Math pedagogy in US highschools

Pre-College Issues High School Life
1

HI all,

I’m very curious to hear what all of you think about this:

I’ve personally always found the approach to proofs in the US math syllabus is slightly strange (and I wish proofs were more emphasized). It seems like you do no proofs before geometry (which makes sense), then get a ton of proofs in geometry, and then do very little after that (other than occasional proofs of trig. Identities in precalc. or derivatives in calc. ) until courses like Real Analysis in college (or sometimes an intro. To abstract math or proofs class).

I’m presuming that there are some great proofs in algebra, precalc. , calc. (for instance, I proved the binomial theorem via mathematical induction on my own which was a lot of fun!!). You can also prove the rational roots theorem by plugging in p/q into a polynomial equations. I know some proofs are inaccessible (for instance, I know you need results from complex analysis to prove the fundamental theorem of algebra, so obviously, you can’t do that in an algebra II class); however, it seems like other proofs would be very doable.

I guess geometry proofs are very intuitive because you can visualize (whereas proving binomial theorem or the fundamental theorem of calculus is not at all intuitive).

To me, the wrong thing is emphasized within US math curricula. Many students ask when they’ll ever use the material, and they’re probably right; they may never ever need it again. That’s why in my opinion even if all classes don’t become intensive, proof based classes, math should be much more about pattern searching than just memorizing. Math was developed based on incredibly simple axioms into an incredible subject. I don’t think it yields much benefit to just teach the final theorems as formulas to memorize and can be off-putting to many students. Instead, we should emphasize less content and much more pattern searching and problem solving (finding conjectures and investigating if not fully proving them).

So some questions for all of you:

  1. Do you agree or disagree with the argument that the US math curriculum needs major change or overhaul?
  2. With what should we replace the current mathematics syllabus? How should we incorporate deeper problem solving instead of formula memorization?

I’m personally incredibly passionate about math (you can probably tell through my username :slight_smile: ) and have been thinking a lot about this. I haven’t been all that challenged via the regular curriculum but have found that math contests don’t appeal that much to me either (too much time pressure in addition to more tricks to memorize which aren’t fully understood). Instead I’m moving on upwards and taking courses on my own (such as linear algebra, intro. to real analysis, etc.). I’m very curious to hear all of your perspectives from both teachers and students.

2

Hm this caught my eye before I called it a night, so brief thoughts:

  1. The current system is flawed, largely due to what you’ve mentioned. I’ve had similar thoughts in the past.
  2. A major problem lies in the “accountability” aspect of teaching. Standardized testing and standards can easily say “know how to solve a quadratic equation” but can’t say “know how to apply general problem solving strategies”.
  3. It’s a positive feedback loop: the current math system creates students and teachers that largely believe that math is memorization and rote application. This in turn leads to resistance to reform. For example, the new Common Core had some more intriguing mathematics in it, but was met with harsh criticism.
3

Being the math nerd and enthusiast that I am, I love proofs and I usually try to prove why things work. Who needs to worry about remembering the quadratic formula or power rule when you can derive them? (Ok, for time reasons you don’t want to derive these constantly, but you get the point.) However, I think that many students would struggle with proofs so it’s better placed in advanced classes.

I mean, idealistically, all students would learn how to write proper proofs and understand the majority of the math they’re learning, but I don’t think that’s realistic. In my geometry class we haven’t done much with proofs-- what we did do with proofs was a joke-- but I heard a lot of students complaining that the proofs were so hard. Granted, my school only has one geometry (no honors, accelerated, pre-AP or any of that stuff), so that’s a factor.

I’ve often thought that it would be nice to have a rigorous, heavily proof-based math track at my school. However, I can easily see a scenario where a lot of honors-track kids want to be in the “best” math so it looks good for colleges, and then either get upset when they don’t get into the class (if it’s hard to enroll in) or upset that the class is hard because of the level of thinking needed (if it’s easy to enroll in).

TL;DR: I wish we had more proofs but I’m not sure it’s realistic. Maybe with a radical change in the way math is taught at the lower levels so more students are prepared.

4

I think the current system is fine as is. We might not realize it if we surround ourselves by intelligent peers, but many, many high school students will struggle to even read and comprehend what you wrote. The students who will benefit from your proposal are few and far between, and I think are better served by providing them with math competitions and special programs rather than by baffling the vast majority of students.

5

I love proofs. But then again, I’m a math person.

I will tell you that they’re brutally difficult to teach-- and I’m in a college prep school. It’s not a question of studying enough or even one of being “smart” enough.

Being good at proofs is like having a good sense of direction. Lots of very, very smart people have a bad sense of direction. (Like my mom. At 84, she’s brilliant. But if the guy on the corner paints his house, my mom will get lost.) It’s something that can’t be taught. It can be… “guided” for lack of a better word. But it can’t be taught. Kids can be guided to the point where they’ll get most of the proof (and the credit) most of the time. But if you can’t see it, you simply can’t see it.

Proofs are sometimes the Great Equalizer. Kids who have been “smart” for their entire academic careers flounder, and have no idea why. It’s not that they’ve suddenly lost their great work ethic; they simply can’t see the flow of the proof. And kids who have struggled for a grade of 70 in math their whole lives can’t understand what the fuss is all about, but all of a sudden they’re pulling 90’s.

It’s Russian Roulette-- there’s no way to determine who will be good at proofs and who won’t.

So I’m going to disagree with the premise of this thread-- even though I LOVE proofs.

As a math teacher for the past 35 years, I would much, MUCH rather see a strong emphasis on verbal problems at the high school level. Once they leave their last math class, they’re not likely to run into a lot of equations again. But they will, time after time in life, run into situations involving numbers-- verbal problems. And if we can teach them to organize and categorize information so that they can form that equation, they’ll have a real shot at solving the real life problem.

And, while I’m on my soap box, I would LOVE more emphasis on basic computation in elementary. I can’t tell you how many honors freshmen don’t know their times tables because someone handed them a calculator when they were in 3rd grade. So they get to factoring and simply can’t come up with the factors. Or someone asks them to estimate a discount, and they have no idea of which numbers interact in what way.

But in order for all of that to happen, we’ll have to pull some things OUT of the curriculum. As much as politicians don’t believe it, there are a finite number of minutes in a school day, a school year, a school year, a school career. Continually dumping more and more into the curriculum simply guarantees that our kids see a confusing array of material. It takes TIME for a good teacher to help them internalize that material.

If we could agree on a curriculum that emphasized the basics-- that’s what Common Core was SUPPOSED to be at the beginning-- then we could have a generation of kids who were capable of handling the upper level stuff when it was presented. But the current approach is to dump topic after topic after topic onto the curriculum, sometimes supplement that with a scripted curriculum, and expect teachers and kids to simply make it happen. The problem, of course, is that the kids in the classroom aren’t the Stepford Students envisioned by the curriculum designers.

And, nope, my school doesn’t follow Common Core. So this isn’t the rant of a frustrated teacher whose job is on the line. But my kids are caught up in it, and the kids I tutor are. And it’s a poorly executed, poorly designed hybrid of what was once a great idea and an awful lot of political agenda.

6

Hi all,

Thanks for your thoughts. @DigitalKing and @DogsAndMath23 , I definitely agree. As much as it claims to, I don’t think the AP Calculus level really does appeal to the kids who are the most mathematically motivated or able. I too wish there was some kind of honors/theoretical math track.

@aldfig0 -yes, I’m definitely curious about what we should do for the majority of students. Most kids probably aren’t able to handle an intensive, proof based curriculum. Still, I feel like something where they’re forced to see patterns and problem solve rather than just memorize is far more beneficial.

@bjkmom -I’m very curious about what you’d like to cut out. I definitely agree that we’re trying to cover too much and at not enough depth. However, it becomes difficult to cut things out when eventually they will be essential. Just for example, most kids will study matrices in Algebra II/Precalc. and at that level not really use them again. Yet for the kids who pursue higher math and take linear algebra and other such courses, matrices will once again become essential. I think cutting from the math syllabus is a great idea, but it’s always so hard to come up with what.

Okay, so one idea of something that I think could be inserted in the regular math syllabus which would be much more beneficial:Mathematical Induction.

My argument in favor of teaching Mathematical Induction is yes, there are some very challenging proofs by induction; however, the basic concept is not that hard to grasp. However, it also requires some manipulation of variables and some creativity/planning in doing the proof. It’s hard to understand initially (I really didn’t understand the principle when I first covered it), but with examples, I don’t think it’s that much harder than what’s already covered. I feel like this is just one example of something which could be covered which would yield enormous benefits in terms of thinking and problem solving.

Definitely, let me know all of your thoughts!

7

That’s pretty much the $64,000 question.

Our country is trying to define a “common core” of information in all subjects that will be useful for all people following all career paths. But the reality is that an art major needs different information in many subjects than a Physics major, and they both need different information from someone who is getting married and being a SAHM right after high school, or someone joining the Marines.

I wouldn’t mind a return to basics. A real emphasis on numbers in elementary. Let’s give kids a good head for numbers. Let’s let them see all the ways those numbers inter-react.

It’s funny, when I teach Heron’s formula in geometry, my kids end up with something like radical (28 * 8 *8 *12). And, without fail, they want to multiply all those numbers together and THEN simplify the radical. I show them to factor each of the numbers and pull out pairs of factors, and they think it’s magic. It’s not-- it’s the principal that the product of a pair of equal factors yields a perfect square. But it takes them so much by surprise. It’s because, while they learned the basics of simplifying radicals, they never had the time to internalize the ideas, to play with them.

Let me turn your question around on you: if we’re going to cover Mathematical Induction before Precalc, then when? And what topic(s) do we pull OUT of the syllabus to make room for it? To teach it to a typical Precalc class, you need a week and half to two weeks. Teaching it to younger kids will take a bit longer. So what do you pull out? Or do you dump yet another topic into the syllabus, diluting the kids’ understanding of what they’re already learning?

8

@bjkmom - I don’t think there are any units I would completely remove; rather, I think I’d take little pieces from each unit, especially those which seem to be merely mechanical knowledge.

For instance, I personally don’t think it benefits kids that much to do large units on function graphing. It seems to be more key to have them realize conceptually what an asymptote is, what a hole is, end behavior/limits, etc. than to memorize a million different function forms. In my opinion, they ought to be familiar with the basic forms (log, trig., polynomial, etc.) but shouldn’t have to memorize much beyond that. Ultimately, kids will be able to just graph on their calculators, so having so much emphasis seems pointless to me.

Things like seeing the binomial theorem as just something to memorize. Proving it via mathematical induction is very worthwhile (although might be too challenging for most classes), but just memorizing the formula and applying it on a test doesn’t have much worth to me. Kids once again should know what the theorem says, have a few examples in class and maybe at least some intuition of where it comes from but should not have to memorize it.

I think the key is not even necessarily removing a ton of content but rather re-engineering the curriculum which is already there. Using an example from calculus, far too often kids just memorize a ton of integration techniques without really thinking about what integration is. I came across a problem on the Oxford University MAT entrance exam (which I’ll be taking next fall) which makes you think about what integration is. It was to find the integral from 0 to n of the greatest integer of 2^x. We have no formula for integrating greatest integer functions, so you have to draw the function, begin adding up rectangles, and actually see a pattern/come to a conclusion. This problem challenges kids to think beyond the formula and problem solve.

More problems like this, even if they aren’t formal proofs, I believe are incredibly worthwhile. For me, there are definitely a few topics (like induction and proofs in general) not in the syllabus which should be. However, there’s far more which is currently in the syllabus which probably ought to remain but definitely needs to be re-engineered so as not to be taught just as formulas.

9

I’m in the middle of getting through school work right now. (It’s a day off for those of us in Catholic schools, following yesterday’s Ascension Thursday holy day.)

I won’t have access to the computer once the contractor gets here to work on the living room floor.

I’ll read and get back to you much later today.

10

We went to the moon first, so the math curriculum is working fine.

In what country do they teach math proofs in K-12 education?

You can get all the proofs you want if you go to MIT.

11

@rhapsody17 - I think many country’s in Asia do in fact teach proofs in K-12 education (from what I’ve heard/researched- I’m not 100% sure though). I’m originally from the UK, and if you take the higher math curriculum there (called Further Math)- they do cover proof by Mathematical Induction as well as some matrix subspace proofs and proofs of trig. identities using DeMoivre’s Theorem, etc.

I am actually going to apply to MIT- it’s one of my reach/dream colleges :slight_smile: . I visited and absolutely fell in love with it in terms of the general atmosphere- it definitely felt like home. However, I think you shouldn’t need to go to MIT in order to get a good pre-university math education which emphasizes critical thinking skills.

12

US Math is geared towards education of all students, not just math geniuses. Proof based rigor will discourage average students, so the current system is good because it caters to the average student and advanced math students have the opportunity to take advanced math classes (at colleges, CCs) to suit their needs.

13

@Mathinduction Indeed; the basic principles are quite simple. However, a slightly easier proof of the binomial theorem (to me at least) is more combinatorial in nature.

I do feel a bit disturbed by the lack of proofs in most HS curricula, and also the fact that proofs are usually only restricted to geometry, and for the most part, the infamous “two-column proof” which virtually no one uses in academia. It really isn’t that difficult to explain why the rational root theorem, for example, holds, and I usually give a proof of it when trying to explain it to someone.

I don’t think we should incorporate difficult Olympiad-level proofs in the general HS math curricula, but I do feel that proofs shouldn’t be restricted to only those 2D-geometry problems of proving segments or triangles congruent. In general, there needs to be a greater emphasis on problem solving.

Of course, there’s the other problem of lack of “number sense,” but that’s a whole different issue.

14

Lots of stuff here.

I agree with rhapsody17-- we’ve got to remember that the typical poster here is NOT the typical high school student. Most students tend to see education through exactly one pair of eyes-- their own, since it’s the only frame of reference they have. But the vast majority of kids in schools would struggle with induction. Lots of kids in Precalc are eventually able to do them, simply because they have strong algebra skills and can memorize the approach. But they never understand the beauty of induction-- probably because their elementary and secondary education was a mile wide and two inches deep.

Mathinduction, you state: " I definitely agree that we’re trying to cover too much and at not enough depth." But, then, a few posts later, you suggest taking little chunks out of the material we’re teaching. So you want to remove that depth.

As someone who has done this for a long time: I’m not in favor of doing most things via calculator unless and until we’ve taught how to do it by hand. Once a kid sees that setting the denominator equal to zero yields a vertical asymptote, and that dividing the leading coefficients yields the horizontal asymptote, then I’m fine with doing it via calculator. But until that point, there’s a huge disconnect with the idea that those “undefined” fractions from algebra and that “no slope” answer in geometry were because the function has an asymptote. They don’t see that there’s a huge difference in a fraction with a higher degree in the numerator as opposed the denominator. If they learn it by hand first, then when they hit that same idea in Limits, it’s time for an “AHA!” moment. Without that basic understanding, the limit become just another thing to memorize without understanding.

You mention binomial theorem. It’s a huge timesaver-- what would you have kids use instead? Their calculator?

Everything we teach is a building block for something down the road. So, for example, when I teach factoring the difference of two squares to my algebra kids, I always mention that THEY can’t factor the sum of perfect squares, but that I can, because I know how to work with imaginary numbers. Invariably someone goes home and looks up imaginary numbers.

The idea behind education is not to teach our kids as much as we can, to cram in more and more and more. It’s to teach them how to learn. It’s to provide the building blocks that they can use down the road, regardless of where life takes them. I’m not just preparing my kids for that Oxford entrance exam. I’m preparing kids to major in Art and Journalism and Political Science and Physics. I’m preparing my kids for life as a parent, as a business owner, as a plumber or a lawyer or a police officer.

Kids in this country who do opt for higher math will see it. But the vast number of kids don’t need induction. It has absolutely no value in their lives, and spending two weeks teaching it means that other things must either be removed from the syllabus or taught in a shorter window of time.

For what it’s worth, I’m starting proofs Monday in my Geometry class. (As a private school, we get to set our own syllabus. We’re playing with putting proofs off until the end of the year, when the kids are already familiar with the theorems. We’ve found that the kids experience a lot more success this way, since the memorization isn’t an issue at this point in the year.)

Monday’s lesson will be a logic puzzle, one I made up. They’ll have to find out who solved the crime, and prove that none of the other suspects is guilty “beyond a reasonable doubt.” From there, we’ll talk about just what a proof IS, what it does and what it doesn’t do.

Then we’ll start some geometric proofs. We’ll get as far as we get before the school year ends. But at least they’ll have a good idea of how a solid proof is built.

15

I feel lost. Some of the stuff that traditional Algebra I, Geometry, and Algebra II students learn are completely different from what Math I, Math II, and Math III students learn at my school. We have the second sequence, and looking at the two curriculums, I see that they’re about 65% aligned. It’s strange, as I don’t know if I’ll be prepared for SAT or anything. What topics do Algebra I students typically learn in your school? I don’t know if this is hijacking, as it’s related.

In Math I, we learned/reviewed:
-Slope (review)
-Functions (notation, vertical line test, etc)
-Arithmetic Sequences
-Systems of equation/inequalities
-Radicals and exponents
-Polynomials
-Quadratics
-Exponentials (Geometric sequences, too)
-Volume of prisms
-Statistics, data

Is this what Algebra I students would learn at your school?

16

Take a look here http://www.regentsprep.org/ for a pretty good rundown on what’s done in NY state.

17

@MITer94 - Yes, I agree. I think the 2 column proof does have SOME validity when you start teaching proofs; however, it should only be kept for a month or two and then later dropped. When you write proofs in two column format, you lose the art of mathematical writing. Kids should learn how to explain their reasoning in proof in a way such that it’s not too verbose (ie. including excessive detail) but also not too concise to follow. That skill is essential for any kid who will pursue higher math.

@bjkmom -Sorry if I wasn’t clear; I meant taking out the concepts that are fairly rote and don’t require high levels of mathematical thinking in order to expand upon content which really pushes kids further. Ultimately, I am advocating for more depth but through lightening the concepts that don’t matter.

I think what you were saying about politicians, artists, journalists, etc. is very true which for me means it’s more essential to teach thinking and reasoning skills than rote equations. They may not need to know how to solve a quadratic equation or graph a sine curve, but they certainly must know how to think. Math, if taught right, is a great platform on which to do this. That’s why I think induction (which is just one example of many) could be a key component for all students as it’s not completely rote, and you have to think through what you’re doing and the expressions you’re manipulating. The integral of greatest integer of 2^x is another example for which you can’t just use a formula but have to know what an integral is at its essence. Knowing how to problem solve benefits kids at any points within their careers.

Your lesson on proofs sounds incredible by the way and like a great way to engage kids in mathematics!! I’m sure they’ll really enjoy it.

Finally, @IrrationalPepsi -yes that does sound very similar to what I learned in Algebra I. I’m sure you won’t be at a disadvantage for the SAT & other standardized tests. However, I’d definitely encourage you to get a prep. book and look through it in order to go over any concepts that you haven’t learned well enough for standardized testing. Khan academy is another great resource (and there are many others out there) if you get stuck on any of the material (you can also definitely get help from your math teachers). However, prep. books are great because they’ll show you what you need to learn.

18

What do others who haven’t commented yet think about this issue of math syllabus?