enhancing third-grade students’ mathematical problem solving with

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Enhancing Third-Grade Students Mathematical Problem Solving WithSelf-Regulated Learning Strategies

Lynn S. Fuchs, Douglas Fuchs, Karin Prentice, Mindy Burch, Carol L. Hamlett,Rhoda Owen, and Katie SchroeterVanderbilt University

The authors assessed the contribution of self-regulated learning strategies (SRL), when combined with

problem-solving transfer instruction (L. S. Fuchs et al., 2003), on 3rd-graders mathematical problem

solving. SRL incorporated goal setting and self-evaluation. Problem-solving transfer instruction taught

problemsolution methods, the meaning oftransfer, and 4 superficial-problem features that change a

problem without altering its type or solution; it also prompted metacognitive awareness to transfer. The

authors contrasted the effectiveness of transfer plus SRL to the transfer treatment alone and to

teacher-designed instruction. Twenty-four 3rd-grade teachers, with 395 students, were assigned randomly

to conditions. Treatments were conducted for 16 weeks. Students were pre- and posttested on problem-

solving tests and responded to a posttreatment questionnaire tapping self-regulation processes. SRL

positively affected performance.

Mathematical problem solving, which requires students to apply

knowledge, skills, and strategies to novel problems, is a form of

transfer that can be difficult to effect (cf. Bransford & Schwartz,

1999; Mayer, Quilici, & Moreno, 1999). One method to promote

mathematical problem solving is to help students regulate their

learning; that is, to become more metacognitively, motivationally,

and behaviorally active in their own learning (cf. De Corte, Ver-

schaffel, & Eynde, 2000; Zimmerman, 1995). The capacity to

self-regulate learning is associated with self-efficacy and intrinsic

task interest (Schunk, 1986, 1996; Zimmerman, 1995) as well as

academic achievement (e.g., Pintrich & DeGroot, 1990; Zimmer-

man & Martinez-Pons, 1986, 1988). That self-regulation covarieswith academic achievement prompts the question of whether in-

struction designed to increase student behaviors associated with

self-regulated learning strategies (SRL) promotes learning.

Theoretically, the use of SRL should promote learning. Goal

setting serves a motivational function, which may mobilize and

sustain effort to achieve objectives (Cervone, 1993). When com-

bined with self-evaluation, goal setting serves an informational

purpose (Schunk, 1994), whereby students may monitor their

progress against a standard and thereby adjust their use of skills

and strategies to increase the probability of goal attainment (Gra-

ham & Harris, 1997), even as students come to appreciate the

value of applying those skills and strategies (Zimmerman, 1995).

For example, Zimmerman and Kitsantas (1999) showed that

self-monitoring the use of a strategy in a single session enhanced

self-efficacy, self-reaction beliefs, and writing skill. Schunk found

that self-referenced progress feedback provided across 3 days led

to higher perceptions of efficacy, persistence, and skill (Schunk,

1982); goal setting implemented across three sessions enhanced

students self efficacy and skill (Schunk, 1985); and self-

evaluation promoted motivation as well as achievement when used

with performance goals across 7 days (Schunk, 1996). Page-Voth

and Graham (1999) showed that seventh and eighth graders with

learning disabilities, who wrote essays in response to goals across

three sessions, produced longer compositions, with more support-

ing reasons, which were qualitatively better than essays written

without goals.

In experiments lasting 3 weeks, Graham and colleagues exam-

ined the effects of SRL when combined with cognitive strategy

instruction (Graham & Harris, 1989; Sawyer, Graham, & Harris,

1992). Sawyer et al. (1992), for example, provided fifth- and

sixth-grade students with learning disabilities with small-group

(2:1) instruction to recite and use a five-step strategy for writing

compositions (look at the story stimulus picture, let your mind be

free, write down the reminder for the three story parts, write down

story ideas for each part, and write your story). In one of twotreatment groups, students also set goals for the number of story

elements in compositions and assessed or graphed the number and

kind of story elements in their stories. Both treatments produced

greater schematic structure posttest scores compared with a prac-

tice control condition, but only the combined treatment, which

included SRL, resulted in greater generalization to performance in

classrooms.

Although experimental work has assessed the effects of SRL in

mathematics (e.g., Schunk, 1982, 1985, 1996), studies are largely

restricted to performance on discrete computational skills, with

limited generalizability to problem solving. This is unfortunate

Lynn S. Fuchs, Douglas Fuchs, Karin Prentice, Mindy Burch, Carol L.

Hamlett, Rhoda Owen, and Katie Schroeter, Department of Special Edu-

cation, Vanderbilt University.

This research was supported in part by U.S. Department of Education,

Office of Special Education Programs Grant H324V980001 and National

Institute of Child Health and Human Development Core Grant HD15052 to

Vanderbilt University. Statements do not reflect the position or policy of

these agencies, and no official endorsement by them should be inferred.

We acknowledge the contribution of and express appreciation to Steve

Graham and Karen Harris in designing the SRL treatment.

Correspondence concerning this article should be addressed to Lynn S.

Fuchs, Box 328 Peabody, Vanderbilt University, Nashville, Tennessee

37203. E-mail: [email protected]

Journal of Educational Psychology Copyright 2003 by the American Psychological Association, Inc.2003, Vol. 95, No. 2, 306 315 0022-0663/03/$12.00 DOI: 10.1037/0022-0663.95.2.306

306


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because mathematics education reform over the past 15 years has

prompted schools to deemphasize computation and focus more on

problem-solving capacity with performance assessments that pose

real-world problem-solving dilemmas and require students to de-

velop solutions involving the application of multiple skills (e.g.,

De Corte et al., 2000; Resnick & Resnick, 1992; Rothman, 1995).

In fact, SRL may be especially relevant for complex problemsolving, which requires metacognition and perseverance in the face

of challenge (De Corte et al., 2000). Causal-comparative studies

(e.g., Lester & Garofalo, 1982; Schoenfeld, 1992; Silver, Branca,

& Adams, 1980) illustrate how self-regulation differs between

weak and skilled problem solvers: Experts spend more time ana-

lyzing problems before initiating solutions, reflect more frequently

on their problem solving, and alter their approach more flexibly.

To examine whether instructional environments might be designed

to foster self-regulation in the service of mathematical problem

solving, De Corte et al. (2000) described four design experi-

ments (Cognition and Technology Group at Vanderbilt, 1997;

Lester, Garofalo, & Kroll, 1989; Schoenfeld, 1985, 1992; Ver-

schaffel et al., 1999; all cited in De Corte et al., 2000) with

promising results. Those design experiments, however, incorpo-

rated multiple innovative principles, with varying levels of exper-

imental control. So, it was not possible to determine whether

effects were attributable to SRL.

In the present study, we used an experimental design to separate

the effects of SRL, including goal setting and self-evaluation, on

mathematical problem solving. We investigated effects on a range

of mathematical problem-solving tasks and on SRL processes, as

assessed with a student questionnaire tapping self-efficacy, goal

orientation, self-monitoring, and effort. Also, we extended the

external validity of previous experiments by relying on whole-

class instruction delivered in general education for 16 weeks and

assessing effects for high-, average-, and low-achieving (HA, AA,

and LA, respectively) students as well as those with disabilities.We examined the contribution of SRL when combined with

problem-solving transfer instruction (Fuchs et al., 2003). Consis-

tent with Salomon and Perkinss (1989) framework, mathematical

problem solving requires metacognition (i.e., decision-making pro-

cesses that regulate the selection of various forms of knowledge;

Zimmerman, 1989). At the same time, metacognition is a critical

process for self-regulation, whereby self-regulated learners set

goals, self-monitor, and self-evaluate their performance (cf. Zim-

merman, 1990). So, one approach for strengthening the metacog-

nitive component of problem-solving treatment is to incorporate

the SRL of goal setting and self-monitoring.

With the present studys SRL treatment, students set goals for

their performance on independent practice tasks during instruc-

tional sessions: Students set goals for their performance, scored

performance in terms of the process of their work and the accuracy

of their answer, and graphed scores on individual graphs. To

prompt transfer, we had students score and graph completion of

homework assignments on a class graph; students also identified

opportunities to apply skills outside of instructional sessions, dis-

cussed those opportunities with partners, reported them to the

class, and graphed their reports on a class graph. We deemed class

graphing and reporting to be a part of SRL because graphing

provided a socially mediated method to help students monitor

progress toward completing homework and identifying transfer

opportunities; reporting offered a socially mediated forum for

increasing motivation to exert effort in completing homework and

identifying transfer opportunities. We were interested in the con-

tribution of SRL on the mathematical problem solving of students

with varying achievement histories because of the possibility that

SRL effects may be mediated by prior achievement status. In early

research, children with cognitive deficiencies experienced diffi-

culty determining how well they used strategies (Borkowski &Buechel, 1983) and failed to make accurate competency assess-

ments (Licht & Kistner, 1986). As Schunk (1996) suggested,

because AA and HA students assess their learning progress more

reliably than remedial students, SRL effects may be weaker for

low achievers. Our design addressed this possibility.

Although the present study shares measures and several other

procedures with Fuchs et al. (2003), the purpose of the two studies

differed. Fuchs et al. assessed the effects of an explicit approach to

teaching transfer; the present study, by contrast, examined the

contribution of SRL to mathematical problem solving. It is also

important to note that the two studies were conducted separately,

during different academic years, with different teacher and student

participants and with teachers in the present study participating

more in the delivery of lessons.

Method

Participants

From six schools in a southeastern urban school district, 24 third-grade

teachers volunteered to participate. Stratifying so that each condition was

represented approximately equally in each school, we randomly assigned

teachers to three conditions (8 per condition): control (teacher-designed

instruction informed by the basal), transfer (Fuchs et al.s, 2003, solution

plus transfer, with some modifications described in the following para-

graphs), and transfer plus SRL. Teacher groups were comparable on

gender, age, education, and years teaching. Students were the 395 children

in these classrooms who were present for each pre- and posttest. Studentgroups were comparable on gender, reduced or free lunch, race, special-

education status, and English-as-a-second-language status. Also, the 395

children in the complete data set were demographically comparable with

the remaining 58 pupils who were absent on one or more pre- or posttest.

At the beginning of the study, on the basis of classroom observations and

scores from the preceding years district accountability testing, teachers

designated childrens initial mathematics achievement status as high, av-

erage, or low (i.e., HA, AA, or LA, respectively). Students were distributed

across HA, AA, and LA status, respectively, as follows: in control ( n

120), 26 (22%), 60 (50%), and 34 (28%); in transfer (n 138), 32

(23%), 68 (49%), and 38 (27%); and in transfer plus SRL ( n 137), 32

(23%), 71 (52%), and 34 (25%).

TreatmentThe 24 teachers followed the districts curriculum, which required them

to address the same content each week of the year. We chose our four

problem types from this curriculum to ensure that control-group students

had instruction relevant to the study. See Fuchs et al. (2003) for informa-

tion on the four problem types and on the basal math program on which

control teachers relied almost entirely.

We supplemented the basalcontrol treatment with two experimental

conditions. Table 1 details which sessions were allocated to which activ-

ities for the transfer and the transfer-plus-SRL treatments. This sequence of

activities (i.e., Sessions 1 6) recurred for each of five units, but the content

for the activities varied according to the problem-solving unit. So, treat-

ment was 30 sessions (6 sessions 5 units) plus 2 cumulative review

307MATHEMATICAL PROBLEM SOLVING


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sessions delivered the week after winter break, for a total of 32 sessions.

Two sessions occurred each week. The transfer treatment comprised teach-

ing rules for problem solution, teaching for transfer, and cumulative

review. Differences between the Fuchs et al. (2003) solution-plus-transfer

and the present studys transfer treatment were as follows. First, we added

a 3-week introductory unit on basic math problem-solving information

(making sure answers made sense; lining up numbers from text to perform

math operations; checking computation; labeling work with words, mon-

etary signs, and mathematics symbols). Second, we dedicated one unit

(instead of two) to the shopping-list problem type. Third, at the end of each

session, students completed one problem independently and checked work

against an answer key. Fourth, students were assigned a problem for

homework, which they returned the next morning to the classroom home-

work collector (a competent classmate).

In the transfer-plus-SRL treatment, SRL components were incorporated

as follows. First, after students completed working the independent prob-

lem of each session, they also scored their independent problem using an

answer key specific to the units problem structure, which provided credit

for the process of the work and the accuracy of the answer. Second,

students charted their daily scores on an individual thermometer that went

from 0 to the maximum score for that problem type. For each unit, students

kept their chart in a personal folder. The chart showed 4 5 consecutive

thermometers, 1 for each of the last 4 5 sessions of the unit. Third, at the

beginning of the next session, students inspected their charts, were re-

minded that they wanted to beat their previous score(s) (or again achieve

the maximum score), and set a goal to beat their highest score on that days

independent problem. Fourth, using an answer key, students scored home-

work prior to submitting it to the homework collector. Fifth, at the begin-

ning of Sessions 1, 3, 4, 5, and 6 of each unit, students reported to the class

examples of how they had transferred the units problem structure to

another part of the school day or outside of school. The sixth activity

involved a class graph, on which the teacher recorded (a) the number of

students who had submitted homework to the homework collector and (b)

the number of pairs reporting a transfer event. In these ways, SRL incor-

porated goal setting and self-assessment referenced to the content of

instructional sessions, including acquisition of problemsolution rules as

well as transfer.

In each experimental condition, research assistants taught the first

problemsolution lesson and the first transfer lesson of each unit; teachers

were always present. Teachers taught the remaining sessions; typically but

not always, a research assistant was present. Each of five research assis-

tants had responsibility for classes in every condition. All sessions were

scripted to ensure consistency of information; however, to permit natural

teaching styles, we had the research assistants study, not read, the scripts.

To ensure comparable mathematics instructional time across conditions,

we scheduled the experimental sessions to occur during the mathematics

instructional block. At the end of the study, teachers reported the number

of minutes per week they spent on math, including this project. Means for

the control, transfer, and transfer-plus-SRL conditions, respectively, were

275.00 (SD 38.17), 263.75 (SD 44.65), and 276.88 (SD 36.82),F(2,

21) 0.15,ns.

Fidelity of treatment was measured as in Fuchs et al. (2003). In each

treatment group, audiotapes of research assistants, classes, units, and lesson

types were sampled equitably, with no tape reviewed more than once. In

the transfer condition, the percentage of key information points addressed

on the tapes averaged 96.90 (SD 9.56) for teachers and 95.70

(SD 4.14) for research assistants; in transfer plus SRL, the percentage of

key information points averaged 96.00 (SD 6.91) for teachers and 94.70

(SD 4.75) for research assistants.

Measures

We used four measures: three transfer measures and a student question-

naire of self-regulation processes. For information on the transfer mea-

sures, see Fuchs et al. (2003). Interscorer agreement computed on 20% of

the protocols by two independent,blindscorers, was .980 at pretreatment

and .968 at posttreatment (number of agreements divided by number of

items) for immediate transfer; .980 at pretreatment and .954 at posttreat-

ment for near transfer; and .950 at pretreatment and .940 at posttreatment

for far transfer. We scored the immediate- and near-transfer measures as

Table 1

Activities for Each Units Six Sessions by Treatment

Activity

Treatment

Transfera Transfera SRL

Solutioninstruction

Transferinstruction

Cumulativereview

Solutioninstruction

Transferinstruction

Cumulativereview

Individual graph inspected; days goal set 2,b 3, 4 6 4, 6Worked examples with explicit explanations 14 56 4, 6 14 56 4, 6Dyadic practice 1,b 24 6 4, 6 1,b 24 6 4, 6Independent problem

Completed 24 6 4, 6 24 6 4, 6Checked against key 24 6 4, 6 24 6 4, 6Scored against key 24 6 4, 6Score graphed on individual chart 24 6 4, 6

HomeworkAssigned 24 56 24 56Collected 1,c 3, 4 56 1,c 3, 4 56Scored 1,c 3, 4 56Number scored graphed on class chart 1,c 3, 4 56

Dyadic debriefing on transfer events:

number graphed on class chart

1,c 3, 4 56

Note. Sessions 1 6 occurred for each unit (15). SRL self-regulation learning strategies.a In Fuchs et al. (2003), the effects of this transfer treatment were assessed (labeled full solution transfer), with solution rules, transfer instruction, andcumulative review. b Depending on unit, Sessions 1 and 5 contained the most new instructional content, precluding independent work and precludingdyadic practice in Session 5 (and on occasion, in Session 1). c Problem from previous unit.

308 FUCHS ET AL.


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product scores (i.e., answers) and as process scores (i.e., methods revealed

in students work). Because results were highly similar for both types of

scores and because of space constraints, we only report product scores here

(for process data or a copy of the measures, contact Lynn S. Fuchs). As

with Fuchs et al., research assistants delivered a 45-min test-wiseness

lesson immediately before pre- and posttesting in all treatment groups.

To assess self-regulation processes, we designed a questionnaire titled,

What Do You Think? with four statements assessing self-regulationprocesses. The first two statements, I know how to transfer skills I learn

to new kinds of math problems and I learned a lot about math problem

solving this year,assessed self-efficacy. The next statement, When I do

math, I think about whether my work is getting better, examined goal

orientation and self-monitoring. The last statement, I worked hard this

year so I could get better in math, assessed effort.

Below each statement, were three response options: true, kind of true,

and not true. Research assistants read the following directions to the

students:

I have some questions Id like you to answer. I will read a sentence to

you. After I read the sentence, you mark your answer right below that

sentence. When youre done marking your answer, put your pencil

down. Do not work ahead. Wait until I read the next sentence before

moving on. As I read the sentence, listen carefully. Think about howtrue that sentence is for you. There are no right or wrong answers.

Your answer tells how you feel. No one will see your answers except

me: not your classmates, not your teacher, not your parents. So, be

honest.

The research assistant then read a practice item, I am a good swimmer:

true, kind of true, not true, and said

Is this sentence true for you? Are you a good swimmer? If this

sentence is true for you, you are a good swimmer, circle true. If this

sentence is kind of true for you, circle kind of true. If this sentence is

not true for you, youre not a good swimmer, circle not true. You can

see that the same answer is not right for everybody. You have to circle

the answer thats right for you. Any questions? OK, here are some

more sentences.

Interscorer agreement was 100%. Responses were coded as true 1,kind

of true 2, and not true 3, making low scores desirable.

Data Collection

See Fuchs et al. (2003) for pre- and posttreatment data collection

procedures on the transfer measures. Student questionnaire data were

collected by trained research assistants at posttreatment in a whole-class

arrangement. Research assistants read each item and ensured that all

students had circled a response before proceeding to the next item. To

avoid familiar research assistants prompting awareness that experimental

conditions might be applicable, research assistants did not collect data in

classrooms where they had delivered lessons.

Data Analysis

Using teacher as the unit of analysis, we conducted a two-factor mixed

model analysis of variance (ANOVA) on each transfer measure and on

each item on the student questionnaire. Condition was the between-

teachers variable; initial achievement status (HA, AA, LA) was the within-

teacher variable. These analyses were applied to pretreatment transfer

scores to examine treatment group comparability, to change transfer scores

to investigate treatment efficacy in promoting growth, and then to post-

treatment self-regulation scores. (Results were analogous using change

scores and analysis of covariance. We opted to report change scores

because their interpretation is more straightforward and because they are

equally acceptable for analyzing two-wave data, each presenting a different

set of problems.) To evaluate pairwise comparisons for significant effects,

we used the Fisher least significant difference (LSD) post hoc procedure

(Seaman, Levin, & Serlin, 1991). For students with disabilities, we ran a

one-way ANOVA on each measure using condition as the factor and using

student as the unit of analysis because of the small sample size and the

uneven distribution of students across classrooms.

To estimate the practical significance of effects, we computed effectsizes (ESs): on transfer measures, by subtracting the difference between

improvement means and then dividing by the pooled standard deviation of

the improvement/square root of 2(1rxy

) (Glass, McGaw, & Smith, 1981);

on questionnaire items, by subtracting the difference between the means

and then dividing by the pooled standard deviation (Hedges & Olkin,

1985).

Results

Table 2 contains means and standard deviations for pretreat-

ment, posttreatment, and improvement scores on the immediate-,

near-, and far-transfer measures; Table 3 includes ESs for the

treatment effects on the improvement scores; Table 4 has means

and standard deviations for student questionnaire items; and Table5 includes demographics and performance variables for students

with disabilities.

Pretreatment Differences Among Treatment Groups

Groups were comparable prior to the study, as manifested by the

lack of significant main effects for condition and interactions

between condition and students initial treatment status on pre-

treatment immediate-, near-, or far-transfer measures. The F(2, 21)

values for the condition main effect on the three respective mea-

sures were 1.46, 0.86, and 0.33; the F(4, 42) values for the

interaction effect on the three respective measures were 0.34, 0.74,

and 0.47. There were significant main effects for students initial

achievement status at pretreatment: For the three respective mea-sures,F(2, 42) values were 63.00, 51.45, and 62.86 (all ps .01).

LSD (using a critical p value of .016) follow-up tests showed that

for each measure, HA students scored higher than AA students,

who in turn scored higher than LA students. These effects pertain

across conditions and, therefore, do not threaten the validity of the

study. Rather, they provide support for the teachersclassifications

of students into initial achievement groups.

Differential Student Learning as a Function of Treatment

Student improvement varied as a function of condition. For

immediate transfer, the condition main effect was significant, F(2,

21) 187.76, p .001. With Fishers LSD post hoc procedure,

the control groups improvement was less than that of the transfer

group, which in turn was inferior to that of the transfer-plus-SRL

group. This main effect was not mediated by students initial

treatment status, F(4, 42) 0.52, ns. See Table 2 for means and

standard deviations; see Table 3 for ESs.

For near transfer, the significant condition main effect, F(2,

21) 46.76,p .001, was mediated by a significant Condition

Initial Achievement Status interaction,F(4, 42) 5.05, p .01.

With Fishers LSD post hoc procedure, the HA control group s

improvement was less than that of the HA transfer group, which in

turn was inferior to that of the HA transfer-plus-SRL group; by

contrast, for AA and LA students, both experimental groups out-



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performed the control group, but the improvement of the transfer

and transfer-plus-SRL groups was comparable. See Table 2 for

means and standard deviations; see Table 3 for ESs.

For far transfer, the condition main effect again was significant,

F(2, 21) 4.03, p .05. With Fishers LSD post hoc procedure,

the transfer-plus-SRL groups outperformed the control group;

however, the improvement of the transfer group was comparable

with control and with transfer plus SRL. This main effect was not

mediated by students initial treatment status, F(4, 42) 0.18, ns.

See Table 2 for means and standard deviations; see Table 3 for

ESs.

(As with pretreatment performance, we found a significant main

effect for initial achievement status; for the three respective mea-sures, F[2, 42] 31.81, 30.18, and 17.23 [all ps .01]. Across

conditions, LSD [using a critical p value of .016] follow-up tests

showed that (a) on acquisition, HA students outgrew AA students,

whose improvement was comparable with LA students, and (b) on

near and far transfer, HA students outperformed AA students, who

in turn grew more than LA students. These effects, which pertain

across conditions, do not threaten the studys validity.)

Self-Regulation Processes as a Function of Treatment

Results for the student questionnaire are shown in Table 4 (low

scores are desirable). On all four questions, we found a significant

effect for condition, which was not mediated by students initial

achievement status. With Fishers LSD post hoc procedure, for

three items,I learned a lot about math problem solving this year,

I worked hard this year so I could get better in math, and When

I do math, I think about whether my work is getting better,

control- and transfer-group responses were comparable but higher

than those of the transfer-plus-SRL group. For I know how to

transfer skills I learn to new kinds of math problems, control-

group responses were higher than those of either treatment groups,

whose scores were comparable with each other. ESs comparing

control against transfer, control against transfer plus SRL, and

transfer against transfer plus SRL, respectively, were as follows:

for I learned a lot about math problem solving thisTable2

Per

formance

byMeasure

,InitialAch

ievement

Status

,Trial

,an

dTreatment

Measure

Initial

status

Trialtreatment

Pre

Post

Improve

Control

Transfer

Transfer

SRL

Control

Transfer

Transfer

SRL

Control

Transfer

Transfer

SRL

M

S

D

M

SD

M

SD

M

SD

M

SD

M

SD

M

SD

M

SD

M

SD

Immediate

High

10.5

2

3

.99

11.6

6

2.7

2

9.9

4

3.6

2

21.2

4

5.2

4

38.3

8

4.7

1

41.06

3.9

5

10.7

1

2.2

7

26.7

2

4.59

31.1

2

1.6

3

Average

5.9

9

1

.55

7.4

7

3.2

8

5.7

9

1.4

7

13.9

6

3.3

6

34.0

2

5.5

5

36.42

4.4

5

7.9

7

2.8

6

26.5

5

3.76

30.6

3

3.5

5

Low

4.2

2

2

.12

4.6

8

4.0

7

2.3

3

1.8

6

6.7

3

2.9

5

23.9

9

7.3

1

24.01

7.9

2

2.5

1

2.5

7

19.3

0

4.03

21.6

8

7.2

9

Across

6.9

1

1

.82

7.9

4

3.0

0

6.0

2

1.6

7

13.9

8

2.1

0

32.1

3

4.2

6

33.83

3.3

6

7.0

6

1.6

2

24.1

9

2.55

27.8

1

2.5

6

Near

High

8.5

5

3

.82

8.9

3

3.4

9

7.0

6

2.8

7

14.4

5

5.4

0

29.3

7

6.4

9

34.19

7.7

8

5.9

0

4.0

5

20.4

4

3.73

27.1

3

6.9

7

Average

3.4

4

1

.77

4.6

6

2.3

8

4.3

1

2.3

7

8.6

2

3.5

4

21.6

8

6.1

3

24.44

6.0

0

5.1

8

3.1

6

17.0

2

4.15

20.1

3

5.5

0

Low

2.5

0

1

.31

2.9

4

3.0

2

1.6

6

0.7

8

6.2

3

2.5

7

14.1

8

6.1

0

14.72

5.0

9

3.7

3

1.8

3

11.2

3

3.85

13.0

6

4.7

5

Across

4.8

3

1

.56

5.5

1

2.4

7

4.3

4

1.0

1

9.7

7

3.3

4

21.7

4

4.6

5

24.45

4.6

8

4.9

4

2.8

8

16.2

3

2.53

20.1

1

4.1

7

Far

High

19.2

3

9

.34

20.8

5

7.2

0

19.9

2

4.5

0

30.5

0

9.1

2

37.2

2

13.2

5

38.88

9.0

5

11.2

7

6.7

8

16.3

6

8.44

18.9

6

8.3

4

Average

9.7

8

3

.83

11.1

6

6.3

0

10.5

2

4.0

8

15.6

6

6.2

1

22.4

7

11.2

0

23.17

9.9

6

5.8

8

5.3

2

11.3

1

6.60

12.6

5

7.4

0

Low

6.7

8

3

.05

7.5

3

6.4

8

4.0

0

4.7

8

10.7

8

4.0

1

14.6

8

3.6

7

12.92

5.8

8

3.9

2

2.6

6

7.6

1

4.70

8.9

1

4.8

8

Across

11.9

5

3

.77

13.1

8

5.6

9

11.4

8

2.5

5

18.9

5

3.7

7

24.7

9

8.0

5

24.99

7.2

1

7.0

2

3.5

6

11.6

1

3.93

13.5

1

6.1

6

Note.

SRL

self-regulatedlearningstrategies;Pre

pretest;Post

posttest;Imp

rove

posttestminuspretest.

Table 3

Improvement Effect Sizes by Measure, Initial Achievement

Status, and Treatment

Measure

Initial

status

Contrast

Control vs.

transfer

Control vs.

transfer SRL

Transfer vs.

transfer SRL

Immediate High 1.91 2.29 1.11Average 1.78 2.47 0.92Low 1.83 2.68 0.33Across 1.98 2.81 1.05

Near High 1.75 2.40 1.04Average 1.22 1.81 0.55Low 1.24 2.18 0.35Across 1.41 2.43 0.89

Far High 0.47 0.87 0.25Average 0.54 0.81 0.12Low 0.69 1.17 0.21Across 0.72 1.18 0.27

Note. SRL self-regulated learning strategies.

310 FUCHS ET AL.


6/10

Table4

Stu

dent

Questionna

ire

Items

byInit

ialAch

ievement

Statusan

dTreatment

Item

Initial

status

Treatment

LSDtreat

LSDinitial

Control

Transfer

Transfer

SRL

Across

F

M

SD

M

SD

M

SD

M

SD

Initiala

Treat

b

Treat

Initialc

IknowhowtotransferskillsIlearn

High

1.3

6

0.3

5

1.3

0

0.3

6

1.1

1

0.2

4

1.2

2

0.3

2

0.7

0

3.8

1**

0.5

1

C

T

T

SRL

tonewkindsofmathproblems.

Average

1.2

0

0.1

0

1.1

5

0.1

8

1.1

1

0.1

9

1.1

5

0.1

6

Low

1.3

2

0.1

9

1.2

2

0.1

8

1.0

8

0.1

3

1.2

1

0.1

9

Across

1.2

6

0.1

8

1.2

2

0.1

6

1.1

0

0.1

0

1.1

9

0.1

6

Ilearnedalotaboutmathproblem

High

1.5

9

0.4

1

1.3

0

0.4

1

1.1

1

0.2

1

1.3

3

0.4

0

3.0

5*

13.3

1***

1.3

2

C

T

T

SRL

H

A

L

solvingthisyear.

Average

1.6

9

0.4

0

1.1

3

0.1

9

1.3

8

0.3

3

1.4

0

0.3

9

Low

1.9

0

0.3

4

1.2

7

0.2

5

1.4

9

0.3

2

1.5

6

0.4

0

Across

1.7

3

0.2

0

1.2

3

0.2

2

1.3

3

0.1

8

1.4

3

0.2

9

WhenIdomath,

Ithinkabout

High

1.7

4

0.4

9

1.6

3

0.1

7

1.3

6

0.2

9

1.5

7

0.3

7

1.4

5

3.8

8**

0.2

1

C

T

T

SRL

whethermyworkisgettingbetter.

Average

1.5

9

0.4

4

1.4

7

0.3

3

1.3

5

0.2

3

1.4

7

0.3

4

Low

1.7

6

0.2

4

1.6

0

0.2

3

1.4

6

0.3

1

1.6

1

0.2

8

Across

1.6

9

0.2

7

1.5

7

0.1

7

1.3

9

0.2

0

1.5

5

0.2

4

IworkedhardthisyearsoIcould

High

1.3

6

0.4

7

1.5

1

0.2

5

1.1

5

0.2

5

1.3

4

0.3

6

1.2

4

5.5

2**

0.5

6

C

T

T

SRL

getbetterinmath.

Average

1.3

4

0.3

1

1.2

8

0.4

1

1.1

1

0.1

5

1.2

4

0.3

1

Low

1.5

4

0.3

0

1.4

5

0.3

3

1.1

4

0.2

2

1.3

8

0.3

3

Across

1.4

1

0.1

4

1.4

1

0.2

7

1.1

4

0.1

4

1.3

2

0.2

3

Note.

Lowscoresaredesirable.

SRL

self-regulatedlearningstrategies;LSD

leastsignificantdifference;C

controlcondition;T

transfercondition;H

high-ach

ievingstudents;A

average-achievingstudents;L

low-achievingstudents.

a

Initial

students

initialachievement

status(df

2,

42).b

Treat

treatment(df

2,

21).c

Treatment

Students

InitialAch

ievementStatus(df

4,

42).

*p

.058.

**p

.05.

***p.0

01.



7/10

year, 0.23, 1.14, and 0.92; for I know how to transfer skills I

learn to new kinds of math problems, 2.38, 2.26, and 0.50; for

When I do math, I think about whether my work is getting

better,0.54, 1.30, and 1.20; and for I worked hard this year so I

could get better in math,0.00, 1.93, and 1.35. (We found a nearly

significant main effect for students initial achievement status onthe question concerning knowledge about transferring skills; with

LSD follow-up, across conditions, HA and AA students scored

comparably but lower than LA students.)

What About Students With Disabilities?

As shown in Table 5, treatment groups were comparable on

demographics. Groups were also comparable on pretreatment

scores: F(2, 37) 1.35 for immediate transfer, 0.93 for near

transfer, and 0.89 for far transfer. By contrast, on improvement

scores, treatment effects were obtained for immediate transfer,

F(2, 37) 6.47, p .01 (Fisher LSD post hoc procedure:

control transfer transfer plus SRL) and for near transfer,F(2,

37) 3.81, p .05 (Fisher LSD post hoc procedure: control

transfer plus SRL; transfer comparable with both groups). On far

transfer, the treatment effect approached significance, F(2,

37) 2.35 (with LSD follow-up, the improvement of the control

group was inferior to that of the transfer-plus-SRL group). Forimmediate-, near-, and far-transfer measures, respectively, the ES

for transfer versus control was 1.07, 0.51, and 0. 24; for transfer

plus SRL versus control, 1.43, 0.95, and 0.58; for transfer versus

transfer plus SRL, 0.23, 0.25, and 0.43.

Discussion

Results strengthen previous work (Fuchs et al., 2003) showing

that mathematical problem solving may be strengthened with

explicit transfer instruction that (a) broadens the categories by

which students group problems requiring the same solution meth-

ods (i.e., promotes a higher level of abstraction) and (b) prompts

Table 5

Demographics and Performance for Students With Disabilities by Treatment

Variable

Treatment

Control (n 12) Transfer (n 13) Transfer SRL (n 15)

n % M SD n % M SD n % M SD

DemographicGender (female) 3 25 3 23 5 33Free or reduced lunch 11 92 10 77 9 60Race (of color) 8 67 8 61 9 60Disability

LD 8 67 9 69 14 93MMR 1 8 0 0 0 0BD 0 0 0 0 1 7Speech 3 25 4 31 0 0

Math IEP 7 58 7 54 12 80Reading IEP 8 67 8 61 14 93Class behavior acceptable 5 42 10 77 5 33

Occasional problem 3 25 1 8 5 33Frequent problem 4 33 2 15 5 33

Reading status

High 0 0 1 8 0 0Average 2 17 2 15 3 20Low 10 83 10 77 12 80

Math statusHigh 1 8 1 8 1 7Average 0 0 4 31 3 20Low 11 92 8 61 11 73

ESL 1 8 0 0 2 13Performance

ImmediatePre 2.50 2.81 2.46 3.77 1.00 1.25Post 2.00 4.61 7.54 6.01 7.20 5.91Improve 0.50 3.00 5.08 5.53 6.20 5.83

NearPre 2.50 3.24 2.40 2.87 1.28 1.71Post 2.83 3.07 4.60 4.54 4.40 3.32

Improve 0.33 2.66 2.19 2.59 3.12 2.63FarPre 4.71 3.90 5.69 4.99 3.40 4.54Post 8.00 5.19 10.31 6.59 12.37 13.12Improve 3.29 5.91 4.61 4.76 8.97 9.49

Note. SRL self-regulated learning strategies; LD learning disability; MMR mildly mentally retarded; BD behavior disorder; IEP individualeducation plan; ESL English as a second language; Pre pretest; Post posttest; Improve posttest minus pretest.

312 FUCHS ET AL.


8/10

students to search novel problems for these broad categories (i.e.,

increases metacognition). On the immediate- and near-transfer

problem-solving measures, students in the problem-solving trans-

fer treatment reliably outgrew those in the control group. ESs were

large, regardless of students initial achievement status (1.91

and 1.98 for HA, 1.22 and 1.78 for AA, and 1.24 and 1.83 for LA)

and similar to those reported by Fuchs et al. On the far-transfermeasure, however, effects did not reliably favor the problem-

solving transfer treatment over the control group, as Fuchs et al.

had found (even though ESs were moderate to large: 0.47 for

HA, 0.54 for AA, and 0.69 for LA).

As results on the far-transfer measure suggest, a need exists to

enhance the strength of the problem-solving transfer treatment.

SRL represents one avenue to accomplish that goal due to its

capacity to strengthen the metacognitive value of the problem-

solving transfer treatment and to increase perseverance in the face

of challenge (Zimmerman, 1995). In fact, the combination of the

problem-solving transfer treatment and SRL promoted reliably

stronger improvement compared with the control group. ESs ex-

ceeded 2.00 standard deviations on immediate transfer, ranged

from 1.81 to 2.40 on near transfer, and fell between 0.81 and 1.17

on far transfer. So, whereas the problem-solving transfer treatment

alone failed to promote reliable effects on the far-transfer measure

(the most novel, and therefore truest, measure of mathematical

problem solving in this study), the combination of problem-solving

transfer and SRL succeeded in effecting this challenging outcome.

Of course, the study design also permits us to estimate the

specific contribution of SRL by comparing the improvement of

students who received the problem-solving transfer treatment com-

bined with SRL with those who received the problem-solving

transfer treatment alone. Results were mixed. On immediate trans-

fer, the contribution of SRL was evident. Children in the combined

treatment reliably outgrew those in the problem-solving treatment

without SRL. Interestingly, although the interaction between con-dition and students initial achievement status was not significant,

ESs were larger for HA and AA students than for LA students.

This suggests the possibility of differential efficacy for SRL,

which Schunk (1996) hypothesized on the basis of research show-

ing that low-performing students may not monitor their perfor-

mance accurately (Borkowski & Buechel, 1983; Licht & Kistner,

1986). Moreover, this suggestive pattern on the immediate-transfer

measure was evident on the near-transfer task, where an interaction

between condition and initial achievement status was significant:

Although HA students in the combined treatment reliably outgrew

those in the problem-solving transfer treatment alone, with an ES

exceeding 1.00 standard deviation, the growth of the AA and LA

students was not statistically significant, with moderate ESs

of 0.55 and 0.35. Finally, on the far-transfer measure, distinctions

between the two experimental treatments were unreliable and

small for all three achievement groups, with ESs ranging be-

tween 0.12 and 0.25. Consequently, as the transfer demands in-

creased across the range of problem-solving measures, the specific

contribution of SRL became less clear.

On the one hand, the combined treatment with SRL promoted

far transfer when the problem-solving transfer treatment alone

failed to effect this challenging outcome. On the other hand, the

specific contribution of SRL, as revealed by comparing the two

experimental groups, was clear only on immediate transfer and, for

HA students, on near transfer. It is therefore instructive to examine

findings on the student questionnaire, which tapped SRL pro-

cesses. As results suggested, the explanation for the superior

growth of the combined treatment may reside with SRL. On three

of four questions assessing self-efficacy, goal orientation, self-

monitoring, and effort, students in the combined treatment scored

better (i.e., lower) than those in the problem-solving transfer

treatment without SRL (and better than those in the control group).For I learned a lot about math problem solving this year, an

index of self-efficacy, the ES comparing the combined treatment

with the problem-solving transfer treatment without SRL was 0.92.

For When I do math, I think about whether my work is getting

better, a question designed to tap goal orientation and self-

monitoring, the ES was 1.20. Moreover, student effort was greater

in the combined condition with SRL, with students in the com-

bined treatment agreeing more strongly with the statement, I

worked hard this year so I could get better in math, compared

with students in the problem-solving treatment alone (ES 1.35).

In this way, SRL may have provided the key mechanism by which

the effects of the combined treatment were realized.

With respect to students with disabilities, a group of children

who receive most of their instruction in regular classrooms (U.S.

Department of Education, 1999) and for whom transfer effects are

most difficult to effect (e.g., White, 1984), both treatment groups

grew comparable amounts on immediate transfer and improved

more than the control group. ESs were large: for transfer versus

control, 1.07; for transfer plus SRL versus control, 1.43. Moreover,

although effects for the combined treatment on measures with

greater transfer challenge failed to achieve statistical significance

for the small sample of students with disabilities, the ESs of 0.95

on near transfer and 0.58 on far transfer are notable. In fact, the ES

for the combined treatment of 0.58, with lessons delivered to the

whole class, are almost identical to the ES reported on the same

far-transfer measure for small-group tutoring that incorporated the

problem-solving transfer treatment without SRL (Fuchs, Fuchs,Hamlett, & Appleton, 2002). So, even for this lowest achieving

group of students, who may experience difficulty setting realistic

goals (Robbins & Harway, 1977; Tollefson, Tracy, Johnsen, Buen-

ning, & Farmer, 1982) and monitoring performance accurately

(e.g., Borkowski & Buechel, 1983; Licht & Kistner, 1986), the

promise of SRL is great. In the future, we might explore the power

of combining SRL with small-group service delivery to increase

the magnitude of effects documented in the present study.

In sum, instruction designed to increase student behaviors as-

sociated with SRL promotes SRL processes as well as learning.

The SRL literature is extended in four ways. First, we experimen-

tally established effects on mathematical problem solving, a do-

main potentially well suited for SRL due to demands for metacog-

nition and perseverance in the face of challenge (De Corte et al.,

2000). Second, we extended the range of SRL transfer effects.

Sawyer et al. (1992) demonstrated SRL effects on generalization

across settings, from pull-out instructional locations to the regular

class. In the present study, we demonstrated effects on a far-

transfer measure for which the format and content varied from

instructional materials. Third, we extended the external validity of

previous work: Our treatments were delivered in whole-class for-

mat to naturally constituted classes over a relatively long duration

of 4 months. Finally, we contributed to the SRL literature by

separating effects for HA, AA, and LA learners as well as those

with disabilities, thereby, documenting effects for the range of



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ment: An overview. Educational Psychologist, 25, 318.

Zimmerman, B. J. (1995). Self-regulation involves more than meta-cogni-

tion: A social cognitive perspective. Educational Psychologist, 30, 217

222.

Zimmerman, B. J., & Kitsantas, A. (1999). Acquiring writing revision skill:Shifting from process to outcome self-regulatory goals. Journal of

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Received October 25, 2001

Revision received December 10, 2002

Accepted December 11, 2002


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