This week, I decided to go through Gauss’s problem: How many ways can you write an integer as the sum of two perfect squares? I watched dad/teacher Mike take his son through this problem (who is a genius!!) and took notes myself along the way. Here is what I gained…
I read e: the Story of a Number for MTH 495 and honestly didn’t really know what to expect from it in the beginning. I was afraid the math was going to be way too old school and over my head, but was surprised to find it relatively accessible. Eli Maor discusses the lives (not simply mathematical, either) of great mathematicians from Newton to the late 19th century. I really enjoyed the fact that I could put the stories he wrote with the theorems I had learned in various math classes, because when we learned them the first time around I had no idea who they were named after and what significance they held. Maor’s stories were surprising entertaining and “literature-like” for lack of a better word. I feel like often times people think that “math” people aren’t creative or capable of story telling, but Maor does an expert job of combining both disciplines to produce an astounding piece of literature. He tells the story of the birth of the letter e in math, but is sure to add in snide remarks and funny stories to make the process more enjoyable. As a side note, I also found it extremely helpful that he put little “side stories” in to show how e is used in real life, or to expand and extend a detail in one of the chapters. It gave more depth to the book overall.
There were, however, a few things I didn’t care so much for in the book. I am a very visual person, so just reading about a mathematical problem and its solution doesn’t really do it for me. I need to have it written out, in pictures, in steps, etc. to fully grasp what is happening. Although there were many instances where Maor did include pictures, there were many where he did not. Surely the book would have been much thicker with these pictures taking up more room, but it really would have helped me understand the math the first time around. Instead, I found myself reading pages over and over and over again. This next critique is going to sound odd, but it’s something that really bothered me while trying to read the book. The font is so dang small!! When I’m doing math and writing about math, you better believe I’m taking up pages upon pages because I want to be able to look at it without becoming overwhelmed. Trying to read through some of his explanations and problems literally made my head hurt at times.
So, after nitpicking a few items, I overall really enjoyed the book. It makes math history much more that tolerable–it makes it interesting! Reading about the mathematicians who invented the theorems we use and take for granted today gives me a whole new appreciation for what it means to invent and discover mathematics.
For this week’s work, I decided to dive even deeper into the rascal triangle. I was just so impressed by the work that these three students put forth and the fact that they weren’t going to let their teacher’s discouraging words bring them down. This just shows that students should still pursue the “wrong” answers, because they never know what discoveries they may make. I am going to include some of the work that I wrote out for my daily work and then elaborate on it via this blog post.
From my research, the values in the first few rows of the rascal triangle support the claim that each element of the nth diagonal is congruent to 1 modulo (n-1). To explain this, in the fourth diagonal, each of 1, 4, 7, 10… is congruent to 1 modulo (4-1)=1 modulo 3. So, in these cases, n=1, 2, 3,… Moreover, we can see that the nth number on the rth row a∨(nr) is the nth number on the (r-n)th diagonal and n, r=1, 2, 3,… and the 0th row is excluded. This observation is fundamental in proving that the rascal triangle consists strictly of integers. Also, the sums of the rows of the rascal triangle are “cake numbers” (I had never heard of these before…). A cake number is the maximum number of regions into which a 3-dimensional cube can be partitioned by exactly n planes.
Another way to prove that all of the entries in the rascal triangle are integers, is by doing what the boys did…
They realized that the NE-SW diagonals exhibit a simple pattern: the mth diagonal (starting the count with m = 0) is an arithmetic progression with m as a difference and 1 as the first term. The entry #n in the diagonal #m is simply nm + 1. The formula to generate the triangle is then
(m + 1)(n + 1) + 1 = [((m + 1)n + 1)(m(n + 1) + 1) + 1] / (mn + 1)
(CTK Insights).
I show this (or at least attempt to) in the picture I included, so that people could put a visual with the written work. I am so inspired by these students, and it makes me wonder how often we as math teachers shut down valid ideas. Not intentionally, of course, but without even realizing that our students could really be onto something. It just makes me think about how I’ll go about teaching in the future.
I had honestly never really given this a second thought. Math is always associated with the sciences, so why wouldn’t it be one? However, turns out that this is much more of a controversial issue than I ever realized. I mean, sure I think of math as its own language, but I’ve never identified it as literature or speech–it’s just always been a science. But why? Here is the dictionary’s definition of the word science: the intellectual and practical activity encompassing the systematic study of the structure and behavior of the physical and natural world through observation and experiment. I’m going to dissect it and see what conclusions I can draw.
“The intellectual and practical activity encompassing the systematic study” If I look at this part first, it is clear to me that the science label fits well with mathematics. It takes active participation and involvement to truly do math. Plugging and chugging numbers into formulas is, I suppose, math, but it’s not the true heart of exploring numbers and patterns and theories. This takes deep thought and actual “doing” to achieve. And as for the systematic study part of this definition, the first thing that comes to my mind is proofs. Sure, proofs give us mathematicians the opportunity to be creative and spread our literary wings, but there is still a systematic approach of sorts we must follow. That is, we can’t just jump from assumption to assumption in order to prove a theorem–we must use only the axioms we know and build upon them to reach the end product. This system is one that is common across all mathematics.
“Study of the structure and behavior of the physical and natural world” Though math may not focus on explaining why polar bears only live in cold climates, it can tell us why 2 polar bears plus 2 more polar bears is 4 polar bears; why 6 polar bears minus 1 polar bear is 5 polar bears, and so on–mathematical abstractions arises naturally from the physical and natural world. Math helps explain how objects exist in the natural world, and the majority of math has numerous real-life application. Areas such as Algebra and Calculus help us understand things such as the basic rules of motion. Math is an integral part of everyday life, so I find it difficult to think of an argument that supports the idea that math doesn’t deal with the structure and behavior of the physical and natural world.
“Through observation and experiment” In my opinion, this is the most obvious part of math. It’s learning by doing and observing. Noticing patterns and seeking out new ones is what mathematicians love to do. It takes numerous rounds of trial and error, critique and communication to solidify a mathematical conjecture as a theorem, but that’s part of what makes mathematics a science. Observing what people have done in the past and building on these ideas is very similar to what scientists do. Though they aren’t an exact mirror of each other, math possesses all the characteristics of the science definition.
Kalie's Student Teaching Adventures 