Contributed by:

This booklet summarizes the mathematics chapter from the Handbook of research on improving student achievement, second edition, published by the Educational Research Service. The Handbook is based on the idea that, in order to succeed, efforts to improve instruction must focus on the existing knowledge base in respect of effective teaching and learning. The Handbook was specifically designed to help school administrators and teachers carry out their evolving instructional leadership roles by giving them a ready source of authoritative yet practitioner-based information about research on effective teaching and learning.

1.
INTERNATIONAL ACADEMY

EDUCATIONAL PRACTICES SERIES–4

OF EDUCATION

INTERNATIONAL BUREAU

OF EDUCATION

By Douglas A. Grouws

and Kristin J. Cebulla

EDUCATIONAL PRACTICES SERIES–4

OF EDUCATION

INTERNATIONAL BUREAU

OF EDUCATION

By Douglas A. Grouws

and Kristin J. Cebulla

2.
The International Academy

of Education

The International Academy of Education (IAE) is a not-for-profit

scientific association that promotes educational research,its dis-

semination, and the implementation of its implications. Found-

ed in 1986, the Academy is dedicated to strengthening the con-

tributions of research, solving critical educational problems

throughout the world, and providing better communication

among policy makers, researchers and practitioners. The seat

of the Academy is at the Royal Academy of Science, Literature

and Arts in Brussels, Belgium, and its co-ordinating centre is at

Curtin University of Technology in Perth, Australia.

The general aim of the IAE is to foster scholarly excellence in

all fields of education. Towards this end, the Academy provides

timely syntheses of research-based evidence of international im-

portance. The Academy also provides critiques of research, its

evidentiary basis, and its application to policy.

The current members of the Board of Directors of the Acad-

emy are:

• Erik De Corte, University of Leuven, Belgium (President);

• Herbert Walberg, University of Illinois at Chicago,

United States of America (Vice-President);

• Barry Fraser, Curtin University of Technology, Australia (Ex-

ecutive Director);

• Jacques Hallak, UNESCO, Paris, France;

• Michael Kirst, Stanford University,

United States of America;

• Ulrich Teichler, University of Kassel, Germany;

• Margaret Wang, Temple University,

United States of America.

2

of Education

The International Academy of Education (IAE) is a not-for-profit

scientific association that promotes educational research,its dis-

semination, and the implementation of its implications. Found-

ed in 1986, the Academy is dedicated to strengthening the con-

tributions of research, solving critical educational problems

throughout the world, and providing better communication

among policy makers, researchers and practitioners. The seat

of the Academy is at the Royal Academy of Science, Literature

and Arts in Brussels, Belgium, and its co-ordinating centre is at

Curtin University of Technology in Perth, Australia.

The general aim of the IAE is to foster scholarly excellence in

all fields of education. Towards this end, the Academy provides

timely syntheses of research-based evidence of international im-

portance. The Academy also provides critiques of research, its

evidentiary basis, and its application to policy.

The current members of the Board of Directors of the Acad-

emy are:

• Erik De Corte, University of Leuven, Belgium (President);

• Herbert Walberg, University of Illinois at Chicago,

United States of America (Vice-President);

• Barry Fraser, Curtin University of Technology, Australia (Ex-

ecutive Director);

• Jacques Hallak, UNESCO, Paris, France;

• Michael Kirst, Stanford University,

United States of America;

• Ulrich Teichler, University of Kassel, Germany;

• Margaret Wang, Temple University,

United States of America.

2

3.
This booklet has been adapted for inclusion in the Educational

Practices Series developed by the International Academy of

Education (IAE) and distributed by the International Bureau of

Education (IBE) and the Academy. As part of its mission, the

Academy provides timely syntheses of research on educational top-

ics of international importance. This booklet is the fourth in the

series on educational practices that generally improve learning.

The material was originally prepared for the Handbook of

research on improving student achievement, edited by Gordon

Cawelti and published in a second edition in 1999 by the

Educational Research Service (ERS). The Handbook, which also

includes chapters on subjects such as generic practices and sci-

ence, is available from ERS (2000 Clarendon Boulevard,

Arlington, VA 22201-2908, United States of America; phone (1)

800-791-9308;fax (1) 800-791-9309;and website:www.ers.org).

ERS is a not-for-profit research foundation serving the

research and information needs of educational leaders and the

public. Established in 1973, ERS is sponsored by seven organiza-

tions: the American Association of School Administrators; the

American Association of School Personnel Administrators; the

Association of School Business Officials; the Council of Chief

State School Officers; the National Association of Elementary

School Principals; the National Association of Secondary School

Principals; and the National School Public Relations Association.

As Vice-President of the Academy and editor of the present

series, I thank ERS officials for allowing the IAE and the IBE to

make available the material adapted from the Handbook to edu-

cators around the world.

The first author of the present pamphlet, Douglas A.

Grouws,is Professor of Mathematics Education at the University

of Iowa. He was the editor of the Handbook of research on

mathematics teaching and learning (Macmillan,1992) and has

a large number of other publications on research in mathemat-

ics education to his credit. He has made invited research pre-

sentations in Australia, China, Hungary, Guam, India, Japan,

Mexico,Thailand and the United Kingdom. He has directed sev-

eral research projects for the National Science Foundation

(NSF) and other agencies in the areas of mathematical problem-

solving and classroom teaching practices. His current NSF work

involves mathematics and technology. He received his Ph.D.

from the University of Wisconsin.

3

Practices Series developed by the International Academy of

Education (IAE) and distributed by the International Bureau of

Education (IBE) and the Academy. As part of its mission, the

Academy provides timely syntheses of research on educational top-

ics of international importance. This booklet is the fourth in the

series on educational practices that generally improve learning.

The material was originally prepared for the Handbook of

research on improving student achievement, edited by Gordon

Cawelti and published in a second edition in 1999 by the

Educational Research Service (ERS). The Handbook, which also

includes chapters on subjects such as generic practices and sci-

ence, is available from ERS (2000 Clarendon Boulevard,

Arlington, VA 22201-2908, United States of America; phone (1)

800-791-9308;fax (1) 800-791-9309;and website:www.ers.org).

ERS is a not-for-profit research foundation serving the

research and information needs of educational leaders and the

public. Established in 1973, ERS is sponsored by seven organiza-

tions: the American Association of School Administrators; the

American Association of School Personnel Administrators; the

Association of School Business Officials; the Council of Chief

State School Officers; the National Association of Elementary

School Principals; the National Association of Secondary School

Principals; and the National School Public Relations Association.

As Vice-President of the Academy and editor of the present

series, I thank ERS officials for allowing the IAE and the IBE to

make available the material adapted from the Handbook to edu-

cators around the world.

The first author of the present pamphlet, Douglas A.

Grouws,is Professor of Mathematics Education at the University

of Iowa. He was the editor of the Handbook of research on

mathematics teaching and learning (Macmillan,1992) and has

a large number of other publications on research in mathemat-

ics education to his credit. He has made invited research pre-

sentations in Australia, China, Hungary, Guam, India, Japan,

Mexico,Thailand and the United Kingdom. He has directed sev-

eral research projects for the National Science Foundation

(NSF) and other agencies in the areas of mathematical problem-

solving and classroom teaching practices. His current NSF work

involves mathematics and technology. He received his Ph.D.

from the University of Wisconsin.

3

4.
The second author, Kristin J. Cebulla, is a mathematics edu-

cation doctoral student at the University of Iowa.She previously

taught middle-school mathematics. Before teaching, she worked

as a research chemical engineer. She received her bachelor of

science degree in mathematics and chemical engineering from

the University of Notre Dame and her master’s degree from the

University of Mississippi.

The principles described in this booklet are derived in large

part from the United States and other English-speaking coun-

tries. Other research also has important implications for the

teaching of mathematics. An example is Realistic Mathematics

Education, initiated by H. Freudenthal and developed since the

early 1970s at the University of Utrecht (Dordrecht, The

Netherlands, Kluwer, 1991). Another example is research on

problem-solving summarized in a recent book by L.Verschaffel,

B. Greer, and E. De Corte—Making sense of word problems

(Lisse,The Netherlands, Swets & Zeitlinger, 2000).

The officers of the International Academy of Education are

aware that this booklet is based on research carried out primar-

ily in economically advanced countries. The booklet, however,

focuses on aspects of learning that appear to be universal in

much formal schooling. The practices seem likely to be gener-

ally applicable throughout the world. Even so, the principles

should be assessed with reference to local conditions, and

adapted accordingly. In any educational setting, suggestions or

guidelines for practice require sensitive and sensible applica-

tion and continuing evaluation.

HERBERT J.WALBERG

Editor, IAE Educational Practices Series

University of Illinois at Chicago

4

cation doctoral student at the University of Iowa.She previously

taught middle-school mathematics. Before teaching, she worked

as a research chemical engineer. She received her bachelor of

science degree in mathematics and chemical engineering from

the University of Notre Dame and her master’s degree from the

University of Mississippi.

The principles described in this booklet are derived in large

part from the United States and other English-speaking coun-

tries. Other research also has important implications for the

teaching of mathematics. An example is Realistic Mathematics

Education, initiated by H. Freudenthal and developed since the

early 1970s at the University of Utrecht (Dordrecht, The

Netherlands, Kluwer, 1991). Another example is research on

problem-solving summarized in a recent book by L.Verschaffel,

B. Greer, and E. De Corte—Making sense of word problems

(Lisse,The Netherlands, Swets & Zeitlinger, 2000).

The officers of the International Academy of Education are

aware that this booklet is based on research carried out primar-

ily in economically advanced countries. The booklet, however,

focuses on aspects of learning that appear to be universal in

much formal schooling. The practices seem likely to be gener-

ally applicable throughout the world. Even so, the principles

should be assessed with reference to local conditions, and

adapted accordingly. In any educational setting, suggestions or

guidelines for practice require sensitive and sensible applica-

tion and continuing evaluation.

HERBERT J.WALBERG

Editor, IAE Educational Practices Series

University of Illinois at Chicago

4

5.
Table of contents

Introduction, page 7

1. Opportunity to learn, page 10

2. Focus on meaning, page 13

3. Learning new concepts and skills while solving

problems, page 15

4. Opportunities for both invention and practice, page 17

5. Openness to student solution methods and

student interaction, page 19

6. Small-group learning, page 21

7. Whole-class discussion, page 23

8. Number sense, page 25

9. Concrete materials, page 27

10. Students’ use of calculators, page 29

Conclusions, page 31

Additional resources, page 35

References, page 39

This publication has been produced in 2000 by the International

Academy of Education (IAE), Palais des Académies, 1, rue Ducale,

1000 Brussels, Belgium, and the International Bureau of Education

(IBE), P.O. Box 199, 1211 Geneva 20, Switzerland.

It is available free of charge and may be freely reproduced and

translated into other languages. Please send a copy of any publica-

tion that reproduces this text in whole or in part to the IAE and the

IBE. This publication is also available on the Internet in its printed

form; see:

http://www.ibe.unesco.org

or

http://www.curtin.edu.au/curtin/dept/smec/iae

The authors are responsible for the choice and presentation of the

facts contained in this publication and for the opinions expressed

therein, which are not necessarily those of UNESCO/IBE and do not

commit the Organization. The designations employed and the pre-

sentation of the material in this publication do not imply the expres-

sion of any opinion whatsoever on the part of UNESCO/IBE con-

cerning the legal status of any country, territory, city or area, or of

its authorities, or concerning the delimitation of its frontiers or

boundaries.

Printed in Switzerland by PCL, Lausanne.

5

Introduction, page 7

1. Opportunity to learn, page 10

2. Focus on meaning, page 13

3. Learning new concepts and skills while solving

problems, page 15

4. Opportunities for both invention and practice, page 17

5. Openness to student solution methods and

student interaction, page 19

6. Small-group learning, page 21

7. Whole-class discussion, page 23

8. Number sense, page 25

9. Concrete materials, page 27

10. Students’ use of calculators, page 29

Conclusions, page 31

Additional resources, page 35

References, page 39

This publication has been produced in 2000 by the International

Academy of Education (IAE), Palais des Académies, 1, rue Ducale,

1000 Brussels, Belgium, and the International Bureau of Education

(IBE), P.O. Box 199, 1211 Geneva 20, Switzerland.

It is available free of charge and may be freely reproduced and

translated into other languages. Please send a copy of any publica-

tion that reproduces this text in whole or in part to the IAE and the

IBE. This publication is also available on the Internet in its printed

form; see:

http://www.ibe.unesco.org

or

http://www.curtin.edu.au/curtin/dept/smec/iae

The authors are responsible for the choice and presentation of the

facts contained in this publication and for the opinions expressed

therein, which are not necessarily those of UNESCO/IBE and do not

commit the Organization. The designations employed and the pre-

sentation of the material in this publication do not imply the expres-

sion of any opinion whatsoever on the part of UNESCO/IBE con-

cerning the legal status of any country, territory, city or area, or of

its authorities, or concerning the delimitation of its frontiers or

boundaries.

Printed in Switzerland by PCL, Lausanne.

5

6.

7.
This booklet summarizes the mathematics chapter from the

Handbook of research on improving student achievement,

second edition, published by the Educational Research Service.

The Handbook is based on the idea that, in order to succeed,

efforts to improve instruction must focus on the existing knowl-

edge base in respect of effective teaching and learning. The

Handbook was specifically designed to help school administra-

tors and teachers carry out their evolving instructional leader-

ship roles by giving them a ready source of authoritative yet

practitioner-based information about research on effective

teaching and learning.

The practices identified in this booklet reflect a mixture of

emerging strategies and practices in long-term use. The authors

briefly summarize the research supporting each practice,

describe how this research might be applied in actual classroom

practice, and list the most important studies that support the

practice. A complete list of references is provided at the end of

the booklet for readers who want to study and understand the

practices more fully.

In most cases, the results of research on specific teaching

practices show only small or moderate gains. In education, we

need to understand, carefully select, and use combinations of

teaching practices that together increase the probability of

helping students learn, knowing that these practices may not

work in all classrooms at all times.

The strongest possibility of improving student learning

emerges where schools implement multiple changes in the

teaching and learning activities affecting the daily life of stu-

dents. For example, if the aim is to improve students’ scientific

problem-solving skills, the school might plan to introduce

training for teachers in (1) use of the learning cycle approach;

(2) use of computer simulations; and (3) systemic approaches

to problem solving. To simultaneously plan for the training

and other provisions needed to sustain all three of these

changes would be no small undertaking, but would hold great

promise for improving the quality of student problem-

The research findings presented in this booklet provide a

starting point for developing comprehensive school plans to

improve mathematics instruction. Teachers and school leaders

will inevitably need time for further study, discussion and other

7

Handbook of research on improving student achievement,

second edition, published by the Educational Research Service.

The Handbook is based on the idea that, in order to succeed,

efforts to improve instruction must focus on the existing knowl-

edge base in respect of effective teaching and learning. The

Handbook was specifically designed to help school administra-

tors and teachers carry out their evolving instructional leader-

ship roles by giving them a ready source of authoritative yet

practitioner-based information about research on effective

teaching and learning.

The practices identified in this booklet reflect a mixture of

emerging strategies and practices in long-term use. The authors

briefly summarize the research supporting each practice,

describe how this research might be applied in actual classroom

practice, and list the most important studies that support the

practice. A complete list of references is provided at the end of

the booklet for readers who want to study and understand the

practices more fully.

In most cases, the results of research on specific teaching

practices show only small or moderate gains. In education, we

need to understand, carefully select, and use combinations of

teaching practices that together increase the probability of

helping students learn, knowing that these practices may not

work in all classrooms at all times.

The strongest possibility of improving student learning

emerges where schools implement multiple changes in the

teaching and learning activities affecting the daily life of stu-

dents. For example, if the aim is to improve students’ scientific

problem-solving skills, the school might plan to introduce

training for teachers in (1) use of the learning cycle approach;

(2) use of computer simulations; and (3) systemic approaches

to problem solving. To simultaneously plan for the training

and other provisions needed to sustain all three of these

changes would be no small undertaking, but would hold great

promise for improving the quality of student problem-

The research findings presented in this booklet provide a

starting point for developing comprehensive school plans to

improve mathematics instruction. Teachers and school leaders

will inevitably need time for further study, discussion and other

7

8.
exposure to what a particular practice entails before deciding to

include it in their school’s plans.

The complexities involved in putting the knowledge base

on improving student achievement to work in classrooms must

be recognized. As Dennis Sparks writes in his chapter on staff

development in the Handbook of research on improving stu-

dent achievement, schools and school districts have a respon-

sibility to establish a culture in which teachers can exercice

their professional competence,explore promising practices and

share information among themselves, while keeping the focus

on the ultimate goal of staff development—the improvement of

student learning.

Improving teacher effectiveness

The number of research studies conducted in mathematics edu-

cation over the past three decades has increased dramatically

(Kilpatrick, 1992). The resulting research base spans a broad

range of content, grade levels and research methodologies. The

results from these studies, together with relevant findings from

research in other domains, such as cognitive psychology, are

used to identify the successful teaching strategies and practices.

Teaching and learning mathematics are complex tasks.The

effect on student learning of changing a single teaching practice

may be difficult to discern because of the simultaneous effects

of both the other teaching activities that surround it and the

context in which the teaching takes place.

Thus, as teachers seek to improve their teaching effective-

ness by changing their instructional practices, they should care-

fully consider the teaching context, giving special consideration

to the types of students they teach. And,further,they should not

judge the results of their new practices too quickly. Judgements

about the appropriateness of their decisions must be based on

more than a single outcome. If the results are not completely

satisfactory, teachers should consider the circumstances that

may be diminishing the impact of the practices they are imple-

menting. For example, the value of a teacher focusing more

attention on teaching for meaning may not be demonstrated if

student assessments concentrate on rote recall of facts and pro-

ficient use of isolated skills.

The quality of the implementation of a teaching practice also

greatly influences its impact on student learning. The value of

using manipulative materials to investigate a concept, for exam-

ple, depends not only on whether manipulatives are used, but

also on how they are used with the students. Similarly, small-

group instruction will benefit students only if the teacher knows

8

include it in their school’s plans.

The complexities involved in putting the knowledge base

on improving student achievement to work in classrooms must

be recognized. As Dennis Sparks writes in his chapter on staff

development in the Handbook of research on improving stu-

dent achievement, schools and school districts have a respon-

sibility to establish a culture in which teachers can exercice

their professional competence,explore promising practices and

share information among themselves, while keeping the focus

on the ultimate goal of staff development—the improvement of

student learning.

Improving teacher effectiveness

The number of research studies conducted in mathematics edu-

cation over the past three decades has increased dramatically

(Kilpatrick, 1992). The resulting research base spans a broad

range of content, grade levels and research methodologies. The

results from these studies, together with relevant findings from

research in other domains, such as cognitive psychology, are

used to identify the successful teaching strategies and practices.

Teaching and learning mathematics are complex tasks.The

effect on student learning of changing a single teaching practice

may be difficult to discern because of the simultaneous effects

of both the other teaching activities that surround it and the

context in which the teaching takes place.

Thus, as teachers seek to improve their teaching effective-

ness by changing their instructional practices, they should care-

fully consider the teaching context, giving special consideration

to the types of students they teach. And,further,they should not

judge the results of their new practices too quickly. Judgements

about the appropriateness of their decisions must be based on

more than a single outcome. If the results are not completely

satisfactory, teachers should consider the circumstances that

may be diminishing the impact of the practices they are imple-

menting. For example, the value of a teacher focusing more

attention on teaching for meaning may not be demonstrated if

student assessments concentrate on rote recall of facts and pro-

ficient use of isolated skills.

The quality of the implementation of a teaching practice also

greatly influences its impact on student learning. The value of

using manipulative materials to investigate a concept, for exam-

ple, depends not only on whether manipulatives are used, but

also on how they are used with the students. Similarly, small-

group instruction will benefit students only if the teacher knows

8

9.
when and how to use this teaching practice. Hence, as a teacher

implements any of the recommendations,it is essential that he or

she constantly monitors and adjusts the way the practice is

implemented in order to optimize improvements in quality.

These cautions notwithstanding, the research findings indi-

cate that certain teaching strategies and methods are worth

careful consideration as teachers strive to improve their mathe-

matics teaching practices. As readers examine the suggestions

that follow, it will become clear that many of the practices are

interrelated.There is also considerable variety in the practices

that have been found to be effective, and so most teachers

should be able to identify ideas they would like to try in their

classrooms. The practices are not mutually exclusive; indeed,

they tend to be complementary. The logical consistency and

variety in the suggestions from research make them both inter-

esting and practical.

The authors wish to acknowledge the following colleagues

who made helpful suggestions: Tom Cooney, Professor of

Mathematics, University of Georgia; James Hiebert, Professor of

Mathematics Education, University of Delaware; Judy Sowder,

Professor of Mathematics, San Diego State University; and Terry

Wood, Professor of Mathematics Education, Purdue University.

9

implements any of the recommendations,it is essential that he or

she constantly monitors and adjusts the way the practice is

implemented in order to optimize improvements in quality.

These cautions notwithstanding, the research findings indi-

cate that certain teaching strategies and methods are worth

careful consideration as teachers strive to improve their mathe-

matics teaching practices. As readers examine the suggestions

that follow, it will become clear that many of the practices are

interrelated.There is also considerable variety in the practices

that have been found to be effective, and so most teachers

should be able to identify ideas they would like to try in their

classrooms. The practices are not mutually exclusive; indeed,

they tend to be complementary. The logical consistency and

variety in the suggestions from research make them both inter-

esting and practical.

The authors wish to acknowledge the following colleagues

who made helpful suggestions: Tom Cooney, Professor of

Mathematics, University of Georgia; James Hiebert, Professor of

Mathematics Education, University of Delaware; Judy Sowder,

Professor of Mathematics, San Diego State University; and Terry

Wood, Professor of Mathematics Education, Purdue University.

9

10.
1. Opportunity to learn

The extent of the students’ opportunity to

learn mathematics content bears directly

and decisively on student mathematics

achievement.

Research findings

The term ‘opportunity to learn’ (OTL) refers to what is studied

or embodied in the tasks that students perform.In mathematics,

OTL includes the scope of the mathematics presented, how the

mathematics is taught, and the match between students’ entry

skills and new material.

The strong relationship between OTL and student perfor-

mance in mathematics has been documented in many research

studies. The concept was studied in the First International

Mathematics Study (Husén), where teachers were asked to rate

the extent of student exposure to particular mathematical con-

cepts and skills.Strong correlations were found between student

OTL scores and mean student achievement scores in mathemat-

ics, with high OTL scores associated with high achievement.The

link between student mathematics achievement and opportu-

nity to learn was also found in subsequent international studies,

such as the Second International Mathematics Study (McKnight

et al.) and the Third International Mathematics and Science Study

(TIMSS) (Schmidt, McKnight & Raizen).

As might be expected, there is also a positive relationship

between total time allocated to mathematics and general math-

ematics achievement. Suarez et al., in a review of research on

instructional time, found strong support for the link between

allocated instructional time and student performance.

Internationally, Keeves found a significant relationship across

Australian states between achievement in mathematics and total

curriculum time spent on mathematics.

In spite of these research findings, many students still spend

only minimal amounts of time in the mathematics class. For

instance, Grouws and Smith, in an analysis of data from the 1996

National Assessment of Educational Progress (NAEP) mathemat-

ics study, found that 20% of eighth-grade students had thirty min-

utes or less for mathematics instruction each day.

10

The extent of the students’ opportunity to

learn mathematics content bears directly

and decisively on student mathematics

achievement.

Research findings

The term ‘opportunity to learn’ (OTL) refers to what is studied

or embodied in the tasks that students perform.In mathematics,

OTL includes the scope of the mathematics presented, how the

mathematics is taught, and the match between students’ entry

skills and new material.

The strong relationship between OTL and student perfor-

mance in mathematics has been documented in many research

studies. The concept was studied in the First International

Mathematics Study (Husén), where teachers were asked to rate

the extent of student exposure to particular mathematical con-

cepts and skills.Strong correlations were found between student

OTL scores and mean student achievement scores in mathemat-

ics, with high OTL scores associated with high achievement.The

link between student mathematics achievement and opportu-

nity to learn was also found in subsequent international studies,

such as the Second International Mathematics Study (McKnight

et al.) and the Third International Mathematics and Science Study

(TIMSS) (Schmidt, McKnight & Raizen).

As might be expected, there is also a positive relationship

between total time allocated to mathematics and general math-

ematics achievement. Suarez et al., in a review of research on

instructional time, found strong support for the link between

allocated instructional time and student performance.

Internationally, Keeves found a significant relationship across

Australian states between achievement in mathematics and total

curriculum time spent on mathematics.

In spite of these research findings, many students still spend

only minimal amounts of time in the mathematics class. For

instance, Grouws and Smith, in an analysis of data from the 1996

National Assessment of Educational Progress (NAEP) mathemat-

ics study, found that 20% of eighth-grade students had thirty min-

utes or less for mathematics instruction each day.

10

11.
Research has also found a strong relationship between math-

ematics-course taking at the secondary school level and student

achievement. Reports from the NAEP in mathematics showed

that ‘the number of advanced mathematics courses taken was

the most powerful predictor of students’ mathematics perfor-

mance after adjusting for variations in home background’.

Textbooks are also related to student OTL, because many

textbooks do not contain much content that is new to students.

The lack of attention to new material and heavy emphasis on

review in many textbooks are of particular concern at the ele-

mentary school and middle-school levels. Flanders examined

several textbook series and found that fewer than 50% of the

pages in textbooks for grades two through eight contained any

material new to students. In a review of a dozen middle-grade

mathematics textbook series, Kulm, Morris and Grier found that

most traditional textbook series lack many of the content rec-

ommendations made in recent standards documents.

United States data from TIMSS showed important differ-

ences in the content taught to students in different mathemat-

ics classes or tracks. For example, students in remedial classes,

typical eighth-grade classes and pre-algebra classes were

exposed to very different mathematics contents, and their

achievement levels varied accordingly. The achievement tests

used in international studies and in NAEP assessments measure

important mathematical outcomes and have commonly pro-

vided a broad and representative coverage of mathematics.

Moreover, the tests have generally served to measure what even

the most able students know and do not know. Consequently,

they provide reasonable outcome measures for research that

examines the importance of opportunity to learn as a factor in

student mathematics achievement.

In the classroom

The findings about the relationship between opportunity to

learn and student achievement have important implications for

teachers. In particular, it seems prudent to allocate sufficient

time for mathematics instruction at every grade level.Short class

periods in mathematics, instituted for whatever practical or

philosophical reason, should be seriously questioned. Of special

concern are the 30–35 minute class periods for mathematics

being implemented in some middle schools.

Textbooks that devote major attention to review and that

address little new content each year should be avoided, or their

use should be heavily supplemented in appropriate ways.

Teachers should use textbooks as just one instructional tool

11

ematics-course taking at the secondary school level and student

achievement. Reports from the NAEP in mathematics showed

that ‘the number of advanced mathematics courses taken was

the most powerful predictor of students’ mathematics perfor-

mance after adjusting for variations in home background’.

Textbooks are also related to student OTL, because many

textbooks do not contain much content that is new to students.

The lack of attention to new material and heavy emphasis on

review in many textbooks are of particular concern at the ele-

mentary school and middle-school levels. Flanders examined

several textbook series and found that fewer than 50% of the

pages in textbooks for grades two through eight contained any

material new to students. In a review of a dozen middle-grade

mathematics textbook series, Kulm, Morris and Grier found that

most traditional textbook series lack many of the content rec-

ommendations made in recent standards documents.

United States data from TIMSS showed important differ-

ences in the content taught to students in different mathemat-

ics classes or tracks. For example, students in remedial classes,

typical eighth-grade classes and pre-algebra classes were

exposed to very different mathematics contents, and their

achievement levels varied accordingly. The achievement tests

used in international studies and in NAEP assessments measure

important mathematical outcomes and have commonly pro-

vided a broad and representative coverage of mathematics.

Moreover, the tests have generally served to measure what even

the most able students know and do not know. Consequently,

they provide reasonable outcome measures for research that

examines the importance of opportunity to learn as a factor in

student mathematics achievement.

In the classroom

The findings about the relationship between opportunity to

learn and student achievement have important implications for

teachers. In particular, it seems prudent to allocate sufficient

time for mathematics instruction at every grade level.Short class

periods in mathematics, instituted for whatever practical or

philosophical reason, should be seriously questioned. Of special

concern are the 30–35 minute class periods for mathematics

being implemented in some middle schools.

Textbooks that devote major attention to review and that

address little new content each year should be avoided, or their

use should be heavily supplemented in appropriate ways.

Teachers should use textbooks as just one instructional tool

11

12.
among many,rather than feel duty-bound to go through the text-

book on a one-section-per-day basis.

Teachers must ensure that students are given the opportu-

nity to learn important content and skills.If students are to com-

pete effectively in a global, technologically oriented society,

they must be taught the mathematical skills needed to do so.

Thus, if problem solving is essential, explicit attention must be

given to it on a regular and sustained basis.If we expect students

to develop number sense, it is important to attend to mental

computation and estimation as part of the curriculum. If pro-

portional reasoning and deductive reasoning are important,

attention must be given to them in the curriculum implemented

in the classroom.

It is important to note that opportunity to learn is related to

equity issues. Some educational practices differentially affect

particular groups of students’opportunity to learn.For example,

a recent American Association of University Women study

showed that boys’ and girls’ use of technology is markedly dif-

ferent. Girls take fewer computer science and computer design

courses than do boys. Furthermore, boys often use computers

to programme and solve problems,whereas girls tend to use the

computer primarily as a word processor. This suggests that, as

technology is used in the mathematics classroom,teachers must

assign tasks and responsibilities to students in such a way that

both boys and girls have active learning experiences with the

technological tools employed.

OTL is also affected when low-achieving students are

tracked into special ‘basic skills’ curricula, oriented towards

developing procedural skills, with little opportunity to develop

problem solving and higher-order thinking abilities. The impov-

erished curriculum frequently provided to these students is an

especially serious problem because the ideas and concepts fre-

quently untaught or de-emphasized are the very ones needed in

everyday life and in the workplace.

References: American Association of University Women, 1998;

Atanda, 1999; Flanders, 1987; Grouws & Smith,

in press; Hawkins, Stancavage & Dossey, 1998;

Husén, 1967; Keeves, 1976, 1994; Kulm, Morris &

Grier, 1999; McKnight et al., 1987; Mullis, Jenkins

& Johnson, 1994; National Center for Education

Statistics, 1996, 1997, 1998; Schmidt, McKnight &

Raizen, 1997; Secada, 1992; Suarez et al., 1991.

12

book on a one-section-per-day basis.

Teachers must ensure that students are given the opportu-

nity to learn important content and skills.If students are to com-

pete effectively in a global, technologically oriented society,

they must be taught the mathematical skills needed to do so.

Thus, if problem solving is essential, explicit attention must be

given to it on a regular and sustained basis.If we expect students

to develop number sense, it is important to attend to mental

computation and estimation as part of the curriculum. If pro-

portional reasoning and deductive reasoning are important,

attention must be given to them in the curriculum implemented

in the classroom.

It is important to note that opportunity to learn is related to

equity issues. Some educational practices differentially affect

particular groups of students’opportunity to learn.For example,

a recent American Association of University Women study

showed that boys’ and girls’ use of technology is markedly dif-

ferent. Girls take fewer computer science and computer design

courses than do boys. Furthermore, boys often use computers

to programme and solve problems,whereas girls tend to use the

computer primarily as a word processor. This suggests that, as

technology is used in the mathematics classroom,teachers must

assign tasks and responsibilities to students in such a way that

both boys and girls have active learning experiences with the

technological tools employed.

OTL is also affected when low-achieving students are

tracked into special ‘basic skills’ curricula, oriented towards

developing procedural skills, with little opportunity to develop

problem solving and higher-order thinking abilities. The impov-

erished curriculum frequently provided to these students is an

especially serious problem because the ideas and concepts fre-

quently untaught or de-emphasized are the very ones needed in

everyday life and in the workplace.

References: American Association of University Women, 1998;

Atanda, 1999; Flanders, 1987; Grouws & Smith,

in press; Hawkins, Stancavage & Dossey, 1998;

Husén, 1967; Keeves, 1976, 1994; Kulm, Morris &

Grier, 1999; McKnight et al., 1987; Mullis, Jenkins

& Johnson, 1994; National Center for Education

Statistics, 1996, 1997, 1998; Schmidt, McKnight &

Raizen, 1997; Secada, 1992; Suarez et al., 1991.

12

13.
2. Focus on meaning

Focusing instruction on the meaningful

development of important mathematical

ideas increases the level of student learning.

Research findings

There is a long history of research, going back to the 1940s and

the work of William Brownell, on the effects of teaching for

meaning and understanding in mathematics.Investigations have

consistently shown that an emphasis on teaching for meaning

has positive effects on student learning, including better initial

learning, greater retention and an increased likelihood that the

ideas will be used in new situations. These results have also

been found in studies conducted in high-poverty areas.

In the classroom

As might be expected, the concept of ‘teaching for meaning’ has

varied somewhat from study to study,and has evolved over time.

Teachers will want to consider how various interpretations of

this concept can be incorporated into their classroom practice.

• Emphasize the mathematical meanings of ideas, includ-

ing how the idea, concept or skill is connected in multiple

ways to other mathematical ideas in a logically consistent

and sensible manner. Thus, for subtraction, emphasize the

inverse,or ‘undoing’, relationship between it and addition.In

general, emphasis on meaning was common in early

research in this area in the late 1930s, and its purpose was

to avoid the mathematical meaningfulness of the ideas

taught receiving only minor attention compared to a heavy

emphasis on the social uses and utility of mathematics in

everyday life.

• Create a classroom learning context in which students

can construct meaning. Students can learn important math-

ematics both in contexts that are closely connected to real-

life situations and in those that are purely mathematical.The

abstractness of a learning environment and how students

relate to it must be carefully regulated, closely monitored

and thoughtfully chosen. Consideration should be given to

students’ interests and backgrounds. The mathematics

13

Focusing instruction on the meaningful

development of important mathematical

ideas increases the level of student learning.

Research findings

There is a long history of research, going back to the 1940s and

the work of William Brownell, on the effects of teaching for

meaning and understanding in mathematics.Investigations have

consistently shown that an emphasis on teaching for meaning

has positive effects on student learning, including better initial

learning, greater retention and an increased likelihood that the

ideas will be used in new situations. These results have also

been found in studies conducted in high-poverty areas.

In the classroom

As might be expected, the concept of ‘teaching for meaning’ has

varied somewhat from study to study,and has evolved over time.

Teachers will want to consider how various interpretations of

this concept can be incorporated into their classroom practice.

• Emphasize the mathematical meanings of ideas, includ-

ing how the idea, concept or skill is connected in multiple

ways to other mathematical ideas in a logically consistent

and sensible manner. Thus, for subtraction, emphasize the

inverse,or ‘undoing’, relationship between it and addition.In

general, emphasis on meaning was common in early

research in this area in the late 1930s, and its purpose was

to avoid the mathematical meaningfulness of the ideas

taught receiving only minor attention compared to a heavy

emphasis on the social uses and utility of mathematics in

everyday life.

• Create a classroom learning context in which students

can construct meaning. Students can learn important math-

ematics both in contexts that are closely connected to real-

life situations and in those that are purely mathematical.The

abstractness of a learning environment and how students

relate to it must be carefully regulated, closely monitored

and thoughtfully chosen. Consideration should be given to

students’ interests and backgrounds. The mathematics

13

14.
taught and learned must seem reasonable to students and

make sense to them. An important factor in teaching for

meaning is connecting the new ideas and skills to students’

past knowledge and experience.

• Make explicit the connections between mathematics and

other subjects. For example, instruction could relate data-

gathering and data-representation skills to public opinion

polling in social studies. Or, it could relate the mathematical

concept of direct variation to the concept of force in

physics to help establish a real-world referent for the idea.

• Attend to student meanings and student understanding

in instruction. Students’ conceptions of the same idea will

vary, as will their methods of solving problems and carrying

out procedures. Teachers should build on students’ intuitive

notions and methods in designing and implementing

instruction.

References: Aubrey, 1997; Brownell, 1945, 1947; Carpenter et

al., 1998; Cobb et al., 1991; Fuson, 1992; Good,

Grouws & Ebmeier, 1983; Hiebert & Carpenter,

1992; Hiebert & Wearne, 1996; Hiebert et al., 1997;

Kamii, 1985, 1989, 1994; Knapp, Shields &

Turnbull, 1995; Koehler & Grouws, 1992; Skemp,

1978;Van Engen,1949;Wood & Sellers,1996,1997.

14

make sense to them. An important factor in teaching for

meaning is connecting the new ideas and skills to students’

past knowledge and experience.

• Make explicit the connections between mathematics and

other subjects. For example, instruction could relate data-

gathering and data-representation skills to public opinion

polling in social studies. Or, it could relate the mathematical

concept of direct variation to the concept of force in

physics to help establish a real-world referent for the idea.

• Attend to student meanings and student understanding

in instruction. Students’ conceptions of the same idea will

vary, as will their methods of solving problems and carrying

out procedures. Teachers should build on students’ intuitive

notions and methods in designing and implementing

instruction.

References: Aubrey, 1997; Brownell, 1945, 1947; Carpenter et

al., 1998; Cobb et al., 1991; Fuson, 1992; Good,

Grouws & Ebmeier, 1983; Hiebert & Carpenter,

1992; Hiebert & Wearne, 1996; Hiebert et al., 1997;

Kamii, 1985, 1989, 1994; Knapp, Shields &

Turnbull, 1995; Koehler & Grouws, 1992; Skemp,

1978;Van Engen,1949;Wood & Sellers,1996,1997.

14

15.
3. Learning new concepts

and skills while solving

problems

Students can learn both concepts and skills

by solving problems.

Research findings

Research suggests that students who develop conceptual under-

standing early perform best on procedural knowledge later.

Students with good conceptual understanding are able to per-

form successfully on near-transfer tasks and to develop proce-

dures and skills they have not been taught. Students without

conceptual understanding are able to acquire procedural

knowledge when the skill is taught, but research suggests that

students with low levels of conceptual understanding need

more practice in order to acquire procedural knowledge.

Research by Heid suggests that students are able to under-

stand concepts without prior or concurrent skill development.

In her research with calculus students, instruction was focused

almost entirely on conceptual understanding. Skills were taught

briefly at the end of the course. On procedural skills, the stu-

dents in the conceptual-understanding approach performed as

well as those taught with a traditional approach. Furthermore,

these students significantly outperformed traditional students

on conceptual understanding.

Mack demonstrated that students’ rote (and frequently

faulty) knowledge often interferes with their informal (and usu-

ally correct) knowledge about fractions. She successfully used

students’ informal knowledge to help them understand symbols

for fractions and develop algorithms for operations. Fawcett’s

research with geometry students suggests that students can

learn basic concepts, skills and the structure of geometry

through problem solving.

In the classroom

There is evidence that students can learn new skills and con-

cepts while they are working out solutions to problems. For

15

and skills while solving

problems

Students can learn both concepts and skills

by solving problems.

Research findings

Research suggests that students who develop conceptual under-

standing early perform best on procedural knowledge later.

Students with good conceptual understanding are able to per-

form successfully on near-transfer tasks and to develop proce-

dures and skills they have not been taught. Students without

conceptual understanding are able to acquire procedural

knowledge when the skill is taught, but research suggests that

students with low levels of conceptual understanding need

more practice in order to acquire procedural knowledge.

Research by Heid suggests that students are able to under-

stand concepts without prior or concurrent skill development.

In her research with calculus students, instruction was focused

almost entirely on conceptual understanding. Skills were taught

briefly at the end of the course. On procedural skills, the stu-

dents in the conceptual-understanding approach performed as

well as those taught with a traditional approach. Furthermore,

these students significantly outperformed traditional students

on conceptual understanding.

Mack demonstrated that students’ rote (and frequently

faulty) knowledge often interferes with their informal (and usu-

ally correct) knowledge about fractions. She successfully used

students’ informal knowledge to help them understand symbols

for fractions and develop algorithms for operations. Fawcett’s

research with geometry students suggests that students can

learn basic concepts, skills and the structure of geometry

through problem solving.

In the classroom

There is evidence that students can learn new skills and con-

cepts while they are working out solutions to problems. For

15

16.
example, armed with only a knowledge of basic addition, stu-

dents can extend their learning by developing informal algo-

rithms for addition of larger numbers. Similarly, by solving care-

fully chosen non-routine problems, students can develop an

understanding of many important mathematical ideas, such as

prime numbers and perimeter/area relations.

Development of more sophisticated mathematical skills can

also be approached by treating their development as a problem

for students to solve. Teachers can use students’ informal and

intuitive knowledge in other areas to develop other useful pro-

cedures. Instruction can begin with an example for which stu-

dents intuitively know the answer. From there, students are

allowed to explore and develop their own algorithm. For

instance, most students understand that starting with four piz-

zas and then eating a half of one pizza will leave three and a half

pizzas. Teachers can use this knowledge to help students

develop an understanding of subtraction of fractions.

Research suggests that it is not necessary for teachers to

focus first on skill development and then move on to problem

solving. Both can be done together. Skills can be developed on

an as-needed basis, or their development can be supplemented

through the use of technology. In fact, there is evidence that if

students are initially drilled too much on isolated skills, they

have a harder time making sense of them later.

References: Cognition and Technology Group, 1997; Fawcett,

1938; Heid, 1988; Hiebert & Wearne, 1996; Mack,

1990; Resnick & Omanson, 1987; Wearne &

Hiebert, 1988.

16

dents can extend their learning by developing informal algo-

rithms for addition of larger numbers. Similarly, by solving care-

fully chosen non-routine problems, students can develop an

understanding of many important mathematical ideas, such as

prime numbers and perimeter/area relations.

Development of more sophisticated mathematical skills can

also be approached by treating their development as a problem

for students to solve. Teachers can use students’ informal and

intuitive knowledge in other areas to develop other useful pro-

cedures. Instruction can begin with an example for which stu-

dents intuitively know the answer. From there, students are

allowed to explore and develop their own algorithm. For

instance, most students understand that starting with four piz-

zas and then eating a half of one pizza will leave three and a half

pizzas. Teachers can use this knowledge to help students

develop an understanding of subtraction of fractions.

Research suggests that it is not necessary for teachers to

focus first on skill development and then move on to problem

solving. Both can be done together. Skills can be developed on

an as-needed basis, or their development can be supplemented

through the use of technology. In fact, there is evidence that if

students are initially drilled too much on isolated skills, they

have a harder time making sense of them later.

References: Cognition and Technology Group, 1997; Fawcett,

1938; Heid, 1988; Hiebert & Wearne, 1996; Mack,

1990; Resnick & Omanson, 1987; Wearne &

Hiebert, 1988.

16

17.
4. Opportunities for both

invention and practice

Giving students both an opportunity to

discover and invent new knowledge and

an opportunity to practise what they have

learned improves student achievement

Research findings

Data from the TIMSS video study show that over 90% of mathe-

matics class time in United States eighth-grade classrooms is

spent practising routine procedures, with the remainder of the

time generally spent applying procedures in new situations.

Virtually no time is spent inventing new procedures and

analysing unfamiliar situations. In contrast, students at the same

grade level in typical Japanese classrooms spend approximately

40% of instructional time practising routine procedures, 15%

applying procedures in new situations, and 45% inventing new

procedures and analysing new situations.

Research evidence suggests that students need opportuni-

ties for both practice and invention.The findings from a number

of research studies show that when students discover mathe-

matical ideas and invent mathematical procedures, they have a

stronger conceptual understanding of connections between

mathematical ideas.

Many successful reform-oriented programmes include time

for students to practise what they have learned and discovered.

Students need opportunities to practise what they are learning

and to experience performing the kinds of tasks in which they

are expected to demonstrate competence. For example,if teach-

ers want students to be proficient in problem solving, students

must be given opportunities to practise problem solving. If

strong deductive reasoning is a goal, student work must include

tasks that require such reasoning. And, of course, if competence

in procedures is an objective, the curriculum must include

attention to such procedures.

17

invention and practice

Giving students both an opportunity to

discover and invent new knowledge and

an opportunity to practise what they have

learned improves student achievement

Research findings

Data from the TIMSS video study show that over 90% of mathe-

matics class time in United States eighth-grade classrooms is

spent practising routine procedures, with the remainder of the

time generally spent applying procedures in new situations.

Virtually no time is spent inventing new procedures and

analysing unfamiliar situations. In contrast, students at the same

grade level in typical Japanese classrooms spend approximately

40% of instructional time practising routine procedures, 15%

applying procedures in new situations, and 45% inventing new

procedures and analysing new situations.

Research evidence suggests that students need opportuni-

ties for both practice and invention.The findings from a number

of research studies show that when students discover mathe-

matical ideas and invent mathematical procedures, they have a

stronger conceptual understanding of connections between

mathematical ideas.

Many successful reform-oriented programmes include time

for students to practise what they have learned and discovered.

Students need opportunities to practise what they are learning

and to experience performing the kinds of tasks in which they

are expected to demonstrate competence. For example,if teach-

ers want students to be proficient in problem solving, students

must be given opportunities to practise problem solving. If

strong deductive reasoning is a goal, student work must include

tasks that require such reasoning. And, of course, if competence

in procedures is an objective, the curriculum must include

attention to such procedures.

17

18.
In the classroom

Clearly, a balance is needed between the time students spend

practising routine procedures and the time which they devote

to inventing and discovering new ideas. Teachers need not

choose between these activities; indeed, they must not make a

choice if students are to develop the mathematical power they

need. Teachers must strive to ensure that both activities are

included in appropriate proportions and in appropriate ways.

The research cited above suggests that attention to them is cur-

rently out of balance and that too frequently there is an over-

emphasis on skill work, with few opportunities for students to

engage in sense-making and discovery-oriented activities.

To increase opportunities for invention, teachers should fre-

quently use non-routine problems, periodically introduce a les-

son involving a new skill by posing it as a problem to be solved,

and regularly allow students to build new knowledge based on

their intuitive knowledge and informal procedures.

References: Boaler, 1998; Brownell, 1945, 1947; Carpenter et

al., 1998; Cobb et al., 1991; Cognition and

Technology Group, 1997; Resnick, 1980; Stigler &

Hiebert, 1997;Wood & Sellers, 1996, 1997.

18

Clearly, a balance is needed between the time students spend

practising routine procedures and the time which they devote

to inventing and discovering new ideas. Teachers need not

choose between these activities; indeed, they must not make a

choice if students are to develop the mathematical power they

need. Teachers must strive to ensure that both activities are

included in appropriate proportions and in appropriate ways.

The research cited above suggests that attention to them is cur-

rently out of balance and that too frequently there is an over-

emphasis on skill work, with few opportunities for students to

engage in sense-making and discovery-oriented activities.

To increase opportunities for invention, teachers should fre-

quently use non-routine problems, periodically introduce a les-

son involving a new skill by posing it as a problem to be solved,

and regularly allow students to build new knowledge based on

their intuitive knowledge and informal procedures.

References: Boaler, 1998; Brownell, 1945, 1947; Carpenter et

al., 1998; Cobb et al., 1991; Cognition and

Technology Group, 1997; Resnick, 1980; Stigler &

Hiebert, 1997;Wood & Sellers, 1996, 1997.

18

19.
5. Openness to student

solution methods and

student interaction

Teaching that incorporates students’

intuitive solution methods can increase

student learning, especially when combined

with opportunities for student interaction

and discussion.

Research findings

Recent results from the TIMSS video study have shown that

Japanese classrooms use student solution methods extensively

during instruction. Interestingly, the same teaching technique

appears in many successful American research projects.

Findings from American studies clearly demonstrate two impor-

tant principles that are associated with the development of stu-

dents’deep conceptual understanding of mathematics. First,stu-

dent achievement and understanding are significantly improved

when teachers are aware of how students construct knowledge,

are familiar with the intuitive solution methods that students

use when they solve problems,and utilize this knowledge when

planning and conducting instruction in mathematics. These

results have been clearly demonstrated in the primary grades

and are beginning to be shown at higher-grade levels.

Second, structuring instruction around carefully chosen

problems, allowing students to interact when solving these

problems, and then providing opportunities for them to share

their solution methods result in increased achievement on prob-

lem-solving measures. Importantly, these gains come without a

loss of achievement in the skills and concepts measured on stan-

dardized achievement tests.

Research has also demonstrated that when students have

opportunities to develop their own solution methods, they are

better able to apply mathematical knowledge in new problem

19

solution methods and

student interaction

Teaching that incorporates students’

intuitive solution methods can increase

student learning, especially when combined

with opportunities for student interaction

and discussion.

Research findings

Recent results from the TIMSS video study have shown that

Japanese classrooms use student solution methods extensively

during instruction. Interestingly, the same teaching technique

appears in many successful American research projects.

Findings from American studies clearly demonstrate two impor-

tant principles that are associated with the development of stu-

dents’deep conceptual understanding of mathematics. First,stu-

dent achievement and understanding are significantly improved

when teachers are aware of how students construct knowledge,

are familiar with the intuitive solution methods that students

use when they solve problems,and utilize this knowledge when

planning and conducting instruction in mathematics. These

results have been clearly demonstrated in the primary grades

and are beginning to be shown at higher-grade levels.

Second, structuring instruction around carefully chosen

problems, allowing students to interact when solving these

problems, and then providing opportunities for them to share

their solution methods result in increased achievement on prob-

lem-solving measures. Importantly, these gains come without a

loss of achievement in the skills and concepts measured on stan-

dardized achievement tests.

Research has also demonstrated that when students have

opportunities to develop their own solution methods, they are

better able to apply mathematical knowledge in new problem

19

20.
In the classroom

Research results suggest that teachers should concentrate on

providing opportunities for students to interact in problem-rich

situations. Besides providing appropriate problem-rich situa-

tions, teachers must encourage students to find their own solu-

tion methods and give them opportunities to share and com-

pare their solution methods and answers. One way to organize

such instruction is to have students work in small groups ini-

tially and then share ideas and solutions in a whole-class discus-

One useful teaching technique is for teachers to assign an

interesting problem for students to solve and then move about

the room as they work, keeping track of which students are

using which strategies (taking notes if necessary). In a whole-

class setting, the teacher can then call on students to discuss

their solution methods in a pre-determined and carefully con-

sidered order, these methods often ranging from the most basic

to more formal or sophisticated ones. This teaching structure is

used successfully in many Japanese mathematics lessons.

References: Boaler, 1998; Carpenter et al., 1988, 1989, 1998;

Cobb, Yackel & Wood, 1992; Cobb et al., 1991;

Cognition and Technology Group, 1997; Fennema,

Carpenter & Peterson, 1989; Fennema et al., 1993,

1996; Hiebert & Wearne, 1993, 1996; Kamii, 1985,

1989, 1994; Stigler & Hiebert, 1997; Stigler et al.,

1999; Wood, Cobb & Yackel, 1995; Wood et al.,

1993;Yackel, Cobb & Wood, 1991.

20

Research results suggest that teachers should concentrate on

providing opportunities for students to interact in problem-rich

situations. Besides providing appropriate problem-rich situa-

tions, teachers must encourage students to find their own solu-

tion methods and give them opportunities to share and com-

pare their solution methods and answers. One way to organize

such instruction is to have students work in small groups ini-

tially and then share ideas and solutions in a whole-class discus-

One useful teaching technique is for teachers to assign an

interesting problem for students to solve and then move about

the room as they work, keeping track of which students are

using which strategies (taking notes if necessary). In a whole-

class setting, the teacher can then call on students to discuss

their solution methods in a pre-determined and carefully con-

sidered order, these methods often ranging from the most basic

to more formal or sophisticated ones. This teaching structure is

used successfully in many Japanese mathematics lessons.

References: Boaler, 1998; Carpenter et al., 1988, 1989, 1998;

Cobb, Yackel & Wood, 1992; Cobb et al., 1991;

Cognition and Technology Group, 1997; Fennema,

Carpenter & Peterson, 1989; Fennema et al., 1993,

1996; Hiebert & Wearne, 1993, 1996; Kamii, 1985,

1989, 1994; Stigler & Hiebert, 1997; Stigler et al.,

1999; Wood, Cobb & Yackel, 1995; Wood et al.,

1993;Yackel, Cobb & Wood, 1991.

20

21.
6. Small-group learning

Using small groups of students to work on

activities, problems and assignments can

increase student mathematics achievement.

Research findings

Considerable research evidence within mathematics education

indicates that using small groups of various types for different

classroom tasks has positive effects on student learning.

Davidson, for example, reviewed almost eighty studies in math-

ematics that compared student achievement in small-group set-

tings with traditional whole-class instruction. In more than 40%

of these studies, students in the classes using small-group

approaches significantly outscored control students on mea-

sures of student performance. In only two of the seventy-nine

studies did control-group students perform better than the

small-group students, and in these studies there were some

design irregularities.

From a review of ninety-nine studies of co-operative

group-learning methods at the elementary and secondary

school levels, Slavin concluded that co-operative methods were

effective in improving student achievement. The most effective

methods emphasized both group goals and individual account-

From a review by Webb of studies examining peer interac-

tion and achievement in small groups (seventeen studies,grades

2–11), several consistent findings emerged. First, giving an

explanation of an idea, method or solution to a team mate in a

group situation was positively related to achievement. Second,

receiving ‘non-responsive’ feedback (no feedback or feedback

that is not pertinent to what one has said or done) from team

mates was negatively related to achievement. Webb’s review

also showed that group work was most effective when students

were taught how to work in groups and how to give and receive

help. Received help was most effective when it was in the form

of elaborated explanations (not just the answer) and then

applied by the student either to the current problem or to a new

21

Using small groups of students to work on

activities, problems and assignments can

increase student mathematics achievement.

Research findings

Considerable research evidence within mathematics education

indicates that using small groups of various types for different

classroom tasks has positive effects on student learning.

Davidson, for example, reviewed almost eighty studies in math-

ematics that compared student achievement in small-group set-

tings with traditional whole-class instruction. In more than 40%

of these studies, students in the classes using small-group

approaches significantly outscored control students on mea-

sures of student performance. In only two of the seventy-nine

studies did control-group students perform better than the

small-group students, and in these studies there were some

design irregularities.

From a review of ninety-nine studies of co-operative

group-learning methods at the elementary and secondary

school levels, Slavin concluded that co-operative methods were

effective in improving student achievement. The most effective

methods emphasized both group goals and individual account-

From a review by Webb of studies examining peer interac-

tion and achievement in small groups (seventeen studies,grades

2–11), several consistent findings emerged. First, giving an

explanation of an idea, method or solution to a team mate in a

group situation was positively related to achievement. Second,

receiving ‘non-responsive’ feedback (no feedback or feedback

that is not pertinent to what one has said or done) from team

mates was negatively related to achievement. Webb’s review

also showed that group work was most effective when students

were taught how to work in groups and how to give and receive

help. Received help was most effective when it was in the form

of elaborated explanations (not just the answer) and then

applied by the student either to the current problem or to a new

21

22.
Qualitative investigations have shown that other important

and often unmeasured outcomes beyond improved general

achievement can result from small-group work. In one such

investigation, Yackel, Cobb and Wood studied a second-grade

classroom in which small-group problem solving followed by

whole-class discussion was the primary instructional strategy

for the entire school year. They found that this approach created

many learning opportunities that do not typically occur in tra-

ditional classrooms, including opportunities for collaborative

dialogue and resolution of conflicting points of view.

Slavin’s research showed positive effects of small-group

work on cross-ethnic relations and student attitudes towards

In the classroom

Research findings clearly support the use of small groups as part

of mathematics instruction. This approach can result in

increased student learning as measured by traditional achieve-

ment measures, as well as in other important outcomes.

When using small groups for mathematics instruction,

teachers should:

• choose tasks that deal with important mathematical con-

cepts and ideas;

• select tasks that are appropriate for group work;

• consider having students initially work individually on a task

and then follow this with group work where students share

and build on their individual ideas and work;

• give clear instructions to the groups and set clear expecta-

tions for each;

• emphasize both group goals and individual accountability;

• choose tasks that students find interesting;

• ensure that there is closure to the group work, where key

ideas and methods are brought to the surface either by the

teacher or the students, or both.

Finally, as several research studies have shown, teachers should

not think of small groups as something that must always be used

or never be used. Rather, small-group instruction should be

thought of as an instructional practice that is appropriate for

certain learning objectives, and as a practice that can work well

with other organizational arrangements, including whole-class

References: Cohen, 1994; Davidson, 1985; Laborde, 1994;

Slavin, 1990, 1995; Webb, 1991; Webb, Troper &

Fall, 1995;Yackel, Cobb & Wood, 1991.

22

and often unmeasured outcomes beyond improved general

achievement can result from small-group work. In one such

investigation, Yackel, Cobb and Wood studied a second-grade

classroom in which small-group problem solving followed by

whole-class discussion was the primary instructional strategy

for the entire school year. They found that this approach created

many learning opportunities that do not typically occur in tra-

ditional classrooms, including opportunities for collaborative

dialogue and resolution of conflicting points of view.

Slavin’s research showed positive effects of small-group

work on cross-ethnic relations and student attitudes towards

In the classroom

Research findings clearly support the use of small groups as part

of mathematics instruction. This approach can result in

increased student learning as measured by traditional achieve-

ment measures, as well as in other important outcomes.

When using small groups for mathematics instruction,

teachers should:

• choose tasks that deal with important mathematical con-

cepts and ideas;

• select tasks that are appropriate for group work;

• consider having students initially work individually on a task

and then follow this with group work where students share

and build on their individual ideas and work;

• give clear instructions to the groups and set clear expecta-

tions for each;

• emphasize both group goals and individual accountability;

• choose tasks that students find interesting;

• ensure that there is closure to the group work, where key

ideas and methods are brought to the surface either by the

teacher or the students, or both.

Finally, as several research studies have shown, teachers should

not think of small groups as something that must always be used

or never be used. Rather, small-group instruction should be

thought of as an instructional practice that is appropriate for

certain learning objectives, and as a practice that can work well

with other organizational arrangements, including whole-class

References: Cohen, 1994; Davidson, 1985; Laborde, 1994;

Slavin, 1990, 1995; Webb, 1991; Webb, Troper &

Fall, 1995;Yackel, Cobb & Wood, 1991.

22

23.
7. Whole-class discussion

Whole-class discussion following individual

and group work improves student

achievement.

Research findings

Research suggests that whole-class discussion can be effective

when it is used for sharing and explaining the variety of solu-

tions by which individual students have solved problems. It

allows students to see the many ways of examining a situation

and the variety of appropriate and acceptable solutions.

Wood found that whole-class discussion works best when

discussion expectations are clearly understood. Students should

be expected to evaluate each other’s ideas and reasoning in

ways that are not critical of the sharer. This helps to create an

environment in which students feel comfortable sharing ideas

and discussing each other’s methods and reasoning.

Furthermore, students should be expected to be active listeners

who participate in the discussion and feel a sense of responsi-

bility for each other’s understanding.

Cognitive research suggests that conceptual change and

progression of thought result from the mental processes

involved in the resolution of conflict and contradiction. Thus,

confusion and conflict during whole-class discussion have con-

siderable potential for increasing student learning when care-

fully managed by the teacher. As students address challenges to

their methods, they strengthen their understanding of concepts

and procedures by working together to resolve differences in

thinking or confusions in reasoning. In a sense, the discussion

becomes a collaborative problem-solving effort. Each individual

then is contributing to the total outcome of the problem-solving

situation. This discussion helps produce the notion of com-

monly held knowledge (public knowledge).

In the classroom

It is important that whole-class discussion follow student work

on problem-solving activities.The discussion should be a sum-

mary of individual work in which key ideas are brought to the

23

Whole-class discussion following individual

and group work improves student

achievement.

Research findings

Research suggests that whole-class discussion can be effective

when it is used for sharing and explaining the variety of solu-

tions by which individual students have solved problems. It

allows students to see the many ways of examining a situation

and the variety of appropriate and acceptable solutions.

Wood found that whole-class discussion works best when

discussion expectations are clearly understood. Students should

be expected to evaluate each other’s ideas and reasoning in

ways that are not critical of the sharer. This helps to create an

environment in which students feel comfortable sharing ideas

and discussing each other’s methods and reasoning.

Furthermore, students should be expected to be active listeners

who participate in the discussion and feel a sense of responsi-

bility for each other’s understanding.

Cognitive research suggests that conceptual change and

progression of thought result from the mental processes

involved in the resolution of conflict and contradiction. Thus,

confusion and conflict during whole-class discussion have con-

siderable potential for increasing student learning when care-

fully managed by the teacher. As students address challenges to

their methods, they strengthen their understanding of concepts

and procedures by working together to resolve differences in

thinking or confusions in reasoning. In a sense, the discussion

becomes a collaborative problem-solving effort. Each individual

then is contributing to the total outcome of the problem-solving

situation. This discussion helps produce the notion of com-

monly held knowledge (public knowledge).

In the classroom

It is important that whole-class discussion follow student work

on problem-solving activities.The discussion should be a sum-

mary of individual work in which key ideas are brought to the

23

24.
surface. This can be accomplished through students presenting

and discussing their individual solution methods, or through

other methods of achieving closure that are led by the teacher,

the students, or both.

Whole-class discussion can also be an effective diagnostic

tool for determining the depth of student understanding and

identifying misconceptions. Teachers can identify areas of diffi-

culty for particular students, as well as ascertain areas of student

success or progress.

Whole-class discussion can be an effective and useful

instructional practice. Some of the instructional opportunities

offered in whole-class discussion do not occur in small group or

individual settings. Thus, whole-class discussion has an impor-

tant place in the classroom together with other instructional

References: Ball, 1993; Cobb et al., 1992; Wood, 1999.

24

and discussing their individual solution methods, or through

other methods of achieving closure that are led by the teacher,

the students, or both.

Whole-class discussion can also be an effective diagnostic

tool for determining the depth of student understanding and

identifying misconceptions. Teachers can identify areas of diffi-

culty for particular students, as well as ascertain areas of student

success or progress.

Whole-class discussion can be an effective and useful

instructional practice. Some of the instructional opportunities

offered in whole-class discussion do not occur in small group or

individual settings. Thus, whole-class discussion has an impor-

tant place in the classroom together with other instructional

References: Ball, 1993; Cobb et al., 1992; Wood, 1999.

24

25.
8. Number sense

Teaching mathematics with a focus on

number sense encourages students to

become problem solvers in a wide variety

of situations and to view mathematics as a

discipline in which thinking is important.

Research findings

‘Number sense’ relates to having an intuitive feel for number

size and combinations, as well as the ability to work flexibly

with numbers in problem situations in order to make sound

decisions and reasonable judgements. It involves being able to

use flexibly the processes of mentally computing, estimating,

sensing number magnitudes, moving between representation

systems for numbers, and judging the reasonableness of numer-

ical results.

Markovits and Sowder studied seventh-grade classrooms

where special units on number magnitude, mental computation

and computational estimation were taught. From individual

interviews, they determined that after this special instruction

students were more likely to use strategies that reflected sound

number sense, and that this was a long-lasting change.

Other important research in this area involves the integra-

tion of the development of number sense with the teaching of

other mathematical topics, as opposed to teaching separate

lessons on aspects of number sense. In a study of second

graders, Cobb and his colleagues found that students’ number

sense was improved as a result of a problem-centred curriculum

that emphasized student interaction and self-generated solution

methods. Almost every student developed a variety of strategies

to solve a wide range of problems. Students also demonstrated

other desirable affective outcomes, such as increased persis-

tence in solving problems.

Kamii worked with primary-grade teachers as they

attempted to implement an instructional approach rooted in a

constructivist theory of learning that is based on the work of

Piaget. Central to the instructional approach was providing sit-

uations for students to develop their own meanings, methods

25

Teaching mathematics with a focus on

number sense encourages students to

become problem solvers in a wide variety

of situations and to view mathematics as a

discipline in which thinking is important.

Research findings

‘Number sense’ relates to having an intuitive feel for number

size and combinations, as well as the ability to work flexibly

with numbers in problem situations in order to make sound

decisions and reasonable judgements. It involves being able to

use flexibly the processes of mentally computing, estimating,

sensing number magnitudes, moving between representation

systems for numbers, and judging the reasonableness of numer-

ical results.

Markovits and Sowder studied seventh-grade classrooms

where special units on number magnitude, mental computation

and computational estimation were taught. From individual

interviews, they determined that after this special instruction

students were more likely to use strategies that reflected sound

number sense, and that this was a long-lasting change.

Other important research in this area involves the integra-

tion of the development of number sense with the teaching of

other mathematical topics, as opposed to teaching separate

lessons on aspects of number sense. In a study of second

graders, Cobb and his colleagues found that students’ number

sense was improved as a result of a problem-centred curriculum

that emphasized student interaction and self-generated solution

methods. Almost every student developed a variety of strategies

to solve a wide range of problems. Students also demonstrated

other desirable affective outcomes, such as increased persis-

tence in solving problems.

Kamii worked with primary-grade teachers as they

attempted to implement an instructional approach rooted in a

constructivist theory of learning that is based on the work of

Piaget. Central to the instructional approach was providing sit-

uations for students to develop their own meanings, methods

25

26.
and number sense.Data obtained from interviews with students

showed that the treatment group demonstrated a greater auton-

omy, conceptual understanding of place value, and ability to do

estimation and mental computation than did students in com-

parison classrooms.

In the classroom

Attention to number sense when teaching a wide variety of

mathematical topics tends to enhance the depth of student abil-

ity in this area. Competence in the many aspects of number

sense is an important mathematical outcome for students. Over

90% of the computation done outside the classroom is done

without pencil and paper,using mental computation,estimation

or a calculator. However, in many classrooms, efforts to instil

number sense are given insufficient attention.

As teachers develop strategies to teach number sense, they

should strongly consider moving beyond a unit-skills approach

(i.e. a focus on single skills in isolation) to a more integrated

approach that encourages the development of number sense in

all classroom activities, from the development of computational

procedures to mathematical problem solving. Although more

research is needed,an integrated approach to number sense will

be likely to result not only in greater number sense but also in

other equally important outcomes.

References: Cobb et al., 1991; Greeno, 1991; Kamii, 1985, 1989,

1994; Markovits & Sowder, 1994; Reys & Barger,

1994; Reys et al., 1991; Sowder, 1992a, 1992b.

26

showed that the treatment group demonstrated a greater auton-

omy, conceptual understanding of place value, and ability to do

estimation and mental computation than did students in com-

parison classrooms.

In the classroom

Attention to number sense when teaching a wide variety of

mathematical topics tends to enhance the depth of student abil-

ity in this area. Competence in the many aspects of number

sense is an important mathematical outcome for students. Over

90% of the computation done outside the classroom is done

without pencil and paper,using mental computation,estimation

or a calculator. However, in many classrooms, efforts to instil

number sense are given insufficient attention.

As teachers develop strategies to teach number sense, they

should strongly consider moving beyond a unit-skills approach

(i.e. a focus on single skills in isolation) to a more integrated

approach that encourages the development of number sense in

all classroom activities, from the development of computational

procedures to mathematical problem solving. Although more

research is needed,an integrated approach to number sense will

be likely to result not only in greater number sense but also in

other equally important outcomes.

References: Cobb et al., 1991; Greeno, 1991; Kamii, 1985, 1989,

1994; Markovits & Sowder, 1994; Reys & Barger,

1994; Reys et al., 1991; Sowder, 1992a, 1992b.

26

27.
9. Concrete materials

Long-term use of concrete materials is

positively related to increases in student

mathematics achievement and improved

attitudes towards mathematics.

Research findings

Many studies show that the use of concrete materials can pro-

duce meaningful use of notational systems and increase student

concept development. In a comprehensive review of activity-

based learning in mathematics in kindergarten through grade

eight, Suydam and Higgins concluded that using manipulative

materials produces greater achievement gains than not using

them. In a more recent meta-analysis of sixty studies (kinder-

garten through post-secondary) that compared the effects of

using concrete materials with the effects of more abstract

instruction, Sowell concluded that the long-term use of con-

crete instructional materials by teachers knowledgeable in their

use improved student achievement and attitudes.

In spite of generally positive results, there are some incon-

sistencies in the research findings. As Thompson points out, the

research results concerning concrete materials vary, even

among treatments that were closely controlled and monitored

and that involved the same concrete materials. For example, in

studies by Resnick and Omanson and by Labinowicz, the use of

base-ten blocks showed little impact on children’s learning. In

contrast, both Fuson and Briars and Hiebert and Wearne

reported positive results from the use of base-ten blocks.

The differences in results among these studies might be due

to the nature of the students’ engagement with the concrete

materials and their orientation towards the materials in relation

to notation and numerical values. They might also be due to dif-

ferent orientations in the studies,with regard to the role of com-

putational algorithms and how they should be developed in the

classroom. In general, however, the ambiguities in some of the

research findings do not undermine the general consensus that

concrete materials are valuable instructional tools.

27

Long-term use of concrete materials is

positively related to increases in student

mathematics achievement and improved

attitudes towards mathematics.

Research findings

Many studies show that the use of concrete materials can pro-

duce meaningful use of notational systems and increase student

concept development. In a comprehensive review of activity-

based learning in mathematics in kindergarten through grade

eight, Suydam and Higgins concluded that using manipulative

materials produces greater achievement gains than not using

them. In a more recent meta-analysis of sixty studies (kinder-

garten through post-secondary) that compared the effects of

using concrete materials with the effects of more abstract

instruction, Sowell concluded that the long-term use of con-

crete instructional materials by teachers knowledgeable in their

use improved student achievement and attitudes.

In spite of generally positive results, there are some incon-

sistencies in the research findings. As Thompson points out, the

research results concerning concrete materials vary, even

among treatments that were closely controlled and monitored

and that involved the same concrete materials. For example, in

studies by Resnick and Omanson and by Labinowicz, the use of

base-ten blocks showed little impact on children’s learning. In

contrast, both Fuson and Briars and Hiebert and Wearne

reported positive results from the use of base-ten blocks.

The differences in results among these studies might be due

to the nature of the students’ engagement with the concrete

materials and their orientation towards the materials in relation

to notation and numerical values. They might also be due to dif-

ferent orientations in the studies,with regard to the role of com-

putational algorithms and how they should be developed in the

classroom. In general, however, the ambiguities in some of the

research findings do not undermine the general consensus that

concrete materials are valuable instructional tools.

27

28.
In the classroom

Although successful teaching requires teachers to carefully

choose their procedures on the basis of the context in which

they will be used, available research suggests that teachers

should use manipulative materials in mathematics instruction

more regularly in order to give students hands-on experience

that helps them construct useful meanings for the mathematical

ideas they are learning. Use of the same material to teach multi-

ple ideas over the course of schooling has the advantage of

shortening the amount of time it takes to introduce the mater-

ial and also helps students to see connections between ideas.

The use of concrete material should not be limited to

demonstrations. It is essential that children use materials in

meaningful ways rather than in a rigid and prescribed way that

focuses on remembering rather than on thinking. Thus, as

Thompson says, ‘before students can make productive use of

concrete materials, they must first be committed to making

sense of their activities and be committed to expressing their

sense in meaningful ways. Further, it is important that students

come to see the two-way relationship between concrete

embodiments of a mathematical concept and the notational sys-

tem used to represent it.’

References: Fuson & Briars, 1990; Hiebert & Wearne, 1992;

Labinowicz, 1985; Leinenbach & Raymond, 1996;

Resnick & Omanson, 1987; Sowell, 1989; Suydam

& Higgins, 1977; Thompson, 1992; Varelas &

Becker, 1997.

28

Although successful teaching requires teachers to carefully

choose their procedures on the basis of the context in which

they will be used, available research suggests that teachers

should use manipulative materials in mathematics instruction

more regularly in order to give students hands-on experience

that helps them construct useful meanings for the mathematical

ideas they are learning. Use of the same material to teach multi-

ple ideas over the course of schooling has the advantage of

shortening the amount of time it takes to introduce the mater-

ial and also helps students to see connections between ideas.

The use of concrete material should not be limited to

demonstrations. It is essential that children use materials in

meaningful ways rather than in a rigid and prescribed way that

focuses on remembering rather than on thinking. Thus, as

Thompson says, ‘before students can make productive use of

concrete materials, they must first be committed to making

sense of their activities and be committed to expressing their

sense in meaningful ways. Further, it is important that students

come to see the two-way relationship between concrete

embodiments of a mathematical concept and the notational sys-

tem used to represent it.’

References: Fuson & Briars, 1990; Hiebert & Wearne, 1992;

Labinowicz, 1985; Leinenbach & Raymond, 1996;

Resnick & Omanson, 1987; Sowell, 1989; Suydam

& Higgins, 1977; Thompson, 1992; Varelas &

Becker, 1997.

28

29.
10. Students’ use of calculators

Using calculators in the learning of

mathematics can result in increased

achievement and improved student attitudes.

Research findings

The impact of calculator use on student learning has been a

popular research area in mathematics education.The many stud-

ies conducted have quite consistently shown that thoughtful

use of calculators in mathematics classes improves student

mathematics achievement and attitudes towards mathematics.

From a meta-analysis of seventy-nine non-graphing calcula-

tor studies, Hembree and Dessart concluded that the use of

hand-held calculators improved student learning. In particular,

they found improvement in students’ understanding of arith-

metical concepts and in their problem-solving skills.Their analy-

sis also showed that students using calculators tended to have

better attitudes towards mathematics and much better self-con-

cepts in mathematics than their counterparts who did not use

calculators. They also found that there was no loss in student

ability to perform paper-and-pencil computational skills when

calculators were used as part of mathematics instruction.

Research on the use of scientific calculators with graphing

capabilities has also shown positive effects on student achieve-

ment. Most studies have found positive effects on students’

graphing ability, conceptual understanding of graphs and their

ability to relate graphical representations to other representa-

tions, such as tables and symbolic representations. Other con-

tent areas where improvement has been shown when these cal-

culators have been used in instruction include function

concepts and spatial visualization.Other studies have found that

students are better problem solvers when using graphing cal-

culators. In addition,students are more flexible in their thinking

with regard to solution strategies, have greater perseverance

and focus more on trying to understand the problem conceptu-

ally rather than simply focusing on computations. However,

with increased use of graphing calculators, students are more

likely to rely on graphical procedures than on other procedures

such as algebraic methods. Most studies of graphing calculators

29

Using calculators in the learning of

mathematics can result in increased

achievement and improved student attitudes.

Research findings

The impact of calculator use on student learning has been a

popular research area in mathematics education.The many stud-

ies conducted have quite consistently shown that thoughtful

use of calculators in mathematics classes improves student

mathematics achievement and attitudes towards mathematics.

From a meta-analysis of seventy-nine non-graphing calcula-

tor studies, Hembree and Dessart concluded that the use of

hand-held calculators improved student learning. In particular,

they found improvement in students’ understanding of arith-

metical concepts and in their problem-solving skills.Their analy-

sis also showed that students using calculators tended to have

better attitudes towards mathematics and much better self-con-

cepts in mathematics than their counterparts who did not use

calculators. They also found that there was no loss in student

ability to perform paper-and-pencil computational skills when

calculators were used as part of mathematics instruction.

Research on the use of scientific calculators with graphing

capabilities has also shown positive effects on student achieve-

ment. Most studies have found positive effects on students’

graphing ability, conceptual understanding of graphs and their

ability to relate graphical representations to other representa-

tions, such as tables and symbolic representations. Other con-

tent areas where improvement has been shown when these cal-

culators have been used in instruction include function

concepts and spatial visualization.Other studies have found that

students are better problem solvers when using graphing cal-

culators. In addition,students are more flexible in their thinking

with regard to solution strategies, have greater perseverance

and focus more on trying to understand the problem conceptu-

ally rather than simply focusing on computations. However,

with increased use of graphing calculators, students are more

likely to rely on graphical procedures than on other procedures

such as algebraic methods. Most studies of graphing calculators

29

30.
have found no negative effect on basic skills, factual knowledge

or computational skills.

In general, research has found that the use of calculators

changes the content, methods and skill requirements in mathe-

matics classrooms. Studies have shown that teachers ask more

high-level questions when calculators are present, and students

become more actively involved through asking questions, con-

jecturing and exploring when they use calculators.

In the classroom

Research strongly supports the call in Curriculum and evalua-

tion standards for school mathematics, published by the

National Council of Teachers of Mathematics, for the use of cal-

culators at all levels of mathematics instruction. Using calcula-

tors in carefully planned ways can result in increases in student

problem-solving ability and improved affective outcomes with-

out a loss in basic skills.

One valuable use for calculators is as a tool for exploration

and discovery in problem-solving situations and when intro-

ducing new mathematical content. By reducing computation

time and providing immediate feedback, calculators help stu-

dents focus on understanding their work and justifying their

methods and results. The graphing calculator is particularly use-

ful in helping to illustrate and develop graphical concepts and

in making connections between algebraic and geometric ideas.

In order to accurately reflect their meaningful mathematics

performance, students should probably be allowed to use their

calculators in achievement tests. Not to do so is a major disrup-

tion in many students’ usual way of doing mathematics, and an

unrealistic restriction because when they are away from the

school setting, they will certainly use a calculator in their daily

lives and in the workplace. Another factor that argues for calcu-

lator use is that students are already permitted to use them in

some official tests. Furthermore, some examinations require the

candidates to use a graphing calculator.

References: Davis,1990;Drijvers & Doorman,1996;Dunham &

Dick, 1994; Flores & McLeod, 1990; Giamati, 1991;

Groves & Stacey, 1998; Harvey, 1993; Hembree &

Dessart, 1986, 1992; Mullis, Jenkins & Johnson,

1994; National Council of Teachers of

Mathematics, 1989; Penglase & Arnold, 1996; Rich,

1991; Ruthven, 1990; Slavit, 1996; Smith, 1996;

Stacey & Groves, 1994;Wilson & Krapfl, 1994.

30

or computational skills.

In general, research has found that the use of calculators

changes the content, methods and skill requirements in mathe-

matics classrooms. Studies have shown that teachers ask more

high-level questions when calculators are present, and students

become more actively involved through asking questions, con-

jecturing and exploring when they use calculators.

In the classroom

Research strongly supports the call in Curriculum and evalua-

tion standards for school mathematics, published by the

National Council of Teachers of Mathematics, for the use of cal-

culators at all levels of mathematics instruction. Using calcula-

tors in carefully planned ways can result in increases in student

problem-solving ability and improved affective outcomes with-

out a loss in basic skills.

One valuable use for calculators is as a tool for exploration

and discovery in problem-solving situations and when intro-

ducing new mathematical content. By reducing computation

time and providing immediate feedback, calculators help stu-

dents focus on understanding their work and justifying their

methods and results. The graphing calculator is particularly use-

ful in helping to illustrate and develop graphical concepts and

in making connections between algebraic and geometric ideas.

In order to accurately reflect their meaningful mathematics

performance, students should probably be allowed to use their

calculators in achievement tests. Not to do so is a major disrup-

tion in many students’ usual way of doing mathematics, and an

unrealistic restriction because when they are away from the

school setting, they will certainly use a calculator in their daily

lives and in the workplace. Another factor that argues for calcu-

lator use is that students are already permitted to use them in

some official tests. Furthermore, some examinations require the

candidates to use a graphing calculator.

References: Davis,1990;Drijvers & Doorman,1996;Dunham &

Dick, 1994; Flores & McLeod, 1990; Giamati, 1991;

Groves & Stacey, 1998; Harvey, 1993; Hembree &

Dessart, 1986, 1992; Mullis, Jenkins & Johnson,

1994; National Council of Teachers of

Mathematics, 1989; Penglase & Arnold, 1996; Rich,

1991; Ruthven, 1990; Slavit, 1996; Smith, 1996;

Stacey & Groves, 1994;Wilson & Krapfl, 1994.

30

31.
This booklet is excerpted from the mathematics chapter of the

Handbook of research on improving student achievement,

second edition. It provides a synthesis of the knowledge base

regarding effective practices for improving teaching and learn-

ing in mathematics. These materials are intended for use by

teachers, principals, other instructional leaders and policy mak-

ers who are undertaking the quest to improve student achieve-

The research findings presented are intended to be used as

a starting point, which can initiate staff development activities

and spark discussion among educators, rather than as a pre-

scription that is equally applicable to all classrooms. As Miriam

Met writes in her chapter on foreign languages in the

Handbook of research on improving student achievement:

Research cannot and does not identify the right or best way

to teach […] But research can illuminate which instructional

practices are more likely to achieve desired results, with which

kinds of learners, and under what conditions. [...] While

research may provide direction in many areas, it provides few

clear-cut answers in most. Teachers continue to be faced daily

with critical decisions about how best to achieve the instruc-

tional goals embedded in professional or voluntary state or

national standards. A combination of research-suggested

instructional practices and professional judgment and expe-

rience is most likely to produce [high student achievement].

Thus, this booklet cannot give educators all the information

they need to become expert in research-based instructional

practices in mathematics. Rather,these materials are designed to

be used as a springboard for discussion and further exploration.

For example, one approach to professional development

might be to distribute the booklet to teachers, find out which

teachers already use certain practices, and then provide oppor-

tunities for them to demonstrate the practices to their col-

leagues. Next, a study group might be formed to pursue further

reading and discussion. Both the extensive reference list on

page 39 and the list of additional resources on page 35 can serve

as a starting point.The study group’s work might lay the foun-

31

Handbook of research on improving student achievement,

second edition. It provides a synthesis of the knowledge base

regarding effective practices for improving teaching and learn-

ing in mathematics. These materials are intended for use by

teachers, principals, other instructional leaders and policy mak-

ers who are undertaking the quest to improve student achieve-

The research findings presented are intended to be used as

a starting point, which can initiate staff development activities

and spark discussion among educators, rather than as a pre-

scription that is equally applicable to all classrooms. As Miriam

Met writes in her chapter on foreign languages in the

Handbook of research on improving student achievement:

Research cannot and does not identify the right or best way

to teach […] But research can illuminate which instructional

practices are more likely to achieve desired results, with which

kinds of learners, and under what conditions. [...] While

research may provide direction in many areas, it provides few

clear-cut answers in most. Teachers continue to be faced daily

with critical decisions about how best to achieve the instruc-

tional goals embedded in professional or voluntary state or

national standards. A combination of research-suggested

instructional practices and professional judgment and expe-

rience is most likely to produce [high student achievement].

Thus, this booklet cannot give educators all the information

they need to become expert in research-based instructional

practices in mathematics. Rather,these materials are designed to

be used as a springboard for discussion and further exploration.

For example, one approach to professional development

might be to distribute the booklet to teachers, find out which

teachers already use certain practices, and then provide oppor-

tunities for them to demonstrate the practices to their col-

leagues. Next, a study group might be formed to pursue further

reading and discussion. Both the extensive reference list on

page 39 and the list of additional resources on page 35 can serve

as a starting point.The study group’s work might lay the foun-

31

32.
dation to plan a staff development programme for the next year

or two that would enable teachers to learn more and become

confident enough to use the selected practices in their class-

Suggestions from users of the Handbook

Since the publication of the first edition of the Handbook of

research on improving student achievement, the Educational

Research Service has asked users how the Handbook and

related materials have helped them in their efforts to improve

instructional practice. Here are a few of their experiences in

using these materials for staff development:

• Some teachers suggested reviewing one practice a month

through the school year at department meetings. The prac-

tice would provide a focus for discussion, with teachers

who already used the practice available as resources and as

mentors for other teachers who were interested in using the

practice in their own classrooms. As one teacher remarked,

‘staff development doesn’t work when teachers are told

what they need—often, they then just go along for the ride’.

• One school reported using the materials as a resource when

teachers met to discuss alternative approaches that might

be used with students who were struggling. The Handbook

‘provided ideas and was a guide to other resources’.

• Curriculum specialists studied the Handbook together, and

then met with teachers in their own content areas to review

both the contents of the subject-area chapter and the ideas

shared among the specialists. Each teacher was asked to

identify one research-based practice that would expand his

or her personal repertoire of instructional strategies and to

introduce its use during the first three months of school.

Follow-up discussions were held by content-area teachers

and specialists, as well as by the specialists who met as a

group to share ideas generated by the teachers with whom

they worked.

• One respondent identified an important use for these mate-

rials: to validate the instructional practices that teachers

already employ. In his words,‘it is as important for teachers

to know what they know as well as what they still have to

learn’.

• Teachers in one district reviewed and discussed the

research findings, then received training and follow-up sup-

port in strategies in which they were interested.

• One principal,while expressing concern about the time that

teachers in her school spent at the photocopying machine,

32

or two that would enable teachers to learn more and become

confident enough to use the selected practices in their class-

Suggestions from users of the Handbook

Since the publication of the first edition of the Handbook of

research on improving student achievement, the Educational

Research Service has asked users how the Handbook and

related materials have helped them in their efforts to improve

instructional practice. Here are a few of their experiences in

using these materials for staff development:

• Some teachers suggested reviewing one practice a month

through the school year at department meetings. The prac-

tice would provide a focus for discussion, with teachers

who already used the practice available as resources and as

mentors for other teachers who were interested in using the

practice in their own classrooms. As one teacher remarked,

‘staff development doesn’t work when teachers are told

what they need—often, they then just go along for the ride’.

• One school reported using the materials as a resource when

teachers met to discuss alternative approaches that might

be used with students who were struggling. The Handbook

‘provided ideas and was a guide to other resources’.

• Curriculum specialists studied the Handbook together, and

then met with teachers in their own content areas to review

both the contents of the subject-area chapter and the ideas

shared among the specialists. Each teacher was asked to

identify one research-based practice that would expand his

or her personal repertoire of instructional strategies and to

introduce its use during the first three months of school.

Follow-up discussions were held by content-area teachers

and specialists, as well as by the specialists who met as a

group to share ideas generated by the teachers with whom

they worked.

• One respondent identified an important use for these mate-

rials: to validate the instructional practices that teachers

already employ. In his words,‘it is as important for teachers

to know what they know as well as what they still have to

learn’.

• Teachers in one district reviewed and discussed the

research findings, then received training and follow-up sup-

port in strategies in which they were interested.

• One principal,while expressing concern about the time that

teachers in her school spent at the photocopying machine,

32

33.
kept a copy of the Handbook by the machine. She reported

that teachers liked the short format, which allowed them to

read quickly about one of the practices.

• Another suggestion made by teachers was the use of the

materials to help less-experienced teachers ‘take the rough

edges off’. More-experienced teachers would work collabo-

ratively with them to help the newer teachers expand and

refine their repertoire of strategies.

The context: a school culture for

effective staff development

Experience has shown that teachers need time to absorb new

information, observe and discuss new practices, and participate

in the training needed to become confident with new tech-

niques. This often means changes in traditional schedules to give

teachers regular opportunities to team with their colleagues,

both to acquire new skills and to provide instruction. As schools

continue the task of improving student achievement by expand-

ing the knowledge base of teachers, the need to restructure

schools will become more and more apparent.

Successful use of the knowledge base on improving student

learning in mathematics, as in the case of all the other subjects

included in the Handbook,relies heavily on effective staff devel-

opment. As Dennis Sparks, executive director of the National

Staff Development Council, says in his Handbook chapter:

If teachers are to consistently apply in their classrooms the

findings of the research described in this Handbook, high-

quality staff development is essential. This professional devel-

opment, however, must be considerably different from that

offered in the past. It must not only affect the knowledge, atti-

tudes, and practices of individual teachers, administrators,

and other school employees, but it must also alter the cultures

and structures of the organizations in which those individu-

als work.

Changes needed in the culture of staff development include an

increased focus on both organization development and individ-

ual development; an inquiry approach to the study of the teach-

ing/learning process; staff development efforts driven by clear,

coherent strategic plans; a greater focus on student needs and

learning outcomes; and inclusion of both generic and content-

specific pedagogical skills.

The contents of this booklet and the Handbook of research

on improving student achievement can provide the basis for

33

that teachers liked the short format, which allowed them to

read quickly about one of the practices.

• Another suggestion made by teachers was the use of the

materials to help less-experienced teachers ‘take the rough

edges off’. More-experienced teachers would work collabo-

ratively with them to help the newer teachers expand and

refine their repertoire of strategies.

The context: a school culture for

effective staff development

Experience has shown that teachers need time to absorb new

information, observe and discuss new practices, and participate

in the training needed to become confident with new tech-

niques. This often means changes in traditional schedules to give

teachers regular opportunities to team with their colleagues,

both to acquire new skills and to provide instruction. As schools

continue the task of improving student achievement by expand-

ing the knowledge base of teachers, the need to restructure

schools will become more and more apparent.

Successful use of the knowledge base on improving student

learning in mathematics, as in the case of all the other subjects

included in the Handbook,relies heavily on effective staff devel-

opment. As Dennis Sparks, executive director of the National

Staff Development Council, says in his Handbook chapter:

If teachers are to consistently apply in their classrooms the

findings of the research described in this Handbook, high-

quality staff development is essential. This professional devel-

opment, however, must be considerably different from that

offered in the past. It must not only affect the knowledge, atti-

tudes, and practices of individual teachers, administrators,

and other school employees, but it must also alter the cultures

and structures of the organizations in which those individu-

als work.

Changes needed in the culture of staff development include an

increased focus on both organization development and individ-

ual development; an inquiry approach to the study of the teach-

ing/learning process; staff development efforts driven by clear,

coherent strategic plans; a greater focus on student needs and

learning outcomes; and inclusion of both generic and content-

specific pedagogical skills.

The contents of this booklet and the Handbook of research

on improving student achievement can provide the basis for

33

34.
well-designed staff development activities. If schools provide

generous opportunities for teacher learning and collaboration,

teachers can and will improve teaching and learning in ways

that truly benefit all students. To achieve that end, professional

development must be viewed as an essential and indispensable

part of the school improvement process.

34

generous opportunities for teacher learning and collaboration,

teachers can and will improve teaching and learning in ways

that truly benefit all students. To achieve that end, professional

development must be viewed as an essential and indispensable

part of the school improvement process.

34

35.
Additional resources

Resources available through the

Educational Research Service

Handbook of research on improving student achieve-

ment, second edition (207 pages, plus appendix). Edited by

Gordon Cawelti, this publication gives teachers, administrators

and others access to the knowledge base on instructional prac-

tices that improve student learning in all the major subject areas

from kindergarten to the end of secondary education, including

mathematics.The Handbook, originally published in 1995, has

been updated by the original authors,who are respected author-

ities in their content areas. Thorough reviews of the recent

research have led to the addition of new practices and

expanded insight into existing practices. An appendix covers

research-based practices in beginning reading instruction.

• Improving student achievement in mathematics (28-

page booklet).This booklet contains the entire mathematics

chapter of the Handbook of research on improving stu-

dent achievement, written by Douglas A. Grouws and

Kristin J. Cebulla. It includes an introduction by Gordon

Cawelti and a section on ideas for expanding teachers’ abil-

ity to use research-based instructional practices.

• Improving student achievement in mathematics (two

30-minute videotapes). These videotapes illustrate each of

the ten instructional practices described in the mathematics

chapter of the Handbook, using classroom scenes and inter-

views with teachers and school administrators in the Cedar

Rapids School District, Iowa, and the Alexandria City

Schools,Virginia.The teachers’ insights based on their actual

experience using these research-based practices can serve

as a springboard for powerful staff development activities

that will spark discussion and further exploration. The ten

practices are presented in self-contained segments, giving

users the option of viewing one practice and then studying

that practice in detail before exploring additional practices.

ERS Info-Files

Each ERS Info-File contains 70–100 pages of articles from pro-

fessional journals, summaries of research studies and related lit-

erature concerning the topic, plus an annotated bibliography

35

Resources available through the

Educational Research Service

Handbook of research on improving student achieve-

ment, second edition (207 pages, plus appendix). Edited by

Gordon Cawelti, this publication gives teachers, administrators

and others access to the knowledge base on instructional prac-

tices that improve student learning in all the major subject areas

from kindergarten to the end of secondary education, including

mathematics.The Handbook, originally published in 1995, has

been updated by the original authors,who are respected author-

ities in their content areas. Thorough reviews of the recent

research have led to the addition of new practices and

expanded insight into existing practices. An appendix covers

research-based practices in beginning reading instruction.

• Improving student achievement in mathematics (28-

page booklet).This booklet contains the entire mathematics

chapter of the Handbook of research on improving stu-

dent achievement, written by Douglas A. Grouws and

Kristin J. Cebulla. It includes an introduction by Gordon

Cawelti and a section on ideas for expanding teachers’ abil-

ity to use research-based instructional practices.

• Improving student achievement in mathematics (two

30-minute videotapes). These videotapes illustrate each of

the ten instructional practices described in the mathematics

chapter of the Handbook, using classroom scenes and inter-

views with teachers and school administrators in the Cedar

Rapids School District, Iowa, and the Alexandria City

Schools,Virginia.The teachers’ insights based on their actual

experience using these research-based practices can serve

as a springboard for powerful staff development activities

that will spark discussion and further exploration. The ten

practices are presented in self-contained segments, giving

users the option of viewing one practice and then studying

that practice in detail before exploring additional practices.

ERS Info-Files

Each ERS Info-File contains 70–100 pages of articles from pro-

fessional journals, summaries of research studies and related lit-

erature concerning the topic, plus an annotated bibliography

35

36.
that includes an Educational Resources Information Center

(ERIC-CIJE) search.

• Math education and curriculum development. Examines the

implementation of the curriculum standards for mathematics,

including models for integrating the standards and related

impacts on students and teachers.

• Math manipulatives and calculators. Describes the use of

concrete objects to teach mathematical concepts. Includes

suggestions for materials, the scope of use of manipulatives,

structuring manipulatives into lesson plans and use of com-

puters. Discusses the rationale for using calculators to teach

mathematical concepts.

• Problem solving in math and science. Reviews effective

methods and strategies for teaching problem solving from

kindergarten to grade 12. Materials include ideas for activi-

ties as well as grading methods.

Additional sources of information

Every child mathematically proficient: an action plan. This

action paper was developed by the Learning First Alliance, an

organization of twelve leading national education associations.

It sets forth recommendations for curriculum changes, profes-

sional development initiatives, parent involvement efforts and

research-based reforms. 24 pages. Price: $3.00. Order from

National Education Association Professional Library order desk:

(1-800) 229-4200.

Improving teaching and learning in science and mathemat-

ics. Illustrates how constructivist ideas can be used by science

and mathematics educators for research and the further

improvement of mathematics practice. 1996. Available from

Teachers College Press,Teachers College, Columbia University,

P.O. Box 20,Williston,VT 05495, USA.

Telephone: (1-802) 864-7626.

Mathematics, science, & technology education programs that

work, and Promising practices in mathematics and science.

Published by the United States Department of Education. The

first volume describes programmes from the Department’s

National Diffusion Network; the second describes successful

programmes identified by the Office of Educational Research

and Improvement. Price: $21.00 for the two-volume set. Stock

No. 065-000-00627-8. Available from Superintendent of

Documents, P.O. Box 371954, Pittsburgh, PA 15250-7954, USA.

Telephone: (1-202) 512-1800; fax: (1-202) 512-2250.

36

(ERIC-CIJE) search.

• Math education and curriculum development. Examines the

implementation of the curriculum standards for mathematics,

including models for integrating the standards and related

impacts on students and teachers.

• Math manipulatives and calculators. Describes the use of

concrete objects to teach mathematical concepts. Includes

suggestions for materials, the scope of use of manipulatives,

structuring manipulatives into lesson plans and use of com-

puters. Discusses the rationale for using calculators to teach

mathematical concepts.

• Problem solving in math and science. Reviews effective

methods and strategies for teaching problem solving from

kindergarten to grade 12. Materials include ideas for activi-

ties as well as grading methods.

Additional sources of information

Every child mathematically proficient: an action plan. This

action paper was developed by the Learning First Alliance, an

organization of twelve leading national education associations.

It sets forth recommendations for curriculum changes, profes-

sional development initiatives, parent involvement efforts and

research-based reforms. 24 pages. Price: $3.00. Order from

National Education Association Professional Library order desk:

(1-800) 229-4200.

Improving teaching and learning in science and mathemat-

ics. Illustrates how constructivist ideas can be used by science

and mathematics educators for research and the further

improvement of mathematics practice. 1996. Available from

Teachers College Press,Teachers College, Columbia University,

P.O. Box 20,Williston,VT 05495, USA.

Telephone: (1-802) 864-7626.

Mathematics, science, & technology education programs that

work, and Promising practices in mathematics and science.

Published by the United States Department of Education. The

first volume describes programmes from the Department’s

National Diffusion Network; the second describes successful

programmes identified by the Office of Educational Research

and Improvement. Price: $21.00 for the two-volume set. Stock

No. 065-000-00627-8. Available from Superintendent of

Documents, P.O. Box 371954, Pittsburgh, PA 15250-7954, USA.

Telephone: (1-202) 512-1800; fax: (1-202) 512-2250.

36

37.
Curriculum and evaluation standards for school mathemat-

ics. Describes fifty-four standards developed by the National

Council of Teachers of Mathematics to ‘create a coherent vision

of mathematical literacy and provide standards to guide the revi-

sion of the mathematics curriculum in the next decade’. 1989.

258 pages. $25.00. Available from National Council of Teachers

of Mathematics, 1906 Association Drive, Reston,VA 20191-1593,

USA. Telephone: (1-703) 620-9840; fax: (1-703) 476-2970.

Eisenhower National Clearinghouse (Ohio State University).

Part of a network funded by the United States Department of

Education, which together with ten regional science and math-

ematics consortia, collaborates to identify and disseminate

exemplary materials, to provide technical assistance about

teaching methods and tools to schools, teachers and adminis-

trators, and to work with other organizations

trying to improve mathematics and science education.

Online: www.enc.org

National Center for Improving Student Learning and

Achievement in Mathematics and Science, Wisconsin Center

for Educational Research, University of Wisconsin-Madison.

Publications include the quarterly newsletter Principled prac-

tice, which examines educators’ observations and concerns

about issues in mathematics and science education. 1025 West

Johnson Street, Madison,WI 53706, USA.

Telephone: (1-608) 265-6240; fax: (1-608) 263-3406;

e-mail: ncisla@mail.soemadison.wisc.edu;

web site: www.wcer.wisc.edu/ncisla

Related websites

• ERIC Clearinghouse for Science, Mathematics, and

Environmental Education

www.ericse.org/sciindex.html

• The Regional Alliance for Mathematics and Science

Education

http://ra.terc.edu/alliance/HubHome.html

• National Council of Teachers of Mathematics

www.nctm.org

37

ics. Describes fifty-four standards developed by the National

Council of Teachers of Mathematics to ‘create a coherent vision

of mathematical literacy and provide standards to guide the revi-

sion of the mathematics curriculum in the next decade’. 1989.

258 pages. $25.00. Available from National Council of Teachers

of Mathematics, 1906 Association Drive, Reston,VA 20191-1593,

USA. Telephone: (1-703) 620-9840; fax: (1-703) 476-2970.

Eisenhower National Clearinghouse (Ohio State University).

Part of a network funded by the United States Department of

Education, which together with ten regional science and math-

ematics consortia, collaborates to identify and disseminate

exemplary materials, to provide technical assistance about

teaching methods and tools to schools, teachers and adminis-

trators, and to work with other organizations

trying to improve mathematics and science education.

Online: www.enc.org

National Center for Improving Student Learning and

Achievement in Mathematics and Science, Wisconsin Center

for Educational Research, University of Wisconsin-Madison.

Publications include the quarterly newsletter Principled prac-

tice, which examines educators’ observations and concerns

about issues in mathematics and science education. 1025 West

Johnson Street, Madison,WI 53706, USA.

Telephone: (1-608) 265-6240; fax: (1-608) 263-3406;

e-mail: ncisla@mail.soemadison.wisc.edu;

web site: www.wcer.wisc.edu/ncisla

Related websites

• ERIC Clearinghouse for Science, Mathematics, and

Environmental Education

www.ericse.org/sciindex.html

• The Regional Alliance for Mathematics and Science

Education

http://ra.terc.edu/alliance/HubHome.html

• National Council of Teachers of Mathematics

www.nctm.org

37

38.

39.
American Association of University Women. 1998. Gender gaps:

where schools still fail our children.Washington,DC,AAUW.

Atanda, D. 1999. Do gatekeeper courses expand education

options? Washington, DC, National Center for Education

Statistics. (NCES 1999303.)

Aubrey, C. 1997. Mathematics teaching in the early years:

an investigation of teachers’ subject knowledge. London,

Falmer Press.

Ball, D. 1993. With an eye on the mathematical horizon:

dilemmas of teaching elementary school mathematics.

Elementary school journal (Chicago, IL), vol. 93,

p. 373–97.

Boaler, J. 1998. Open and closed mathematics: student

experiences and understandings. Journal for research in

mathematics education (Reston, VA), vol. 29, p. 41–62.

Brownell, W.A. 1945. When is arithmetic meaningful? Journal

of educational research (Washington, DC), vol. 38,

p. 481–98.

—. 1947. The place of meaning in the teaching of arithmetic.

Elementary school journal (Chicago, IL), vol. 47,

p. 256–65.

Carpenter, T.P., et al. 1988. Teachers’ pedagogical content

knowledge of students’problem solving in elementary arith-

metic. Journal for research in mathematics education

(Reston, VA), vol. 19, p. 385–401.

—. 1989. Using knowledge of children’s mathematics thinking

in classroom teaching: an experimental study. American

educational research journal (Washington, DC), vol. 26,

p. 499–531.

—. 1998. A longitudinal study of invention and understanding

in children’s multidigit addition and subtraction. Journal for

research in mathematics education (Reston, VA), vol. 29,

p. 3–20.

Cobb, P.;Yackel, E.;Wood,T. 1992. A constructivist alternative to

the representational view of mind in mathematics educa-

tion. Journal for research in mathematics education

(Reston, VA), vol. 23, p. 2–23.

Cobb, P., et al. 1991. Assessment of a problem-centered second-

grade mathematics project. Journal for research in mathe-

matics education (Reston, VA), vol. 22, p. 3–29.

39

where schools still fail our children.Washington,DC,AAUW.

Atanda, D. 1999. Do gatekeeper courses expand education

options? Washington, DC, National Center for Education

Statistics. (NCES 1999303.)

Aubrey, C. 1997. Mathematics teaching in the early years:

an investigation of teachers’ subject knowledge. London,

Falmer Press.

Ball, D. 1993. With an eye on the mathematical horizon:

dilemmas of teaching elementary school mathematics.

Elementary school journal (Chicago, IL), vol. 93,

p. 373–97.

Boaler, J. 1998. Open and closed mathematics: student

experiences and understandings. Journal for research in

mathematics education (Reston, VA), vol. 29, p. 41–62.

Brownell, W.A. 1945. When is arithmetic meaningful? Journal

of educational research (Washington, DC), vol. 38,

p. 481–98.

—. 1947. The place of meaning in the teaching of arithmetic.

Elementary school journal (Chicago, IL), vol. 47,

p. 256–65.

Carpenter, T.P., et al. 1988. Teachers’ pedagogical content

knowledge of students’problem solving in elementary arith-

metic. Journal for research in mathematics education

(Reston, VA), vol. 19, p. 385–401.

—. 1989. Using knowledge of children’s mathematics thinking

in classroom teaching: an experimental study. American

educational research journal (Washington, DC), vol. 26,

p. 499–531.

—. 1998. A longitudinal study of invention and understanding

in children’s multidigit addition and subtraction. Journal for

research in mathematics education (Reston, VA), vol. 29,

p. 3–20.

Cobb, P.;Yackel, E.;Wood,T. 1992. A constructivist alternative to

the representational view of mind in mathematics educa-

tion. Journal for research in mathematics education

(Reston, VA), vol. 23, p. 2–23.

Cobb, P., et al. 1991. Assessment of a problem-centered second-

grade mathematics project. Journal for research in mathe-

matics education (Reston, VA), vol. 22, p. 3–29.

39

40.
—. 1992. Characteristics of classroom mathematics traditions:

an interactional analysis. American educational research

journal (Washington, DC), vol. 29, p. 573–604.

Cognition and Technology Group. 1997. The Jasper Project:

lessons in curriculum, instruction, assessment, and pro-

fessional development. Mahwah, NJ, Erlbaum.

Cohen, E.G. 1994. Restructuring the classroom: conditions for

productive small groups. Review of educational research

(Washington, DC), vol. 64, p. 1–35.

Davidson, N. 1985. Small group cooperative learning in mathe-

matics: a selective view of the research. In: Slavin, R., ed.

Learning to cooperate, cooperating to learn, p. 211–30.

New York, Plenum Press.

Davis, M. 1990. Calculating women: precalculus in context.

Paper presented at the Third Annual Conference on

Technology in Collegiate Mathematics, Columbus, OH,

November.

Drijvers, P.; Doorman, M. 1996.The graphics calculator in math-

ematics education. Journal of mathematical behavior

(Stamford, CT), vol. 15, p. 425–40.

Dunham, P.H.; Dick,T.P. 1994. Research on graphing calculators.

Mathematics teacher (Reston,VA), vol. 87, p. 440–45.

Fawcett,H.P. 1938. The nature of proof: a description and eval-

uation of certain procedures used in senior high school to

develop an understanding of the nature of proof. 1938

Yearbook of the National Council of Teachers of

Mathematics. New York, Columbia University, Teachers

College.

Fennema, E.; Carpenter,T.P.; Peterson; P.L. 1989. Learning mathe-

matics with understanding: cognitively guided instruction.

In: Brophy, J., ed. Advances in research on teaching,

p. 195–221. Greenwich, CT, JAI Press.

Fennema, E., et al. 1993. Using children’s mathematical knowl-

edge in instruction. American educational research jour-

nal (Washington, DC), vol. 30, p. 555–83.

—. 1996. A longitudinal study of learning to use children’s

thinking in mathematics instruction. Journal for research

in mathematics education (Reston, VA), vol. 27, p. 403–34.

Flanders, J.R. 1987. How much of the content in mathematics

textbooks is new?’ Arithmetic teacher (Reston, VA), vol. 35,

p. 18–23.

Flores, A.; McLeod, D.B. 1990. Calculus for middle school teach-

ers using computers and graphing calculators. Paper pre-

sented at the Third Annual Conference on Technology in

Collegiate Mathematics, Columbus, OH, November.

40

an interactional analysis. American educational research

journal (Washington, DC), vol. 29, p. 573–604.

Cognition and Technology Group. 1997. The Jasper Project:

lessons in curriculum, instruction, assessment, and pro-

fessional development. Mahwah, NJ, Erlbaum.

Cohen, E.G. 1994. Restructuring the classroom: conditions for

productive small groups. Review of educational research

(Washington, DC), vol. 64, p. 1–35.

Davidson, N. 1985. Small group cooperative learning in mathe-

matics: a selective view of the research. In: Slavin, R., ed.

Learning to cooperate, cooperating to learn, p. 211–30.

New York, Plenum Press.

Davis, M. 1990. Calculating women: precalculus in context.

Paper presented at the Third Annual Conference on

Technology in Collegiate Mathematics, Columbus, OH,

November.

Drijvers, P.; Doorman, M. 1996.The graphics calculator in math-

ematics education. Journal of mathematical behavior

(Stamford, CT), vol. 15, p. 425–40.

Dunham, P.H.; Dick,T.P. 1994. Research on graphing calculators.

Mathematics teacher (Reston,VA), vol. 87, p. 440–45.

Fawcett,H.P. 1938. The nature of proof: a description and eval-

uation of certain procedures used in senior high school to

develop an understanding of the nature of proof. 1938

Yearbook of the National Council of Teachers of

Mathematics. New York, Columbia University, Teachers

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46

47.
47

48.
The International

EDUCATIONAL PRACTICES SERIES–4

Bureau

of Education—IBE

An international centre for the content of

education, the IBE was founded in Geneva

in 1925 as a private institution. In 1929, it

became the first intergovernmental organiza-

tion in the field of education. In 1969, the

IBE joined UNESCO as an integral, yet

autonomous, institution with three main

lines of action: organizing the sessions of the

International Conference on Education; col-

lecting, analysing and disseminating educa-

tional documentation and information, in

particular on innovations concerning curric-

ula and teaching methods; and undertaking

surveys and studies in the field of compara-

tive education.

At the present time, the IBE: (a) manages

World data on education, a databank pre-

senting on a comparative basis the profiles

of national education systems; (b) organizes

courses on curriculum development in

developing countries; (c) collects and dis-

seminates through its databank INNODATA

notable innovations on education; (d) co-

ordinates preparation of national reports on

the development of education; (e) adminis-

ters the Comenius Medal awarded to out-

standing teachers and educational

researchers; and (f) publishes a quarterly

review of education—Prospects, a quarterly

newsletter—Educational innovation and

information, a guide for foreign students—

Study abroad, as well as other publications.

In the context of its training courses on

curriculum development, the Bureau is

establishing regional and sub-regional net-

works on the management of curriculum

change and developing a new information

service—a platform for the exchange of

information on content.

The IBE is governed by a Council com-

posed of representatives of twenty-eight

Member States elected by the General

Conference of UNESCO.

http://www.ibe.unesco.org

EDUCATIONAL PRACTICES SERIES–4

Bureau

of Education—IBE

An international centre for the content of

education, the IBE was founded in Geneva

in 1925 as a private institution. In 1929, it

became the first intergovernmental organiza-

tion in the field of education. In 1969, the

IBE joined UNESCO as an integral, yet

autonomous, institution with three main

lines of action: organizing the sessions of the

International Conference on Education; col-

lecting, analysing and disseminating educa-

tional documentation and information, in

particular on innovations concerning curric-

ula and teaching methods; and undertaking

surveys and studies in the field of compara-

tive education.

At the present time, the IBE: (a) manages

World data on education, a databank pre-

senting on a comparative basis the profiles

of national education systems; (b) organizes

courses on curriculum development in

developing countries; (c) collects and dis-

seminates through its databank INNODATA

notable innovations on education; (d) co-

ordinates preparation of national reports on

the development of education; (e) adminis-

ters the Comenius Medal awarded to out-

standing teachers and educational

researchers; and (f) publishes a quarterly

review of education—Prospects, a quarterly

newsletter—Educational innovation and

information, a guide for foreign students—

Study abroad, as well as other publications.

In the context of its training courses on

curriculum development, the Bureau is

establishing regional and sub-regional net-

works on the management of curriculum

change and developing a new information

service—a platform for the exchange of

information on content.

The IBE is governed by a Council com-

posed of representatives of twenty-eight

Member States elected by the General

Conference of UNESCO.

http://www.ibe.unesco.org