Implications of Constructivism for Teaching Math to Students with Moderate to Mild Disabilities

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1994 28: 290J Spec EducCecil D. Mercer, LuAnn Jordan and Susan P. Miller

DisabilitiesImplications of Constructivism for Teaching Math to Students with Moderate to Mild

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IMPLICATIONS OF CONSTRUCTIVISM FOR TEACHING MATH TO STUDENTS WITH MODERATE

TO MILD DISABILITIES

This article defines constructivism and examines the theory in terms of the view of the learner, the content, teacher-student interactions, moti-vation, and assessment. The standards generated by the National Council of Teachers of Math-ematics are reviewed in terms of their sensitivity to students with moderate to mild disabilities. Constructivistic teaching principles are abstracted from the constructivism and learning strategy literature and research. Nineteen instructional

Cec i l D. Mercer

LuAnn Jordan

University of Florida

Susan P. Mi l ler

University of Nevada, Las Vegas

components are identified and discussed in terms of teacher behaviors, teacher modeling of ex-plicit strategies, teacher-student interactions, instructional content, and learning factors. An analysis of these components reveals that most constructivists adopt an exogenous construc-tivistic approach to teaching math to students with moderate to mild disabilities. Finally, obstacles to applying research-based construc-tivistic components in classrooms are identified.

In the spring o f 1992, I (the senior author) had the opportunity to work on a

math project with second-grade teachers and their students in a rural Florida

school. During the last week of school I attended the end-of-the-school-year pic-

nic. I was sitting at a table enjoying an ice cream cone when I heard a confident

but quiet voice say, "I know ninety-nine times zero." Standing beside the table was

a second grader named Matt, with his hands in his pockets, waiting for my

response. I decided to have some fun. I said, "How could you possibly know

ninety-nine times zero and still be in the second grade? That must be at least a

third-grade skill." Matt quickly replied, "Ninety-nine times zero is zero." With a

surprised expression, I said, "Lucky guess!" Matt looked directly at me and said,

"I'll c o m e back." Then he turned and walked away. A few minutes later, Matt

approached the table with several of his second-grade friends. Upon their arrival,

one o f Matt's friends said, "I know one thousand times one." I responded, "You're

kidding. That problem is impossible for a second grader." T h e friend blurted out,

"One thousand." More second graders jo ined our discussion. They continued to

share with me their knowledge o f multiplication—a million times one, six thou-

sand times zero, nine times eight, and so on. T h e group o f 8 to 10 students

consisted of normally achieving students and students with learning problems.

They displayed their knowledge o f rules, multiplication facts, and problem-

Address: Cecil D. Mercer, Department of Special Education, G315 Norman Hall, University of Florida, Gainesville,

FL 32611.

at Monash University on December 4, 2014sed.sagepub.comDownloaded from


solving skills. I praised them for their knowledge and smartness. I was surprised

that the students continued to discuss math when they had the opportunity to be

playing. That afternoon while driving home, I realized that I had learned a lesson

about empowered students.

Unfortunately, this scenario of students excited about their math learning is

uncommon. For many students, math problems often result in school failure and

lead to much anxiety. Although deficiencies in reading are cited most often as a

primary characteristic o f students with learning disabilities, Mastropieri, Scruggs,

and Shiah ( 1 9 9 1 ) noted that deficits in math are as serious a problem for many

of these students. Research confirms that many students with learning problems

are unable to compute basic number facts (Fleischner, Garnett , & Shepherd,

1982; Goldman, Pellegrino, 8c Mertz, 1 9 8 8 ) . Fleischner and her colleagues found

that sixth-grade students with learning disabilities computed basic addition facts

no better than nondisabled third graders. Other studies indicate that math defi-

ciencies of students with learning disabilities emerge in the early years and con-

tinue into high school. Cawley and Miller ( 1 9 8 9 ) reported that the mathematical

knowledge of students with learning disabilities progresses approximately 1 year

per 2 years o f schooling. Warner, Alley, Schumaker, Deshler, and Clark ( 1 9 8 0 )

discovered that the math progress of students with learning disabilities reaches a

plateau after the seventh grade. T h e students they studied achieved only one

more year's growth in math from the 7th through the 12th grades. Both studies

reported high fifth-grade mean performances among students with learning dis-

abilities in the 12th grade.

McLeod and Armstrong ( 1 9 8 2 ) surveyed students with learning disabilities in

the sixth grade and above and found that two of every three students were receiv-

ing special math instruction. In a survey of e lementary and secondary teachers

of students with learning disabilities, Carpenter ( 1 9 8 5 ) found that teachers used

one-third of their instructional time to teach math. T h e importance o f providing

quality instruction for students with math problems is apparent; however, the

challenge intensifies when one examines the reforms being considered in math

education. F o r example , the National Council o f Supervisors o f Mathematics

( 1 9 8 8 ) and the National Council o f Teachers of Mathematics ( N C T M ) ( 1 9 8 9 ) are

promoting reforms that call for higher standards of math achievement. F o r exam-

ple, some states already have raised their high school diploma requirements to

include the successful completion o f Algebra I. Reforms that produce higher

standards are certain to frustrate teachers and students who are struggling with

current standards and traditional curriculum.

Given the poor math progress o f students with learning problems and the

likelihood of higher standards, a need clearly exists to improve math instruction.

Without better math instruction, these individuals will continue to face debilitat-

ing frustration and failure. This article examines the potential role o f construc-

tivism for improving instruction to students with math problems.

Constructivism

Teacher behavior and its effect on student learning have been the focus o f

investigation for years. During the 1970s and 1980s, researchers developed quan-

titative instruments to provide objective descriptions o f c lassroom activities.



Because researchers examined the relationship between teacher behavior and

student learning outcomes, this body of literature is identified as the process -

product research on teaching.

The process-product research provides educators with a wealth of information

on instructional methods, classroom management , lesson presentation, organiza-

tion and management o f instructional time, and seatwork management (Englert,

Tarrant, 8c Mariage, 1 9 9 2 ) . Although this quantitative look at teacher behavior

and student outcomes has been informative, many educators claim that a quali-

tative examination of teaching is essential. More information about the quality of

instruction and the nature o f teacher-student and student-student interactions

and their effect on learning outcomes is needed. For example, a qualitative look

at teacher-student interactions may provide insight into when it is parsimonious

for the teacher to teach new information explicitly, provide prompts and cues

that guide the student to new information, or present the student with challenges

and encourage the discovery o f new information. Constructivism, with its empha-

sis on teacher-student interactions, appears promising for extending knowledge

about teaching.

Constructivistic instruction is based on the premise that the student is a natur-

ally active learner who constructs new personalized knowledge via linking prior

knowledge and new knowledge. Authentic knowledge provides the content for

the instructional process, which involves an interactive and collaborative dialogue

between the teacher and the student. T h e teacher orchestrates the instruction

within the student's "zone of proximal development" (Vygotsky, 1 9 7 8 ) by provid-

ing assistance when the learner seems inefficient or frustrated. This zone refers

to the instructional area between where the learner has independence (mastery)

and what can be achieved with competent assistance (potential) . Constructivists

differ concerning the degree o f help the teacher should provide; however, some

c o m m o n instructional practices of the teacher include modeling cognitive pro-

cesses, providing guided instruction, encouraging reflection about thinking, giv-

ing feedback, and encouraging transfer. T h e teacher focuses on guiding the

student to achieve success and become a self-regulated strategic learner.

View of Learner. Constructivists view the l earner as a proactive participant in

the learning process. T h e learner is a naturally active, self-regulating participant

who intelligently responds to a perceived world (Poplin, 1 9 8 8 ) . T h e student is

an apprent ice l earner—not a sponge waiting to absorb information passively.

During the learning process, the student connects prior o r existing knowledge

with new knowledge to construct o r transform a new, personalized knowledge.

This construct ion of knowledge is believed to result in ownership and a deep

understanding o f the new information (Pressley, Harris , 8c Marks, 1 9 9 2 ) .

View of Content. The content o f instruction springs from authentic and pur-

poseful contexts . For example , rather than only teaching students to use algo-

rithms to solve equations, the students are presented with an authentic word

problem that can be solved by a single variable equation. Within this context ,

the purpose o f solving equations becomes apparent . For many constructivists,

instruction in subskills and mechanics remains important but occurs within a



content that enables the learner to exper ience the whole cognitive endeavor

and the usefulness o f the new knowledge. In essence, instruction focuses on

accomplishing whole tasks through simultaneous whole and part instruction

(Pressley et al., 1 9 9 2 ) .

To date, learning strategy instruction has incorporated many principles of a

constructivist teaching approach (Harris & Pressley, 1991; Lenz, 1992a; Pressley

et al., 1 9 9 2 ) . Pressley et al. reported that there is widespread agreement that

"strategies are goal-directed, cognitive operations employed to facilitate perfor-

mance" (p. 4 ) . At the Center for Learning at the University of Kansas, Deshler,

Schumaker, Lenz, and their colleagues defined learning strategies as techniques,

principles, or rules that enable a student to learn, to solve problems, and to

complete tasks independently (Lenz, 1 9 9 2 a ) . Lenz noted that learning strategy

instruction concentrates on teaching students how to learn and how to demon-

strate mastery of knowledge in performing academic tasks. With the growth of

strategy instruction, educators are hypothesizing that the learning of multiple

strategies interrelates to create strategic or self-regulated learners across tasks and

settings.

View of Teacher-Student Interactions. T h e nature of teacher-s tudent interactions

is viewed differently among constructivists (Moshman, 1982; Pressley et a l . , 1 9 9 2 ) .

T h e endogenous constructivist believes the interaction should be s tructured so

that students discover new knowledge without explicit instruction from the

teacher. Students are presented with challenges and then allowed to explore

and self-discover new knowledge. T h e exogenous constructivist believes the

teacher engages in m o r e direct instruction. Interactions are character ized by

the teacher providing explicit instruction through the use o f describing, explain-

ing, modeling, and guiding pract ice with feedback. T h e position o f the dialec-

tical constructivist is between the endogenous and exogenous constructivists.

These interactions are collaborative, with the t eacher providing instruction

(e.g., offering metacognitive explanations, model ing cognitive processes, asking

leading questions, and providing e n c o u r a g e m e n t ) as needed to guide student

discovery.

Many terms (socratic dialogue, collaborative discussions, interactive discourse,

reciprocal teaching, scaffolding) are used to describe these interactions. Reid and

Stone ( 1 9 9 1 ) described scaffolding as a teacher collaborating with a learner to

provide the learner with whatever is needed for meaningful participation. Paris

and Winograd ( 1 9 9 0 ) maintained that a key feature of scaffolded instruction is

a dialogue between teacher and student that provides the learner with just enough

support and guidance to enable the student to achieve a goal that would be

impossible without the assistance.

View of Motivation. Implicit in constructivistic teaching is the recognit ion and

development o f metacognitive strategies or self-regulation skills. Borkowski ( 1 9 9 2 )

noted that self-regulation skills form the basis for adaptive, planful, and thoughtful

learning that enables the student to read and problem solve across academic

domains. In the beginning, these self-regulation skills help a student size up

tasks and select an appropriate problem-solving sequence. As the student's self-



regulation skills develop, they serve to help the learner monitor, evaluate, and

revise strategies during the learning process. Borkowski, Estrada, Milstead, and

Hale ( 1 9 8 9 ) reported that every important cognitive act has motivational con-

sequences. In turn, these consequences p r o m o t e future self-regulatory actions.

Paris and W i n o g r a d ( 1 9 9 0 ) c la imed that the persona l n a t u r e o f such

metacognitive processes as self-appraisal and self-management motivates the

learner. Harris and Pressley ( 1 9 9 1 ) reported that an individual's motivation to

learn results from an intrinsic need to reflect on one's self, behavior, and knowl-

edge. As the learner develops self-regulation skills and begins to value the impor-

tance o f being a strategic learner, feelings o f self-efficacy emerge . In essence,

the awareness that self-regulation skills lead to academic success helps the stu-

dent b e c o m e a proactive learner.

View of Assessment. Because instruction should o c c u r within the zone o f proxi-

mal development (Vygotsky, 1 9 7 8 ) , the teacher assesses the prior knowledge or

preskills o f a student before beginning instruction. F o r example , knowledge o f

basic math skills is an important preskill for solving algebra equations. More-

over, the student's needs and motivation are considered. O n c e instruction is

initiated, assessment becomes a dynamic and continuous process. During col-

laborative dialogues (scaffolding), the teacher monitors the student's strategies,

attitude, and progress. Whenever a student needs assistance, the teacher navi-

gates the l earner in a productive direction via leading questions, prompting,

cueing, providing feedback, modeling strategic processes, o r isolating a specific

step or c o m p o n e n t for direct instruction. Often, the teacher asks the student to

share his or her thought processes while problem solving o r to reflect about

thinking (Garnett , 1 9 9 2 ) . Thus, assessment is on-line and spontaneous through-

out t eacher-s tudent interactions that are orchestrated to help the student con-

struct or discover new knowledge and b e c o m e a self-regulated l earner (Harris

& Pressley, 1 9 9 1 ) .

Peterson, Carpenter, and Fennema's ( 1 9 8 9 ) finding that the teacher's knowl-

edge o f an individual student's problem-solving skills predicted math achieve-

ment better than the teacher's knowledge o f problem solving or number fact

strategies underscores the need for teachers to be aware of the learning charac-

teristics of their students. This learner-specific knowledge was positively corre -

lated with the teacher presenting problems for students, questioning students

about methods they used to solve problems, and listening to students. Knowl-

edge o f a student enables the teacher to interact m o r e prescriptively with a stu-

dent to enhance achievement and motivation (e.g., asking appropriate questions,

modeling cognitive processes, monitoring progress, providing feedback and

encouragement ) .

Given that many students with learning disabilities use qualitatively different

strategies in math than students who are high achievers, it is important for teach-

ers to ascertain the strategies that students are using. This information is critical

to designing instruction because evidence indicates that different students need

different types of strategy instruction (Swanson, 1 9 9 0 ) . Moreover, when the strat-

egy instruction and a student's strategies do not match, the instruction may inter-

fere with some students' strategic performance (Montague, 1 9 9 3 ) . In essence,



specific strategy instruction needs to be considered in relation to a student's

knowledge base and capacity.

NCTM Standards

Curriculum and Evaluation Standards for School Mathematics, published in 1989 by

the National Council o f Teachers o f Mathematics (NCTM, 1 9 8 9 ) , is shaping

current reform efforts in math education. Spurred by poor performances o f

American youth on national math tests, this document urged professionals to

adopt the NCTM standards. Specifically, the NCTM called for practit ioners and

researchers to embrace problem solving as the basis for math instruction and

minimize meaningless rote drill and practice activities. According to the authors,

We are convinced that if students are exposed to the kinds of experiences outlined in the Standards, they will gain mathematical power. This term denotes an individual's abilities to explore, conjecture, and reason logically, as well as the ability to use a variety of mathematical methods effectively to solve nonroutine problems. This notion is based on the recognition of mathematics as more than a collection of concepts and skills to be mastered; it includes methods of investigating and reason-ing, means of communication, and notions of context. In addition, for each individual, mathemati-cal power involves the development of personal self-confidence. (NCTM, 1989, p. 5)

Thomas Romberg, the chair o f the commission that produced the standards

document , hailed the standards as a "rallying flag" for math teachers (Romberg,

1 9 9 3 ) . He noted that "mathematical power means having the exper ience and

understanding to participate constructively in society" (p. 3 7 ) .

Upon initial inspection, the standards are very appealing. T h e goals o f the

standards enjoy widespread acceptance . Closer inspection, however, reveals some

serious limitations of the standards as they pertain to the education o f students

with disabilities. T h e following limitations center around the modest attention

given to student diversity and the rigid adherence to select instructional para-

digms:

1. T h e standards make only modest reference to students with disabilities

(Giordano, 1993; Hofmeister, 1993; Hutchinson, 1993b; Rivera, 1 9 9 3 ) .

2. T h e standards are not based on replicable, validated, instructional programs.

Research-supported instructional programs for students with moderate to mild

disabilities are especially lacking (Hofmeister, 1993; Hutchinson, 1993b; Rivera,

1 9 9 3 ) . For example, Hutchinson ( 1 9 9 3 b ) stated that there is "no evidence to

support the claim that exposure to the proposed content and experiences will

result in mathematical power for students with disabilities" (p. 2 0 ) .

3. T h e s tandards p r o m o t e an e n d o g e n o u s constructivist a p p r o a c h (self-

discovery) for teaching math to all students. This position ignores the wealth of

teaching practices generated from the process -product research (Englert et al.,

1992) that have proven effective with students who have moderate to mild dis-

abilities. Moreover, strict a d h e r e n c e to endogenous constructivism does not

recognize the promising findings being generated regarding exogenous con-

structivism and dialectical constructivism.

Hutchinson ( 1 9 9 3 b ) noted that the authors o f the s tandards imply that

constructivism is opposed to practice o f skills and that student construction o f

knowledge is the antithesis o f instruction. She noted that these are false dichoto-



mies. Basically, many constructivists do not believe that discovery alone promotes

understanding (Isaacson, 1 9 8 9 ) .

Mercer, Harris, and Miller ( 1 9 9 3 ) r e c o m m e n d e d that mathematical reform be

guided by contributions across paradigms. Specifically, they r e c o m m e n d e d that

specific loyalty to one paradigm be replaced with an intellectual honesty that

recognizes the contributions of various paradigms. (For detailed reactions to the

NCTM standards regarding students with disabilities, see the N o v e m b e r /

December, 1993 , issue o f Remedial and Special Education.)

Applying Constructivism to Math Instruction

A sampling of the literature and research on constructivistic teaching and strat-

egy instruction for students with moderate to mild disabilities provides perspec-

tives on how constructivism is being interpreted and applied. This literature

reveals that constructivism has been interpreted and applied in many ways (e.g.,

Baroody 8c Hume, 1991; Borkowski, 1992; Englert et al., 1992; Harris & Pressley,

1991; Hutchinson, 1993b; Lenz, 1992b; Mastropieri et al., 1991; Mercer 8c Miller,

1992; Montague, 1992; Paris 8c Winograd, 1990; Pressley et al., 1992; Reid 8c

Stone; 1991; Rosenshine & Meister, 1992; Wong, 1 9 9 2 ) . Table 1 displays a list o f

instructional components and the number o f articles that r e c o m m e n d each com-

ponent . Inspect ion o f these instruct ional c o m p o n e n t s indicates that

constructivism has implications for teacher behaviors, instructional content, and

learning factors.

Teacher Behaviors. Since teacher-directed instructional components (e.g., com-

ponents 1, 4, 6, 11 , and 15) are frequently listed, it is apparent that the majority

o f constructivists favor exogenous or dialectical constructivism when students

with moderate to mild disabilities are the target population. This position is

understandable when the characteristics o f these learners are considered. To

expect students who have a history o f problems with automaticity, metacognitive

strategies, memory, attention, generalization, proactive learning, and motiva-

tion to engage in efficient self-discovery learning (i.e., endogenous construc-

tivism) is not plausible. Thus, teacher-directed instruction is a primary component

for teaching math to students with disabilities (Borkowski, 1 9 9 2 ) . Cobb, Yackel,

and Wood ( 1 9 9 2 ) e laborated on the need for directed instruction in helping

students to learn mathematical concepts and relationships. They question whether

it is possible for the students to recognize mathematical relationships that are

developmentally m o r e advanced than their current internal representations.

Further, they stated that "it would seem necessary to consider the teacher's role

in helping students construct replicas o f the mathematical relationships pre-

sented to them in an easily apprehensible form" (p. 5 ) . Finally, Cobb et al.

maintained that much teacher assistance is necessary if learners are to c o m e to

a conventional wisdom about math that took thousands o f years to evolve.

Teacher Modeling of Explicit Strategies. A promising feature from the constructivist

l iterature entails the substance o f t eacher modeling. Instead o f the traditional

t e a c h e r mode l ing o f m a t h e m a t i c a l a lgor i thms in a meaning less c o n t e x t ,



TABLE 1 I N S T R U C T I O N A L C O M P O N E N T S D E R I V E D F R O M C O N S T R U C T I V I S M

A N D L E A R N I N G STRATEGY L ITERATURE A N D R E S E A R C H

Number of sources recommending use in

Instructional component the 14 articles reviewed

1 . Model target strategy 12 2. Engage in interactive dialogue (i.e., scaffolding, socratic

teaching, and collaborative discussion) 9 3. Encourage metacognition and self-regulation 8 4. Provide prompts and guidance 8 5. Focus on authentic learning and rationales for learning 7 6. Use graph to monitor progress and provide feedback 7 7. Teach to mastery 7 8. Use goal setting 7 9. Teach for transfer 6

10. Focus on understanding and helping students link previous knowledge with new knowledge 6

11. Provide explicit instruction 5 12. Teach mnemonics 4 13. Encourage reflection and discussion 4 14. Use student "think-alouds" 4 15. Use teacher scripts 4 16. Teach in zone of proximal development 4 17. Check preskill development prior to teaching 3 18. Use verbal rehearsal 3 19. Promote peer collaboration 2

constructivists are exploring the modeling o f problem-solving strategies (i.e.,

cognitive and metacognit ive) to solve meaningful problems and develop self-

regulation processes. Woodward ( 1 9 9 1 ) noted that cognitive task analyses are

being conducted to enable teachers to model thinking processes involved in

math problem solving. Explicit t eacher modeling of cognitive and metacognit ive

strategies in solving word problems has yielded encouraging results in teaching

students with learning disabilities (Hutchinson, 1993a; Mercer & Miller, 1992;

Montague, 1 9 9 2 ) .

Hutchinson's ( 1 9 9 3 a ) intervention for teaching adolescents with learning dis-

abilities included explicit teacher modeling o f strategies for solving algebra word

problems. T h e strategies consisted o f self-questions for representing and solving

algebra word problems. The intervention also featured scripts to guide instruc-

tion, teacher prompts, encouragement , corrective feedback, student think-alouds,

guided practice, independent practice, and the use o f graphs to monitor student

progress. Selected self-questions for representing algebra word problems included

the following: (a) Have I read and understood the sentence? (b) Do I have the

whole picture, a representation, for this problem? and (c) Have I written the

representation on the worksheet? T h e students displayed improved per formance

on algebra word problems, and maintenance and transfer o f the problem-solving

strategy were evident.



Mercer and Miller ( 1 9 9 2 ) used strategy instruction to help e lementary students

with learning problems to compute math facts and solve word problems. In addi-

tion to explicit teacher strategy modeling, their intervention included guided

practice, independent practice, corrective feedback, student-generated word prob-

lems, mnemonics , mastery learning, and graphing of student progress. Results

indicated that the students made substantial gains in math achievement. More-

over, they acquired, maintained, and generalized the strategies. T h e strategy used

to help students represent and solve word problems was FAST DRAW:

FAST

Find what you're solving for.

Ask yourself, "What are the parts of the problem?"

Set up the numbers.

Tie down the sign.

DRAW

Discover the sign.

Read the problem (i.e., the equation) .

Answer, or draw and check (i.e., answer the problem from m e m o r y or repre-

sent the equation via drawings and check work).

Write the answer.

To teach problem-solving strategies to middle school students with learning

disabilities, Montague ( 1 9 9 2 ) used explicit modeling of cognitive and meta-

cognitive strategies. The intervention also included verbal rehearsal, corrective

and positive feedback, guided practice, and mastery checks. T h e specific cogni-

tive strategies included (a) read for understanding, (b) paraphrase in your own

words, (c ) visualize a picture or diagram, (d) hypothesize a plan to solve the prob-

lem, (e) estimate or predict the answer, (f) compute or do the arithmetic, and (g)

check to make sure everything is right. Montague reported that the students readily

learned these strategies and applied them successfully in solving word problems.

These pre l iminary findings (Hutch inson , 1 9 9 3 a ; M e r c e r & Miller, 1 9 9 2 ;

Montague, 1 9 9 3 ) suggest that specific strategy instruction in math holds signifi-

cant promise for students with moderate to mild disabilities. These findings sup-

port Zawaiza and Gerber's ( 1 9 9 3 ) position that "successful strategy instruction

. . . requires modeling o f competent strategy use, sufficient and appropriate

exemplar problems, ample opportunity to practice and receive correct ion on

strategy use, and adequate opportunities for students to describe and evaluate

how effectively they are employing newly learned strategies" (p. 6 7 ) .

Teacher-Student Interactions. Because the interactive nature o f t eacher- s tudent

dialogue is a major factor o f constructivism, it is not surprising that it was

ment ioned as an important instructional c o m p o n e n t in 9 o f the 14 articles

reviewed. A focus on teacher - s tudent interactions provides a timely opportunity

for educators to focus on improving the quality o f instructional discourse for

teaching math. Product -process research mainly has stressed the role of the

teacher in providing instruction to cover content, whereas constructivism research

focuses on the dynamic nature o f the dialogue between the student and the



t eacher to develop conceptual understandings. In this dynamic process, the

teacher is constantly adapting the dialogue according to student needs. Teach-

ers are encouraged to time their interactions prescriptively so that they know

when it is appropriate to provide direct instruction, give guided instruction, ask

questions, challenge, offer correct ive feedback, encourage , let the student work

independently, reflect with the student, set instructional goals, model a cogni-

tive o r metacognitive strategy, discuss rationales for learning new declarative or

procedural knowledge, or discuss transfer. W h e n teachers prescriptively interact

(i.e., base interactions on student behavior) during instruction to ensure that

students develop conceptual understandings, students are treated as active agents

in their own learning.

Teacher Knowledge and Understandings. E lmore ( 1 9 9 2 ) noted that teachers need

extensive help to learn and apply the ideas o f c u r r e n t research on teaching. H e

claimed that it is patently foolish to expec t teachers to accomplish this by them-

selves. Apparently, t eacher educat ion and commerc ia l materials have not helped

teachers to teach conceptual understandings. Most materials present information

that describes how to use algorithms to solve math problems. This algorithm-

driven approach provides little or n o help to teachers who desire to teach the

conceptual underpinnings implicit in math. To illustrate the lack o f understand-

ing o f math concepts among teachers or adults, consider some o f the following

questions:

1. Why do you get a smaller n u m b e r when you multiply fractions?

2. Why do you get a larger number when you divide fractions?

3. Why does a negative number times a negative n u m b e r yield a positive

number?

4. Why does the algorithm "invert and multiply" yield the c o r r e c t answer for

dividing fractions?

5. Can you write an authentic word problem for 6 x + 6 = 30?

To help teachers model strategies and teach understanding o f math concepts ,

several researchers have provided scripts or sample dialogues (e.g., Hutchinson,

1993a; Mercer & Miller, 1992; Montague, 1 9 9 3 ) . These scripts o r sample dia-

logues provide the teacher with an initial guide on how to model metacognitive

strategies explicitly and how to lead the student to conceptual understandings of

math concepts (declarative knowledge) and apply declarative and procedural

knowledge to solve word problems and math equations. These scripts serve as a

springboard for helping teachers engage in productive teacher-s tudent discourse.

As teachers gain confidence and experience with these interactions, the sample

dialogues are not needed. Researchers (e.g., Harris, 1992; Mercer, Enright, &

Tharin, 1994; Mercer 8c Miller, 1 9 9 3 ) are field-testing a sample dialogue that

guides the teacher through lessons and features the following steps: give an

advance organizer, describe and model the skill or strategy, provide guided prac-

tice and interactive discourse, provide independent practice to mastery, provide

elaborated feedback, and teach generalization and transfer. T h e describe-and-

model step features explicit teacher modeling of strategies. T h e guided practice

step parallels scaffolding as the teacher guides the students to conceptual under-



standings and independent work. T h e independent practice step incorporates

working alone and with peers to gain mastery. T h e elaborated feedback step is

used to recognize successes and relate them to learning goals and to use errors

in math for teaching and learning opportunities. T h e generalization and transfer

step encourages the students to reflect on strategy uses and create their own word

problems. Table 2 provides a beginning format for developing sample dialogues

and practice activities.

Instructional Content. As noted in component 5 in Table 1, many constructivists

r e c o m m e n d that math instruction focus on problem solving within an authentic

context . O n e way to accomplish this is to introduce math concepts and opera-

tions within the context o f a word problem. F o r example , solving division equa-

tions in algebra could be introduced with the following word problem: "Cindy

plans to give 3 0 coupons for free pizza to 6 o f her friends. How many coupons

will each friend receive?" T h e teacher explains that this can be represented and

solved via simple division: 30 coupons 6 friends = coupons per friend.

Then the teacher demonstrates how the problem can be solved via algebra by

giving the unknown a letter name and moving it to the left side o f the equation:

30 coupons 6 friends = becomes 6 friends x c (coupons per friend) =

30 coupons. Further, the teacher uses the students' prior knowledge about the

multiplication and division relationship to solve the problem (i.e., what number

multiplied by 6 equals 3 0 ? ) . As the lessons progress, the teacher guides and

encourages students to create their own word problems that can be solved via

division equations.

If mathematical content is to be relevant to learning, it is imperative that it be

presented in a real world context . For example, if the instructional content fails

to relate 6y + 2y + 6 = 4 8 to a pragmatic word problem, then students are memo-

rizing meaningless procedures for obtaining answers. In a study with techno-

logical math interventions with students with learning problems, Bottge and

Hasselbring ( 1 9 9 3 ) found that contextualized learning was a key factor. Finally,

Miller and Mercer ( 1 9 9 3 ) provided a graduated sequence for teaching word

problems that involves the concrete-representational-abstract (CRA) sequence and

culminates in the students creating their own word problems.

Several o f the instructional components relate to the content o f instruction

and increase the likelihood o f good instructional matches for students with

moderate to mild disabilities. Mnemonics ( component 12) and verbal rehearsal

( component 18) are instructional components that help students with memory

problems acquire, remember, and apply specific math strategies. Also, the CRA

teaching sequence has been found to facilitate the math learning o f students with

moderate to mild disabilities (Harris, 1992; Mercer & Miller, 1 9 9 2 ) . Implicit in

this method of instruction is an emphasis on enabling students to understand the

concepts o f math prior to memorizing facts, algorithms, and operations. Accord-

ing to the CRA sequence, instruction begins at the concrete level, where the

student uses three-dimensional objects to solve computat ion problems. After

successfully solving problems at the concrete level, the student proceeds to the

representational level. At the representational level, students use two-dimensional

drawings to solve computation problems. After successfully solving problems at



TABLE 2

F O R M A T F O R D E V E L O P I N G S A M P L E D I A L O G U E S

Step 1: Give an advance organizer

Link lesson to previous learning or lesson. Identify the target skill. Provide a rationale for learning the skill or strategy and discuss relevance of the

new knowledge.

Step 2 : Describe and model skill or strategy

Procedure 1 : Teacher asks a question and teacher answers question. The students hear and observe the teacher think aloud whi le modeling metacognitive strategies.

Procedure 2 : Teacher asks a question and students help provide answer. The students participate by answering the questions and solving the problem. The teacher and the students perform the strategy together and the teacher con-tinues to provide modeling.

Step 3 : Conduct guided practice and interactive discourse

Procedure 1: The teacher guides the student through problem-solving strategies without demonstration unless it is essential. Gu idance is provided as needed and the following supportive techniques are used: (a) Teacher asks specific leading questions and models if necessary (e.g., W h a t

is the first step in solving a problem?). (b) Teacher provides prompts regarding declarative knowledge (e.g., use a

variable [letter] to represent the unknown in the word problem). (c) Teacher provides cues regarding procedural knowledge (e.g., remember to

isolate the variable in solving the equation). Procedure 2 : The teacher instructs student to do task and reflect on the process

and product. The teacher provides support on an as-needed basis and uses fewer prompts and cues. The student is encouraged to become more indepen-dent.

Step 4 : Conduct independent practice to mastery

Student is encouraged to reflect (i.e., estimate, predict, check, create) and work without teacher assistance. Activities include peer tutoring, cooperative learning, instructional games, self-correcting materials, or computer-assisted instruction.

Step 5: Provide elaborated feedback

This procedure, based on research of Kline, Schumaker, and Deshler (1991) regarding feedback routines, guides the teacher to give feedback on correct responses and use incorrect responses as teaching and learning opportunities.

Find the score. Explain the grade. Enter the score. Use a graph and goal setting and make it meaningful. Evaluate the score in terms of the goal. Determine errors by examining the pattern. Begin error correction. Teacher models similar problem. Ask student to apply the correction procedure. Close out the session by giving positive feedback on correction. Kick back and relax!

Step 6 : Teach generalization and transfer

Reflect on applications of new knowledge across settings and situations. Encourage students to create meaningful math problems related to new knowledge.



this level, the student begins to work at the next level, the abstract level. At the

abstract level, the student looks at the computation problem and tries to solve it

without using objects or drawings. The student reads the problem, remembers

the answer, or thinks o f a way to compute the answer, and writes the answer. No

objects or drawings are used in the computation unless the student is unable to

answer a problem. Because success in math requires the ability to solve problems

at the abstract level, student mastery at this level is essential. T h e CRA sequence

seems to be especially useful in helping students who have deficits in representing

or reformulating math from word problems to equations and vice versa, equations

to objects and vice versa, and pictures or drawings to equations and vice versa.

Because the CRA sequence requires students to represent math concepts and

operations with objects and drawings, math concepts (e.g., addition, place value,

multiplication, fractions, equations) are understood.

Learning Factors. Teaching to mastery ( c o m p o n e n t 7) refers to teaching a

skill to a level o f automaticity, which usually is obtained when an individual

continuously responds to math problems without hesitating to think about com-

puting the answer. Reaching mastery on a skill provides numerous benefits,

including improved retention and ability to c o m p u t e or solve higher-level prob-

lems. Other benefits include finishing timed tests, complet ing homework faster,

receiving higher grades, and developing positive feelings about math. Before

mastery instruction or techniques are used, it is essential that the student pos-

sess the preskills and understand the concept related to the targeted skill. O n c e

an understanding of a skill is achieved, mastery-level instruction becomes appro-

priate. Independent pract ice is the primary instructional format used to acquire

mastery. O n c e a mastery level is achieved, the teacher and student are able to

move to the next level skill with appropriate prior knowledge and m o r e confi-

dence .

Teaching for transfer ( component 9) refers to the per formance o f the targeted

behavior in different, nontraining conditions (i.e., across subjects, settings, people,

behaviors, or time) that do not involve the same events that were present in the

training conditions. Students with learning problems typically have difficulty

generalizing skills. A lack o f instruction aimed at teaching these students to trans-

fer math skills has contributed to their generalization problems. Selected instruc-

tional practices to help students generalize math skills include the following:

1. Develop motivation to learn. It is believed that students who desire to learn

a skill or strategy are most likely to generalize it. Motivation helps students feel

responsible for their own learning and helps establish the independence needed

to apply the new skill in settings without teacher support.

2. T h r o u g h o u t the instruct ional process , hold per iodic discussions with

students about the rationale for learning the math skill and in which situations

(e.g., homework, recreational activities, shopping) it is useful.

3. Throughout the instructional process, provide students with a variety o f

examples and experiences.

4. Teach students to solve problems pertinent to their daily lives. This connects

the skill to functional uses and promotes motivation and the need to generalize.

The teacher's effort to achieve an instructional match between student and

task characteristics results in mutual goal setting. Thus, appropriate instructional



goals are based on careful assessment of a student's learning needs and input.

Basically, goals provide the basis for instruction. T h e r e is growing support for the

premise that teachers tend to set goals that are too easy for students with learning

problems (Anderson & Pellicer, 1990; Clifford, 1 9 9 0 ) . Clifford reported that stu-

dents need a challenge rather than easy success, and that tasks involving moder-

ate risk taking provide the best level o f difficulty in setting goals. She

recommended that instructional environments feature e r r o r tolerance and reward

for e r r o r correct ion. A substantial research base (Locke & Latham, 1990) docu-

ments the premise that difficult but attainable goals lead to higher effort and

achievement than do easier goals.

Closing Perspectives

According to Mtetwa and Garofalo ( 1 9 8 9 ) , students believe that (a) math is a

set o f rules that requires memorization and rote practice in order to succeed,

(b) computation problems are always solved by using algorithms, ( c ) problems

always have one correc t answer, and (d) people who use mathematics are geniuses.

Given the views of students about math, it is imperative that they be taught to

become proactive learners.

Many of the constructivist instructional applications have m u c h appeal for

helping students with moderate to mild disabilities acquire and use math fluently

in their daily lives. Many constructivistic practices also seem to be promising for

helping teachers improve the quality and effectiveness o f their math teaching. If

the potential o f constructivism's encouraging practices is to be realized in our

nation's schools, several obstacles must be overcome. Some o f these obstacles

center around commercial materials, teacher education, and paradigm dogma.

Algorithm-driven instructional materials must be replaced with math materials

that (a) have been validated and replicated in various school settings prior to

publishing; (b) correspond to the academic school year (i.e., many materials have

4 0 to 6 0 more daily lessons than there are days in the school year); ( c ) guide the

teacher to use authentic content, model explicit metacognitive strategies, use

instructionally prescriptive interactive dialogues, use elaborated feedback, and

use transfer o f learning techniques; and (d) recognize the strengths o f various

paradigms.

Teacher educators and academicians must stop advocating math reform prior

to gathering supportive and generalizable findings. As a profession, math educa-

tion would be better served by adopting a refining rather than a reforming

posture. Reform implies that the prior knowledge in the profession has been

unacceptable, whereas refinement implies that prior knowledge can be connected

with new knowledge to construct new and improved knowledge. Perhaps research-

driven changes would decrease the need to have a major reform movement in

math every 20 years (Carnine , 1992) and result in a growing knowledge base

about how to learn and teach math.

The potential new knowledge that constructivism can provide is exciting if

educators take the time and effort to apply and test it systematically with teachers

and students. Given the heterogeneity o f learners, it is highly probable that

deductive and inductive learning are both important for many individuals as they

acquire, maintain, and generalize knowledge. Carnine ( 1 9 9 2 ) reported that rigid



adherence to a constructivistic paradigm (endogenous constructivism) has resulted

in five major reform cycles in mathematics since 1900 . He acknowledged that the

National Council of Teachers of Mathematics admits that its recommendat ions

are largely untested. As professionals, educators are much m o r e concerned with

what works with youngsters than with paradigm allegiance.

If educat ional researchers can scientifically tap the potential benefits o f

constructivistic principles and if teacher educators and publishers o f commercial

materials can place the products o f these findings in the hands o f teachers,

educators will have an opportunity to improve significantly the math learning o f

students and the math instruction of teachers. Perhaps student beliefs about

learning math will change, and more math teachers will be sitting quietly on a

playground and experience the satisfaction and joy o f having a student approach

them and say, "I know how to solve 9x - 6 x + 12 = 24" . . . and the fun begins.

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Implications of Constructivism for Teaching Math to Students with Moderate to Mild Disabilities