The way textbooks may position position mathematical learners

 This text highlighted some key examples of how different textbooks represent mathematical ideas differently. The ones that stood out to me were how absolute values, the binomial coefficient, and derivatives were all somehow represented with different lenses. It is expected that the arts will have different biases, but it is not expected for bias to emerge in STEM subjects such as math. Because bias is not expected, its presence is even more insidious.

I have talked about textbooks at great lengths with my SAs. In order to teach the current grade 10 curriculum, I require 3 different textbooks. This is because the curriculum keeps changing and the textbooks quickly become out of date. Because textbooks are static by nature they are not good teaching tools for the changing environment of learning.

My final thoughts on textbooks is that they teach material a certain way, and it might not line up with the way I view that topic of math. I will be biased in my own teaching, but I think that as an educator, it is better to teach well with your own bias than to teach a choppy, inauthentic lesson with someone else's bias. Students learn best from teachers who are passionate about the material, and that passion might disappear if I was forced to teach math in an inauthentic way.

Dave Hewitt & mathematical awareness

 In the videos we watched in class, the three things that made me stop and think were:

1) Hewitt's ideas that students will naturally find the correct answers, they just needed guidance to get there faster. That if left to their own devices, mathematical thinking comes naturally, it is only that the pace needs to be quickened

2) The idea of a group choir makes a lot of sense. I especially enjoyed that for tricky answers the unison would break and then the group as a whole would understand that further thought had to be put into this question.

3) I loved that Hewitt did not tell his students what was right and what was wrong, he was only a guide. He was constantly asking if the students thought they were right. In honesty, I remember being a student and being decently upset with teachers who did this. I remember feeling cheated, why was I doing their job for them? But i recognize now the significance of framing the relationship with students in this way.

I think that the fraction problem is a great inquiry problem that doesn't have any correct answer, at least not at the start. It is obvious that Hewitt has a particular fraction in mind, but by framing the question in terms of multiple steps, students feel much more agency. Just finding a fraction in between 5/7 and 3/4, there are infinite answers. By making the problem more concise, he is controlling which fraction the students give as an answer, but it still does not feel like agency is removed from the student.

I have already incorporated group choir into my teaching because I felt like it is a good way for students that have self doubts to more easily play along; they are just one voice in a group of 30. It make me happy to see that someone so professional as Hewitt approves of this teaching style.

Hewitt Arbitrary vs Necessary

 After reading this article, issues that I ran into during my practicum were voiced. I was tasked with teaching the equations of lines, but ran into difficulties quickly. I thought that I would have been able to just share my knowledge with my students. However, I ran into needing to know a bunch of different forms of the lines. For example, I needed to teach that "General Form of a line is 0 = Ax +By +C where A B C are all integers and A is non-negative."

I marked tests and found that many students did not exactly follow these rules, and felt bad that I had to take marks away for these mistakes. I am happy that I now know that these rules were arbitrary, and this exemplifies why there was so much friction between me and this issue.

The most annoying aspect of this experience is that students were so busy learning these arbitrary rules, that they did not understand the necessary concepts of lines being a relationship between x and y. Students would remove x and y just in order to make the equation look more like General Form.

When I teach my classes, I am going to focus on necessary knowledge and prioritize it. After students understand necessary concepts, I will teach them the arbitrary rules.