National Council of Teachers of Mathematics 2012 Research Presession

Please note: The NCTM conference program is subject to change.

120- Open Questions Generate a Culture of Sense-Making Reasoning and Proving

Wednesday, April 25, 2012: 1:00 PM-2:30 PM
Salon I/J/K/L 4 (Philadelphia Marriott Downtown)
Principles and Standards for School Mathematic (NCTM, 2000) calls for mathematical reasoning and proof to play a role at all grade levels.  Since this publication, Focus on High School Mathematics: Reasoning and Sense Making (NCTM, 2009) has described a classroom culture where reasoning and sense making are intertwined across the continuum from informal observations to formal deductions.

Unfortunately, outside of high school geometry students experience little if any exposure to proving (Ball, et al, 2002; Knuth, 2002b; Wu, 1996), and considerable literature has shown that students often leave their high school geometry course with a narrow understanding of the role of proof (Chazan, 1993; Knuth, 2002a; Martin and Harel, 1989; Martin and McCrone, 2001).  Moreover, school proving experiences, when they occur, are often formal and do not connect well with other types of mathematical study such as exploration, problem solving, and discovery, to name a few.

Yet authors such as Hanna (2000) and deVilliers (1999) describe numerous roles proof can play in school mathematics, including: verification, explanation, systematization, discovery, communication, construction, and exploration of mathematics. Weber (2009) discusses how proving is related to problem solving. A disheartening finding of Knuth (2002) is that many high school teachers view proof in a limited pedagogical way—that it is a topic of study primarily for verifying truth.  This novice’s view of proof may come from the teachers’ own proof experience. Schoenfeld (1985) argues that "[f]or most students proof lacks purpose: students engage in argument only to confirm something that is intuitively obvious, in which case proof is redundant, or to verify something they are told in which case proof is arcane."  Other researchers have noted this problem.  Martin and McCrone (2001) assert that when students are asked only to prove statements they already believe to be true, they fail to develop an appreciation for the learning and explanatory role of proof.

To address this problem, the author of this talk has conducted three iterations of a teaching experiment to find ways to engage pre-service teachers in a task that expands their understanding of the role of proving, reasoning, and sense making in mathematics and particularly in the mathematics classroom. By presenting an open question, providing the right amount of scaffolding, and giving students more than a month to address in the question, numerous roles of proving and reasoning discussed in the literature emerged.

From the three iterations of this type of task (here “task” is defined to be all the student actions centered on addressing the question posed), the author has learned a great deal about common themes that emerge. A great deal has also been learned about such tasks’ potential for expanding learners’ understanding of the importance, power, and potential of reasoning, proving, and sense-making in mathematical thinking. Moreover, a great deal has been learned about how informal reasoning and formal reasoning intertwine to support mathematical investigation and problem solving.

Data; including student collective discourse from an online environment, videotapes of classroom discourse, and students’ written work; will be presented. Analysis of this data reveals how students:  

  1. Developed a collective understanding the question posed.
  2. Developed an answer to the question.
  3. Determined what support (e.g. axioms, definitions, and other prior results) were needed to justify their responses.
  4. Developed a variety of approaches: recursive, algebraic, pattern recognition.
  5. Addressed impossibility theorems as well as possibility theorems.
  6. Used inductive reasoning to move toward deductive reasoning.
  7. Develop a self-motivated level of rigor, e.g. created a social-culture nature of proof (Stylandies (2007)).

Participants and the presenter will then engage in collect reflection and discourse on the following questions:

  1. How can the data collection process and subsequent analysis be refined?
  2. How can the structure and delivery of the teaching experiment be improved?
  3. What is the potential of the approach and findings for broader impact?
Lead Speaker:
David A. Yopp


Description of Presentation:

The speaker will present findings from an experiment that engaged preservice teachers in tasks that expanded their understanding of the roles of proof and reasoning. An open question, the right amount of scaffolding, and giving students more than a month to address the question brought out numerous roles discussed in the literature.

Session Type: Poster Session

See more of: Poster Session
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