Our 19th-Century Math and Science Curriculum

In My Humble Opinion
1

Those of us who teach math are used to hearing two things when we inform anyone of our profession: “I hated math when I was in school” and “I was no good at math when I was in school.” Sometimes both. I sympathize. Math as it’s currently taught deserves to be hated and doesn’t deserve to be learned.

By the time a typical student graduates from high school, he or she has solved hundreds of quadratic equations, graphed scores of hyperbolas, used the phrases “alternate exterior angles” and “alternate interior angles” countless times, and drawn 30-60-90 triangles until his or her fingers were ready to fall off. Why do we make students do so much of these particular topics?

Some people probably think that it’s a natural decision: that’s just what math is, isn’t it? But there are vast landscapes of math that our current curriculum avoids entirely: topology, logic, combinatorics, graph theory, group theory, and many more. Most folks who don’t major in math in college will probably never know that most of these fields exist. They are not too hard to teach to children. Indeed there are children’s books out there which give introductions to all of these topics at levels that children can understand. There are even a few private schools that teach kids this stuff, starting in elementary school.

So why does the mainstream education system remain stuck with parabolas and polynomial long division? Bluntly, that’s what was chosen generations ago, and it hasn’t changed since then. (It’s changed a little, I suppose, but not much.) In the industrial revolution, people who worked with their brains needed to solve polynomials by hand and do other suck tasks, so we taught kids how to do that. Today technology has changed everything and the demands and scientists and engineers are totally different. Entire new fields, such as computer programming, have come into existence. We still prepare kids to meet the needs of the 19th century.

Some will say that it’s still important to teach kids how to do everything by hand, even if we have technology that can do it for them. Supposedly the skills are still important. But as a broad statement, this simply isn’t true. Take polynomial long division, the biggest waste of time in any pre-calculus class. Even if a child truly understands what it is, what it’s supposedly to accomplish, and how to do it, it’s still a dead end that does not lead the way to any other useful skills in math.

Math could be and should be an immense amount of fun. It would be if we expanded the curriculum to include other areas of math. Even among students not inclined towards enjoying math, the mere presence of variety and absence of endless repetition would spark their interest.

2

So, let me understand—you teach math, but math as it is taught deserves to be hated and doesn’t deserve to be learned. Am I conclude that you are required to teach things that you yourself don’t believe should be learned, and forbidden from teaching things that should be learned?

What level do you teach at, and who mandates what you have to teach? I’d appreciate a cite for some of your claims. This is the best cite I could find on what math gets taught in high school nowadays.

Are you saying that “we” make them do these things way more times than it actually takes to learn them, and that they’ve actually mastered solving quadratic equations after the first couple dozen but we then make them do a couple hundred more? Or what?

So, in the past several generations, nobody (besides yourself) has thought to update the content of mathematics education?

Nice typo!

3

The fact that you find this even remotely surprising or puzzling reveals that you have absolutely no clue about how modern educational systems work. Yes, high school teachers have very little say about what should be taught in their classes. (University teachers would have a good deal more autonomy, but far from absolute even there.)

I expect a lot of people have, but have run into various sorts of political resistance and bureaucratic inertia. There have certainly been attempts to reform the curriculum, such as the notorious “new math”, but they have run into a lot of pushback, usually largely from parents and other laypeople who had little or no knowledge of either math (beyond what they themselves were taught in school) or math education.

4

This is basically the “new math” fad from the 60’s. The fact that that effort died off pretty quickly and people reverted back to more or less the old math curriculum to me that the “19th century” curriculum exists for reasons beyond simple intertia.

5

Not really surprised or puzzled, and I probably came off sounding more belligerent than I should. I really do want to understand where ITR Champion is coming from: What do you yourself teach, and how, and why?

6

There have been multiple new maths since. One of the current ones is the University of Chicago “Everyday Math” curriculum that my kids have gone through. Maligned by many mathematicians because it spirals around concepts instead of every getting kids to develop full skill and confidence with any algorithms. The result in my house was that I often couldn’t help my kids with their math until they got to High School … algebra, trig, geometry, calc, I could help with … how to solve a division problem using whatever array method (and counting “points”?) they were on about, I had no idea.

I lived through that 60s “new math” and still recall that doing problems in base 7 was was fun but not sure what it taught me for later.

OTOH I feel strongly that having to do proofs in geometry taught me a whole bunch about deductive reasoning and questioning how we know what we know and what we accept axiomatically. It prepared me for my later science education better than memorizing facts in science class did.

The question ITR is what learning math through HS level is supposed to accomplish and how a different curriculum would accomplish that better.

IMHO:

It should provide students with the skills to function as informed basic consumers and workers in a technical world - able to make change, to figure out what fraction of an hour a certin number of minutes are, to figure out what product is a more cost effective buy, to understand how interest and yeilds and risks work, and hopefully to be able to understand enough math to deal with probabilities and to stem off pseudoscience. Those who do not go past HS need enough math to be able to be skilled technical workers dealing with complex machines else they never get past working at McDonald’s. Topology does not help with that, cool and fun though it may be.

It should produce some fraction who are prepared to go into the STEM fields with the basics needed for the sort of maths they’ll be learning and using in their higher education.

It should train them in habits of mind that serve them in their other education endevours (like I believe geometry and proofs did for me but that I don’t think base 7 did so much so).

Does the level of exposure you could cover in these grade levels of topology, group theory, logic, etc. etc. etc., do those things better than mastery of basic skills with perhaps less calculus and more probability and statistics integrated with critical evaluation of what sorts of statistics are appropriate for what sorts of questions and how to critically evaluate news reports, be it polls or science reporting?

Not so sure that fun is the main goal of education rather than one tool that helps accomplish the goals.

(Yes, I read Number Devil to my kids and did extra stuff about sequences and believe it has value … but it was supplemental stuff that I did for fun with them, not the core curriculum.)

I’m not arguing against your point; I am really asking both what you think the goals of math education should be and how you think your proposal better meets those goals. I am open.

7

Clearly we needto something better.

ITR allow me to share with you my personal vent. We first get kids to learn to read and then have them read to learn; very little occurs in the math world in same form The other curriculum should be having the kids USE the math skills they have mastered over the previous and current year as part of their assignments in appropriate ways to drive in that math is a tool for all subjects to the same degree that reading the chapter is. Again things like Bayes thereom, fractions, basic stats, etc., are key to be able to apply to all subjects. To state that in the math class and then not apply that in the other courses seems to prove to the kids (incorrectly) that the claim is bullshit.

8

Not really. Even university math curriculums have to adhere to some sort of standard either by mandate or by default (textbooks).

You don’t really get any semblance of autonomy until the higher level electives.
I hear what the OP is saying but there is a reason why we teach math the way we do other than just tradition. Algebra lends itself to geometry which lends itself to graphing which lends itself to modeling. That’s just what happens.

You need the algebra and the parabolas in order to do the science. If you don’t make algebra mandatory and let kids choose logic and set theory without nailing down the polynomial/logarithmic/trigononometric coursework then they’ll be completely crippled when it comes to the sciences.

My beef with the mathematical education system is that the 3rd-8th grade years are squandered. There’s no reason why a 6th grader can’t handle algebra but the standard is to have Algebra 1 begin in 9th grade.

9

Base 7 math in itself, probably not. However, I found that my early lessons in base-N math contributed to my fundamental understanding of math and–more concretely–gave me a major head start in dealing with binary and hexadecimal, which proved very useful in college and in my current profession.

10

At the third grade level, I’ve been arguing for a few years that we need to put some effort into teaching kids how to program (using Logo or Tynker or other kid-friendly programming environments) and how to work a spreadsheet. I’m not sure what we ought to remove from the third-grade math curriculum, though.

11

I teach calculus and pre-calculus in a private school. That’s what the head of school has decided that I should teach. Next year I can discuss with him and we can choose to offer other math classes if we choose to do so. That’s the advantage of being in a private school.

In public schools, in most states, the state education department sets the guidelines and it’s been that way for years. Now many states, though not all, are adopting “common core”, which will give individual teachers even less freedom to decide what to cover. In grades 1-8, the decision about what almost every child in the country should learn in a given year will now be determined at the national level. In the high school grades, there might be a very limited amount of freedom to offer math electives, but certainly the No Child Left Behind act and the standards by which schools and teachers are currently judged don’t offer any motivation for anyone to expand beyond the basics.

12

Different sciences require different math. Graph theory and logic would be extremely useful for preparing kids for computer science. They’d be much less useful for preparing kids for physics, obviously. Parabolas and hyperbolas and sines and cosines are useful in physics, but not in computer science.

Ideally there should be some balance here. There should also be opportunities. Kids who lean more towards physics should get math lessons appropriate for that, and kids who are into computer programming should get math lessons appropriate for that. Right now, through 12th grade and even the first year of college, we give them only the physics-related math and none of the computer science-related math.

Our current math curriculum was designed before computers existed. Only minor changes have occurred during the 70-some years when computers have been around.

I entirely agree.

13

You can go pretty far in computer science without formal logic or graph theory. And as far as actual programming goes, you don’t really need them at all. I work at a college, and I haven’t really heard any computer scientist bemoan incoming Freshmans lack of knowledge of graph theory (and they would if it were even a bit of an issue, because bemoaning incoming freshmans lack of knowledge of things are science profs favourite item of small-talk).

Physics (and most other hard sciences) on the other hand, require a fair amount of calculus, analytic geometry and trig as pre-requisites. Most physics 101 courses require Calc I as a co-requisite, so if prospective students have to take college algebra, trig and then Calc, they won’t be able to take the foundational class in their desired major till Sophmore year.

I think there are pretty strong reasons that the basic math curriculum has remain standardized, despite several efforts to change it.

14

As a parent I can’t tell how much I think that idea stinks.

High School is not the time to begin tracking to particular vocational training paths. There is time for that. Okay, I can see senior year diverging for the high end math kids, some getting AP Calc, some getting AP Stats, some Graph Theory and Logic … but that’s it.

Sorry to repeat my question, but what do you think learning math through HS level is supposed to accomplish? Create some small number to work with computers and another separate group to become engineers? Or prepare all to function in today’s world with functional numeracy even if they never take another math class while also preparing a few in a more advanced fashion to go in a variety of STEM directions?

Our educational system right now is creating a population that is functionally pretty math illiterate and even those who have done well in math classes are ill-prepared to apply any of their maths to other subjects and endeavors. What solutions do you propose for that majority of students (and I am sure they are the majority in your school as well as any other)?

I am asking this sincerely - Canada, Korea, and Finland all well-outperform America SES for SES (especially at the high end) on math; what do they do differently and is it something that can be done here?

15

I happen to think the common core is an excellent idea. I’m not a teacher or anything but I did grow up in Texas and since I left I’ve seen a bit of the stigma that goes along with getting a Texas education. Potential employers have never said a word, of course, but I can’t tell you the number of times I’ve heard, “Did they teach you evolution in science class?” or other inquisitorial digs at my education. It especially became clear once I had a baby and we decided to move out of state to put her in the best possible schools we could afford. Any time we discussed it with anyone who knew my history we heard some variation of the joke, “Looking for the best schools? Well then you are* obviously* going to move back to Texas, right? Hah!”

This leads me to believe that, although potential employers have never said anything, that I’ve probably missed out on some job opportunities because of the implication that going to school in Texas means you are undereducated and probably insanely religious to boot. I imagine that people who grew up in Alabama, Mississippi and other states low on the educational totem pole have similar experiences. If common core is able to level that playing field a little bit it can only be a good thing as far as I’m concerned.

16

Measured by high school graduation rates or standardized test scores, the public schools in Texas are in the middle, compared to other states. The attitudes that you’ve encountered show a need to counter anti-Texas bigotry.

17

Yes, the current high school math curriculum prepares students for a college freshman physics course. It does not provide any support for any computer science courses. Colleges and universities are aware of this and they shape their curriculum accordingly. Beginning physics courses are designed for students who have relevant math background. Computer science classes, on the other hand, have to teach the math background as they go.

But why should it be this way? It doesn’t have to be. If we changed the high school math curriculum, we could have students with some preparation for computer science and some for physics, rather than solely preparation for physics. If so, then the college curriculum would change as well. Computer science departments wouldn’t need to spend as much time teaching math; they could do more of other things. Physics departments would need to spend more time on mathematical background, and thus would have to trim their curriculum elsewhere. There are always trade-offs.

But why should physics rule the roost? Physics was around in the 19th century, and helped establish the curriculum. Computer science wasn’t and didn’t. You say that computer scientists don’t complain about their students’ lack of knowledge while other scientists do. Everybody is used to the curriculum as it currently stands, and we rarely imagine that it could be different. But it could be different.

18

I don’t think there is a singular explanation for why Canada, Korea, and Finland collectively out-perform the US. Each one probably has a couple of explanatory factors that contribute. It’s also comparing apples and oranges when it comes to little countries vs big countries.

I’ve taught math and tutored math on and off for HS and College over the past decade and I can say in my opinion a big reason why people struggle with math is that they are bad at mental math. That’s not a failing on their part. It’s mostly genetic. They can’t hold numbers in their heads. Then there’s a unspoken shame-stigma that goes along with it. Kids start thinking they’re bad at math because they can’t hold numbers in their heads, or solve equations without writing down all the steps. Then the shame spirals and kids just say “I don’t get math” when really it’s “I can’t hold numbers in my head and so I’m a step slower than someone who can.” It gets worse when the kids who can hold numbers in their head shout out the answer quickly. Then doubly worse when teachers (who almost always can hold numbers in their head) work quickly and teach “shortcuts” without putting pencil to paper (I’d like to see the number of teachers that teach factoring with a formalized method vs explaining it as reverse-foiling and just guess numbers that fit. Or the quadratic formula before completing the square). Then it gets super-worse when the kids who can’t keep numbers in their head compensate with a calculator rather than pencil/paper. All of this snowballs when really it could be cut off in the beginning with a little bit of patience and an extra couple of seconds spent writing out the steps.

19

That’s one problem with education reform efforts in the USA. We don’t start by asking what the education system is supposed to accomplish. We talk constantly about “performance”, but that can be defined in a vast number of ways. Schools have a general purpose of “education” which includes many smaller purposes, including all the ones that you listed. Somehow the school system must accommodate all to some degree.

One thing I’m sure of is that we need to do a better job of awakening intellectual curiosity in students. Some students will get that from their parents, but some never get it at all. Math is surely one of the subjects that seems most deathly dull to a great many students at the moment. If we freed the math curriculum from its current narrow confines, we would surely awaken more interest. I’ve seen plenty of kids books that try to do that, but they’re generally not used in school.

20

I am not a computer scientist, but I did take several CS courses in college, and I don’t agree with you about what math is required to prepare one for computer science.

Certainly, one would have to be comfortable with variables and formulas and functions—both the concepts behind them and the details of working with them. One would have to be able to think algorithmically—to formulate step-by-step procedures for solving certain sorts of problems. All of this can be developed and reinforced in a good algebra or calculus class. Furthermore, I certainly used “parabolas and hyperbolas and sines and cosines” when working with computer graphics.

As I said, IANACS. But if you want to hear from people who are, I found this discussion: Should certain math classes be required for a Computer Science degree?