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!!!