Response to "To Know and to Teach: Mathematical Pedagogy from a Historical Context"

This week, I read the article "To Know and to Teach: Mathematical Pedagogy from a Historical Context" by Frank Swetz. In the article, the author notes that particular techniques for teaching mathematics, such as the use of instructional discourse, a logical sequencing of problems and exercises from simple to complex, and the use of manipulatives is much the same as it was thousands of years ago. He goes on to expand on each of these by pulling examples from different times in history, including some geometry problems from the Old Babylonian period approximately 3000 years ago! I actually worked through some of the problems myself and found them to be quite fun.

The authors' section on mathematical exercises was particularly intriguing to me, as I was constantly thinking of current school mathematics texts and the structure of exercises in those book. What struck me the most was the total amount of exercises. Within these old texts, the number of exercises seems to be capped around 20. In regards to current calculus texts, one section could have at least 80 problems!! Most of the problems are essentially the same, with some slight variations here and there. In regards to some Chinese problems from 100 BC, the author states that in order to solve them "thinking is required." This made me stop to think. In the current school system, how much "thinking" do our students actually do? Even if problems in textbooks go from simple to complex, how much thinking are students doing on these problems?

Moreover, do current mathematical textbooks encourage students to think critically? I am tempted to say no for the most part. Of course, I'm running off of my own personal experience at this point. I remember the majority of my homework in math class being the same problem over and over, tests being variations of those problems, and this evolved to the point where I "didn't have to think about it." If I am remembering correctly, there was a Schoenfeld article I read recently about students' beliefs about mathematics. He was observing a high school geometry class and when the teacher was referring to studying geometric construction problems for the state exam, he told the class "they shouldn't even need to think" (or something along those lines). Unfortunately, I think I'm also guilty of this as a teacher, in telling my students that certain types of problems should be essentially automatic. This is certainly not what I want for my students, but it is what expected in the system.

Does performing 40 variations of the same problem enhance student learning and understanding of particular mathematics concepts? Why did texts evolve in such a way? I am left with many questions as to why textbook exercises are the way they currently are. I look forward to the discussion on Wednesday.

Reflections on "Learning angles through movement"

This week, I read Smith, King, and Hoyte's 2014 article "Learning angles through movement: Critical actions for developing understanding in an embodied activity." In the study, the authors worked with students in grades three and four, while they explored a task developed for a Kinect for Windows program. The program detected angles that students made with their bodies, while projecting an abstract visual of the angle on the screen. The abstraction was done in four phases. The first associated only colour to different classifications of angles. That is, if the student made an acute angle in any orientation, the screen would appear pink. Obtuse, right, and 180 degree angles were associated to their own colours as well. The second phase projected the color as well as two line segments (of equal length) connected at a point that mimicked the angle. Since research has shown that many students seem to make decisions about the size of an angle based on the length of the line segments, the third phase projected two line segments of non-equal length with the corresponding background colour. Finally, the fourth phase took all the aspects from the third, but added a protractor that changed with the orientation of the angle.

The initial sample consisted of 20 students who took a pre-test, participated in the task-based interview, and then were administered a post-test. The authors report that half of the students had increases in test scores from pre- to post-test, two scores decreased, and eight scores stayed the same. The authors also found that increase in test scores was statistically significant. Besides presenting this quantitative data, the authors also presented a case study of two students who had low pre-test scores. Ian, who scored low on the pre-test but scored 100% on the post-test, was very engaged in the task. He had a strong connection between the movement of the arms with the visual images on the screen, as well as an openness to exploration. Kara, on the other hand, seemed quite reluctant with the task. She was not as open to exploration and did not have a strong connection between the movement and the visual images. I do wonder how Kara was feeling coming into the interview. Just from the pictures of her in the article, her gestures seemed quite timid, especially compared to Ian's large, confident gestures. Could this have attributed to her openness to the task?

This paper was my first exposure to any of the embodiment literature, so I came in a bit skeptical. I have to say that I was very excited by the study once I started to read. Movement has been a large part of my life from a very young age. Even in my teaching, some colleagues have described me as "a mover." I love moving in my entire teaching space, using my arms and hands to demonstrate certain concepts, as well as just using my hands for speech in general. The authors concluded that it appears exploration with the body can facilitate exploration of the mind. When I put this into the context of my ballet training, I couldn't agree more. I, like Ian, often experimented with my body, particularly when it came to turning. I realized that shifting my weight slightly in one direction helped my balance immensely, so I stored that into my mind for the next time I needed to try it. Slight adjustments to my body, assisted in my overall understanding of how to execute a movement. As a side note, I was very open to making these adjustments. I really didn't care if I fell, if anything I learned something from it. On the other hand, many of my friends were very timid when it came to pirouettes and turning, since they didn't want to slip and fall. This brings me back to the openness of Ian and Kara and how this might have affected their involvement in the task.

As an extension to this study, I wonder how this could be related to students' understanding of radians. In the calculus curriculum at UBC, degrees are used sparingly. In fact, I avoid them entirely within my lectures and I will go as far as to say that most people in the department do as well. I have found that the majority of my North American students prefer to use degrees and generally avoid radians. Could a similar technology be used to help students understand radians for angle measurement?

Response to FLM Issue #1

For my textual analysis, I thought that an interesting first step might be to do a super quick skim through the journal. I essentially scrolled through as quickly as possible to get a visual sense of the journal. My first impression was that there were a significant number of figures and pictures throughout the issue. When I moved on to look at the table of contents to consider the titles of the articles within the issue, my observation of the large number of pictures within the article made sense! Three out of the nine articles were focused on geometry. The images in these articles varied significantly from the figures in the other articles though, which focused on students' mathematical learning processes. One common feature amongst these figures though was that many of them seemed to be flow charts of sorts. I suppose the idea of using a flow chart is an interesting model when trying to "map out" a child's thinking. What did surprise me though, was the total number of these flow charts. Was such modeling popular in the late 70's and early 80's when this issue was published?

I then went to consider the references in each of the articles. What I found a bit strange was that the articles either had a pretty large number of references or none at all. Something that I have noticed in FLM compared to other journals, is the number of expository articles by particular authors. Expository in the sense of the author reflecting on their own life experiences in the hopes that the reader might be able to relate. I personally think that this is a very nice touch to the journal. Sometimes it can be a bit overwhelming to read so many technical and/or theoretical articles, that having a narrative article of sorts can be very enjoyable, and sometimes feel a bit more applicable for someone working in a classroom. Of course, the expository articles in the issue are the ones with few or zero references. On the other hand, "Student Errors in the Mathematical Learning Process: A Survey" had over a page of references in the two column format! All of the articles seemed to be appropriate for all grade levels; even the article on memorizing or mastering multiplication tables is relevant for older students, even more so in this day and age when calculators are taking over the classroom earlier in students' academic careers. The last article, a narrative by a professor teaching a differential calculus class, was specifically targeted at university mathematics, but as it was a narrative, I think it was appropriate for anyone reading the journal.

What I found most intriguing though, was the cover picture of this article. Upon considering the number of geometry articles and "flow charts" in the article, the uniform tessellation of the plane seemed oddly appropriate.

Response to "Why You Should Learn Geometry"

Walter Whitely's article "Why You Should Learn Geometry" was a response to an LA Times article titled "Why You Should Learn Algebra" which brings forth the value of mathematics as a classical training of them mind. When I saw the title "Why You Should Learn Algebra," I was brought to remember a NY Times article that was essentially titled "Why You Should NOT Learn Algebra." I'll try to find a link to it and post it here.

Whitely elaborates on the unnecessary equivalence between that is often associated between algebra and mathematics. That is, if you excel at mathematical thinking, you excel at algebraic thinking. He brings forward the case of Michael Faraday, who was dyslexic. Although Faraday did not do his work algebraically, he was very geometrically and visually inclined. I have always considered myself to be a visual learner of sorts, but I have always had a great deal of respect for those who can "see" the answer; an ability I do not have. But, as Whitely notes, many students with a strong geometric intuition are often pushed aside for those who are algebraically competent. He argues that geometry and algebra both play significant roles in developing mathematical reasoning and warns that we should not narrow the mathematical landscape.

When I first started dating my boyfriend, one of the first things we talked about was our experience in school. Although I wasn't involved in the education field yet, I was particularly intrigued by the way mathematics was taught in British Columbia. In the United States, students typically study "algebra" in grade 8, "geometry" in grade 9, "algebra II" in grade 10, and the mathematics course of their choice in grade 11 and 12. During high school, I was under the impression that I was "better" at algebra than I was at geometry. Somehow, they seemed like separate quantities to me. My boyfriend, on the other hand, just saw the material as "math." He told me that he never thought "I'm better at geometry than algebra" perhaps because the material was not presented to him as such. It was just math class; all the topics were intertwined and played important roles in the development of his mathematical knowledge. So, I've since wondered if the majority of students who go through such a system feel this way.

As a side note, I was a little bit bothered during Whitely's mention of Faraday, when he stated that "he did not reason with formulae (algebra)." The term "algebra" as a mathematical field is not about formulae. I honestly do not understand why the term algebra was coined for purely arithmetic manipulations. Really, this "kind" of "algebra" is just advanced arithmetic. In fact, it's quite surprising how many students do not see this connection.

Response to "Proofs as bearers of mathematical knowledge"

I was looking forward to reading this article, as I hope my intended research to have proof at the forefront. Unfortunately, I was a bit underwhelmed by this article. The authors, Gila Hanna and Ed Barbeau, base their investigation on Yehuda Rav's paper "Why do we prove theorems?". In this paper, Rav states that proof should be the focus of mathematical interest, because within a proof lies the development of mathematics as a field. He contends that far too much emphasis is put on the importance of theorems (product), rather than the proof (process). While Rav focuses on the field of mathematics, Hanna and Barbeau attempt to extend his insights to mathematics education.

When I entered into a mathematics major during my undergrad, the concept of "proof" was essentially completely unfamiliar. The "proof" that I had seen in high school consisted of trigonometric identity manipulations, and most proofs I saw in calculus were either derivation type proofs, or very brief conceptual proofs of about two lines. I continually wondered why it took until my third year in university to see a rigorous mathematical proof, from which I learned a great deal; not only the final "fact", but the process in between that came in as useful later on. Throughout my teaching, I have always enjoyed presenting proofs to my students, as I feel that it gives a great deal more insight into the problem at hand, and how very simple statements, such as the Intermediate Value Theorem, have very non-trivial proofs. Proofs give rise to the importance of assumptions, something that many students tend to forget about. One of my favourite examples is that of fixing a car; if you have a specific part that only works for one specific type of car, you probably don't want to put it in a different model car; something bad might happen!

My disappointment in the article started when the authors began their "case study" of particular proofs that are apparently seen in most secondary mathematics classrooms. First of all, there is nothing I despise more than hearing from a professor "you should have seen this in high school (and/or first year)" in regards to something non-trivial. I've had this happen to me in graduate classes, and I can't express how degrading it feels in the moment. As someone relatively mathematically confident, I can't imagine what that feels like to a student who is not as confident in their mathematical ability. I felt this tone in much of the paper, there seemed to be an assumption that most students (and teachers) know these facts and their proofs inside out, when I would have to argue that most do not.

The authors consider the cases of roots of a quadratic function and the case that an angle inscribed in a semi-circle is a right angle. I have done a lot of tutoring of students working with quadratic functions, and I know from experience that 1) students have difficulty with quadratics when the coefficients are defined real numbers and 2) that this difficulty increases exponentially as soon as the coefficients are arbitrary. The authors recommend that the proof of quadratic roots be explained first by considering "nice" quadratics like x^2 - k and then moving onward to develop the idea of completing the square; a technique that the students will be able to use on quadratics in the future, as well as extend the notion to cubics and quartics.

While I do believe of  the importance of conceptual proof and a lack thereof within the secondary mathematics curriculum, there are a number of issues that need to be attended to. First of all, the teachers working with such material need to be very flexible in their mathematical knowledge in order to attend to the requests of Hanna and Barbeau. Moreover, what are secondary school teachers conceptions of proof and their usefulness in the classroom? Next, the request to have substantial, conceptual proofs within the curriculum starting in secondary school might be a bit of a slap in the face to many students. If there is to be such a large curricular change in upper grade levels, shouldn't there be consideration of the importance of proof in elementary and middle school grades?

I should probably stop here. I really apologize to the people in my reading group for the length of my response. I think I could go on about this paper and the issues it brings up for hours......

Response to Models and Maps from the Marshall Islands

For this week's paper on ethnomathematics, I read Marcia Aschers' analysis of navigation devices used by the people of the Pacific Marshall Islands. This article was of particular fascination to me considering my experience in my undergraduate degree in meteorology and a brief exposure to oceanography. The Marshall Islands lie in the middle of the Pacific ocean, between Hawaii and Papua New Guinea. The islands embedded within a ring shaped coral reef surrounded by ocean, which enclose a lagoon. Here's a picture of the Bikini atoll in the Marshall Islands:

Marshall Islanders constructed what the author refers to as meddo, rebbelith, and mattang out of palm ribs and coconut fibres for use as training devices for navigators of the area. With over 1,000 islands and inlets in the Marshall Island system, the interactions of land, sea, and air were of vital importance to the navigators. Just as Western Navigators used compasses, maps, and charts for navigation, the people of the Marshall islands utilized the meddo, rebbelith, and mattang for navigation. Of particular interest to the author was the mattang, a device used in the training of navigators (a duty passed on from generation to generation) to describe the "complicated and distinctive interaction of modified swells that are the landmarks which the Marshallese navigators learn to read and interpret". The curves on the mattang pictured below (with a little description) represent swells approaching land masses. Straight pieces of palm rib represent the direction of the prevailing wind, an important factor to consider during navigation. It is an intricate device which the author spends 10 pages describing, so for a better description, take a look at the article. It should be noted however, that these devices were not carried with the navigators on their voyages! All of the information and cues from the mattang were taken to memory!

The ways in which the Marshallese conducted their navigation compared to Western traditions is an interesting comparison. While there are agreements on how to sail in regards to wind directions, the Marshallese generalize the system with respect to earth, air, and atmosphere, which all interact into one system. The Marshallese were also very adamant that the mattang were relevant to oceans and land masses anywhere and everywhere. It is interesting to compare this to our idea of "science" where we consider the theories to be applicable anywhere and everywhere. What was particularly striking to me was that the Marshallese described oceanographic phenomena without an explicit mention of mathematics. It is indeed difficult, having grown up in western culture, to imagine physical phenomena without the use of our mathematics. Of course, the question of why mathematics be necessary in the descriptions of physical phenomena has been of great philosophical debate throughout the years, and the author notes this as well.

Having never been exposed to articles on ethnomathematics, I have to say that I'm even more intrigued than I was before. I've always thought that the idea of the field could bring a really beautiful, human aspect to mathematics, that is often lost in modern culture. After reading this article, and the quite intricate mathematical analysis (using "our" math), I am quite eager to learn more!

Respose to "In Fostering Communities of Inquiry, Must It Matter That the Teacher Knows "The Answer"?

I was eager to read this week article, as I've always been particularly intrigued by Schoenfeld's work. This piece took a more expository stance, as he used his teaching experience in a problem solving classroom and related it to his work within his research cohort. In his research cohort, although he is the "leader" of the group, he is often unsure of what his data means or where it will take him. In contrast, he states that in his problem solving class, where he is also an authoritative figure, not only does he "know all the answers", but he also has a particular "lesson image" for how the class will go. His "lesson image" is so detailed that he can predict the questions he and his students will ask, as well as the responses that he will provide. My immediate reaction was "well, that must be kind of boring." For me, I enjoy the unique questions and inquiries that come from my students when we are solving problems, but I can certainly relate to his statement (particularly when I reflect on group based problem solving sessions I've assisted in). He goes on to state that his students are probably unaware of his ability to provide such detailed descriptions of what they will do, but that what ultimately matters is that he is "reacting directly and honestly to what the produce."

His ultimate goal in the course is for student's not to see him as the mathematical authority, but the mathematics as the authority. For me, this took awhile to develop. I don't think it was until much later in my mathematical career that I realized mathematics could "speak." I knew whether my argument was flawed or not, since the mathematics I used told me so. I didn't need a professor to tell me if it was "the right answer," but rather their reaction to my argument and advice on perhaps making it a bit more elegant. By the end of the course, this is the type of "community of inquiry" that exists in Schoenfeld's classroom. Students no longer request his "certification" at the end, as they have realized on their own, through the mathematics, that they understand the problem and it's solution. If at the start of the course a student would ask him "is this right?", he would react with " don't convince me; convince the class. Do you folks buy it?". He admits that students in his class have occasionally  discovered results that were new to him and he claims that it didn't really matter to the students that it was new to him - that they were "simply doing mathematics."

I found his story compelling, but it has brought many questions to mind. As an instructor in a University mathematics course, I often struggle with the idea of not being able to answer my students questions. Especially as a graduate student, the amount of "authority" and "experience" I have, I suspect, is significantly lower than that of a tenured professor. Through my classes, I make a strong attempt to make the mathematics a joint effort between myself and my students, so that I am not perceived as the knower of all. Even still, I get emails referring to me as "Professor Radzimski"! A question I would ask Schoenfeld is, as a graduate student instructor not much older than most of my students, would you think that it might be easier to foster a community of inquiry within the classroom? Particularly in a non-problem solving based class, filled with theorems and definitions, what action can we take to foster inquiry among students in general?