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")
and
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.
Conclusion
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.
See the 13 Comments below.
10 May 2011 at 4:03 am [Comment permalink]
In the Bulgarian language the numbers are organized almost perfectly, similar to the Japanese system you mentioned above.
In Bulgarian the digits are: "edno", "dve", "tri", "chetiri", "pet", "shest", "sedem", "osem" and "devet". For "ten" we say "deset". And then it continues:
11: "edno nad deset" (one over ten) spelled together "edinnaddeset"
12: "dve nad deset" (two over ten) spelled "dvenaddeset"
13... etc
Then it comes 20 as "dva deset" (two tens) spelled in one word as "dvadeset". And continues:
21: "dva deset i edno" (two tens and one) spelled "dvadeset i edno"
...
26: "dva deset i shest"...
... etc
It continues similar. For example 58 is "petdeset i osem" (five tens and eight), etc.
When we reach a hundred (100) we call it "sto". And it continues the same way. For example 179 is "sto sedemdeset i deved" (a hundred with seven tens and nine).
Etc. 1000 is "hiliada". 10000 is "deset hiliadi" (ten thoudands). 100000 is "sto hiliadi" (hundred thousands). 1000 000 is a "milion" like in English.
Final example 1368184 is spelled as "edin milion trista shestdeset i osem hiliadi sto osemdeset i chetiri" (one milion three hundred and sixty tens and eight thousands and a hundred with eight tens and four).
10 May 2011 at 9:01 am [Comment permalink]
Thanks for the insights, Philip. Do you think this has a positive impact on learning math, compared to the number system used in English?
10 May 2011 at 2:05 pm [Comment permalink]
Absolutely yes. The children see the logic in maths from the very beginning - learning numbers. It is very easy after that to move on to building the base. This is what I call a "flying start". By the way in Bulgaria the children are traditionally much better in maths compared to other subjects.
I can give you a counter example from a Slavic language which is very close to ours. I do have friends mathematicians in Russia, which are teaching in the kinder garden and first classes in schools. The grammatical numerals in Russia are a big problem for them. For example counting years is:
1 god (one year)
2 goda (two years)
3 goda
4 goda
5 let (five years)
6 let
7 let
8 let
9 let
10 let
11 god (eleven years)
21 goda (twelve years)
...
24 goda
25 let (twenty five years)
...
101 god
102 goda
...
108 let
...
Ok, the inflected language makes it very beautiful and rich - I totally agree. But on the viewpoint of maths - I do think that it does not help much. Especially the "exceptions" in the language system are a killer for math thinking. There are no exceptions from the rules in math, yes!
10 May 2011 at 2:40 pm [Comment permalink]
Why is the decimal point so little. It has more impact on value than any number so it should be bigger
10 May 2011 at 2:44 pm [Comment permalink]
Good point, Bruce. The "multiplication dot" used in the US is usually bigger than the decimal point (and usually aligned above the baseline), so is fairly easy to discern.
In many fonts, and especially with small font sizes, the comma is also quite hard to read.
Hey, let's have new symbols for both. Maybe a bullet (•) for decimal point, and semi-colon (;) for the 3-digit spacer (thousand, million, etc) may work better?
10 May 2011 at 2:48 pm [Comment permalink]
In Bulgaria we do not put any periods in numbers. It is a sign for multiplication. For example the students will recognize 100.10 as the result 1000. We use comma for your "decimal point" (we call it "desetichna zapetaia" or "decimal comma" in English).
When we have large numbers we put spaces. For example a billion (we call it "milliard") will be written as "1 000 000 000".
10 May 2011 at 2:53 pm [Comment permalink]
Furthermore for the language side of the problem - when we have a "decimal comma" we pronounce it the following way: 10,45 is "deset cialo i chetireset i pet" (ten whole and fourty five). The word "cialo" indicates that any number before it will be a whole number before the "decimal comma". Everything after it is after the "decimal comma". Students understand it very easy.
I am unsure how you will spell 10,45 in English. Can you enlighten me? I am sorry, my English is not perfect (as you may see above) 🙂
10 May 2011 at 2:58 pm [Comment permalink]
That's interesting - it's almost like "cialo" has a similar function to Paula's "capital" - that is, it alerts the listener that the more "important" number has finished, and a less important one comes afterwards.
Actually, "10,45" doesn't really exist as a number in English. If it was not in a context (of money, say, especially with a euro symbol in front), then it would be just confusing. We wouldn't know if a number was missed at the end (10,456 is a "real" number), or if the comma was put in the wrong place (1,045 makes sense).
10 May 2011 at 3:05 pm [Comment permalink]
Yes, "cialo" (whole) indicates explicitly that the important number ends.
Well, OK sorry, I meant "10.45" (with decimal point). We write it with comma here instead. How do you pronounce it? 🙂
P.S. I think the easiness of the math notations and language constructions in Bulgaria are coming due the fact that our country was a long time under the Otoman Empire slavery and practically we did not have even normal schools. The rest of the world was making evolutions in the math while we was not having any math education at all. Practically our serious math books started to be written after 1890. The biggest boost in science (plus some language reconstruction together) came with the communism after 1945. That's why probably we did "fix" some notation and linguistic flaws".
10 May 2011 at 6:42 pm [Comment permalink]
Philip:
If I were to read 10.45 I would normally say "ten point four five". If I were reading it as a price I would say "ten forty-five". If it needed clarification, "ten dollars and forty-five cents".
The actual formal way of saying it is "ten and forty five one hundredths", which would be used in school but probably not by anyone who was using numbers for practical purposes because it's too cumbersome.
One more way you might hear it is "ten dot four five". I am not sure when I started saying numbers like this. I think it might have to do with dealing with ip addresses like 10.0.0.1, where we are used to saying 'dot' for the decimal point.
10 May 2011 at 6:45 pm [Comment permalink]
One thing that might be of interest--in order to make large numbers easier to read, the perl programming language lets you use underscores in the number, so you can write ten million as 10_000_000.
10 May 2011 at 7:45 pm [Comment permalink]
I wish all kids would have more time to discover things, or at least enough time to explore and work out what is being taught them.
My 5 year old daughter was recently 'teaching' her toy cats their letters and numbers. She said, "I'll write 100 really BIG, because it's a big number, and 1000 even BIGGER. And then a quadrillion!" She's a very visual girl, drawing all the time, always measuring, comparing sizes, so this makes sense from her.
10 May 2011 at 8:07 pm [Comment permalink]
Thanks msouth.
The example from Perl is very good. I was thinking from a long time to make a translator for the C programming language which will change notations.