The importance of proportional reasoning has been highlighted in NCTM Standards and most recently, in the Common Core Standards (2010). One of our goals was to support pre-service teachers’ explorations of how the ideas associated with proportional reasoning underlie heretofore seemingly separate topics– “mixture,” direct and indirect variation, similarity, and trigonometry—through the balance metaphor (Flores, 1995).

Theoretical Framework for the Research

The theoretical perspective underlying this work is Sfard’s (2007, 2008) commognitive view of learning.  This framework  dissolves the distinction between mathematical thinking (which was generally assumed to be “in the head”) and mathematical communication (which can be observed and recorded).  According to Sfard (2008), mathematical thinking can be studied in terms of students’ participation in discursive routines and commognitive conflicts..  In other words, thinking can, to some degree, become apparent to the outside observer by documenting changes in students’ participation in discourse practices and their interpretations of discursive “rules”.  We found Sfard’s (2008) outline of a model for the development of word use to describe how students come to develop fluency with word use (moving from passive use, to routine-driven use, to phrase-drive use to object-driven use) to be particularly helpful.

Methods

This study was conducted in one mathematics class with 38 prospective middle school mathematics teachers at a large university in the southwestern US.  We focus on four consecutive ninety-minute class sessions.  The basic research question we addressed was: What changes in student discussions and descriptions of proportional reasoning occurred as a result of students’ activities with the sequence of balance problems[SN1]?

The first author of this paper (the instructor of the course), and the second and third authors (who took field notes and videotaped) met on a daily basis to reflect on the current lesson and make revisions for the next day’s instruction. Our modes of inquiry combined quantitative analyses -a pre-post test of Learning Mathematics for Teaching Items (Hill, Schilling, and Ball, 2004 - and qualitative analyses (a commognitive analysis of changes in the discursive routines that occurred over the four days of instruction).

Subjects and Data Collected

The data set includes digital images of student work, videotaped classroom lessons, video-based interviews with students working on the tasks, results from pre-post LMT items and student solutions on the final exam.

Results: 

Results from the paired t-test conducted on pre-post proportional reasoning items of the LMT indicated that the increase in average score was significant at the p<.001 level.  The students in the class did learn how to identify and solve proportional reasoning situations, as well as pedagogical knowledge.  The qualitative analysis helped us understand how these pre-service teachers developed.

Day 1: The balance sequence was enacted over four consecutive instruction days, but was based on prior instruction during which the class had  established a working definition of “proportional quantities” (Lobato, Ellis, Charles and Zbiek, 2010) Students were challenged to imagine equivalent ratios modeled on a balanced see-saw.  Students identified the two quantities (weight and distance from fulcrum) and used either additive or multiplicative reasoning to justify their solutions.

Day 2: The authors noted that the communication routines tended to remain at what Sfard might call an “active use” level of the term “proportional”. That is, the students were able to engage in a description of their work, but not all had developed a sense of how a balance actually worked, or how two weights might use distance from the fulcrum to balance. Therefore, the authors introduced an applet that enabled students to model a variety of balance situations such as mixture problems and average rate.  The applet had some built-in constraints that were designed to support the students’ effort to re-unitize their activities.  For example, the applet only spanned 10 feet in each direction so that students had to adjust for quantities requiring greater than 10 feet.

Day 3: During this class, the students were asked more formally to justify why the relationships were multiplicative. Excerpts from this discussion will be presented during the presentation.  Argumentation during this discussion indicated students engaging in phrase-driven use of the word “proportional”.

Day 4: During the final day of the sequence, the instructor introduced the distinction between direct and indirect variationDuring this class graphical representations (Sfard’s  “realizations of these signifiers”) were created and contrasted, but the balance metaphor was not mentioned.

Final exam: The final exam included an item that  asked whether it was possible to use the balance metaphor to solve an indirect variation situation. Results indicated that while approximately ¼ of the students did engage in this routine, the remainder were thinking/communicating in ways that did not allow them to consider the use of the balance metaphor as a realization of multiple signifiers.

Importance of the Research:? This research explores the teaching and learning of mathematics in the context of small group and whole-class discussions and brings the role of symbolizations (both individual and computer-based) into play and highlights their roles within the scheme of communication development.

References

Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. Retrieved from http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf

Flores, A. (1995). Connections in proportional reasoning: Levers, arithmetic means, mixtures, batting averages, and speeds. School Science and Mathematics, 95, 423–430.

Hill, H., Schilling, B., & Ball, D. (2004). Developing measures of teachers' mathematics knowledge for teaching. The Elementary School Journal, 105(1), 11-30.

Lobato, J., Ellis, A., Charles, R., & Zbiek, R. (2010). Developing essential understanding of ratios, proportions, and proportional reasoning for teaching mathematics in grades 6-8. Reston, VA: National Council of Teachers of Mathematics.

Sfard, A. (2007). When the rules of discourse change, but nobody tells you: Making sense of mathematics learning from a commognitive standpoint. The Journal of the Learning Sciences, 16(4), 565-613.

Sfard, A. (2008). Thinking as communicating: Human development, the growth of discourses, and mathematizing. Cambridge, UK: Cambridge University Press.