Section D7.2 Developing a mathematical inquiry independently: Methods

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Methods course documentation
Developing a mathematical inquiry independently (D7.2)
Logistics and scheduling information (by Raffaella Borasi)

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Choice of scope and format for this activity

Scheduling of this activity within the course

Materials needed to implement this activity

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Choice of scope and format for this activity:

Although I think that this activity is extremely valuable for prospective teachers, I have not always chosen to include it in the Methods course because of time constraints -- since it requires about 3 weeks of instruction plus considerable time at home. Within the limits of a semester course (which is all the coursework in mathematics education elementary pre-service teachers would take in the teacher preparation program offered at my Institution), I have often opted for using this time instead to make participants aware of resources that they could be using to make their teaching of mathematics more meaningful and effective (see Sections D7.3 and D7.4).

When I chose to include this experience for secondary math pre-service teachers (as it was the case in Implementation B of the Methods course), I have tried to promote and support the participants' independent inquiry through a number of important initiatives (as explained in more detail in the Facilitator's plan for this activity), including:

engaging the participants as a group in some thought-provoking activities about taxicab-geometry -- so as to stimulate their interest and questions about this and/or other "alternative geometries" (in case they did not already have a math topic they were interested in exploring) and also to "model" how one could go about initiating an independent inquiry; note that, although there are many other topics that could have been chosen for this beginning phase, I chose taxicab geometry because I expected it to be new to most participants and I knew from past experiences that this topic could generate a lot of worthwhile directions for inquiry that could be conducted without too much technical background;
providing structures and class time for individual participants to share their preliminary findings -- so as to create some intermediate deadlines and, most importantly, to enable enquirers to gather feedback that would enable them to enrich their inquiries;
expecting participants to report on the final results of their inquiry in both oral and written form -- since I wanted to use this activity also to enable them to experience as learners the implications of using these two alternative media to communicate the results of one's math inquiry.

Note that I encouraged the participants to engage in their inquiry by themselves -- mostly because I wanted to make sure each person would pursue a topic that truly interested him/her -- although I would not have objected to a pair working together if they had a good rationale for doing so.

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Scheduling of this activity within the course

Whenever this activity was implemented with secondary math pre-service teachers, I scheduled it right after this group of participants began to meet separately from the elementary pre-service teachers, and before the reflective activities that concluded the course (see Section D8). The activity required 3 class periods (the first to conduct the preliminary activities on taxicab-geometry, the second to give students an opportunity to share in pairs preliminary ideas and results, and the last class for the final presentations) and most of the homework assigned during those weeks.

The following excerpt from the schedule of Implementation B of the Methods course provides some additional details about the scheduling of key components of this activity (indicated in boldface) in relation to other parts of the section of the course that was specific to secondary math pre-service teachers:

SECONDARY MATH pre-service teachers ONLY:

(Remember that Lessons 1-9 had been jointly conducted with the elementary pre-service teachers)

Hw. 9 (sec.):

Readings on an inquiry approach
MIDTERM take-home exam (i.e, Analyze in writing a classroom vignette from an inquiry perspective) due

Lesson 10 (sec.):

Setting the stage for an independent inquiry experience:
- Thought-provoking activity on taxicab geometry
- Reflections on the process
- Brainstorming possibilities for inquiry on alternative geometries

Hw. 10 (sec.):

Independent inquiry experience assignment:
- Choose topic for individual inquiry
- Begin inquiry and be ready to report on preliminary ideas and results
- (suggested) Readings on reading and writing in math

Lesson 11 (sec.):

Independent inquiry -- part II:
- Large group sharing of topic chosen and process followed
- Pairs share and elaborate on preliminary results

Hw. 11 (sec.):

Independent inquiry experience assignment:
- First draft of written report due (for instructor's feedback)
- Prepare oral presentation on results of the inquiry
- (suggested) Readings on reading and writing in math

Lesson 12 (sec.):

Presentations and discussion of independent inquiry projects

Hw. 12 (sec.):

Identify teaching strategies modeled in the course
readings on reading and writing in math
Final written report on inquiry experience due

Lesson 13 (sec.):

Follow-up discussion on reading and writing in math instruction
Reflecting on teaching strategies modeled in the course

Hw. 13 (sec.):

"What have I learned" paper (participants' written final reflection)

Lesson 14 (sec.):

Course evaluation (forms + open discussion)

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Materials needed to implement this activity:

Readings, manipulatives, etc. necessary to conduct the preliminary "thought-provoking activity"
books/articles on alternative geometry to show as possible resources:
- Kline M. (1980). Mathematics: The loss of certainty (chapter on non-Euclidean geometries)
- (MAYBE) Krause (1986). Taxicab-geometry (but don't let students use it at this stage, or it could give much of their possible inquiries away)
- Abbott's Flatland
- (MAYBE) Sphereland
Overhead projector and/or whatever other audio-visual equipment is needed for the final presentations (make sure that individual participants communicate these needs to the instructor at least a week in advance).

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