3 Levels in Mathematics Education

In a recent mathematics conference, I had the honour of attending a talk by Japanese educators. Prof Yoshinori Shimizu shared about the different stages of mathematical development and how students can master a particular topic. In particular, we can structure learning that is interesting and insightful. The following are 3 levels where we can structure a mathematic topic:

Level 1: 守 (Shu) Follow the form.

Following the form requires the student to apply standard techniques to standard questions that are probably similar to what they have encountered.

Level 2:破 (Har) Break the form.

Breaking the form requires the student to apply standard techniques to solve problems that they have not encountered before. This requires students to apply higher order thinking skills. Moreover, they must possess creativity and problem solving skills in order to apply prior knowledge to a new situation. This will assess their metacognition and habits of mind.

Level 3: 离 (Ri) Extend away from the form.

This may require students to extend what they have learnt to deeper concepts. For instance, if they have only learnt how to solve quadratic equations, students can question how they can extend this to cubic or quartic equations? Is there a standard formula to solve quartic or quintic as well? Also, another view is how students can extend what they have learnt to other disciplines and how the concepts can be applied to authentic situations.

In many cases, teachers often complain that students are competent to solve math problems that they have done before. However, when faced with new problems, students either avoid the question or simply raise their hands in defeat. By structuring lessons in the above 3 forms, students should be able to master their content using problem solving skills.