# Teaching for understanding and teaching procedures

Many critics of the current state of mathematics education take issue with asking students to explain their reasoning. They’d rather students just apply an algorithm and get the answer.

The following is quoted from QED: The Strange Theory of Light and Matter, where Richard Feynman describes how he’s going to explain for a lay audience the techniques behind quantum mechanics that earned him a Nobel Prize. (By the way, I highly recommend this book.)

How am I going to explain to you the things I don’t explain to my students until they are third-year graduate students? Let me explain it by analogy.

The Maya Indians were interested in the rising and setting of Venus as a morning “star” and as an evening “star” – they were very interested in when it would appear. After some years of observation, they noted that five cycles of Venus were very nearly equal to eight of their “nominal years” of 365 days (they were aware that the true year of seasons was different and they made calculations of that also). To make calculations, the Maya had invented a system of bars and dots to represent numbers (including zero), and had rules by which to calculate and predict not only the risings and settings of Venus, but other celestial phenomena, such as lunar eclipses.

In those days, only a few Maya priests could do such elaborate calculations. Now, suppose we were to ask one of them how to do just one step in the process of predicting when Venus will next rise as a morning star – subtracting two numbers. And let’s assume that, unlike today, we had not gone to school and did not know how to subtract. How would the priest explain to us what subtraction is?

He could either teach us the numbers represented by the bars and dots and the rules for “subtracting” them, or he could tell us what he was really doing: “Suppose we want to subtract 236 from 584. First, count out 584 beans and put them in a pot. Then take out 236 beans and put them to one side. Finally, count the beans left in the pot. That number is the result of subtracting 236 from 584.”

You might say, “My Quetzalcoatl! What tedium, counting beans, putting them in, taking them out – what a job!”

To which the priest would reply, “That’s why we have the rules for the bars and dots. The rules are tricky, but they are a much more efficient way of getting the answer than by counting beans. The important thing is, it makes no difference as far as the answer is concerned: we can predict the appearance of Venus by counting beans (which is slow, but easy to understand) or by using the tricky rules (which is much faster, but you must spend years in school to learn them).”

To understand how subtraction works – as long as you don’t have to actually carry it out – is really not so difficult.

That’s my position: I’m going to explain to you what the physicists are doing when they are predicting how Nature will behave, but I’m not going to teach you any tricks so you can do it efficiently. You will discover that in order to make any reasonable predictions with this new scheme of quantum electrodynamics, you would have to make an awful lot of little arrows on a piece of paper. It takes seven years – four undergraduate and three graduate to train our physics students to do that in a tricky, efficient way. That’s where we are going to skip seven years of education in physics: By explaining quantum electrodynamics to you in terms of what we are really doing, I hope you will be able to understand it better than do some of the students!

In the same way, I want students in 2nd and 3rd grades to understand what they are really doing when they subtract, and not just mindlessly follow a procedure to get an answer that they do not really understand.

Where I tend to agree with most critics of the Common Core is that students are asked to write miniature essays to explain their reasoning, and that’s probably a bad idea. Even though I want students to understand why subtraction works, 2nd and 3rd graders are still learning how to write complete sentences and can get easily frustrated with explaining their reasoning in paragraph form. I think there are better ways (like drawing pictures) of assessing whether young children really understand subtraction that is more developmentally appropriate.