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.