How do you remember all that stuff?

topA common question in a first or second year university math class is “how do you remember all this stuff?” The answer is a little complicated, but the solution is simple — the students who ask this question need to change strategies. They are usually trying to memorize math rather than remember it..

Most students try to cram only the math they need to pass a test or quiz into their heads just before they need it. This may or may not work — depending on skill and aptitude — but it grants almost no ability to remember the math. This is often deadly because math is cumulative. It also absolutely maximizes the amount of work you need to do. When you memorize something — especially in a hurry and only for a particular purpose — you are putting a pile of individual facts into your head in a semi-organized fashion so that they are all available. This is difficult and the individual nuggets are often not durable. Remembering math, on the other hand, means that you have used the math enough that not only the individual parts, but the relationships between the parts, are familiar to you. Remembering math is actually easier that memorizing it, but you need to get to the state where you can remember it through practice. That is what this post is about.

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What is Graph Theory?

tcOccupy Math has done several posts that use a part of mathematics called graph theory. Occupy Math was trained in graph theory and his university is finally getting a course in graph theory which he gets to teach next winter! In the last ten years Occupy Math has taught reading courses in graph theory four times, because students needed the material to do their research. This is done as an overload, extra work with no comp time or added pay. Graph theory is really useful, showing up in everything from urban planning to particle physics, and getting to teach it to a whole class again will be wonderful.

A graph is just dots (called vertices) with some pairs of dots connected by lines (called edges). The graph at the top of the post is remarkable, for reasons we will get into a little later. A list of Occupy Math’s posts that use graph theory appears near the end of this post. Graph theory is one of the simplest types of advanced math to learn. In Cookies or Calculus, Occupy Math argues that graph theory is easier than calculus and — except for STEM students — more useful than calculus. Graph theory includes the study of networks, like contact networks in an epidemic, or influence networks in social or business situations. Graph theory is useful in some types of conflict resolution. This post is the first in a series that will introduce the power and beauty of graph theory.

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Making guess-the-next-number puzzles

topOne problem with math education is that access is very uneven. In theory, the internet can level things out a bit. This post is the next in Occupy Math’s series of activities that parents and teachers can use for enrichment and enhancement. For free.

Here is an example of a “guess-the-next-number” puzzle — “what is the next number in the sequence 2,5,8,11,14,?” The answer is 17. The student should figure out that the terms of the sequence increase by three every time, and 14+3=17. This post is about constructing this sort of puzzle with some notes on how to make harder and easier puzzles. Puzzles like this are good arenas to practice math skills. They can be structured as contests which is motivational, and with the information in this post, they can be tuned to your student’s needs.

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Student Evaluations: Customer Service Universities?

topAn end-of-semester ritual in most university classes is filling out evaluations for your professor. One of the justifications for this is the assertion that students are the university’s customers. This reasoning is badly flawed, even though the students — or their parents — are shelling out serious money to pay for university. The key difference is this. The quality of customer service when you are buying a new coat or having a meal in a restaurant has almost no consequences for the customer in the future. Even if the winter coat is flawed, purchasing another one is not difficult, and it is often possible to return the bad coat. You cannot return a flawed education. In fact, you probably will not notice the flaws until much later. A better model would be that of the relationship between a doctor or dentist and their patient — trust from the student and professionalism by the professor are required. Treating evaluation of the quality of university instruction based solely on student evaluations turns out to do damage to both the students and the professors.

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Goodhart’s Law

topIn an earlier blog, Let Teachers TEACH, Occupy Math spoke a bit about Goodhart’s Law, named after Charles Goodhart, pictured at the top of the post. This post goes into a lot more detail, examining different forms of the law. The fundamental idea of the law is this: if people are aware of rules intended to shape their behavior, they may try to exploit — rather than comply with — those rules. The place where this matters in mathematics education is in the realm of standardized testing, but there are many applications of the principles of Goodhart’s Law.

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Guess My Number — An Introduction to Algorithms

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Occupy Math just got back from Schloss Dagstuhl, a castle in the woods in Germany not too far from Luxembourg, where there are week-long research meetings. This meeting was on using artificial intelligence for games. One of the outcomes of this meeting was a simple game used to explore cooperation between humans and artificial intelligence — which Occupy Math thought had potential as an activity to introduce the mind set for programming. This activity is in today’s post. One of the groups was looking at making artificial intelligence to cooperate with human beings. They invented what they thought was the simplest possible coordination game. The game works like this.

  1. Both players pick a number from 1-100 and write the numbers down secretly.
  2. The numbers are revealed. If they are the same, the game is done.
  3. If the number are not the same, the players write numbers again and repeat step 2.
  4. Other than revealing numbers, the players are not allowed to communicate.

The goal of the game is to coordinate — get to the same number — in the smallest number of steps. A play of the game that takes 8 steps is shown at the top of the post. In the rest of the post, we give a couple of ways to use this game as an activity, and we also explain why the researchers do not think, after having people play it, that this game is simple.

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Sorting Fractions, an Activity

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This post is intended for the teachers of students who are just starting out with fractions, or who need to improve their understanding of fractions. The basic idea is simple: sort a list of fractions into ascending order. This can be done a number of ways, from reducing the fractions to decimal numbers and then sorting those, to using the cross-multiplication trick shown near the bottom of the post. There are also several special purpose shortcuts that solve parts of the problem, making this an exercise in problem solving as well, especially if the fraction sorting is done as a race or under time pressure.

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Being gifted can be a curse, without help.

topThis post begins Occupy Math’s fifth year of publication and it’s on a serious topic: support for gifted kids. The post is motivated by a confrontation at a school board meeting in which the parent of a child with severe learning disabilities took the parent of a gifted child to task. The parent of the disabled student was obscenely and profanely certain that money spent on the gifted student ensured that her child would not get the help they needed. They also felt that being gifted was a privileged state and so support for the gifted was robbing the needy to help those who were already ahead. There are a number of problems with that argument, and that is the subject of this post. Occupy Math has been an advocate for gifted children for decades and so this matter is near to his heart.

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And now for something completely different … heh, heh!

Occupy Math and his editor debated this post quite a bit. If you get through this post with no issues, without needing to re-read paragraphs, there is a decent chance you are a natural mathematician. More people should be able to get through the post, however, because Occupy Math’s editor made him revise this post more times than any other, ever. If you’re not feeling up to a challenge today, read the last paragraph and go make some hot chocolate or whatever your comfort beverage is.

We are used to multiplying numbers, as in 5×7=35, but there are many different kinds of multiplication in mathematics. That immediately causes a serious problem in understanding: people are so strongly conditioned to thinking of multiplication as something you do with, and only with, numbers that multiplying other types of objects is just weird. In math, multiplication can be any way of taking two objects A and B and getting back a third object C. We use the same notation: C=A×B but the symbol × has thousands of different possible meanings. The good news? This post is only looking at one of them. Occupy Math picked out this one, particular type of “multiplication” because it shows up all over both abstract mathematics and in natural science.

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Profit Mad Publishers Poisoning Math Instruction!

topIn this post we look at several ways traditional textbook publishers are poisoning math instruction — charging insane prices for books, writing lectures for professors (bad ones), and doing a bad job of generating problems for practice, homework, and examinations. We begin with the price issue. A while back, Occupy Math announced he had written a calculus book, Fast Start Calculus for Integrated Physics. The material from the book is being republished as the three books pictured at the top of the post. This is part of Occupy Math’s war on outrageous textbook prices. These books are distributed at low cost to a university and zero cost to students at participating universities.

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