Wednesday, January 07, 2009


Should Children Learn Math by Starting with Counting? by Keith Devlin

Thanks to Rob (blog) for putting me onto the Devlin's Angle essays. Still reading. He is well researched, discusses fundamental issues of maths pedagogy and is making me think.

This particular essay made me think about Number and how it is taught in Schools. To tell the truth I had to go back and review my own foundational understanding of number. The way that it is taught in our Schools is like a drip feed but for most students and teachers the "big picture" is never put together. It's like a jigsaw puzzle that is seldom completed.

Hence my need for an overview even though this terminology is not definitive, eg. see Wolfram for a more systematic approach:

natural number (also called counting number) can mean either an element of the set {1, 2, 3, ...} (the positive integers) or an element of the set {0, 1, 2, 3, ...} (the non-negative integers)

whole numbers: 0, 1, 2, 3 ...

integers ... -2, -1, 0, 1, 2 ...

algebraic numbers: integers, roots of positive numbers, fractions

rational numbers: 42, -23/129 (integers, fractions)

irrational numbers: pi, sqrt (2) infinite decimals

real numbers are the numbers that can be written in decimal notation, including those that require an infinite decimal expansion ... OR points on a infinitely long number line

imaginary numbers (unreal): i, 4th root(-9)

complex numbers (mixture of real and unreal): 2 + 3i

At any rate, Devlin points out that in the USSR due to the influence of Vygotsky and his followers, such as Davydov, that the curriculum from the beginning focuses on real numbers, it does not begin with natural numbers, as we do in Australia (or in Devlin's case the USA). He gives a broad overview of that curriculum in his article.

Here is a quote from Devlin which provides a rationale for the validity of this approach:
Humans have not only a natural ability to abstract discrete counting numbers from our everyday experience (sizes of collections of discrete objects) but also have a natural sense of continuous quantities such as length and volume (area seems less natural), and abstraction in that domain leads to positive real numbers.

In other words, from a cognitive viewpoint (as opposed to a mathematical one), the natural numbers are neither more fundamental nor more natural than the real numbers. They both arise directly from our experiences in the everyday world. Moreover, they appear to arise in parallel, out of different cognitive processes, used for different purposes, with neither dependent on the other. In fact, what little evidence there is from present-day brain research suggests that from a neurophysiological viewpoint, the real numbers - our sense of continuous number - is more basic than the natural numbers, which appear to build upon the continuous number sense by way of our language capacity
Fascinating. We need to explore fundamental knowledge deeply. This is probably the most important part of improving education.


rob said...

also some nice maths sound bites from Devlin here

less detailed, but a nice media presentation of maths

Peter William Lount said...

Really interesting.

I wonder what Alan Kay and his group of researchers who work/worked with children have to say on this?

I've had to put together the number system picture for myself since the school systems that I was in were scattered too in their teaching methods.

Also, I've thought about this for sounds and letters. Why is it that we are only taught the English letters in schools? Ok, Spanish or French or whatever, it's always a limited set that's taught (at least to most people that I've ever met and talked about language learning).

Why not teach the full set of letters and sounds from all the human languages to kids when they have the ability to learn it the best?

Powel Janulus, a man who knew 14 languages as he grew up told me that it was easiest as a child. Later on he went on to learn 50 languages fluently and another 30 partially - enough for basic conversations. Unfortunately his methods are basically lost as he seems to have passed away a few years ago. To him learning the sounds of each language, while important wasn't the most important aspect but it was up there as he attempted to speak fast and sound like a native while getting the "concepts" across. He listened well for the sounds and any words or other hidden changes as the sentences he was testing were altered slightly. Notably he didn't care about grammar.

I wonder if that's also true for math, that not worrying about "math grammar" but focusing on the concepts first is better. The grammar can come after.

In that sense would real numbers being a super set of natural numbers be a more general concept to learn first thus making the other number categories easier?

It was also very interesting that Powell said that after the 25th language it became childs play to him. I wonder if learning to be fluent in many languages is like learning real numbers first before other simpler number concepts? Is there a full set of language atoms/concepts that can be taught to enable people to learn languages better? Powell thought so. I miss his lessons and wish I had learned more from him while there was time.

Anonymous said...

algebraic numbers: integers, roots of positive numbers, fractions

Strictly speaking, an algebraic number is a root of a polynomial over the field of integers (ie: the coefficients of the polynomials are integers).

In the most commonly recognized cases, algebraic numbers are, as noted, rational products of roots of positive integers. However this is not always the case. it is commonly known that e is a transcendental (non-algebraic) number, and that e^pi is also transcendental.

The last time I studied number theory, nothing was known about pi^e (it could be either algebraic or transcendental), though it does not fit the familiar form described above.


Tony Forster said...

"continuous number - is more basic than the natural numbers, which appear to build upon the continuous number sense by way of our language capacity"

Thanks I had a real Aha! moment reading this. It was as if I could feel my brain hemispheres swapping as I conceptualised reals (speed, force, distance) and naturals (One Two Three).

So maybe maths is better conceptualised with real-world reals, (ie multiplication as force*time or speed*time)?

But then again, maths is a language of symbols as well as a set of relationships, it lives somehow in both hemispheres of the brain. Interesting, thanks.