One problem that Occupy Math has in teaching his first year courses is that many of the students have been trained, by their high school experience, to believe that a math class is a game that is scored in points with the goal of a grade. If it weren’t actually true in some of their high school classes, it would be nonsensical, and the whole notion is counter-productive to the goal of getting an education. Every year, Occupy Math has some students who are trying to do just enough work to pass the course, no more. Many of them flunk because a math course builds technique upon technique. What appears to be the correct level of effort for a D near the beginning of the term is actually preparation for an F or an F–. These grade-management tools from high school are also used as fear-management tools; instead of engaging with math, the student tries to scam a passing grade and so avoids the math. This has all sorts of bad downstream effects. In this week’s Occupy Math, we want to look at the issues of effort, fear, and effective teaching and learning.
Today’s post looks at the following problem. Color a plane (an infinite flat surface) so that any two points that are one unit apart are also different colors. The picture above is an example of such a coloring, with two caveats. The black borders are there to help you see what is going on (remove them to get the actual solution) and you have to continue the pattern indefinitely. The goal is to use as few colors as possible. This smallest number of colors that meet the goal is called the chromatic number of the plane. The formal name of this problem is the Hadwiger-Nelson problem. This problem is famous, in part, because much of the progress on it has been made by amateur mathematicians. The professors ended up needing a lot of help on this one. We also still don’t know the final answer to this problem. Occupy Math will go over what we do know.
This week in Occupy Math, we proudly announce a book published by Dr. Andrew McEachern, GAME THEORY: A Classical Introduction, Mathematical Games, and the Tournament. Game theory is a formal structure for studying and resolving conflict and encouraging cooperation that rephrases cooperation and conflict as a game. Andrew developed and taught a course in game theory taken by advanced students from many programs while he was at Queens University. This book is a text based on the course he taught and it is part of an effort to bring textbook prices under control. The book introduces the classical analysis used in game theory — his exposition of The Lady or the Tiger is wonderful — but Andrew also introduces material outside of the standard game theory fare. These include the math behind the fraction teaching game that Dr. McEachern and Occupy Math are developing and techniques for designing fair, balanced tournaments for anything from Prisoner’s Dilemma to Basketball. The book is a text for a course for non-majors that nevertheless has a solid mathematical foundation. We now ask Dr. McEachern a few questions about his book.
The factorial of a number is what you get when you multiply that number and those smaller than it (down to one) together. That means that five factorial is 5x4x3x2x1=120. The mathematical notation for factorial is to use an exclamation point: 5!=120. Occupy Math was teaching a course that used factorials to count things and one of the sharper students kept getting problems wrong. Occupy Math wrote “5” on the board and asked “what number is that?” The student replied “five”. Occupy Math added an exclamation point to get “5!” and again asked the student what the number was. The student replied “FIVE!” This was a third-year university student — hence this educational post. This week’s Occupy Math looks at what factorials do (e.g.: they count things). Factorials also provide an example of something that grows faster than exponentially.
This week we look at one of the big achievements in math, figuring out the minimum number of colors needed to permit adjacent countries to be shown in contrasting colors on any map, and connect it with a type of conflict resolution. Applications include efficient timetables for meetings, relatively peaceful assignments of students to cars for a field trip, and even putting as many types of fish as you can into the display window of a store that sells tropical fish. All of these applications have two steps: make a graph of the situation and then color that graph properly. We will explain the terms in italics in the rest of the post.
The fascinating mathematical fact here is that all these applications use exactly the same math.
This week, Occupy Math looks at math tests — and some other tests — from the perspective of fairness. It turns out that questions that test the same skills can have extremely adjustable difficulty levels. There is also the issue of tests designed for failure. For that, we will look at some examples of cosmically unfair questions. On the issue of math tests, this post discusses the differences between easy and hard questions for the same topic. Occupy Math can probably dial the average grade on a test across a range of 20% by playing with the way questions are phrased. All this will give you some perspective on how to survive a test (it helps to be able to spot structurally hard questions) — but mostly the message is this.
Fairness is largely an illusion and enforcing it is close to impossible. Hope for competent teaching and mercy instead.
In this week’s Occupy Math we look at a math problem that can be explained in less than ten minutes — that still stumps every mathematician who has ever looked at it. This problem is also a marvelous place for students to explore and look for patterns (some problems for students are near the end of the post). The problem is based on this rule: “If a number is odd, triple it and add one, but if it is even, divide it in half”. Not a hard rule, but it is the basis of an unsolved problem: “if you start with any positive whole number, do you eventually get to the number 1?” This question has several names. One of the most used is the Collatz conjecture (follow the link for many other names including the very common “3n+1 problem”). A number of prominent mathematicians, including the incredible Paul Erdős, have expressed the opinion that this problem is too hard for us in our current mathematical infancy. The theory also floated briefly, after multiple math departments were consumed by this problem, that it was created by the KGB (the Soviet Union’s spy agency) to stymie mathematics research in the west.
How can such a simple problem be so incredibly hard?
Teaching is not an easy job to do well. Math teaching is somewhat more difficult than teaching many other subjects because its the bogey-man of school topics. This suggests that we should work as hard as we can on making teachers jobs less difficult and, maybe, we should spend a little more effort on cutting teachers who have math in their portfolio a break. Logical and obvious, right? Instead what happens is that we assume problems with teaching are the fault of the teachers, punish them, and try and create rules that force them to do a good job. Occupy Math is a professor of mathematics with a research specialty in the creation of sets of rules for games and controlling systems — he is sure this cannot work. A set of rules that forces good behavior does not exist in a situation more complex than a game of Candy Land.
Evidence suggests that simply treating teachers as professionals and letting them teach will yield better results.
You’ve probably been lied to. It might be a direct and intentional lie, as with high school students Occupy Math met at a girls math conference (not Occupy Math’s name for it). Teachers told these women that they couldn’t do math — often before grading any of their work. The lie might be implied, as when the person teaching you math in fourth grade was clearly scared of it herself. We also have a distressing cultural context which paints mathematical ability — or even just being smart — as being anti-feminine. Men are not immune to this sort of idiocy either. Occupy Math has encountered far too many students who, when faced with a difficult concept in math said “I can’t do math, I’m not smart!” With many of these students it turns out they could do the math and they were pretty smart. For some reason, however, running away was a much more comfortable proposition than, oh, saying “can you explain that again a different way?”
Occupy Math finds the term “math person” pretty frustrating. If you think you’re not a math person then you’re taking a whole hamper full of issues and putting a two-word sticky note on it. Do you mean you will never win a prize for mathematical research or do you mean you can never learn to add? “Not a math person” lumps together a huge spectrum of ability levels and training in one small place. Here’s the scoop: unless you meant the “not winning a prize for original research in math” extreme, you’re probably wrong about not being a math person. I don’t mean you’re good at math right now or that you’re not afraid of it. I mean that it is incredibly unlikely that you cannot do math at some basic (but useful) level and probably you can do more. In this week’s post we will look at both fear and the case for trying to overcome that fear.
At this point some of Occupy Math’s readers are thinking “But I don’t want to be a math person!”
This week Occupy Math pays off on a promise to look at Trump administration appointee Betsy DeVos. Now confirmed — by the lowest margin possible and the lowest margin ever for a cabinet level post — Ms. Devos is in charge of the federal government’s education policies. While Secretary of Education is not as big a deal as Secretary of State or Attorney General, this is an important government post with a good deal of influence on education in the United States. Today’s post looks at her potential impact on education, including math education.
This post examines what we know about possible impacts of Ms. DeVos proposed programs, based on their partial implementation in her home state of Michigan.