Battle ground schools response

I had to stop at the table of dichotomies that you acknowledged were false. I focussed on each row, but while trying to choose either conservative or progressive, I almost always found myself identifying with both sides. The conservative side frequently viewed math as a means to an end, whereas the progressive side viewed it as an end in itself. The table’s category of “conservative” seemed to represent elitist views, which don’t map onto the current American use of conservatism that has become politically anti-intellectual and against academic elitism. I found myself disagreeing with both sides regarding assessment. I personally prefer individual derivations as the predominant method of assessment, rather than multiple choice that doesn’t test for rigour or process thinking that underemphasizes the final result. Perhaps I hope to assimilate the population into the elite, which is why I prefer high school assessment to mimic university assessment. You later clarified that these camps often affirm values emphasized by the other side, notably progressives requiring fluency and conservatives requiring understanding. Thus, I am not alone in asserting both sides of some table rows.

I also stopped at your mention of non-specialist teachers teaching mathematics at the lower secondary level due to a shortage. I have volunteered in a classroom with a teacher who admitted she wasn’t strong enough mathematically to teach above Math 9, and she was a specialist math teacher. I greatly appreciated volunteering for her, but I was surprised that she had survived as a teacher that long with such a small teaching range. This profession can provide refuge for those who didn’t find work elsewhere, but that can come at the cost of students when those most suitable to teach never choose the profession.

I was heavily taken back by the Bourbaki group’s decision to ban teaching geometry. Even if analytic geometry allows all geometric problems to ultimately be reduced to set-theoretic statements in a formal language, the original statements that are being reduced are geometric in nature and therefore have inherent geometric meaning that differs from the non-geometric meaning of the statements that they are reduced to. How did this group expect students to solve any geometric problem if they were fully deprived a geometric education? Set theory is useless to students if they cannot translate their problems into that formal form.

Fibonacci Math Art Project Group Reflection

 

Curricula reading response

 I was surprised by the insinuation that the separation of classes by ability was really a statement of honour. Naturally students feel good about being placed in the higher class, but is this really a reward for moral virtue? The students in the bottom class aren’t being punished. At least I have heard it classified as sorting students according to ability so that each class proceeds at an appropriate pace. This way no student gets held back or left behind.

I was also surprised that the time of day for a subject like art could impact how we signal importance. Even if such signalling is deliberate, I doubt these scheduling decisions take place in our modern crowded schools. To not use the art room is a waste of space. I wonder if our schools have reduced the second/third curricula that they teach, as we fixate more on the first curriculum.

We might expand our idea of curriculum from referring to subject specific facts to a way of functioning in the world. These ideas about curriculum go as far as life skills. These BC curriculum has gone in a more concept oriented direction away from facts, but these lifeskill abilities discussed haven’t been formalized into BC’s curriculum. They remain unstated, but students will pick up on them nevertheless.

Good and bad teachers

 One of my high school teachers was very good. He taught the entire syllabus well, and gave us plenty of opportunities to think for ourselves as we solved problems. He got us to work through problems in groups of two on the whiteboards quite often, and he varied class activities quite frequently. He was very competent in his understanding of mathematics, and I felt well equiped for my subsequent studies after the training he provided. As I look to become an effective teacher, I can deploy his wide variety of class activities that kept us students on our mathematical feet.

One of my professors was a very poor teacher. He had a surface level understanding of the applied mathematics that he taught, and he specifically lacked an understanding of the pure mathematics that underlaid the constructs that he used. On more than one occasion, when we asked him probing questions, it became clear to us that he hadn't mastered the material that he was teaching. It seemed like he was simply guessing answers to our questions, and we saw right through this. In order to avoid his error, it is important first and foremost that I plan my lessons well and think about a variety of questions that I could be asked. I will need to demonstrate competence in every topic that I teach. It is inevitable that I will receive questions that I haven't heard the answer to before, so I must be honest about my ignorance.

The locker problem

This was a fun puzzle that reminded me of the math contest puzzles that I used to write as a student; that training certainly helped. While reading through the problem in the UBC garden, I constructed the solution in my mind.

I write my thoughts as follows:

1. Each locker is flipped several times, and since it starts open, it will end open if and only if the number of flips is even.
   [Open] ---Flip---> [Closed] ---Flip---> [Open] ---Flip---> [Closed] ---Flip---> [Open]
   [Open] ---Flip---> [Closed] ---Flip---> [Open] ---Flip---> [Closed]
   
   
   
2. Every student has some index n, and they only flip the lockers that are multiples of n.
   Student 4 flips the underlined lockers:   1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ,11, 12
   
   
3. Thus, each locker is flipped by students with indices that are factors of the locker number
   Locker 12 is flipped by the underlined students:    1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ,11, 12
   
   
4. The final “openness” of the locker is the “evenness” of the number of flips, which is the “evenness” of the number of factors of the locker number.
   This follows directly from (1) and (3).
   
   
5. Each factor has a paired factor, and if this paired factor is distinct, the pair of distinct factors does not contribute to the “evenness” of the number of factors.
   All factors of 12 form distinct pairs: (1, 12), (2, 6), (3, 4).
   
   
6. Only factors that are paired with themselves can result in an odd number of total factors, and this self-pairing requires the repeated factor to be the square root.
   Factor pairs of 16 aren't all distinct: (1, 16), (2, 8), (4, 4).
   
   
7. Thus, locker numbers that are squares finish closed and non-square locker numbers finish open.

If I were to write this solution up as a formal proof I would do so with much greater rigour, although I wouldn’t show it to my students. I would use functions mapping from {1, …, 100} to {-1, 1} to represent each student’s flipping operation and after multiplying these functions together I would arrive at the final locker state. I would need to translate all the statements above into equations. There are certain logical technicalities I would need to handle to make the proof rigorous, which I deliberately omitted above for brevity.

The level of rigour I use when explaining to students will vary with grade level. For an IB HL math course I would show part of the formal proof because induction is part of the syllabus. But for most students I will provide the list of steps I previously mentioned. The explanation above uses mathematical terms that match what is taught in the BC curriculum. For the IB courses I teach, I will use algebraic expressions to denote multiples and factors.

Response to Skemp article

 Response to Skemp article (Sorry for missing the 9am Deadline)

I was in full agreement when Skemp answers "Yes; Relational" as he considered relational understanding to be better than instrumental understanding, especially since relational understanding contains instrumental understanding within it. I was taken back by "the existence of a large body of experienced teachers and of a large number of texts belonging to the other camp."

I expected the arguments Skemp lists under Devil's advocate, notably that instructional understanding is simpler and therefore could take less time to teach. I still find myself aligning more with the relational approach, and I will need to prepare to deal with students who prefer the instructional approach.

Hello World

 Hello everyone!

I'm looking forward to learning with you.

Josh