What Does Good Math Instruction Look Like? - PSD

from the ERS
Supporting
Good
Teaching
Series
What Does Good Math
Instruction Look Like
Research-based knowledge about mathematics education has
increased dramatically in recent years, contributing greatly to our understanding
of effective mathematics instruction. While a variety of instructional
strategies and methods has been found to improve student
achievement, a general theme predominates throughout the research:
high-quality mathematics instruction requires good teaching.
This issue in the Supporting Good Teaching series looks at the mathematics
skills students need to develop, the instructional techniques that
research indicates are most effective in helping students develop these
skills, and the key role of good teaching.
Balanced Math Instruction
What is the most effective method for helping students learn mathematics
While this question has been—and continues to be—the subject of
debate, standards developed by the National Council of Teachers of
Mathematics (NCTM) advocate approaches that are more consistent
with the developmental needs of children than traditional mathematics
instruction.
The two teaching approaches are clearly different. For example, in traditional
mathematics instruction, teachers focus on computation and recall
of facts. In the second approach, teachers encourage students to
explain how they arrived at a solution and to consider more than one
way of solving a problem.
Ideally, teachers should strive for a balance between the two approaches
by teaching procedural competence within a mathematics curriculum
promoting conceptual understanding and knowledge construction. By
providing a mixture of experiences that encourage computational, conceptual,
and strategic competence skills, teachers can help more of their
students achieve proficiency in mathematics.
Sequencing of Skills and Knowledge
For mathematics instruction to be effective, topics must be presented in
a sequence and manner appropriate for the developmental level of the
students (Reys et al. 1999). Although the rate at which children develop
“First and foremost,
effective mathematics
provides high-quality
instruction focused
on three important
areas: teaching for
conceptual
understanding,
developing children’s
procedural literacy,
and promoting
strategic competence
through meaningful
problem-solving
investigations”
(Shellard and Moyer
2002, 8).
Issues in the Supporting Good Teaching Series are designed to help teachers successfully
address challenges present in every classroom. Each of the issues includes a synthesis of research,
best practice, and questions intended to encourage teacher self-reflection as well as discussion among
school staff members.
© 2004 Educational Research Service

2
Using Research and Best Practice to Support Good Teaching
“Research suggests 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” (Grouws 2004, 168).
between such processes as addition and multiplication,
and subtraction and division. Instruction should
introduce multiplicative reasoning, equivalence, and a
variety of methods for computation. Instruction at this
level also should focus on developing children’s interest
in mathematics.
• Middle school students in grades six through eight
are forming conclusions about their mathematical
abilities, interest, and motivation that will influence
how they approach mathematics in later years.
Instruction at this level should build on their emerging
capabilities to think hypothetically, comprehend cause
and effect, and reason in both concrete and abstract
terms. Algebra and geometry form a large part of the
recommended curriculum during these years.
• At the high school level, students should be “adept at
visualizing, describing, and analyzing situations in
mathematical terms … they need to be able to justify
and prove mathematically based ideas” (2001,
online). Oftentimes, it is helpful for students at this
developmental level to supplement their fundamental
mathematics knowledge with additional courses,
such as computer science or statistics.
An important key to developmentally appropriate mathematics
instruction, at any age or grade level, is achieving
balance between teaching for conceptual understanding
and teaching for procedural fluency. When
students learn procedures without meaning, they are
really only memorizing discrete pieces of information
that are much more difficult for them to remember.
Students should develop an understanding of the conmathematically
varies from child to child, the NCTM
(2001) has developed a general timeline for students’
mathematical skills development and instruction identified
as appropriate for each level. According to this
timeline:
• From pre-kindergarten through second grade, children
are developing a mathematical foundation by
building beliefs about what mathematics is and what
it means to know and do mathematics. Instruction
should be provided that helps them understand patterns
and measurement and develop a solid understanding
of the numeration system.
• Building on the inquisitive nature of children in grades
three through five, students should be encouraged to
develop and investigate solutions to everyday problems.
Instruction should focus on the relationship
Basic Components of Mathematical Literacy
1. Problem solving. Students must learn to solve problems by applying previously learned or
acquired knowledge to new situations.
2. Communicating mathematical ideas. Students need to learn the language and notation used
in mathematics. They should be able to communicate mathematical ideas through the use of
manipulatives, drawings, writing, and speaking.
3. Applying mathematics to everyday situations. Students should be able to translate daily
experiences into mathematical representations, such as graphs, tables, and diagrams, and interpret
and explain the results.
4. Focusing on appropriate computational skills. Students must gain proficiency in using the
following mathematical operations: addition, subtraction, multiplication, and division, as well as
with using whole numbers, fractions, and decimals (Mercer and Miller n.d.).

What Does Good Math Instruction Look Like 3
cepts they are studying before they apply these ideas
to procedural strategies.
To determine readiness, teachers can pose questions
to assess student levels of understanding at different
points during the lesson, before students are asked to
apply skills. Good mathematics instruction will ensure
students “see” mathematics as a body of connected
ideas building on one another.
“The student should be the
reference point for addressing the
complex issue of who should learn
what mathematics and when”
(Sutton and Krueger 2002, 4).
Providing Instruction for a Range of
Knowledge and Skill Levels
While the research on grouping and tracking for mathematics
instruction is mixed, some studies suggest:
Expectations placed on students differ according
to their assigned ability group or track . . .
[and there are] crucial differences in the kinds of
instruction offered…. Instruction in the lower
tracks tends to be fragmented, often requiring
mostly memorization of basic facts and algorithms
and the filling out of worksheets.
Although some higher track classes share these
traits, they are more likely to offer opportunities
for making sense of mathematics, including discussion,
writing, and applying mathematics to
real life situations (Sutton and Krueger 2002, 4).
An alternative, especially in the elementary and middle
grades, is a differentiated approach to instruction. Using
this approach, the teacher makes intensive use of information
gathered through formal and informal assessment—much
of which is embedded in instruction—to
decide which skills and knowledge individual students
need at a specific point in time to help them move
toward competence in a particular standard. Although
ability grouping is used in addition to whole-class instruction,
the groups are fluid and membership changes
often, depending on the specific instruction needed to
move specific students to higher levels of learning.
Effective Mathematical Environments
Teachers should work toward improving the effectiveness
of their mathematics instruction by creating classroom
environments that support students’ developing
mathematical skills and nurture their interest in mathematics.
Both teacher behaviors and instructional characteristics
are important.
Teacher Behaviors
In effective classrooms, teachers project a positive attitude
about mathematics, use questioning techniques to
facilitate learning, and encourage students to develop
divergent solutions and inventive ideas for presentation
to and discussion with the whole group.
Acceptance of Students’ Divergent Ideas
Throughout a lesson, the teacher should accept varied
responses from students, challenge them to think more
deeply about the problems they are solving, and ask
them to explain the solutions. An accepting and
encouraging atmosphere is essential so students do
not develop mathematics anxiety. It is important that
teachers’ comments and questions help students
develop confidence in their own abilities to do mathematics.
In addition, discussion helps the student explain
his or her approach to gain an even firmer grasp of the
concept and provides other students with alternative
ways to solve a specific problem.
Teacher Questioning
Teachers can influence learning by posing challenging
and interesting questions. Comments and questions for
students should demonstrate a sound understanding of
the mathematics behind the lesson. Rather than tell
students the answers, teachers should present questions
that stimulate students’ curiosity and encourage
them to investigate further. Teachers may give students
suggestions and hints, and ask additional questions
that lead them to search for different solutions. The
questions should encourage students to rely on one
another and themselves for ideas about mathematics
and problem solving.
Teacher Attitudes
Student motivation and success in mathematics are
greatly influenced by teacher attitude, which should communicate—through
speech and actions—what material is
important and how to learn it. To create an environment
that supports mathematical learning, teachers must communicate
to students that mathematics is an important

4
Using Research and Best Practice to Support Good Teaching
activity in which their efforts are valued. A teacher’s attitude
toward mathematics should demonstrate enthusiasm
for the content and a belief that all students are
capable of learning the material, with lessons designed to
encourage curiosity and interest as well as skill building.
Instructional Characteristics
In addition to teacher behaviors, certain instructional
characteristics also are associated with effective mathematics
instruction. By embedding the following
approaches in instruction, teachers can promote student
learning and motivation:
• Students are actively engaged in doing mathematics.
• Students are solving challenging problems.
• Interdisciplinary connections and examples are used
to teach mathematics.
• Students are sharing their mathematical ideas while
working in pairs and groups.
• Students are provided with a variety of opportunities
to communicate mathematically.
• Students are using manipulatives, calculators, and
computers.
Being Actively Engaged in Doing Mathematics
Students should not be sitting back watching others
“do mathematics”; rather, everyone should be engaged
in the investigations. This encouragement might take
the form of working on an interesting problem, using
manipulatives to find several solutions to a problem, or
using a computer graphing program to represent mathematical
data. What is important is that all students are
involved in finding the solutions.
Solving Challenging Problems
Mathematics is a stimulating and interesting field generating
new knowledge every day, and students should
be exposed to this excitement and challenge. Problems
that involve various steps or that model mathematics
problem solving in the real world are more authentic and
meaningful for students. Students should be provided
with opportunities to practice using many different skills
embedded in solving an interesting problem.
Use of Small Groups in Mathematics Instruction
Research findings clearly support the use of small groups as part of mathematics instruction. This
approach can result in increased student learning as measured by traditional achievement measures as
well as other important outcomes, such as improvement in student ability to communicate, resolve differences,
and get along with others. When using small groups for mathematics instruction, teachers should:
• Choose tasks dealing with important mathematical concepts and ideas.
• Select tasks appropriate for group work.
• Consider having students initially work individually on a task and then engage in group work to share
and build on their individual ideas and work.
• Give clear instructions to the groups, and set clear expectations for each group.
• Emphasize both group goals and individual accountability—for example, indicate that an individual
from each group will report for the group, but withhold his or her name until group work has been
completed.
• Choose tasks students find interesting.
• Ensure the group work has closure, where key ideas and methods are brought to the surface either
by the teacher, the students, or both.
Finally, as several research studies have shown, teachers should not think of small groups as an approach
that must always be used or never be used. Rather, small-group instruction should be seen as an appropriate
instructional practice for certain learning objectives, and as an approach that can work well with
other organizational arrangements, including whole-class instruction (adapted from Grouws 2004).

What Does Good Math Instruction Look Like 5
In addition, students should be asked to do more than
the application of mathematical procedures. Research
that compares the way in which mathematics is typically
taught in many U.S. classrooms with the approach
used in other countries with higher mathematics
achievement suggests U.S. students should be provided
with more opportunities to “make connections.” For
example, “in asking students to solve for x in the equation
2x+3=11, teachers might also have asked if it
would be acceptable to divide each side by a specific
number, changing the process from a simple use of
procedure to a more complex examination” of the
problem (Association for Supervision and Curriculum
Development 2003, online).
Using Interdisciplinary Connections and Examples
Using literature as a springboard to mathematical investigation
is a useful way to introduce a problem in context
and encourage meaningful mathematics investigations.
A mathematics investigation placed in the context
of a story provides an authentic problem-solving situation,
sometimes with “messy” results that model realworld
problems. This engages students in connecting
the language of mathematical ideas with numerical representations
and develops important skills that support
students’ abilities to solve word problems. Teachers
may also connect mathematics with other content
areas in the school curriculum, such as science, social
studies, or language arts, to show its relevance to
problem solving in different disciplines.
Providing Opportunities for Group Work
Research shows students who work in groups on
problems, assignments, and other mathematical investigations
display increased achievement. Opportunities
for students to work in pairs and small groups give
them the chance to share their ideas and solution
routes with peers. As students listen to and talk with
one another, they begin to see mathematical relationships
that build on their previous notions and conjectures.
Giving an explanation to a peer is also positively
related to increased student achievement.
Allowing Opportunities for Student Communication
During a lesson, students should have many opportunities
to communicate their ideas. They may draw their
ideas in pictures or write them in mathematics journals.
After students work on a challenging problem, the
teacher should ask individuals or groups to share their
solution routes by presenting them on chart paper or a
transparency. Whole-class discussions, which are effective
for sharing and explaining solutions, give students a
chance to challenge and evaluate the validity of other
students’ ideas in an environment of respect and
understanding.
Using Manipulatives
The long-term use of mathematics manipulatives is
positively related to student achievement and attitudes
about mathematics. It is not enough, however, to simply
provide students with manipulatives: they must be
taught how to use these materials. When used correctly,
manipulatives are “conducive to the concrete kinds
of learning that lay a sufficient foundation for abstract
thought” (Ross and Kurtz 1993, 256).
Several steps can be taken to ensure students benefit
from a lesson involving manipulatives. First, the teacher
must make certain the manipulatives support the lesson’s
objectives. Second, before allowing students to
handle the materials, the teacher must demonstrate use
of the manipulatives and procedures for handling them.
And third, the lesson design must support the active
participation of all students (Ross and Kurtz 1993).
Teachers should provide students with a wide variety of
manipulative materials for use in mathematics activities.
Students should use manipulatives to represent mathematics
in concrete form, and then proceed to using pictorial
models and abstract symbols. There should be
items for drawing and writing about mathematics during
problem solving, as well as geometric shapes and other
materials for building and exploring.
Using Calculators and Technology
Another important tool for the mathematics classroom
is technology. As with manipulatives, the way a teacher
uses technological tools determines the tools’ effectiveness
for conveying mathematical ideas. The computer
environment provides many visual representations—
graphs, diagrams, geometric figures, and moving
images. The sheer graphic power of technological tools
allows students to explore and manipulate many different
visual representations of mathematical models.
Virtual manipulatives are now available for investigating
the characteristics of geometric shapes (Moyer,
Bolyard, and Spikell 2002). A child as young as kindergarten
age can explore complex repeating patterns
using virtual manipulatives on the computer screen.
Because technology use is one key to language and
mathematical literacy, students need different opportunities
to learn to use it effectively as a learning tool.

6
Using Research and Best Practice to Support Good Teaching
National Council of Supervisors of Mathematics: Components of Essential Mathematics
1. Problem Solving. Learning to solve problems is the principal reason for studying mathematics. Problem solving
is the process of applying previously acquired knowledge to new and unfamiliar situations. Solving word problems
in texts is one form of problem solving, but students also should be faced with non-text problems.
2. Communicating Mathematical Ideas. Students should learn the language and notation of mathematics. For
example, they should understand place value and scientific notation. They should learn to receive mathematical
ideas through listening, reading, and visualizing. They should be able to present mathematical ideas by speaking,
writing, drawing pictures and graphs, and demonstrating with concrete models. They should be able to discuss
and ask questions about mathematics.
3. Mathematical Reasoning. Students should learn to make independent investigations of mathematical ideas.
They should be able to identify and extend patterns and use experiences and observations to make conjectures
(tentative conclusions). They should be able to distinguish between valid and invalid arguments.
4. Applying Mathematics to Everyday Situations. Students should be encouraged to translate everyday situations
into mathematical representations (graphs, tables, diagrams, or mathematical expressions), process the
mathematics, and interpret the results in light of the initial situation. They should be able to solve ratio, proportion,
percent, direct variation, and inverse variation problems. Students should see how mathematics is applied in the
real world, as well as observe how mathematics grows from the world around them.
5. Alertness to the Reasonableness of Results. In solving problems, students should question the reasonableness
of a solution or conjecture in relation to the original problem. Students must develop number sense to determine
if results of calculations are reasonable in relation to the original numbers and the operations used. With
society’s increasing use of calculating devices, this capability is more important than ever.
6. Estimation. Students should be able to carry out rapid approximate calculations by using mental arithmetic and
various computational estimation techniques. Students should acquire simple methods for estimating measurements
such as length, area, volume, and mass (weight). They should be able to decide when a particular result is
precise enough for the purpose at hand.
7. Appropriate Computational Skills. Students should gain facility in using addition, subtraction, multiplication,
and division with whole numbers and decimals. In learning to apply computation, students should have practice in
choosing the appropriate computational method: mental arithmetic, paper-pencil algorithm, or calculating device.
8. Algebraic Thinking. Students should learn to use variables (letters) to represent mathematical quantities and
expressions, and they should be able to represent mathematical functions and relationships using tables, graphs,
and equations. They should understand and correctly use positive and negative numbers, order of operations, formulas,
equations, and inequalities. They should recognize the ways in which one quantity changes in relation to
another.
9. Measurement. Students should learn the fundamental concepts of measurement through concrete experiences.
They should be able to measure distance, mass (weight), time, capacity, temperature, and angles. They should
learn to calculate simple perimeters, areas, and volumes. They should be able to perform measurement in both
metric and customary systems using the appropriate tools and levels of precision.
10. Geometry. Students should understand the geometric concepts necessary to function effectively in the threedimensional
world. They should have knowledge of concepts such as parallelism, perpendicularity, congruence,
similarity, and symmetry. Students should know properties of simple plane and solid geometric figures.
11. Statistics. Students should plan and carry out the collection and organization of data to answer questions in their
everyday lives. Students should know how to construct, read, and draw conclusions from simple tables, maps,
charts, and graphs. They should be able to present information about numerical data such as measure of central
tendency (mean, median, mode) and measures of dispersion (range, deviation). Students should recognize the
basic uses and misuses of statistical representation and inference.
12. Probability. Students should understand elementary notions of probability to determine the likelihood of future
events. They should become familiar with how mathematics is used to help predict election results, business forecasts,
and outcomes of sporting events. They should learn how probability applies to research results and to the
decision-making process (excerpted from National Council of Supervisors of Mathematics 1989, 44-46).

What Does Good Math Instruction Look Like 7
Calculator use in early-grades mathematics instruction
has caused great debate within the mathematics community
and among the general public (Battista 1994;
Calvert 1999; Goldenberg 2000; Mercer 1992).
Calculator use can affect not only how students learn but
also what they learn. As with other mathematics tools, it
is how the tool is used for teaching and learning—not
the tool itself—that causes the debate. Advocates for
calculator use in the early grades propose students use
calculators in school mathematics to explore patterns
and relationships with numbers. For example, young children
can examine patterns among very large numbers
with the support of a calculator or create a graph of a
large data set using graphing programs.
This does not mean that students should use calculators
when they are working to develop computational
fluency. During problem solving or when working with
large numbers, however, calculators can be used to
relieve the burden of complex computations and to
help students focus on the concepts being presented.
Opportunities to use calculators as part of the curriculum
throughout the year help students to recognize
when calculator use is appropriate.
Good Teaching Is Key
Of course, effective mathematics instruction begins
with effective teaching. No lesson, no matter how well
planned, can be successful if the elements of effective
teaching are not in place. Douglas Grouws, editor of
the Handbook of Research on Mathematics Teaching
and Learning and author of the mathematics chapter in
Handbook of Research on Improving Student
Achievement, discusses instructional practices identified
by research as having a positive impact on student
learning and then mentions the role of the teacher:
The quality of the implementation of a teaching
practice also greatly influences its impact on
student learning. The value of using manipulative
materials to investigate a concept, for
example, depends not only on whether manipulatives
are used, but also on how they are used
with the students. Similarly, small-group instruction
will benefit students only if the teacher
knows when and how to use this teaching
practice (2004, 162).
Finally, effective teachers of mathematics must be
effective teachers overall, exhibiting effective classroom
management skills, especially in classrooms using differentiated
instruction; actively engaging their students;
and making efficient use of instructional time. A mathematics
lesson cannot succeed if the other elements of
teaching—classroom management, a logical progression
of lessons, an effective use of assessment, and
time management—are not in place.
References
Association for Supervision and Curriculum Development. (2003,
October). International comparisons of mathematics instruction.
Research Brief. Retrieved from
http://www.ascd.org/publications/researchbrief/volume1/v1n21.html
Battista, M.T. (1994). Calculators and computers: Tools for mathematical
exploration and empowerment. Arithmetic Teacher (March
1994), 412-417.
Calvert, L.G. (1999). A dependence on technology and algorithms
or a lack of number sense Teaching Children Mathematics
(September 1999), 6-7.
Goldenberg, E.P. (2000). Thinking (and talking) about technology in
mathematics classrooms. Newton, MA: The K-12 Mathematics
Curriculum Center, Education Development Center, Inc. Retrieved
from http://www2.edc.org/mcc/iss_tech.pdf
Grouws, D.A. (2004). Chapter 7. Mathematics. In G. Cawelti (Ed.),
Handbook of Research on Improving Student Achievement (3rd edition),
162-181. Arlington, VA: Educational Research Service.
Mercer, J. (1992). What is left to teach if students can use calculators
The Mathematics Teacher (September 1992), 415-417.
Mercer, C.D., & Miller, S.P. (n.d.). Teaching students with learning
problems in math to acquire, understand, and apply basic facts.
Retrieved from
http://www.enc.org/print/professional/learn/equity/articles/document.shtminput=ACQ-111397-1397_1
Moyer, P.S., Bolyard, J.J., & Spikell, M.A. (2002). What are virtual
manipulatives Teaching Children Mathematics (February 2002),
372-377.
National Council of Supervisors of Mathematics. (1989). Essential
mathematics for the 21st century. Arithmetic Teacher (September
1989), 44-46.
National Council of Teachers of Mathematics (NCTM). (2001).
Principals and standards for school mathematics. Retrieved from
http://www.nctm.org/standards/overview.htm
Reys, R.E., Suydam, M.N., Lindquist, M.M., & Smith, N.L. (1999).
Helping children learn mathematics (5th edition). New York: John
Wiley and Sons, Inc.
Ross, R., & Kurtz, R. (1993). Making manipulatives work: A strategy
for success. Arithmetic Teacher (January 1993), 254-257.
Shellard, E., & Moyer, P.S. (2002). What principals need to know
about teaching math. Alexandria, VA: National Association of
Elementary School Principals and Educational Research Service.
Sutton, J., & Krueger, A. (2002). EDThoughts: What we know about
mathematics teaching and learning. Aurora, CO: Mid-continent
Research for Education and Learning.

8
What Does Good Math Instruction Look Like
Questions for Discussion and Reflection
• How do you ensure your students truly understand the math concepts they are studying rather than
merely memorizing procedures Share and discuss any specific techniques with your colleagues.
• Revisit the section titled “Effective Mathematical Environments” and glance through the section subheadings.
Are your personal teaching behaviors and instructional characteristics consistent with
those recommended How might you improve the environment in your classroom
• Discuss the information from the National Council of Supervisors of Mathematics provided in the
text with your colleagues. Do you consider the components listed when developing lesson plans
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Problem Solving in Math and Science (#NT-5292)
Reviews effective methods and strategies for teaching problem
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Assesses ability grouping in math instruction and the effects of
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