ERIC Identifier: ED463953
Publication Date: 2000-12-00
Author: Grouws, Douglas A. - Cebulla, Kristin J.
ERIC Clearinghouse for Science Mathematics and Environmental Education
Improving Student Achievement in Mathematics, Part 2: Recommendations for the Classroom. ERIC Digest.
The number of research studies conducted in mathematics education over the
past three decades has increased dramatically (Kilpatrick, 1992). 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. Research findings
indicate that certain teaching strategies and methods are worth careful
consideration as teachers strive to improve their mathematics teaching
practices. To readers who 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 interesting and practical.
For a summary of the research findings on which these recommendations are
based, please see the companion to this Digest, "Improving Student Achievement
in Mathematics, Part 1: Research Findings" (EDO-SE-00-09)
1. The extent of the students' opportunity to learn
mathematics content bears directly and decisively on student mathematics
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-
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.
Focusing instruction on the meaningful development of important mathematical
ideas increases the level of student learning.
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.
Students can learn both concepts and skills by solving problems.
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.
Giving students both an opportunity to discover and invent new knowledge and an
opportunity to practice what they have learned improves student achievement.
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 procedures.
Teaching that incorporates students' intuitive solution methods can increase
student learning, especially when combined with opportunities for student
interaction and discussion.
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.
Using small groups of students to work on activities, problems and assignments
can increase student mathematics achievement.
When using small groups for mathematics instruction, teachers should:
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
* 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.
Whole-class discussion following individual and group work improves student
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.
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.
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 problem-solving.
Long-term use of concrete materials is positively related to increases in
student mathematics achievement and improved attitudes towards mathematics.
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.
Using calculators in the learning of mathematics can result in increased
achievement and improved student attitudes.
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
American Association of University Women.
(1998). "Gender gaps where schools still fail our children." Washington, DC:
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.
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,