When Are We Ever Gonna Use This?

Raise your hand if you have ever heard this question, “When are we ever gonna use this?” When I was a young teacher, I tried hard to answer. I used to give my students (junior high and high school) examples of math problems from various occupational fields. I bought a large poster that listed many occupations along the top and many mathematics topics along the side with black dots showing exactly which occupations use which topics.

Years passed. Film projectors gave way to Youtube videos. Mimeograph machines gave way laser printers. Whole new field of occupations emerged. I metaphorically threw up my hands in exasperation. When the inevitable question arose, I answered that I had no idea how they were going to use this information. I had no idea how their interests would develop, or which occupations they would pursue, or what the jobs of the future would be. All I could do was teach them a little bit of what had taken thousands of years for people to discover about math. My students were not always satisfied.

Then Paul Lockhart came along and wrote “A Mathematician's Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form.” https://www.maa.org/external_archive/devlin/LockhartsLament.pdf

Now I had an answer that captured their imaginations:

“In any case, do you really think kids even want something that is relevant to their daily lives? You think something practical like compound interest is going to get them excited? People enjoy fantasy, and that is just what mathematics can provide -- a relief from daily life, an anodyne to the practical workaday world….People don’t do mathematics because it’s useful. They do it because it’s interesting … The point of a measurement problem is not what the measurement is; it’s how to figure out what it is.”

The question of the usefulness of any particular subject stems from the mutual internalization of both the teacher and students of a questionable, yet unexamined assumption.

“To say that math is important because it is useful is like saying that children are important because we can train them to do spiritually meaningless labor in order to increase corporate profits. Or is that in fact what we are saying?”

Thus instead of teaching real mathematics, we teaching “pseudo-mathematics,” or what I have often called non-math, and worse, we use math class to accomplish this miseducation (See https://schoolcrossing.blogspot.com/2012/11/tricks-and-shortcuts-vs-mathematics.html and others). According to Lockhart, we teach math as if we think “Paint by Number” teaches art.

“Worse, the perpetuation of this “pseudo-mathematics,” this emphasis on the accurate yet mindless manipulation of symbols, creates its own culture and its own set of values….Why don't we want our children to learn to do mathematics? Is it that we don't trust them, that we think it's too hard? We seem to feel that they are capable of making arguments and coming to their own conclusions about Napoleon. Why not about triangles?

Math is like playing a game. As with any game, it has rules to be sure. However, it is more fun and more elegant than all other games because it is literally limitless.

Physical reality is a disaster. It’s way too complicated, and nothing is at all what it appears to be. Objects expand and contract with temperature, atoms fly on and off. In particular, nothing can truly be measured. A blade of grass has no actual length. Any measurement made in the universe is necessarily a rough approximation. It’s not bad; it’s just the nature of the place. The smallest speck is not a point, and the thinnest wire is not a line. Mathematical reality, on the other hand, is imaginary. It can be as simple and pretty as I want it to be. I get to have all those perfect things I can’t have in real life. I can never hold a circle in my hand, but I can hold one in my mind. […] The point is I get to have them both — physical reality and mathematical reality. Both are beautiful and interesting… The former is important to me because I am in it, the latter because it is in me.

Mathematics offers infinite possibilities for storytelling. I tell many stories as I teach math. My students are positively enchanted and remember them forever. One of my favorites is the kimono story.

I tell my students how in old Japan, servants helped geisha to put on the multiple layers of kimono. Each layer has to arranged and offset just so in order to reveal the colors of each layer. I tell them we are going to start with a geisha like 1/3. First we put on the 2/2 layer. 1/3 x 2/2 = 2/6. Notice that the geisha looks a little different, but underneath it is the same geisha. How about another layer, maybe 3/3. Okay 2/6 x 3/3 = 6/18. How about another 2/2 layer. 6/18 x 2/2 = 12/36. We can take off the layers one-by-one as well. This is called “simplifying a fraction.” Simplifying a fraction is simply a process of finding out which geisha is at the bottom of all those layers. If we are in a hurry, we can remove all the layers at once. How would we do that? In the case of our geisha, dividing by 12/12. The students love it.

The most elegant math story is the proof.

A proof is simply a story. The characters are the elements of the problem, and the plot is up to you. The goal, as in any literary fiction, is to write a story that is compelling as a narrative. In the case of mathematics, this means that the plot not only has to make logical sense but also be simple and elegant. No one likes a meandering, complicated quagmire of a proof. We want to follow along rationally to be sure, but we also want to be charmed and swept off our feet aesthetically. A proof should be lovely as well as logical.

The Candle Problem: How to Damage Motivation

Herbert Kohl says we are missing the boat, motivation wise, in an open letter to Arne Duncan, Secretary of Education.

Now the mantra is high expectations and high standards. Yet, with all that zeal to produce measurable learning outcomes we have lost sight of the essential motivations to learn that moved my students. Recently I asked a number of elementary school students what they were learning about and the reactions were consistently, “We are learning how to do good on the tests.” They did not say they were learning to read.

Mr. Kohl sees a fundamental contradiction between what we say we want and what we are doing to get it.

It is hard for me to understand how educators can claim that they are creating high standards when the substance and content of learning is reduced to the mechanical task of getting a correct answer on a manufactured test.

What, for Mr. Kohl, motivates learning, at least for learning to read?

...reading is a tool, an instrument that is used for pleasure and for the acquisition of knowledge and information about the way the world works. The mastery of complex reading skills develops as students grapple with ideas, learn to understand plot and character, and develop and articulate opinions on literature.

Nowhere does Mr. Kohl mention extrinsic rewards. Teachers have observed, and Robert Slavin's research has confirmed the dissipating effect of extrinsic rewards.

Robert Slavin's position--that extrinsic rewards promote student motivation and learning--may be valid within the context of a "facts-and-skills" curriculum. However, extrinsic rewards are unnecessary when schools offer engaging learning activities; programs addressing social, ethical, and cognitive development; and a supportive environment.

Not only do extrinsic rewards fail to motivate, except in limited cases, but research has also found that extrinsic rewards actually sabotage motivation.



So what's with the ubiquitous classroom token economies? Why must teachers have jars ofmarbleson their desks? Are we deliberately sacrificing long-term learning benefits for short-term classroom management? How about pay-for-performance or merit pay? First. And foundationally, EVERYONE deserves to be paid FAIRLY. “Getting the issue of money off the table,” as Dan Pink says.

If our society want to motivate the highest performance from teachers, then give them:

Autonomy
Mastery
Purpose

NOT merit pay.

Merit pay is inherently unfair. The bug-a-boo with merit pay is that teachers have so little control over the factors that impact student achievement. What do we say, for example, about the student who actually scored worse after his first year with me only to leapfrog three grades the second year with me.  Should I have lost pay the first year?  I was still the same great teacher.  I had no idea his alcoholic uncle moved in with him and his mom that first year. What do you do if you are a great teacher in an environment where just about everything seems to be conspiring against the kids? And what if you are lucky enough to teach in a school where kids have all kinds of advantages and their scores show it regardless of who is their teacher? Policy-makers have not figured out any equitable mechanism for awarding merit pay.