Abstract
In formal education, learning mathematics is typically done by receiving direct instruction within the confines of a classroom. From first grade through graduate school, students are expected to learn mathematics primarily by being taught by instructors with previous knowledge of the subject. Research mathematicians, on the other hand, must rely on other methods; the mathematics they are trying to understand may not, as yet, be known to anyone else. Hence, they learn primarily through experimentation, self-directed study, and collaboration with peers. In recent years, these methods have been expanded to use modern tools and ideas. Research mathematicians initiated several successful large-scale online collaboration projects, such as the Polymath project and the MathOverflow website. In this chapter, we discuss these two projects, along with various other examples of online collaborative learning of mathematics. Our primary motivation is captured in the following question: why aren’t we all learning math this way? While a complete answer is beyond the scope of this work, we hope to at least stimulate a debate among a wide audience. The major part of our discussion is thus informal; we defer the contextualization of these examples within modern education research until the end of the chapter.
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- 1.
For now, we will consider both “collaborative” and “cooperative” online learning, without worrying too much about the differences between the two; we will come back to this issue and discuss these differences briefly at the end of the chapter.
- 2.
It would be an interesting study indeed, to examine effects of the collaboratively-edited (via search-engines) results from such communities versus ones provided by “authoritative sources.”
- 3.
We are compelled to note that “democratic” in this context certainly does not mean that one should decide the veracity of mathematical propositions by vote.
- 4.
Of course, there are technical exceptions, such as the loss of a person who is running the server/software; in such a case, a strong community would presumably simply reform elsewhere on the Web.
- 5.
One should be careful here not to confuse single authorship of papers (which is quite common in mathematics) with working in complete isolation (which is exceptionally rare.)
- 6.
Some projects of this kind already exist, e.g., the Global Nomads Group and iEarn; neither of these focus specifically on mathematics.
- 7.
In this context “polymath” really means “many mathematicians” rather than the usual definition, i.e., “a person of wide-ranging knowledge or learning.” Rather than depending on the powers of a single rare and remarkable individual of the mold of Newton or Einstein, the Polymath projects depend on the combined strength of a number of more ordinary mathematicians.
References
Alagic, M. (2003). Technology in the mathematics classroom: Conceptual orientation. Journal of Computers in Mathematics and Science Teaching (JCMST), 22(4), 381–399.
Alagic, M., Gibson, K., & Doyle, C. (2004). The potential for autonomous learning through ICT. In R. Ferdig, et al. (Eds.), Proceedings of Society for Information Technology & Teacher Education International Conference 2004 (pp. 1679–1684). Chesapeake, VA: AACE. Retrieved from http://www.editlib.org/p/14667
Baez, J. C. (2010, March). Math blogs. Notices of the AMS (opinion). Retrieved August 30, 2011, from http://www.ams.org/notices/201003/rtx100300333p.pdf
Benkler, Y. (2006). The wealth of networks: How social production transforms markets and freedom. New Haven, CT: Yale University Press.
Cannatella, H. (2000). Autonomous learning and teaching. Retrieved September 20, 2003, from https://www.ilt.ac.uk/iltac2000/cannatella/Autonomy.rtf
Cheung, A. C. K., & Slavin, R. E. (2011). The effectiveness of educational technology applications for enhancing mathematics achievement in K-12 classrooms: A meta-analysis. Baltimore, MD: Johns Hopkins University, Center for Data-Driven Reform in Education.
Corno, L., & Mandinach, E. B. (1983). The role of cognitive engagement in classroom learning and motivation. Educational Psychologist, 18(2), 88–108.
Dalsgaard, C., & Paulsen, M. F. (2009). Transparency in cooperative online education. The International Review of Research in Open and Distance Learning, 10(3), 1–22. http://www.irrodl.org/index.php/irrodl/article/view/671/1267
Dewey, J. (1916/2010). Democracy and education: An introduction to the philosophy of education. Los Angeles, CA: Indo-European Publishing.
Dron, J., & Anderson, T. (2007). Collectives, networks and groups in social software for e-learning. In Proceedings of World Conference on E-Learning in Corporate, Government, Healthcare, and Higher Education, Quebec. www.editlib.org/index.cfm/files/paper_26726.pdf
Engel, S. (2011, March 14). Let kids rule the school. The opinion line. New York Times. http://www.nytimes.com/2011/03/15/opinion/15engel.html?_r=3&src=ISMR_HP_LO_MST_FB, page A35 of the New York edition.
Freire, P. (1970/2006). Pedagogy of the oppressed, (M. B. Ramos, Trans.). New York: Continuum.
GeoGebra. (2011). Retrieved July 7, 2011, from http://www.geogebra.org/cms/
Geraschenko, A., & Morrison, S. (2009, October 14). Math overflow. Retrieved May 15, 2012, from http://sbseminar.wordpress.com/2009/10/14/math-overflow/
Gibbons, M. (2004). Pardon me, didn’t I just hear a paradigm shift? Phi Delta Kappan, 85(6), 461–467.
Goodyear, P., Banks, S., Hodgson, V., & McConnell, D. (Eds.). (2004). Advances in research on networked learning. London: Kluwer Academic Publishers.
Gowers’s Weblog. (2011). Retrieved June 5, 2011, from http://gowers.wordpress.com/2009/01/27/is-massively-collaborative-mathematics-possible/
Greeno, J. G., & Hall, R. P. (1997, January). Practicing representation: Learning with and about representational forms. Phi Delta Kappan, 1997, 361–367.
Hannula, M. (2002). Attitude toward mathematics: Emotions, expectations and values. Educational Studies in Mathematics, 49, 25–46.
Hidi, S., & Harackiewicz, J. M. (2000). Motivating the academically unmotivated: A critical issue for the 21st century. Review of Educational Research, 70(2), 151–179.
Hurme, T.-R., Merenluoto, K., & Järvelä, S. (2009). Socially shared metacognition and feeling of difficulty in the process of mathematical problem solving in CSCL context. Journal of Educational Research and Evaluation, 15, 503–524.
Jonassen, D., & Reeves, T. (1996). Learning with technology: Using computers as cognitive tools. In D. H. Jonassen (Ed.), Handbook of research for educational communications and technology (pp. 693–719). London: Prentice Hall International.
Jones, C. R., Ferreday, D., & Hodgson, V. (2008). Networked learning, a relational approach: Weak and strong ties. Journal of Computer Assisted Learning, 24(2), 90–102.
Knapper, C. (1988). Technology and lifelong learning. In D. Boud (Ed.), Developing student autonomy in learning (2nd ed., pp. 91–106). London: Kogan Page.
Koschmann, T. (1996). Paradigm shifts and instructional technology: An introduction. In T. Koschmann (Ed.), CSCL: Theory and practice of an emerging paradigm (pp. 1–23). Mahwah, NJ: Lawrence Erlbaum Associates.
Litz, I. R. (2007). Student adoption of a computer-supported collaborative learning (CSCL) mathematical problem solving environment: The case of the math forum’s virtual math teams (VMT) chat service. Graduate School of Computer and Information Sciences, Nova Southeastern University. Unpublished Dissertation.
Lois, J. (2006). Role strain, emotion management and burnout: Homeschooling mothers; adjustment to the teacher role. Symbolic Interactions, 29(4), 507–530.
Markoff, J. (2010, August 16). Step 1: Post elusive proof. Step 2: Watch fireworks. The New York Times Reprints. Retrieved August 31, 2011, from http://www.cs.rice.edu/∼vardi/nyt81610.pdf
Math Forum. (2011). Retrieved June 5, 2011, from http://mathforum.org/
Math Stack Exchange Q&A. (2011). Retrieved June 5, 2011, from http://math.stackexchange.com/faq
Mathematics – Stack Exchange. (2011). Retrieved June 5, 2011, from http://math.stackexchange.com/
MathOverflow. (2011). Retrieved June 5, 2011, from http://www.mathoverflow.net
MathOverflow Contributors. (2011). Retrieved August 31, 2011, from http://www.quora.com/What-proportion-of-living-research-mathematicians-contribute-to-MathOverflow
Meta Mathoverflow. (2011). Retrieved August 31, 2011, from http://meta.mathoverflow.net/discussion/64/where-to-keep-track-of-mathoverflow-success-stories/#Item_0
Meta Mathoverflow Discussions. (2011). Retrieved August 31, 2011, from http://meta.mathoverflow.net/discussion/985/woman-in-mathoverflow/#Item_0
Molenaar, I., van Roda, C., Boxtel, C., & Sleegers, P. (2012). Dynamic scaffolding of socially regulated learning in a computer-based learning environment. Computers in Education, 59(2), 515–523.
Nason, R., & Woodruff, E. (2004). Online collaborative learning in mathematics: Some necessary innovations? In T. Roberts (Ed.), Online learning: Practical and theoretical considerations (pp. 103–131). Hershey, PA: Idea Group Inc.
Nielsen, M. A. (2010). Introduction to the polymath project and “Density Hales-Jewett and Moser Numbers”. Bolyai Society Mathematical Studies, 21, 651–657. Retrieved from http://www.springerlink.com/content/v160qt6727441m82/fulltext.pdf
Panitz, T. (2003): Collaborative versus cooperative learning – A comparison of the two concepts which will help us understand the underlying nature of the interactive learning. Retrieved July 4, 2011, from http://home.capecod.net/~tpanitz/tedsarticles/coopdefinition.htm
Perkins, D. N., & Salomon, G. (1992, September 2). Transfer of learning: Contribution to the international encyclopedia of education (2nd ed.). Oxford, England: Pergamon Press.
Polymath. (2011). Retrieved June 5, 2011, from http://www.springerlink.com/content/v160qt6727441m82/fulltext.pdf
Polymath Blog. (2011, March 9). Polymath discussion at Institute for Advanced Study [Web log message]. Retrieved from http://polymathprojects.org/2011/03/09/polymath-discussion-at-ias/
Roberts, T. S. (Ed.). (2004). Online collaborative learning: Theory and practice. Hershey, PA: Information Science Publishing.
Rockwood, H. S., III. (1995). Cooperative and collaborative learning. The National Teaching & Learning Forum, 4(6), 8–9.
Roschelle, J., & Teasley, S. D. (1995). The construction of shared knowledge in collaborative problem solving. In C. O’Malley (Ed.), Computer supported collaborative learning (pp. 69–97). Berlin, Germany: Springer.
Salmon, G. (2003). E-moderating: The key to teaching and learning online (2nd ed.). London: Taylor & Francis.
Sarvate, D., Wetzel, S., & Patterson, W. (2009). Analyzing massively collaborative mathematics projects (Viewpoint). The Mathematical Intelligencer, 33(1), 9–18. http://www.springerlink.com/content/p8t49074lh144278/fulltext.pdf
Schank, R. (2011). Teaching minds: How cognitive science can save our schools. New York: Teachers College Press.
Schoenfeld, A. (1989). Explorations of students’ mathematical beliefs and behavior. Journal for Research in Mathematics Education, 20(4), 338–355.
Schugurensky, D. (2000). The forms of informal learning: Towards a conceptualization of the field (WALL Working Paper No.19, 2000). Retrieved July 5, 2011, from https://tspace.library.utoronto.ca/bitstream/1807/2733/2/19formsofinformal.pdf
Serow, P., & Callingham, R. (2011). Levels of use of interactive whiteboard technology in the primary mathematics classroom. Technology, Pedagogy and Education, 20(2), 161–173. doi:10.1080/1475939x.2011.588418.
Siemens, G. (2005, January). Connectivism: A learning theory for the digital age. International Journal for Instructional Technology and Distance Learning. Retrieved May 15, 2012 from http://www.itdl.org/Journal/Jan_05/article01.htm
Snow, R., Corno, L., & Jackson, D. (1996). Individual differences in effective and conative functions. In D. Berliner & R. Calfee (Eds.), Handbook of educational psychology (pp. 243–310). New York: Macmillan.
Stahl, G. (2009). Studying virtual math teams. New York: Springer.
Stahl, G. (2010). How I view learning and thinking in CSCL groups. Research and Practice in Technology Enhanced Learning (RPTEL). Retrieved May 15, 2012 from http://GerryStahl.net/pub/rptel2010.pdf
Stahl, G., Koschmann, T., & Suthers, D. (2006). Computer-supported collaborative learning: An historical perspective. In R. K. Sawyer (Ed.), Cambridge handbook of the learning sciences (pp. 409–426). Cambridge, UK: Cambridge University Press.
Tao, T. (2008). Structure and randomness: Pages from year one of a mathematical blog. Providence, RI: American Mathematical Society.
Tao, T. (2011, July 15). What’s new [Web log message]. Retrieved from http://terrytao.wordpress.com/
The Future of Science. (2011). Retrieved August 31, 2011, from http://michaelnielsen.org/blog/the-future-of-science-2/
The Institute Letter. (2010, Fall). Institute for Advanced Study. Retrieved August 31, 2011, from http://www.ias.edu/files/pdfs/letter-2010-fall.pdf
The Problem. (2011). Retrieved July 17, 2011, from http://math.stackexchange.com/questions/51505/line-segment-joining-two-centers-of-circles-is-perpendicular-to-line-segment-join
Vygotsky, L. S. (1978). Mind in society: The development of higher psychological process. Cambridge, MA: Harvard University Press.
Warschauer, M., & Matuchniak, M. (2010, March). New technology and digital worlds: Analyzing evidence of equity in access, use, and outcomes. Review of Research in Education, 34, 179–225.
Watters, J. J., Rogers, G., Gibson, K., Alagic, M., & Haack, C. (2004). From Kansas to Queensland: Global learning in preservice elementary teacher education. In Proceedings American Educational Research Association Annual meeting, San Diego. http://eprints.qut.edu.au/1700/1/1700.pdf
Widaman, K. F., & Kagan, S. (1987). Cooperativeness and achievement: Interaction of student cooperativeness with cooperative versus competitive classroom organization. Journal of School Psychology, 25(4), 355–365.
Winne, P., & Hadwin, A. (2008). The weave of motivation and self-regulated learning. In D. Schunk & B. Zimmerman (Eds.), Motivation and self-regulated learning: Theory, research, and applications (pp. 297–314). New York: Taylor & Francis.
Wolters, C. A., Pintrich, P. R., & Karabenick, S. A. (2003, March). Assessing academic self-regulated learning. Paper presented at the conference on Indicators of Positive Development: Definitions, Measures, and Prospective Validity, Washington, DC.
Zheng, R. (Ed.). (2008). Cognitive effects of multimedia learning. Hershey, PA: Information Science Reference/IGI Global Publishing.
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Alagic, G., Alagic, M. (2013). Collaborative Mathematics Learning in Online Environments. In: Martinovic, D., Freiman, V., Karadag, Z. (eds) Visual Mathematics and Cyberlearning. Mathematics Education in the Digital Era, vol 1. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-2321-4_2
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