanation for the misunderstanding in the first place. The argument against the increased chance, as I would use, works both ways and that is much more apparent the smaller the initial chance is. But I only intuitively used it to defend my position.
A sample space is the set of all possible outcomes of a random experiment. An event is a subset of a sample space.
Let S be the finite sample space with equally likely outcomes in it. Let E be an event. Then the probability of E is the number of outcomes in E over the total number of outcomes in S.
Now consider 52 cards in a deck. The sample space of outcomes here are the 52 cards. What is the event that the chosen card is a black face card? E = {J-club,Q-club,K-club,J-spade,Q-spade,K-spade}. What is the probability that the chosen card is a black face card? 6/52 by the formula above.
Monty Hall problem can be solved by using the above as a guide.
Sample space here : {switch, switch, stay-with-your-original-choice}. By the formula above, the probability of switching is 2/3 and the probability of staying put is 1/3.
If you're doubting the formula, I can provide you with a proof.
As I mentioned in another comment, it's important to note that Monty Hall is guaranteed to open a door with a goat behind it (a losing door). If he chose randomly and just happened to reveal a goat, the answer would be different.
Indeed. This is the major reason I don't like the Monty Hall problem. When using stats and probability (at least in science), the universe is not your adversary.
I think there is an easy and intuitive explanation:
Two thirds of the time, you pick the wrong door on your first choice. So far, so obvious. Now, you picked a wrong door, Monty opens the other wrong door, and there are only three doors. Obviously the remaining door has the prize.
Every time you pick the wrong door, and then switch, you win. That means two thirds of time, if you switch, you win. That makes switching the right strategy.
The key is that Monty Hall is guaranteed to show you a losing door. If that weren't the case -- if he just picked one of the remaining doors at random -- then the fact that he happened to pick a losing door would count as evidence that your door is a winner, leaving you with a 50/50 shot. I think that's one source of intuitive discomfort, and even of lasting confusion if the scenario isn't specified precisely.
The problem is, it's hard to understand statistics without knowing at least basic calculus. As soon as you move from coin flips to probability density, you need derivatives and integrals.
These days you can solve many practical statistical problems with Monte Carlo, but understanding why those curves look the way they do is still very useful.
Though i think pdfs are useful, they may be a bit advanced.
Perhaps linking integrals/derivatives into stats could be done from a more discrete approach as opposed to a continuous one first.
For example:
Starting with a representation of an integral as sum of areas of rectangles (something most students can do) and then saying what happens if the width of all the rectangles gets smaller and smaller. A la Riemann
This could then tie into the pdf quite reasonably?
I had kind of a strange experience. While in college, I worked as a math tutor. Since I had taken one semester of stats, the tutoring center decided I was qualified to run a tutoring session for the non-calculus stats course. I was happy for the money, and had been reasonably successful with other tutoring assignments.
When we got to continuous distributions, we kind of hit a wall. Here's the puzzle. Suppose the data are recorded to two decimal places, and are reasonably bounded, for instance they are between 0 and 100. That's a discrete distribution. All of the data in the examples and problem sets were fixed-point.
I literally couldn't justify to someone who had never taken calculus, why a wholly different kind of distributions with its own collection of formulas, was necessary or useful. The teacher had never explained it either.
The chance of choosing 0.5 from the interval 0 to 1 is 0. So if you want the change of choosing between 0.4 and 0.6 in that interval, you can't just sum all chances between 0.4 and 0.6 you'd have infinity x 0 which isn't 0.2 .
Instead, to deal with this 'a lot of really small things' we need a slightly different approach. Then draw a pdf, and talk about area under the curve. Perhaps relate this to the discrete case where the curve is just really blocky.
That's a good point. I'm not an educator and it's been a while so I can't recall the sequence of stats 101, calc 1, more stats, more calc, etc. It all blurs together at some point.
Until calc is needed in the statistics. On which level do you even compare this, space or time complexity? I mean, are you saying, the complexity of the sum of the stochastic in use exceeds that of calc?
I think you gave an unfounded statement, without proof. i'm guessing you are biased to think statistics is harder, because you were less exposed to those classes and now you have remorse. If it was the other way around, you'd want more calc.
The way curricula are laid out, I'm getting the impression it is not so much about what is taught, school is rather a benchmark of abilities and a day care.
If we can't have better math instruction, let's at least have more relevant math instruction. Calculus is very useful in some disciplines, like physics, but startlingly useless elsewhere. Cool, yes. But useless.
The problem with trying to teach statistics, to anyone really, is that once you get past the very basics, you have to stop every time the lecturer/textbook invokes measure theory and figure out how to explain the concepts without just making everyone sit through two to three semesters of real analysis.
Agreed, there is definitely a huge gap between doing, for example, a t-test and deriving the t-test (or giving any substantial justification for its form). You need a class in analysis, a class in measure-theoretic probability, then a mathematical statistics class to bring you back to the topic. It's a tough square to circle.
Most who teach stats at the undergrad or hs level have little choice but to defer a lot of things to a later course.
In my experience, just knowing discrete probability and statistics is profoundly unuseful, especially when you walk out of class thinking you know the subject, sign up for a next course that involves real-valued quantities, and bomb everything.
One issue is statistics can be super boring if taught incorrectly. I am an engineer too, loved probability but found statistics to be 'shoot me in the head' boring.
the class really needs to be taught well for people to understand it. also. i think it should include probability too.
I agree, but I thought most lower level classes were often titled things like "Intro to Probability and Statistics." Further, it's been my experience meeting and being around a lot of people who thought they knew stats that they only learned some basics which included but not limited to some basics such as: factorials, normal distributions, coin flips, t-test, permutations, combinations, etc.
I think a good stats education requires heavy emphasis on doing real problems with what you learn. It's so easy to do and apply to the world around you. Unfortunately, most books that even touch on this have absurd text and problem sets that revolve around pulling jelly beans out of a bag or rolling a dice (I mean yeah, but super literal). Mostly the professors and teachers I knew that taught statistics at the lower levels always seemed bored and wanted to be elsewhere. A shame, really.
Linear algebra is also a lot more useful in real world applications than calculus. As it stands calculus seems as much like some kind of college hazing ritual as it does an essential part of anybody's education.
At my high school (Texas public school), Calculus and Statistics were both optional AP courses.
You can usually choose whether or not to take whichever combination you want. Some people took neither. I took both.
If high schoolers had to take one advanced class, I think it should be discrete math. Keeps their algebra skills sharp, introduces them to proofs, and still introduces them to basic probability. Obviously the availability of this offerring makes this impractical at the moment.
One of my degrees is in statistics, and the others are in computer engineering, finance, and computer science. As such I not only took a ton of math, but studied and worked in the contexts you describe where stats is relevant.
The problem with statistics education is that it is generally heavily watered down, particularly the classes at the university level for business, psychology, and other majors that need the knowledge but aren't math or engineering majors. People heavily misunderstand the usage, difficulty, and nuances of stats education if they are only exposed on this level. It's better than nothing, but leads to serious issues that remind me a bit of the novice arrogant programmer vs. the grizzled veteran smart senior programmer. Generally, most people I see with limited stats education come out of it thinking mean, median, normal distributions, and at best linear regression are the whole of stats.
Another problem that also related to the parent with regard to stats vs. calculus is that calculus is actually a pre-requisite for most useful statistics work. It is true there are plenty of topics in stats you can cover without knowing calculus, but you are at a severe disadvantage and won't really understand the "why" and just be regurgitating the memorized method. I feel a lot of statistics is a combination of the "why" and "how" because so much of it is applying math at a more discretionary and opinionated level, like is done in science. Really learning stats requires a lot of math, from advanced algebra to diff eq possibly to matrices to calculus. To do good work in stats, you need a lot of tools to draw from, otherwise you're stuck with tools that often don't work (common example is people who only are familiar with normal distributions and nothing else).
Yet another problem with stats is that you can really make up whatever you want when you conduct statistical applications such as an analysis. The rule is that as long as you provide justification and proof for what you are doing - i.e. how and why, it is acceptable along with some text to explain. This is why good statisticians do things like describe and reveal their sampling methods, explain what did and did not work, and why they selected various tools. Obviously most dubious work can be quickly disputed and identified by someone who knows what they are doing, but the reality is that so many people never follow-up, fact check, or even read anything but the conclusion.
Stats is in so many ways about showing the process and justifying your answers than the quantitative answers. This is similar to financial forecasts as well for example which also use a lot of applied math, and people just believe what they read even though the evidence to disprove the report is right there. The science comparison applies here too. Just like with research studies, a lot of people don't check, verify, and re-run the results so people just believe what they read or rubber stamp it. I recall I had a very mean professor that once gave a final exam question that actually could not be solved at all using the methods we learned because the types of analysis and models we were asked to try to apply were invalid for the data. The simplest version of this familiar to many people is trying to create a best fit linear equation to non-linear data - simply draw it on paper against a data plot and you'll see it makes no sense.
If there's one important thing I learned in stats, it's to be skeptical and prove things with evidence. Nearly every time I watch the news, read an article, etc., I go nuts because they are making huge conclusions without giving you a way to understand who, what, where, why, and how about the study. The presidential elections in the US are an obvious example I see a lot with things like "Survey Monkey" passed off as scientific data in some contexts. Even represented as a casual, unscientific process, such things can be dangerous because people still accept statistics at face-value usually.
In summary, yes, people should learn statistics but it requires a lot of hard work to develop the tools and analytical skills to do it right. Like calculus, it can require many courses which can equate to years of study. Like anything, the skills involved also tend to get better with time and increased knowledge. At least an extra course or two beyond what is taught now would go a long way, but people really need to emphasize more than stats as repeating some processes of functions, formulas, proofs, and such, and more as a way of thinking and set of tools. Stats should teach you to be very paranoid and skeptical, and verify what you see in the world around you as best you can before you believe it. Once you do that, like the parent says, you will see so much clearer what is true, false, dubious, could be better, etc. in the world around you.
For students who aren't going to learn calculus, I wonder if a better introduction to prob and stats (and perhaps math in general) would be through simulation and visualization rather than learning formulas by rote.
I had the opposite problem, which was that I learned stats as a bunch of proofs with practically no applications. I only use fairly basic stats today, but I often test my conclusions by running some sort of monte carlo analysis. I graph everything.
It sounds like a lot of what you're recommending could be described as teaching general quantitative scientific methodology.
This is the general problem with math education in general. Just equations. Solving equations is relatively easy. Coming up with them is where the real challenge and innovation is.
Where I went to school there was an undergrad stats sequence for engineers, which was mathematically more intense than the general stats sequence. But it still didn't use much, if any, calculus. And I think in general I'm on board with most here that believe that for most people who aren't going to be engineers, a year of stats is more valuable than a year of calculus.
Oh I agree with that notion, it's just hard to get a lot out of stats without a large volume of it and a lot of other types of math as tools. Some are pre-requisites, while others are simply useful for applying in conjunction/situationally.
I still feel like calculus is one of the more valuable types of math. It really does teach you a lot, but like anything, it requires a good teacher. I find it all ties together for me in the end, like when doing graphics programming and games, I use a lot of trig, geometry, algebra, and calculus even, and stats some (and even more for business-type stuff).
I don't think it's possible to make the assertion that one branch of math is more relevant than another, given that the set of students have diverse interests.
From a computer science, more specifically programming, background I find linear algebra, logic, calculus and the combination thereof to be more insightful when handling abstract ideas.
I fear that if too much focus is put on statistics then we many end up with another generation of heavy statistical machine learning (and programs that consume high amounts of power to function).
But doesn't understanding statistics require a solid understanding of calculus anyway?
Or do you mean statistics as a bunch of applied methods, where you follow some recipes without understanding the math behind them? It's taught that way even at university level, but I can't see how this could be helpful in the long run.
I think we teach Calculus wrong in a deep fundamental way. We should incorporate more high math concepts earlier into education BUT people are always afraid of anything past 4th grade math!
"A minority of students then wend their way through geometry, trigonometry and, finally, calculus, which is considered the pinnacle of high-school-level math.
But this progression actually “has nothing to do with how people think, how children grow and learn, or how mathematics is built,” says pioneering math educator and curriculum designer Maria Droujkova."
...
"“Calculations kids are forced to do are often so developmentally inappropriate, the experience amounts to torture,” she says. They also miss the essential point—that mathematics is fundamentally about patterns and structures, rather than “little manipulations of numbers,” as she puts it. It’s akin to budding filmmakers learning first about costumes, lighting and other technical aspects, rather than about crafting meaningful stories."
Why is calculus the pinnacle when I actually feel that pre-calculus would be better students even make it into middle school? These concepts of patterns and structures is the foundation of not only math but also of language and reading.
IME the #1 problem is the failure to teach anything WELL. I took AP Calculus in high school, and today, as a 35-year-old, I'm still continually surprising myself with the problems I realize must be related to Calculus-- but nobody bothered to tell me. Things like the 'ol speed vs acceleration vs jerk kind of problems.
All calculus ever taught me was to apply these formulas to those problems (for some reason) and that dx/dy meant derivative of x with respect to derivative of y.
It boggles my mind that nobody simply bothered to tell us what a goddamned derivative was; what the word meant, and what real-life examples were. The "with respect to" was the most confusing and least useful concept. If we knew what the things were in the first place, the fact that one was "with respect to" the other would have been obvious. And "with respect to" could have been replaced with "relative to" in our minds, because we'd know what the things were.
Calculus seemed to be entirely repeating nonsensical mantras and applying impenetrable methods to problem sets and receiving an answer out the other end.
I'll also echo calls for more time on algebra. Surely the problem with algebra was that I wasn't a devoted enough student to repeating problem sets ad nauseum, but I just never got the 'knack' for it. I understood what I was supposed to be doing, but I was never satisfied with the answer "you'll start getting a feel for what to simplify to what after you do enough problems" as an explanation. It was frustrating to not feel like there was a concrete process to it all.
This made later Calculus and Statistics hard for me, because many exams would simply be 2 enormous problems ; the calculus I could get through but I couldn't simplify the problems quickly enough to be able to apply calculus or statistics at all.
When calculus is taught badly no one knows how to apply it even when the problems are staring them in the face.
I TA'ed physics at one of the top Ivy League universities in the US, and what I found amazing was that we didn't require calculus for our introductory mechanics course. You don't get into these universities without taking all the highest level courses in high-school, which means that almost everyone in the class must have taken at least one semester of calculus. But we still avoided the simple v = dx/dt calculations, the logic being that physics was too difficult to combine with the math they'd all learned years before.
My friends in economics told similar stories: "Don't worry, we don't have to use calculus to calculate marginal cost, we can use 'the midpoint method'", followed by 20 minutes of explaining an arcane nondeterministic procedure to approximate a derivative.
Math should be motivated by science and modeling, otherwise it's just an exercise in diddling numbers.
> Math should be motivated by science and modeling, otherwise it's just an exercise in diddling numbers.
I like a lot of what you said, but I do disagree with that point. Immediate application of mathematical concepts may be gratifying, but making it the _only_ motivating factor behind maths can result in students who conflate the underlying machinery with its application [1].
Taking a pure maths course in undergrad with related subject matter prior to one of my Controls classes made the class substantially easier to understand. Learning the abstract concepts beforehand let me see the common applications more quickly than others.
I think the biggest difference is that when I learned some Calculus through Physics, it was a little more difficult to go from a concrete basis to a more general one. My understanding ended up being based on analogies to other concepts until I went back and covered the theory again.
[1] This is all my own opinion, I'm not an educator so the most I can do is pull from my own pedagogical experience :)
Dude, the "with respect to" clause is a pretty deep idea in calculus that really matters in multi-variate but not so much in uni-variate.
I keep seeing how society needs better math education, whether it be statistics or calculus. Well no shit. Unfortunately those are extremely hard subjects to teach in high school. I'd argue statistics is harder to teach than calculus. In calculus you usually derive things by following the formulas. In statistics there isn't always a formula, logic, or path you can follow.
I took both AP Calc BC and AP Stats in HS and I'll tell you I learned a lot of calculus then and am learning a lot of statistics now (I'm 32).
See my other post, but I have to agree stats is incredibly hard to teach at the high school level. I also see the problem with grandparent's failures at applying math.
The huge problem as I mentioned elsewhere is that stats is really applied math. You do learn a lot of regurgitation-style stuff - formulas, proofs, blah blah, but it's all useless without applying it. Moreover, stats draws from so many different areas in math and requires analytical skills that are closer to many of the things scientists emphasize in learning to conduct studies and experiments. This makes it almost unsuitable to teach the more valuable parts of stats at lower levels because the students can't possibly be prepared.
At best, I think you can teach some general things about stats and the mentality I described of being analytical and skeptical. Unfortunately, stats is just really hard for anyone to properly and comprehensively learn who isn't going to be able to invest a lot of time both learning pre-requisites and then all the different areas of stats. It's like learning to be a carpenter and only understanding how to work a hammer, but not a saw, measuring tape, or anything else.
>BUT people are always afraid of anything past 4th grade math!
Then the solution is simple! We teach calculus to 3rd graders, then no one will be afraid of it!
But seriously, I agree that calculus is taught way to late. I wasn't introduce to calculus until college, and if I had been even by senior year of high school, I would probably be a mathematician right now.
In the US, problem #1 is that it is very hard to find people who understand Mathematics sufficiently well to teach it effectively, who are willing to work for the salary typically paid to a teacher.
For example, my 8th grader is being taught "Higher School Algebra" by a teacher who last year taught 5th grade and by all accounts has to leave the classroom often to get tips on the material from the 7th grade math teacher.
Spot on. Both of my parents were high school math teachers (for most of their career). They essentially forbade me from going into public education, which is what I thought I wanted to do when I was in high school.
Boy, am I ever glad!
I am much more financially comfortable, enjoy the hell out of my job, and don't have to deal with the annoyances and politics of being a high school teacher. Would I be technically capable? Absolutely. Would I be good [enough] at it? Probably. Would I do it for a mid (or even high) five-figure sum, even with the good retirement package? Not on your life...
Yeah, the line in the article about demand for calc teachers outstripping the supply suggests an obvious remedy...
Unfortunately, the fragmented (ie, redlined) us public school system means that higher teacher play is impossible in lower income areas. So private schools will pay more for the calculus teachers, and the low income public schools will continue multi-classing pe teachers for the task. And in another twenty years we will continue to lament the increasing lack of social mobility in this country...
> Unfortunately, the fragmented (ie, redlined) us public school system means that higher teacher play is impossible in lower income areas.
The lowest-income school districts around here pay higher salaries than the nicest districts by a solid 20-30%. Most teachers with options still won't work there because the environment is frustrating, unrewarding, dangerous, and toxic.
the other criteria that kicks some people out are the requirements for a K-12 "teaching certificate". if you don't get a degree in education, it can be difficult and time consuming to get a teaching certificate. a friend of mine some years back desperately wanted to teach high school math, and had a bachelor's in pure math, tutored for years and was well loved, but couldn't surmount the teaching certificate hurdles. so now he works in software QA.
I think that a student not well versed in how to perform calculations will not do well with the theory. Last night in the calculus class I am currently teaching a student could not correctly translate the fact that f is increasing where f' > 0. This on a problem where f' was linear. He just stared blankly at the inequality 2x-3>0. Anyone well versed in this sort of calculation is not going to fumble around at this problem. He thinks calculus is the problem when it's his knowledge of algebra that hinders him.
I'd be very interested to hear your opinion about a book I've been working on. It's a 3-in-1 combo, math+mech+calc, I wanted to teach mechanics and calculus in an integrated manner and I realize, from 15 years of private tutoring, I need to review all the high school math material because many students don't know the basic.
Please drop me a line if this sounds interesting. Email in profile.
"“Calculations kids are forced to do are often so developmentally inappropriate, the experience amounts to torture,” she says. They also miss the essential point—that mathematics is fundamentally about patterns and structures, rather than “little manipulations of numbers,” as she puts it. It’s akin to budding filmmakers learning first about costumes, lighting and other technical aspects, rather than about crafting meaningful stories."
I used to think this until I had kids and started teaching them math. When you know math you can abstract it away and really focus on the concepts. But you don't realize how many simple calculations are done when abstracting these concepts.
> BUT people are always afraid of anything past 4th grade math!
The problem is that elementary and middle school math teachers can't teach except out of a book. Your child's math teacher isn't teaching your child math, she is just the proctor for their math textbook, which isn't teaching them anything at all really.
This is not a dig on (all) teachers. These workbook-driven curriculum are a band-aid for bad teachers, and all they really do is harm good teachers and students.
As an alien to the US education system, I am always completely confused by these 'pre-' course designations. What on earth is 'pre-trig' or 'pre-calc'? I mean, I got all the way up to undergraduate level in mathematics and I never needed to learn 'pre-' anything.
They're not always required, but they try to bridge what you already know (or are supposed to know) with what is to come and move at a slower pace. It's also sometimes like an extra option for the less mathematically inclined who are still required to take such a number of math courses. On the other hand the only 'pre-' course I ever took (I tested out of pre-algebra somehow) was 'pre-calculus with honors'. I don't remember it much but I think it was mostly reviewing the harder topics from intermediate algebra aka 'algebra 2', which included trig (no separate trig course in my system), introduction to limits, and toward the end differentiation and maybe discrete summation. The 'honors' variant was also geared toward students who were planning on taking the more comprehensive 'AP Calculus BC' variant rather than the 'AB' variant.
Yes. In Alberta, all you care about is the rate of change of the the price of oil and computing the volume of manure, so only the basics about derivatives and integrals are needed. In British Columbia, you have to worry about how to fit your oddly-shaped modern furniture into your tiny Vancouver apartment and find the region of convergence for your mortgage, so more advanced techniques are necessary.
Hadn't thought of it like that before. :) A bit more usefully than the "one is harder one is easier" distinction, you can kind of think of the letters corresponding to semesters in a college course. A would be an introductory level (and possibly the only exposure a student might get if they're not interested in math but have to take a course), B would be a semester of Calculus 1, and C would be a semester of Calculus 2. Since high school the class goes year round you're still at a slightly slower pace than college, but if you do well on the AP BC test at the end many colleges will waive Calc 1 and Calc 2 from your requirements, whereas the AB test would only let you waive Calc 1. (A few other AP tests have different letter levels -- for physics only the C level seems to count and that's just basic Newtonian physics with very little calculus, though there is an optional E&M portion I never got to that may approach the Maxwell equations by the end... For computer science they used to have an A and an "AB", but they've dropped the "AB" since I took it. The "AB" basically got into actual algorithms and data structures, the A portion was basic programming and OOP heavy. All in Java.)
So I went to school in a country that is notorious for forcing kids to specialize early - the UK. In my age cohort, at age sixteen we had to pick just three subjects to study for the next two years. I chose 'pure and applied mathematics' as one of those three subjects. Some people chose 'pure mathematics' and 'applied mathematics' as Two of their subjects. I guess, vaguely, that doing the expanded applied maths would have included more calculus (not sure what that might have entailed - more second order differential equation stuff maybe?) - but there was definitely no point where I was faced with the choice of 'should I do a bit of calculus, or a lot of calculus?'. 16 year olds are not good at knowing how much calculus they need to learn. I was pretty happy just choosing 'do I want to learn some mathematics, no mathematics, or a lot of mathematics?' - and leaving it up to the school to figure out how much calculus to include. Based on my career since, the amount of calculus included was, it turns out, 'enough'.
Worth noting that the Scottish education system is different - people do 5 or 6 highers at 16/17 and those are used for university entrance rather than 3 A-levels at 17/18.
Scottish first degrees are 4 years rather than 3 in England - presumably to allow for this.
The idea of a "pre-" course is that you cover ideas needed for the later course. For example, a "pre-calc" course typically covers the sort of algebra that you would need in calculus but not actually integrating or differentiating anything. It's a matter of naming. You could call it "Algebra II" or "pre-calc".
I'm already baffled by the distinction between algebra, trig, and calculus.
Where I live, it was just called 'math'. The first time I encountered algebra was in university, and that started with groups (a set and an operation on pairs of items in the set)
As to the 'pre': I have a masters in math, and lots of what we did was in some sense pre.
For example, calculus on real variables showed you how to define differentiation and integration and how to rigorously prove lost of theorems (pro tip: if you can't make heads or tails of an exam question, at least write down "let epsilon > 0")
Next, we basically went through the same set of theorems but in multiple dimensions, but showed that those rigorous proofs had glaring holes, but hey, now you know better. Also: infinities are weird (pro tip: the area under a 2D curve can depend on the order in which you add up its parts).
Next, calculus on complex numbers extended that and showed that those rigorous proofs had glaring holes, but hey, now you know better (pro tip: every contour integral you compute at an exam has a value of plus of minus 2n pi, with n typically being plus or minus one)
For math majors, the next step was measure theory: that definition of integration we started with? Forget it. Lebesgue came up with something that's better. And by the way, those proofs we told you about have glaring holes (let's spend a few hours proving that cutting a rectangle across a diagonal gives you two parts that each have half the rectangle's area). A few of the functions that we proved cannot not be integrated can be, if you use this new definition of how integration works.
That seems a waste of time, but I don't think it is. Very few people can jump in at the deep end and survive; the first iterations are necessary to prepare one's mind for the next steps.
Given the above, I think you will have had various 'pre-' things, even though they weren't named so.
Same experience here. I went to international school, so I often ended up with American friends who'd talk about Algebra 1/2/3, pre-calc, trig, geometry, etc.
Makes no sense to me. All those things are related. There's a picture that shows you why Pythagoras' Theorem is true. And why the difference of two squares is never prime.
You'll have had a whole summer holiday between trig and algebra if they're separate things. Surely they are taught as a kind of whole?
I tried to post this yesterday but was getting rate-limited.
After Geometry we had Algebra 2 (quadratics, etc.), then Advanced Math (trig, pre-calc). To my detriment I made up "pre-trig" to illustrate being thrust back into a cold curriculum after (what I felt were) warmer concepts in Geometry.
We should teach early calculus along with entry-level SR/GTR instead of rehashed Newtonian crap. Unite math and physics for a while, and kids might actually see the point of it.
Exactly. Physics becomes so much simpler with calculus. Once you realize e.g. the relationship between position, velocity and acceleration you don't need to memorize 20 different equations for all the different flavours of problems that are basically down to integrating or differentiating ...
I held off taking physics in college until my senior year because I dreaded the 3 hour labs. By the time I got there I had 3 levels of calculus (and linear algebra, and plenty of proof based CS classes) under my belt, and it turned out to be one of the easiest classes I've taken.
Like you said, I didn't have to bother memorizing anything, because everything could pretty much be done with simple integration or differentiation.
But "you need to learn the hard way everybody else did it before X formula/equation was developed!"
I agree it's nice to know the logic and background that is the foundation of how we got _here_, but sometimes the older details aren't really as important and take away time from students learning the modern, more useful techniques and concepts.
One of the big things about kids and math these days is that you have less and less of their attention. If they see it's getting too hard, most don't take the challenge and they just put less effort in.
I couldn't disagree more. It gets too hard when you're asking students to memorize arbitrary equations that make no sense to them. It gets too hard when you give them 30 problems that look identical and they do 3 or 4 and figure that's good enough because they know how to run numbers through the meat grinder of a formula.
I don't see how students can truly learn if they don't understand the formulas they're applying and their relevance to the real world.
Yeah, I might have been guilty of some hyperbole. I remember a nasty integral on the Linear Systems midterm. It probably would have taken an hour or so if I didn't make any mistakes; but you could transform it and solve it as a linear equation in 10 seconds.
That was the point of the class, that there area two kinds of problems, linear problems and problems you can't solve.
I'm not in the least suggesting that an engineer shouldn't go through three semesters of Calculus, Linear Algebra+DiffEq and a Discrete Math course.
The first sentence in Your last statement in your conversation to your daughter is profoundly wrong. Your daughter was not in any way made to better understand fractions by this interaction. By the way, .3333 is a rational number and therefore equal to a fraction.
Your idea did not translate into English. I think you are trying to say that most decimals we see in real life are approximations, not exact measurements, while fractions are exact?
I have a 5 year old, and I was talking to some acquaintances who also have a kindergartener. The kids are learning math, by playing with manipulatives. It's great; they're learning mathematical concepts, and they have no idea anyone in the world doesn't like "math". They don't know what "math" is, they are just starting to discover that the world is made up of interesting but identifiable shapes, and that numbers have some really interesting patterns as well.
These parents, however, don't see anything they recognize as "math". They're afraid their kid is going to fall behind, so they're giving their kid worksheets at home focusing on two-digit addition and subtraction. Their kid doesn't like math already, because at home math is just writing things on a piece of paper with little meaning. This is sad to me, because there are so many interesting things you can do with your kids at home to help them develop their mathematical understanding.
Our educational issues with math start young, and they come from adults. I'm fortunate to work in a small high school where I get to treat each kid individually. It's wonderful to meet kids where they're at, and help them move forward and start to enjoy math again.
I've often thought about inventing math toys, e.g., group theory for toddler: S_3 = triangle toy, S_4 square toy, etc. I'm sure there are many advanced math topics that could be turned into toys. No symbols, no equations, just toys, but subtly you're getting kids familiar with important math structures.
AP Calculus was the most useful course I took in high school. It was rigorous, and I scored high enough on the exam to get two free college classes and start freshman year at the sophomore level, which left room at the end of my college career for more specialized courses.
There should be some vetting process for the students who want to take AP classes. A lot of the students in my AP Calculus class had no business being there. One guy got a 4 out of 50 on a test once, another guy didn't even take the AP exam, and another guy just put his head down on the desk during the AP exam. But this is really a problem for individual schools and has nothing to do with College Board.
> another guy didn't even take the AP exam, and another guy just put his head down on the desk during the AP exam.
I don't know anything about these people but there's two "good" reasons for this:
* Some colleges don't take certain AP credits e.g. the college I went to didn't take AP Bio credit, so I didn't take the exam for it when I took the class
* Some high schools give GPA boosts for AP classes. For example, an A in an AP class was worth a 5.0 vs a 4.0 in a regular class. Some students may think it's worth it even if they don't do that well
In general I agree with you because my AP Calc class wasn't rigorous or interesting at all. It was "here's how you do X, now do X 100 times for homework and take a test on it".
> A lot of the students in my AP Calculus class had no business being there
Maybe I'm cynical but this was the case at my college Calculus class as well. The entire back row was basically people cheating. There'll always be lazy people in freshman and sophomore courses.
> Tuition only covers a portion of a student's cost to a university. I think it's around 30% at mine.
Where are you getting that from? Are you just dividing the tuition revenue by the university's budget? Because the university is there for a lot more than educating undergrads.
Most universities aren't primarily focused on research.[1] They are there to educate students. Research does occur at every university, but looking at their research budget as a fraction of their whole budget will show what sort of institution it is.
Even if you subtract the research budget out first you will still arrive at a figure less than 50% for most universities.
> Most universities aren't primarily focused on research.
Where are you getting "many" (rather than "some")? The page you link to discusses colleges in addition to universities, and it's well known that colleges exist primarily to educate students.
> Even if you subtract the research budget out first you will still arrive at a figure less than 50% for most universities.
Universities are composed of colleges in USA. You seem to have more of a 'Canadian' definition, where college means something like "community college" in USA.
I am not using such a definition. I am an American, and I am thinking of American places like Kenyon College, Hamilton College, Harvey Mudd College, Union College, Claremont College, etc., etc.
> Universities are composed of colleges in USA.
This is pretty rare terminology in the US, being mostly found in Ivy league or other old schools. Most "colleges" in the US are independent institutions.
You continue to not back up your claim. That link talks about "all post secondary institutions", not universities. Why don't you just edit your original comment to remove the claim?
I am using statistics that I know to be true about my university (University of Alaska Anchorage). As far as I know UAA is typical for state run universities.
I am having a hard time finding aggregate data online that supports (or contradicts) my claims on a national level beyond what I have already provided. Unfortunately the edit timer has already expired for my comments so I can't delete or alter them. If you can find reliable data that contradicts my claims reply to my root post and I'll be happy to upvote it.
At my university, they would usually let students sit in on any lecture classes they wanted. But students taking classes they aren't qualified for isn't the most efficient use of educational resources
>another guy just put his head down on the desk during the AP exam
That's too bad. Either his parents were paying the cost of the test (I think $86 per test when I was in school) or he was getting a paid-for test from some scholarship or another.
> There should be some vetting process for the students who want to take AP classes.
Is this not usually the case? My high school used to make you fill out a short form and make the case for why you should be allowed in the AP class, and my senior year I had to go in and speak with a counselor who was concerned that I was taking too many. Sure, it was a pointless hurdle for me personally, but I can see how it would be a good idea for students who don't realize what they're getting into, or who are trying to take the course because their parents want them to.
Depends on the school district and size of the school. For example, Texas high schools are for the most part huge and can accommodate anyone who wants to take AP classes without question. We did check with our counselor but they were more than happy students took AP classes.
It wasn't a size thing as much as for "protecting" students from classes they couldn't really handle. I can't speak to whether that was necessary or not; it's always worked out very well for me to try things that I didn't think I could achieve, but maybe that doesn't generalize to everyone.
Same here. I took both AP Stats and AP Calc BC in high school and I'd be more worried about stats being taught wrong to a bunch of teenagers than calculus.
I think linear algebra is a much more valuable and general-purpose math tool. I'm not sure how feasible it would be to teach linear algebra topics "properly" to high schoolers, but even a half-assed linear algebra would be more useful than calculus.
The connections to geometry, general systems thinking, formal math methods, and countless applications of linear algebra are just the kind of thing students need to get them interested in learning more math.
Anyone interested in LA an its applications should check out my upcoming book, the No bullshit guide to linear algebra availale on pre-order here https://gum.co/noBSLA (it's almost finished; just beefing up the problem sections).
They taught me about matrices and vectors and common matrix/vector operatoins in pre-calculus and even earlier in a shitty public school.
Calculus needs to happen in high school so that people are more prepared to take it again in college. For me, calculus 2 was the hardest class in college. This is coming from someone who went to a math grad school.
Some people need to study way more for calculus than for any other class in high school or college.
LA wouldn't be any harder to teach than Calculus. In fact, if you wanted to teach proof based math that isn't Geometry, it's probably the easiest subject area.
I don't think you can get rid of Calc in HS though. It's just too useful. I think room can be made for both subjects, though.
> Too many students experience a secondary-school calculus course that drills on the techniques and procedure that will enable them to successfully answer standard problems but are never challenged to encounter and understand the conceptual foundations of calculus,” he said.
As I get older I've realized that I learn better when I understand the concepts about things. I then get excited to apply what I've learned into practice.
I think that if I learned about "why calculus", what problems did it solved at the time it came around, and how we use it in practice I would have been able to grok it quicker and deeper. I didn't end up getting above a 3 on the AP calculus exam.
I did however get a 5 on the US government exam. I had a teacher who did a great job throughout the course having us work out essays and testing arguments in class to help solidify our knowledge as opposed to memorizing facts.
Conceptual background is so important for understanding something. I was lucky to have a great teacher for BC calc, and I got an easy 5 on the exam. The concepts make sense, they are consistent, and they build on each other, so with the right progression of ideas students can get a lot farther than they would otherwise.
I remember once when when driving I was pleased when it occurred to me that I should be seeking to minimise the third derivative of distance with respect to time for the comfort of my passengers. That was the sum total of calculus's contribution to my adult life. Not sure it was worth it.
One of my calculus classes in college involved calculating the required spline for a curve of a theoretical portion of interstate in order to meet the interstate requirements/regulations. It was pretty interesting.
Don't quote me on this since I don't remember why I believe it, but I think it's part of the legally mandated design requirements for the US interstate system.
No I'm sure there are requirements, but sometimes there are also space or budget constraints (I assume) since some curves are definitely safer and easier than others for the same highway (I-95 for me).
For lower level math, my kids seemed to revisit the same subjects over and over from k-9th grades. For example, mean, median and mode, year after year. The first few years it was just robotic problem solving. After a few years, they started to get a broader understanding. Maybe some people need to take several passes through the higher level stuff too.I guess the problem is that there is not enough time.
What infuriates me, looking back on it, was simply how much of my time the educational system was allowed to waste. Thousands and thousands of lost hours, some of them filled with pointless busy-work, but many more just squandered on inanity. At least some of the teachers would let you finish the daily penance quickly, and sit quietly in the corner reading something interesting. Too few, though.
I look at this way: early education focuses on learning to read and write and do arithmetic. The basic purpose is to give you tools whereby they can continue to educate yourself throughout your life. So basically, our early education aims to put our future education into our own hands.
Let me qualify the above a little. You might ask, well what are the purpose of high-school teachers or university lecturers? Once we are given these tools shouldn't we just be given the books to progress on our own? Hopefully, teachers are people who can take a complex subject and guide us towards an understanding of that subject. To me teachers are like guides through a jungle of knowledge. We need a guide so that we don't get lost. For me, a teacher is part guide and part "composer of problems". I consider teaching to be guided problem-solving. When I am teaching a course, I pose problems, whether it be in Math or Chemistry, and guide the students as they try to solve those problems.
And as for your comment about "wasted time", I am not sure what to think about this. The difficulty with learning is that we need repetition and reinforcement. Yes, I see a lot of pointless repetition, but we do need some repetition. I am presently struggling with improving my chess skills, as well as trying to learn to program an Arduino. Mastering these skills require a lot of repetition in order to cement that knowledge. Wish me luck.
I believe there's decent support for the idea that you could just teach them nothing until they reached the age of understanding, and theyd pick it up pretty much just as quick without the years of rote.
A U.S. elementary school principal in the 20th C. got his school to try leaving out any math in the first three grades (iirc), and the kids reportedly came out fine. I can't remember his name, and I'd like to be reminded too.
Interesting. I've been thinking about education and a few dynamics of it, including human cognitive development and individual differences.
1. Young children seem to have the biggest capability in learning languages.
2. There's the John Stuart Mill example -- homeschooled by his father in the classics and philosophy. Interesting biography (Wikipedia has a fair account).
3. There are children who are naturally skilled in art, storytelling, maths, sciences, music, etc. An opportunity to work to those strengths might be useful.
I've also noticed that a short period of instruction at the appropriate time often seems disproportionately effective.
> 1. Young children seem to have the biggest capability in learning languages.
Is there any evidence that this is actually due to their capability? I would say it's far more likely it's their environment. IE, people are constantly talking to young kids. They simplify what they say to make themselves understood. And children have a huge incentive to learn to communicate - to get what they want from parents, or just understand the world around them. The same isn't true for a grown adult trying to learn a second language, in isolation, that they don't really need in their day to day lives.
I'm also aware of a few other age/skill type relations.
Musical ability frequently manifests early -- by age 4-5 in numerous historical cases.
Talents requiring physical coordination -- ballet and gymnastics -- around age 9-10 by Russian tradition.
Maths and scientific curiosity perhaps early adolescence.
Complex logic and activities requiring a fully developed inhibitory function (avoiding reflex and rash actions) possibly early 20s.
I'm suspecting other forms of more synthetic (as in: assembling from parts) logic might not develop fully until the 30s or 40s. Other skills might similarly not develop until later.
My guess is that math teaching can be a benefit to anyone at any age, but the usual curriculum on average mainly teaches people to hate math. (That's what's happening with my niece in grade school right now.) For some kids arithmetic may be boring, but at least it's easy, and they suffer less of this damage. If you start it later when they're readier, more of them will be open to learning a little actual math, not just gritting through baffling seemingly-arbitrary procedures.
Montessori schools are said to be better on this score, though I haven't looked into them.
I've been looking at John Cleese's presentations on creativity, and suspect he's very much on to something with the idea of play and humour as key to real learning. I also think that there are approaches which take this too far (video games and such), though I've learned a number of things from unexpected places.
Neal Stephenson's Cryptonomicon, The Baroque Cycle, and Anathem, for example, are all explorations of ideas. The first two of business, banking, and money, in contemporary and historical times, respectively. The latter covers a great deal of historical thought.
Better authors of topics often incorporate humour in their writing -- someting the past few generations of academic literature would do well to consider. OTOH, it's a cure for insomnia.
I have a 5th and a 6th grader. I'm going to buy them an educational copy of Maple soon, so they can have a beautiful platform to plot 2-D and 3-D graphs, solve equations, do derivatives and integrals, and factor algebraic expressions.
I want them to do all this in high school, whether or not they take Calculus, so they can get an intuition for it. Then they can memorize the trig rules in college.
You should also consider SymPy, available as a webapp http://live.sympy.org/ or in jupyter notebook form.
Plotting function graphs is a bit harder (matplotlib) but overall I find SymPy to be much more logical and easy to grok than Maple's weird syntax. Here is a short tutorial on how to do basic math using SymPy: https://minireference.com/static/tutorials/sympy_tutorial.pd...
I love Jupyter. I set up pydot in a notebook, and made high level functions to create and print trees and graphs. My 10 yr old picked it up quickly while we killed time at his brother's swim meet.
But, I'd love for them to learn to use a CAS engine. I appreciated, in college, being able to quickly factor a polynomial, solve an equation, and find derivatives.
Sample size of one: I think I understand calculus, the needs of calculus, and how it is profoundly meaning to understanding things around me. Can I do calculus? no. I couldn't do calculus because I really couldn't do algebra. I think that if I had been taught the concept and relevance of calculus earlier, and only had to worry about the execution of calculus later, I would have been much better off.
understanding what derivatives and integrals and trig functions mean should come before we are forced to execute their use, or learn proofs, or even really go beyond basic notation.
I'm skeptical that a person can really understand why x^(2/3) is not differentiable at x=0 but has an absolute minimum at x=0 without understanding the algebra. I think one can get an idea of the concepts without knowing algebra but I doubt one can understand calculus without knowing algebra.
We always teach the meaning of a given mathematical concept before doing using them or doing calculations. It's just that true understanding comes from the doing the calculation part.
Not the parent, but I had the same problem. I understood that kind of algebra but could never figure out how to simplify things well, which hamstrung me in Calculus. Also, they never bothered to explain the purpose of any of the equations in calculus.
Personally, I have always wondered about the prominent position trigonometry and geometry get.
As it is typically taught, it is the most useless thing, the time spent there would be much better spent on algebra, calculus.
From basic theorem of algebra flows trigonometry and a lot of number theory, while from calculus with some linear algebra flows geometry.
For a lot of people, geometry is their primary or perhaps only exposure to proofs. Not that everyone needs to understand proofs in their day-to-day lives, but it's a good mental discipline to at least be exposed to, and geometry proofs are fairly straightforward and relate to things that many people will have a good intuition to convince them that the thing proved by each correctly-constructed proof is really true.
Also, geometry gives students something concrete to visualize, and so might be easier for some to understand than more abstract kinds of math.
Maybe trig is over-emphasized at the high school level, but it is pretty useful in a lot of real-world situations, and it's easy to apply without having to know a lot of other stuff.
From what I can tell, proofs are gone. I got my daughter all excited about geometry, by promising her that she could do proofs. The curriculum was almost exclusively focused on problem solving based on remembering formulas. She went ahead and did all of the proofs anyway.
IMHO, high school math misses the one course that would be most useful to the students. Statistics. While there are a LOT of statistical models that aren't especially useful in every day life, knowing the most common ones can be a big help.
Of all the math classes I took in college the only one I use on a regular basis is statistics. Thinking back on high school I spent a whole lot of time on basically calculus prep and in the end I hardly ever take a derivative or integral of anything.
I always found trig and geometry to be the most interesting and useful part of math. It maps pretty well onto real things that you might have to do. Maybe this is more relevant in the rural area I grew up in, but geometry and trig is really useful when you're trying to do carpentry, or figure out if you've got room enough to chop down a tree, pathfinding on a map, etc.
Trig is a pretty useful subject for the skilled trades. Framing a house, as mmmpop notes, but also things like plumbing and cement work. If you don't go on to college, or you don't major in engineering or math, calculus is pretty useless. I've never once used calculus outside work, but I have used trig.
So I guess you've never framed up a house have you? Basic trig is so incredibly important to anyone that ever desires to build a thing. Don't you remember learning the unit circle in Calc 2??
trig has seemed extremely important to me. specifically using sin and cos. helps a lot in physics as well as computer graphics and many other applications.
>This is not true for students who take Advanced Placement calculus
I hated BC calc. Bad teacher. I learned the same material at the same time in AP Physics, and loved it.
I think a related issue to the author's point, though, is that students are taught in a way that implicitly suggests that calculus is the pinnacle of mathematics. Which is blatantly wrong.
I have had the pleasure of pushing many female students into taking calculus but my success rate was about 50%. Each one of my children's friends LOVED STEM and were thinking about going into engineering. All of them said that they were the only girl in the class and were intimidated and the other said it was REALLY boring. Calculus should be easily made fun just exploring the actual concepts in real life similar to a good physical science teacher.
She was a firm believer in "teach to the test", "do these exercises and follow the magic rules of differentiation, shut up and just do it don't ask why it works".
I think this is just another example of standardized testing (specifically AP course exams in this story) almost creating a niche-like education for students.
Maybe that's an odd way to put it, but some students wind up spending hours and hours studying AP/SAT/ACT/etc.-specific materials (literally 500+ page textbooks for each type of standardized test). Sure this method of studying helps them achieve great scores on the standardized tests, but my impression as a recent student in academia (finishing college now) is that this creates an issue where students are almost "overly" dependent on more specialized areas.
I don't know if it's an issue of not having time to learn "street smarts" (so to speak) because standardized studying takes up all their time, or if it's something else. But there's definitely some improvement that can be made.
A pattern that you see almost everywhere is: imperfect incentives/objectives + strong competition = perverse outcomes
Over the last decades, we're increasingly seeing this in academia (publish or perish), the corporate world (boost the bottom-line at the expense of everything else), politics (win the election even if you have to burn your own party to the ground), and many other walks of life, as the world has gotten more competitive.
To fix this, you can either fix the objective (which may be very difficult), or you can reduce the level of competition.
This also causes some odd results. If the MIT/Caltech/Harvard/etc student bodies are generally less intelligent than you might expect and often boring people.
You see the same thing at most national level high school competitions. The surface details may seem very diverse, but underneath that they tend to have a lot in common.
Schools should adopt more dynamic curriculum, instead of a one size fits all, for individual students. Not everyone will grow up to be an engineer, but that doesn't mean those who have an interest in advanced mathematics should be held back. Encourage students to pursue their academic passions, and don't limit certain subjects depending on grade level.
A personal anecdote: I didn't start liking math until Calc AB, as subjects beforehand were presented as dry and were relatively easily so I didn't individually pursue mathematics outside of school. But by that time, I was already a junior and as a result, I often wonder what better math teachers and a non static curriculum would've done for my education.
So let me get this straight: kids who didn't study enough math in middle school have difficulty doing harder math later on, so they rote learn it to get into college, then drop out of math. Okay?
And the kids that studied math, e.g. because they enjoy it and/or have good teachers early on, take AP math and go on to do well in math in college.
Is this a Captain Obvious moment? When will people learn that there is no shortcut and they actually have to do the work, and pay a living wage to teachers?
It reminds me of the 'coding in schools' thing. Who exactly is going to take a massive pay cut to teach computing? And do we just give them a walled garden to code in?
It sounds like the issue is there are too many high schools offering classes in AP calculus which don't do a decent job of actually teaching the concepts to the students. Having a good teacher for these classes makes all the difference. Additional testing at the college level would help assess a student's ability level but it does not solve the problem of time wasted when the course must be retaken.
Another issue this brings up is it causes a larger gap to form between trigonometry and calculus 2. Where memorization of hundreds of trig identities builds on top of the identities learned in trig. If a student takes Trig -> pre-calc->AP calc -> Calc 1 -> Calc 2
They will have a much harder time in Calc 2. That is what happened to me. I had to take Calc 2 a couple times before I got it sorted out.
I think the answer is to get more qualified teachers in the schools teaching the AP level classes. There is a program in Alaska called ANSEP (Alaska Native Science and Engineering Program) who have this figured out. Middle school students who enter the program come to the university during the summer time and take math classes from university professors. Many of the students are completely finished with their math track (for engineering programs) before they start university their freshmen year. These kids are all the way through partial differential equations at 17.
The graduation rate for ANSEP students who enter engineering programs is around double the national average. There are many other important aspects to it. Including mandatory study sessions and living with other students who are also taking the same classes. Students are also given well paid internships in the industry during the summer time. When looked at as a whole, there is no doubt that they have figured out a far more effective means for getting students through STEM programs and jobs once they graduate.
Full Disclosure: I am good friends with a few of the people running this program.
I feared calculus, right up until circumstances forced me to not only take calc in high school but to take the hardest level of calc my school offered (AP BC Calculus, counted as 2 semesters of college calculus). I was greatly benefited by an amazing teacher.
I've always felt that calculus is one of the most important branches of mathematics for the simple insight it should hammer home time and again: An infinite amount of infinitely small things can and will add to infinity. All too often, humans fail to see the cumulative effect of an integral.
Every branch of math has these types of insights built into them. Calc the power of instantaneous change. Geometry shows the beauty of a proof - and if you scratch a little bit, the shattering epiphany that triangles are literally the same everywhere, always and forever. The reality is that many of these concepts are simply absorbed into our culture, and we fail to realize it until someone explains that it took centuries of work to formulate the concept of 0.
When it comes to education, I believe the struggle has far more to do with testing standards than the material. It's easy to teach to process, much harder to teach insight and further harder to test. In reality, none of this matters until your institution decides and commits to truly educating its students.
tldr - None of this matters until your institution decides to truly educate its students.
Short recap: Mathematics is not bunch of dull rules. Education system puts way too much emphasis on memorization of dull rules instead of problem solving and developing strong intuition. Calculus overrepresented and discrete math underrepresented in education system.
As a person who currently is rediscovering math from scratch, I find these articles very insightful. I rediscover math from problem solving/intuitive point of view rather than beating my head against the wall of formal definitions.
We can teach intuitive integral calculus in kindergarden:
Have lots and lots of ping pong balls handy, and try to measure the volume of everyday things in ping pong balls.
Exact results are not a requirement and in fact they don't matter at all for this early age, only the intuitive visualization.
We can teach the concepts of squares, cubes and prime numbers using the ping pong balls as well: Get for example nine balls and form a square shape. Squares are numbers that can form that shape. Similar for cubes. Factorization in two factors is arranging the balls in a rectangle. A prime number is any number that can't be arranged in a rectangle.
All this is intuitive, children can understand it easily, and it will help them in the future.
And all this is before they learn about equations and algebra, it's even before fractional numbers.
I have a specific problems with the HS math curriculum I experienced (as opposed to the general problems with the whole math curriculum I had before my junior year of college). Mainly, there was just so much wasted time it was unbelievable.
I actually cannot remember a single thing that I learned in Algebra II because I either relearned it in Pre-Calc or never used it again. Similarly, we spent the whole first semester of Calc BC redoing all of Calc AB. What a waste!
If I could, I would replace all that wasted time with Linear Algebra (with proofs) and Probability (throw in statistics if you want). Calculus should stay, but teachers should motivate it better (this goes back to the general problems I alluded to). Those two classes are immensely useful and teach a whole different modes of thought compared to the regular pre-college curriculum.
As someone currently teaching college calculus on the side this semester & works in aerospace engineering, I agree. The two math subjects that I notice most people are deficient in post-college are statistics, and linear algebra. In my opinion, we teach way, way too much calculus.
I grew up 3 years ahead of my classmates in math. I found everything up until calculus to be easy but elegant - algebra, geometry, probability, trigonometry. Much like programming, there were often multiple ways of the solving the same problem, and deriving a formula was feasible and logical.
In contrast, I found calculus to be mind-numbingly boring. It required memorizing and reciting formulas upon formulas. It was the first time I felt that math was tedious, and pushed me away from majoring in math in college.
Terrible teachers making difficult subjects impossible for students.
It's the same experience for my kids that I recall from my time. Industry can pay more, and outside the rare person who has come to teaching to give back, the instructors are those who can't.
Even when I was at Cal Poly, the only instructors in eng/sci who I recall being helpful were those from JPI that would come in and teach a class or two. One taught us three classes worth of material in course when he realized how much we had missed from our previous instructors.
We're building out an online solution over the next year to help prepare final year high school students for exams - with an initial focus primarily on Maths.
Would love to hear from people with thoughts around Math tools they feel would be valuable - especially taking into account current resources such as KhanAcademy, and how we could potentially provide something complementary.
Either comment here, or hit me up - louis@connecteducation.com.au
I have no idea why calculus gets such exalted status in the annals of highschool nightmares. No I am not bragging, humble or otherwise. I found calculus to be simple neat and logical, and not so mind bending. Linear algebra on the other hand was far less trivial; not the mechanics, the aha moment bit. Geometry.. that for me was genuine mind bending pleasure pain.
The best thing about learning calculus is that you see how everything you have learned so far - algebra, geometry, trigonometry - everything comes together and becomes equally useful and important. This is, in fact, what mathematics is all about.
I am not sure how much of this is specific to the American education system, as I am Canadian, and have not experienced it, but my own take after going through high school to my current graduate studies is that
1) We teach algebra, geometry, and their relationship wrong in many ways
2) We don't teach enough calculus at lower levels of mathematics, instead opting to prefer a "poor man's" algebra.
I'm sure there's too much to write on for one HN comment, but a lot of 1) comes down to how we teach "solve for x" type problems which are more interested in re-ordering and simplifying equations than we are about giving people a fundamental understanding of what our relations and equations actually mean. Often I notice that many students couldn't graph or draw or even begin to reason about what they're trying to solve for. If you give someone a graph and ask them to find the maximum point of a line on a graph, they can very easily point it out. The next step is doing it numerically, but a lot of times the relationship between what we can determine visually and what we can determine computationally is lost.
Which is sort of what prompts 2). You can't learn Calculus without learning limits, and Calculus places explicit meanings to relations that we can graph, and how those relations change according to specific variables or unknowns. Finding the maximum of a relation using Calculus is much easier than strictly using algebra for this reason, because we can make the relationships between a curve and it's derivative explicit. Most students cringe at the thought of Calculus because we place it on a pedestal and students just assume Calculus is the peak of mathematics, but we really need to introduce Calculus as more "normal" math earlier on, IMO. I'm talking basic tasks like derivatives finding the equation of a line tangent to a curve, or finding the limit as we approach a point on a curve, or even just (re)factoring equations so that we can plot them easier and / or take their derivative easier.
In almost every example I can think of where I learned Calculus, it was easier than building intuition for Algebra because there is less robotic crunching and more reasoning about what specific problems mean. What does it mean to take the derivative? Why are volume and area and perimeter related? I know in many ways it seems like I'm just advocating for thinking spatially or visually about mathematical problems, but that's part of what (I think) makes Calculus more approachable. You can get a lot farther with a basic understanding of Algebra and a very small amount of Calculus than you can with just Algebra alone. Should our curriculum be entirely Calculus? No, that's ridiculous; however we should ease up on the systems-of-equations type problems and obtuse word problems that require students to produce or remember all manner of expressions and formulae, and instead focus on building that first intuition of how we can use Calculus as a tool for problems that are much, much harder without. I expect that it would give Calculus a more realistic reputation, and would probably put off less students who are already giving up on math.
>Finding the maximum of a relation using Calculus is much easier than strictly using algebra for this reason
I recently-ish helped my stepdaughter with an algebra assignment where she was asked to find the maximum of some polynomial. I totally blanked on how to do this with only the tools that she was supposed to have. I didn't know how to do it or explain it without calculus.
Use a tool like Symbolab. It shows you the steps for different methods. Very useful.
Further, we should all remember to draw a graph as our first step. Symbolab, Desmos and Geogebra (my favourite) are all fantastic graphing software.
Also, for a quadratic if you know the zeroes (roots), it's half-way between them. So for the equation -(x-1)(x-5) = 0, the roots are x=1 and x=5, and the mid-point (maximum) is at x=3. Your graphing will confirm this.
Your stepdaughter might also have learnt that the mid-point of a quadratic ax^2 + bx + c = 0 is x=-b/2a. (This can be derived by setting the derivative equal to zero, but is generally just given to the students.) Her teacher may be expecting her to use that formula.
Alternatively the students might be required to determine and plot various data points using, say, Excel, and find the maximum this way.
The other point is that solutions often do not spring to mind immediately. That is why I ask students to send me their problems before our tutoring sessions. I often need time to think about them.
The fun of math is this exploration to find an answer. So try to get problems and give yourselves time (days) to explore them together.
If it was a quadratic, you could (in effect) complete the square and thereby get it to a form like
a(x - b)^2 + c
At that point, you get the answer by inspection. I.e., in this case, if a is positive, then the function is minimized when x = b, and the minimum value is c.
In lucky cases, fourth-order polynomials could be put in such a form. Odd-degree polynomials will not have a unique maximum.
I came to the US (where I have children in school) from Scotland (where I obtained my education). US K-12 Mathematics education drives me nuts. Peeve #1 is the idea that the subject should be divided into these seemingly separate subjects : Algebra, Geometry, Calculus and so on. These are not different subjects -- they're all deeply interconnected, and furthermore : observing the connected nature of seemingly disparate areas in Mathematics is one of the key insights to be had.
To be honest I don't really care what kind of Mathematics they teach: I just want it to be taught with passion and rigor such that the students gain insight and a better understanding of how the world is put together; how to solve problems and express ideas.
Maybe logic should be the first mathematical discipline to be taught in middle school. The notions of and / or and the representation of language in mathematical form?
Statistics is relevant to more fields than calc. It matters a lot more in business and in all the jobs where you use calc you will also have to use statistics. It's also useful for citizens to understand statistics better - and educating citizens is truly one of the purposes of public education.