Learning to understand mathematical statements

You'll notice that in the last section, you had the opportunity to re-read the mathematical statement several times throughout the section. Why was this important?

The fact is- even if you were an experienced mathematician, you might not have understood a mathematical statement like the one in section 3.6 the first time you read it. Mathematicians don't expect to understand mathematical statements the first time they read them. They know that they often need to read a mathematical statement several times through, carefully, before they start to understand it.

So, if you read a mathematical statement once, and you don't understand it fully, or even at all, don't be concerned. That's perfectly normal, and mathematicians expect it to happen. The problem is that most people don't know that this is perfectly normal, and they panic when this happens. Then they decide that they can't do math.

Most mathematicians, however, know that it is perfectly normal to not understand something very general and abstract the first time through, so they immediately begin to employ various strategies that will help them to understand the mathematical statement.

Strategy One: Read the problem several times through (at least three times), slowly.
Strategy Two: Draw a diagram
Strategy Three: Think of a specific example, or create a specific example.

Let's briefly discuss each of these strategies.

Strategy One: Read the problem several times through (at least three times), slowly.

You may have noticed that each time you read the statement section 3.6, it seemed you understood it a tiny bit more. This strategy will work even if you don't do any additional thinking about the statement in between readings. That's because each time you read the statement, your brain is able to get a little bit more information out of it, which your brain can then use to get just a little bit more information on the next pass. It's a bit like opening a stuck drawer by gradually moving it a little bit forward on one side and then a little bit forward on the other.

Strategy Two:

It is very useful to draw pictures of the situation the statement is describing. In the case of a statement about lines, you can actually draw the lines. In other situations you need to get more creative- perhaps drawing boxes that represent different parts of the problem and connecting them with lines to show how they relate to each other

Strategy Three:

Since a general statement is true about a particular object that fits that description, you can take a particular object and try out the statement on that object. In the case of the statement in 3.6, for example, you might have tried laying out a piece of string, and measuring it. Then you might have tried picking a point on the string, and cutting the string in half with a pair of scissors at that point. Then you could have measured the two new pieces of string and added the lengths of the two of them together to see what the result was. Making an actual physical model using string, or modelling clay, or any other material you have at hand, also has the advantage of letting you actually explore the problem in a hands on way. Some people find this way of problem solving easier than just drawing diagrams or writing statements about the problem on paper.

 

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Copyright Jen Schellinck, 2006