Promising Practices in Mathematics Education
Terese A. Herrera and S. Asli Özgün-Koca
This discussion of promising practices in mathematics education synthesizes the results of
several national evaluations of science and math programs and curriculum materials. The
evaluations were conducted by the U.S. Department of Education, the National Science
Foundation, the American Association for the Advancement of Science, the Eisenhower National
Consortia, the National Diffusion Network, and the Northwest Regional Educational Laboratory.
Identified practices focus on those that have direct instructional impact and therefore immediate
relevance for K-12 teachers who are looking for effective ways to improve classroom learning.
The classroom practices that follow are derived from the criteria used in the national evaluations
and from descriptions of exemplary programs:
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Problem-Solving Approach |
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Experimental Approach |
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Development of Mathematical Reasoning |
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Cooperative/Group Learning |
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Illustration of Mathematical Connections |
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Use of Technology |
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Communication of Mathematical Ideas |
The following discussion defines the practices and provides curriculum resources that support
each of the practices in the classroom setting.
Problem-Solving Approach
Teachers introduce content by presenting a problem. As children grapple with making sense of
the problem, they are taught to model the situation mathematically, to formulate processes to deal with that model, and finally, to generalize those processes into useful algorithms. It is critical that teachers:
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Engage children in problems that embody significant mathematical concepts. |
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Create and sustain a classroom environment that supports problem solving--for example,
help children cope with frustration and honor all their attempts to work through the problem. |
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Assist children in seeing the mathematics embedded in the problem and in creating
models that represent the mathematical situation. |
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Assist children as they move from the concrete model to the more abstract formulation of
their problem-solving processes into algorithms. |
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Teach children to constantly monitor and reflect on their strategies. |
Children engaged in a problem-based curriculum build mathematical knowledge through their
work with and discussion of problems. Children explore, report on their findings, explain and
talk about their varied solutions, invent ways to represent their thinking, and generalize their
solutions to other areas.
Experimental Approach
The experimental approach owes much to the scientific method: forming a hypothesis, collecting
data to test it, analyzing the data, and reaching a conclusion. In the math classroom,
experimentation begins with an open-ended question. Children then make predictions, plan
strategies to explore the question, and collect and compare data. What counts is for kids to get
their hands on the problem, either literally, by using manipulatives or some other concrete model,
or figuratively. There is a real spirit of play involved in such investigation, but also purpose.
Teachers direct experimentation as they:
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Set out the question to be investigated, or recognize and accept one from the class. |
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Motivate children to engage in the investigation. |
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Teach children methods of collecting data, representing data (tables, charts, graphs,
algebraic expressions), and drawing conclusions. |
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Provide children with tools for exploring (manipulatives, calculators, graphing software,
microworlds). |
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Help children see the significance of their work. |
The environment described here is supported in research by constructivist learning theory, which
states that children actively construct knowledge from their experiences.
Development of Mathematical Reasoning
Mathematical reasoning can be defined as "a part of mathematical thinking that involves forming
generalizations and drawing valid conclusions about ideas and how they are related" (O'Daffer
and Thornquist, 1993, p. 43). Such thinking occurs not only in the high school geometry
classroom, but also in all math classrooms, regardless of topic and age level.
Teachers foster mathematical reasoning in the classroom when they:
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Encourage questions, even doubt. This is the opposite of the children's usual acceptance
of the teacher or the printed word as the arbiter of "right or wrong" in a mathematical argument. |
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Expect children to explain and justify their answers. |
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Assist children in considering and evaluating several solutions to a problem. |
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Teach high school students formal methods of proof (that is, deductive reasoning) as well
as statistical arguments, and the difference between the two. |
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Above all, create "a classroom environment where students feel comfortable questioning,
challenging, suspending judgment, and demanding reasons and justification as they deal with
mathematical and real-world content" (O'Daffer and Thornquist, 1993, p. 42). |
The habit of reasoning does not develop in a specific math course but instead throughout the
school curriculum. As stated in Curriculum and Evaluation Standards for School Mathematics
(National Council of Teachers of Mathematics, 1989), "Mathematical reasoning cannot develop
in isolation. . . . [T]he ability to reason is a process that grows out of many experiences that
convince children that mathematics makes sense" (p. 31).
Cooperative/Group Learning
When kids work in small groups to solve a problem, they have the opportunity to give and
receive help in a comfortable, nonthreatening environment. The difference between the
teacher-centered classroom and the cooperative learning classroom can be visualized as pictured
in the figure to the right.
Although research has documented positive results for cooperative learning strategies in all grade
levels and all subjects (Artzt and Newman, 1997), several elements are required for this approach
to be an effective classroom practice: positive interdependence (children's sense of sinking or
swimming together), individual accountability (each child has to contribute and learn),
interpersonal skills (communication, trust, decision making), and willingness to reflect on the
team process (for more information, see the Cooperative Learning Center at the University of
Minnesota at http://www.clcrc.com).
The teacher, then, is essential in making this approach productive. It is the teacher who:
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Assigns students to groups, deciding the appropriate number and mix of children and then
monitoring group interaction. |
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Designs problems that require the input of each group member. |
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Motivates children to collaborate, in part through well-designed incentives. |
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Holds children accountable, individually and as a group, through varied assessments. |
Illustration of Mathematical Connections
The aim of this practice is to illustrate the relationships and interconnections among different
math topics in such a way that children see math not as a set of isolated topics but as a "coherent
whole." Moreover, by incorporating everyday experiences or real-world situations into the
curriculum, math teachers can show kids real-world applications of math and can help them use
it in context.
To implement this practice, teachers can:
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Refer to and build on children's previous experiences, bringing their world into the math
classroom. |
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Create opportunities to relate the current topic to other math topics. |
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Use multiple representations such as tables, graphs, and equations for a given function to
build bridges among different mathematical ideas. |
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Collaborate with teachers of other school subjects, such as art and biology. |
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Design environments that require children to apply math in a real-world scenario. |
Use of Technology
The use of technology in the classroom not only prepares children to live and work in a high-tech
society but also helps them understand mathematical concepts in powerful ways. For instance,
many graphing tools decrease the drudgery of graphing and save time for children to focus on the
interpretation of graphs and give meaning to them. Scientific or graphing calculators, computer
software, spreadsheets, the Internet, or calculator-based or microcomputer-based laboratories--even videotapes--are examples of technological tools used in math classrooms.
Teachers successfully incorporate technology into the classroom when they:
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Continually improve their own command of the available technology, including its use in
mathematics education. |
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Facilitate exploration of mathematical ideas through technological means; this teaches
children that the teacher is not the only source of information in the classroom. |
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Focus the class on the mathematical concepts rather than on the technological tools
themselves. |
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Expose children to those mathematical representations available through technology, such
as online manipulatives, graphing software, and spreadsheets. |
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Make use of multimedia in their lessons to support the multiple intelligences of students;
for example, teachers can show videos that involve children in real-world scenarios. |
Communication of Mathematical Ideas
Communicating mathematically through oral or written tasks is another way for children to
engage in the learning process. In classrooms where communication is active, children
rationalize and justify their thinking through discussion. Teachers create environments in
which children have opportunities to discuss, write, and listen to one another in the classroom.
Role playing, drawing pictures, and presenting information to the class are activities that may
encourage kids to communicate mathematically. Teachers can incorporate this practice by:
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Asking questions that clarify and extend children's ideas. |
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Assisting children as they try to model mathematical situations and develop multiple
representations (oral, written, pictorial, graphical, algebraic, or physical). |
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Exploring and utilizing mathematical language and notation with children. |
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Encouraging children to listen and to share their mental concepts of mathematical ideas. |
References
Artzt, A. F., and C. M. Newman. 1997. How To Use Cooperative Learning in the Mathematics
Class. Reston, VA: National Council of Teachers of Mathematics.
National Council of Teachers of Mathematics. 1989. Curriculum and Evaluation Standards for
School Mathematics. Reston, VA: Author.
O'Daffer, P. G., and B. A. Thornquist. 1993. "Critical Thinking, Mathematical Reasoning, and
Proof." In P. S. Wilson, ed., Research Ideas for the Classroom: High School Mathematics (pp.
39-56). New York: Macmillan Publishing Company.
| Terese A. Herrera is the Mathematics Resources Specialist at the Eisenhower National
Clearinghouse for Mathematics and Science Education at The Ohio State University in
Columbus, Ohio.
S. Asli Özgün-Koca is the Mathematics Education Analyst and an AskERIC Specialist at the
ERIC Clearinghouse for Science, Mathematics, and Environmental Education at The Ohio State
University in Columbus, Ohio. She is also a doctoral student in mathematics education at the
university.
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 Best Practices in Science Education
 Table of Contents
 Minorities in Science and Mathematics: A Challenge for Change
This page was updated on Fri Nov 2 19:14:42 GMT 2001
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