APA Citation:

Burton, L. (1984). Mathematical thinking: the struggle for meaning. Journal for Research in Mathematics Education,15(1), 35. doi:10.2307/748986

Summary:  

Answers three questions: 

• “What is mathematical thinking?
• What does it have to do with mathematical content?
• Can it be taught?” (p. 35)

Mathematical Thinking:

 “…mathematical thinking is a natural means by which we classify, combine, relate and transform information…” (p. 44)

As the scientific method does not cover any content in science, mathematical thinking is not focused on the content, but how to do math.  In general, the four processes below are central to mathematical thinking. Specializing – examining a particular example. Examples provide concrete, tangible elements.

1. Conjecturing – after a series of examples, what is noticed about the relationships, what is the connection, the pattern?
2. Generalizing – can this pattern be generalized to other instances?
3. Convincing – the generalization must be tested until it is convincing, not just for the one doing the activity, but for the outside world as well.

What Does Mathematical Thinking Have to do With Mathematical Content?  

An example given on p. 42-43, there was a discount of 20% offered on an item and 15% tax on the item.  Would it be better for the purchaser to first calculate the discount or the tax? The initial conjecture was that applying the discount would make the base lower and then apply the tax, so some examples were used to test this theory.

Trying it with $100 dollars and the discount first leaves $80, then a 15% tax is $92.  When tried the tax first, $115, then a 20% discount is $92. This was tried with other amounts $65, and again, it did not matter which was calculated first, the discount or the tax.

With mathematical thinking, one is able to engage – openly inquire and make meaning.  Although learning just the math, this gives a closed answer, an algorithm, it can be used to confirm or deny our conjecture.

Can Mathematical Thinking be Taught?

In order to learn mathematical thinking, one must ask questions, challenge and reflect.  Questions such as, “Why do I think that?”, or “Is there another way?” are just a couple. One also needs to be ready for struggling to find meaning.  

Things Learned:

Mathematical thinking and math are not the same, but math can be used to validate one’s mathematical thinking.

My Question:

This answers my question about mathematical thinking as the author shares examples of what mathematical thinking is – finding patterns and looking for structure.

APA Citation:

Challoner, J. (2010, October 18). How Mandelbrot’s fractals changed the world. Retrieved July 4, 2019, from https://www.bbc.com/news/magazine-11564766

Summary:  

The term fractal was coined in 1975 by a Polish-born mathematician Benoit Mandelbrot.  Fractal geometry is complex and beautiful – to get an idea of what a fractal is, look at nature.  Fractals are found in clouds, mountains, coastlines, among a multitude of other places. Examining a fractal, one can find the same shape, but smaller, repeated again and again.  A pine tree and notice that it is composed of branches, and each branch is composed of smaller branches, and so on.

The patterns in fractals are not regular (Circles, cones, spheres, etc.) but reflect the irregularity found in the world – “Mandelrot referred to it as ‘roughness’”.  These patterns are used to describe the world and useful when looking into medicine, engineering, genetics, art, as well as computers and describing financial markets.

Things Learned:

Fractals are artistically pleasing, and interesting to look at the world with; I was intrigued to learn that they can be applied to describe financial markets.

My Question:

This answers my question about mathematical thinking as fractals are exploring the patterns and finding the logic in what seems to be chaos.