CE2. Constructing: Best practices for fostering mathematical thinking
(A CE in which students learn as much as possible about how mathematical thinking is presented and promoted by others.)

Imagine a continuation of the book in CE1: a section that aims to help readers appreciate the idea that everyone can think mathematically and to help them help others appreciate that idea. The end-product of this CE are drafts of entries to this section of the book, which might take the form of text, maps, schemas, mp3s, problem sets, or something else (adding up to at least 1200 words or its page-equivalent, in one or more entries). These entries should introduce and organize key resources from how mathematical thinking is presented and promoted by others, i.e., key concepts, issues and debates, references to research, quotes or paraphrases from those references, interactive activities and personal habits, people and organizations to take note of, appropriate stories. (Do not be concerned about whether your entries overlap with anyone else's.)

Some questions that might stimulate your inquiries:

• How much have well-worn sources from the 80s and 90s been superseded by more recent research and writing; how much do old sources hold up?
• Could the mathematical thinking process be thought of less as adding rule-bound (or algorithmic) practices and more as recognizing and removing obstacles that have come into place and obscured natural mathematical thinking? What authors have promoted the latter approach?
• How much does the mathematical thinking process need to involve individuals seeking or creating supportive "context," e.g., arranging sounding boards or establishing one's surroundings as a "studio" to make a space where mathematical thinking comes easier? What is known about how spaces for mathematical thinking, communities and historical periods came together? What does mathematical thinking mean in different fields of work?
• To the extent that the mathematical thinking process like the creative thinking process involves the capacity to manage, seek out, even welcome risk, struggle and failure, how can we feel more comfortable and supported in allowing "failures" to happen... of letting go of positions we once held strongly to?
• What is there to support, or contradict, the idea that everyone can think mathematically? In guiding those who believe that they are not mathematical thinkers, what steps might be taken to encourage them to at least explore the possibility?
• How is improvement in mathematical thinking assessed? How are different tools and activities to foster mathematical thinking evaluated?

The process towards the end products should involve reading and digesting as much as you can in the time available, guided by some of the questions above that interest you. The assumption (is this justified?) is that your experience undertaking CE1 before you look at how mathematical thinking is presented and promoted by others will help you to choose topics that most grab your interest and be engaged in learning about them. In any case, there is no expectation that you think like a textbook writer who has to cover every topic. Instead, you should identify a theme that can govern what your writing focuses on. Entry points for readings are given by:

• National Council on Education and the Disciplines (2003), "Why numeracy matters," https://www.maa.org/sites/default/files/pdf/QL/WhyNumeracyMatters.pdf (viewed 7 Sep 17)
• the syllabi from CCT courses in Mathematical Thinking (2013, 2006, 1999 and http://www.cct.umb.edu/courses.html#650);
• The abundant resources of NCTM, https://www.nctm.org/
• "8 alternatives to traditional mathematics education," http://davidwees.com/content/8-alternatives-traditional-mathematics-education/ (viewed 8 Sep 17)
• CCT syntheses addressing mathematics
• Videos bookmarked in diigo group on Mathematical Thinking (but always search for critiques of ideas enthusiastically presented so as to put the videos in perspective)

(Steps to undertake and when.)

Steps

• Class 5: Autobiographical stories, retold in relation to topic of CE 2
• Before Class 6: Notes on inquiries pursued, posted on wordpress site (using Category "Work or Thinking in Progress"); ditto before Class 7. (One written assignment C for CE2)
• By Class 8: Two bibliography contributions with paragraph-length annotations, drawn from readings assigned or encountered during CE, posted on wordpress site (using Category "Bibliography Entry"). (The other written assignment C for CE2)
• Class 6: 5-phase Dialogue process (aka Dialogue Hour) to share and clarify what we are inquiring into regarding the case.
• Class 7: Work-in-progress presentations, each followed a few minutes of time to write Plus-Delta feedback
• Class 8: Dialogue Hour for Taking stock of this Collaborative Exploration
• By Class 8, Draft of your CE 2 product submitted; by class 9, Comments on another student's draft; by class 10, Revision in response to comments, posted to wordpress.