Week+3+questions

Type in your questions on the readings for Week 3 here. Add your initials in the parentheses at the end of each of your questions.

In the Pirie and Kieren article, they mention "'don't need' boundaries" on page 173. By this they mean that "one does not need to be constantly aware of inner level of understanding" (last sentence of paragraph one). They specifically mention that "a third 'don't need' boundary occurs between observing and structuring " (first sentence of third full paragraph). While I understand the point that they are trying to make, I am not sure that I completely agree with these statements. How could a student give a convincing proof about mathematical structures if they "do not have the meaning that is brought to it by any of the inner levels"? (CZ)  Regarding Dubinsky (1991), it seemed to me that the analytical framework explained and exampled in the chapter was more complex than it needed to be. What is the framework's value? (JE)

Sfard and Linchevski's article (1994) explored in depth students' conceptions (and misconceptions) with algebra. This article concentrated on the "learning of mathematics." How can this work be transfered to the "teaching of mathematics?" (quotes mine, not the articles) (JE)  What are the precise ways in which reflective abstraction (Dubinsky, 1991) differs from reification (Sfard and Linchevski, 1994). In what ways are these differences productive? In other words, what can we learn from the differences? How could one use the differences in the theories to inform choice of theoretical framework? (RK, 01/25)

With respect to the Dubinsky piece, should we refrain from introducing actions on objects until we are certain that our students have encapsulated the necessary processes? Assuming that the answer is "no, we cannot wait until we are certain that 25 of 25 students have encapsulated the original process," how will we make sure that the slowest student, in spite of being faced with tasks for which he has no objects in his schema, is able to continue learning? (RK, 01/25)

Since we are considering the slow student, what are the implications of Dubinsky's (and in that case, Sfard and Linchevski's) thoughts for stronger students? For example, is it possible to use these ideas to design tasks for "differentiated instruction"? Perhaps, "having completed today's tasks (which called for process or object conceptions), give the following a try (a task that calls for object or process conceptions, respectively). Is this a reasonable application? Are there others for strong students? (RK, 01/25).

Sfard and Linchevski talk about the need for "flexibility," which is an ability to move easily between process and object conceptions. Can this flexibility be supported or encouraged through written reflection? If so, what goes into a good writing assignment for a mathematics course? (RK, 01/25).

Finally (should I apologize?), who do you think is the intended "end-user" of the concepts presented in this week's articles? Curriculum designers? Teachers of school mathematics? Mathematicians and teachers of collegiate mathematics? Would there be value in talking about some version of these ideas with undergraduates? Perhaps in the name of self awareness or metacognition? (RK, 01/25)

How do these theories compare in terms of the “time” theorized for change or growth in learning? For example, in Davis (1997), Wendy’s transformation in listening was a gradual process. Is it possible for the process of learning to be instantaneous or must it occur over time? (NF, 1/25/10)

What are the problematic features or pitfalls of Sfard and Linchevski’s (1994) theory of reification? In particular, can we explain the title of this article—the gains and pitfalls of reification? (NF, 1/25/10)

What have we learned about meaningful learning? For example, in the case of Sfard and Linchevski, meaningful learning in algebra “comes with the ability of ‘seeing’ abstract ideas hidden behind the symbols” (p. 224). What are some examples you’ve picked up on from other theories? (NF, 1/25/10)

In Pirie and Kieren (1994), I noticed that Richard's 'mapping' went out until the ring of structuring, while Katia's 'mapping' only went as far as formalising. Is this because of the difference in the levels of the problems each of them were dealing with? Are there any other factors than the difference of the problems? (JH, 1/25/10)

In Sfard and Linchevski (1994), they talked about undertaking a teaching experiment in which their suggested didactic ideas are "put to the test" (p. 225). I wonder what the outcomes of this experiment were? (JH, 1/26/10)

How does "folding back" while learning a new concept fit with building on prior learning? In some cases, isn't it hard to tell the difference between folding back and recalling something that was previously learned? (CZ)

Do the theories presented in the three articles encompass all of which occurs in mathematical learning? Are there different ones? (JH/ 1/26/2010)

 Last week we saw that children selling candy and those involved in grocery shopping learn mathematics by active engagement in cultural activities. Do the growth of their understanding of mathematics in these activities follow Pirie and Kieren’s (1994) eight levels? (NA) How different or similar is reflective thinking to Pirie and Kieren’s (1994) eight levels of growth in mathematical understanding? (NA)

Sfard and Linchevski discuss psudostructral approaches where symbols become the object of a semantic operation instead of the mathematical construct being the object of a mathematical operation (page 116-8). How widespread is this phenomenon? We have all encountered the common perception that "math is hard!" How much of this perception is due to the actual difficulty of math, and how much is due to the artificial difficulty imposed by these misunderstandings? In other words, would students find math easier or harder if they were somehow blocked from forming these psudostructral devices? (I know that this is impossible - and probably undesirable; I mean it as a hypothetical exercise) (JDS)