The last blog post (Mathematical Pedagogy from a Historical Context)

In this article, Swetz gives a brief history of math textbooks. He contrasts those that are mostly a series of problems (to be accompanied with oral instruction by a teacher) and those that contain instruction. He also discusses how authors used diagrams and manipulatives to accompany texts, and what these might mean about how the authors hoped their texts might be understood. 

In my relatively short teaching career, I have already observed two curriculum changes (one in Ontario and one in BC). As a result of this (and of changing schools multiple times in Vancouver,) I have been involved with many different department meetings selecting new textbooks (or workbooks) within the financial constraints of our current educational system. Most of the newer texts tend to spend most of their pages constructing knowledge with detailed explanations and investigative questions. This often comes at the cost of practice questions, and I find myself generally sending students home with worksheets to practice concepts.

Certainly, the construction of knowledge featured in these books is important, but students still need practice to master a concept. In addition, these textbooks don’t take into account that few students read them. I find students are more likely to follow links I provide (such as purplemath.com) to get an explanation for a topic they find challenging than looking a concept up in their textbook. I have always felt that my job is to help students construct knowledge and give clear explanation of topics – All I hope for from a textbook is interesting and challenging practice questions, which are rarely featured in newer ministry approved textbooks.  

How Multimodality Works In Mathematical Activity: Young Children Graphing Motion

In this article, Ferrera analyzes primary students' learning of graph motion by using CBR devices (which measure and make a graph out of motion) and assesses how this affects the way they present their understanding of mathematical relationships (mostly through distance-time graphs). In the first part of the paper, she discusses how the network of perception (body and imagination) affect student learning, and then goes on to describe a particular students' understanding as developed through graphing motion. 

Two things struck me while reading this article:

Firstly, at the school I started teaching at, we had a class set of CBR devices - I had an activity where I had students walk towards and away from a wall and look at the resulting distance-time graph to have them analyze how slope increased/decreased as they moved faster or slower. Sometimes, if I was feeling ambitious, we would try to analyze the graph of a ball bouncing or other movements using the CBR devices. CBRs were clunky technological things that were hard to read and often malfunctioned. Still, the kids were usually impressed by the technology that could record their movements. I haven't seen CBR devices since leaving that school, but I'm sure there's an app that could be downloaded to an iphone, probably for free, that would do a much better job than the CBR did, yet I'd also be willing to bet that the kids now would not be as impressed by it as they were by CBRs 9 years ago.

The other thing that struck me was the idea that doing math can feel differently depending on how you are doing it. Is solving an activity on a board the same as solving it on paper?  I knew a teacher who had a great deal of success giving his students erasable markers and having them write all over the walls and windows in the class - he found them much more motivated to solve problems and to work together. I know I lament the loss of my blackboard (in exchange for a whiteboard) as it's somehow not as rewarding to compute equations unless it's a bit gritty. I'd be interested to learn more about why this feels different for us cognitively.

The first issue of FLM

I'm always pleased when a profesor assigns a reading from FLM as the articles tend to be more meaningful to me than articles from some other journals. The articles tend to also be of manageable  length and use language I can readily understand. Given all the above, I wasn't surprised to find the claim from the very first issue that the journal is "intended for math teachers," that the articles should be limited to 2500-5000 words (which I think is as long as most articles ever need to be!). The journal has a good international representation amongst editors, and seems to focus on issues relevant for math educators. The articles seem aimed for people with a mathematics background - not too much focus for elementary teachers - and most topics seem to be around secondary or early universtiy math.

A peculiarity of the journal is that it doesn't feature abstracts. I question this decision on behalf of the editors as I think abstracts are pretty important when I decide if an article will be relevant to what I am interested in researching at the time. I suspect the editors hoped that people would treat the journal as more of a magazine to subscribe and flip through, but when examining back issues for certain topics, it is useful to have an abstract.

week 7: why study geometry?

My reading for this week included two short letters explaining reasons to study geometry. Both point out that geometrical reasoning is another form of reasoning with mathematical concepts, and that, while it may not train students directly for the job market, it can train students how to think.

IN a similar vein, Last summer, a (younger) friend of mine got her first teaching job, and was a bit worried as she was teaching a block of Math 8, and wasn't too comfortable with math. Nonetheless, as a new teacher, she was super-keen to try to do a good job, and sent me an email asking for advice. In her email, she told me she had always hated math as it never seemed at all interesting, relevant or useful to her, and that she thought she should spend most of the Grade 8 year focusing on the stats and probability units as these were the only ones she saw any real life connections to. This was my response to her...

you can't really think of all the math we teach in schools as being useful for the regular person in the real world. some is, some isn't. studying math and learning the problem solving, patterning etc. generally makes people more intelligent - builds brain cells allowing them to be more adept at problem solving, following patterns, understanding logic in various aspects of life. furthermore, things like linear relations, integers, fractions etc. are important for future math. at a grade 8 level, you're still focusing on a general education - (most) 13 year old kids aren't mature enough to decide where they want to specialize, so some of the things we teach are just to give background for future math.

Week 6: Learning from Learners: Robust Counterarguments in Fifth Graders Talk about Reasoning and Proving

In this article, Zack outlines a problem and erroneous solutions she gave her class. The 'chessboard' problem asks students to find all the squares in a 4x4 grid, then extends to a 5x5, 10x10 and 60x60 grid. She then gives a solution that a 10x10 grid has 385 squares, which you can multiply by 6 to get 385 X 6 = 2310 squares. Her students adeptly identify that this is incorrect, and she outlines 5 counterarguments given by nearly every student in her class.

I had a lot of trouble accepting Zack's arguments as, based on my teaching experience, her results seemed nearly impossible! The language Zack gives in her counterarguments is clearly her own, and does not convince me that it is what a Grade 5 student would think about the problem. I would imagine that a 60x60 grid is quite an abstract concept for a 5th grader, and to concieve of numbers of squares in the thousands (apparently, the correct answer is 73 810 squares) seems to be quite a tall order. I felt there must have been quite a bit of coaxing from the teacher/interviewer to encourage the students to come up with these detailed counterarguments - I was not convinced that these would represent Grade 5 students' independant thinking.

Having criticized Zack's methods, I do appreciate her analysis on the value of counterexamples, and teaching these from a young age. She says "Having a diverse assortment of counterarguments is beneficial. The counter-arguments tell us different things about the mathematics in the task." I think encouraging various counterarguments is a great way to help students consolidate different areas of Math across topics.

Week 5: Models and Maps from the Marshall Islands

In this article, Ascher describes the Marshall Islands, and the way that people were able to navigate the oceans around there. In particular, she describes a device called a Mattang which is used to measure swells and interpret the ocean landmarks based on these. She makes the point that without knowledge of western mathematics which would normally be used to navigate the ocean (trigonometry, ocean winds and so on), Marshall Islanders have found a highly scientific oral tradition which allows them to navigate their waters.

In her conclusion, Asher states

Geometric and then algebraic representations of physical systems have been the hallmark of modern science. The linkage is so tight it is hard to conceive of the study of physics without the involvement of mathematical ideas and mathematical descriptors... these discussions have included speculation on whether this is the nature of the universe, or the nature of the human mind, or simply the scientific tradition of Western culture. (pg. 366)

This is certainly not the only occurrence where a people without knowledge of western mathematics have performed amazing feats of engineering (ancient structures like Machu Pichu and the Pyramids come to mind). Western mathematics has found many efficient and effective ways of describing phenomenon, but this is really only one 'way of knowing' how the world around us interacts, It's interesting in the reading to understand how an isolated culture has used very intense powers of observation to solve the problem of navigating around complicated waters. It's also interesting to consider how these ways of gaining knowledge result in a different world view.

week 4: A Linguistic and Narrative View of Word Problems in Mathematics Education

Susan’s article attempts to “establish a description of the mathematical word problem as a linguistic genre, particularly considering its pragmatic structure... [she hopes] to find clues to the unspoken assumptions underlying its use and nature as a medium of instruction.” While I don’t have enough background in linguistics to understand much of the article (see the L-tense and M-tense section!), I appreciated some new perspective she brought to how we see word problems.

In particular, Susan points out that since word problems generally don’t refer to real life situations, they could be considered to be works of fiction. The section in which she points out how authors of word problems try to convince students to believe the underlying facts they will use to approach the problems is quite humorous. Susan wonders (as many students often do) what the purpose is of these works of fiction (aside: what is the purpose of any work of fiction?)

Last year, we decided the midterm exam for our Grade 8s should be a task based exam. The most revealing task we gave them was also the simplest. We gave students different sized boxes and measuring tapes, and they had to find their surface area. We watched on with surprise as our top students who could have easily have found surface area of a rectangular prism with a formula struggled to use the measuring tape carefully and correctly. This was a meaningful problem solving task that students can connect to “real life” (wrapping a present, painting a room) much more directly than in a fictional (Kevin is painting a room…) problem. Perhaps if students were given the time to undertake more tasks such as these, they would develop meaningful problems to them instead of invented stories when asked to come up with their own math problems. 

Week 3: Victoire sur les Maths

For the Americans in my group: the title is French, and translates to Victory over Maths. In this article, the author reflects on a book written by Lusiane Weyl-Kailey, who is a therapist (and ex-math teacher) charged with teaching Math in a clinic to students with special learning difficulties as well as psychological problems. Through a few case studies, the point is made that students’ abilities to understand Mathematical relationships are directly related to the comfort and safety they feel in their human relationships. Weyl-Kailey seems to have found success using this approach in both directions: helping students feel more confident in Math by first helping to heal their emotional struggles; and helping students feel more emotionally stable by improving their mathematical ability.

I think a lot of pressure is put on students to be good at Math in school. A common conception is that those who are good at Math are smart and will be successful in life (and the opposite). I can imagine that this can be emotionally damaging to students who don’t ‘get it.’ Certainly, I have found the most success while teaching in Special Needs programs by removing the pressures – allowing the students to work at their own level and pace without tests until they are ready. I have definitely observed that these emotionally fragile students perform better in Math class when they are feeling stable in their lives outside of the classroom. I have been lucky for the past two years to have the freedom to do this in my classrooms as I work with small groups and am not tied by the bounds of curriculum. However, it’s much more difficult to apply to mainstream classrooms where a certain pace must be maintained, and those that can’t keep up fall further and further behind. This reminds me of some advice a colleague gave me in my first year of teaching – she suggested I not feel bad about students who were not succeeding, but that students had a time they were ready to learn and receive knowledge (somewhat similar to Piaget’s stages), and that it could not necessarily be rushed. I think that emotional instability can cause interruptions and delays in these stages, which is sometimes most obvious to the Math teacher as it may be harder to face mathematical understanding than it is in other courses.

Week 2: From Intended Curriculum to Written Curriculum: Examining the Voice of a Mathematics Textbook

Firstly - to Vanessa and Kerri, very sorry for this late post - I got a bit distracted over the weekend. Hopefully it wasn't too inconvenient for you

The author critically examines a disconnect between the NCTM's professional standards and the way that textbook authors present information. Some of the points she makes that I found most interesting:

• When asking "questions", textbooks use imperatives (Do, Make, Study, Organize) to encourage them to discover mathematical concepts. There aren't really questions at all.
• Texts make assumptions about student knowledge (eg. In your earlier work, you saw that linear relationships can be described by equations of the form y = mx + b). Is this really different from just stating mathematical facts?
• If a teacher changes something in the textbook, it may diminish the authority of the textbook, but also imply that the teacher has knowledge about students' actual abilities and prior knowledge

This article brought to mind a lesson I learned from a department head in my first year of teaching. He reminded me that we do not teach textbooks; we teach curriculum. While this might be obvious to me now, and to many studying math education (and reading this blog). It was not obious to me as a first year math teacher, who observed most of my colleagues following textbooks very closely. I find many new textbooks overly conceptual and 'wordy' causing unnecessary confusion in students. Personally, I feel it is my job to create and find meaningful resources and explorations which will provide students with the opportunity to discover and engage with the prescribed mathematical subjects; all I hope for from a text is relevant and meaningful practice questions.

If I were to ask the author a question (which I suppose I may do,) I would be interested in her opinion on flipped classrooms - this is a model that is being explored in some schools where students are encouraged to do their learning at home, and their 'homework' at school. Students engage in a text or online resource to learn the material, and do the practice work in class with support of peers and teachers. This emphasized the importance of the text on initial learning, but allows students to have more support while attempting to analyze and expand upon their learning.

Response to Week 1 Reading 2

Response to A Research Programme for Mathematics Education

The Author, David Wheeler sent a callout to a number of Mathematics Educators asking how they would envision a research program for Mathematics Education. Three responses are published. After reading the introduction, I assumed the respondents would suggest quantitative studies which contrasted student results being taught math concepts through traditional rote learning as compared to being taught using manipulatives and a more problem solving approach.  I was not too far off. Nesher responds with a suggestion that Math research should focus on the processes involved in the acquisition of math as a language system. Ball would like to instill a “programme of teaching experiments where the aim is to achieve a discrete permanent change in the level of understanding of individuals in a key conceptual task” He suggests research using manipulatives and meaningful examples to create meaningful understanding would provide us more evidence of how students learn math. Finally, Gattegno suggests a research program that starts from the recognition that mental structures required by math are the ‘endowments’ of perceptive people being becoming aware of their own surroundings.

Ball and Gattegno’s responses seem to support each other in that they believe an effective math education system should start with meaningful problems, and provide students with the motivation to find ways to reconcile these problems. Nesher’s reponse implies that she believes the main stumbling block for math understanding comes from students not understanding the ‘language’ of mathematics – not only the syntax but also ‘conceptual frameworks’ required for math understanding. I particularly liked Ball’s quote that “we know that most of what is taught is not learned” and that math research should focus on ways to identify what is learned and how we can learn from what is learned. It is interesting to me that this article, published in 1981 identifies the same issues and stumbling blocks to math education which we still struggle with today, 34 years later. It will be interesting in this course to examine how researcher’s work has progressed into curriculum changes, and what the results are of these changes as far as students’ math understanding in the last 30 years. 

Hello World!

Hello World!!!