A year level coordinator once told me about an English teacher who made the students repeat a poem until they could recite it by heart. That took place during an excursion. When they were back at school, many students commented that they had never realised one could remember something if one repeated it over and over! These were 15 year olds who had not developed strategies for memorisation. I once read about Western hostages in Beirut, and how they kept themselves sane by reciting their favourite poems.

The other day I was describing a project on the Pythagorean Theorem to my nephew, a French educated 17-year old from the middle east. I first asked him if he knew the theorem. He thought for a couple of seconds and said: "Dans un triangle carré, Le carré de l'hypoténuse est égal à la somme des carrés des deux autres côtés" (In a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides).

Did the fact that he knew it by heart mean that he understood little of it? Not at all. It gave him the necessary vocabulary to describe his understanding. As we walked on, I described the geometric proofs that my students had to describe as part of their project. He had never seen those proofs before, but we could discuss them abstractly, without having the pictures in front of us. I could use words like "somme" (sum), "surface" (area) and "longueur" (length), with which he was entirely comfortable.

I then mentioned the concept of a proof by induction, something taught in the first year of a science degree at a typical Australian university. He said, "yes, I know what that is. Induction is the opposite of deduction in that you begin with a particular case and generalise. In deduction, you apply a general rule to a particular case."

Let me temper all this by stating that I am not a "back to basics" teacher. I teach algorithms and shortcuts only when absolutely necessary. I do believe that mathematics needs a context, and that understanding is paramount. My complaint is that we seem to have thrown the baby out with the bath water. We often speak as though learning by rote is a poor alternative to learning with understanding. I think that we can use some rote learning to support understanding. Let me know what you think.

Elias.

Categories: Education

## 7 comments:

yes, i agree totally.we have got the "rote vs understanding" out of balance in my opinion.

In combination, there is powerful learning to be had!

Here here, Elias!

Always good to see your comments, TB. Thanks for your contributions to this blog.

Elias.

To the extent that mathematics is like another language, it requires of students that they learn it as such.

To learn another language, students first need to manipulate individual sounds and words in that language with only a limited understanding of those elements. This is how we all learn to speak our languages.

And to learn to fluency requires, in my opinion, a far greater focus on "mindless" manipulation of words and ideas than it does a focus on objective, deep understanding of the language.

Your post reminded me of this paper by Wu Basic Skills Versus Conceptual Understanding* A Bogus Dichotomy in Mathematics Education

Thanks for your comments, Mr. Person and Dr. P. The University of Melbourne's website used to carry a paper by Professor David Clarke in which he argues against other false dichotomies, such as telling vs not telling and teacher-centred vs student-centred classrooms.

Elias.

I am a student of a Buddhist teacher who thinks rote learning is the way we should be taught. I have no problem withrote learning if at some point there is follow-up, but what I find on a consistent basis is people "parrot" and there is no understanding of what they have said. If one doesn't understand what is parroted, the person cannot explain it as they have no understanding or real knowledge. It starts to seem like our current corporate culture with it's slogans about ethics, customer care, team work etc., but no one reallyhas any understanding of what they are repeating. Conversations withpeople are starting to portray this, people repeat things without any thought and if you question it youfind silence or stumbling stupidity. Roter learning must be followed up with an explanation. It this were the way to go then there would be no problem with education inthe u.s., as everyone would know the answers to the questions on standardized test. The reasonthey don't is that they have memorized the question, answer and if you ask with a different word order, theydon't recognize what you are saying. I have been told too many times my answer is wrong, because it wasn't answered EXACTLY as one remembered. They also are unable to explain in their own words because they haveno idea what it means.

You're quite right, Mr. Elias. I often tell my students that knowing their multiplication tables is important because it makes one totally familiar with the _concept_ of multiplication, which is important when you start dealing with algebra. A lot of algebraic misconceptions occur when students forget that things are being multiplied.

This is just one example. Most people come to understand something by relating it to things they're already familiar with. To build up a stock of familiar facts and concepts, you need good grounding and practice in "the basics".

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