Occupy Math has recently become the chair of his department. Getting more mail is one of the side effects of this. Recently he received a rather tense letter from the mother of an eleventh-grader; she pointed out that her son was not getting as much math instruction as he should and wanted to know what the university was going to do about it. Occupy Math regretfully told her that we were going to do nothing about it, because we were losing resources, and thus had no people to assign to remedial education for incoming students. This is somewhat tempered by the fact that we already have some measures in place, a help center and a packet of summer review questions that at least serve as a warning about what a student is supposed to know. Occupy Math looked into the matter: a tense letter from a parent can fall anywhere between prophetic warning and arrant nonsense. This one was spot on. The rest of this post looks at what happened to math education during the pandemic, discusses why the pandemic adaptations of math education are very bad for learning math, and suggests some things a parent might do to compensate.
Activities and Materials
Announcing Occupy Math’s New Book
This post contains a fairly detailed description of Occupy Math’s new book on evolutionary computation, A Course in Evolutionary Computation. The basic notion of evolutionary computation is to create a computer program that implements a simplified version of evolution, where the evolving “creatures” are solutions to a problem. The technique is most useful when you do not understand the problem too well, but can recognize good solutions. Evolutionary computation has been mentioned in number of posts like the one on finding rules to make cavern maps and even evolving fractals.
Figuring out how and why evolutionary computation works is at the core of Occupy Math’s research. Occupy Math’s last book on evolutionary computation, an introductory text titled Optimization and Modeling with Evolutionary Computation, was published in 1996 and a lot has changed in the field since then. For that matter, Occupy Math himself has learned a lot, and it is time for him to write a new introduction to the field. Occupy Math teamed up with his wife for this effort — and the production quality of the book is much higher because of this. An important point to make is what is new and not new in this publication; the topics from the original book have been both revised and redivided so that they are scattered about in the new book. There is relatively little in common between the two books — the new book is not a second edition of the original one. This post will go through the topics in the book and explain a good deal about evolutionary computation. The book is available from Amazon:
Number Sentence Puzzles, Part I
Today’s post explores one of several sorts of number sentence puzzles; an example appears at the top of the post. Occupy Math will do posts on other sorts of puzzles later. The puzzle is the line of symbols and the answer is given below it (since this post is for parents and teachers, we will give answers with the puzzles). Each symbol represents a digit; when the symbols are the same, the digits are supposed to be the same. This post contains examples of such puzzles. It also explains the technique used to generate them, a technique which is really stupid in one sense, but also involves some clever math. Notice that the example puzzle uses seven sixes — the clever part of the problem generator ensures lots of repeated digits, which makes the problem both a bit easier and more interesting.
Can this number of neighbors happen? A problem factory.
A problem factory is a set of mathematical principles that leads to a large, often infinite, collection of problems. Typically only a finite number of these problems are reasonable to assign to students, but a good problem factory will generate a large number of reasonable problems. This week we look into the question of the pattern of numbers of neighbors in a contact network. The picture at the top of a post represents a contact network with 32 people (the dots) each of which is in contact with four other people, as shown by the lines in the network.
The problems we will look at in this post take the following form. Given a sequence of numbers, can those numbers be the numbers of neighbors in a contact network? The picture at the top of post shows that 3,3,3,…,3 (32 threes) can be the number of neighbors in a contact network — the picture shows this. We call these sequences the contact numbers of the network. These questions make a good problem factory because, although there are many sequences that are the numbers of neighbors in a contact network, there are also many sequences that are not, making the questions challenging and real.
When chocolates make problems
This Occupy Math is the next in our series on problem factories. Problem factories are mathematical structures that give rise to a large number of problems. Occupy Math and his collaborators hit on the idea while attending a conference on mathematics education. Many of the presenters had favorite problems, including a version of the chocolate box problem, but they just assumed that a clever teacher could make up their own problems. Well, they probably can, but Professors of Mathematics have a lot more time than overloaded middle school and high school teachers. These problems are also useful to parents who may not remember all their math education from, *ahem*, a while ago.
The chocolates at the top of the post are our prop to set up the problem. Suppose you have a number of goods with different prices, like the chocolates above, and you are making up gift boxes with specified prices. The goal might be to make an assortment of different boxes with the same price. The math underlying this problem has some interesting features that we cover in the post — it turns out that finding problems that can be done is exceptionally easy in this case. The punch-line is this: usually there are many ways to make a medium-sized box with a specified price. Occupy Math also notes that this is actually a problem that comes up, in retail practice, in a variety of forms.
A geometry practice problem factory
The area of a triangle is one-half the length of its base times its height; the area of a rectangle is just its length times its width. This week’s post is about an activity that gives students puzzles that can be fairly challenging, but which are built on these two simple rules. Beyond that, after working some of these puzzles, students can create their own puzzles and challenge one another. This post is the next in our series on problem factories.
Discovery Learning in Appropriate Doses
Occupy Math has posted a number of times on problems with math education up to and including parental sabotage. This post starts with an anecdote from the parental sabotage post about the discovery learning fad.
In sixth grade, Occupy Math’s editor was given the problem of finding the ratio of the distance around a circle to the distance across — an example of discovery learning. The answer is, of course, the universal constant pi. Occupy Math’s editor came to her parents who made semi-helpful strategic comments and eventually she got a value of just above three. Almost every other parent simply told their kids a modern approximate value for pi, pretty much destroying the “discovery” part of the exercise. This post explains the value of such exercises, including letting the student do the exercise on their own, and also gives an example of such an exercise. Many teachers are leery of or actively dislike discovery learning — probably because it is supposed to be used quite sparingly and often is not.
There is a second significant part of the pi anecdote. Occupy Math’s editor was in tears by the end of the several hours it took her to complete the assignment (this may be part of what was motivating the parents that just gave away the answer). The way we teach math — broken into tiny “Knowledge McNuggets” — makes students think that, if the answer is not immediately obvious to them, they are stupid. Taking three hours to figure out something that Babylon and Egypt spent decades on made a good student feel stupid. One long term goal of the projects post series is to model the more correct view that mathematics is often slow, tentative, and can contain trips down blind alleys. Occupy Math has been working on some problems for thirty years and has only modest progress to show. People who have been trained to give up after ten minutes are not really what we need coming out of mathematics classes.
Information keyed mazes
Reputed to be the greatest hero of ancient Athens, Theseus is most famous for the adventure where he slew the Minotaur, a monster that devoured sacrificial youths and maidens each year. The Minotaur lived in a great maze. Theseus managed to defeat the Minotaur through strength of arms and the maze by unrolling a thread behind him as he advanced into it. This post is about a way of creating a mental “thread” that lets you navigate a maze perfectly. The mental thread works by embedding the thread in the design of the maze: designing mazes that have a mental thread or key is the topic of the post. The post also gives a simple way to build one of these mazes, as a notebook, so it can be used with your friends or in a classroom.
Lining up in Hyperspace
This post reveals a method that Occupy Math uses to create large symmetric maps — not maps like a map of the United States, but rather connection diagrams. The easiest way to explain the techniques uses a lineup as the core of describing the process. The rooms or destinations in the map are represented by orders of the people in the lineup and the connections between the destinations are created by rules for how to change the orders of the lineups. Considering the order of people in a lineup is one way to do a kind of abstract algebra called group theory.
Finding a match for everyone
This Occupy Math presents a new puzzle — a solved version of the puzzle appears at the top of the post. This version of the puzzle has 8 pieces which you can cut out of heavy paper or just draw with a pencil.
The goal is to place the pieces to fill a 4×4 square so that every number is adjacent to the other copy of itself — this is the match for everyone feature. The numbers can be sideways or upside-down, we just created all the pictures with the numbers right side up to make them easy to read. The post also explains how to build these puzzles, including how to get huge numbers of additional puzzles once you have one of them. We call these things Orthogonal Layout Puzzles.
Occupy Math 