Volume 1 | Issue 1 | Year 2011 | Article Id. IJMTT-V1I1P6 | DOI : https://doi.org/10.14445/22315373/IJMTT-V1I1P6

Recent research on problem solving explores its potential as a pedagogical practice. This emphasis rejuvenates the interest in problem solving as a learning activity. This paper presents the practice of using a selected problem together with its variants in a single lesson. The practice was implemented in middle school classroom settings with gifted students and with mixed ability students as well as in teacher education classrooms. Experiences from practice are used to illustrate that the use of a set of closely related problems is likely to make students more eager to share their solutions, to generate several solution strategies, and to show various connections among the ideas involved. The shift toward exploring multiple math strategies and representations, and big ideas is at the center of innovative and successful approaches to teaching mathematics. The paper is guided by an evaluation of literature that considers teaching through problem solving and of literature on complex professional tasks.

[1] ZDM, “The International Journal on Mathematics Education (2007). A
special issue on Problem solving around the world: Summing up the
state of the art,”. ZDM, vol. 39, pp. 5-6, 2007.

[2] A. H. Schoenfeld, “Problem Solving in the United States, 1970-2008:
Research and theory, practice and politics,” ZDM Mathematics
Education, vol. 39, pp. 537-551, 2007.

[3] D. Clarke, M. Goos, and W. Morony, “Problem solving and working
mathematically: An Australian perspective,” ZDM Mathematics
Education, vol. 39, pp. 475-490, 2007.

[4] F. K. Lester, and R. L. Charles, Teaching mathematics through
problem solving: Prekindergarten-grade 6. Reston, VA: NCTM, 2003).

[5] S. Crespo, and N. Sinclair, “What makes a problem mathematically
interesting? Inviting prospective teachers to pose better problems,”
Journal of Mathematics Teacher Education, vol.11, pp. 395-415, 2008.

[6] G. Brousseau, and P. Gibel, “Didactical handling of students’
reasoning processes in problem solving situations,” Educational
Studies in Mathematics, vol. 59, pp. 13-58, 2005.

[7] Y. Bar-Yam, Dynamics of complex systems, Reading, MA: AddisonWesley, 1997.

[8] E. Simmt, “Mathematical knowing in action,” Unpublished doctoral
dissertation, University of Alberta, 2000.

[9] G., Gadanidis, J. Hughes, R. Scucuglia, and S. Tolley, “Low floor,
high ceiling: performing mathematics across grades 2-8,” In Swars, D.
W. Stinson, and S. Lemons-Smith, (Eds.), Proceedings of the 31st
annual meeting of the North American Chapter ofthe International
Group for the Psychology of Mathematics Education. Atlanta, GA:
Georgia State University, 2009.

[10] G. Polya, How to solve it: A new aspect of mathematics, 2nd ed, New
Jersey: Princeton University Press, 1945/1973

[11] G. Flewelling,, and W. Higginson, Realizing a vision of tomorrow’s
mathematics classroom: A handbook on rich learning tasks. Kingston:
MSTE, 2001.

[12] A. Sfard, “On the dual nature of mathematical conceptions: Reflections
on process and objects as different sides of the same coin,”
Educational Studies in Mathematics, vol. 22, pp. 1-36, 1991.

[13] M. Lampert, Connecting mathematical teaching and learning, E.
Fennema, T. P. Carpenter, and A. J. Lamon, Eds. Integrating research
on teaching and learning mathematics (pp. 121-152). Albany: State
University of New York, 1991.

[14] C. T. Fosnot, and M. Dolk, Mathematicians at Work: Constructing
Multiplication and Division. Portsmouth, NH: Heinemann, 2001.

[15] I. K. Namukasa, and G. Gadanidis, (2010) Mathematical tasks as
experiential therapy for elementary pre-service teachers. International
Journal for Mathematics Teaching and Learning. [Online]. Available:
http://www.cimt.plymouth.ac.uk/journal/default.htm.

[16] C. Fernandez, “Lesson study: A means for elementary teachers to
develop the knowledge of mathematics needed for reform-minded
teaching?,” Mathematical Thinking and Learning, vol.7, pp. 265-289,
2005.

[17] T. P. Carpenter, E. Fennema, M. L. Franke, L. Levi,, and S. B. Empson,
Children’s mathematics: Cognitively guided instructions.Portsmouth:
Heinemann, 1999.
[18] OECD: The PISA 2003 Assessment Framework. Mathematics,
Reading, Science and Problem Solving Knowledge and Skills. Paris:
OECD, 2003.

[19] English, L. D. Integrating complex systems within the mathematics
curriculum. Teaching Children Mathematics, 15(1), 38, 2008.

[20] Nelson, Nelson Mathematics Grade 6 by Kelleher, H., Kubota-Zarivnij,
K., Milot, P., Morris, B., and Super, Doug. Thomson, 2005.

[21] McGraw-Hill Ryerson, Principles of Mathematics 9 by Darling,
Erdman, Ferneyhough, MCCudden, Meisel & Speijer. McGraw-Hill
Ryerson, 1999

[22] OME. (2006). Targeted Implementation and Planning Supports for
Revised Mathematics Continuum and Connections: Patterning to
Algebraic Modelling (TIPS4RMCC) Grades 7 & 8. A Ministry of
Education, Ontario resource. [Online]. Available:
http://www.edu.gov.on.ca/eng/studentsuccess/lms/files/tips4rm/TIPS4
RMccpatterns.pdf

[23] R. Leikin, and A. Levav-Waynberg, “Solution spaces of multiplesolution connecting tasks as a mirror of the development of
mathematics teachers’ knowledge.” Canadian Journal of Science,
Mathematics and Technology Education, vol. 8, pp.233-251, 2008.

[24] M. L. Blanton, and J. J. Kaput, “Characterizing a classroom practice
that promotes algebraic reasoning,” vol. 35, pp. 412-446, Nov. 2005.

[25] J. W. Stigler, P. A. Gonzales, T. Kawanka, S. Knoll, & A. Serrano,
“The TIMSS videotape classroom study: Methods and findings from
an exploratory research project on eighth-grade mathematics
instruction in Germany, Japan, and the United States,” Washington,
DC: U.S. Department of Education, National Center for Education
Statistics., 1999

[26] M. Cordy, (2012). Mild, Medium and Spicy: The mathematics of
Chopsticks. [Online]. Available:
http://publish.ed.uwo.ca/george.gadanidis/Michellel/index.html

[27] M. Small, Big Ideas from Dr. Small: Creating a comfort zone with
teaching mathematics, Grades 4-8. Toronto: Nelson, 2009.

[28] Ontario Ministry of Education. A guide to effective instruction in
mathematics: Kindergarten to Grade 6, volume 2. Toronto: Ontario
Ministry of Education, 2009..

[29] NCTM (2011). Supreme Court welcome. [Online]. Available:
http://illuminations.nctm.org/LessonsDetail.aspx?ID=L630.

[30] M. L. Blanton, and J. J. Kaput, Developing Elementary teachers’
“Algebra eyes and ears.” Teaching Children Mathematics. Reston:
NCTM, 2003.

[31] A. Kajander, Big ideas for small mathematicians: Kids discovering the
beauty of math with 22 ready-to-go activities. Tucson, AZ: Zephyr,
2003.

Immaculate K. Namukasa, Elena Polotskaia, "Teaching through Mathematics Problems: Redesigned for a Focus on Mathematics," *International Journal of Mathematics Trends and Technology (IJMTT)*, vol. 1, no. 1, pp. 50-56, 2011. *Crossref*, https://doi.org/10.14445/22315373/IJMTT-V1I1P6