My Updated Unit Plan

 Here is a link to my updated Unit Plan:

https://drive.google.com/file/d/19u0DcQmrspQBU75cpXKx9MLYe_UKQiuQ/view?usp=sharing

I have updated the following:

- I have included instructions for the project (including a list of suggested topics and draft rubric) and modified it so that students will work in groups of 3

- I have included an exit slip template in the closure sections of all three lessons

- I have included worksheet links (there are hyperlinks saying "Worksheet Document" in place of the pictures that were there initially).

- I attempted to clarify the C=6x+4y example

Course Reflection Post

Image source: https://blogs.pugetsound.edu/cesblogs/files/2016/01/Square-kitten-reflection.gif

i) What I have learned

 I learned something from every assignment we were asked to complete. Some caused me to look back on my own school experience with a different perspective and realize there were some helpful things teachers did that I hadn't recognized as a student. Other activities made me aware of concepts and areas of research I wasn't familiar with (e.g. Richard Skemp's instrumental vs. relational understanding, the Math Wars, hidden curriculum that schools teach etc.). Overall, I believe I learned how to solidify some of the methods I wanted to apply to my classroom through this course. For example, in my letter from a future student, I defined as aspect of successful teaching to be adding variation to the classroom setting. From the ideas we explored during our first assignment with looking at the Bridges Conference art, covering different homework problems in class with the intent of looking at them from a student's perspective, and various other discussions/added resources during the course, I believe I now have many tools to implement variation in my classroom.

ii) How my ideas changed

 I think the main difference between now and the beginning of the course is that I have more awareness of how to achieve the goals I have for my classroom (such as added variation, as discussed in i)), and issues (such as the hidden curriculum) that may stand in the way of those goals, or are aspects of teaching I hadn't realized existed.

iii) Suggestions for Improvements

I have no suggestions for improvements to the course at this time. I found everything very useful and I learned a lot through the readings, activities, and discussions we had in this course.

My Favourite Math Things

 For the purposes of sharing in class, this choreography puzzle might be a fun one to try. I hadn't realized I was literally embodying math through dance since I was nine and then actually figuring these puzzles out for myself when I started choreographing more at 16. I've done this lesson four times on my short practicum/subsequent visits and it's interesting how there were new approaches to it even by the fourth time it was presented!

 

 

In terms of math puzzles/problems that are my favourite, I tend to like math contest word problems. A lot of the AMC problems are fun for me to solve so I chose one at random (#19) and actually went the wrong way with it but found my way back eventually. I thought about how sometimes students will be able to identify exactly what to do and exactly what they need right from the start but some need that extra time and further exploration before they can know what exactly is happening and what needs to be done. If I had access to scissors (I'm currently self-isolating in an apartment above my family's store and can't touch things they need), I would have physically cut out this shape and tried to figure out the properties of this cylinder and what it would look like to tape two sides together. 

One puzzle that stands out from my math journey is actually a rather simple one that I think I first came across during elementary math contests. It's the one where there are multiple triangles within larger triangles where the trick is not to just count the inner triangles but recognize that there are multiple ones built up of smaller triangles (or you could describe it as many triangles contained in one big triangle). The reason this was memorable was that I ended up being in a summer camp in which I was often doodling, and one day I was trying to recreate that puzzle from memory. I wrote down "There are 23 triangles" or something, and one of the tutors running the camp came over and said, "Oh okay so there are 23 triangles here? I thought there were only 16." I started to explain, "See we have to count the triangles that are made up of triangles-"And as I was explaining, I realized I missed some so I said, "Oh no, I missed some. But basically that's what we have to do." And he just went, "Ohhhh that's interesting. Cool!" He was a math tutor and he had absolutely seen that puzzle before, which was why he asked me about it - so I would explain my process and could catch my mistake on my own. I ended up getting tutored by him for a few years after that, and eventually working with him a few years later at that same company. He was definitely a great role model for me and I have always striven to be as inspiring and engaging as he has and continues to be when teaching/tutoring students.

Unit Plan Draft

Below is a link to my unit plan draft:

https://drive.google.com/file/d/19u0DcQmrspQBU75cpXKx9MLYe_UKQiuQ/view?usp=sharing

Edit: After taking my classmates' suggestions into consideration, I have updated my unit plan:

https://drive.google.com/file/d/19u0DcQmrspQBU75cpXKx9MLYe_UKQiuQ/view?usp=sharing

The major changes are:

- breaks have been written in

- the unit project timeline has changed to allow those who wish to get started on the project early the opportunity to do so

- There was a thought that these lessons may be too long for the allotted time, which may very well be the case. However, I have left them for the time being with the thought that it is better to have more to choose from and to adjust my schedule accordingly if I sense we are not going to get through all the material listed, rather than running out of things to do. Please let me know if this is the wrong approach.

- There was another suggestion to arrange for a library visit, which is an idea I like (I even checked with my practicum school and received permission to do so if I want to during my long practicum). However, I believe the nature of this project is to have students browse the different applications of linear programming, much of which can be found online, so I don't think it is necessary or very time-effective to have them search through books for these very specific pieces of information (although I would definitely allow them to visit the library during the work period if they wanted to). I will try to find another project/opportunity to arrange for a library visit during my long practicum because it's a great opportunity to move away from placing math in its own separate box and instead moving towards interdisciplinary connections.

Homework: Dave Hewitt on Arbitrary vs. Necessary

 During my practicum, I observed a teacher implementing some of the techniques Dave Hewitt talked about in terms of allowing students to derive understandings from a given problem or exercise. In the activity this teacher put forth, she drew a central inequality (e.g. 2 < 5), circled it, and had lines coming out of the circle labelled "divided by 2," "multiplied by -4," "+ 2" etc. and had her students fill in the answers at the end of those lines on their own. Afterwards, she had them call out their answers and asked if anyone noticed a pattern. A student responded but I knew most of them had reached that conclusion as I walked around to observe them and listen to the conversations they had with their neighbours about their findings. I think it was a great activity and would like to do something similar in my class. 

The examples in figures 9 and 10 on page 8 are also good examples of this. During a math workshop one of my university classes held in a high school, we had a set list of word problems that we proposed to students which included something similar to this example but on a higher level which involved factoring - I think it was something like, "A rectangle has side lengths (x + 7) and (x - 3) with an area of 39. What are the two side lengths?" However, this group of students hadn't covered their factoring chapter yet so we hadn't expected anyone to answer it. Instead, a lot of students ended up with the right answer because they did some guessing and checking. Some happened upon the answer by systematically plugging numbers in for x until they got the desired area, but others recognized that the side lengths were 10 units apart so they looked for factors of 39 that were 10 apart and found 3 and 13. Thus, they skipped right over finding x and went straight to answering the question. Other students found that x had to be bigger than 3 and less than 39, but because numbers closer to 39 would multiple to make very high numbers, they figured it would be closer to less than 15, and narrowed their scope from there until they got x = 6 and side lengths 3 and 13. So for those of us who are used to seeing these types of questions, we would have jumped straight to solving for x in whichever method we preferred, but for these students, they had to develop a method to solve the question, and used important mathematical thinking and reasoning along the way. I think another important note here was to have students verbally explain their process to us and their peers, because it forced them to reflect on their process instead of solving and moving on.

A computer science prof I once had used to have us try problems in class on topics we hadn't yet covered, and would walk around to observe our work. His reasoning for this was, "If you're not struggling, you're not learning," which I believe is a similar sentiment to "If I'm having to remember ..., then I'm not working on mathematics." We were presented with a problem and no pre-determined algorithm for how to solve it so we often took unique paths to solve the same question, much like the students in the math workshop I talked about. I would like to attempt this with my students as well by presenting them with warm-up puzzles (possibly from websites like the ones Asha shared with us) or exploration exercises like the teacher whose class I observed. Another idea I had was to maybe put math terms with their definitions on the board (the "arbitrary" aspects of math), give students a problem to work on in groups (the "necessary" aspect), and ask them if they are able to incorporate those terms in their explanation to me/the class (bridging the two). Some of the possible hurdles to this would be students looking the terms up on their phones and finding the answer to the problem by copying an existing example of an algorithm, or not engaging with the activity for a variety of reasons. However, hopefully all it takes is some adjustments such as asking an extending problem that they can't look up, offering some hints so the problem becomes within reach etc. to realign the students to a state of "flow" and so they gain some useful understanding through the exercise.

Entrance Slip # 9: TPI Results

 1)

From my results, it seems that nurturing is my dominant perspective while social reform is the perspective I identify with the least. The perspective with the most inconsistency is transmission, where it seems that I apply and believe in it, but don't really intend to. 

2) I was most surprised by the height of the developmental perspective because, although that is one of my goals as a teacher, I don't think I have done enough to achieve 12 points for action (my highest action points across all categories) - I would have expected it to be closer to the transmission range, with apprenticeship and nurturing being my top two perspectives. I wasn't too surprised at the results for social reform because it is one I am still grappling with. 

 3)  Considering that for my top two perspectives, my beliefs/intentions are higher than my actions, I wonder how I could bridge this gap and align them so they are more or less on the same level. In encouraging us to look into the reason for these differences, the TPI website asks if it's a matter of job constraints, philosophical inconsistencies or lack of clarity about departmental expectations. At this stage, only having taught a few lessons in a school setting during my practicum, I think it's partially due to the fact that I haven't experienced the full job yet. For instance, if I wanted to raise my nurturing actions to the same level as my intentions/beliefs, I could implement certain assessment strategies that facilitated learning instead of causing students extra stress with standardized tests. Similarly, I would have more opportunity to explore apprenticeship as I was in charge of more lessons.

Leaving social reform to the side because I still need to struggle with that perspective and find out where I stand, my second lowest is transmission, which I intend to work on over the years anyway so there also isn't much for me to say about that. However, for the developmental perspective, I think I need to work a lot at effective questioning. I would like to follow Asha's advice in this regard by taking simple questions and taking out information to make it more interesting, but I would have to try my hand at it before I could say where my skill level on that lies.

Entrance Slip # 8: Thinking about math textbooks

Image source: https://edcp34220.blogspot.com/2020/11/your-homework-reading-for-sunday_16.html

As a former student, I don't know that I necessarily thought a lot about the language used in terms of using first/second person in problems. However, I do think the way problems were structured as absolute fact (such as the equations for the male/female height-femur relationship) shaped my attitude towards math. In contrast, I remember complaining about physics problems that told us to "assume friction is negligible," because I thought it was silly for us to be doing a "real-life problem" in a vacuum setting. I admitted I understood it was for simplicity in calculation but it still bothered me. Now I realize math had the same problems, but because the questions were stated as fact while the physics questions admitted their modifications, I hadn't questioned it. I think it also influenced the way I viewed math as more stable than physics or other sciences because we focused more on math "rules" and less on the experimental and uncertain aspects surrounding the situations in the problems we worked on.

As a teacher, I think textbook questions are often manufactured to fit a lesson instead of finding an application of the lesson content and modifying it to fit the skill level of the class. Of course, as mentioned on page 12, textbooks have to make certain assumptions of student knowledge and can't completely cater to a class, but they could still try to base their examples in real-life applications, or give better context. Going back to the femur/height example, I don't know a lot about proportions of the human body and these examples seem like generalizations made up for the sake of tests a math skill, which either should have been stated as such so students knew this wasn't some steadfast rule, or if it is, then that rule should have been explained more to spark interest in students.

In my opinion, textbooks are good for skill building questions, but less helpful in meaningful word problems or activities. In my opinion, word problems should be formed by the teacher so that the teacher can tailor them more to the students, make the problems more applicable to the environment around the students (i.e. where they live, what their context is with respect to the world etc.), and bring some of their own interests and ideas to those problems as this will likely spark more interest and inspiration in students than if the problems seem to come from a faceless source. The textbook may also offer further/alternative explanation to math ideas but I believe students generally turn to the internet for video explanations and easy-to-follow explanations instead of reading through the textbook, which may contain information that is either not being covered in class or is more difficult to follow. However, from what I have observed in classrooms and from my own experiences, workbooks with answer keys are almost better than textbooks because they provide the student a means to make sure they are on the right track. Thus, I think textbooks or workbooks can be useful for testing basic skills and as an additional source of knowledge, but I think classrooms can survive without them if the teacher is able to provide students with strong notes, their own problems that are more applicable, and perhaps use something like Khan Academy as a supplement to skill building problems from the textbook. One of the disadvantages to using something like Khan Academy is if students don't write their work down in a notebook, then they won't be able to refer back to their process at a later date.