For February 1 - Brown and Walter - Problem Posing in Math Education

The authors write about problem in posing in mathematics by focusing on what they define as "sensitivities" or categories to consider when working in a problem-solving setting and asking problems. The five sensitivities they pose (titles directly copied) are:

1) An Irresistible Problem Solving Drive: We have a (trained?) sense of wanting to find an answer to a problem, even if the problem is poorly posed and may be interpreted in multiple ways.

2) Problems and Their Educational Potential: Posing problems can lead students to pose new problems and may require (and breed) creativity in solving problems.

3) Interconnectedness of Posing and Solving: Problems often require “restructuring” (pp. 21), resulting in the ability to connect different content to a seemingly unrelated problem.

4) Coming Up With Problems: Rather than accepting a set of conditions and finding only one answer, more successful and meaningful problems can be created when students are asked to make generalizations, broadening their scope.

5) The Social Context of Learning: Collaborating solutions can help students explore personal understanding and extend their knowledge of problem solving.

I appreciated the authors’ acknowledgement of social interaction and the view of mathematics as an independent venture. While a great deal of mathematics struggling can happen on a student’s independent time, a Vygotskian method of problem solving can expose students to a variety of solving methods as well as unite students in a common effort. It makes me wonder, however, if problem posing can be open-ended without losing students along the way. For example, I posed the following problem to a grade twelve student in a physics class which had been assigned a work period while the teacher was away:

In a long hallway are 1000 closed doors. People begin to run through the hallway and to do the following: if a door is open, they close it. If a door is closed, they open it.

These people run in a very specific manner:

Person #1 changes every door.

Person #2 changes every second door.

Person #3 changes every third door.

This pattern continues until 1000 people have run through the building. How many doors remain open after the last person has run through the hallway?

This is a problem that some of my grade 8 and 9 students have successfully solved, yet they did not need the pre-requisites that the grade 12 student today thought. In fact, his response when I suggested that he use a different approach was, “Oh, so don’t do math, just solve it?” While he still did not provide me with a solution (nor did I provide him with hints beyond sketching the first few cases), he gave me some insight into his mind. I wonder if he meant that calculations are mathematics and problem-solving is a procedure, or if solving without the use of numbers is no longer mathematical? Perhaps his comment represents a hybrid of the two? I would have liked to listen to him work on the problem with a classmate. The single interaction I managed to observe was when his colleague asked him of the problem “Is it a riddle or an actual math problem?” to which the student responded, “A math problem.”

            Perhaps students decide on the nature of a question first before deciding how to answer it – and, perhaps, whether or not it is worth their time to answer it. This makes problem 4, which discussed more generalized questions, seem to be even more important. If students cannot generalize a math problem beyond the immediate scope of the question and immediately related content they are solving, perhaps it is not worth solving at all.

For January 25 (Sfard article) - "On the Dual Nature of Mathematical Conceptions: Reflections on Processes and Objects as Different Sides of the Same Coin"

         I was annoyed by some concepts introduced in this article, as well as some of the underlying statements made. This article, before I proceed with my sentiment, involved the discussion of mathematics learning as made up of what author decided to call "operational" versus "structural" conceptions. Although she admits she does not know how to define "understanding", the author writes that true understanding only comes from moving from beyond the operational conception of mathematics (requiring the five senses) to structural conception of mathematics (at an abstract level).

        The author outlined that the three stages of developing concepts are:
1) interiorization (in which a learner becomes familiar with a concept)
2) condensation (in which a learner conducts more work in manageable units, which requires concept synthesis)
3) reification (the sudden ability to se something familiar in a totally new light).

         One of my issues in this article is the lack of development/emphasis on how the author suggests we proceed from moving from condensation to reification. Although the process she described around waiting for a sudden understanding of a concept is one to which many mathematicians can attest, there is very little detail on how to come about this sudden revelation. However, it is this part of the process for which educators seem to come under fire in the article, since students are still stuck on the operational (if even that), while little/no structural understanding is developed.

         This article demands that we answer not only what mathematics is for, but also for whom does mathematics exist, and who does it serve (or, who does it fail to serve)? How many students are willing to stare at a problem and work it to exhaustion until they finally reach the "click"? How many have the confidence to continue manipulating the problem over and over until they finally have a revelation? Can "the click" be taught? Does it even need to be taught in a high school setting?

         It is also odd that the assumption in this article is that we begin operationally and then proceed to structural conception; Sfard uses the example of a child's counting being operational. I disagree that the set of three objects being counted aloud in place of a single answer yields an operational conception over a structural one, however: if I were asked to count fifty-seven objects, I certainly would not hesitate to count aloud in some form to fifty seven, even though I know that it is the product of two primes (which I assume would be the abstract form of understanding? However, this was a conclusion I made procedurally by being able to subtract three from a known multiple of three, 60, then knew that a number three smaller would also be a multiple).

        Finally, there were certain judgments about ability in the article that bothered me. For instance, I found the following quote irritating: "Being capable of somehow 'seeing' these invisible objects appears to be an essential component of mathematical ability; lack of this capacity may be one of the major reasons because of which mathematics appears practically impermeable to so many 'well-formed minds' " (pp. 3). Why is this an essential component? Does it always have to exist? How do you know how good your "understanding is", and perhaps Sfard could have defined what this is before claiming that having it is an essential ability in doing mathematics?

        Looking forward to discussing this in class.

For Jan 18 - Eisenmann, "Strong is the Silence"

Dr. Herbel-Eisenmann's article focus on mathematics teacher educators and their role in tackling topics of privilege and oppression provided me with the stark realization that my undergraduate mathematics-education courses never once discussed privilege/oppression. The course layout at UBC made me think that social justice issues were covered in one or two courses, predominantly in the "Aboriginal Education" course, by which time pre-service teachers were exhausted and becoming increasingly negative as a whole towards various aspects of teaching. Many of the same concerns that Dr. Herbel-Eisenmann outlined were prevalent in the attitudes of many of the students, including hesitation to discuss one's own privilege and a shallow level of engagement because "I already know that this is a problem" (personal conversation, 2013). I am grateful that I was able to avoid some of the anger around these issues in my own teacher education programme, and I think this is for two reasons. First of all, this material was continuously (accidentally) re-introduced multiple times over the three years during which I was taking courses in education. My program was revised halfway through, leading to three courses in which we discussed privilege/oppression, rather than just one.

I am surprised that there was no mention of teacher placement in this article with respect to their pre-service placements. I am currently in the middle of reviewing Wassell & Stith's "Becoming an Urban Physics and Math Teacher" for JUME; one of the points they make is that mentor teachers lacked confidence in teaching because of a fear that "new teachers could [not] control their students" (pp. ix, 2007), as there is "a widespread perspective that [teachers should be trained] to establish and maintain tight control over students" (pp. vii, 2007). This perspective establishes a post-colonial reign over children of all identities and reinforces the achievement gap (or education debt, as is suggested in the article by Ladson-Billings, 2006). I disagree that changing rhetoric around the "achievement gap" to "education debt" will make much of a difference; the negative opinions around children of lower socioeconomic status has done its damage. I argue, instead, that there should be a focus on positive stories coming from the urban/inner-city classroom; there is already a lot of negative narrative on the topic. Celebrating positive stories and examining student success (both in pre-service training and with a mentor teacher) may open dialogue as to why other students may not be as fortunate.

This is not the only step towards a positive solution, however; as Dr. Herbel-Eisenmann writes, there must also be an understanding of personal identity that is formed by teachers, in the context of privilege and oppression. Without understanding the manner in which teachers contribute to systemic oppression (through reflection and action, as Freire suggested in "Pedagogy of the Oppressed), there is little hope for change.

(For January 11) Wheeler article - Response

After reading the first column, I predict that the article will set a framework for publishing in the “mathematics education” field(s) as we know it (them) today, compiled from three responses by Pearla Nesher, Alan Bell, and Caleb Gattegno, written for a call for manuscripts from David Wheeler.

It is likely that the rest of the article will involve responses concerning the epistemological nature of qualitative data collection in a field that depends on a “research programme”. Other concerns likely involve the nature of data collection, number of scholars researching, and establishing a reputable and beneficial field.

After having read the entire article, the three main persons’ responses are broken down as follows:

1)   Pearla Nesher raises concerns that the research programme proposed is in response to the negative reactions toward the “New Math” movement, explaining that the research programme should be a revision aimed at improving mathematical understanding as a language system as opposed to using research for responding publicly to a failed programme. The use of other disciplines will be important in developing a competency in mathematics, rather than simply understanding cognitive processes

2)   Alan Bell notes that mathematics education research should be used to achieve a better understanding of conceptual tasks long-term, after tricks and memory aids have been forgotten.

3)   Caleb Gattegno explains the need for understanding the structure of numbers in classical mathematics, in that children must be able to consciously create mental structures so as to be able to produce the current existing (topological, algebraic, order) structures. Algebra, according to Gattegno, was a pre-existing body formalized by humans through the senses into a mathematical structure.

Nesher, Bell and Gattegno’s views toward mathematics education and mathematical knowledge seemed largely positivist, which I was surprised to find. It makes sense, given New Math hangover of the 1960s, which placed a high value on science and technology development. In their views, mathematics is pre-existing and must be broken down in structure and explored scientifically. For example, Nesher writes that mathematics education must be researched in a “rigorous” manner, which often refers to the use of scientific method (perhaps this is only my interpretation?). Alan Bell’s consideration of mathematics education similarly casts doubt on a whole-class teaching method comparison, and says that “results” (rather than results, without quotation marks) were obtained through classroom observations. 

My understanding of the current field is that there is a great deal of concern over validity of results. While obtaining results and considering validity is always warranted, Bell’s one statement discredits the value of qualitative data as being inferior to the quantitative. In a similar fashion, Gattegno explains the connection between algebra and order, that these things are “out there”, to be learned; it seems, according to Gattegno, that children learn mathematics from scratch, rather than having a pre-existing knowledge of mathematics – knowing we were all once children, this implies humans are external to mathematics until it is bridged for our access.