# Technology and the Development of Mathematical Creativity in Advanced School Mathematics

## Abstract

The availability of sophisticated computer programs capable of complex symbolic computations has created challenges for mathematics educators working with mathematically motivated students. Whereas technology may be praised for enabling educators to bridge the gap between the past—when only some students were able to do mathematics, and the present—when an average student is able to enjoy finding an answer to a difficult problem using a computer, it can also put a barrier in the way of developing students’ creative mathematical skills. This dichotomy between positive and negative affordances of technology in the teaching of mathematics calls for the development of new curriculum enabling the outcome of problem solving not to be dependent on students’ ability to simply enter correctly all data into a computer. Towards this end, the chapter proposes a way of modifying traditional problems from advanced mathematics curriculum to be both technology-immune and technology-enabled in the sense that whereas software can facilitate problem solving, its direct application is not sufficient for finding an answer.

## Keywords

Affordances of technology Teacher education TITE problems Problem reformulation Einstellung effect## References

- Abramovich, S. (2014a).
*Computational experiment approach to advanced secondary mathematics curriculum*. Dordrecht, The Netherlands: Springer.CrossRefGoogle Scholar - Abramovich, S. (2014b). Revisiting mathematical problem solving and posing in the digital era: Toward pedagogically sound uses of modern technology.
*International Journal of Mathematical Education in Science and Technology,**45*(7), 1034–1052.CrossRefGoogle Scholar - Abramovich, S. (2015). Mathematical problem posing as a link between algorithmic thinking and conceptual knowledge.
*The Teaching of Mathematics,**18*(2), 45–60.Google Scholar - Abramovich, S., & Cho, E. K. (2013). Technology and the creation of challenging problems.
*Mathematics Competitions,**26*(2), 10–20.Google Scholar - Abramovich, S., Easton, J., & Hayes, V. O. (2014). Integrated spreadsheets as learning environments for young children.
*Spreadsheets in Education,**7*(2), 3.Google Scholar - Abramovich, S., & Leonov, G. A. (2011). A journey to a mathematical frontier with multiple computer tools.
*Technology, Knowledge and Learning,**16*(1), 87–96.Google Scholar - Advisory Committee on Mathematics Education. (2011).
*Mathematical needs: The mathematical needs of learners*. London, UK: The Royal Society.Google Scholar - Angeli, C., & Valanides, N. (2009). Epistemological and methodological issues for the conceptualization, development and assessment of ICT-TPCK: Advances in technological pedagogical content knowledge (TPCK).
*Computers & Education,**52*(1), 154–168.CrossRefGoogle Scholar - Arnold, V. I. (2015).
*Experimental mathematics*. Providence, RI: The American Mathematical Society.Google Scholar - Avitzur, R. (2011).
*Graphing Calculator*(Version 4.0). Berkeley, CA: Pacific Tech.Google Scholar - Baumert, J., Kunter, M., Blum, W., Brunner, M., Voss, T., Jordan, A., et al. (2010). Teachers’ mathematical knowledge, cognitive activation in the classroom, and student progress.
*American Educational Research Journal,**47*(1), 133–180.CrossRefGoogle Scholar - Beghetto, R. A., Kaufman, J. C., & Baer, J. (2015).
*Teaching for creativity in the common core classroom*. New York, NY: Teachers College Press.Google Scholar - Bers, M. U. (2010). When robots tell a story about culture … and children tell a story about learning. In N. Yelland (Ed.),
*Contemporary perspective on early childhood education*(pp. 227–247). Maidenhead, UK: Open University Press.Google Scholar - Bers, M. U., Ponte, I., Juelich, K., Viera, A., & Schenker, J. (2002). Teachers as designers: Integrating robotics in early childhood education.
*Information Technology in Childhood Education Annual,**2002*(1), 123–145.Google Scholar - Borwein, J., & Bailey, D. (2004).
*Mathematics by experiment: Plausible reasoning in the 21st century*. Natick, MA: AK Peters.Google Scholar - Brooks, J. G., & Brooks, M. G. (1999).
*In search of understanding: The case for constructivist classrooms*. Alexandria, VA: Association for Supervision and Curriculum Development.Google Scholar - Collins, A., & Ferguson, W. (1993). Epistemic forms and epistemic games: Structures and strategies to guide inquiry.
*Educational Psychology,**28*(1), 25–42.CrossRefGoogle Scholar - Conference Board of the Mathematical Sciences. (2012).
*The mathematical education of teachers II*. Washington, DC: The Mathematics Association of America.CrossRefGoogle Scholar - Conole, G., & Dyke, M. (2004). What are the affordances of information and communication technologies?
*ALT-J. Research in Learning Technology,**12*(2), 113–124.CrossRefGoogle Scholar - Dahlstrom, E., & Bichsel, J. (2014).
*ECAR Study of Undergraduate Students and Information Technology, 2014*. Research report. Louisville, CO: ECAR, October 2014. Retrieved from http://www.educause.edu/ecar. - David, F. N. (1970). Dicing and gaming (a note on the history of probability). In E. S. Pearson & M. G. Kendall (Eds.),
*Studies in the history of statistics and probability*(pp. 1–17). London, UK: Griffin.Google Scholar - Dittert, N., & Krannich, D. (2013). Digital fabrication in educational contexts—Ideas for a constructionist workshop setting. In J. Walter-Herrmann & C. Büching (Eds.),
*FabLab: Of machines, makers and inventors*(pp. 173–180). Bielefeld, Germany: Transcript Verlag.Google Scholar - Ellis, W. D. (Ed.). (1938).
*A source book of Gestalt psychology*. New York, NY: Harcourt, Brace.Google Scholar - Epstein, D., Levy, S., & de la Llave, R. (1992). About this journal.
*Experimental Mathematics,**1*(1), 1–3.CrossRefGoogle Scholar - Freiman, V., Kadijevich, D., Kuntz, G., Pozdnyakov, S., & Stedøy, I. (2009). Technological environments beyond the classroom. In E. J. Barbeau & P. J. Taylor (Eds.),
*Challenging mathematics in and beyond the classroom*(pp. 97–131). Dordrecht, The Netherlands: Springer.CrossRefGoogle Scholar - Freire, P. (2003).
*Pedagogy of the oppressed*(with an introduction by D. Macedo). New York, NY: Continuum.Google Scholar - Feurzeig, W., & Lukas, G. (1972). LOGO–A programming language for teaching mathematics.
*Educational Technology,**12*(3), 39–46.Google Scholar - Gibson, J. J. (1977). The theory of affordances. In R. Shaw & J. Bransford (Eds.),
*Perceiving, acting and knowing: Toward an ecological psychology*(pp. 67–82). Hillsdale, NJ: Lawrence Erlbaum.Google Scholar - Gurung, R. A. R., Chick, N. L., & Haynie, A. (2009).
*Exploring signature pedagogies: Approaches to teaching disciplinary habits of mind*. Sterling, VA: Stylus.Google Scholar - Hadamard, J. (1996).
*The Mathematician’s mind: The psychology of invention in the mathematical field*. Princeton, NJ: Princeton University Press.Google Scholar - Hodgson, C. F. (1870).
*Educational times*. London, UK: CF Hodgson & Son.Google Scholar - Hughes, J., Gadanidis, G., & Yiu, C. (2016). Digital making in elementary mathematics education? In
*Digital experiences in mathematics education*(pp. 1–15). New York, NY: Springer, May 27. Retrieved from http://link.springer.com/article/10.1007/s40751-016-0020-x. - Isaacs, N. (1930). Children’s why questions. In S. Isaacs (Ed.),
*Intellectual growth in young children*(pp. 291–349). London, UK: Routledge & Kegan Paul.Google Scholar - Kadijevich, D. (2002). Towards a CAS promoting links between procedural and conceptual mathematical knowledge.
*The International Journal of Computer Algebra in Mathematics Education,**9*(1), 69–74.Google Scholar - Kaput, J. J. (1992). Technology and mathematics education. In D. A. Grouws (Ed.),
*Handbook of research on mathematics teaching and learning*(pp. 515–556). New York, NY: Macmillan.Google Scholar - Kaput, J. J., Noss, R., & Hoyles, C. (2008). Developing new notations for a learnable mathematics in the computational era. In L. D. English (Ed.),
*Handbook of international research on mathematics education*(pp. 693–715). New York, NY: Lawrence Erlbaum.Google Scholar - Kennedy, G. E., Judd, T. S., Churchward, A., Gray, K., & Krause, K. L. (2008). First year students’ experiences with technology: Are they really digital natives?
*Australasian Journal of Educational Technology,**24*(1), 108–122.CrossRefGoogle Scholar - Kieran, C., & Drijvers, P. (2006). The co-emergence of machine techniques, paper-and-pencil techniques, and theoretical reflection: A study of CAS use in secondary school algebra.
*International Journal of Computers for Mathematical Learning,**11*(2), 205–263.CrossRefGoogle Scholar - Kirkwood, A., & Price, L. (2005). Learners and learning in the twenty-first century: What do we know about students’ attitudes towards and experiences of information and communication technologies that will help is design courses?
*Studies in Higher Education,**30*(3), 257–274.CrossRefGoogle Scholar - Krutetskii, V. A. (1976).
*The psychology of mathematical abilities in school children*. Chicago, IL: University of Chicago Press.Google Scholar - Kvavik, R.B., & Caruso, J.B. (2005).
*ECAR study of students and information technology, 2005: Convenience, connection, control, and learning*(Vol. 6). Boulder, CO: EDUCAUSE. Retrieved from http://www.educause.edu/ecar. - Lakatos, I. (1976).
*Proofs and refutations: The logic of mathematical discovery*. Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar - Leonov, G. A., & Kuznetsov, N. V. (2013). Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits.
*International Journal of Bifurcation and Chaos*,*23*, 1330002 (2013) [69 pages]. https://doi.org/10.1142/s0218127413300024. - Lillard, P. P. (1996).
*Montessori today: A comprehensive approach to education from birth to adulthood*. New York, NY: Shaken Books.Google Scholar - Lingefjärd, T. (2012). Mathematics teaching and learning in a technology rich world. In S. Abramovich (Ed.),
*Computers in education*(Vol. 2, pp. 171–191). New York, NY: Nova Science Publishers.Google Scholar - Luchins, A. S. (1942). Mechanization in problem solving: The effect of Einstellung. In
*Psychological Monographs*,*54*(6, whole No. 248). Evanston, IL: The American Psychological Association.Google Scholar - Luchins, A. S. (1960). On some aspects of the creativity problem in thinking.
*Annals of the New York Academy of Sciences,**91,*128–140.CrossRefGoogle Scholar - Luchins, A. S., & Luchins, E. H. (1970).
*Wertheimer’s seminars revisited: Problem solving and thinking*(Vol. I). Albany, NY: Faculty-Student Association, SUNY at Albany.Google Scholar - Maddux, C. D. (1984). Educational microcomputing: the need for research.
*Computers in the Schools,**1*(1), 35–41.CrossRefGoogle Scholar - Maddux, C. D., & Johnson, D. L. (2005). Type II applications of technology in education: New and better ways of teaching and learning.
*Computers in the Schools,**22*(1/2), 1–5.Google Scholar - Mason, J. (2000). Asking mathematical questions mathematically.
*International Journal of Mathematical Education in Science & Technology,**31*(1), 97–111.CrossRefGoogle Scholar - Mayer, M. (1965). Introduction. In M. Montessori (Ed.),
*The Montessori method*(pp. xxiii–xli). Cambridge, MA, Robert Bentley.Google Scholar - Mehan, H. (1979).
*Learning lessons*. Cambridge, MA: Harvard University Press.CrossRefGoogle Scholar - Meyer, H. B. (2016).
*Eratosthenes’ sieve*. Retrieved from http://www.hbmeyer.de/eratosiv.htm. - Ministry of Education Singapore. (2012).
*N(T)-level mathematics teaching & learning syllabus.*Curriculum Planning & Development Division. Retrieved from http://www.moe.gov.sg/education/syllabuses/sciences/files/normal-technical-evel-maths-2013.pdf. - Montessori, M. (1965).
*The Montessori method*. Cambridge, MA: Robert Bentley.Google Scholar - National Governors Association Center for Best Practices, Council of Chief State School Officers. (2010)
*Common core state standards (Mathematics)*. Washington, D.C.: National Governors Association Center for Best Practices, Council of Chief State School Officers. Retrieved from http://www.corestandards.org/Math/. - Niess, M. L. (2005). Preparing teachers to teach science and mathematics with technology: Developing a technology pedagogical content knowledge.
*Teaching and Teacher Education,**21*(5), 509–523.CrossRefGoogle Scholar - Noss, R., & Hoyles, C. (1996).
*Windows on mathematical meanings: Learning cultures and computers*. Dordrecht, The Netherlands: Kluwer.CrossRefGoogle Scholar - Pea, R. D. (1993). Distribted intelligence and designs for education. In G. Salomon (Ed.),
*Distributed cognitions—Psychological and educational considerations*(pp. 47–87). New York, NY: Cambridge University Press.Google Scholar - Peschek, W., & Schneider, E. (2001). How to identify basic knowledge and basic skills? Features of modern general education in mathematics.
*The International Journal of Computer Algebra in Mathematics Education,**8*(1), 7–22.Google Scholar - Pólya, G. (1954).
*Induction and analogy in mathematics*(Vol. 1). Princeton, NJ: Princeton University Press.Google Scholar - Pólya, G. (1957).
*How to solve it*. New York, NY: Anchor Books.Google Scholar - Pólya, G. (1965).
*Mathematical discovery: On understanding, learning, and teaching problem solving*. New York, NY: Wiley.Google Scholar - Prensky, M. (2001). Digital natives, digital immigrants.
*On the Horizon,**9*(5), 1–6.CrossRefGoogle Scholar - President’s Council of Advisors on Science and Technology. (2010).
*Prepare and inspire: K–12 education in science, technology, engineering and math (STEM) for America’s future*. Washington, D.C.: Author. Retrieved from https://www.whitehouse.gov/sites/default/files/microsites/ostp/pcast-stemed-report.pdf. - Schoenfeld, A. H. (1988). Users of computers in mathematics instruction. In D. A. Smith, G. Porter, L. Leinbach, & R. Wenger (Eds.),
*Computers and mathematics: The use of computers in undergraduate instruction*(Vol. 9, pp. 1–11). Washington, D.C.: The Mathematical Association of America.Google Scholar - Shulman, L. S. (2005). Signature pedagogies in the professions.
*Daedalus,**134*(3), 52–59.CrossRefGoogle Scholar - Sivashinsky, I. H. (1968).
*Zadachi po matematike dlja vneklasnyh zanjatii (Mathematical problems for after school activities)*. Moscow, Russia: Prosveschenie. (In Russian).Google Scholar - Stark, H. M. (1987).
*An Introduction to number theory*. Cambridge, MA: MIT Press.Google Scholar - Stern, C., & Stern, M. B. (1971).
*Children discover arithmetic: An introduction to structural arithmetic*. New York, NY: Harper & Row.Google Scholar - Sugden, S., Baker, J. E., & Abramovich, S. (2015). Conditional formatting revisited: A companion for teachers and others.
*Spreadsheets in Education,**8*(3), 2.Google Scholar - Tall, D., Gray, E., Ali, M. B., Crowley, L., DeMarois, P., McGowen, M., et al. (2001). Symbols and the bifurcation between procedural and conceptual thinking.
*Canadian Journal of Science, Mathematics and Technology Education,**1*(1), 81–104.CrossRefGoogle Scholar - Tchekoff, A. (1970).
*Russian silhouettes*(Translated from Russian by M. Fell). Freeport, NY: Books for Libraries Press.Google Scholar - Wertheimer, M. (1959).
*Productive thinking*. New York, NY: Harper & Row.Google Scholar - Wertheimer, M. (1938). Gestalt theory. In W. D. Ellis (Ed.),
*A source book of Gestalt psychology*(pp. 1–11). New York, NY: Harcourt Brace.Google Scholar