First Reading (Sep 9)--Skemp on two approaches to teaching and learning mathematics

 

In this article, Skemp mentions two types of understanding in terms of Mathematics teaching and learning. They are "relational understanding" and "instrumental understanding". I personally prefer relational understanding in learning Mathematics, and I believe with "relational understanding" students could learn how to apply the knowledge and theories in different cases, and the methods of determining and solving problems. However, I know in some circumstances, instrumental understanding is also necessary. For example, if students could understand instrumentally Sine is the ratio of the opposite side of a right triangle to the hypotenuse, they could solve the problem quickly and reliably.  

 

When I was reading the article, three parts of the article resonate me:

1. " The other mis-match, in which pupils are trying to understand relationally but the teaching makes this impossible, can be a more damaging one."

 

Skemp uses "damage" to describe this situation. I think the biggest risk in this case is the children's curiosity will be gradually killed, and they will get lost on the way of pursuing knowledge. In addition, for some children, they need to know the reason why first, then remember, otherwise they cannot remember and solve the problem. I used to help a child with long division. He couldn't do it until I explained to him why we need to start from highest digit, and why we need to do subtraction.

 

2. The example about set. The teacher asked the class to pay attention to write their answers exactly like the example in the book.

 

This is the wrong way of using examples in teaching. Teachers use examples to help students better understand individual concepts. Encouraging students to learn examples could make the teaching more relevant and more understandable to the students. Examples could foster students' relational understanding. How to best use examples is critical. If a teacher asks the students to write their answers exactly like an example, I am sure that students could not get higher level of understanding on the concept behind this example. It's because they cannot connect the information in this example with what they already learned or experienced.  

 

3. "Difficulty of assessment of whether a person understands relationally or instrumentally."

 

Skemp suggests " In a teaching situation, talking with the pupil is almost certainly the best way to find out" whether he understands relationally or instrumentally, " but in a class of over 30, it may be difficult to find the time"(p.12) I totally agree with his suggestion. Communication is always a good choice to understand our students. On the other and, students' understanding could transfer from instrumentally to relationally. We need to give them more time to digest what they have just learnt.