How to teach or learn mathematics?

Today’s Occupy Math is triggered by an attempt to improve math education in Ontario. This headline in the Toronto Star is one of many that have appeared recently. The problem is that the province’s standardized math test scores have been slipping for twelve years. This problem has a lot of moving parts. Probably the largest is a complete confusion about the goals of teaching mathematics. Which of the following are actually priorities of the current program?

  1. Teach basic math skills (this is a goal but FAIL).
  2. Teach problem solving (actively discouraged by the teaching methods).
  3. Teach algorithms for specific tasks, e.g. long division (de-emphasized).
  4. Teach students how to discover mathematics for themselves (this is a goal but FAIL).

Discovery learning, the core of the government’s curriculum update, is really hard to do well. The person supervising it and prodding it along has to know the subject and its history to the hilt. This knowledge is needed to guide discovery into productive channels, to reward good discoveries with praise, and to help add the missing bits to partial discoveries. This need for substantial scholarly ability spits in the teeth of remarkably absent teacher training by the government. When a good month of training is needed to help people learn discovery learning techniques, the proposal is to add one day of training a semester.

All of Occupy Math’s complaints need to be taken with a grain of salt – there are good teachers who can make the current system work – but we need a system that will let an average teacher do a good job. Look at Data Management. This course, in theory an introduction to statistics, was dropped out of the sky with completely inadequate teacher training. The most applied math taught in the high schools became the math for people that were not going to use any math in their careers. The math used to evaluate claims and defend people from con artists became the course for people that didn’t really need math.

The big problem with math education in Ontario is that it’s being run by micro-managers with no understanding of how to effectively teach math.

The government is planning to drop an additional 60 million dollars on math education. Occupy Math has read the entire announcement twice and cannot tell what sort of math teaching is going to be emphasized. Alarmingly, there is talk of shifting more strongly to discovery based learning, at which the province has already demonstrated substantial incompetence. A study in Forbes found that More Spending Doesn’t Lead To Improved Student Learning. The Forbes conclusion – that we probably shouldn’t spend more money seems wrong. Instead, we should stop wasting money on things that don’t help.

Discovery learning is based on the (correct) premise that what you discover for yourself, you remember and understand more completely.

So how could discovery learning be a bad thing? The basic problem with using discovery learning on basic math skills is that they took about a thousand years (not a joke ™) to figure out the first time and cannot be re-discovered in any reasonable amount of time without some drill-style education and a modest amount of proscriptive guidance. What the province is doing now does not work and they are planning to do it much harder.

One of the best ways to do discovery learning is to state a problem and then devise a diverse collection of solutions. The old joke, how do you divide five apples among six people is a wonderful starting points because “make applesauce” is only one of several workable solutions. Occupy Math’s colleagues with children in school are being asked to figure out algorithms for basic operations like finding the median or average of a collection of numbers in abstract without the context of a problem to solve. This is an excellent way to be confusing while making math seem pointless.

Occupy Math thinks that some drill in the basics blended with an emphasis on connections with other disciplines and applications to problems might work wonders.

Math is often taught in a vacuum, without reference to its appearance all through creation. One of Occupy Math’s goals is to connect math to snowflakes, fractals, games, puzzles, and mysteries like the number of petals on a flower. Given that the low-level directions to the entire universe are written in solid math, it seems a plot of the dark one that math is viewed as boring and pointless.

Occupy Math has had three children go through high school math education in Ontario. During that time, the ministry mandated texts did not exist at least once, because the ministry published the specifications six months late. All three of Occupy Math’s children were marked off for solving problems “the wrong way”. It was the most prescriptive, uniformatarian math program Occupy Math had ever seen. The province also got rid of the last year of high school without consulting with the universities, which meant that a lot of kids were showing up without skills the university classes assumed would be in place.

Having moved to Canada from Iowa where parents and teachers are in charge of the methods for teaching math, Occupy Math is sure that having a ministry dictate how math will be taught is a dumb idea. If individual districts construct their own teaching methods – with the government supplying only standards for what the students must be able to do at the end – then the local districts will adapt to the cultural needs of their students and to any pockets of brilliance among their teachers. It is certain that, on average, they will do a far better job that the current Soviet Central Planning model. The idea that there is a single correct way to teach math is itself pernicious! Got an idea for better math teaching? Let Occupy Math know by commenting or tweeting!

I hope to see you here again,
Daniel Ashlock,
University of Guelph,
Department of Mathematics and Statistics

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