----

##  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 achievement. 
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. 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. 
3. 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. 
4. 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. 
5. 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. 
6.                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 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                   following individual and group work improves student                            achievement. 
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. 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. 
9. 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. 
10. 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 geometric ideas. 
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. REFERENCESAmerican             Association of University Women. (1998).  "Gender gaps where schools            still fail our children."  Washington, DC: Author. 
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. 
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 
An expanded version of the ideas presented in this             Digest is available online at                                                   http://www.ibe.unesco.org/Publications/Practice/prac04e.pdf 
-----             
This digest was funded by the Office of Educational Research and              Improvement, U.S. Department of Education, under contract no.                   ED-99-CO-0024. Opinions expressed in this digest do not necessarily             reflect the positions or policies of OERI or the U.S. Department of             Education.                                                                   **Title:** Improving Student Achievement in Mathematics, Part 2:                           Recommendations for the Classroom. ERIC Digest.                              **Note:** For part 1, see SE 064 317.                                                  **Document Type:** Information Analyses---ERIC Information Analysis Products (IAPs) (071); Information Analyses---ERIC Digests (Selected) in Full Text (073); **Available From:** ERIC/CSMEE, 1929 Kenny Road, Columbus, OH 43210-1080. Tel:                      800-276-0462 (Toll Free); e-mail: [email protected]; Web site:                     http://www.ericse.org.                                                       **Descriptors:**  Academic Achievement, Calculators, Concept Formation, Elementary               Secondary Education,  Mathematics Education, Problem Solving                 **Identifiers:**
###
[Return to ERIC Digest Search Page]