RESEARCH PAPER
Generalizing and Proving in an Elementary Mathematics Teacher Education Program: Moving Beyond Logic
 
 
More details
Hide details
1
Oakland University, Rochester, Michigan, USA
Online publish date: 2019-04-12
Publish date: 2019-04-12
 
EURASIA J. Math., Sci Tech. Ed 2019;15(9):em1740
KEYWORDS
ABSTRACT
Generalization and proof are a foundation of mathematical practice and as such should be integral to K-12 mathematics instruction. However, if generalizing and proving are to become popular within K-12 mathematics classrooms, we must consider how to effectively enlist pre-service teachers (PSTs) into supporting these activities in their prospective students. Some researchers have suggested that the major challenges to generalization and proving for students in mathematics lie in developing and explicating logical statements. These researchers indicate that children do not have access to these forms of reasoning until adolescence. Those holding such views typically do not advocate for the training of PSTs in elementary education to support mathematical generalizing and proving. This study characterizes the views of and engagement with mathematical generalizing and proving of those principally involved in elementary mathematics education: mathematics faculty, mathematics education faculty, and PSTs. These views and this engagement are analyzed from survey responses and participation in a problem-solving session. Few PSTs provided descriptions of proving as a generalized explanation and demonstrated explicit generalization and proving infrequently. The results suggest that a mathematical focus on logic may be an impediment to proving and generalizing.
 
REFERENCES (44)
1.
Balacheff, N. (1988). Aspects of proof in pupils’ practice of school mathematics. Mathematics, teachers and children, 216, 235.
 
2.
Ball, D. L., & Bass, H. (2003). Making mathematics reasonable in school. A research companion to principles and standards for school mathematics, 27-44.
 
3.
Ball, D. L., & Cohen, D. K. (1999). Developing practice, developing practitioners: Toward a practice-based theory of professional education. Teaching as the learning profession: Handbook of policy and practice, 1, 3-22.
 
4.
Ball, D. L., & Forzani, F. M. (2011). Building a common core for learning to teach: And Connecting professional learning to practice. American educator, 35(2), 17.
 
5.
Barkai, R., Tsamir, P., Tirosh, D., & Dreyfus, T. (2002). Proving or Refuting Arithmetic Claims: The Case of Elementary School Teachers.
 
6.
Burger, W. F., & Shaughnessy, J. M. (1986). Characterizing the van Hiele levels of development in geometry. Journal for research in mathematics education, 31-48. https://doi.org/10.2307/749317.
 
7.
Byrnes, J. P., & Overton, W. F. (1988). Reasoning about logical connectives: A developmental analysis. Journal of Experimental Child Psychology, 46(2), 194-218. https://doi.org/10.1016/0022-0....
 
8.
Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic and algebra in elementary school. Heinemann, 361 Hanover Street, Portsmouth, NH.
 
9.
Chazan, D. (1993). High school geometry students’ justification for their views of empirical evidence and mathematical proof. Educational studies in mathematics, 24(4), 359-387. https://doi.org/10.1007/BF0127....
 
10.
Dubinsky, E. (1986). Teaching mathematical induction I. Journal of Mathematical Behavior, 5, 305–317.
 
11.
Dubinsky, E. (1990). Teaching mathematical induction II. Journal of Mathematical Behavior, 8, 285–304.
 
12.
Dubinsky, E., & Lewin, P. (1986). Reflective abstraction and mathematics education: The generic decomposition of induction and compactness. Journal of Mathematical Behavior, 5, 55–92.
 
13.
Ellis, A. B. (2011). Generalizing-promoting actions: How classroom collaborations can support students’ mathematical generalizations. Journal for Research in Mathematics Education, 42(4), 308-345. https://doi.org/10.5951/jresem....
 
14.
Hanna, G. (1991). Mathematical proof. In D. Tall (Ed.), Advanced mathematical thinking (pp. 54-61). Dordrech, The Netherlands: Kluwer Academic Publishers.
 
15.
Hanna, G. (1995). Challenges to the importance of proof. For the Learning of Mathematics, 15, 42-49.
 
16.
Harel, G. (2002). The development of mathematical induction as a proof scheme: A model for DNR-based instruction. In S. Campbell, & R. Zaskis (Eds.), Learning and teaching number theory: Research in cognition and instruction (pp. 185–212). New Jersey: Ablex Publishing Corporation.
 
17.
Herbst, P. G. (2002). Engaging students in proving: A double bind on the teacher. Journal for research in mathematics education, 176-203. https://doi.org/10.2307/749724.
 
18.
Herbst, P., & Brach, C. (2006). Proving and doing proofs in high school geometry classes: What is it that is going on for students? Cognition and Instruction, 24(1), 73-122. https://doi.org/10.1207/s15326....
 
19.
Piaget, J., & Inhelder, B. (2013). The growth of logical thinking from childhood to adolescence: An essay on the construction of formal operational structures. Routledge. https://doi.org/10.4324/978131....
 
20.
Jeannotte, D., & Kieran, C. (2017). A conceptual model of mathematical reasoning for school mathematics. Educational Studies in Mathematics, 96(1), 1-16. https://doi.org/10.1007/s10649....
 
21.
Kilpatrick, J., Swafford, J., & Findell, B. (2001). The strands of mathematical proficiency. Adding it up: Helping children learn mathematics, 115-118.
 
22.
Knuth, E. J. (2002). Secondary school mathematics teachers’ conceptions of proof. Journal for Research in Mathematics Education, 33, 379–405. https://doi.org/10.2307/414995....
 
23.
Lakatos, I. (1976). Proofs and refutations: The logic of mathematical discovery. Cambridge, UK: Cambridge University Press. https://doi.org/10.1017/CBO978....
 
24.
Lehrer, R., & Lesh, R. (2003). Mathematical learning. Handbook of psychology. https://doi.org/10.1002/047126....
 
25.
Lehrer, R., Kobiela, M., & Weinberg, P. J. (2013). Cultivating inquiry about space in a middle school mathematics classroom. ZDM, 45(3), 365-376.
 
26.
Movshovitz-Hadar, N. (1993). The false coin problem, mathematical induction and knowledge fragility. Journal of Mathematical Behavior, 12, 253–268.
 
27.
National Council of Teachers of Mathematics (NCTM) (2000). Principles and Standards for School Mathematics, Commission on Standards for School Mathematics, Reston, VA: NCTM.
 
28.
National Council of Teachers of Mathematics (NCTM) (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.
 
29.
National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common core state standards for mathematics. Washington, DC: Authors. Retrieved from http://www.corestandards.org/a....
 
30.
Piaget, J., & Inhelder, B. (1958). The growth of logical thinking from childhood to adolescence. Cambridge, MA: Routledge.
 
31.
Selden, J., & Selden, A. (1995). Unpacking the logic of mathematical statements. Educational Studies in Mathematics, 29(2), 123-151. https://doi.org/10.1007/BF0127....
 
32.
Selden, A., & Selden, J. (2013). Proof and problem solving at university level. The Mathematics Enthusiast, 10(1/2), 303.
 
33.
Schauble, L. (1996). The development of scientific reasoning in knowledge-rich contexts. Developmental Psychology, 32(1), 102. https://doi.org/10.1037/0012-1....
 
34.
Schauble, L., Glaser, R., Duschl, R. A., Schulze, S., & John, J. (1995). Students’ understanding of the objectives and procedures of experimentation in the science classroom. The journal of the Learning Sciences, 4(2), 131-166. https://doi.org/10.1207/s15327....
 
35.
Stylianides, A. J. (2007a). Introducing young children to the role of assumptions in proving. Mathematical Thinking and Learning, 9(4), 361-385. https://doi.org/10.1080/109860....
 
36.
Stylianides, A. (2007b). Proof and proving in school mathematics. Journal for Research in Mathematics Education, 38(3), 289-321.
 
37.
Stylianides, A. J. (2007c). The notion of proof in the context of elementary school mathematics. Educational Studies in Mathematics, 65, 1-20. https://doi.org/10.1007/s10649....
 
38.
Stylianides, A. J., & Ball, D. L. (2008). Understanding and describing mathematical knowledge for teaching: Knowledge about proof for engaging students in the activity of proving. Journal of mathematics teacher education, 11(4), 307-332. https://doi.org/10.1007/s10857....
 
39.
Stylianides, G. J., & Stylianides, A. J. (2009). Facilitating the transition from empirical arguments to proof. Journal for Research in Mathematics Education, 314-352.
 
40.
Stylianides, G. J., Stylianides, A. J., & Philippou, G. N. (2007). Preservice teachers’ knowledge of proof by mathematical induction. Journal of Mathematics Teacher Education, 10(3), 145-166. https://doi.org/10.1007/s10857....
 
41.
van Hiele, P. M. (1986). Structure and insight: A theory of mathematics education. Academic Press.
 
42.
Ward, S. L., & Overton, W. F. (1990). Semantic familiarity, relevance, and the development of deductive reasoning. Developmental Psychology, 26(3), 488. https://doi.org/10.1037/0012-1....
 
43.
Zazkis, R. (2005). Representing numbers: Prime and irrational. International Journal of Mathematical Education in Science and Technology, 36(2-3), 207-217. https://doi.org/10.1080/002073....
 
44.
Zazkis, R., & Sirotic, N. (2004). Making Sense of Irrational Numbers: Focusing on Representation. International Group for the Psychology of Mathematics Education.
 
eISSN:1305-8223
ISSN:1305-8215