Curriculum Review

 The first thing that caught me off guard while reviewing the curriculum was the variety of courses offered to fulfil requirements. This is less to do with the specific content or philosophy of the curriculum, but I think it is emblematic of the approach that the Department of Education is taking with regards to student choice and embracing all elements of mathematics. It's possible that these courses have been offered for many years, but I know that my own high school in BC only offered two courses at the Grade 11 level and three at the Grade 12 level. Obviously, there are also administrative constraints, but I was very impressed by the inclusion of things like History of Mathematics and Statistics, neither of which were offered at my school. 

The second surprise (also pleasant) was the heavy emphasis on teaching math for citizenship. In the past, I have seen and heard strong focus on teaching math for personal use and use for employment. These have some merit, but generally are persuasive to only a fraction of students. The emphasis on numeracy and citizenship is I believe a much stronger argument which applies to all students. Especially in the data-rich climate that we live in, understanding the use of statistics and number to inform opinion is crucial. Seeing the curriculum specifically draw attention to the "ethical use of mathematics" was especially shocking, as I think of math in school as working very hard traditionally to avoid such realms.

Charting Mathways in BC

 In looking at the graduation requirements for the Dogwood Diploma, it is clearly outlined that a student must have taken at minimum one Math 10 course (which could be either Workplace Mathematics or Foundations of Math and Precalculus) as well as one Math 11 or Math 12 course. I have not included grade 8 in the below diagram because there is only a single option which then leads directly to Math 9, so it has no bearing on the variety of pathways. This means that there is an extraordinary variety of potential pathways that a student could follow in the BC math curriculum. 

One might be inclined to point out that not taking some courses precludes you from take a related course (for example Precalculus must be taken before Calculus) but this is not technically true. While it is strongly recommended that this be the case, students are free to enroll in whatever class they choose provided they have support. I know this to be true because I myself took Calculus 12 and Precalculus 12 concurrently, having only had that option due to some scheduling conflicts which arose from trying to take advanced art classes as well as calculus (they had apparently not planned for such a possibility). 

Geometric Circle Puzzle

 My initial instinct for this puzzle was to examine it by summing opposite pairs. For example, I wondered if all opposite pairs summed to 31. This is obviously not correct, but I think that my mind went there because some similar geometric puzzles I have solved in the past can be approached by that method. I wanted to record that thinking here because it was an example of how my past experience in similar puzzles had a direct and immediate impact on my approach to problem-solving. 

When I actually sat down to solve the problem, it became much clearer to me how I might go about it. I know that a clock (which has 12 points) has a difference of 6 between each point. This makes sense, since 6 is half of 12 and the opposed points have equal circumference between them in either direction. I can use this knowledge to extend to a larger circle of 30, knowing that opposed points must all be different by 15. To show this more rigorously, we can add in modulo 30 and know that the difference between points a and b must be d, such that 

a + d = b

and

b + d = a

in mod 30. Thus, d is 15 and 7 is opposite 22.

In terms of extensions, I am very curious about how one might use shapes other than circles to ask similar questions. Equilateral triangles are the first that come to mind, marking different places on one side and looking at the rotationally symmetrical points. In that case, it would be crucial that the number of points be a multiple of 3, as opposed to a multiple of 2 in the case of the circle.

I don't have a strict definition of what a geometric puzzle is, but my intuitive understanding is one in which the simplest proof is a visual one. in this case, one could simply draw the given circumstance and arrive at the answer with no algebraic or logical work whatsoever. 

Eisner Response

 My first 'stop' in this reading is when Eisner says "it is possible to create a school environment in which the taking of initiative becomes an increasingly important expectation..." (pp. 88). I am reminded of my experience as a summer camp leader, training new staff members. The most oft-mentioned buzzword in performance reviews is "initiative." It's an extremely difficult skill to teach, requiring a learner to make bold and occasionally wrong choices. In school, those wrong choices are harshly punished, either socially or by grading. Our current system is not well designed to foster initiative, at least by my own estimation. A shift towards inquiry-based learning will certainly help provide opportunities for more students to take initiative and I would also like to see more focus on building initiative on areas of education that are less content-based. For example, I remember having a math teacher in high school who asked for student input on when assignments should be due based on how much content they contained. That was revolutionary to me as a student and hugely increased my buy-in to the class.

My second 'stop' was when Eisner discussed the work of Lepper et al. (1978) on the perils of a rewards-based system. I stopped here because another of the classes I am currently enrolled in was just discussing the educational approach of behaviourism. As far as I can tell, the course took an uncritical view of behaviourism, provided it was approached in a humane fashion. In my own experience, though, the perils that Lepper describes are very real. I think that we do a disservice to the very enjoyable and meaningful activity of learning when we assume that students will require a reward in order to participate. I am not so naive as to believe that all students will joyfully participate immediately upon being given an activity, but I think that a systematized reward schema will lead to system-motivated students. That is to say: when we give students a framework for what to expect and what is valued, we should not be surprised when they adhere rigidly to this framework and do not creatively strain against its limitations.

Another stop (albeit brief) was at the mention of curved grading systems. I find the idea of a curved grading system immensely frustrating, both from the perspective of an educator and a student. If the quality of work/understanding is at an A level, it shouldn't matter what the understanding of the other students in the class is. Grade curving can lead to some truly illogical behaviours: I remember my mother telling me a story about a teacher she had who would give a student a 98% if they got very question right because they needed to learn that "nobody's perfect." The great irony is that students would then try to drop only a single mark on large tests so that they could get 99% - a better score than if they'd gotten the question correct. Obviously this is a wild example, but it also illustrates what I spoke to above about students living within the provided framework.

As another minor stop, I want to just note how much I love the term "convivial" in this context. Overall, I really resonated with this reading and its critiques of industrialized schooling! And what a powerful couplet to end the poem! I myself am something of a poet and found that a phenomenal mantra for educators.

Finally, with reference to current curriculum, I always find curriculum documents a bit of a paradox. If everything they aspired to were to happen, then our schools would be much better, I think. The problem is that they are often vaguely aspirational but lack the will to change our null curriculum. In another of my classes, we talked about the three (often conflicting) goals of school: Traditional, Progressive, and Socializing. We simply cannot meet all of these goals at once, which is something that curriculum tries to do. In my mind, the factory school system is not root cause but a victim of a culture which both venerates and sublimates the individual. I don't have a solution for that, but I am glad to read other people with more experience than myself thinking about similar things.

Microteaching Lesson Plan

 My apologies for the late lesson plan post: I was travelling Friday and ended up being without wifi for longer than I anticipated (Ferries, ugh!). Here's the plan:

Goal: Teach students how to play Knock Euchre, a popular and simple partner game. Euchre can be tricky to get the hang of, but provides a great vehicle for learning simple card playing strategy in a trick taking game. It is also very flexible between contexts, from casual games to serious tournaments. 

INTRO & SURVEY: 1-2 min

Begin with personal anecdote - I spent all my childhood years going to summer camp and getting to know my cabinmates through mini euchre tournaments. I have great memories of light-hearted competition and banter and am excited to teach new people the game.

Next, survey the students:

• What experience do they have with card games?
• Have they played a trick-taking game before (i.e. hearts, screw your neighbour, bridge)?
• Have they played a game with trump before (i.e. bridge, rook, wizard)?

Based on these answers, the lesson plan will vary. If there is a wide range of experience between students, consider asking experienced students to provide explanations of basic rules elements to keep them engaged.

TEACHING THE PLAY: 3-7 mins (based on student backgrounds)

Then, explain rules. This will be heavily lecture-based, but brief. I cannot think of a way to present this specific technical information through a discovery-based approach. First, the play:

1. Euchre is played with only the cards 9-A of each suit (Aces high)
2. Euchre is a trick-taking game, which means we each play one card and the highest card wins
1. Exception to this is following suit
3. Trump is the strongest suit, determined each hand separately
1. In Euchre, trump cards follow a slightly different order

Let's play out a hand! I will have constructed an open hand using an applet online (https://playingcards.io/jd7g8d). We can play through to check for understanding of rules of play.

TEACHING THE BIDDING: 5 mins (if time allows)

1. After dealing, the dealer turns up a card and players opt to choose that trump or pass
1. If they choose this, the dealer keeps the revealed card
2. If everyone passes, go around again with any trump option available
3. If you declare trump, you and your partner are "on the hook" for 3/5 tricks
4. Brief explanation of scoring (<1 min)
5. Optional rules: Canadian Loner, Stick the Dealer

Now we put that bidding to practice with another open hand!

If time allows (especially if students are already familiar with rules like trick-taking and trump) we can play some closed hands using the same applet.

REFLECTION:

I addressed this to some degree in my comments below, but I definitely put too much content into the lesson on the assumption that students would have more understanding than they already did. When I ran the timings on my own, I got through everything within 10 minutes, but this relied on an estimation of how much time demonstrations and interactive elements would take. I had originally hoped to test my timings with a group of friends, but unfortunately was unable to coordinate that in time. 

 I am quite pleased with how our presentation on the mathematics of beehives and nets turned out. I was initially concerned about the amount of math present in the presentation, but as I dug around to enrich that aspect of the presentation, I was pleasantly surprised by the amount of depth involved in nets! I think it would be very interesting to further explore the patterns between geometric shapes and isometric nets.

Something I learned from the project that I will hold onto is the amazing way in which beehives are constructed, or rather the uncertainty in the scientific community of just how it is done. Trying to work out what might happen, even just as idle speculation, engaged a few different parts of my brain, from the chemical to the psychological.

If I were to do anything differently in this assignment, it would be to dive deeper into the math of net construction, as I noted above. I though that the presentation was smooth, but really only very introductory. Part of the issue is that we bounced around a bit between context, math, and design elements. Obviously, all presentations are a mix of these things, but I think a more intentional flow between them could have helped us make better use of our limited presenting time.

Battleground Schools Response

 My first "stop" was at the table outlining the different modes of viewing math education and its goals. Most of these were familiar divides to me and I knew quite clearly which side I landed on. The last point, though, about "teacher-proofing" curriculum, was not something I have heard before. It really hammers home to me how many challenges arise from the factory model of schooling. This is something I've had discussions about in other courses, so I am glad to be able to make another connection to it here.

The second part of the reading, that which focuses on complicating factors, is essentially the subject of my undergraduate thesis project. Rather than recreate all of the connections that I have, both personal and academic, to the section, I have instead linked the artist statement (which addresses the academic grounding for the artwork) here. If you have a moment, I'd love to hear anyone's thoughts. One connection in particular that I would like to highlight is that between this article and the work of Sheila Tobias, author of Overcoming Math Anxiety. Much of her work deals with a more individual scale, but it touches on the same problems and systems of thought.

As I continued to read, I was struck first by some amount of despair at the thought that the battle between conservative and progressive mathematics education has gone on for so long, despite seeming wins for the progressive camp, when it seems as though every progressive win slides back into conservativism. After thinking that, though, it occurred to me that my own math experiences (at least in secondary school) were largely progressive, inquiry-based, and exploratory. Noticing this in my own thinking, I wondered what had given me the impression that the conservative model still dominated. The answer, I think, is that this model and perception of mathematics is still our cultural attitude towards math, even if the practice in classrooms has shifted. There is more inertia, I think, in our culture than in our classrooms. I'd love to hear what other students' experiences in the math classroom were like to get a bit of a survey (albeit biased) of the current state of the field of battle.

The Dishes Problem

 My solution using conventional algebra:

My solution without symbolic algebra:

If 2 people eat one dish of rice, 3 people eat one dish of broth, and 4 people eat one dish of meat, then each person eats 1/2 a dish of rice, 1/3 a dish of broth, and 1/4 a dish of meat. Each person then needs 1/2+1/3+1/4 dishes, or 13/12. There are 65 dishes and 13/12 dishes per person, so there must be 65/(13/12) people, or 60. 

I'm not certain whether or not this method would still be considered algebra, since the mathematical procedure is in many ways the same, but it seemed like a reasonable approach provided the solver has an understanding of proportions and adding fractions. I think that the context of a story puzzle can sometimes feel contrived, but placing a puzzle in its historical context avoids that feeling. Something about using the same story that was originally used lends it validity, at least in my eyes. I think it is hugely important to offer problems (and solutions) from many cultures and traditions in mathematics. Students are primed through social dialogue and prevailing educational culture to think of mathematics as an activity done by only a small subset of a single group (that being White, European men). This perception means that students who don't fall into similar demographics are being told that math isn't for them, a message they probably hear far too much as it is.

As for enjoyment, I think that I do get some additional satisfaction in approaching the problem because of the story, provided it doesn't feel too contrived, as I addressed above. I believe wholeheartedly in the motivating and imaginative power of story, and I especially think that story can be helpful in prompting students to look for other ways of solving the problem. I'm reminded of a study that was done with child street vendors (I can't remember in what country, but somewhere in Central America). The children were sat in a math classroom and given tests on simple arithmetic, on which they did very poorly. Then, the students were given the same problems but in the context of making change and purchasing goods (something they did daily). In this new context, the children performed far above what we would expect at their age level. Context is everything.