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.

"Set" Card Game

I came across this card game called "Set" which is used for cooperative learning in class. It is a simple card game where a group of students try to identify sets of cards from a deck with different attributes, such as shape and color. A "set" of cards is one where either each attributes is the same or different.

Try this Flash tutorial to understand the mechanics of the game.
http://www.setgame.com/set/index.html

There is alot of potential in this game for teaching and learning in the following areas:

Permutation and Combination
Eg: What is the total number of sets that can be formed from the deck?
Set Theory
Eg: If A is the set of red cards and B is the set of diamond shaped cards, what does the set A union B mean?
Probability
Eg: What is the probability that the game will end with 3 cards?
Problem Solving
Eg: What is the maximum number of cards that can be picked before a set must be found?

More information can be found in the following two URLs:
http://www.setgame.com/set/article_nctm.htm
http://www.setgame.com/set/index.html

Whatcha Know About Math?

Kinematics and Calculus Made Easy

Geometry from the Incas

This is one of the most comprehensive sets on geometry that I 've seen. It contains dozens of examples of different results in geometry from Marion Walter's Theorem to Menalus' Theorem. A wonder site for exploration in the world of Geometry.


The URL is
http://www.gogeometry.com/index.html

A paradox in logic

Mr Lee, the teacher, announces to the class that there will be a surprise test next week. It can occur on any day from Monday to Friday afternoon. Alex, the smartest student in the class, stood up and pointed out the following to Mr Lee.


"Mr Lee"

"Yes, Alex?"

"Well there can't be a spring test on any day next week."

"Why not Alex?"

Alex begins to explain his argument:

"Let's assume you are telling the truth and the test is indeed a surprise. If there is no test by Friday morning, then it is no surprise that the test will be on Friday afternoon. Hence the test cannot occur on Friday, so it must occur between Monday and Thursday. However, if the test does not occur on Friday, then we would have known by Thursday morning that the test must be on Thursday afternoon, which again is no surprise. After ruling out Thursday and Friday, we can apply a similar argument to the rest of the days. Hence there cannot be a surprise test from Monday to Friday"

"Very well Alex, we shall see how it goes next week."

In the following week, a test was given out on Wednesday when the students did not expect it. It was indeed a surprise test.

Analyse Alex's argument using Paul's Wheel of Reasoning:

1. What is the problem in the question?
2. What information is given by Mr Lee?
3. What are the assumptions made by Alex?
4. What are the implications and consequences of his assumptions?
5. Whose perspectives should be considered?
6. What are the main concepts in this paradox?