Best Practices in Mathematics
This section of the Frameworks provides information
for teachers about the Best Practices in teaching mathematics.
Research-based Top Ten Strategies for Mathematics Achievement
Research findings indicate that certain teaching strategies
and methods are worth careful consideration as teachers
strive to improve their mathematics teaching practices.
The following ten instructional practices are from "Improving
Student Achievement in Mathematics: Part 1: Research Findings",
by Douglas A. Grouws & Kristin J. Cebulla; December
2000 (Updated January 2002). Published by ERIC
- Opportunity to Learn
- Focus on Meaning
- Problem Solving
- Opportunities for Invention and
- Openness to Student Solutions
and Student Interactions
- Small Group Learning
- Whole-Class Discussion
- Focus on Number Sense
- Use Concrete Materials
- Use Calculators
1. Opportunity to Learn
The extent of the students’ opportunity to learn
mathematics content bears directly and decisively on student
mathematics achievement. Opportunity to learn (OTL) was
studied in the First International Mathematics Study (Husén,
1967), where teachers were asked to rate the extent of
student exposure to particular mathematical concepts and
skills. Strong correlations were found between OTL scores
and mean student achievement scores, with high OTL scores
associated with high achievement. The link was also found
in subsequent international studies, such as the Second
International Mathematics Study (McKnight et al., 1987)
and the Third International Mathematics and Science study
(TIMSS) (Schmidt, McKnight, & Raizen, 1997).
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. Teachers
should use textbooks as just one instructional tool among
many, rather than feel duty-bound to go through the textbook
on a one-section-per-day basis.
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 of Women study (1998) showed that boys’
and girls’ use of technology is markedly different.
Girls take fewer computer science and computer design
courses than do boys. Furthermore, boys often use computers
to program and solve problems, whereas girls tend to use
the computer primarily as a word processor. 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.
2. Focus on Meaning
Focusing instruction on the meaningful development of
important mathematical ideas increases the level of student
There is a long history of research, going back to the
work of Brownell (1945,1947), on the effects of teaching
for meaning and understanding. 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.
- Emphasize the mathematical meanings of ideas, including
how the idea, concept or skill is connected in multiple
ways to other mathematical ideas in a logically consistent
and sensible manner.
- Create a classroom learning context in which students
can construct meaning.
- Make explicit the connections between mathematics
and other subjects.
- Attend to student meanings and student understandings.
3. Problem Solving
Students can learn both concepts and skills by solving
Research suggests that students who develop conceptual
understanding early perform best on procedural knowledge
later. Students with good conceptual understanding are
able to perform successfully on near-transfer tasks and
to develop procedures and skills they have not been taught.
Students with low levels of conceptual understanding need
more practice in order to acquire procedural knowledge.
There is evidence that students can learn new skills and
concepts while they are working out solutions to problems.
Development of more sophisticated mathematical skills
can also be approached by treating their development as
a problem for students to solve. 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.
4. Opportunities for Invention
Giving students both an opportunity to discover and invent
new knowledge and an opportunity to practice what they
have learned improves student achievement.
Data from the TIMSS video study show that over 90% of
mathematics class time in the United States 8th-grade
classrooms is spent practicing routine procedures, with
the remaining time generally spent applying procedures
in new situations. Virtually no time is spent inventing
new procedures and analyzing unfamiliar situations. In
contrast, students at the same grade level in typical
Japanese classrooms spend approximately 40% of instructional
time practicing routine procedures, 15% applying procedures
in new situations, and 45% inventing new procedures and
analyzing new situations.
Research suggests that students need opportunities for
both practice and invention. Findings from a number of
studies show that when students discover mathematical
ideas and invent mathematical procedures, they have a
stronger conceptual understanding of connections between
Balance is needed between the time students spend practicing
routine procedures and the time they devote to inventing
and discovering new ideas. Teachers need not choose between
these; indeed, they must not make a choice if students
are to develop the mathematical power they need.
To increase opportunities for invention, teachers should
frequently use non-routine problems, periodically introduce
a lesson 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
5. Openness to Student Solutions
and Student Interactions
Teaching that incorporates students’ intuitive solution
methods can increase student learning, especially when
combined with opportunities for student interaction and
Student 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
Structuring instruction around carefully chosen problems,
allowing students to interact when solving problems, and
then providing opportunities for them to share their solution
methods result in increased achievement on problem-solving
measures. These gains come without a loss of achievement
in the skills and concepts measured on standardized achievement
Research results suggest that teachers should concentrate
on providing opportunities for students to interact in
problem-rich situations. Besides providing appropriate
problem-rich situations, teachers must encourage students
to find their own solution methods and give them opportunities
to share and compare their solution methods and answers.
One way to organize such instruction is to have students
work in small groups initially and then share ideas and
solutions in a whole-class discussion.
6. Small Group Learning
Using small groups of students to work on activities,
problems and assignments can increase student mathematics
Davidson (1985) reviewed studies that compared student
achievement in small group settings 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 measures of student performance.
In only two of the 79 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 99 studies of cooperative group-learning methods,
Slavin (1990) concluded that cooperative methods were
effective in improving student achievement. The most effective
methods emphasized both group goals and individual accountability.
When using small groups for mathematics instruction, teachers
- Choose tasks that deal with important mathematical
concepts and ideas.
- Select tasks that are appropriate for group work.
- Consider having students initially work individually
on a task and then follow with group work where students
share and build on their individual ideas and work.
- Give clear instructions to the groups and set clear
expectations for each (for each task or each group?).
- 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.
7. Whole-Class Discussion
Whole-class discussion following individual and group
work improves student achievement.
Research suggests that whole class discussion can be
effective when it is used for sharing and explaining the
variety of solutions 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 (1999) 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. Students should be expected
to be active listeners who participate in discussion and
feel a sense of responsibility for each other’s
It is important that whole-class discussion follows student
work on problem-solving activities. The discussion should
be a summary of individual work in which key ideas are
brought to the 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 diagnosis
tool for determining the depth of student understanding
and identifying misconceptions. Teachers can identify
areas of difficulty for particular students, as well as
ascertain areas of student success or progress.
8. Focus on 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.
Number sense relates to having an intuitive feel for
number size and combinations, and the ability to work
flexibly with numbers in problem situations in order to
make sound decisions and reasonable judgments. It involves
mentally computing, estimating, sensing number magnitudes,
moving between representation systems for numbers, and
judging the reasonableness of numerical results. Markovits
and Sowder (1994) studied 7th-grade classes where special
units on number magnitude, mental computation and computational
estimation were taught. 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. In a study of second graders,
Cobb (1991) and his colleagues found that students’
number sense was improved by a problem-centered 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 increased persistence in solving problems.
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 instill 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
9. Use Concrete Materials
Long-term use of concrete materials is positively related
to increases in student mathematics achievement and improved
attitudes towards mathematics.
In a review of activity-based learning in mathematics
in kindergarten through grade 8, Suydam and Higgins (1977)
concluded that using manipulative materials produces greater
achievement gains than not using them. In a more recent
meta-analysis of sixty studies (kindergarten through postsecondary)
that compared the effects of using concrete materials
with the effects of more abstract instruction, Sowell
(1989) found that the long-term use of concrete materials
by teachers knowledgeable in their use improved student
achievement and attitudes.
Research suggests that teachers use manipulative materials
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 materials to
teach multiple ideas over the course of schooling shortens
the amount of time it takes to introduce the material
and helps students 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.
10. Use Calculators
Using calculators in the learning of mathematics can result
in increased achievement and improved student attitudes.
Studies have consistently shown that thoughtful use of
calculators improves student mathematics achievement and
attitudes toward mathematics. From a meta-analysis of
79 non-graphing calculator studies, Hembree and Dessart
(1986) concluded that use of hand-held calculators improved
student learning. Analysis also showed that students using
calculators tended to have better attitudes towards mathematics
and better self-concepts 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 graphing calculators has also
shown positive effects on student achievement. Most studies
have found positive effects on students’ graphing
ability, conceptual understanding of graphs and their
ability to relate graphical representations to other representations.
Most studies of graphing calculators have found no negative
effect on basic skills, factual knowledge, or computational
One valuable use for calculators is as a tool for exploration
and discovery in problem-solving situations and when introducing
new mathematical content. By reducing computation time
and providing immediate feedback, calculators help students
focus on understanding their work and justifying their
methods and results. The graphing calculator is particularly
useful in helping to illustrate and develop graphical
concepts and in making connections between algebraic and
In order to accurately reflect their meaningful mathematics
performance, students should be allowed to use their calculators
in achievement tests. Not to do so is a major disruption
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.
Brownell, W.A. (1945). When is arithmetic meaningful?
Journal of Education Research, 38, 481-98.
Brownell, W.A. (1947). The place of meaning in the teaching
of arithmetic. Elementary School Journal, 47, 256-65.
Cobb, P, et al. (1991). Assessment of a problem-centered
second-grade mathematics project. Journal for Research
in Mathematics Education, 22, 3-29.
Davidson, N. (1985). Small group cooperative learning
in mathematics: A selective view of the research. In R.
Slavin (Ed.), Learning to cooperate: Cooperating to learn.
(pp.211-30) NY: Plenum.
Hembree, R. & Dessart, D.J. (1986). Effects of hand-held
calculators in pre-college mathematics education: A meta-analysis.
Journal for Research in Mathematics Education, 17, 83-99.
Husén, T. (1967). International study of achievement
in mathematics, Vol. 2. NY: Wiley.
Kilpatrick, J. (1992). A history of research in mathematics
education. In Grouws, D. A., (Ed.), Handbook of research
on mathematics teaching and learning. (pp. 3-38) NY: Macmillan.
Markovit, Z., & Sowder, J. (1994). Developing number
sense: An intervention study in grade 7. Journal for Research
in Mathematics Education, 25, 4-29.
McKnight, I.V.S., et al. (1987). The underachieving curriculum.
Champaign, IL: Stipes.
Schmidt, W.H., McKnight, C.C., & Raizen, S.A. (1997).
A splintered vision: An investigation of U.S. science
and mathematics education. Dordrecht, Netherlands: Kluwer.
Slavin, R.E. (1990). Student team learning in mathematics.
In N. Davidson (Ed.), Cooperative learning in math: A
handbook for teachers. Boston: Allyn & Bacon, (pp.
Sowder, J. (1992a). Estimation and number sense. In D.A.
Grouws (Ed.), Handbook of research on mathematics teaching
and learning. (pp. 371-89) NY: Macmillan.
Sowder, J. (1992b). Making sense of numbers in school
mathematics. In R. Leinhardt, R. Putman, & R. Hattrup
(Eds.), Analysis of arithmetic for mathematics education.
(pp. 1-51)Hillsdale, NJ: Lawrence Erlbaum.
Sowell, E.J. (1989). Effects of manipulative materials
in mathematics instruction. Journal for Research in Mathematics
Education, 20, 498-505.
Suydam, M.N. & Higgins, J. L. (1977). Activity-based
learning in elementary school mathematics: Recommendations
from research. Columbus, OH: ERIC Clearinghouse for Science,
Mathematics, and Environmental Education.
Wood, T. (1999). Creating a context for argument in mathematics
class. Journal for Research in Mathematics Education,
Other sources of information about best practices:
NCTM Illuminations (http://illuminations.nctm.org/index2.html)
National Center for Improving Student Learning and Achievement
in Mathematics and Science (http://www.wcer.wisc.edu/ncisla/)
Eisenhower National Clearinghouse for Mathematics and
Science Education (http://enc.org/)
SE 064 317
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Title: Improving Student Achievement in Mathematics —
Part 1: Research Findings — ERIC Digest
Author: Douglas A. Grouws & Kristin J. Cebulla
Publication Date: 2000
Publisher/Institutional Source: ERIC Clearinghouse on
Science, Mathematics, and Environmental Education
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