APA Citation:

Drvcourt. (2016, July 09). What Is mathematical thinking? Retrieved July 7, 2019, from https://drvcourt.wordpress.com/2016/07/08/what-is-mathematical-thinking/

Summary:  

Mathematical thinking is looking at the basic structure or patterns of things.  “Math is about patterns”. Math is about describing something that happens consistently, all the time. And mathematical thinking is not only for math class, it can be used to understand the world – the structures in nature or society too.

The process of thinking mathematically includes these steps – often they are done together:

• Break task into components
• Identify similar tasks that may help
• Identify appropriate knowledge and skills
• Identify assumptions
• Select appropriate strategy
• Consider alternative approaches
• Look for a pattern or connection
• Generate examples

Thinking mathematically combines being creative as well as the technique.  When this is done, and a solution is found, it’s time to test it. If your solution is valid, how can this be proved?  If it is not, what changes can you make – if it is unsuccessful, try again. The root of math is asking good questions, thinking deeply, making connections and extending it to further ‘what if’ situations to find, and prove your solution.

Things Learned:

It is important to be persistent – if the solution does not work, go back to the drawing board examine the steps, and see if you can devise a new solution.

My Question:

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

APA Citation:

Bradley, T-D. (2017, January 17). What is category theory anyway? [Web log post]. Retrieved July 1, 2019, from https://www.math3ma.com/blog/what-is-category-theory-anyway

Summary:  

Category theory is a way of taking “birds-eye-view” of the mathematical landscape.  From a high vantage point, patterns can be seen and methods of finding solutions shared.  It is a tool to solve math problems. Move a problem from one discipline (perhaps, algebra) to another discipline (perhaps, geometry), and see the problem with a different angle and find new tools to solve the problem.

The key in category theory is not looking so much at the actual objects but at the relations between them, “connections between seemingly unrelated things.”  Finding these relationships – how collections of objects relate to each other sensibly, is the root of category theory.

Things Learned:

Category theory is a way to look at links between types of math problems – how they can be seen with other eyes and solved using other tools.

My Question:

This answers my question about mathematical thinking as it describes category theory and how it logically looks for relations and patterns to solve problems.

Davidson, J. (n.d.). What is calculus? Retrieved June 27, 2019, from https://www.sscc.edu/home/jdavidso/mathadvising/AboutCalculus.html

Summary:  

“What is Calculus” explores the creation (or discovery) of the branch of maths.  Calculus is not as tough as it’s reputed to be – with a good understanding of algebra then understanding calculus is the next step.  

A little history – math, arithmetic and geometry have been around since ancient times.  The idea of algebra was around, but the tools (the operations and a number system) had not yet been discovered.  It was not until the 9th century AD that these were available and used by a Persian, al-Khwarizmi, to develop algebra.  But algebra and geometry were not well linked.  

In the late 16th century, Rene Descartes used a graph with two axis (one vertical and the other horizontal) to name the position of objects on the plane.  The position was described using a pair of numbers, the horizontal position was the x’s and the vertical was y’s.  With this, Descartes was able to use algebra to represent geometric objects.

Later, in the 17th century, calculus was developed independently by two mathematicians (Newton and Leibnitz) in the 1670s as a need to understand constant change: the change in the speed of an accelerating object, or the effects of gravity on a falling object – not an average, but the speed at a specific time.  Today, calculus is the foundation for sciences, engineering, and advanced maths.

  

Things Learned:

Three things needed to be successful in calculus are: the ability to apply algebra skills, understand the concepts (not memorize), and dedication.

My Question:

This answers my question about mathematical thinking by learning about the beginning of calculus and how it was developed through the use of mathematical thinking.  These two mathematicians looked for patterns of objects in the and described how they changed using logic and patterns.