This post is the second on the game FRAX and spends some time explaining how the game works and where it came from. The first post was on Occupy Math’s sister blog Dan and Andrew’s Game Place. FRAX seems, based on our initial testing, to be a fun game in addition to giving the players practice with fraction arithmetic. To get the rules to FRAX (and if you’re interested in testing the game), click the link! We are giving away FRAX sets to people who will help us with the play testing. FRAX is a card game, not a computer game, though we have some thoughts in that direction.
This week’s Occupy Math is on a familiar topic, the effective teaching of math, but it was sparked by a number of recent media reports on why students in Occupy Math’s part of Canada are doing worse into the teeth of increased spending and emphasis on math. The problem has two parts. The first is the elementary teachers are not required to learn math themselves and are often afraid of it. The second is they are being handed a teaching strategy that cannot work and then getting no help trying to make it work. This is a big multi-part problem and Occupy Math hopes we can all dive in and help.
For the past three years Occupy Math has been working on a calculus text for the integrated first-year math and physics course he co-designed. The regular books did not have the right topics in the correct order. This post announces that the book is ready. It is the second edition (now with far fewer errors!) We used the book last year and found a need to revise and extend. The book is not a standard calculus book and the rest of the post is about that. The big points are: we got the cost down to less than half that of the book it replaces. It covers all the topics of a standard first year calculus course for science majors. The presentation is different based on decades of experience with what does and does not work.
One of the things a math teacher needs is a supply of good problems. Occupy Math and his collaborator Andrew McEachern have coined a new term in connection with this need: Problem Factories. A problem factory has two parts. The first is a basic understanding of a mathematical fact, the second is a general type of problem or puzzle based on that fact. Ideally the understanding of the math will specify which versions of the problem can be done and give an idea of how hard they are. From this point, the way forward lies in an example.
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.