Does Schema-Based Instruction and Self-Monitoring Influence Seventh Grade Students’ Proportional Thinking?
Asha Jitendra, University of Minnesota
Jon R. Star, Harvard University
John Woodward, University of Washington at Puget Sound
Abstract
Tasks
Results
Proportional reasoning exemplifies the kind of multiplicative thinking that is
central to middle school mathematics. This study examined in greater depth
the impact of the SBI-SM intervention in the Jitendra et al. (2008) on a
representative sample of students from different ability groups. Unlike the
multiple choice test used as the main dependent measure in the larger study,
the represented sub-sample of students was individually interviewed on three
performance tasks that varied in complexity. Each task was administered as
a pre and posttest, and all of the interactions between the interviewer and the
student were recorded and transcribed. It was hypothesized that a detailed
analysis of the student performance on these measures would yield a more
comprehensive explanation of why the academically low achieving students
did not do as well as their more capable peers in the larger study. It was also
hypothesized that the protocols for the different performance tasks would
offer a richer picture of what makes proportional thinking difficult for all
students at the middle grades, particularly when the topic is reviewed in a
compressed manner as it was in this study. Results elucidate the
comparative differences in thinking between high, average, and low ability
students. They also document the limited ways in which students were able
to relate proportional reasoning to other kinds of mathematical topics.
Implications of the study regarding test preparation practices and the amount
of teaching time allocated to complex topic such as proportions at the middle
grade level will be discussed.
The first task (see Figure 1) required students to
identify which of two plants grew more during two
months and to provide a reason for their response. As
shown in Figure 2, the second problem required
students to identify two cards out of six that
represented the same ratio of the number of soccer
balls to skateboards. The final task (see Figure 3) was
a word problem involving a recipe for making cookies.
How did performance on individual performance assessments vary across problems and ability groups?
Plant 1:
2 months ago
8 in.
Plant 2:
2 months ago
As shown in Figure 4, significant effects were
found for the total score on the three tasks (p
< .001, d = 1.18), on the picture matching
task, (p < .01 d = 0.91), and recipe task (p <
.001, d = 1.49). Pretest-posttest differences on
the plant growth task were not significant (p =
0.26). For student ability level status, results
indicated large effect sizes for posttest scores
when compared with pretest scores on the
picture matching and recipe tasks for high and
average achieving students. In contrast, low
achieving students showed improvement from
pretest to posttest (d > 0.90) on all three
tasks.
Plant 1:
now
11 in.
Plant 2:
now
Figure 4. Means on individual performance assessments for total sample and by ability level. Lighter colors represent
pretest mean scores and darker colors represent posttest mean scores.
P lant G rowth
Total
Rec ipe
Participants were twenty-four (17 female) 7th grade students assigned to the
treatment group in the SBI-SM intervention in Jitendra et al. (2008). Six
students were chosen from each of four sections of seventh graders. Sections
represented classrooms of students tracked on the basis of their mathematics
performance: high ability (academic), average ability (applied), and low ability
(essential). Of the six students from each class, two high, two average, and
two low ability students were chosen based on their teachers’ judgments.
Three performance measures were employed as both pretest and posttest to
assess student understanding of proportional reasoning. An examiner read
each problem aloud and instructed the student to solve the problem and show
all work. Then examiners used both open-ended and specific questions to
probe students’ proportional thinking that focused on processes and
strategies (e.g., How did you figure out the problem?), conceptual
understanding (e.g., Show how you would represent this problem
mathematically), connections (e.g., Was this problem like any other you have
solved before?), and communication (e.g., How did you check your answer to
see if it was correct?) standards articulated in Buschman (2003).
High
Average
Low
T otal
High
Average
T otal
High
Average
Low
Pretest
2
1.67
0.83
2.33
1.92
1.33
2.67
2.5
1.83
Posttest
1.83
1.75
1.67
3.5
2.92
2.33
3.67
3.33
3
Pretest
1.54
2
1.67
0.83
1.88
2.33
1.92
1.33
2.38
2.67
2.5
1.83
Posttest
1.75
1.83
1.75
1.67
2.92
3.5
2.92
2.33
3.33
3.67
3.33
3
SBI-SM
Control
1.2
Raw Score
1
0.8
0.6
0.4
0.2
0
P retest
P osttest
W ord Proble m S olving Me asure
2
Raw Score
SBI-SM
Control
1
P retest
P osttest
Transfe r Me asure
28 in.
25 in.
How did students conceptualize each performance task?
As shown in Figure 5, many students
recognized the plant growth problem, but few
perceived it as a ratios and proportions task.
Explanations were often confusing or involved
additive relationships. Students tended to be
unfamiliar with the picture-matching task
during the pretest phase. Interestingly,
students were most likely to use some kind of
proportion vocabulary (e.g., ratio, proportion,
rate) to relate it to other problems that they
had seen previously. Many students
recognized the recipe task during the posttest
phase. There was a greater tendency to
describe the relationship between eggs and
cookies in a quantitative manner.
Figure 1. Pictorial representation of the percent
change problem
Figure 5. Student conceptualization of each performance task on four criteria. Light blue represents the
pretest phase and dark blue represents the posttest phase.
Figure 2. Sample of matched squares
with the same ratio
How did you figure out this problem?
A recipe that makes 40 sugar cookies calls for
two eggs.
a) How many cookies can be made with 1 egg?
Method
High
Average
Low
Rec ipe
P lant G rowth
Introduction
Recent research by Jitendra, Star, Starosta, and Sood (2008) examined the
impact of a 10-day intervention that taught seventh grade students multiple
strategies for solving proportion problems using schema based instruction
and self monitoring strategies (SBI-SM). The SBI-SM intervention was
contrasted with a control condition. Results of this study indicated significant
differences favoring the SBI-SM condition, though performance of low
achieving students in the SBI-SM condition was comparable to that of low
achieving students in the control condition. This study examined in greater
depth the impact of the SBI-SM intervention in Jitendra et al. (2008) on a
representative sample of students from different ability groups from the
intervention condition. Students were assessed using three different
performance assessment tasks. It was hypothesized that a detailed analysis
of the student performance on these measures would yield a more
comprehensive explanation of why the academically low achieving students
did not do as well as their more capable peers in the larger study. It was also
hypothesized that the protocols for the different performance tasks would
offer a richer picture of what makes proportional thinking difficult for all
students at the middle grades, particularly when the topic is reviewed in a
compressed manner as in this study.
High
Average
Low
High
Average
P ic ture M atc hingLow
P ic ture M atc hingLow
b) How many cookies can be made with 3 eggs?
Pretest
Highability
“I counted up the skateboards and went to the boxes and counted “Because I saw that this diagram had 1 soccer ball and 3
them up in their boxes and the amount of skateboards were each skateboards and this had twice as many.”
three apart and the amount of soccer balls were three apart.”
Averageability
“Uh well I would say it had 3 skateboards two soccer balls and
then in the other box we had three skateboards and six soccer
balls.”
“I added up the soccer balls and I tried to see if any of them had the
same number of soccer balls… For each picture, I would put the
number of soccer balls and the number of skateboards and
compared each one to the other ones.”
Lowability
“Because like I went back and counted how many soccer balls
were like in each, and none of them seemed to be equal to any of
them. Except like those one, the soccer balls they are equal in this
one, but if you look at the skateboards this one, the skateboards
weren’t like the same as like the one that actually was equal.”
“Because the skateboards there’s 4 and the soccer balls there’s 4
and then there’s 4 more skateboards.”
“There’s 3 skateboards, 3 soccer balls, and then 3 more
skateboards.”
Figure 3. Representation of the recipe problem
Measures
Four research assistants were trained to administer
and score each of the ratio and proportion
performance assessments. All data were collected
through 15-20 minute individual interviews in which
students were administered all three performance
assessments. A 5-point rubric was used to score each
assessment. Inter-rater reliability for this analysis was
.87, based on a sample of about 60 percent of the
problems used in the study. The final set of questions
concerned how students conceptualized each
performance task. Inter-rater reliability for this analysis
was 0.79.
Posttest
Figure 6. Excerpts of student responses for the picture-matching task at pretest and posttest
Conclusion
It was understandable that the plant task proved difficult for students before and after the intervention, since percent
change was part of the previous year’s instruction and not part of the 10-day intervention. However, there were
significant shifts in thinking on the matching and recipe performance tasks, which reflected the content of the 10-day
intervention. The way students thought about proportion problems in a general sense, that is, how problems related
to other kinds of math topics and the degree to which the problems were similar to other problems, informed
instruction.