University of Massachusetts at Boston

College of Advancing & Professional Studies

Critical and Creative Thinking Program

Mathematical Thinking

CrCrTh 650

Course description

This course explores several types of mathematical thinking in the context of number theory, algebra, geometry, and introductory calculus, and relates them to critical and creative thinking skills. Developmental and experiential factors in learning and teaching mathematics are considered, as well as techniques for determining a learner's mathematical abilities and learning styles. Readings, discussion, research, and problem-solving are used to provide a historical context, and to suggest connections with other disciplines. Individual and small-group projects are adapted to student interests. No formal mathematical background beyond high school algebra and geometry is required.

Fall 2017 Syllabus


Components of the syllabus:

I. Quick access to key information and links that should be bookmarked on your browser
followed by
II. Information to get started, orient yourself, and refer back to from time to time.
III. Contract: What is expected overall.
IV. Schedule of classes: What is expected each session and why -- how each session contributes to the unfolding of the course (starting with list of links to specific sessions).
V. Bibliography
POST-IT the start of each component in your printed version of this syllabus

Instructor
Peter Taylor, Critical & Creative Thinking Program and Science in a Changing World track
Email:
peter.taylor@umb.edu
Phone
617-287-7636 (note: email gets faster response)
Office
Wheatley 4-170
Office hours (http://bit.ly/peter_taylorhangout or in office):
Tuesday 1.40-3.40pm www.faculty.umb.edu/peter_taylor/PTOfficeHours.html, or by arrangement
Class time & location
Wednesdays 6.30-9pm, 9/13-12/13; by Hangout
URL for hangouts
http://bit.ly/650hangout BOOKMARK THIS!
http://bit.ly/CCTbreakout1, http://bit.ly/CCTbreakout2, http://bit.ly/CCTbreakout3
Blog
http://crcrth650mathematicalthinking.wordpress.com BOOKMARK THIS!
Report glitches in online materials
using this form
Syllabus
www.faculty.umb.edu/peter_taylor/650/Syllabus.html, with a menu of useful links at the top right BOOKMARK THIS! (backup copy at http://www.faculty.umb.edu/peter_taylor/650-17.pdf)
Public g+ community
http://bit.ly/SICWgplus, for public sharing of final products (optional)

II. Information to get started, orient yourself, and refer back to from time to time
Pointers about the preparation assumed for this course
(in lieu of formal prerequisites): CrCrTh 650 is appropriate for any student with a commitment to the personal development of themselves and others in the area of mathematical thinking. You will find it helpful to be familiar with the university's library and research services. You should be prepared to make time outside class--at least 6.5 hours/week--for undistracted work on the course and to view each assignment and each session in relation to the unfolding of learning during the course. (That is, do not expect the syllabus and online links to allow you to simply cut to the chase about what to do for the following day's class.)

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:

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.

Course Objectives
By the end of the semester, you will have:

Texts and Materials
Readings for the course consist of
Recommended (available from online retailers):

Technical set-up

Writing Support
For graduate students, see http://www.cct.umb.edu/writingsupport.html.

Accommodations
Sections 504 and the Americans with Disabilities Act of 1990 offer guidelines for curriculum modifications and adaptations for students with documented disabilities. The student must present any adaptation recommendations to the professors within a reasonable period, preferably by the end of the Drop/Add period.

Code of Conduct
The University's Student Code of Conduct (https://www.umb.edu/life_on_campus/policies/community/code) exists to maintain and protect an environment conducive to learning. It sets clear standards of respect for members of the University community and their property, as well as laying out the procedures for addressing unacceptable conduct. Students can expect faculty members and the Office of the Dean of Students to look after the welfare of the University community and, at the same time, to take an educational approach in which students violating the Code might learn from their mistakes and understand how their behavior affects others.

Students are advised to retain a copy of this syllabus in personal files for use when applying for certification, licensure, or transfer credit.

This syllabus is subject to change, but workload expectations will not be increased after the semester starts. (Version 10 September 2017)


III. Contract: What is expected overall

Written assignments and presentation (2/3 of grade)
Participation Items (1/3 grade)
Rubric
For each of the following 10 qualities, * [= "fulfilled very well", 2 points], OK [= "did an OK job, but room for more development/attention", 1 point], or - [= "to be honest, this was not my strength in this course", 0 points]

Plagiarism: Using another person's ideas or material you did not write without citing the source is plagiarism and is unacceptable (seelibrary guide and Academic Honesty policies).

IV. Schedule of classes: What is expected each session and why -- how each session contributes to the unfolding of the course

1. 9/13, 2. 9/20, 3. 9/27, 4. 10/4, 5. 10/11, 6. 10/18, 7. 10/25, 8. 11/1, 9. 11/8, 10. 11/15, no class 11/22, 11. 11/29, 12. 12/6, 13. 12/13

The Sessions have two parts of 60-75 minutes, with a 10-minute break between them: 1) The CE component; 2) Activities around a key concept in the field.

Session 1
Introductions and When do people use (or need) mathematical thinking?
Preparation:
Get set up on Technical matters
View video Bennett: "Why Math Instruction Is Unnecessary.
Session Exercises:
• Introduction to Collaborative Explorations.
• Quick preview of syllabus and tasks to get set up.
• Take stock of the session (Critical incident questionnaire)
Follow-up:
CE1Shifting: Changes in the ways people need to think mathematically
(A CE in which students identify a range of ways in which changes in work, technology, commerce, and social life have changed our needs and capacities for mathematical thinking, find patterns in them, and consider implications for education, including lifelong learning.)

As work, technology, commerce, and social life change, our needs and capacities for mathematical thinking shift. For example, at primary school before conversion to metric in Australia, I was assigned problems involving weights in units of tonnes, hundredweight, quarters, pounds, and ounces. After weights became measured in grams and kilos and metric tons, addition was so simple teachers found others ways to occupy students' time in class. The arrival of hand calculators helped diminish further the need for arithmetic. That said, my ability to do mental arithmetic applying multiplication tables means that I am quicker at seeing what ball park an answer should be in and thus detecting when someone has made a mistake inputting figures into their calculator. That said, class time for me as a child centered on quiet solitary solving of multiple problems of the same kind so there was little scope for collaboration, peer-to-peer support (except in the form of cheating), or for discussion of alternative approaches.

One side in the "Math Wars" maintains that computational "skills should be memorized and practiced, using time-tested traditional methods until they become automatic" (Wikipedia, n.d. *). However, instead of arguing a position in the polarized arena of the Math Wars (e.g., did the traditional methods actually work back in their time?), let us imagine a book that treats the audience as capable of addressing the complexities of change in work, technology, commerce, and social life as it relates to shifts in our needs and capacities for mathematical thinking. Then let us identify patterns in past changes, with a view to helping readers think about the implications for formal education as well as for the ways each of us continues to learn in response to ongoing change over our lifetimes. (*Wikipedia, n.d. "Math wars," https://en.wikipedia.org/wiki/Math_wars (viewed 4 Sep 17))

The end goal for the CE is that the class as a whole produces thought-supporting, constructive 1200 word entries for this hypothetical book. A premise for this book is that it would be unlike other mathematical thinking texts. Indeed, it may be more like a combination of provocations and resources for people—not only teachers—who want to foster ongoing development of people's mathematical thinking. In this spirit, this CE is an experiment—it is not clear in advance what a "pattern" is or what ways you will invent to "help readers think about the implications." (Steps to undertake and when.)
Session 2: Shifting: Changes in the ways people need to think mathematically (CE1) + Spreadsheet as a tool that extends and constrains our thinking
Preparation:
Session Exercises:
Follow-up:
Work due by the first day of this session:

Session 3: Shifting: Changes in the ways people need to think mathematically (CE1) + Simple rules generate complex behaviors
Preparation:
Session Exercises:
Follow-up:
Work due by the first day of this session:

Session 4: Shifting: Changes in the ways people need to think mathematically (CE1) + Big data allows micro-targeting
Session Exercises:
Follow-up:
Work due by the first day of this session:
CE2Constructing: 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:

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 having looked 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:
(Steps to undertake and when.)
Session 5: Constructing: Best practices for fostering mathematical thinking (CE2)+ Inquiry-based learning: problem-posing, problem-solving, and persuasion
Preparation:
Session Exercises:
Follow-up:
Work due by the first day of this session:

Session 6: Constructing: Best practices for fostering mathematical thinking (CE2)+ Correlation, causation, and consequences
Session Exercises:
Follow-up:
Work due by the first day of this session:

Session 7: Constructing: Best practices for fostering mathematical thinking (CE2)+ Traps in thinking about probability
Preparation:
Session exercises:
Follow-up:
Work due by the first day of this session:

Session 8: Constructing: Best practices for fostering mathematical thinking (CE2)+ Tools to extend thinking by considering feedback loops
Session exercises:
Follow-up:
Work due by the first day of this session:
CE3Framing and Practicing: Ongoing development beyond the course, CE3 (classes 9-12)
(A CE in which students, building on CEs 1 & 2, formulate specific plans for how to continue your own development as a mathematical thinker and, as a result, be able to foster the same among colleagues or students in your work/life/teaching situation.)

Books such as Julia Cameron's The Artist's Way provide readers with a program for developing one's creativity, but what is the equivalent for developing one's mathematical thinking? In any case, given that a mark of creativity is to develop one's own program, not follow someone else's, what would your program—or framework—for mathematical thinking look like? This said, all invention involves borrowing, so the challenge is really to synthesize elements from sources encountered during and before this course. These syntheses should be selected and organized in a framework so as to inspire and inform your efforts in extending mathematical thinking beyond the course. For a brief introduction to the experience of past students who prepared frameworks (called "manifestos"), not for mathematical thinking, but for critical thinking, see section 2 of Taylor (2002). For the full manifestos from a 1999 critical thinking class, see Readings. These frameworks might make up a 3rd section of the book from CE1 and CE2.

Corresponding to your framework, what is your plan for practice to develop your ability to foster the development of others as mathematical thinkers in your work/life/teaching situation? The plan should demonstrate how and when you plan to put into practice the skills and tools from the course - in your work situation or community, and/or how you could adapt and practice using those tools for opportunities in the future. You should include a plan for evaluating the outcome so you learn from experience and practice. (Test score results are not the only measure of improvement in mathematical thinking!) For examples of Plans for Practice from different CCT courses, see Readings and google+ community (search for #601). (Steps to undertake and when.)
Session 9: Framing and Practicing: Ongoing development beyond the course (CE3) + Prediction
Preparation:
Session Exercises:
Follow-up:
Work due by the first day of this session:

Session 10: Framing and Practicing: Ongoing development beyond the course (CE3) + Designing a critical thinking activity using spreadsheets
Session Exercises:
Follow-up:
Work due by the first day of this session:

Session 11: Framing and Practicing: Ongoing development beyond the course (CE3) + Critical assumptions at the foundations of statistical analysis
Preparation:
Session Exercises:
Follow-up:
Work due by the first day of this session:

Session 12: Framing and Practicing: Ongoing development beyond the course (CE3) + Visualization of inequalities
Session Exercises:
Follow-up:
Work due by the first day of this session:

Session 13: Heterogeneity + Taking Stock of the Course: where have we come from and where are we headed?
Session Exercises:
Follow-up:
Work due by the first day of this session:

V. Bibliography
(Link to password-protected pdf's of readings)
Blastland, M. and A. Dilnot (2009). The Numbers Game: The Commonsense Guide to Understanding Numbers in the News, in Politics, and in Life. New York, Gotham.
Cadwalladr, C. (2017) "The great British Brexit robbery: how our democracy was hijacked," https://www.theguardian.com/technology/2017/may/07/the-great-british-brexit-robbery-hijacked-democracy (viewed 7 Sep 17)
Chase, Allen. (1976). “A Few False Correlations = A Few Million Real Deaths: Scientific Racism Prevails Over Scientific Truth.” The Legacy of Malthus. The social costs of the new scientific racism. New York, Knopf.
Daniel, D., C. Fauske, P. Galeno, and D. Mael. (2001). Take Charge of Your Writing: Discovering Writing Through Self-Assessment. Boston, Houghton Mifflin.
García-Barrios, L. et al, (2016) "Azteca chess: Gamifying a complex ecological process of autonomous pest control in shade coffee," Agriculture, Ecosystems and Environment 232: 190–198
Gilovich, T., R. Vallone and A. Tversky (1985). "The Hot Hand in Basketball: On the Misperception on Random Sequences." Cognitive Psychology 17: 295-314
Hacker, D. (2000) A Pocket Style Manual. Boston, Bedford/St. Martins
Krimmel, K. and K. Rader (2017). "Opposition to Federal Spending Is Driven by Racial Resentment," Harvard Business Review, https://hbr.org/2017/09/research-opposition-to-federal-spending-is-driven-by-racial-resentment (viewed 10 Sep 17)
Meffe, G. K., A. H. Ehrlich and D. Ehrenfeld. "Human population control: The missing agenda." Conservation Biology 7, no.1 (1993): 1-3
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)
Peterson, N. S. and J. R. Jungck (1988). "Problem-posing, problem-solving, and persuasion in biology." http://www.bioquest.org/note21.html (viewed 7 Sep 17)
Rainey, J. (2017), "Predicting Irma's Path Is Giving Supercomputers a Challenge," https://www.nbcnews.com/storyline/hurricane-irma/predicting-irma-s-path-giving-supercomputers-challenge-n798961 (viewed 8 Sep 17)
Silverman, D. (2013) "How to Learn Board Game Design and Development," https://gamedevelopment.tutsplus.com/articles/how-to-learn-board-game-design-and-development--gamedev-11607 (viewed 7 Sep 17)
Taylor, P. J. (1997). How do we know we have global environmental problems? Undifferentiated science-politics and its potential reconstruction. Changing Life: Genomes, Ecologies, Bodies, Commodities. P. J. Taylor, S. E. Halfon and P. E. Edwards. Minneapolis, University of Minnesota Press: 149-174
Taylor, P. J. (2000), “How do we know there is a population-environment problem?” http://www.faculty.umb.edu/peter_taylor/popdialogue.html (view 7 Sep 17)
Taylor, P. J. (2002), “We know more than we are, at first, prepared to acknowledge: Journeying to develop critical thinking,” Working Papers in Critical, Creative, and Reflective Practice, http://scholarworks.umb.edu/cct_ccrp/1/
Taylor, P.J. (2008) "Why was Galton so concerned about 'regression to the mean'?—A contribution to interpreting and
changing science and society" DataCritica, 2(2): 3-22. http://www.faculty.umb.edu/peter_taylor/taylor07dGalton.pdf
Taylor, P.J. (2010) "Who can act? Critical assumptions at the foundations of statistical analysis: Explaining differences among means – What can that mean?" manuscript pdf
Taylor, P. and J. Szteiter (2012) Taking Yourself Seriously: Processes of Research and Engagement Arlington, MA, The Pumping Station
Wikipedia, n.d. "Math wars," https://en.wikipedia.org/wiki/Math_wars (viewed 4 Sep 17)
See also annotated bibliography entries on the course blog, https://crcrth650mathematicalthinking.wordpress.com/category/bibliography-entry/