Graphs of Linear Functions: Making Mathematical Principles Explicit For Fifth Graders

This proposal presents results of tutorial study involving Grade 5 students engaged in graphing algebraic functions. Algebra has long been a difficult domain for middle and high school students (Kieran, 2007), and in response, researchers investigated ways of promoting algebraic reasoning with elementary school students in order to anticipate and preempt later difficulties (Cai & Knuth, 2011; Carraher & Schliemann, 2007; Carraher, Schliemann, Brizuela & Earnest, 2006; Kaput, Carraher, & Blanton, 2008). Early algebra (EA) research has produced promising findings that elementary students can productively engage in algebraic reasoning. This project is concerned with students' understanding of representations of continuous quantities—in particular, graphs of algebraic functions. Graphs have been repeatedly cited as core to EA (Carraher & Schliemann, 2007; Carraher, Schliemann & Schwartz, 2008; Janvier, 1987; Kaput, 1999, 2008), yet much remains unknown about how young students ascribe meaning to the various representational forms in the context of algebraic functions.

Perspective

Any mathematical representation has no inherent meaning in and of itself.  Rather, representational forms take on particular meanings through interactions in activity. This research takes this perspective that mathematics is a social practice.  To capitalize on this, the tutorial introduces a communication game in which players—with their views occluded from one another—must construct number lines and graphs (see Saxe et al., 2010).  Breaches in communication (or, the child's incorrect constructions) result in the generation of a written 'agreement,' by which players agree to construct graphs in particular ways in order to achieve identical constructions. Agreements (Table 1) are then used to regulate and justify subsequent game play. 

Table 1: Agreements used in the tutorial

Methods

Data come from a larger, two-part dissertation study involving (1) a 32-item pretest with Grade 5 (n=126) and Grade 8 (n=131) students; and, building off these results, (2) a tutorial intervention involving a subset of Grade 5 students (n=40).  This proposal reports only on Part 2.

Participants. Participants included 40 Grade 5 students from the San Francisco Bay Area. Using identical pretest scores, students were matched to create 20 pairs, with partners randomly assigned to tutorial or comparison group.

Tutorial. The tutorial design builds on an integer number line study that led to substantial learning gains with a large effect size when contrasted to a control (Saxe et al., 2010). The tutorial included two key design features:

(1) Communication game structure. The tutor engaged each tutee with a card game that required their coordinated action. A game card instructed players to perform some action on a number line or graph printed on either paper (tutee) or transparency (tutor). A screen occluded players' work from one another (Figure 2). Removing the screen, they compared point placement by overlaying the tutor's transparency over the tutee's paper. Discrepant point placement resulted in players writing 'agreements' in order to regulate subsequent game play.

Figure 2: Communication game structure for tutorial

 (2) Written agreements. Through a structured design of problem sets ordered to scaffold learning, tutor and tutee co-constructed number line 'agreements'—explicit mathematical principles written in language friendly to a fifth grader—that served both to repair breaches in communication when players' respective placements are misaligned and to anticipate subsequent repairs on upcoming problems. When the constructions did not match, the tutor initiated and supported reflection on the sources of discrepancy by asking, "How can we do this next time to make our work the same?"  Through dialogue, the tutor encouraged the use of the agreements to support justifications.  Tutorial and control students were then administered a posttest identical to the pretest.

Data sources include students' pre/post assessments and written work produced during the tutorial, all of which were scanned and entered into a database.  All tutorials were videotaped, and StudioCode software was used to analyze video.

Results

Results of pre- and post-test scores are presented here, along with some patterns revealed in qualitative analyses. Figure 3 displays pretest and posttest performance of the tutorial and control groups.  At pretest, groups showed a similar distribution of scores as expected.  Mean scores at pretest were 13.10 (sd=2.99) and 13.05 (sd=2.98) out of 32 items for the tutorial and control groups, respectively.  The tutorial group had much greater gains on posttest.  For the tutorial group, the gain was 10.15 points (sd=4.36) whereas for the control, the mean gain was 1.20 points (sd=5.21).  For the tutorial group, the gain represented a shift of 2.33 standard deviations.  A 2 x 2 ANOVA revealed an interaction between group and pre-to post assessments (F(1, 38)=55.980, p<.0001), and follow-up analyses revealed a significant difference between tutorial students' pre- and posttest performance (t(19)=17.058, p<.0001), but no difference between pre- and posttest for the control. The analysis suggests that students learned relevant content as a result of intervention.

Figure 3: Bar graph of pre- to post-test performance for control and tutorial groups

Qualitative analyses were conducted to understand patterns across tutorial students.  A brief summary of some patterns includes:

á       Almost all students struggled when first introduced to linear functions; with tutoring, 15 out of 20 successfully solved the particular task.  Analyses suggest (1) the need to model the problem context, and (2) that with practice, young students may reasonably construct and interpret graphs of algebraic functions;

á       'Agreement' usage was highly correlated with success on tutorial problems and posttest;

á       Students' initial graphing efforts often do not coordinate inscriptions in the plane space with values along the axes;

á       Gridlines were a salient representational form that often had bearing on students' construction of graphs in ways usually inconsistent with mathematical principles.

Educational Importance

                  This study contributes important findings on (a) the manner in which young students approach function graphs; and (b) the use of explicit mathematical principles to facilitate the learning and teaching of this content.  With EA research still relatively new, the field needs to understand patterns that emerge across students as they engage in this content, as well as promising approaches to teaching such content.  Such research is necessary in order to ultimately support students' learning and teachers' instruction of algebra in elementary school.

References

Cai, C., & Knuth, E. (2011). Early algebraization: A global dialogue from multiple perspectives. Springer.

Carraher, D. & Schliemann, A.D. (2007).  Early algebra and algebraic reasoning.  In F.K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (Vol. 2, pp. 669-705). Greenwich, CT: Information Age Publishing.

Carraher, D., Schliemann, A.D., Brizuela, B., & Earnest, D. (2006).  Arithmetic and Algebra in Early Mathematics Education.  Journal for  Research in Mathematics Education, 37(2), 87-115.

Carraher, D.W., Schliemann, A.D. & Schwartz, J. (2008). Early algebra is not the same as algebra early. In J. Kaput. D. Carraher, & M. Blanton (Eds.), Algebra in the Early Grades.  Mahwah, NJ, Erlbaum, pp. 235-272

Janvier, C. (1987). Problems of representation in the teaching and learning of mathematics. Hillsdale, NJ: Erlbaum.

Kaput, J. (1999). Teaching and learning a new algebra. In E. Fennema & T. Romberg (Eds.), Mathematics classrooms that promote understanding (pp. 133-155). Mahwah, NJ: Lawrence Erlbaum Assocaiates.

Kaput, J. (2008).  What is algebra?  What is algebraic reasoning?  In J. Kaput, D. W. Carraher, & M. Blanton (Eds.)  Algebra in the early grades.  Mahwah, NJ: Erlbaum.

Kaput, J.J., Carraher, D.W., & Blanton, M.L. (2008).  Algebra in the early grades. Mawhaw, NJ: Erlbaum.

Kieran, C. (2007). Learning and teaching algebra at the middle school through college levels: Building meaning for symbols and their manipulation. In F.K. Lester (Ed.), Second handbook of research on mathematics teaching and learning. Greenwich, CT: Information Age Publishing.

Saxe, G.B., Earnest, D., Sitabkhan, Y., Haldar, L.C., Lewis, K.E., & Zheng, Y. (2010). Supporting Generative Thinking about Integers on Number Lines in Elementary Mathematics.  Cognition and Instruction, 28(4), pp. 433-474.

Saxe, G.B., Gearhart, M., Shaughnessy, M.M., Haldar, L.C., Earnest, D., & Sitabkhan, Y. (2009). Integers on the Number Line: Students' Understanding of Linear Unit. Poster Presentation at the 2009 Research Pre-session of the National Council of Teachers of Mathematics Meeting.  Washington, D.C.