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.
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