APA Citation: 

Thinking mathematically. (n.d.). Retrieved July 5, 2019, from https://nrich.maths.org/mathematically

Summary:  

Reading about math, hearing about it or watching video can give a good background, but to truly understand it, one must do the math.  Nrich Maths provides wonderful opportunities to interact with math and mathematical thinking. Activities are designed to build and develop mathematical thinking as well as mathematical habits of mind.

Things Learned:

How does one know if they have found all of the possible answers?  The answer is by working systematically.  

My Question:

This answers my question about mathematical thinking as this it is practice of what Boolean Algebra is and how it operates.  To truly understand math, it needs to be worked and used.

APA Citation:

Fractals are SMART: science, math and art! (n.d.). Retrieved July 4, 2019, from http://fractalfoundation.org/

Summary:  Fractals are complex, repeating and never-ending geometric patterns.  They are found in nature, art, science and technology. The Koch Curve (pictured below) exhibits how a fractal grows in surface area yet requiring limited space for this.  Fractals have been applied in the design of many things, including antenna for cell phones, cancer research and computer chips.

In abstract math, fractals can make abstract math visual – invoking curiosity instead of fear.  

Things Learned:

Fractals have a finite area and an infinite perimeter.  This allows lungs to maximize the surface area in a small amount of space; this same concept is found in tree leaves.  Fractals have been applied in the design of many things, including antenna for cell phones, cancer research and computer chips.

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.

http://nandgame.com/

APA Citation:

Build a computer from scratch. (n.d.). Retrieved July 5, 2019, from http://nandgame.com/

Summary:  

Nand is an exercise in thinking logically – mathematically.  One is given a task and series of Boolean gates to complete the task which begin at the basic level of understanding and applying the logic to very complex designing applying logic to solve the challenge.

As one explores the even the basic levels, it is clear that there is more than one way to solve the problem.  The longer a problem is examined, and solutions tried, a more efficient way to solve the challenge emerges – and that is the key to math, finding the easiest way to solve a problem.

Things Learned:

My Question:

This answers my question about mathematical thinking as this it is practice of what Boolean Algebra is and how it operates.  To truly understand math, it needs to be worked and used.

APA Citation: James, J. (2014, June 07). Boolean algebra explained part-1. Retrieved July 2, 2019, from https://www.youtube.com/watch?v=2zRJ1ShMcgA

Summary:  

Boolean Algebra is centered on three operators, or gates: Not, And and Or gates – uses 1 (true) & 0 (false). 

“Not” gates takes only one input and gives the opposite of what is put in (if 0 is input, 1 is returned, and vice versa).  In Boolean notation this is written A.

Input	Output
1	0
0	1

“And” gates gives one output from two inputs (input 0, 0 -> output 0; input 0,1 -> output 0; input 1, 0-> output 0; input 1, 1 -> output 1).  In Boolean notation this is written Q = A ∙ B. To have an output of 1, both inputs must be 1. Three inputs are allowed and in order to get an output of 1, all three inputs must be 1.  

Input 1	Input 2	Output
1	1	1
1	0	0
0	1	0
0	0	0

“Or” gates can have two or more inputs.  (input 0, 0 -> output 0; input 0, 1 -> output 1; input 1, 0-> output 1; input 1, 1 -> output 1).  In Boolean notation Q = A + B. To have an output of 1, one of the inputs must be a 1. Three inputs are allowed and in order to get an output of 1, only one if the inputs must be 1.

Input 1	Input 2	Output
1	1	1
1	0	1
0	1	1
0	0	0

Things Learned: 

Boolean operators are based on one of two outcomes and they are well defined: yes or no, true or false, on or off.  This can be applied to think logically and make decisions. 

My Question:

This answers my question about mathematical thinking as logic is needed when using the gates to take a limited number of inputs and use them to find the one output desired.

APA Citation:

Pierce, Rod. (5 Apr 2018). Introduction to calculus. Retrieved 24 Jun 2019 from http://www.mathsisfun.com/calculus/introduction.html

Summary:  

Calculus has two methods to measure change: differential calculus and integral calculus. Differential calculus breaks something into small pieces to see how it changes and Integral calculus takes small pieces and puts them together to see how much there is; differential and integral calculus are inverse operations.  Differential calculus can be used to find the speed at any given point in time, not just the average speed for the trip. Integral calculus can be used to find the area under a curve.

Things Learned:

Calculus can be used to find the speed of an object at any point along the journey.  Calculus can be used to find the changing volume of an object.

My Question:

This answers my question about mathematical thinking as exploring calculus and learning how it is used to find patterns by breaking things apart or putting them together is thinking logically and looking for patterns – mathematical thinking.