Week Two, Day 1: Visual Algebra

VISUAL ALGEBRA
Today we discussed Visual Algebra, using graphic representations to illustrate alegrbaic concepts or proofs.
The Greeks and Romans did not use letters like we are accustom to using in Algebra (Algebra is influenced by Islamic mathematics). The Greek used words. During class we used visuals, rectangles and squares, to illustrate the concepts we discussed.  We worked on three equations in class:
(a-b)^2 = a^2 – 2ab +b^2
(a+b)^2=a^2 + 2ab +b^2
a^2  +b^2 = 2ab + (a-b)^2

Math Blocks

http://www.lcdgankara.org

 This is a visual presentation of (a-b)² = a² – 2ab + b²
What the equation is expressing is finding area of the shaded square given the lengths of the big square (a) and the side of rectangle inside it (b). Since the area of a square is its side squared, we need to know the length of the side of the shaded square, which we can find by subtracting the length of the side of the rectangle (b) from the side of the large square (a). This length squared is the area of the shaded square (a-b)². 

This is the same as subtracting the non-shaded areas from the larger square. The nonshaded area is the area of the 2 rectangles[ab] less the area of the smaller square [b²] formed from their intersection or 2ab – b²  then subtracting this from the larger square a²- (2ab – b²) which is the same as a²- 2ab + b²

 Mathematical Interest

Our professor also talked about why some have lost their interest in math. Although math ability can have an intrinsic skill level, for most people learning has to do with their interest in the subject. Some have had bad experiences with math and consequently lost their interest. Professor Gottlieb brought up how this class is an opportunity to refresh our lost interest and to get beyond mental blocks for math. We will be continuing to use visuals and games to help us understand concepts. We will learn that math can be about being playful and curious in searching for understanding.  Additionally, this class will help us see that math has real life repercussions.
 

 

 

 

Math and Health

Prof Gottlieb described how mathematical inquiry has contributed to the evolution of health practices. In a study of the practice of bloodletting, it was determined that those who had been treated with the practice of bloodletting were more likely to due than those who were not treated with such a practice.

From: georgewashington.si.edu

George Washington and blood letting
“History tells us that General Washington was a physically impressive man, measuring 6 feet 3 inches in height and weighing 230 pounds. Knowing his size we can calculate that his total blood volume should be about seven quarts. The removal of more than half of his blood volume within a few hours led to contributing causes in his death. The fact that General Washington appeared physically calm just before his death may have been due to profound hypertension and shock induced by the blood letting.” (www.revolutionarywararchives.org/washdeath.html)
 
The measure of all things sounds like an interesting book that mixes history, politics and a mathematical quest. From Professor’s Gottlieb description, it sounds like it may touch on ethical and political dynamics related to the search for empirical understanding as well as how the ethical and political context of scientific pursuits frame the meaning and moral dimensions of math research.