I’ve been thinking about what are the essentials of mathematics education, summarized in a few lines? [This has been motivated by preparing for the AIS talk on Curriculum Issues and Geometry that I am giving on Monday.]
I’ve decided that good mathematics education requires a three-way balance between
1) conceptual understanding
2) algorithmic computation
3) problem solving.
Suppose we are learning about arithmetic with fractions, as so many of us do around the age of 10 (and for many years afterwards, if we don’t understand it properly to begin with!) The conceptual understanding is to have a good feeling for what a fraction is, maybe several different ways of thinking about them: parts of a pie, a point on a number line, the result of the division of one natural number by another, and how these different view relate. The algorithmic computation skills are being able to add, subtract, multiply or divide any two fractions automatically. The problem solving is more open-ended, and tackling a variety of problems we have not seen before; by first putting them into a conceptual framework, perhaps then applying some computation, and then properly interpreting the results. For example, that might be: if twelve people divide a bill totalling $165, and each person has only twenty cent coins (yes rather unlikely!) then how many coins must each person front up?
[Note that you can tell this is an Australian problem: we have twenty cent coins here, not quarters. Our twenty cent coins are bigger, though.]
After chatting about this topic with David Houghton, a high school teacher who is spending the year teaching here in the School of Mathematics and Statistics at UNSW, I’ve realized that a big part of the problem with mathematics education around the world might be that: the second piece of the picture—algorithmic computation—is a lot easier to teach than either the first—conceptual understanding—or the third—problem solving.
So students get more of that second slice, and not enough of the first or third parts of the pie. It’s like getting too much cheese with the pizza, and not enough crust, or other toppings. This can’t be a very original idea, but sometimes admittedly simplistic categorization like this can help us structure our understanding.
How does it explain what it is going on? Common sense suggests that there will be a natural inclination for weaker, or less prepared, teachers to concentrate more on algorithmics (how do you divide 16/3 by 8/5?) rather than concepts (what does 16/3 divided by 8/5 actually mean?) or problem solving (if an alien requires 8/5 of a kilo of asparagus a day to survive, how long will she survive on 16/3 kilos of asparagus?)
Assessing algorithmics is more straightforward than assessing understanding or problem solving; for example it can be done by multiple choice tests, or even with these on-line automatic maths learning programs. Students conspire in this too: they are reassured by the cut-and-dried aspect of an algorithmic approach, and it limits the amount of effort they have to put in. They, and some of their teachers, might even come to think that this is all that mathematics is.
It has many sensible functions everywhere in the world,
in numerous fields and disciplines, and can be utilized in day to day life as well.
https://math-problem-solver.com/ . This project
is lead by the Australian Association of Mathematics Teachers and the Australian Academy of Science.
I don’t believe teachers teach algorithms either. They teach “memorise this table and then apply magic over there”. From a computational perspective there are /far/ more complete algorithms out there, some of which are even suitable for human use. For example, diminished complements allow one to do all additions without needing to memorise a table. But more important than not memorising a table is the fact that addition (and consequently subtraction) can be defined in a base independent manner. Need to suddenly do addition/subtraction in base 16? No problem, do /exactly/ the same thing as you used to in base 10 except for changing a couple of constants.
Given my lack of education (which I’m currently fixing) I have yet to come up with satisfactory formulations of multiplication and division, but similar treatments may be possible.
As someone who has had to reinvent half the planet’s wheels due to current “math education” I unfortunately cannot agree that teachers get even algorithms correct.
Prof. N J Wildberger
For comprehension of PI
Please contact Dr. Gay Bradshaw
Please request interview, i would love to watch/listen