History is a fundamentally interesting subject. It is our story told through so many different lenses and we learn more about our place in the world. The characters are fascinating and the more one learns, the more profound the study becomes. I was fortunate to take a wonderful survey course on the History of Science in college and left with a far greater appreciation for that realm of history. I wish I could have taken one on math history but the interwebs provides more resources than I can consume. As I read and watch lectures on the weird folks in mathematics and the profound discoveries of different societies, I have tried to process a way to teach mathematics through its history. Maths History is a fantastic resource to help one both better understand the progression of mathematical thinking and find ways to teach appropriate concepts within their historical context.
The first lesson demonstrates a unit I plan on doing in my 7th grade Prealgebra class that follows the history of the Pythagorean Theorem and how the Greeks understood the areas as numbers. Students can examine similar shapes as areas of triangles. The Pythagoreans focused on squares on the side of triangles but the theorem works for any similar shapes (semi-circles would be readily done). We can examine the relationships by constructing different triangles and comparing the area of the squares on the three sides. By changing the triangles, we can look at congruence and similarity, as well as the nature of right triangles. This part does not follow the Greek version but will allow us to work with measurements, fractions, and scaling. Once students come to a Pythagorean conclusion, we will move to studying the classic proof of the theorem and finally applications that can develop to the Distance Formula.
The Pythagoreans were an interesting group and they believed that the world should be explainable by rational numbers. They developed harmonies based on ratios of Natural Numbers. My own understanding of music is too slim but we could have a great small project examining harmonies as balances of rational numbers.
Rational numbers were great and wonderful until they studied a square. The ratio of the side to the diagonal of a square is the square root of two. The classic proof of this number’s irrationality is based on the fundamental theorem of arithmetic.As we study this proof, we can study prime factorization which leads to integer exponents, GCF’s, LCM’s and basic number theory. This element also leads to the most interesting part of the history, the story Hippasus. He was rumored to have either discovered the proof or revealed it to an outsider and so the Pythagoreans drowned him at sea.
The proofs are perhaps the most challenging element but it is important for students to see how the process works for proofs are the heart of mathematics. These are not the two column proofs of geometry class but real logical arguments based on reasoning that lead to new understanding. These proofs are challenging and provide wonderful avenues for visible thinking routines. If a student can explain these proofs and then apply them to a variety of problems then he or she will have built a strong foundation for mathematical reasoning.
The unit will take several weeks rather than the single lecture. How is that and why would it be interesting? A 7th grade prealgebra class really needs to cover common core standards for 7th and 8th grade. This unit addresses parts of 3 of the main objectives of 7th grade and 2 of the primary objectives for 8th grade.
Along the way, history provides a narrative arch and allows for many more avenues to explore and keep the material interesting.
I will add more to this list and look forward to hearing from y’all.
Next up, history of numbers…