One by one (Regina Bittencourt) - Group Write-Up

 Group members: Alina, Emily K, Keith, Tara

Regina Bittencourt: One by one (The original version)

Our remake of the original

In remaking the artwork, we chose to use acrylic paints on paper. During this process, we wondered about why the artist stopped at a maximum of 9 rows (before descending). Exploring this informed our group activity of attempting to make a 10th row, observing that we no longer have the same palindromic pattern, and discussing why that happens. We tried adding a row ourselves and soon found that the “place holders” get in the way and the pattern breaks. Inspired by the lesson on Babylonian place holders, this prompted us to wonder if there was a maximum number of rows for other bases as well. After trying out a few it became clear that this maximum is one minus the base you’re working in. For example, base 10 has a maximum 9 rows while base 2 has only 1 row before the pattern breaks!

Our extension

The extension that multiplies 9x9, 99x99, 999x999, … does not follow quite the same pattern as with the 1s, and so we worked to find out why this one has a pattern of its own. Of course the painting does not have 90-degree symmetry like the original, so we decided to adapt it slightly, “flipping” the bottom half to produce an image with 180-degree symmetry to give it more visual appeal. In making the extension, we had to make decisions regarding whether the color representative of each number should be the same as the original. We ultimately decided to keep the colors consistent between both works, in the hopes that it would make it easier for viewers to easily switch back and forth between viewing. Since 0 does not appear in the original, we chose the color to represent 0 (burgundy) based on what we thought would look best with the other colors in the piece.

In designing a class activity, we wanted to offer a way to further explore the idea of the One by one piece. At first we considered using sticky-notes or tiles, but decided this would be difficult to manage in the short time for presentation. We decided to create a visual handout with the images of our artworks on one side and an activity on the reverse. Providing this visual handout means the class can see the detail of the painting (the paintings are on 12” x 12” paper, so it would be difficult to see the colours from the back of the room).

For the activity, we ask the class to consider what comes next. That is, what would happen if we attempt to add a tenth row to the painting? We’ll work this out together on the paper and on the whiteboard. Then, we ask the class to consider what would happen if the painting were done in a number system other than base 10.

No action shots available of our painting because we were too in the moment. 🙂 

To set the vibe we played the September 17th NTS Breakfast Show w/ Flo while painting and learned a lot more about each other's interests and hobbies.

Response to Gerofsky, "Battleground Schools" (Mathematics Education)

 Part of the enjoyment of EDCP 342 so far for me has been the opportunity to contextualize my own secondary mathematics education, since some of the articles we have read date from roughly the time of my high school years in the U.S. In this piece, though written later (in 2008), I have again been led to consider my experience, this time in the context of the history of mathematics education.

My first "stop" was at Table M.1 on pgs. 392-393. While I have been peripherally aware of some discussions regarding mathematics pedagogy, I never considered that there were two "camps." I assumed (naively, I now realize) that all math teachers worked with a more or less traditional approaches to the subject as a baseline while wisely incorporating "newer" pedagogical techniques. Similarly, before this course, it never occurred to me that mathematics education could have different goals (something we explored in an earlier blog entry). So for the first time, I found myself analyzing my secondary mathematics experience in light of "conservative" and "progressive" characteristics, all while asking myself: "Where do I instinctively fall in this table as a teacher?"

I am happy to report that both in my experience as a student and my current instincts as a teacher, I find myself in neither column. In my years as a high school math student, I experienced teaching that was a blend of conservative and progressive approaches. For example, when considering the "attitudinal goals of math education," I can recall instructional moments that emphasized the importance of precision and correctness (conservative), but also many other moments (in the same class!) that emphasized original thinking and problem solving skills. In this case, I saw the two approaches work together in a complementary way: learning some facts and algorithms deductively became the basic equipment we could use when presented with new and unfamiliar material. However, there are a few areas where the instruction I received was clearly more conservative. These are: modes of mathematics teaching (it was presented to us, usually), assessment (we never had group work), and the locus of mathematical knowing (it was supposed to be in my head!). While I hope my own teaching would be even better "blended" than what I experienced (for example, I'd love to find ways to elicit more mathematical thinking rather than always presenting), I find my instincts fall along similar lines.

My second "stop" is in the consideration of the New Math of the 1960s. This is something I knew nothing about before reading this summary - which is precisely why I stop here. As Gerofsky notes on pg. 398, "By the early 1970s, the New Math program was being denounced" and was considered to be a "misguided experiment, and the movement quickly came to an end." I can personally vouch for the accuracy of that statement, since in the 1980s, when I began my mathematical learning, nothing of the New Math approach seems to have remained. The first I ever heard of set theory, linear algebra, and abstract algebra was in university coursework I completed in order to pursue the B.Ed. program! Abstraction was not emphasized. I marvel that such an experiment in pedagogy could so rapidly "fall apart," but I also wonder if perhaps its demise was a case of "throwing the baby out with the bath water." While I cannot imagine the New Math approach being beneficial for all (or even most) students, it seems we have all but completely lost any trace of it in secondary education for even the brightest math pupils. There is a big gap between secondary mathematics and university level mathematics; at times they seem like entirely different worlds. Of course they aren't different worlds at all, which makes me wonder if there could be some elements of the New Math approach that are carefully incorporated in (upper level?) math coursework. I would have benefitted enormously, for example, to have encountered some set theory before arriving at university. And perhaps giving high school students at least a taste of some of these topics would entice some to consider studying math in their post-secondary years.

My third "stop" is in the connection of conservative attitudes towards mathematics education being adopted by right-wing political groups in the 1990s as part of a backlash against standards-based curricular reforms. I confess I struggle to understand how the approach to teaching math could become a politically charged issue, with traditionalist approaches being "affiliated with a patriarchal, authoritarian, fundamentalist religious agenda" (p. 399), and yet, seeing where we are today (especially in the U.S.), perhaps I shouldn't be. The polarization of our society around political identities has been a defining characteristic of our time. What is the solution? Certainly as a teacher, I hope my approach to teaching math will draw from the strengths of both conservative and progressive methods. But I wonder if we might also need a new John Dewey to come along. We face in this day (again, particularly in the U.S.) a political movement that would very much like there to be more "blind obedience to authority and lack of independence of mind" (p. 396) among our fellow citizens. Teachers need to be conscious of how their approach to teaching - even teaching mathematics! - either furthers such agendas or stands against them. 

Response to Eisner, "The Three Curricula That All Schools Teach"

 The experience of reading Elliot Eisner's chapter, "The Three Curricula That All Schools Teach," when one is some 27 years beyond the completion of high school is a somewhat surreal one. From what I could find, Eisner wrote the book (at least the original edition) in 1994, the year before I entered high school in the U.S. So "spot on" for me are his reflections that he might as well have been talking about my own school and educational experience.

My first "stop" when reading Eisner's chapter came in his discussion of competitiveness as part of the implicit curriculum that schools often teach, specifically as that competitiveness is "fostered by the differentiation of classes into ability groups" (p. 91). This is precisely how my high school was structured. Being a capable student, I was in all the "honours" classes. And Eisner is precisely correct: I felt "honoured" by being in the honours classes. My parents and teachers celebrated this fact, not just because I had worked hard, but because it seemed to mean I was worth more as a person. The best teachers at our school had the honours sections. The newest calculators were available for the honours sections. Special privileges were afforded to the honours sections (no "hall passes" required for us to go to the bathroom - I'm not kidding). We even met in designated areas of the school that had nicer classrooms! Why? Because I was "worth it" as a student. I was trusted and thought of as more responsible because I was "smart" - or at least, more able to achieve in a way that impressed my teachers. And while my school didn't have the weighted GPA system Eisner describes, one could not be considered for valedictorian or salutatorian unless one had taken certain upper-level honours classes. All of this impacted me deeply; I came to view myself as "better than" those students who were in the "normal" sections. Not just smarter, but better. I have had to spend years of my life undoing this way of thinking. (And as a result, it is my hope as a math teacher to work with students who have chosen the "workplace" and "foundations" tracks in the BC curriculum, not just those who are in the "pre-calculus" track. I'm thankful it's not called the "honours" track, but I'm sure some of the same associations are present.)

My second "stop" is also in the part of the chapter address implicit curriculum, this time when Eisner discusses school architecture and the design of school furniture. He writes, "Most school rooms are designed as cubicles along corridors and have a kind of antiseptic quality to them...They speak of efficiency more than they do of comfort" (p. 96). How accurate this comment is in my own experience! I can remember no colours being present in my high school. There were few green plants anywhere. Our desks chairs were hard plastic and "one size fits all." I can still feel the sound the chairs made scraping against the flooring. And yet, Eisner is again correct, it seems to me: the message all of this sent may have been unintentional, but neither was it questioned. We were there to learn, and learning was disconnected from our physical and even our emotional selves (Eisner takes this issue up when he discusses the "null curriculum" as well). "Comfort" seemed like an indulgence, perhaps even an inhibition to learning. Any serious school would eschew such frivolous considerations so as to train its pupils to focus the mind! Needless to say, my views have radically shifted over the years. Eisner writes, "Schools are educational churches, and our gods, judging from the altars we build, are economy and efficiency. Hardly a nod is given to the spirit" (p. 97). The way I would put it is this: we are to be about educating wholistically. If we fail to see our students as whole human persons and build/design our classrooms to support them as whole persons, we will fail to deeply educate them. I am left pondering this question: what can I do as an educator to create a space that supports my students as whole persons, even if I find myself in one of the "sterile" contexts Eisner describes?

I do not yet know enough about the BC curriculum to assess how it addresses - or even if it addresses - some of the implicit and null curriculum issues Eisner raises in this chapter. But even if it does move in some of those directions, it seems to me that the main takeaway from this reading is that as teachers (and in some cases future administrators), we cannot rely on a formal curriculum to address all that it means to educate our students. Perhaps we need to define "curriculum" less along the lines of content to be taught and more along the lines of the impact of what is learned in a given high school setting. As Eisner points out, what is learned is far more than the academic content on offer. It will include some subset of that academic content, to be sure, but perhaps a school "curriculum" is something more like a worldview, or a value system. Nothing a school does and no decision a teacher makes could then be considered outside of the "curriculum" received by its students. 

I am grateful to have the chance to reflect on this question as a teacher candidate!

The Locker Problem

 

My Best (and Worst) Math Teachers

 I can say with little hesitation that, on the whole, my math teachers were the best teachers I had in high school (by which I mean grades 9-12 -- I went to high school in Pennsylvania, where grades 7-8 were considered "junior high school"). While I had a couple great non-math teachers in those years, it was always the math classes that I looked forward to the most! 

But while all my math teachers in high school were quite good, there is one who stands out as my favourite. His name was Mr. B and he taught me both algebra I (in grade 9) and calculus (in grade 12). Mr. B was a great teacher in my view for many reasons, including:

• He was a clear and engaging math "storyteller." Every day (or nearly every day, unless we had a work day or a test or something) we would take out our spiral notebooks (laptops didn't exist in my high school days!) and Mr. B would write. In chalk. On a blackboard. And we would all copy down every single thing he wrote. Sounds boring, right? But it wasn't because Mr. B was a math "storyteller." He made connections to what we already knew, he drew our attention to new things we could discover if we just knew where to look, and he illuminated a path that would lead us together to the new learning of the day. His course/class notes, which we wrote out in full in our own hand over the course of the year (and which we had to submit at exam times for completion marks), were like a narrative. His explanations were concise and his examples well-chosen. It was simply the best math teaching I've ever experienced.
• He was himself. By this I mean Mr. B was comfortable being who he was with his students. He was a thin, wispy-haired, soft-spoken, incredibly intelligent, kind of nerdy guy who made corny jokes in class. Yet he also coached the high school soccer team (he had the respect of the athletes!) and he loved music. He showed up at all the school band and choral concerts. He was the organist at his Catholic parish. He loved to eat salads every day for lunch. None of this made him a great math teacher. He didn't overshare about his life - about all we knew of his personal life was that he was not married - but we all felt we knew him as a person because he was just himself and we liked him. And so he could teach us...and we listened. 
• He was available to help, he understood math, and he was kind. In "study hall" (what they used to call it), we knew we could ask to go talk with Mr. B for math help if we needed it, even if the help we needed wasn't for his class! I remember bringing some confusing trigonometry work to him one day and watching and learning as he worked through trig identities with me. He knew his stuff and was glad to help. He didn't talk down to you when you didn't get it yet. He just found other ways to explain it because he understood the subject well enough to do that. And he was kind to every student, regardless of their mathematical abilities.

It wouldn't be until just a couple of years ago, when I took a math course at my local university to help fulfill the prerequisite requirements for the B.Ed., that I would discover that not all math teachers are good ones! The course was in linear algebra. The professor was Dr. G. Perhaps the best way to describe Dr. G's teaching was to contrast it to the points above about Mr. B:

• He was not often clear and his teaching was disconnected. Connections to previously learned material were not often made. Dr. G sometimes simply presented new theorems as facts with little effort put into explaining why the math worked as it did. Our notes were seemingly random examples - often with mistakes made that necessitated reworking them later ourselves!
• He was impersonal. Dr. G seemed unwilling to share anything of himself with his class. He had little personality that he revealed to his students. I don't think we really even knew if he loved the subject he was teaching or not!
• He was not often available for help. Dr. G simply seemed uninterested in his students as people. Since his lectures were usually unclear, we all learned to rely heavily on the textbook. By the end of the term, less than half of the students were even bothering to show up for class...

We are all shaped by teachers we have had in the past. For my future math teaching, Mr. B stands out as exemplary and worthy of emulation in many ways. I may not have exactly the same "storytelling" approach as he did, but from him I have learned the importance of clear and careful preparation, of "showing up" genuinely for my students, and of being supportive and kind. I am grateful for the example of Mr. B and his influence in my life!

Response to Skemp, "Relational Understanding and Instrumental Understanding"

What is mathematics? Or, what does the word "mathematics" mean to a high school teacher and to their students? Like Skemp himself, the meaning of the term is not something I had ever seriously considered. So I found myself stopping early in the article when Skemp discusses the difference between "relational understanding" and "instrumental understanding." He writes (p. 2) that instrumental understanding "is what I have in the past described as 'rules without reasons', without realising that for many pupils and their teachers the possession of such a rule, and ability to use it, was what they meant by 'understanding'." This is the first time I have ever considered that there are many teachers for whom this is the case. I suppose it speaks to the quality of my own secondary math teachers, but I have always assumed that while there will be (many) students who just want to know how to get the right answer, such an approach to "mathematics" would never be the goal or approach of the teacher! Skemp has helped me realize my assumption is incorrect. In discussing why a teacher might "make a reasoned choice to teach for instrumental understanding," Skemp writes that "to make an informed choice of this kind implies awareness of the distinction, and relational understanding of the mathematics itself...One has to face the fact that this is absent in many who teach mathematics; perhaps even a majority" (p. 11). I would not have thought this possible.

Having been made aware of the distinction between relational and instrumental mathematics, another point at which I found myself stopping was when Skemp writes "The most important thing about an activity is its goal" (p. 14). I found this statement to be profound in thinking about the doing and teaching of mathematics, because this stretches me even further to consider what the "goal" of my own teaching will be. I realized at that moment that even though I was taught to understand mathematics relationally, there was still a sense in which the "goal" was quite focused on getting the right answer rather than "building up a conceptual structure (schema)," as Skemp puts it. As a teacher, what is my goal for my math students? I am challenged to more carefully consider this question.

I certainly find myself siding with Skemp as he argues (politely) that "mathematics" ought to be used for relational mathematics only. I cannot imagine myself teaching purely instrumental mathematics as Skemp has described it. And yet, I do have a further wondering. In calculus, we usually follow the pattern of learning how to "do the math" before we then take analysis to more fully grasp "why the math works." Is there a place for a thoughtful approach to teaching some math content more "instrumentally" as a way to build up a sense of familiarity with the material, after which the relational understanding can be better grasped when taught? In other words, I want to think more about whether there might be an "instrumental-relational" option that would work well for the high school classroom. Sometimes answering the question "Why does what you just learned how to do work?" would seem to be an effective approach.

Hello world!

 Hello future math teachers!