TPI Response

 

My scores on the TPI were quite polar, especially when contrasted with those of some other students that I've had the opportunity to look at on their own blogs. I scored particularly strongly on Nurturing and Developmental, which doesn't surprise me. I also expected that I might score Recessive for the Transmissions index. I think the truest sign that the test is accurate is that I could have predicted it very well, which I suppose only means that I am consistent, but at least that's something!

Generally, I feel good about my results. Despite the perhaps negative connotation of the term "recessive," I genuinely do believe that material transmission is one of the least important aspects of teaching, so I am not concerned to see that belief reflected in the TPI. This does mean, especially as a new teacher, that I will likely have to justify some practices to administration and the like, considering the fact that our current model of assessment places emphasis on this pillar, but I feel strongly enough that I am looking forward to these discussions rather than afraid of them (well, maybe a little afraid). 

Thinking About Math Textbooks

 I found this article very thought-provoking. It did a good job of denaturalizing elements of textbooks that I hadn't considered before such as pronoun use. There is a sense of formality to the language of textbooks that I think makes them harder to engage with as a student and also lends them an air of authority. this comes from using language like "must," and other strong modal verbs that the reading called out directly. It really separates math as something one knows from math as something one does.

I recall at the beginning of a class in my undergrad on complex numbers having the professor (Dr. Asia Matthews, whom I think you may remember, Susan) ask all the students if they personally knew any mathematicians. The room was entirely quiet for quite a while. Eventually, I put up my hand and mentioned the name of someone whom I'd met at CMESG. In retrospect, I know many mathematicians, by any stretch of the definition, but somehow I found myself struggling in the moment to apply that label to real people in my life. 

This disconnect between doing and learning math is one that I think a large majority of students (and even teachers) feel. It's easy to think of math as an already determined set of facts and procedures, especially in a high school context. Much of my own personal philosophy about mathematics revolves around humanizing the discipline, so I tend to fall on the side of avoiding direct textbook use. I think they can be quite helpful in developing banks of problems, but even in that case I would endeavor to modify them to a degree and bring them into more of a "thinker" context rather than the "scribbler" default that so many seem to exist in.

The Scales Problem

 

This problem is one with which I am already familiar, so I am going to begin with my solution and then spend the majority of time on my thoughts. The weights must be: 1g, 3g, 9g, 27g. With these four (and only these four) you could weight out any integer weight from 1g to 40g. The key observation here is that you can place a weight on either side of the scale, either opposite the herbs or on the same pan as them. This means that you can functionally add or subtract the weight from the total balance you are trying to create. 

Rather than list out every method to sum the numbers 1 to 40, I think it is more elegant to demonstrate why these numbers work. In constructing the solution, we can begin by imagining each weight as a distance on a number line that we can reach. For example, the 1g weight can move us 1 step in either direction on a number line, depending on which pan it is on. For maximum efficiency, we want there to be only one way to sum to each number. Otherwise we have an extraneous weight. Thus, each combination of adding a weight, subtracting a weight, or omitting a weight should give us a unique number. What I am functionally describing is a tertiary counting system, base 3. 

This requires some imagination, but the analogy is straightforward. Rather than using the digits, 0,1,2, let us imagine using -1,0,1. This does not change the number of unique numbers we can create, only their values. Technically, this method also includes numbers from -40 to 0, but we can disregard them when making our counts. Given that we are working in a tertiary system, we then know the values of each "place." They must be powers of three (e.g. 1,3,9,27). This is not a proof, exactly, but I think it a compelling justification.

An interesting follow-up to this reasoning is asking the question: "what if we couldn't subtract?" For example, what if there was a practical reason that we couldn't place a weight on the same pan as the herbs? Maybe they are loose and would stick to it, for instance, or be contaminated. We can imagine lots of possible reasons. The question is then: "how many weights do we need (and what are they)?"

Happily we can solve using the same logic, only with a binary system rather than a tertiary one. This means our most efficient weights are all powers of two (1,2,4,8,16 in this case). Something that I have used to introduce binary to students is a 'magic trick' I learned from my father. You consider the numbers 1 to 63. On a flash card, write out all the numbers which have a 1 in the 1s place within that range (all the odd numbers). Then, on a new card, write out all the numbers which have a 1 in the 2s place (in binary, that is). Continue this pattern until you have a 1s card, a 2s card, a 4s card, an 8s card, a 16s card, and a 32s card. Then ask a volunteer to pick a number and hand you all the cards which contain that number. You will be able to instantly tell what number they chose by summing the first numbers on every card you are handed. Being handed a card is a 1, not being handed it is a 0. You can use this trick without understanding binary, which is why it is a good intro, but knowing why it works requires the binary understanding. At some point, I would love to develop a tertiary version of the trick.

Group Microteaching Reflection

 On the whole, I think that our presentation went very smoothly. Obviously, having a group of students already familiar with the material means that we could cover ground faster than we would in a real Grade 11 classroom, so I'm not certain if our pacing was suitable, but otherwise I thought that our use of examples and connection to real life was appropriate and engaging.

I was especially pleased with how well it was received when we asked students to brainstorm constraints on the optimization. This is, in my opinion, the key mathematical skill of optimization, so I was glad to see that the lesson clearly encouraged the behaviour. The act of creating an optimization function is also important, but it is a mechanical enough operation that I was never concerned that students would struggle greatly with it. Without calculus, optimization is really just a relatively simple exercise in algebra.

I found this topic very interesting to discuss and to think about teaching because I don't recall discussing it in my own high school experience until the very end of high school in Calculus 12. That makes some sense, since I know the curriculum has changed quite a bit, but I really like the idea of introducing basic optimization principles earlier, since it is a clear use of mathematics for decision-making, something we often gesture towards in our justifications for curriculum but students are sometimes unconvinced by.

Campbell's Soup

 

To begin, let's solve the problem given! I did some research to determine that a soup can is about 11.6cm tall and 8.7cm in diameter. I also determined that the bicycle in the image (assuming that it was sized for an adult) was approximately 1.73m long. Obviously, there are a number of estimates being made here, but it looks as though the can is about three bicycles long, putting its 'height' at 5.18m. If we apply the ratio of the standard Campbell's soup can dimensions, we find that its radius should be 1.94m. We can then use the formula for volume of a cylinder (although there are some dents we can't account for) to determine that the volume of the water tank is approximately 61200000 cubic centimeters, or 61.2 kilolitres. 

My research suggested that an extremely large house fire would require 20000 gallons of water to control, or 76 kilolitres. The water tank, then, wouldn't quite be able to handle a massive house fire, but considering that Hornby Island is not full of sprawling mansions, it should be able to deal with a regular-sized house fire.

The extension that I would propose for this problem is to ask students why cans are sized the way that they are. A classic optimization problem (one that can be solved with graphing only) is to determine the optimal dimensions of a can to minimize surface area. Students can be directed to determine the optimal dimensions to hold the same volume as a Campbell's soup can and then, when they discover that the can is not optimal, ask furthering questions about why that might be. Surely someone at Campbell's has done this calculation! Why not use optimal dimensions?

Microteaching Preparation

 Slides: 

https://docs.google.com/presentation/d/1DK2AGiEvZzxQZS_0X4mRnUEykQXDi5tX0PZsoPfb5QU/edit?usp=sharing

Lesson Plan: 

https://docs.google.com/document/d/1IXNpOCY_sDZmOao_Id_BpyAROaxSJ9mNXes5ucN4X4g/edit?usp=sharing

See you all tomorrow!

Mihalyi Csikszenmihalyi's Flow

 I was first introduced to the concept of "flow" from an educational context at Quest University. Specifically, I was in a class that was discussing Peter Liljedahl's applications of flow in thinking classrooms. I think that it is a very powerful idea, to consider reflectively the states that we are in, and I think (or hope) that most people have some experiences in a state of flow. I specifically resonated with Csikszenmihalyi's description of ecstasy and can recall vividly experiencing that state in a number of contexts myself, one of which is mathematics.

Something that Csikszenmihalyi claimed which I disagreed with was the idea that flow could only be achieved by a person with ten years of experience in a field. He contradicted himself to a degree later, with regards to being in a state of flow while watching TV, but I don't think flow is unique to people with real expertise in a subject. Liljedahl's graph of flow which includes an activity where a person has low skill and approaches a small challenge resonates more with me. 

Of course we as teachers would want to build flow in our classrooms, since either boredom or anxiety make it harder for students to learn. What Liljedahl describes in his slides is a very active teacher role in not only creating but maintaining flow in a problem, building that staircase up to high skill and high challenge. Part of the difficulty in doing this in a classroom is that every student has a different level of skill, so no one challenge is likely to be appropriate for every student. There are, I believe, some "unicorn problem" counterexamples to this, but by and large a single problem isn't appropriate for everyone. 

The key skill, then, is adaptation. Adjusting the challenge or the skill of the group as the problem is progressing is the most important teacher ability for maintaining flow. As a part of this, teachers must also be adept at recognizing when students are in a state of flow and when they stray across the boundaries to boredom or anxiety.