Never say “I’m not a math person”!

Today’s post was inspired by a story in the Washington Post entitled Want kids to learn math? Level with them that it’s hard. The post will discuss how math can be perceived as hard or easy, depending on who you are; it will also deal with the myth that one is or is not a “math person” by nature. There is a widespread notion that some people are just naturally good at math and others are just naturally hopeless at it. This is untrue. Like any other learned skill, you get better at math with practice. While there is some degree of natural talent, anyone who does not have one of a very few rare disabilities can master basic mathematics.

One fairly large problem is that most people teaching math are usually pretty good at it themselves. To them math really is easy, at least at the level they are currently teaching it. If too few of the people charged with explaining math to the rest of us are sufficiently self-aware to realize how difficult and strange math can be to those encountering it for the first or second time, there’s a problem. Worse, many students who are supposed to be learning math are instead actively avoiding learning math while looking frantically for a way to tunnel under the actual work. This leads to the tragicomic sight of people doing huge amounts of work to avoid moderate amounts of work — largely based on a terrifically inaccurate estimate of the amount of work required to achieve basic competence.

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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.

All teachers need to know basic math.

One of the big problems that Occupy Math faces is that the high school students coming into his first year classes do not really know the math that their transcripts say they do. This problem was covered in the earlier post School Math: Epic Fail. More recently, a report found that the percentage of sixth graders meeting the provincial math standards has been declining for a decade. The province is moving to address this problem by requiring teacher candidates to pass an exam in math skills and pedagogy, which is causing concern and protest. The teachers’ union is against this test but has not proposed another plausible way to reverse the decline in math scores. This post is a discussion of the issues surrounding this situation.

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.

Timed Exams are Toxic

One nice thing about being a professor, at least at the universities where Occupy Math has worked, is that you have a great deal of latitude in choosing your textbook(s), method of instruction, and especially your method of evaluating the students. Occupy Math has already posted about fair tests. Today’s post is about some related topics. The most basic one is to question the point of having tests at all as they are nearly useless for what happens after school and are often corrosive to learning and the student-teacher relationship. Because there is no chance we will give up on tests — in spite of the obvious damage they do to students and education — the post will also look at the alternatives in designing a test, including the mis-named cheat sheet. Occupy Math has an endless, low-level conflict with his students about this name.

“Can we have a cheat sheet?”

“No.”

“But you said…”

“No, I absolutely forbade cheat sheets. Cheating is not allowed. Also, if ‘I said’ why are you asking again? For that matter, what I said has no weight. What does it say in the course outline I’ve told you to read more than five times?”

(different student)

“We can bring a page of notes to the test.”

Occupy Math smiles and nods.

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.

I Can’t Do My Kid’s Math Homework!

This post looks at two things, the way we keep changing the curriculum in mathematics and the impact this has on parents. Occupy Math has watched the province of Ontario require the use of textbooks based on curricula they had not yet finalized. The situation in the United States is better. More than forty states have adopted the Obama era Common Core standards, but even these are the subject of suspicion in some quarters. The second issue in this post is that of supporting parents when the curriculum changes. Parental involvement is strongly correlated with student success. It is pretty clear that ambushing parents with a new curriculum is a bad way to help them stay involved in their children’s education. An ambushed parent often objects, “this isn’t the way I learned math,” in a suspicious (or hostile) tone of voice. The case for the new methods of education absolutely needs to be made, including making the case to the parents, and material to support the parents coping with the change would be helpful. In the rest of the post, we will look at why we keep changing the way math is taught and brainstorm about ways to help parents.

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.

Adversarial Grading Encourages Cheating

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.

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.