Reflection on the course

I would divide what I've learned from this course into two categories: the practical, which mostly came from preparing and presenting the microteachings, and the theoretical, which mostly came from class discussions. Of course the theoretical content is very applicable to the classroom as well, but it seemed more focused on the broad philosophies of how to teach rather than the particulars.

From the practical section, I learned how to structure a lesson and how much content to teach in a given amount of time, which are skills I value a lot and hope to keep developing. At the start of the semester I was definitely erring on the side of too much content. The experience presenting was also very helpful; I always feel like I need to get better at speaking.

The unifying theme behind what I learned from the theoretical section was to teach math as a creative endeavour, not just as a set of procedures. This hearkens back to the first blog post, on relational vs. instrumental understanding, and was developed throughout the course, and I definitely feel that I have a better understanding about the many ways in which to approach this. The ideas about the thinking classroom particularly interested me.

As to how to improve the course, I felt like the balance between practical and theoretical was a little heavy on the theoretical. I think it would have been helpful to have class discussion time about how we design our lesson plans or make decisions about unit structure. These are the things that I felt underprepared for when going into practicum, and it would have been nice to have some guidance about them beforehand.

Hewitt - the necessary and the arbitrary

 
This distinction between necessary and arbitrary fits very nicely into the thinking classroom, as described by Peter Liljedahl (and Asha Padmanabhan). Most of the time, in a lesson, the 'arbitrary' content is quite small and can be covered by the teacher in a short amount of time, and the 'necessary' content is what takes up most of the instructional time. So if we allow students to do group discovery work on the 'necesssary' content, that discovery becomes the focus of the class, and students are able to see how the math works rather than being told.

Getting students to characterize for themselves what is 'arbitrary' and what is 'necessary' is another activity that prompts them to look at mathematics deeply, and really understand what doing math is all about. In itself it's quite a metamathematical process - the axiomatic methods of foundational mathematics are really just the same process, but more formal. Learning how to differentiate between the arbitrary and the required is takes some mature critical thinking, and that's something we should really be striving for in our classrooms.

Unit plan - final

Here is the final submission for Assignment 3.

There are two copies of the unit plan and the lesson plans; please read the ones without the .odt extension. I originally uploaded the .odt files and would like to keep them there to be safe, but the upload introduced some formatting issues that I fixed on Google Docs, so the other ones are nicer to read.

My favourite math things - and a bonus

 Ultimate Tic-Tac-Toe is a brilliant twist on tic-tac-toe that's got a lot of really cool strategy and mathematical properties. I love it because it takes something that everyone is familiar with and turns it into something mind-stretching.

The blue eyes logic puzzle is a riddle that introduces mathematical induction in a very clever way, and gets a surprising result.

Finally, this isn't super math-y, but I wanted to plug my own game design: Overgrowth is a 2-player strategy game that you can play with a standard deck of 52 cards. In my biased opinion, it's pretty good.

Unit plan

Here is my unit plan.

I would still like to incorporate more non-traditional elements into it; most of my class plans still have some traditional elements because I am teaching multiple topics per day (I only have 6 instructional days total for this unit). I mostly designed my activities with minimal interpersonal interaction because of COVID-19, but depending on the situation I would love to incorporate more.