Summer Help in Math

** Do your children need outside help in math?
Have them take a free placement test
to see which skills are missing.

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.

Note from Laurie Rogers: If you would like to submit a guest column on public education, please write to me at . Please limit columns to about 1,000 words, give or take a few. Columns might be edited for length, content or grammar. You may remain anonymous to the public, however I must know who you are. All decisions on guest columns are the sole right and responsibility of Laurie Rogers.


Geozel said...

This is nonsense. Mental calculations are performed in a way different from the one used in written calculations by means of the so called standard algorithms (in fact, this is a misleading name because from the point of view of math there is absolutely nothing standard about them).

The main difficulty in mental calculations is the neccesity to keep in memory several different numbers at a time. To minimize that, the strategy in mental multiplication is to subdivide the factors into chunks as big as possible and in such a way that multiplying them mentally is easy. Hence, the subdivision 26=25+1. Surely, a good command of the times table is necessary

The written algorithms like the traditional long multiplication were invented specifically to eliminate the difficult memorization by utilizing the structure of the decimal notation instead. The interim results are written down instead of remembering. Further simplification is achieved by performing only the one digit multiplication on every step followed by carrying to the next rank if necessary (in the not so common these days lattice multiplication introduced in Europe by the famous Leonardo di Pisa even the carrying is eliminated during the multiplication steps and only appears during the final addition).

So, strategies in mental multiplication and written algorithms are totally different, and proficiency with the "standard" algorithm is hardly necessary. Its only usefulness possibly shows in that hopefully a person skillful in long multiplication has learned the times table, and that's it.

I can testify from my personal experience. Before learning the written algorithms my classmates and I were intensively trained to performed mental calculations (of the appropriate level of difficulty, of course) including rational subdivision of large numbers into appropriate, convenient for mental handling chunks. It worked fine, I still can do it...

Geozel said...

A minor correction: in my previous comment I subdivided 26 into 25 and 1 when it should be 20 and 6. While writing the comment I forgot what the author used.

C.M. said...

The beauty of the standard algorithm is that it is exactly what you did in mental math: you multiplied the ones place and the tens place and added them together.

From the standpoint of the donut shop cashier, writing the multiplication out on the bag gave her proof of her calculations to her customers (that they would understand--because it is the Standard method) and a written placeholder for herself if they bought other items. Part of proofs is to be able to show your thinking. As you have pointed out in the past, when students do not have proper experience with the standard algorithms or they have not memorized the facts, then they easily get lost (or it takes them much longer) to understand the thinking of others.

Laurie H. Rogers said...

One of most egregious problems I see with reform math is the lack of structure. This leads to errors, and there is no way to backcheck oneself. Naturally, many of our school districts are all about mental math, and it's one of the first things I work on changing when I tutor.

I learned good process as a child, and it has stuck with me. When I subtract something in a checkbook, I automatically backcheck my subtraction by adding things back. I can do that because the work is there in front of me. Everyone makes errors. The key in math is to eventually be correct. Reformers are wrong that "deeper understanding" is more important than correct answers. No, it isn't.
You want to get a right answer before you stop calculating so that you don't accidentally overspend your bank account and cause bank fees. Or build a bridge that doesn't carry a truck's weight. Or accidentally head off for Mars instead of the Moon.

When I work out a problem, I follow a specific process, and it's what I teach my students:
Write the equation. Be sure it says what I want it to say and that the answer is the answer I want. Sometimes this doesn't happen on the first try, but I can eventually see that because I can see it in front of me.
Fill in what I know to be true.
Simplify where possible.
Solve for the variable (s).
Answer the question.
Check my work.
Do it vertically so it's all there in front of me, not scribbled off to the side where I might lose it.

Mental math is good for a limited number of things -- knowing one's multiplication tables, for example. But when it comes to actual work, mental math is counterproductive for children. They cannot see the work in front of them, cannot check it, they forget pieces of it, they get signs backward, they forget parens … and the solution cannot be checked. If they check their answer by doing it all over again, what happens when they get a different answer? Which answer was correct? Who knows? They will have to do it a third time, and they don’t know if they got one right answer and two wrong ones, or three wrong answers.

In our classrooms, in reform math, in discovery learning, in mental math -- children are told they can just plunk down a number. Work is scribbled off to the side, hardly readable, sometimes erased. There is no way to know where things went wrong. The child doesn't know. The teacher doesn't know. The parent doesn't know. Consequently, there is no way to correct what might be a slight misunderstanding, or it might be an arithmetic issue, or it might be a forgotten negative sign… No way to know. And this leads to wrong answers and poor understanding.

Standard algorithms provide structure, understanding, a way to check work, fewer incorrect final answers …

How many children would be good in math if errors didn't cloud their understanding? If the work allowed teachers to see where things went wrong? If parents could see that the process used to get there was counterproductive?

Perhaps that's one reason why proper process is discouraged. No one wants parents to see children calculate 29 x 37 by adding "29" 37 times. Reform math specifically de-emphasizes standard algorithms, filling in with inefficient, ineffective processes. No one wants to alert and alarm the parents …

It's true that once proper process is ingrained, sometimes mental math will flow naturally from that. My daughter has learned proper process, and her quick brain sometimes leaps ahead to the answer before she knows why. But she still does proper process anyway because sometimes her brain forgets something. She knows that and now appreciates a clear way to check herself. My students come to see that with proper process, they make fewer errors. As my husband - a longtime gun pilot - says, "Smooth is fast." If you do it right the first time, you don't have to do it again.

Proper process prevents poor performance. Standard algorithms provide proper process.

CM said...

Can you imagine the cashier writing out the calcuations on your bag using the lattice method?!! LOL