# Ideas for high-school proof class?

I have a math degree and have been hired to teach a proof class at a summer program. Our goal is to help the students learn the material they need for school (they take an algebra class separately) while also helping them improve their problem-solving skills more generally and find joy in doing math. They are mostly 16-17 years old and test into Algebra I or II in the program.

I've observed and helped with this course in previous years, but the curriculum is not well-defined and there aren't many past resources for me to use. So, my question is: what are your ideas for discussing proof, logical reasoning, etc. with a group of students who may struggle with math and don't necessarily like it very much? I will be focusing on the development of mathematical self-confidence and new ways of thinking, more so than specific content knowledge.

• Your question would be better if you told us the level of the children involved (very smart and advanced? age?). Also what sort of algebra will they take? – guest Jun 7 '18 at 4:59
• Thank you, I added that. – R.B. Jun 7 '18 at 5:04

Thanks for the description of the kids level (below average).

I actually think their time would be better spent on either doing remediation or advancement in their core topics or in just some useful use of the summer (archery, building stuff, whatever). This is NOT to discourage you, though.

My advice would be to keep things light and fun. NOT rigor city. Brain teasers, properties of numbers. Polya (and maybe even easier than his stuff). Simple number theory and even properties of numbers (divisible by 3, etc.).

In terms of direct applicability to algebra, the one key proof topic is mathematical induction and the classic problems in any general algebra 2 text that are under that chapter. If you want to do something useful, covering this, even if it is pre-covering, might help them. I worry though that this is a topic that is slightly hard even for the average student within their algebra 2 class. But perhaps you could really water it down or do it gently or the like. And not make it a hill to die on but just more of giving some exposure.

Actually as I write this (sorry for stream of consciousness), I think more and more that proof ability is extremely unimportant for these kids (that probably lack computational skills). Proof classes are something that the superstars of math contest kids take in HS. Not your guys. However, you do have an alternate mission which is to awaken some interest in math. So I would let them call it a proof class but concentrate on fun activities.

One fun area is symmetry and point groups. Don't take a group theory approach to it, but instead just list the type of symmetry elements (rotations, mirror planes, etc.) and then have them catalog them on objects. If possible, have physical objects to pass around. (If you don't have enough, have the kids do teams...this is the sort of thing where teams actually makes sense, unlike computational problem solving.) You can do 3d and 2d. For 2D, you might do something with wallpaper patterns (space group) but don't do anything with 3d space groups (too hard). And by do something, I don't mean mastering it, but coloring in the birds on some Escher print or listing the rotations and planes and such that exist in some prints.

Also there was another thread on here (can't recall title) about experimental (i.e. physical) math projects for high school. Stuff like dropping the needle and finding pi. I would do some things where the kids can use their hands and such. It is summer after all. Plus really, we are physical creatures (all of us, even Andrew Wiles). And these kids are obviously not abstract thinking superstars.

Bottom line: lots of "sugar" and not much "medicine". Keep the emphasis on fun rather than accomplishment and you will end up with more accomplishment (than if you torture them with proof strategies).

Dana Ernst has a fantastic collection of problems which he bases a course around at Northern Arizona University. Here is a link to the course:

http://danaernst.com/teaching/mat220s18/

This is a collection of puzzles which do not require any background knowledge. So the focus can be purely on working on problem solving skills, communicating mathematics effectively, employing a variety of different modes of reasoning, etc. I am using this for half of'' a discrete mathematics course right now and it is very fun. I am not sure if the level of the questions would be appropriate, but I think it is at least worth a shot.