Monday, September 1, 2014

A lament for geometry proofs

One of many casualties of Reform Math--along with simultaneous equations involving more than two variables, rational expressions involving polynomial denominators with more than two terms and of degree > 3--are traditional geometry proofs like this one

Proofs are gone for many reasons. Some people find them soul-killing. Here, for example, is what Paul Lockhart's (in)famous Lament has to say about this particular proof:
In other words, the angles on both sides are the same. Well, duh! The configuration of two crossed lines is symmetrical for crissake. And as if this wasn’t bad enough, this patently obvious statement about lines and angles must then be “proved.”
Instead of a witty and enjoyable argument written by an actual human being, and conducted in one of the world’s many natural languages, we get this sullen, soulless, bureaucratic formletter of a proof. And what a mountain being made of a molehill! Do we really want to suggest that a straightforward observation like this requires such an extensive preamble? Be honest: did you actually even read it? Of course not. Who would want to?
(Everyone I know loved to construct these proofs--precisely because they forced you to abandon intuition and work things out logically, building up from Euclid's postulates, in the wonderfully exotic, unnatural language of math.)

Others, I suspect, find these proofs too difficult to teach--especially in the era of Reform Math, with incoming geometry students having spent so little time with problems that demand any kind of rigorous, multi-step logic in the unnatural language of math.

Finally, there are the Common Core Math Standards. The only mention of geometry proofs anywhere in the math standards in the Introduction to the High School Geometry section:
During high school, students begin to formalize their geometry experiences from elementary and middle school, using more precise definitions and developing careful proofs. Later in college some students develop Euclidean and other geometries carefully from a small set of axioms.
Within this introduction, the only other mention of proof is in a subsection entitled Connections to Equations, where the type of proof under discussion appears to be algebraic rather than geometric:
Geometric shapes can be described by equations, making algebraic manipulation into a tool for geometric understanding, modeling, and proof.
Nowhere in any of the specific goals for geometry do we find the word "proof."

And yet, surely the Common Core authors, with their obsession with higher-level thinking and deep understanding, would use the powers vested in them by Bill Gates, The National Governor's Association, and Education Industrial Complex, to revive this dying art?

Or perhaps we can come up with a proof as to why this isn't happening:

Geometry proofs don't lend themselves to standardized tests.
Proofs can't be captured in multiple choice format; only expensive experts could grade them.

The Common Core authors don't like goals that aren't easily measured by standardized tests.
Most of them are affiliated with the educational testing industry (see here).

Or, as Lockhart would put it, in terms that are so much less bureaucratic, and in one of the world's many natural language: "Well, duh!"


Auntie Ann said...

Geometry was always my favorite math class. I loved doing the proofs. I'm waiting to see what my niece's geometry will be this year; she claims they still do proofs at her school.

Anonymous said...

So sad. I also found the introduction to proofs through geometry to be mind-expanding learning.

But, though I share the lament, I also think that the "core" learning can't remain stuck in any point in time. As the world changes, we may find other things we need to teach that might even teach similar critical skills, that should function as replacements. For geometry proofs, and the logic and language that I value from them, programming and algorithms might be just as important.


Anonymous said...

Lockhart has a Ph.D. in number theory, so I seriously doubt that he finds proofs "soul-killing". I have read the Lament, and while I don't agree with everything Lockhart says, I understand his point about high school geometry proofs, namely, they do not resemble at all the way real mathematicians communicate real mathematics.

High school geometry proofs are much closer in style to Russell and Whitehead's Principia Mathematica proofs, which might be fine if you are hoping for everyone to become a future Ph.D. in set theory and logic, but not for most everyone else.

No one writes proofs in this ridiculous "statement-reason" format. For instance, I can write a proof of the proposition as follows:

"Let x = measure of angle APC, y = measure of angle APD, z = measure of angle BPD. Then since angles APC and APD form a straight line, x + y = 180 degrees. Similarly, y + z = 180 degrees. Equating, we have x + y = y + z, and subtracting y gives the result." This is how human beings explain things to each other.

Barry Garelick said...

"This is how human beings explain things to each other."

In teaching proofs in the beginning, the statement/reason format is used as a simple way to organize the structure of the proof, but more importantly to get students used to be able to justify steps. In textbooks that still require students to do substantial amounts of proofs (very few do), students depart from the statement/reason two column format to the paragraph style.

Lockhart's Lament does pose an interesting question. For those who complain that traditional math ed is boring to students because it isn't "relevant" to their lives, why are his explorations (such as relationship between area and perimeter) going to suddenly be of interest to all students?

Anonymous said...

Barry, you said,

"In textbooks that still require students to do substantial amounts of proofs (very few do), students depart from the statement/reason two column format to the paragraph style."

Well, this wasn't the case for me. I took high school geometry circa 30 years ago, and most of our textbook still focused on proofs. However, we never were allowed to venture past the "statement-reason" format into natural language. This posed a real stumbling block for me when I got to a class like linear algebra later, when you're expected to express yourself that way. So if what you say is true, then that's a good thing.

"For those who complain that traditional math ed is boring to students because it isn't 'relevant' to their lives, why are his explorations (such as relationship between area and perimeter) going to suddenly be of interest to all students?"

In my experience, when students say they want math to be "relevant", what they really mean is, they want is to be easy. They say they want math that "they're going to use somewhere", but as soon as you present them with actual, real-world problems, and all the ambiguity, messiness, complications, and critical thinking that go along with it, suddenly they yearn for drill-and-kill all over again.

As Lockhart says, learning and doing real mathematics, at any level, is intellectually taxing. Students want short-cuts, tricks, and memorization, and by and large, teachers take the easy way out and give them what they want.

Barry Garelick said...

"Real mathematics" requires learning basic procedures and skills which Lockhart holds in disdain. I have taught algebra and have taught procedures as well as word problems. If by "short-cuts" you mean things like FOIL, I started off with the "long way" first and then showed the short-cut. I also showed the derivation of the quadratic formula and gave extra credit to students who could show its derivation on a test I gave. I certainly did not take the "easy way out" and I don't know too many other teachers who do that either.

As far as the statement/reason template for proofs, I tended to relyon that format more than the paragraph proof, to get me on my feet. I don't recall proofs being a stumbling block later on in college math courses because of that, so it varies on the individual I guess.

Anonymous said...

I can't speak for Lockhart, but I didn't get the impression from him that he holds learning basic procedures and skills in disdain. The main point I got from his essay is that imaginative, interesting problems have been stricken from the curriculum. Basic procedures and skills are certainly important, but teaching these without imaginative, interesting problems, is like teaching music theory without allowing students to compose or listen to music.

The choice between "procedural skills" and "conceptual understanding" is a false dichotomy. Both should be developed simultaneously, reinforcing the other. The introduction of calculators in the elementary school classroom was one of the worst developments ever. But just teaching procedures without context or motivation is harmful, too.

Maybe I should give you a bit of my own background, so you know where I'm coming from. I have a Ph.D. in math, and have been teaching mostly remedial and lower-division coursework at community colleges and state colleges over the past 10 years.

My remark about teachers "taking the easy way out" mostly comes from my experience with teaching remedial students. And so perhaps my opinion is based on selection bias, as I teach the students that most likely had poor teachers themselves. And it probably applies more to elementary school teachers than high school teachers.

By "taking the easy way out", I mean teaching process without any understanding or intuition at all. This is what I see in my remedial students all the time. They have learned to mimic. But then they do things that betray an utter lack of understanding.

A few examples:

1. In an elementary algebra class a few semesters ago, we were reviewing arithmetic with decimals. Many students think things like (0.2)*(0.3) = 0.6, since 2*3 = 6, or that 0.17 > 0.8, since 17 > 8, etc. We came to the point where I carefully explained why the pencil-and-paper algorithms for multiplication and division work (in terms of keeping track of powers of 10, and place value). A few students came up to me after the class and told me that no math teacher had ever given them this explanation of why the algorithm works; they were simply told what steps to do, how to manipulate the decimal points, and just to memorize the steps.

Anonymous said...


2. In an intermediate algebra class this past fall, I covered powers of i. Rather than just showing how the algebraic rules worked on powers of i, (which I certainly did do), I also explained the intuition behind the pattern, drawing essentially the circle of powers of i in the complex plane (minus the axes), getting them to see how the power could be found from the remainder after division by 4, and then asking how this line of thinking might work for negative powers. This led to a general discussion of what it might mean to say a negative number leaves a remainder, and we drew the number line, using 4 different colors (a sort of introduction to congruence arithmetic). It was at this point that one student angrily shouted, "Why are we doing this? Do we need to know this for the test? Can't we answer these problems without knowing any of this? None of my previous math teachers wasted time like you do! Why can't you just be like them, and just tell us what to do?!" This aligns well with what other students tell me, that almost all their math instruction is geared toward passing standardized tests.

3. When I teach absolute value equations and inequalities, students constantly mix up the different types of problems, because they have absolutely no geometric intuition about the number line, the origin, or absolute value. (In fact, in one algebra class, I gave them 5 very simple signed numbers, and asked them to order them from smallest to largest. Less than 1 in 8 could do so correctly.) We go through the steps of say, |x - 3| < 5, and other similar problems. Then I put on the board, |x - 3| > -5, and ask them what are the solutions. Almost all of them just try to mimic what was done in the previous problem by putting -5 in for 5, switching the inequalities, etc., instead of thinking about the meaning of absolute value. When I ask remedial students what "absolute value" means, almost all of them say "take away the minus sign".

I think the problem starts with elementary school teachers. There's a class taught at many universities with a funny-sounding name, it usually goes by the name of "Math for Elementary School Teachers" or something similar. The idea is that it's supposed to teach them how to teach math to elementary students. In reality, they just end up teaching remedial math to future elementary teachers who can't do math themselves. I know, because I've seen the exams from the classes, and my wife took the class. One of the exams just had the students show they could do fraction arithmetic by hand! Both the teachers and my wife said the vast majority of these future elementary school teachers could not do basic pencil-and-paper arithmetic with fractions, decimals, and percents, nor did they understand place value, the number line, or word problems. They hated math, they were scared of math, and they couldn't do math. And we wonder why students can't do math when they reach high school?

Anonymous said...

Reading my last two comments, I noticed that it might appear that I come across as being from the "fuzzy reform" camp, which isn't the case at all. I also stated the introduction of calculators in elementary classrooms was a bad, bad mistake.

There are 4 major weaknesses I see in my students:

1. Lack of algorithmic, procedural, and symbolic fluency. This is probably the worst weakness, and the most important, because you can't get conceptual understanding off the ground without some level of procedural skills. I've actually been told by fellow college math profs that it's unrealistic to expect my remedial math students to do pencil-and-paper arithmetic problems with multiple mixed fraction or decimal operations. This goes all the way up to calculus, where students are hampered by the fact they can't manipulate rational and radical expressions, fractional exponents, and logarithms.

2. When algorithmic, procedural, or symbolic fluency is learned, a lack of conceptual understanding that leads to misapplication of this fluency. Many students do learn process, but since they don't know how or why it works, they use a procedure when they shouldn't, they use the wrong procedure, or they mindlessly alter a procedure in an invalid way.

3. Retarded problem-solving skills, usually limited to solving single-step or two-step problems. Students seem to have trouble keeping track of anything beyond one or two steps in a problem. This causes a lot of issues doing word problems or anything in a STEM class.

4. Poor English (reading comprehension) skills. Many students have reading abilities so low, they don't know common English vocabulary, and they can't parse sentences, or separate out information, or understand what a question is asking for. This is a major stumbling block for statistics students.

If I had to surmise about the causes of these weaknesses, I would say:

1. Over-reliance on calculators in K-8, and lack of emphasis on standard pencil-and-paper algorithms. This goes not just for whole number arithmetic, but through algebra and calculus.

2. Teaching to standardized tests, which leads students to memorize "cookbook techniques", and then try to match their memorized techniques to test problems, instead of developing general problem-solving abilities.

3. Elementary K-6 school teachers who don't understand math, can't do math, don't like math, and are scared of math. They pass along their attitudes and misunderstandings to their students.

4. Lack of follow-up of algorithmic, procedural, and symbolic skills with conceptual understanding. Yes, procedural skills probably should come first most of the time, but at some point in the future, sometime, explanations should be given.

I have a 4-year old just about to start through the school system in a couple years. So, I guess I'll find out if my opinions are validated.

Okay, I've probably said enough.

Barry Garelick said...

The main point I got from his essay is that imaginative, interesting problems have been stricken from the curriculum. Basic procedures and skills are certainly important, but teaching these without imaginative, interesting problems, is like teaching music theory without allowing students to compose or listen to music.

His lament is more a whine, and what he thinks are dull, uninspiring problems, are not to others, as Katharine made clear in her post. He comes up with another way to prove the proposition that he feels is much more eloquent and to the point. Yet, the two-column proof held in disdain by everyone and his brother these days seems to open up a world of reasoning that would otherwise go unnoticed and unappreciated.

I work with remedial students as do you. I offer contextual explanations when I feel they will be absorbed, but I start with the procedural and go from there.