Math and the Teaching Thereof

Firday's quick and sarcastic post came about because I thought the Dean Dad and his commenters had some interesting points in regard to high school math requirements, but we were spending the afternoon driving to Whitney Point so I could give a graduation speech. I didn't have time for a more detailed response.

Now that we're back in town... well, I still don't have time, because SteelyKid has picked up a bit of coxsackie virus, meaning that nobody in Chateau Steelypips is happy. But I did want to offer at least a partial response to some of the comments both here and elsewhere.

To start off, Tom raises a good point about correlation and causation: it may be that, as the Dean Dad notes, students who took four years of math did so because they do well in math, and that math aptitude is the reason for both their extra preparation and their success in college math.

There's probably something to this. Unfortunately, ability to do well in math is only imperfectly correlated with desire to pursue courses of study for which math is a prerequisite. And, more importantly, I think there are different ways to do four years of math classes.

This is also my response to the comment by Jim at the Dean Dad's (the comment permalinks are broken-- it's at 10:20 am, about halfway down):

Math, like Foreign Languages, but unlike almost every other high school subject, builds. If you just scrape through Algebra I, you start Algebra II behind the eight-ball. Algebra II expects you to know the stuff from Algebra I. In contrast, if you barely scrape through Earth Science, you start Biology even.

There's some truth to this, but it seems to me that this is a problem that can be solved by offering different flavors of Algebra II (or whatever)-- students who just scrape through Algebra I should get a version of Algebra II that spends a significant amount of time reviewing and reinforcing the material from Algebra I, while students who breeze through Algebra I go directly into Algebra II.

The Math department at Union does something like this with calculus. The "default" calculus sequence is two courses (two of our three academic terms, so not quite a full year) covering differential and integral calculus. Students who have good-but-not-great AP scores get a one-term version of the same material, while students whose preparation is a little weaker can take a three-term version of the same thing. This creates a few problems with classes like intro physics because our prerequisite enforcement is nonexistent, but from a straight math perspective, it's a good system.

Of course, that would never really work at the high school level, because the overhead is too great. Not only would you need more teachers to cover the different versions of the same course, you would create some administrative overhead in trying to keep track of who took which version, and it would play hell with the notion of standardized testing of all students. In an ideal world, though, I think that would be the way to go-- set a minimum standard, but provide tracks for students with different levels of math aptitude to meet that standard in different amounts of time.

Rather than the current version of the requirement where a student gets through Math II in four years by taking Math I twice, and Math II twice, have a four-year sequence covering the material of Math I and Math II, which would give them a better chance of retaining some of what they're supposed to be learning.

I've switched to Math I and Math II in the above paragraph, because there's also a valid point raised by harlan, and also in this TED talk by Arthur Benjamin (via... Jennifer Ouellette on Facebook, I think), namely that there's no particular reason why the curriculum has to be structured the way it is. As convenient as it is for physicists to have the entire high-school math curriculum aimed at calculus, which is really about physics, there's no reason why things have to be that way. And you could make a good case that teaching every high school student probability and statistics would be more useful to society as a whole than teaching everybody calculus. If you're going to start imagining pie-in-the-sky reforms of math education, you might as well do it from the ground up.

The final frequently-raised point is one that almost doesn't deserve a response, namely "None of these students are ever going to use {math topic}, anyway..." (sometimes in the "I took {math topic} back in the day, and I've never used a bit of it" variant). In addition to the standard annoyed responses ("Yeah? Well, then, take English Literature and stick it up your ass, 'cause they're never gonna need that, either..."), though, there's a response in the Dean Dad thread (Mikey at 7:58 am) that's worth highlighting:

The fact of the matter is that most people don't retain all of the math that they've ever learned. For someone in a developmental math course it might be something like 50%. For a grad student, it might be something like 80% (as a former math grad student, I can tell you that I retained significantly less than that). So if we need someone to be very comfortable with arithmetic to function in society, stopping at the end of arithmetic won't be enough for them to retain it. They'll need a bit more in order to retain everything.

Since math does build on itself, we can start teaching these students algebra. In using algebra, students will be challenged to use their arithmetic skills in many different ways, forcing them to actually own arithmetic. It will give them the practice of not only performing arithmetic, but also knowing when it's appropriate to use different operations.

Finally, the problem solving and even logic skills one employs in an algebra class are really a dimension higher than that in an arithmetic class. Being introduced to a different way of thinking (using variables to represent unknown quantities, as well as using symbolic manipulation) is immensely helpful in any sort of problem solving arena, whether or not it involves math. Everyone needs problem solving skills: if jobs didn't require them, everything would be run 100% by computers.

The problem-solving skills bit is familiar, but the retention argument is new to me. I like it, though, and I think it's a good point. And so, I mention it here.

And that's about all I have on math education at the moment. So, I'm off to check on SpottyKid...


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A related post was published on Sciencewomen earlier this month. I did a quick bit of research and found the following in a report from the Center for Study of Mathematics Curriculum:

11 states already require 4 years of math for students to simply graduate from high school, while 24 require 3 years. Only 7 states require 2 years; the others either make decisions at a local level or have varying requirements for different programs.

How math is taught is a different issue and those problems start well before high school. It is very disheartening to hear so many elementary school teachers in my graduate research classes make the statement "I hate math!"

"I think there are different ways to do four years of math classes.".

True. Here in MIchigan four years of 'math' are required for HS graduation, but math is a loose term - courses in which math is used are sneaked in - business applications, (surprisingly, since this was going on in the 70s when I was in high school) some shop classes, etc, can be counted as math in certain places. I am not implying that those other classes are worthless, or blow-offs, but it does mean that some digging is required to determine what a young man or woman means when he or she says "I had four years of math in high school".

Two related points.

To this: "And you could make a good case that teaching every high school student probability and statistics would be more useful to society as a whole than teaching everybody calculus",

I say huzzah!

Second, I've often been frustrated by the "But Algebra I/II are essential for teaching problem solving skills" mantra. I mean, this seems to me to be a blatantly post hoc rationalization for teaching Algebra. The real reason we insist on focusing on Algebra in high school can be found in your previous post:

Algebra (high school algebra) is to a large extent the language in which higher math and physics is conducted in. Personally, I've never really found the subject all that interesting in and of itself. Useful, sure. But useful mostly in doing higher math and physics. It's like learning to spell and to use decent grammar.

It has always seemed disingenuous to high school students to argue that Algebra has some special significance for the conveyance of problem solving skills. It doesn't. In fact, if we want to teach rational, logical thought and problem solving, probability, statistics, sentential/predicate logic, basic computer programming skills and of course geometry would all be much better topics for that purpose.

Some of it may have to do with the way early math and algebra are studied. I was shocked to find out many European classes do not rely on large numbers of examples in class and for homework to teach.

The typical class I have seen here has you solve forty or fifty problems a day. Then the class goes over the forty or fifty problems and solves them in exactly one way. By the time this is done it is on to another homework assignment.

The various methods of solving any problem, why there are multiple methods, and the advantages/disadvantages of each are not covered. The linkages between methods and concepts are mostly unspoken and understood, if they are ever understood at all, only implicitly through concatenation of use. It is easy for many students to 'learn' to do a particular sort of problem by rote by creating a simple mental template without absorbing the understanding of what they are doing. Of course rote learning the template is the kiss of death for understanding because as soon as the formula is shifted around the template no longer works and the student doesn't understand the concepts well enough to work around it.

I was shocked to learn that in some classes in Europe they may spend an entire period on two or three problems. Comparing and contrasting various methods for solving them. Showing equivalencies and digging into the concepts. Mastering the options, methods and ideas behind the math and avoiding allowing students to get by by simply learning templates for how to solve a particular presentation of a problem.

However, Furday is the day before Caturday.

avoiding allowing students to get by by simply learning templates for how to solve a particular presentation of a problem.

If they don't teach them how to take standardized tests by rote, how do they ever graduate? How does the school maintain its funding?

By D. C. Sessions (not verified) on 30 Jun 2009 #permalink

I guess my comments over at DD's just don't rate because my PhD is being wasted by teaching physics at a CC rather than publishing several dozen more papers in a field you don't care about.


If you were paying attention to all of the comments, you would know that the problem is just as bad where they require four years of math because most of that "math" is junk designed only to keep them in school. Further, there is no impediment to stretching out algebra in HS. It is already being done on an industrial scale. There are plenty of high schools where Algebra I is split across two years to accommodate kids who are unprepared for it due to the effects of K-5 and Middle School math teaching. They find it quite easy to implement, and they also have no trouble at all using "honors" to designate courses that follow a normal curriculum so they don't have to say what the non-honors courses actually are. There is plenty of evidence at our end (advising incoming students) that the level of algebra below "honors" is (not)Algebra, where everything is done with calculators simply to get them past the graduation test.

Seriously, if you are concerned about this issue, you really need to pay attention to Middle School and the reality that is a typical middle school teacher of mathematics. That is where the problem is, but you are unlikely to come across any of the victims of that environment where you teach - and hopefully not where you live. They do, however, make up the majority of students entering a CC, and students entering a CC are a majority of HS grads who go to college. You only see the tip of the tip of the iceberg, even among the non-science majors.

(The problem in Middle School math is incompetence, which is tied to supply and demand problems at the going wage. I encountered one student who had 5 different teachers in one month in 7th grade math. Can you imagine what it takes to get fired from a public school? Years later ze was still dealing with the trauma inflicted by one of those math teachers.)

I'll close with one final observation: How do you teach synthetic division to a student who was never taught long division in elementary school because they use one of the "reform" programs that eschews algorithms? Think about that, and the remediation required to deal with it within the scheme proposed in the comment you quoted. Ah, the stories my math colleagues can tell. I'm just fortunate that I teach the exponential tail of the distribution that finally makes it to calculus.

How do you teach synthetic division to a student who was never taught long division in elementary school because they use one of the "reform" programs that eschews algorithms?

Pretty straightforward if you ask me. First you dig up the faulty foundation ...