1. Cooperative Learning and
the Transition to Proof
A dissertation proposal
Martha Byrne
University of New Mexico
2. To what extent can working in a
Cooperative Learning environment
affect students’ acquisition and
development of proof skills?
3. Proof Skills
• Proof Construction
• Proof Validation
• Concept Image (of Proof)
• Attitudes about Mathematics
• Attitudes about Proof
4. Proof Construction
• Roles of proof
– Dependent on audience
– Proofs that convince vs. Proofs that
explain
• Where the students are
– Empirical evidence
– Authoritarian proof schemes
Almeida (1995), Harel and Sowder (1998)
5. Proof Validation
• Structure vs. Detail
• Formal vs. Informal
• Conviction vs. Validity
Segal (1999), Selden and Selden (1995, 2003), Powers et. al. (2010)
6. Concept Image (of Proof)
• Concept Image vs. Concept Definition
• Concept Image of Proof
– Imprecise definition
– Need for examples and non-examples
– Proof process
Tall and Vinner (1981), Weber (2001)
8. Attitudes about Proof
• Conviction vs. Validity
• Need for Proof
• Lack of Exploration
Almeida (2000), Maclane (1994), Sowder and Harel (2003)
9. Cooperative Learning (CL)
• Positive Interdependence
• Personal Accountability
• Appropriate Grouping
• Student Interaction
• Attention to Social Skills
• Teacher as Facilitator
Cooper (1990), Cooper and Robinson (1994), Millis (1992), Springer (1998)
10. To what extent can working in a
Cooperative Learning environment
affect students’ acquisition and
development of proof skills?
18. Works Cited
• Almeida, D. (1995). Mathematics Undergraduates’
Perceptions of Proof. Teaching Mathematics and Its
Applications, 14(4), p. 171-177.
• Almeida, D. (2000). A survey of mathematics undergraduates’
interaction with proof: some implications for mathematics
education. International Journal of Mathematical Education in
Science and Technology. 31, #6, 869-890.
• Cooper, J. (1990). What is Cooperative Learning? In Cooper,
J., Robinson, P., and Ball, D. (Eds.) (2009). Small Group
Instruction in Higher Education: Lessons from the Past, Visions of
the Future. New Forums Press, Inc.
19. • Cooper, J., and Robinson, P. (1994) FIPSE-Sponsored CL
Research at Dominguez Hills and Community Colleges. In
Cooper, J., Robinson, P., and Ball, D. (Eds.) (2009). Small Group
Instruction in Higher Education: Lessons from the Past, Visions of
the Future. New Forums Press, Inc.
• Harel, G., & Sowder, L. (1998). Students' proof schemes.
Research on Collegiate Mathematics Education, Vol. III. In E.
Dubinsky, A. Schoenfeld, & J. Kaput (Eds.), AMS, 234-283.
• Millis, B. (1992). How Cooperative Learning Can fullfill the
Promises of the "Seven Principles." In Cooper, J., Robinson, P.,
and Ball, D. (Eds.) (2009). Small Group Instruction in Higher
Education: Lessons from the Past, Visions of the Future. New
Forums Press, Inc.
20. • Powers, R. A., Craviotto, C. & Grassl, R. M. (2010). Impact of
proof validation on proof writing in abstract algebra.
International Journal of Mathematical Education in Science
and Technology, 41(4), 501-514.
• Segal, J. (1999). Learning about mathematical proof:
conviction and validity. Journal of Mathematical Behavior, 18,
191-210.
• Selden, A. & Selden, J. (2003). Validations of proofs written as
texts: Can undergraduates tell whether an argument proves a
theorem? Journal for Research in Mathematics Education, 36
(1), 4-36.
• Selden, J. and Selden, A. (1995) Unpacking the logic of
mathematical statements. Educational Studies in
Mathematics, 29(2), 123-151.
21. • Sowder L., Harel G. (2003) Case Studies of Mathematics
Majors' Proof Understanding, Production, and Appreciation
Canadian Journal of Science, Mathematics and Technology
Education 3(2)
• Springer, L. (1998). Research on Cooperative Learning in
College Science, Mathematics, Engineering, and Technology.
In Cooper, J., Robinson, P., and Ball, D. (Eds.) (2009). Small
Group Instruction in Higher Education: Lessons from the Past,
Visions of the Future. New Forums Press, Inc.
• Tall, D. and Vinner, S. (1981) Concept image and concept
definition in mathematics with particular reference to limits
and continuity. Educational Studies in Mathematics, Vol.12
(No.7). pp. 151-169.
22. • Weber, K. (2001). Student difficulty in constructing proofs: The
need for strategic knowledge. Educational Studies in
Mathematics, 48(1), 101-119.
