Unit Outline (updated 12 December)

 Link to Unit Outline

https://docs.google.com/document/d/16zYGzTdHSPeCid_gwV12hDqYhizC8LL8/edit?usp=sharing&ouid=103256016962961193862&rtpof=true&sd=true

Response to Flow

From Czikszentmihalyi's talk about flow, I took away a couple of ideas.

First of all, I was thinking about how to create the conditions that can help to foster this concept. Initially, I wondered about what level of competence is needed to create flow; however, I'm beginning to think that the threshold is much lower than I initially anticipated. 

Secondly, I was considering times when I wonder whether I've experienced some aspect of flow. I can think of writing essays and not knowing where time goes and how the ideas percolate. I can think of reading a book and missing my bus stop (is that even flow?). I am unsure whether I relate flow to mathematics. I enjoy it. I like thinking about questions. But I do not have any specific math-flow memories. I remember time in math classes being gone in a flash (learning and teaching!), but I don't know whether I've been in a math flow. Maybe a math jam. But I'm not prepared to say a math flow. I wonder if it has to do with a level of creativity -- one that I do not always associate with mathematics, at least in the same way I do when I make a craft.

Also, I wondered whether (or how) flow could be induced. As a TC, I'm looking at how to foster a sense of purpose in the classroom and manage student experience and behaviours. I am also interested in how to find activities that hit that 'sweet spot.' Also, I wonder how I could possibly widen a student's flow band (for want of a better phrase). This really seems to be finding either low-floor high-ceiling tasks or finding a good way to build in assessment and finding strengths upon which to build. So, I wonder how well I can do this and how hard it might be. But, I feel that -- if flow is a possibility -- the investment would be worth it.

Locker Problem

In considering the locker problem, I've concluded that, once all 1000 students go through, 31 lockers will be closed. All other lockers will be open. The lockers that I claim will be closed are the perfect squares that are less than (or equal to) 1000, i.e., lockers 1, 4, 9, 16, ..., 900, and 961.

Soup Can Response

Response to the questions asked

First of all, I responded to the question asked on the blog. I concluded that there would be enough water to put out a house fire.

Teacher bird, student bird

In responding to the question, I noticed early on that I had a plan. The questions about dimensions and volume are straightforward enough; however, the openness in the question is where a lot of ambiguity lies.

I first realised that I needed to establish a scale of some sort. This is where the assumptions (and research) started. I looked up bikes (and asked someone who knows a lot more about bike sizes than I do) and chose to work with 26" wheels. I measured the picture to get a reference of 3 cm for the 26". So in establishing a scale, I needed to research, make an assumption, and measure relevant pieces of information. I also needed to choose which dimension to use in my calculation. I chose the 22 cm height of the cylinder, as I'd assumed that dimension changed the least, given the damage to the not-so-circular cylinder.

Then came the research on the proportions of the Campbell's Soup can. In conjunction with information that was provided in the question (and assumed to be true), I could use this information to establish dimensions of the cylindrical water tank. Then volume. Then capacity.

At that stage, some quick searches were done to determine how much water it takes to put out a fire on the average-sized house on Hornby Island.

So I feel the main areas for research (that I determined as I went through the question) had to do with lots of averages: bike size, can of soup, average house size, how much water is used.

The things that I knew or could determine included the steps I needed to follow, how to determine a scale, how to use proportional reasoning to provide a reasonable estimate for what I was trying to calculate, unit conversions. I didn't find that I got too stuck. For me, actually using the average values was the challenging part. I think I wanted actual numbers. Like in a textbook. Because word problems are exactly like real life!

(a) Extending the puzzle

This tension leads to the extension that I would pursue for the question itself. A question like this has many variables. And those variables contribute to ranges of information.

For example, setting aside the soup can proportions could lead to providing a minimum and maximum water capacity of the tank (it's cylindrical at one end, but the cross-section morphs into something more elliptical at the other). This could prompt a student to consider a range of values for the actual capacity. Further, questions along the lines of which value would you work from in a situation where firefighting was concerned (Will you make decisions based on the maximum projected value the tank holds? the minimum? an average?) This range strategy could be applied to most of the researched information: bike size, average house size on Horby Island (maybe consider median house size?), even the soup can sizes vary. So, I would try to develop a project where students needed to prepare a proposal for the town council regarding how much water is actually available and whether they may be prepared for a bad fire season.

Response to 'arbitrary' and 'necessary'

One afternoon I wrote 15-8-2=9 on the whiteboard, just waiting for my grade 7s to come in and get started. Of course they were very keen to tell me that I was wrong. We started the conversation about what could be different if this statement were true. All sorts of ideas were shared: add brackets, change the meaning of the equal sign, change the meaning of the numbers, change the order of operations. This got us to discussing the idea that some things in math are decided or chosen, but some things aren't. We did not frame these findings in terms of arbitrary or necessary knowledge, but we did highlight that some things we just need to know the meaning of if we are going to communicate using mathematics.

Hewitt distinguishes between two main kinds of facts in mathematics: arbitrary and necessary. The arbitrary facts are those that need not be true and cannot be deduced, as such. Arbitrary things are mathematical facts that need to be memorised. Hewitt contrasts this with facts in mathematics that have to be true. These aspects of mathematics can be deduced by someone. The former, arbitrary things are those that need to be memorised: word meanings, symbols, conventions. The latter, necessary things are those which have the potential to be deduced by someone: relationships, properties. 

Considering this division in mathematics has the potential to influence how a teacher teaches. Teaching definitions, symbols, or conventions needs to be shared by teachers and memorised by students. Committing these arbitrary facts to memory will facilitate participation in mathematics, according to Hewitt. Where the mathematics lies is in uncovering the relationships and propositions that hold true despite the words or symbols that are used. (It might be worth noting that arbitrary elements of mathematics have the potential to enable understanding of necessary elements; however, that is a distinct conversation.) These necessary aspects of mathematics have the potential to be uncovered once they are in the 'awareness' of a learner. (The 'awareness,' as Hewitt calls it, seems akin to a learner's PZD.) 

This means that good task design -- perhaps an inquiry-based task -- has the potential to encourage a learner to uncover the relationships that really characterise mathematics. There is no amount of inquiry that can enable a learner to arrive at a mathematically chosen convention; however, relationships and propositions are there to be uncovered if a learner is in an appropriate cognitive state. The specific implications for lesson and unit design are that different activities -- e.g., direct instruction, inquiry tasks -- will be appropriate for different kinds of knowledge. It is a teacher's responsibility to share the arbitrary knowledge through appropriate tasks, and it is a teacher's responsibility (pace Hewitt) to create different tasks that enable/allow for learners to learn necessary knowledge. These tasks will be different in nature because they have different aims and modes. 

Hewitt has the criticism that necessary knowledge is treated like arbitrary knowledge or at least stored as arbitrary knowledge (by some teachers and students). This is a disservice to mathematics and to one's mathematical understanding. While I tend to sympathise with much of what Hewitt claims in the article, I have some reservations regarding graded mathematics education. It can be a challenge for teachers to support a variety of students in a variety of ways. And the propagation of received wisdom may feel like one way to support what could look like student success. However, a criticism of this is that students always take away different things from a lesson. The perceived control of direct instruction is an illusion, and more appropriate task design that honours the role of the learner in their own construction of understanding might not be letting go of any type of control that a teacher thinks they have. Further, with appropriate tasks, students can learn how to participate in mathematics in a more desirable way that supports the competencies that, for example, the BC curriculum aims to develop.