The TPI

Here are my results:

I was not surprised that my Transmission score was low and my Development score was high, since I feel like those results align with my teaching philosophy. I was a little surprised that my social reform score was quite low, since I care quite a bit about teaching conscientiousness and responsibility. It does make sense with the way I answered the questions though - since most of the questions asked about my current teaching habits, and I haven't introduced those topics into my practicum teaching yet, I said I rarely take activist stances in my teaching. I expect that to change a little as I get more comfortable teaching though.

One of the topics on the quiz that I was unsure of was apprenticeship - in particular, is it necessary for a teacher to be an expert in their subject area? My initial reaction was yes, at least to some degree. I don't think teachers need to have, say, PhD's, but they should be able to answer just about any question a student asks, and be able to contextualize and extend the material in interesting ways. In my experience, students also get more excited about learning when they know their teacher has a very good knowledge of the subject area. I do think it's possible to be a very good teacher while not knowing a huge amount beyond the course content, so I'm hesitant to say that it's necessary, but it's always better for the teacher to have more knowledge.

Math textbooks and their language

 In all of elementary, high school, and university, I don't remember ever having a math textbook that I found effective. As a teacher, I really appreciate the attention to linguistic detail put forth in this article - particularly the analysis of the use of first/second/third person, and the discussion of how these details affect whether the reader sees math as absolute or more subjective. On the other hand, though, as a student I don't think that these details can, on their own, make a textbook something I want to learn from. There are many other concerns with textbook design, such as problem design and exposition style, that I find more important when learning from a textbook.

In general I'm inclined to not use a textbook, except possibly for practice questions. I think they are good to give ideas about structuring units, but I value adapting my lessons to my individual style and my individual classes more than the ease of use that comes from a textbook. From my experience, students also dislike textbooks, so it's more likely that they'll be disengaged if I use them.

As an aside, th article did make me consider the common use of "we" in math literature. As the standard pronoun used in proofs, I appreciate the way it makes written mathematical content seem like a collaboration between writer and reader, as if they are exploring the ideas together. I want to make "we" the standard pronoun in my classroom as well, to bring students into the process, even when I'm teaching from the front.

Weighing herbs

 One set of weights that works is 1 gram, 3 grams, 9 grams, and 27 grams.

The first thing I did was translate the problem into mathematical language: weights added to the pan without the herbs means we add those weights, and weights added to the pan with the herbs means we subtract those weights. So, the problem is to find four numbers that we can combine with addition and subtraction to get all numbers from 1 to 40.

One of my first strategies was noticing that, since we can use weights to both add and subtract from a given weight (by placing them on either side of the scale), if we divide the numbers from 1-40 into 3 sections 1-13, 14-26, and 27-40, and include 27 as one of our weights, than we only need to be able to find a set of three weights which can measure any quantity from 1 to 13. This is because we can add or subtract the numbers 1-13 from 27 to get the numbers 14-40. I used the guess and check method here, and was unsuccessful for a while, until I realized I could use the same strategy. By dividing 1-13 into 3 sections 1-4, 5-8, and 9-13 and including 9 as one of our weights, we only need to find two weights which give us the numbers 1-4, for the same reason. This was easier, and gave me the four numbers 1, 3, 9, and 27.

After I solved the problem, I realized something interesting. Each weight can be in one of three positions: on the pan with the herbs, on the pan without the herbs, or not on the scale at all. Therefore, with 4 weights, there are 3^4=81 possible configurations. Removing the configuration with no weights on either side, we have 80. But they come in pairs: given a configuration that measures a positive amount of herbs, we can get a configuration that measures a negative amount of herbs by switching the weights between the pans. Since we aren't interested in configurations that measure a negative amount of herbs (or equivalently, negative sums) there are really only 40 possible valid configurations, which is exactly the number of distinct quantities we need to measure. Therefore, we need to choose our weights in a way that no two configurations add to the same weight. This might be another way to approach the problem.

For students, a way to scaffold this problem might be to ask a similar problem without the possibility of subtraction (in this case, we would get the series 1, 2, 4, 8, ...). A similar but more difficult problem I saw a long time ago, and don't have a good solution for yet, is this:

It is your job to design a new set of bill denominations as currency. Your goal is to create a set of denominations that can express any integer dollar value from 1 to 100 in at most four bills. What is the smallest number of bill denominations you need to achieve this, and what are their dollar values?

Group teaching reflection

I was pleased with how this lesson went - I thought our explanations were clear and the energy was good. My main issue was that we tried to pack too many things in, which meant we had to sacrifice depth in some parts. In particular, I would've liked to have had a longer discussion about percentages in the news, and how they communicate various quantities. It would also have been good to give more time when the class was working on example problems. Having too much content has been a pretty consistent theme in my presentations, so it's something I'll really need to work on.

Otherwise, I felt pretty comfortable giving my own portion of the lesson, and it seemed the rest of the group did as well. Audience engagement is hard to judge online, but I think the class was pretty engaged throughout. If I was teaching a lesson like this in person I would try to give some group exploration time, but that's also made much more difficult by the circumstances (and is hard to fit in a 15 minute lesson).

Group teaching lesson plan

We will be teaching a lesson on percentages, from the grade 8 curriculum.

Lesson plan

Slides

The Hornby Island soup can, and an elevator problem

 My first thought upon seeing this problem is that it is about proportional reasoning. It's stated in an interesting way: no specific values are given, and instead it's asking about a general solution process.

We need to determine the diameter and length of the tank. We can't really figure out the length right away, but from the picture it looks like the diameter is about three times the height of the bike.

Now, since we know the diameter of an actual soup can, we can calculate (height of bike)/(diameter of soup can) to figure out how many times taller the bike is than the soup can is wide. Since the width (diameter) of the tank is three times the height of the bike, the tank is 3*(height of bike)/(diameter of soup can) times bigger than an the actual soup can. This is our scale factor, so the length of the tank is (length of soup can)*3*(height of bike)/diameter of soup can).

Now, my first thought was to use these dimensions to calculate the volume, but we can also just use the scale factor directly. Since we're dealing with volume (scaling the can in three dimensions), the volume of the tank would be the volume of the can, multiplied by the cube of our scale factor.

The next question, whether it holds enough water to put out an average house fire, changes things up for two reasons. First, up until now, I haven't attached any numbers, and now I'll need to (which will involve doing some estimating). Second, I don't know how much water it takes to put out an average house fire, so I'll need to do some research.

My research on can sizes returned mixed results. Eventually I found out that the standard Campbell's tomato soup can holds 10.75oz, but Wikipedia's chart of standard tin can sizes doesn't have any size with that volume. The nearest one is the No.1 (Picnic) can, so we'll go with that. It has dimensions roughly 6.8 cm by 10.2 cm. The bike in the picture looks like it's a pretty normal size, and most bikes are about 1 m tall. That means the tank has a diameter of 3 m, and since can has a diameter of 6.8 cm, our scale factor is 3*100/6.8, or about 44.1. 10.75oz is 305 mL, so the capacity of the tank is 305*44.1^3 mL, or roughly 26,158 L. Based on the size of the tank, that seems reasonable to me.

The consensus I found online regarding how much water it takes to extinguish a house fire was "it depends on a lot of things", but [this firefighter] estimated that 500 gallons (2273 L) would be enough to put out 85% of house fires. That number seems small to me, and this is a random answer from the internet, but given that our tank holds ten times that amount, I think it's reasonable to say that the water in the tank could put out a house fire. This answer seems even more reasonable when you consider that if the tank didn't hold enough water to put out a fire, the Hornby Island fire authorities would probably build a bigger tank.

This problem was interesting because it tested mathematical reasoning in a way that a lot of math problems don't. It was much vaguer and had less information provided than usual, and it was up to me, the problem solver, to make reasonable assumptions and estimates. I think this is one of the most important mathematical skills in the real world, and we don't focus on it enough in schools.

One time when I was visiting my parents, who live in an apartment building, one of the two elevators had broken down, and I noticed that the wait times for the elevators were much longer - certainly more than double. That got me wondering: what is the relationship between the number of elevators and wait times? In a building with 10 floors, what effect will removing one of the elevators have?

This seems to me like a pretty difficult problem, and in order to make progress one would have to make quite a few assumptions about the behaviour of the elevator-goers and the rate at which they leave/enter the building, but I think it would be a fascinating problem to study, with a huge amount of depth. It could be presented to strong high school students as a long-term inquiry project.

Flow state in math classrooms.

 The first lesson we need to take from this is that flow comes from creative activity, not from worksheets or lectures. We need to provide students with stimulating and creative tasks that they can "lose themselves" in. Since flow occurs at the intersection of high skill and high challenge, we need to present them with problems that are difficult, but also belong to an area that they are familiar with. It is easy to set a very difficult and stimulating problem that students do not know how to engage with, and for them to get frustrated and simply give up. This is why proper scaffolding is important: it brings students to the required level of challenge, while helping them feel comfortable with the amount of skill needed.

On the other hand, we shouldn't be too disappointed if we don't always succeed in getting our students to this flow state - as Csikszenmihalyi said, most of these experiences happen in people who have at least 10 years of experience in the area they are working in. The flow state is the end goal, but most of the time, high school students will not have the level of skill or familiarity with math that Csikszenmihalyi argues it requires. But his version of flow, the complete immersion and loss of self-awareness he describes, is a high bar, and I think we can give students opportunities for experiences that are similar, if not quite as intense.