ERIC Identifier: ED463952
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 1: Research
Findings. ERIC Digest.
The number of research studies conducted in mathematics education over the
past three decades has increased dramatically (Kilpatrick, 1992). Research
findings indicate that certain teaching strategies and methods are worth careful
consideration as teachers strive to improve their mathematics teaching
practices. For the classroom implications of the research findings summarized
here, please see the companion to this Digest, "Improving Student Achievement in
Mathematics, Part 2: Recommendations for the Classroom" (EDO-SE-00-10)
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 (Husen, 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).
Focusing instruction on the meaningful development of important mathematical
ideas increases the level of student learning.
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
Students can learn both concepts and skills by solving problems.
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
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 mathematical ideas.
Teaching that incorporates students' intuitive solution methods can increase
student learning, especially when combined with opportunities for student
interaction and discussion.
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 in mathematics.
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 tests.
Using small groups of students to work on activities, problems and assignments
can increase student mathematics achievement.
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.
Whole-class discussion following individual and group work improves student
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 understanding.
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.
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
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
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.
Husen, 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,"
McKnight, I.V.S., et al. (1987). "The underachieving curriculum." Champaign,
Schmidt, W.H., McKnight, C.C., & Raizen, S.A. (1997). "A splintered
vision: An investigation of U.S. science and mathematics education." Dordrecht,
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. 69-102).
Sowder, J. (1992a). Estimation and number sense. In D.A. Grouws (Ed.),
"Handbook of research on mathematics teaching and learning." (pp. 371-89) NY:
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," 30, 171-91.
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
An expanded version of the ideas presented in this Digest is available online