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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.
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INTERNATIONAL ACADEMY
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.
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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.
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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
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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.
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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
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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
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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.
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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.
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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
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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.
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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
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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.
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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
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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.
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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.
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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.
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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
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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.
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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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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
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Aubrey, C. 1997. Mathematics teaching in the early years:
an investigation of teachers’ subject knowledge. London,
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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
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Brownell, W.A. 1945. When is arithmetic meaningful? Journal
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Carpenter, T.P., et al. 1988. Teachers’ pedagogical content
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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.
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—. 1992. Characteristics of classroom mathematics traditions:
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Cohen, E.G. 1994. Restructuring the classroom: conditions for
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