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. 

Assignment 2 Curricular Microteaching Lesson Plan

Response to Eisner "What is meant by 'curriculum'?"

(1) The first place I paused in this article related to the claim that most people will not have jobs in their lives that will be intrinsically interesting or provide the intrinsic motivation required to participate in that activity (pp. 90-91). If I'm honest, I share(-d?) that belief in some capacity. In reconsidering this belief, I do think that there is some truth to it. There are many jobs that people just do because they need money. And these jobs do not (in themselves) foster a rich, intellectual experience. However, I got sad when Eisner claimed that school prepares people for this experience. That I didn't/don't want to believe. While I can see that the schooling experience can privilege delayed gratification, it saddens me to think that all of a student's learning experiences are extrinsically motivated. Maybe this is because I enjoy learning and think that this stuff can be fun; however, it is unfair and a bit arrogant to presume that my liking something is grounds enough for other people to like something too.

As a teacher candidate, this idea encourages me to build on an idea from our assessment class. How can I use assessment to motivate students? I would like to explore this idea more and learn how to motivate students in an effective, authentic, non-star-chart way. (Or whether I can even do that!).

Also, I cannot help but wonder how technology can/will/does affect the content of this claim. More and more jobs that can get automated do seem to get automated. However, they are not automated out of respect for human flourishing (I presume it's a money thing). Maybe this should be a good motivator for teachers and teacher candidates to adapt the learning experience. I am not sure how that might look or what it should look like; however, I cannot help but think that this is an opportunity -- an invitation? -- to reprioritize student experience in their 12,000 hours of state-mandated education.

(2) The next place I stopped was when Eisner concluded that "the cultivation of imagination is not a utopian aspiration." (p. 100). This, I feel, is given a lot more attention now -- and in some cases, I grant, lip service -- than it might have in the past. Nonetheless, this is an area where I feel that growth has been made in formalised schooling.

A few examples have made me think that imagination and cross-hemisphere activities are taking place. This might look like mandatory music lessons in schools or mandatory design classes in middle school (this was the case in some Australian independent middle schools as well as the school where I had my first school visit). What used to be considered electives, these schools do claim to find value in them and force students to engage in these activities. (And, yes, it might be worth noting that in making it mandatory that other values are being taught.) Furthermore, I look at the MYP framework and its interdisciplinary units. These units are designed to promote creative thinking and synthesise information from seemingly different areas.

This second stop actually leads me into my response about how the BC curriculum connects with Eisner's ideas. First of all, yes, schools teach all sorts of things: said, unsaid, and intentionally unsaid. The local curriculum does not (cannot?) escape those features. Not only is the curriculum considered what we explicitly teach content-wise in our classrooms, the curriculum also enforces cultural and behavioural expectations (the 'hidden' and 'null' curricula). As Eisner mentioned in the article, not all of these things are bad. But it is important to be aware of them and that they exist. This means that Curriculum is much broader than might have been anticipated. 

This connects to the BC curriculum in a few other ways. One, schools have their unique cultures. This can have varying effects. In one instance, I've operated in a school culture that is demanding, where students border on disrespectful. In another, I've operated in a school culture that is demanding yet supportive. Each place has its own culture and, I might argue, varying aspects of its own curriculum. (They all had schedules and mandatory classes and the like, so those are similarities.) Secondly, the primacy in the BC curriculum of the curricular competencies wants to speak to some of Eisner's ideas. The explicit curriculum is requiring students to develop thinking skills and engage in activities that explore place, stories, and First People practices. These seem like attempts to shift the focus (or broaden the focus) from the what of learning to how these ideas are expressed and practiced. 

Response to "Battleground Schools"

 (1) The first place I stopped was at Table M1. 

I was reading through the different positions regarding math learning, and a few things came to mind. In the first instance, I was trying to figure out my position on each of those Areas of Interest. I didn't approach the table thinking that my thoughts would align with one column or the other; rather, I tried to see where my initial thoughts landed. (I did, however, in my head replace "conservative" with "old school.") What I noticed is that I too am probably a (closet?) math Platonist. But, slightly to my surprise, I found that my views oscillated between conservative and progressive, and I tried to find the merits in each position. I am interested in this regarding the consistency of my own beliefs regarding math education. I am also prompted to consider whether, in some cases, both outcomes can be produced, even if one outcome may be privileged over another. Furthermore, in trying to find merits in each of the positions, I am encouraged to consider how one could teach and/or assess so that each of those respective positions could be demonstrated.

(2) Another place that I stopped to take a breath was in response to the shift of what mathematics education is for (p. 396). The article introduces Dewey's ideas about privileging inquiry as a way to promote more fully the democratic ideals that were becoming prevalent at that time. Dewey's ideas really did shift what learning could and should look like. I would think that many people may believe that communities of inquiry might be suitable for some aspects of education, but I could understand how there would be pushback from people who think of mathematics as a static, removed set of axioms and ideas. How does inquiry look in mathematics education? It isn't up for debate whether Pythagoras' Theorem is true. And how could we make any progress in mathematics education if each student were asked to 'discover' how a number system worked? In response to this, I want to reply that even in a scenario of a conservative classroom, the teacher still doesn't have so much control over what is actually learned. To teach something and to learn/understand something are different skills. So, by recognizing that people do create the contexts in which they learn, there may be a better discussion for how a democracy or community of inquiry can work in the context of a math education. And, which can often be overlooked, inquiry is not unguided. Providing a rich environment is still a big responsibility of an educator; however, it is worth noting that in this context an educator is not the person who delivers the knowledge. It seems that an educator is more about the person who curates an environment and tests out the resulting ideas. But a potential benefit of adopting some of these ideas is that it places primacy on some skills that will better suit people to participate more fully in a democracy.

(3) The third stop isn't so much a stop as it is an overall response. Who knew that mathematics education could be so politically motivated? In one sense, it really does make sense though. What, after all, is education for? (Maybe Lockhart is right -- that is, maybe math should be an elective.) Every decision that is made in the classroom, even if a teacher or student or parent, is some reflection of some policy decision about something being important. I noticed this after going through an American education system and then teaching in Australia. Different things were taught in Australian classrooms than in American classrooms. That is not to say that fractions were different or that collecting like terms or the FToC were any different. However, some things were very much left absent in different places. Upon reflection, I have found it interesting what is considered important in mathematics education. I also consider the IB math courses. In that programme, math is mandatory. So, it is interesting to see what the curriculum developers feel is important for everyone to know (including those who would not consider themselves mathematically inclined). Furthermore, in a course where, say, graphics calculators are mandatory, it places an emphasis on what skills are promoted. So, in retrospect, it seems initially like a math education is something that might be static; however, a look at other places and times quickly undermines that notion.

Response to Lockhart's Lament

In reading Lockhart's "A Mathematician's Lament," I found that I agreed not so much with one thing or statement; rather, I found that I agreed with the sentiment that (creative) aspects of real mathematics have been removed from mathematics education. In this regard, mathematics and mathematics education are two different things. At times, I feel like teachers try to share this sentiment with their students through opportunities like open-ended questions or comparing approaches/responses to questions. However, this can be a challenge when, as Lockhart cried, politicians step in and tell teachers what to teach, or, for example, in the case of AP or IB programmes, that they are so content-heavy and assessment-driven.
In saying this, I do feel that it is not as bad and gray and lifeless as Lockhart's rhetoric might suggest. Classrooms -- not all of them, I grant -- can be places for flourishing. Mathematical ideas and relationships might just be present, sought, and celebrated. There are instances where real mathematics is being done. So, in defense of the practitioner, sometimes these ideals are fostered (even if it might not look like the creative utopia that Lockhart imagines).
Two more thoughts. The first regards the use of AI. This tool may really help to promote mathematical ideas in classrooms/mathematics education as Lockhart sees it. Do people have to know how to balance a chequebook? (1) What's a cheque? (2) No. There are other tools to do that, and, in a practical sense, as long as people can appropriately interpret the information that is provided by that tool, then the "mere" calculations might not be of the utmost importance to the human. And AI might just be one of those tools that really give teachers (maybe politicians/policymakers/curriculum developers?) the impetus that they need to focus on mathematics-as-flourishing.
Consequently (and secondly), I feel this ties in with Skemp's views on relational mathematics as a priority over instrumental mathematics. Skemp is interested in the relationships present within mathematics and maybe the relationship that we have with mathematics. Skemp doesn't think that understanding necessarily arises from the rote application of algorithms or the memorization of facts. Lockhart, I feel, would agree with this statement by saying that the former is not even really mathematics.

Reflections on Microteaching I

 

My first feeling after completing my lesson was relief. Then my thoughts focused on what I could have done differently. But first, I will consider the feedback from my colleagues. Then I will return to my thoughts in light of their input.

For the most part, the feedback was positive and constructive. My start was shaky -- which affected my teacher presence -- but it seemed to improve as the activity progressed. There was a challenge to the sequencing of the activity, to which I will respond shortly, and the content of the activity seemed to fit the timeframe of 10 minutes.

As for personal reflections, I would have adjusted a couple of things. First of all, I would have closed the activity more definitively and tied it better to the opening scenario. The opening scene was tying the game to its place in Australia's public identity, and I wished I had closed by tying it back to that thought. I also agree that I started off nervous; however, I do feel that I was able to recover throughout the lesson and work on aspects of presence that conveyed a level of confidence that promoted the idea that the teacher knows their stuff. I also shared the goals of the lesson, so I felt that was a benefit to the learners. During my pre-class, in-head rehearsals, I tossed up how to present and sequence the activity. Instead of talking at everyone for the couple of minutes that it may have taken, I decided to get everyone in straight away. This could have prompted a level of uncertainty in the students; however, the rules of the game (I decided) were straightforward enough to learn through demonstration (not recitation). So, this was an intentional decision on my part: to introduce by showing (not telling). I do recognise that not everyone was comfortable with this decision. I also need to grant that there are probably better ways to lead this activity. However, I think each person made progress toward the established lesson goals, and I appreciate that my colleagues were willing to support me as I practice reaching.