Monday, February 27, 2012

Math education: Arguing over false choices

By Dr. Joseph Ganem, professor of physics, Loyola University
"The astronomer may speak to you of his understanding of space, but he cannot give you his understanding.The musician may sing to you of the rhythm which is in all space, but he cannot give you the ear which arrests the rhythm nor the voice that echoes it. And he who is versed in the science of numbers can tell of the regions of weight and measure, but he cannot conduct you thither. For the vision of one man lends not its wings to another man."----Khalil Gibran, "On Teaching," The Prophet

Imagine a football coach who does not spend practices drilling his team and running plays. Instead, players watch videos of football games, analyze and diagram the actions, discuss the reasons that some plays work and others don't, and plan strategies for upcoming games. His reason for this approach is that drill work is tedious, repetitive, and exhausting. Players will enjoy practice much more if they can study the underlying strategies and concepts of football, have engaging discussions, and learn to think like a professional football player.

We would call such a coach delusional, not because of what he is doing, but because of what he is not doing. Obviously everything he is doing needs to be done, but his team will not stand a chance on an actual football field without putting in hours of tedious, repetitive, and exhausting drill work.

For an activity that has a kinesthetic component it is immediately obvious that learning it will only be possible through repetitive drill work. No one would entertain the notion that they could learn to play tennis by watching the Wimbledon tournament on television, learn to play piano by attending a concert at Carnegie Hall, or learn to dance by going to a performance of the New York City ballet. But, if the activity lacks a kinesthetic component somehow, what should still be obvious no longer is.

Consider the debates on math education that have run on for decades. Should students be taught standard algorithms for operations such as multiplication and division and focus on getting correct answers, or should students be taught conceptual thinking and focus on discovering mathematical knowledge on their own? Educators have argued both sides of  this issue, but in reality it is a false choice.

Without a conceptual understanding of math the subject is of little use. Applying math to real-world problems and knowing if the results of a mathematical analysis make sense requires an understanding of the concepts. But, it is not possible to have a conceptual understanding without with the extensive practice, memorization, and drill work needed to achieve computational fluency.

I tell my students that expertise in any subject, math or otherwise, has three components - facts, skills, and understanding. Each of these components is learned in a different way. Facts are static and must be memorized. Skills are actions that must be practiced in order to become proficient. Understanding evolves and comes only through experience and reflection.

This way in which I think about learning is different than the widely influential Bloom's taxonomy. Bloom saw learning as a hierarchical process, while I see it as an iterative process. Bloom saw separate learning domains - cognitive, affective, and psychomotor - that each had their own hierarchy, while I see the iterative learning process as being much the same in each of the different learning domains.

In Bloom's taxonomy, first published in 1956, the hierarchy in the cognitive domain from the bottom up is: knowledge, comprehension, application, analysis, synthesis, and evaluation. In this model of learning, comprehension (or understanding in updated terminology) is necessary before students can actually do something with their new knowledge. Hence many educational reform movements in the decades following the taxonomy have emphasized "conceptual" learning over practice. However, I disagree with the idea that a conceptual understanding is necessary before higher order activities, such as application, analysis, and synthesis can take place, because understanding is an ongoing process.

Chess as Example

For example, consider learning chess. It is an activity without a kinesthetic component hence it would fall under Bloom's cognitive domain of learning. But no one would believe that the game could be mastered without practice, or that novice players could discover the principles of strong play on their own.

To learn chess an aspiring player must memorize the names and movements of the pieces, and the object and rules of the game. These are what I refer to as facts. But the acquisition of skill in playing the game requires a program of study and practice. In order to improve, players must read texts on chess tactics and strategies and attempt to implement those ideas by playing actual games. There is no substitute for practice, but at the same time players must learn additional facts (acquire more knowledge).

However, an understanding of chess evolves in time. A novice, a skilled player, and a grandmaster can all look at the same chess position. The novice will see individual pieces. The skilled player will see groups of pieces. The grandmaster will see the entire position.

But if the grandmaster articulated his understanding of the entire position to the novice, the narrative would be of limited use. The novice would not have the knowledge base and the skills necessary to make sense of most of what a grandmaster would say about a given chess position. But that does not mean that the novice is incapable of applying, analyzing and synthesizing chess ideas. Those ideas might be relatively crude, and obvious to the grandmaster, but the process is necessary to reach a high level of understanding. It is for these reasons that I view learning as an iterative process.


Experts are experts because they do think about their subject of expertise differently than novices. But those thought processes cannot be transferred directly to a student, they must develop through study and practice, and there is no shortcut to that development. This should be especially obvious in a subject such as math but apparently it is not.

Many years ago, before calculators and optical scanners had been invented, I made a purchase at a bakery counter tended to by a young woman who had to pencil in prices on the bags of pastries being sold. I asked for 5 donuts priced at 26 cents each. She placed them in a paper bag and on the outside of the bag she computed 26 x 5 using the standard algorithm for multiplication that I, and countless other students, had learned in grade school. She of course was very proficient at multiplication problems using this method, because throughout the day, everyday, a steady stream of customers patronized the bakery counter.

Before she could write out the problem, I said to her: "It's $1.30." She completed the problem, writing all the steps on the bag, and the result was $1.30. Startled by my seeming clairvoyance, she looked at me for an explanation. She knew of no other way to multiply but the standard algorithm, and that process required time and writing. How could I multiply the numbers instantly in my head and arrive at the correct answer?

I said to her: "If the donuts were 20 cents each how much would 5 cost?"
She replied:" A dollar."
I said: "And what is 5 times 6?"
She understood immediately what I had done, but only because she was already proficient at multiplication. If I tried to teach my methods for doing mental math to people not already proficient in the use of standard algorithms, my explanations would lead to confusion rather than enlightenment.

Real learning is iterative, not hierarchical, and it doesn't matter whether the subject is, to use Bloom's terminology, in the cognitive, affective, or psychomotor, learning domains. However, the desire of educators to systematize learning often leads to rigid ideologies riddled with false choices. The argument over whether math instruction should focus on concepts or computation is in many ways analogous to the argument on whether reading instruction should focus on phonics or whole language. Fluent readers use and understand both approaches.

Likewise, learning math is an iterative process that cycles between concepts and computation. Experts in math are proficient in both because it is impossible to master one without the other.


Joseph Ganem, Ph.D., is a professor of physics at Loyola University Maryland, and author of the award-winning book on personal finance: The Two Headed Quarter: How to See Through Deceptive Numbers and Save Money on Everything You Buy. It shows how numbers fool consumers when they make financial decisions. For more information on this award-winning book, visit His article is reprinted here with permission of Dr. Ganem. This article was previously published on The Daily Riff.

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