28 October, 2005

Conditioning our students not to learn

I have a long, painful history of involvement in teaching. I have tutored for years and years, including charity work; I have taught high school, I have been a teaching assistant, and now I am a professor at a small, private, "liberal-arts" college. I make the following observation not so much to cast stones at others, but to direct my future work and choices.

Today I was teaching remedial algebra students how to solve compound inequalities. It was hard not to shake the conviction that this was, by and large, a colossal act of fraud. I'm taking their money and they're learning very little. Most of them aren't even capable of understanding what I'm talking about.

It's not that I'm talking at too high a level; I've taught long enough that I know how to modulate things down to the level that these students can understand. However, I can say with the utmost confidence that these students don't have the intellectual resources to solve compound inequalities. Why? most of them failed spectacularly the test on solving basic linear equations such as 2(x+1)=0. I'm not exaggerating; the median score was 44.5. More than half the students scored below 50%.

Solving compound inequalities requires the exact same skills as solving equations, with a few extra considerations to boot. A student who can't even solve equations has no hope of learning how to solve inequalities.

It would be the highest flattery to say that these students shouldn't possess a high school diploma. My 9 year old son is now solving linear equations such as 2x+4=6 in the fourth grade. These 18- and 19-year old "college" students shouldn't even have gotten out of grade school.

I'm not being cynical; I'm being honest. I recently introduced a one-hour homework lab in order to give these students time to work under my supervision. This is charity on my part; the students pay nothing, and the college certainly doesn't pay me for it. I do this because I want the students to succeed, because I know that they can succeed if they would only put their mind to it.

Unfortunately, our educational practices have conditioned them to believe that they don't have to put their mind to it. The remainder of this post will highlight some of these damaging practices.

Multiple-choice tests
My first gripe is with multiple-choice math tests. I am quite in favor of high-stakes tests that determine whether students meet certain standards, but math tests should not be scored as multiple-choice tests. Multiple-choice tests teach the following lessons:

  • the answer is more important than the process;
  • you don't have to know the right answer, since often enough you can guess it;
  • you don't have to worry about how readable, organized, or thought-out your work is.

For example, I love the math questions on the SAT. Most are beautiful, thought-provoking questions. Yet the multiple-choice nature of the SAT renders it so vulnerable to manipulation that the Princeton Review has for decades made a fortune from selling books that openly mock the SAT as a measure of scholastical aptitude. People who read a Princeton Review book succeed in raising their score by learning nothing more than some simple test-taking strategies. Most of these books' effort is expended on tricks like this:
  • try the answer for (A) in the equation;
  • if it works, great! you've found your answer; darken the bubble for (A);
  • if it doesn't work, try the answer for (B);
  • repeat as necessary.
People pay hundreds of dollars to attend a seminar that teaches them nothing more than things like this, and they'll pay it because their SAT score rises. This is not a measure of whether a student knows how to solve equations.

The SAT is probably the best-researched multiple-choice test available in the world; they have expended countless millions of dollars in trying to find ways to avoid manipulation. The result has been the introduction of questions that do not have multiple choice; this year they're even including a written essay as part of the English SAT. If the SAT has given up on multiple choice, you can bet your bottom dollar that most multiple-choice tests are parodies of whether the student has learned mathematics. I am no longer surprised that my students wail when I tell them, "There are no multiple-choice tests in this class. I grade you by your thinking, not your luck."

(I'm not exaggerating. One class literally wailed.)

The curve
My second gripe is with curving grades. This is based on the false assumption that a histogram of students' grades should resemble the standard normal distribution. The reality is that this technique is a rationalization for grade inflation; have you ever heard of a teacher who curved grades down to fit the bell curve? I haven't; if a teacher had done so to one of my classes, I would have joined my fellow students in rioting, and I was one of those goody-goodies who always deferred to the teacher.

Curving grades is thus dishonest at least in practice, but it's dishonest in theory, too. To understand this, you have to consider that curving grades simply moves the scores to a point that they resemble a predetermined outcome, where the mean is a 75 (usually), because that's where the average student "should" be. By this logic, a grade reflects only how a student stands in relation to other students.

Let me repeat that for emphasis. The logic of curving scores implies without exception that a grade reflects only how a student stands in relation to other students. It does not reflect how well a student understands a topic. A student can master almost no material, yet "earn" a passing grade simply because no one else mastered the material, either. The teacher's curve moved him to an "average" grade.

The natural outcome is this class of remedial students who find it inconceivable that they will fail my class. I recently warned them that I have awarded F's to 2/3 of a class before, and they gasped audibly with the sudden, sinking realization that things had changed. I followed that up by pointing out that I have awarded passing grades to everyone in a class before, too. The diference being that the second class learned the material, while the first class tried to play the rules in their favor.

A better alternative for the high-stakes tests mandated by the federal government would be this: students give free-response answers. Tests are collected from one school and sent to another school in another part of the state (or another part of the country, even). There, a team of teachers grades the problems according to certain standards: accuracy, strategy, correctness, organization, legibility. This is a much more accurate assessment of what the student has learned.

The drawback to this is that someone would have to pay the team of graders, and that costs a lot more than running multiple-choice tests through a scanner.

Mathmetics is hierarchical: it builds on itself logically. When it's taught properly, it's actually quite beautiful. Textbook publishers must not be run by mathematicians, however, since they do everything possible to hide the innate hierarchy and logical flow of mathematics.

The text we are using right now for these remedial students is a prime example. The notion of a function is introduced in chapter two, section two. It is not introduced as a relation; its definition appears at the beginning of the section. The homework problems ask you to determine which "relations" are functions, so you have to find the definition of a relation, which occurs four pages later. This is counter-intuitive; a function is a special kind of relation, so the definition of a function should come after the definition of a relation, and usually it does.

After this introduction, functions are sprinkled throughout the rest of the textbook as considered necessary, but in very non-obvious ways. The proper understanding of functions requires one to be able to solve for the domain of the function, but good luck finding out how to do it! If I had to pass over the solution of compound inequalities in chapter 4, then I would never notice that the last paragraph in that section discusses how to find the domain of functions with a square root. Students who haven't seen the domain of a function for weeks are suddenly supposed to recall what the domain of a function is.

What's really irritating about placing the problem of domain here, is that one doesn't even need compound inequalities to describe the domain of a function with a variable under a square root. Logically, this should have appeared in the previous section on solving simple inequalities; instead it appears illogically here, as the last example a in a subsection that counts more as a footnote than as an explanation.

To show you how common this practice is in mathematics textbooks, the sales representative of a competing publisher spoke with respect of the textbook I'm using. I have no idea why, except that it probably makes a lot of money due to successful marketing.

Students pay $100-$120 for mathematics textbooks. For $120, they deserve something that is carefully structured and contains good explanations. More, even: for $120 they deserve novel explanations that provide insights never before seen. Nearly all the material in the vast majority of undergraduate mathematics textbooks has been known for three or four centuries; there is nothing new, not even the explanations. For a few topics we'd have to admit the 1950s, but only a very few.

I have looked at algebra books from the 1980s, and the only difference between them and modern texts is that the modern text has larger type and color pictures. Ninety-five percent of students will not read the textbook even if you paid them by the hour to do so, and why should they? Both the textbook and the entire K-12 curriculum presents math as mechanics, not as problem-solving with the mechanics necessary for that. The only purpose textbooks serve is as a resource for problems that teachers want the students to solve, and (maybe) as a resource of example problems to imitate mindlessly. Students should not have to pay $120 for a resource on practice problems with a bunch of pretty, useless pictures; Schaum's Outline series does the same for much less.

Actually, that gives me an idea.

Word problems
My son is now solving word problems. He gets them right every time. Why shouldn't he? His encounters with word problems take the following form: he has a sheet full of 20 or 30 arithmetic problems (multiplication, for example). At the bottom of the page, he has two word problems, each of which requires the same operation as the previous 20 or 30 problems. To solve the word problem, my son doesn't have to understand what is multiplication's real-world application; he doesn't have to ask himself which numbers are relevant information and which are spurious; he doesn't have to decide which operation corresponds to the question. He needs only to pick out the two numbers and apply the same operation that he applied on the rest of the worksheet.

This teaches our children neither how to think, nor how to solve problems; it teaches them how to operate their brains as assembly-line machines. When the time comes to discriminate solving word problems that require some thought, such as deciding which operation to use, or performing more than one step, our students become hopelessly lost. They haven't learned to read mathematics, so of course they can't understand what to do.

This is one of the advantages of the Saxon series of textbooks. For all its flaws, Saxon's philosophy uses constant review, so that the problems are constantly mixed up. When students encounter a word problem, they can't be sure if they are to add, subtract, multiply or divide. They can't just pick two numbers and apply the same operation that they applied in the other 20 or 30 problems, because the other 20 or 30 problems weren't identical. Students working with Saxon texts actually have to think about it. Keep that in mind if you ever hear a math teacher disparage Saxon texts because of their skill-and-drill approach. (A valid criticism, by the way, until you consider what the math teaching establishment considers to be good textbooks.)

This list of issues outlines ideas with what's wrong, and points to ways I think they could be fixed. With time, research, and cooperation with others, I might be able to reform my own teaching; you can see that I've already made some effort by introducing an optional homework lab for my remedial students. God willing...


Steven said...

Dear Jack,

You may not have encountered this at the universities you've been at, but there is another more self-protective reason for grading on the curve. In some places evaluation and tenure is predicated on having a bell-like distribution in your class. I will speculate on the reasons because no one was ever able to tell me why:

It represents a "fairness" and a "justness" to grades.

It shows that you are a competent teacher because your students fit the curve (even though you had to force fit them).

Honestly, I haven't the slightest idea why. But there was at least one professor denied tenure and eventually dismissed because his student population did not fit the beautiful bell-like distribution.

In fact, in most classes I've been in the student population was rather more like a bimodal Gaussian curve--with those who really, really wanted to be there and who were earning their way through and very goal oriented clustered toward the top and a Second group just dead set on passing. And extremely long tails toward the "shallow" end of the pond.



jack perry said...

Dear Steven,

You know, I wasn't worried about this myself until you mentioned it. Thanks for instilling me with a sense of terror :-)

On the upside, I was told when they interviewed me here that they have very weak students. Weak students I don't mind; my beef is that they have no interest in learning anything at all. They are going to college not because they are excited to learn something that will be useful for the future, and not because they are hopeful for the opportunities that a degree opens up for them; they are going to college because they get to play football, or they get to avoid life a little longer, or...

sigh. I could keep complaining, but it's not very Christian of me to do so, is it?

Anyway, my classes here are unimodal (except for the Linear Algebra class). The mean and median are nearly identical, and they're unspeakably low. The outliers aren't on the low end of the scale; they're on the upper end of the scale. I have never, ever dealt with such a disaster before. Sometimes it's an exciting challenge, but usually it's depressing.

Steven said...

Dear Jack,

Perhaps I merely had the advantage of dealing with Science which is marginally less frightening to most than is math. More than that I dealt with a, pardon the pun, very concrete science (Rocks for Jocks, Paleontology) that were seen as "easy courses." I didn't notice as much lack of interest, but then, I didn't pay that much attention in the "required" courses. Those who were in things like graduate Paleontology, Micropaleontology, Analysis of Strata, etc. People only do these things because they really want to or they see careers opening before them. They aren't really a prelude to anything else.

But even so--I suspect that science is probably more reasonably embraced than math.

Too bad I'm not there to be your student. I would probably drive you batty (as I did my Math prof. I was a freshman English Major taking differential equations and analysis and the teacher asked me nearly every other class why I was there if I didn't want to continue in math. AFter the first test when I blew the math majors out of the water he stopped asking and almost begged me to become a math major. My response was that I enjoyed the math because I didn't have to do it, and it was my enjoyment that made success possible. Not possible under any other scenario. Continued with him through three or four higher-level math classes--and I did notice the skew you indicated toward the upper end when people were interested.)

AND personally, I love abstract, nonfunctional math--"How many dimensions dance of the head of a beach?" But I HATED statistics. It was boring, dull, and utilitarian. Our teacher made us memorize the definitions for all those dismal measures to the word and the "work" was a dismal cranking of numbers through a more or less opaque black box.

So much better when we arrived at eigenvectors and eigenvalues and nonparametrics of all sorts. Saw less practical use for it--therefore more a mental exercise--more fun.



jack perry said...

Heh, that's funny, because eigenvectors and eigenvalues are in fact highly applicable.

I HATED statistics. It was boring, dull, and utilitarian. Our teacher made us memorize the definitions for all those dismal measures to the word and the "work" was a dismal cranking of numbers through a more or less opaque black box.

I'm kind of in the same boat. I require the students to know definitions, but I don't require them to know the exact wording; a different wording would work if it was precise. The "work" is, as you aptly describe, the dismal cranking of numbers through a more or less opaque black box.

To understand the black box, of course, these students would have to take calculus and matrix theory (for linear regression) or understand something about the 2-norm as a measure (standard deviation). But, these non-math majors can barely grasp the idea of divide and conquer; you can imagine how unwise it would be to subject them to infinitesimals, improper integrals, and metrics...

Steven said...

Dear Jack,

"Heh, that's funny, because eigenvectors and eigenvalues are in fact highly applicable."

Don't I know it! Part of my dissertation was a projection in 72 dimensional space of the populations of 28 different paleocommunities. But that was 10 years off and I was at the time of study an English major. There was no conceivable purpose to an eigenvector or an eigenvalue, it was simply beautiful.

Then there was the other problem I had with Stats. The professor refused to engage the question of randomness. All of these things they were showing me were predicated on random sampling of some population, but the reality I was aware of was that populations were always skewed in one way or another, there was never perfect randomness. The teacher's answer was that there was another set of statistics for that and that we weren't there to consider reality but theory and these measures worked (by and large) in the real world.

Needless to say, I retain a deep suspicion of all parametric statistics including even measures of central tendency because a measure of central tendency at a outlier population is going to give you a skewed picture of the entire population.

But then I'm preaching to the choir. I love math for its beauty. While I acknowledge its utility, I'm often not terribly interested in it. One exception is non-linear dyamics and fractal geometry.

But enough. Nice to chat about enthusiasms.