As I general rule I find thinking about math pedagogy deeply rewarding. Teaching a technical and beautiful discipline like math is difficult to do well. Students come from all sorts of backgrounds, the material can be challenging, and there are tons of moving parts in a course. It’s a challenge that I find exhilarating.

On the other hand, I find the act of reading the scholarship of math education to be dreadful and unpleasant. It is filled with jargon and hero-worship.

That being said, I’ve been extremely lucky to have great mentors and colleagues to bounce ideas off of. I’ve collected some of this advice in a Reddit post, which I’ll recreate here.

## Concepts that I’ve found useful

Here is some vocabulary that is commonly used when discussing math pedagogy, or pedagogy in general. In general the literature is pretty annoying and frustrating; there’s lots of jargon and lots of stuff is too-high level.

**Active learning**. Is the primary activity in your classroom listening and writing, or discussing and thinking? This could be anything from students working on calculations in-class, to group discussions.**Teaching math is not about convincing your students that you the instructor know the material**, it’s about helping the students learn the material.**Scaffolding**. This is the technique of “building up” exercises or assignments through many small but achievable steps. E.g. You might ask a student to compute (1) the slope of a line between two points, then (2) the full equation of that line, then (3) the tangent line of a specific parabola at a specific point, then (4) the general derivative of a parabola.**Think-pair-share**. Give a question to the class, then (1) let each person*think*about it on their own, then (2) they can discuss it in*pairs*, then (3) any group that wants can share with the entire class. At first this sounds super cheesy, but it’s extremely useful at getting students involved, empowering them and lowering anxiety.**Inverted/Flipped classroom and Peer Instruction**. This is the idea that lecturing is largely ineffective, so classroom time is spent with a tiny bit of lecturing (for definitions/motivation) and most of the class time is spent discussing, writing, calculating, experimenting, arguing, making and testing hypotheses, etc. This is related to the Moore method, and Eric Mazur’s Peer Instruction.**VNPS- Vertical Non Permanent Surfaces**. The name is a joke, but it’s a good idea. This is related to the idea that we should be getting students up and working together at chalkboards/flip charts/windows. The energy in the classroom changes a lot when people are standing up. Source.**Signature pedagogies**. This name is*misleading*, but the idea is to remember to teach students not just the material, but how to use the material as a scientist/mathematician/statistician/programmer would use it. At some point we need to teach people curiosity, problem solving, how to make and test a hypothesis, precise writing, oral communication, algorithmic thinking, etc.**RUME and SOTL**. These stand for “Research in undergrad mathematics education” and “Scholarship of Teaching and Learning”. These are the two major movements for peer-reviewed research into teaching and education that are relevant to math teaching. There’s a recent push to inject good science into teaching with controlled (ethical) experiments, backed up with data. I find these papers excruciating to read because there is a lot of jargon and hero-worship in them. They also tend to not be written for a mathematician audience. Sometimes though you can find useful things here, but it’s rare. RUME starting point.**“Try, Fail, Understand, Win.” and “Productive failure/struggle”**. Source. This is the ethos of the effective math student. It stems from the method of Inquiry based learning (IBL) a method where students discover math on their own through guided exercises and questioning. This is rooted in the idea that students learn much, much better by doing rather than listening, and by struggling rather than having the answers given. See this amazing and persuasive exercise.

## Some ideas I find useful, that don’t have jargon-y names associated to them

**Everything should serve a well-defined purpose.**Decide on the goals of the course (jargon: learning outcomes, learning objectives) and build everything towards that goal. The structure of the course, how you deliver material, the content of the labs, the types of assignments,*everything*should work towards that goal.**Test the thing you want to test**. When writing tests it’s natural to want to include “clever” questions or questions with many moving parts. One issue with this is that students can potentially stumble on an early part of the question and not even have a chance to show off what they know. Be as direct as you can be.**We call on men to answer questions more frequently than women**. Be aware of this bias.**Collaboration over competition**. When possible, set up your class so that people can build each other up, instead of pushing each other down. In practice math is very collaborative. This has the additional nice benefits of lowering anxiety and encouraging women.**There is more to math than just Western Europe**. When including history or historical exercises try to draw from places other than just Western Europe. For example, India (invention of 0), Iran (Astronomy, Geodesy, Optics), Egypt (Rhind Papyrus) and Mesopotamia (first recording counting) all have deep, interesting math history associated to them. Representation matters, and helps students find heroes. MacTutor is a great resource.**Have many entry points and perspectives**. Give students a variety of reasons to care about a topic: historical interest, practical application, theoretical interest, beauty, application to a specific domain, etc.. The goal is to make sure that each student has at least one thing they care about. Case-in-point: the comic Far Side often talked about hyper-specific domains of science and most people didn’t really care, but those that did care*cared a lot*and were really invested. Do stuff that*someone*loves rather than doing something that everyone finds acceptable but boring.**Student evaluations do not reflect how good of an instructor you are.**Source. The way to get high marks is to: make the course easy, give the students past-exams and make your exams similar, show that you care about them, be engaging and to be an attractive dude. The way to get bad numerical scores is to: try something new, challenge the students, get them to do mathematics and grapple with questions, give unexpected questions on tests (even if the questions are easy).**Talk to students as a human**. Find opportunities (before class, before tests) to talk to students as a fellow human. Talk about music you like, or the new avengers movie. This has really helped me connect with students, and in some cases helped me find good summer research students.

## Some other advice

**Talk to your colleagues**. Find people you trust that you can bounce ideas off of. Go for lunch frequently. Discuss pedagogy. Laugh about the jargon. Find potential pitfalls. Complain about students for your sanity.**Find a mentor or two**. You’ll need someone to guide you through your career path and challenge you to improve. Respect their time and find ways to benefit them.**Be a mentor and a good supervisor**. Encourage your TAs and junior colleagues. Treat them with respect and dignity. You were in that position once. Value their time and find ways to benefit them. Challenge and encourage them.**Take measured risks**. At this stage of your career you should be trying lots of different things to find what works. Some of it will go well, and some of it will flop, but it’s all okay!**Make sure important people see you teach**. At some point you’ll ask for a reference letter and it’s really important that they’ve seen you teach.

## Other Resources

- First Year Math in Canada is a (brand new) collection of resources for teaching first-year classes available to instructors.
- MathedMatters. A blog that the “Try, Fail, Understand, Win” post comes from. Lots of great stuff here.
- Academy of IBL. Resources for IBL.