Occupy Math has recently become the chair of his department, which means that he must review and sign every academic misconduct finding filed by members of the department. He also gets copies of how the Dean’s office decided to resolve the accusations. During the pandemic, the amount of cheating has gone up considerably, but it turns out that is just the first part of the awfulness. Another duty of the chair is giving faculty a sympathetic ear when they are having a difficult time. The students who are trying to cheat are mostly amateurs — they get caught because the faculty have set traps, but also because they cheat in a terribly obvious fashion. This post may help cheating students kick up their game, but that is not its intended purpose. A silver lining of the pandemic is that it highlights things we need to fix. The explosion of cheating highlights that the adversarial system of determining grades, always toxic, is extra-toxic right now. Surviving a disaster is something human beings do well, but they accomplish it by cooperating, something that an adversarial student-instructor relationship interferes with. This post looks at a number of aspects of academic dishonesty during the pandemic.

# Image of the Week #256

The summerflake generalized Sierpinski fractal.

# Set theory contains real and unreal worlds.

This post is part of the series on visiting the islands of mathematics. A **set** is a collection of objects in which no object appears twice. **Set theory** is the mathematics you get by thinking about sets really hard. Occupy Math has already introduced a standard method of diagramming the relationship between sets, the Venn diagram, a humorous example of which appears at the top of the post. Set theory turns out to be a way to establish an intellectual foundation for the rest of mathematics — and this leads to a mildly awful anecdote about the interaction between mathematical research and mathematics education, near the middle of the post. Occupy Math has published a book on set theory and regularly teaches a course in the subject for second-year students.

# Image of the Week #255

This week, a Newton’s method starship, ready to take off.

# 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 Mathâ€™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.

# Image of the Week #254

This week we have another Newton’s method fractal, a giant spreading fan.

# Symmetry in Math, Nature, and Society

Symmetry is one of the mainstays of mathematics; it is also one of the points where math intersects with art and architecture. In nature, symmetry is common and beautiful, as in the plant at the top of the page. For physical objects, symmetry is built out of three basic types of transformations: rotation, reflection, and translation. A flower often exhibits rotational symmetry, the wings of a butterfly exhibit reflective symmetry, and the repeating units of a crystal exhibit translational symmetry. In mathematics, with access to abstract spaces and higher dimensions, far more types of symmetry are possible. We will look at all of this in this post, as well as looking at the ability to use lack of symmetry as a warning or sentinel in both nature and society.

# Image of the Week #253

This week’s image is a generalized Sierpinski fractal based on two rings of six points and a center point.

# 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.

# Image of the week #252

The image for this week is a deep zoom into the Mandelbrot set with nicely textured colors.