Textbooks and their readers

How do I respond to the examples provided here -- as a teacher and a former student?

In reading this article and reflecting on my experience as a learner, a few things come to mind.

First of all, I trusted what the textbook said, depending on the language used. For example, considering the examples offered in the article about the length of a femur of a man or the image of the Vesuvius-like man and the equation/measurement offered, I would have taken that as true. Done. I may have even stashed that tidbit away for playing Jeopardy! later. What I cannot say that I did, even if I took the information as true, was put too much weight on it. I guess what I mean by that is that I didn't really evaluate the claims that the textbook made -- especially when it came to word problems or questions with a context. They were 'just' math problems. What I did do was think that the textbook's job was to help me learn math, so the numbers, situations, etc. were way more contrived so that I was forced to do practice a particular concept. So, in that regard, the textbook did not really strike me as a socially embedded educational artifact. I do recall looking at old textbooks from teachers' classrooms. I think we found it more amusing to look at how the kids in them dressed or the technology that was referenced.

Secondly, which is a mild consequence of the point above, maybe a lot of the textbook didn't strike me as odd because I did participate in the dominant culture. I identified with it. I might have even found it odd that someone in the text was named Jamal. I didn't know anyone by that name. I almost felt like the writers were trying 'too' hard to be inclusive by only changing the names of people. Nonetheless, that may be an issue for me and could indicate that I was situated where the textbook was situated.

As a teacher candidate, I recognise that these questions are written by people who are situated and may be part of the dominant voice, and that they may have inherited their views of textbook writing from what they themselves were exposed to as students/teachers. So, I take textbooks now as little more than someone writing a situation not for the sake of sharing information, but rather for the sake of doing math. The real world just happens to be there.

What are my thoughts about the reasons for using/not using textbooks, and the changing role of math textbooks in schools?

I have mixed feelings about using textbooks. In one respect, textbooks offer an all-in-one response to the curriculum. All of one's potential resources could be there. This might be a limited view, however. Teaching to the textbook has long been criticised by curriculum administrators (While I think it's true, I do not have a lot of support for that statement. That was always a potential criticism of math teachers by NSW authorities during registration/auditing time.) But that is also considering that resource as a common one can really be a benefit too. Often times textbook developers do have experience teaching and have tried to make sense of what they feel people should learn and what the local curriculum expects. It also means that teachers are not sourcing every single resource.

Restricting oneself to a textbook is just that: restricting. That is why I feel it's important to look outside of the text. Not just to other texts, but to anything where a person has to (or can!) use math to interpret something about the world. So, I feel that this is more the role of a textbook now: a support or starting place.

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.

Reflection on Assignment 2

I am very glad that we got to try this experiment with Pythagoras' theorem with a group that is supportive! 

Given the feedback on the lesson plan from Susan, I was interested to see how this might play out. My experience with grade 8 students is mixed. The first time I taught Pythagoras' theorem, I did the usual (or what I thought was the usual) cut out unit squares from a 3-4-5 triangle template that I found in a resource book. I found that this only encouraged the students to want to draw squares on every triangle and did not really support the cognitive shift from actual squares to the relationship between the areas, let alone the sides. So, when we went to actually use Pythagoras' theorem in class, they told me it felt separate from the cutting of the squares. So, that meant that the next year I tried a different visual proof for Pythagoras' theorem. Similar. They thought it was cool, but that didn't link the algebraic statement. One year, I tried blocks and setting up tables and triangles. They built towers. The next year, I found this video. And, in that particular instance, it was a game-changer. This is my recollection of Johnny T's outburst (a kid in the back row of the bottom Year 8 class who hadn't volunteered a lot up until then).

    JT: Woah. They're the same.
    Me: What's the same?
    JT: The squares. The small ones with the big one.
    Me: What about the squares is the same?
    JT: The space inside.
    Me: The space inside. We have a math word for that.
    Another student: Area.
    ...

So, in that instance, these 43 seconds set up the entire unit. I did not expect that. For us, it worked. Therefore, I am grateful to my team members for considering this video to share as an introduction to test out in this micro-teaching.

I am also thankful for the feedback of the group members. A lot of the feedback looked at where there could be some confusion in the video. That I appreciate, because it is important to see where these ideas can arise and how they affect the possible plan (and whether to place it elsewhere in the lesson or to set it aside altogether).

Apart from that, I feel other features of the lesson were reasonably chosen. I'm glad we got to try a visual thinking strategy: See. Think. Wonder. This is something that could be followed up on more substantially in a longer lesson. In this specific case, leaving that aside may have helped our timing insofar as Sissie may not have felt rushed at the end; we could have engaged with ideas of learners more as they pertained to Pythagoras; and we could have devoted more time to Pythagoras' theorem as he shared it. I am not too worried that we didn't get to the algebraic representation of it. I was most interested in the relationships. And, Pythagoras didn't offer the statement as we know it now. So, while that was also a critique from one of the learners, I feel that would have come at a later stage in the lesson (it was actually our next slide!). Nonetheless, we arrived at the outcome stated at the beginning of the microteaching event of arriving at Pythagoras' theorem (not our current representation of it).

My group members were very supportive during this process. I was worried that I wasn't contributing as much as I needed (I knew I was sick. I found out afterwards that I had pneumonia -- crazy!), so I am thankful for their support.