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Is there a place for invention in math?

By Murray Bourne, 09 May 2011

Eminent child psychologist Jean Piaget said:

Each time one prematurely teaches a child something he could have discovered for himself, that child is kept from inventing it and consequently from understanding it completely.

Have you ever considered the fact that all math notation (and many math concepts) was invented by someone at some point in history? This includes the numbers we use, symbols for "equals", "plus", "more than", algebra, graphs, tables, and so on.

Unfortunately, mathematics is most often taught with the assumption that there is one correct answer and one correct way of writing that answer. It's unfortunate because it gives the wrong idea about math and relegates most math lessons to being a case of "this is the right way" and "that is the wrong way".

It's fascinating the way childtren make sense of the world, and how they connect what they know from other areas to the task at hand. In the case of math, that means a lot of invention goes on until they eventually figure things out.

But something happens around the early teenage years (the worst possible age to impose ideas), the time students first come across algebra - math becomes more dogmatic and the message becomes "it must be done this way only, and written this way only."

Some interesting case studies

I recently read Mathematical Development in Young Children - Exploring Notations by Barbara M. Brizuela.

Brizuela's delightful short book allows us to see inside the heads of 5 children, ages 5 to 8, as they grapple with their first experiences of the base-10 number system, pre-algebra concepts, graphs and tables.

Here's a summary of some of the case studies in the book.

George - numbers

We first meet 5 year-old George, who is coming to grips with our base-10 number system, and the way it is written. He writes "ten" as "01", "nineteen" as "90" and "seventeen" as "70".

There is logic there: 19 = 9 + 10, so "90" follows, especially considering the words "ninety" and "nineteen" are very similar.

George writes "forty one" as "401". This makes sense to me.

Of course, we all need to end up (more or less) on the same page in math eventually. We can't have everyone inventing inconsistent systems or there would be chaos.

But for children at this age (and for older ages, I would argue), we beed to give space for "sense making". Insisting on the "correct" answer too soon removes thinking, removes observation and removes the joy of discovery.

Paula's "capital" numbers

Paula is also 5 years old and she has invented a neat concept. From language, she has learned that we start a sentence (and people's names) with a capital (upper case) letter.

So when describing the number "3", she calls it "normal", but the 3 in the number "31" is called a "capital 3". Her understanding of number is strongly influenced by her understanding of the written language. She's giving the leading 3 more importance than the 1, and in reality, that is the case.

Like many children, Paula can see a connection between "4" and "40", and "5" and "50" (since they are fairly logical extensions) , but understandably struggles with the connection between "one" and "eleven".

Aside: In this aspect, Chinese and Japanese number system are easier for children since the words used are more closely aligned with their meanings, compared to English. For example, in Japanese, 2 is "ni" and 10 is "juu". Number names represent either sums or products of numbers. So we get:

12 = 10 + 2 (pronounced "juu ni")


20 = 2 × 10 ("ni juu")

In Japanese, the number "22" is "ni juu ni" or 2 × 10 + 2. Some researchers believe that Asian children are generally quicker to pick up math concepts compared to English-speaking chiildren because there is less congnitive load involved in the concepts. (See Why East Asians do well in math.)

Here's a video showing Japanese numbers.

Thomas' periods and commas

Six year-old Thomas also brings his knowledge of writing English to his understanding of mathematics.

He knows that "period" means "stop" and that "comma" means "pause" when writing sentences. He is also quite comfortable with how money is written (from repeated exposure and experience) and he recognizes "9.91" as $9.91.

Thomas writes "ten thousand" as 10.000 (with a period, not a comma) and "one hundred thousand" as 100.000 (also with a period).

Interestingly, he writes "seven thousand and forty" as 70,40. That almost makes sense, when most "thousand" numbers have the comma at the point where we would say "thousand".

Thomas is on the road to "getting it" and as long as he is given space, all will become clear.

Here's one of the many places where math notation is actually not all that consistent. In Europe, the comma is used as the decimal place indicator (where we use period, or "decimal point"). This is because the French were already using a period to make writing Roman numerals easier, so they used the comma instead. There's some interesting background on this here: Decimal mark.


Am I advocating a constructivist-only approach to teachiing math? Unfortunately, most schools have very packed schedules and if we allowed students to "discover" all the things they need to know, the time would be insufficient.

However, we need to allow students to figure out some of what they need to know. They need to make connections by themselves from time to time. The activities need to be well-designed, so that students have a good chance of discovering things efficiently - and enjoyably.

Next time you are about to teach students something, try to think of an activity where they will come to the desired conclusion by themselves. You are designing for the "ah-ha" and "oh yeeeaah" moments - and these are very satisfying for both student and teacher.

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