You keep hearing about Mathematical thinking. What is it? How do you develop a Mindset for Math?

Khalid has a nice post on How to Develop the Math Mindset.

Math uses

made-up rulestocreate modelsandderive relationships.When learning, I ask:

- What
relationshipdoes this model represent?- What real-world items
share this relationship?- Does that relationship
make sense to me?

I will add a couple of more:

- How does one develop a mind-set for thinking beyond mere numbers, formulas and low level concepts.
- How can we take these insights that come out of that mindset and apply to real problems

One of the slides I used to have on my “Thinking About Thinking” talks was to ask the audience (mostly CS students) to do a few of the following multiplications, mentally.

19 X 21

25 X 15

Very few actually find the simple algebraic patterns, till you point them out.

That brings us to one more insight (not my own):

- A lot of problems can be solved by looking for patterns and applying some existing knowledge

The more mental models we build, the easier it is to apply them.

What’s the algebraic pattern to the math questions you posted?… I learn by memorizing. I’ve always had a great memory. This allowed me to learn to read early as I remembered the sound of the words. Eventually I could discern what word should sound like based on what I already know…. I will be taking the GMAT for applications to business school that in January and I’m trying to find a way to solve the problem that irecognize I have which is that my mind finds it difficult to keep the relationships of mathmaticvariables straight in my mind. I constantly jumble it in my mind. I find it difficult to solve problems that are worded slightly differently fromthe ones I’ve done before. Data sufficiency questions involving inequalities with variables baffle me cause there are too many factors to consider. I have to think about whether x and y could be positive or negative, fraction or integer and where one relationship would yield one result and another relationship would yield another result, I find it hard to keep all that information straight. I start getting confused and losing track of the relationships. I’m great when it’s memorization but ask me to think and I get confused. I assume it’s because I don’t fully understand relationships and patterns but I don’t know how to start to see these patterns of which you speak on your post.

Mack,

I think Math is about abstractions and not words. While words help you understand a problem, you need to abstract it into a model. Let me try with a couple of simple arithmetic problems:

when you give some one money (like paying a bill), the amount you have is reduced by the amount paid.

this is very similar to

When some water is taken out of a bucket (with a mug), the quantity of water is reduced.

If you represent what you have as X and what is taken out is Y, then the quantity before the event is

X and after the event (paying or removing water) is x-y where y is the amount taken out.

In the first case x and y represent money

In the second case x and y represent quantities (of water) in gallons/litters

The words are not similar but the basic abstraction of existing quantity, removal operation, remaining quantity are the same in both.

Does that make sense? I think almost anything in Math can be modeled. This model can be reused again and again. I think that is why there are hundred different problems (using different wordings) to illustrate some of the basic principles.

Math is not really about memorization but more about application of what you have learned. That is why it is so different from many other subjects. So pick a problem space (personal finance or physics) and start applying what you are learning with reference to the problem space. It may be good to have more than one.

Does this make sense?

As I mentioned in my commentary on Khalid’s link, I see that Math is some knowledge (algebraic equation (a+b) and (a-b)), seeing a pattern where you can apply that knowledge and getting a result faster by using a simplification. For example the product of 21 and 19 can be thought of as (20+1) and (20-1) and you can see that it (a+b)*(a-b) and so results in a**2 – b**2 which is 20**2 – 1** resulting in 400-1 = 399. This is much faster than trying to multiply 21 with 19.

You may want to look at a few books on mental math that show you some of these patterns and you can discover others on your own.

How to Solve It – by G. Polya gives excellent insight into what patterns are observed by Master mathematicians. This is a process which happens very intuitively in their mind. G. Polya has taken immense trouble to patterns in the thought processes happening in their brains. These patterns are known as heuristics and can be used by anybody in problem-solving like backtracking, solving-similar-problem etc.

This is an amazing slim little book which really shows what real analysis is about – finding simplicity in complexity.