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Mathematical Thinking

CCT611 (Theme: Making Sense of Numbers), F01, CCT650, F17)

Initial Goals
Early in the summer I changed the theme of my Fall '01 Critical Thinking seminar from Science in Society (see 4.) to Making Sense of Numbers for three reasons: a) there was a number of math. teachers among new or prospective CCT students, but no science teachers; b) CCT650, Math. Thinking Skills would not be offered in the fall; and c) experienced graduate students from this course might qualify as teachers for an equivalent undergraduate Quantitative Reasoning seminar in the Spring '02 and beyond.

Like CCT611 in Sp 99 and CCT640 (see 4. and 7. above), the course will operate on three levels, the first level in this case being "learn[ing] a variety of tools for quantitative reasoning and how to interpret their application to situations of social significance." My training and scientific work has involved much quantitative/mathematical work and many of the classes build on activities developed in other contexts. Nevertheless, the course is explicitly experimental and my goal is to model the ongoing pedagogical development I expect of the students. I particularly look forward to leading a three-week unit of full-blown Problem-Based Learning (drawing on Nina Greenwald's expertise) and coaching the students to compile their Personal/Professional Development workbooks (see comments on CCT640 earlier).

Challenges and Responses
The students are all new or prospective CCT students, so I have to get them comfortable with journaling; learning through activities, not lectures; my revise and resubmit system; and other CCT-style practices. The bigger challenge, however, has been adjusting to the turmoil and stress after the September 11 attacks. On the 12th. I introduced them to pairwise constructivist or supportive listening, but ten minutes of this was barely enough for me to focus on the rest of the class. The following week, I led a discussion on "What stops people asking why" as the fundamental question of critical thinking and then asked them to prepare on index cards their own critical thinking questions in this situation concerning numbers. I plan to continue to acknowledge in different ways that we cannot readily return to life/work as usual. Meanwhile, for this and other situations, we have to examine why the numbers we need are difficult either to find, to make sense of, or to get attention paid to their implications.

Future Plans
In Fall 2017, the format of the course has two strands, taking up half the time of each session.
The first strand is centered on 4-week "collaborative explorations" (CEs), a variant of project-based learning (PBL) that begin from a scenario or case in which the issues are real but the problems are not well defined, which leads participants to shape their own directions of inquiry and develop their skills as investigators and teachers (in the broadest sense of the word). The basic mode of a CE centers on interactions in small groups (online or face-to-face) over a delimited period of time in ways that create an experience of re-engagement with oneself as an avid learner and inquirer--as this quote from a student in a PBL course evokes:
"This course is a gift – the chance to be open – open-ended in design, open to process, open to other perspectives, open to changing your ideas, and open to sharing. Of course this means it's risky too – you won't always know when you're coming from or where you are going – you might think you aren't sufficiently grounded by the course. But you have the freedom to change that – and being on the other side of it now, I see it works out beautifully. The attention to process provides you the tools to grow and by the end you're riding the wave of your earlier work..."
The CE format is designed to allow each student to
a) undertake intensive reading in the area of mathematical thinking and learn from other students through their annotated bibliography entries, presentations, and written products;
b) shape a path and final products for each CE that link closely with your personal interests; and
c) see yourselves as contributors to ongoing development of the field, especially by sharing of products with future students on the blog and (optional) with the wider public on a google+ community (and eventually perhaps a book).
The second strand will involve activities or discussion based on shared readings around key concepts or issues in the field. Each activity promotes a way to improve mathematical thinking, but allows for insights about one's thinking to emerge in its own way. Plus-Delta feedback at end of most activities fosters the formation of these insights as well as future improvements of the activity for future offerings of the course. Indeed, the instructor, whose mathematical thinking was formed in the 1960s and early 70s, is looking to students' inquiries in the CEs as well as feedback on the activities to help him clarify what are the most important ways that people's needs and capacities for mathematical thinking have shifted since then.

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