No right or wrong answers, just better

What if there were no right or wrong answers in maths? What if we learned maths like we learn life? Just better guesses, or closer approximations to the "truth".

Here's an approach to learning that turns students into investigators/good guessers and makes them more responsible for their own learning, in the same way that workers on the factory floor in many industries are now responsible for their own productivity/quality improvements.

It's an approach that has helped the Japanese motor car and electronics industries become extraordinarily successful. But they had a lot of help from a mentor, Edwards Demming, one of the father's of the quality movement around the world. He was more famous in Japan, where his methods were adopted, than America, where he lived.

He developed methods for the collection and statistical analysis of manufacturing and process errors to improve product and service quality. But his most important new idea was to give the workers responsibility for finding errors and working out what to do, rather than management going around with a stop watch or micrometer and apportioning blame.

We first tried this error detection method in the late 1990s with a group of 12 secondary school students in years 7 through 10 who had a shaky knowledge of multiplication tables. They used a team learning system to collect their responses anonymously (they could not see each others' contributions) to 100 multiplication problems e.g. What's 5 x 9?

When the contributions were revealed, the students looked for patterns of errors, brainstormed ideas for eliminating them and applied their ideas to new problems.

They noticed that most errors occurred with 7x, 8x, 9x, and 12x tables, and generally with the higher numbers in these series. They also realized that difficult to remember multiplication tables could often be inferred from a lower (or higher) table that was easier to remember, for example 9 x 10 is easy to remember, but 9 x 9 = 81 is difficult. So start with 90 and subtract 9.

They shared techniques they had learned individually. For example, add a zero to a number to multiplying by 10, e.g. 8 x 10 = 80. To calculate numbers multiplied by 11, repeat the number, e.g. 9 x 11 = 99, at least up to 9. They also trialed a method to become familiar with the numbers in each multiplication table, by completing both ascending and descending series, for example 7, 14, 21 etc. and 99, 96, 93, etc.

At the start of the four session trial the error rate was 37.5% and at the end 7.5%.

So here's the method:

7 x 7
8 x 3
9 x 2
6 x 4
...more tasks
8 x 8
12 x 11
5 x 9
Thinking about our performance, where did we perform well?
Thinking about our performance, where did we make most of our mistakes?
What happens in your mind when you get it right? What do you feel?
What happens in your mind when you are having difficulties? What do you feel?
Describe a trick/technique you use to remember?
How could we improve our performance?
Apply new rules to: 10 x 10
6 x 3
12 x 6
7 x 9
12 x 8
....more tasks...