It should be easy to read - after all I am part of a team right now that is struggling with how to interpret what we see during our data structure interviews. In this paper, it does exactly what our situation is:
A one-one encounter between researcher and student where the student is solving a problem selected for her by the researcher. The problem consists of capturing unorganized information and creating an information organization. The researcher is observing the entire process and gathering the think aloud comments of the student. The researcher later attempts to analyze these comments to see if the students's solution confirms some idea that the researcher went in with. This is the exact process that Confrey describes.
The long initial section on comparing and contrasting the discovery learning theories and constructivist theories, is dense and most of her references are from the area of education research that is too theoretical for me. After four failed attempts to proceed past the first section, I decided to go to the end of the paper and work my way back from there. Unbelievably, that seems to have worked. (Though there are still sections that are hard and who is Lakatos(I think I sort of know), and Hegel and Popper these are names I recall from my classmates college Philosophy course!). Ok there is a lot to be learned here and it is worth the hard slog!
Trudging on - I found five assumptions that Confrey scatters throughout the paper that we maybe can use as a framework to analyze participant data:
- Constructivists view mathematics as a human creation, evolving within cultural contexts. They seek out the multiplicity of meanings, across disciplines, cultures, historical treatments, and applications. They assume that through the activities of reflection and of communication and negotiation of meaning, human beings construct mathematical concepts which allow them to structure experience and to solve problems. Thus, mathematics is assumed to include more than its definitions, theorems and proofs and its logical relationships - included in it are its forms of representation, its evolution of problems and its methods of proof and standards of evidence.
- In examining a student’s understanding of a mathematical concept, a constructivist seeks to represent how a student approaches the mathematical content. S/He expects diversity - and idiosyncratic rationality. The interviewer’s knowledge of the mathematical content, complete with multiple representations, competing interpretations, various applications - guides the inquiry, but his/her intent is to examine the student’s use of examples, images, language, definitions, analogies etc. to create a model which may well transform the interviewer’s own understanding of the mathematical content in fundamental ways.
- Problems serve a crucial role in the construction of knowledge. Problems reside in the mind of the student - not in textbooks or in the mathematics. Problems are felt discrepancies, roadblocks to where a student wishes to be and therefore catalysts for action. To accept a problematic an individual must believe that it is capable of being solved - and act as though the problem and solution were preexistent. The cycle of identifying (noticing) problematics, acting and operating on them and then reflecting on the results of those actions is emotionally charged, motivating and demanding. It is this process of knowledge construction which is the critical site for constructivist researchers/teachers.
- Problem solving as enacted in interviews or constructivist instruction is an interactive process. The interviewer selects a task for its potential to invite students to engage with a particular mathematical idea. The task will yield to multiple interpretations and resulting approaches. The interviewer must seek out an understanding of the students’ problematic, choices of actions and means of reflection. The interview setting will itself promote more self-reflection and a stronger approach to knowledge construction. The definition of the problem, of what concepts are related and of what constitutes an appropiate answer will evolve over the course of the interview.
- Students’ responses which deviate from our expectations as research- ers/teachers can appear to he reasoned and well-considered to the student. They may be entirely legitimate - as an alternative perspective, or be effective for a limited scope of application. We must encourage students to express their beliefs, keeping in mind that deviations provide precious opportunities for us to glimpse the students’ perspectives.
Some other glimmers that glow for me:
- the constructivist is engaged in a processs of invention - invention of his/her own models for explaining students’ actions and words.
- s/he(the researcher) begins with the assumption that what a student does is reasonable and then seeks to describe it from the student’s perspective.
- A problem is an intellectual desire ... and like every desire it postulates the existence of something that can satisfy it... (so sez Polanyi)
- Labeling a student’s model as a misconception fails to take in consideration the perspec- tive of the student, for whom the belief may explain all instances under consideration and fail only in cases to which s/he is not privy.
- Much of the success of the constructivist instructional or research model depends on how willingly the teacher 1) seeks to imagine how the student might be viewing the problem; 2) hears mathematical notions which differ from her/his own but possess internal consistency; 3) examines his/her own mathe- matical beliefs and 4) witnesses and describes the student’s choice of operation (action) and method of evaluation and recording (reflection).
- fit vs match: a conception must fit an experience not necessarily match the researcher's conception
- It is at points of contact, at moments of discrepancy, that we have the highest probability of gaining insight into another person’s perspective.
- Finally, I argued that in examining students’ problems and methods of solutions, one has an opportunity to reconsider the mathematics involved.
On the whole, I made a lot of progress when I decided to skip the arduous theoretical descriptions and the long description of "Suzanne" creating a powers of ten timeline. I read the part focusing on Suzanne later and it was much easier to read once I was no longer looking for deeper meaning.
