A Good Math Instruction Manual

Contributed by:
Sharp Tutor
Research indicates that it takes more than a good instructor to teach math. Outlined in this article are recommendations from the National Council of Teachers of Mathematics, as well as other sources, which offer insight on what truly makes math instruction effective.
1. research report
What Does
Good Math
Look Like?
Nancy Protheroe
It involves good teachers,
an effective math environment,
and a curriculum that is
more than a mile wide
and an inch deep.
ur research-based knowledge about good math IN BRIEF
instruction, although not as extensive as that focused Research indicates that it takes more
than a good instructor to teach
on reading instruction, has increased in recent math. Outlined in this article are
years. It now provides a solid base of information for educa- recommendations from the National
Council of Teachers of Mathematics,
tors to use as they identify mathematics skills students need as well as other sources, which offer
to develop, as well as teaching strategies and instructional insight on what truly makes math
instruction effective.
approaches that best support the development of these skills.
What Gets Taught that should be addressed in each grade (see Web
When considering content knowledge and skills, resources section on how to access more informa-
it is obvious that schools must look first at the state tion about the report). For example, NCTM iden-
standards that students are expected to master. tifies these math focal points for second grade:
However, research comparing math instruction
in the U.S. and other countries has pointed to an n Number and Operations. Developing an under-
underlying problem with many of our standards- standing of the base-10 numeration system and
based systems. Typically, these systems address too place-value concepts;
many standards for each grade level—encouraging n Number and Operations and Algebra. Developing
the development of a curriculum that has been quick recall of addition facts and related subtrac-
characterized as “a mile wide and an inch deep.” tion facts and fluency with multidigit addition and
In contrast, the National Council of Teachers subtraction; and
of Mathematics (NCTM) has developed “Cur- n Measurement. Developing an understanding of
riculum Focal Points,” a report that identifies linear measurement and facility in measuring
three broad—but critical—mathematical concepts lengths.
SadIk dEmIROz/SuPERSTOck Principal n September/October 2007 51
2. NCTM suggests that state boards Instruction at this level should also
“… Research comparing
of education and other groups devel- focus on developing children’s
oping standards use the focal points math instruction in the interest in mathematics.
as a “clear organizational model for n Students in grades 6 through 8 are
establishing a mathematics curriculum U.S. and other countries forming conclusions about their
from pre-kindergarten through grade mathematical abilities, interest, and
has pointed to an
8” (NCTM 2006). At the local district motivation that will influence how
and school levels, teacher conversa- underlying problem with they approach mathematics in later
tions and staff development could be years. Instruction at this level should
organized around the focal points. many of our standards- build on their emerging capabilities
Encouraging teachers from several to think hypothetically, comprehend
based systems.”
grades to participate in such a setting cause and effect, and reason in both
ensures discussions are also focused on concrete and abstract terms. Algebra
the linkage of math instruction from What should effective mathematics and geometry form a large part of the
grade to grade. instruction look like? Shellard and recommended curriculum during
Moyer (2002) identify three critical these years.
How It Gets Taught components: “Teaching for conceptual
During the reading wars between understanding, developing children’s An important key to developmental-
proponents of a whole language procedural literacy, and promoting stra- ly appropriate mathematics instruction,
approach and those favoring skills- tegic competence through meaningful at any age or grade level, is achieving
based instruction, educators found problem-solving investigations.” balance between teaching for concep-
that a careful and intensive review of Also, topics should be presented in a tual understanding and teaching for
research revealed the importance of sequence and manner appropriate for procedural fluency. When students
using a combination of both approach- the developmental level of the students learn procedures without meaning,
es. Similarly, there are at least two (Reys et al. 1999). Although the rate they are only memorizing discrete
camps prominent in the discussion of at which children develop mathemati- pieces of information that are diffi-
how math should be taught. The two cally varies from child to child, NCTM cult for them to remember. Students
teaching approaches have clear differ- (2001) has developed a general time- should develop an understanding of
ences. In skills-based instruction, teach- line for students’ mathematical skills the concepts they are studying before
ers focus on developing computational development and instruction identi- they apply these ideas to procedural
skills and recall of facts. In the second fied as appropriate for each level. strategies.
approach, teachers encourage students According to this timeline:
to explain how they arrived at a solu- Good Teaching Is Key
tion and to consider more than one n From pre-kindergarten through Of course, effective mathematics
way of solving a problem. second grade, children develop a instruction begins with effective teach-
Ideally, teachers should strive for a mathematical foundation by build- ing. No lesson, no matter how well
balance between the two approaches. ing beliefs about what mathematics planned, can be successful if the ele-
Doug Grouws (2004), recently hon- is and what it means to understand ments of effective teaching are not in
ored for his long-time contributions and “do” mathematics. Instruction place. Grouws (2004) discusses the
to mathematics education with the should be provided that helps them instructional practices that research
NCTM Lifetime Achievement Award, understand patterns and measure- has shown to have a positive impact on
talks about this: ment and develop a solid under- student learning and then mentions the
standing of the numeration system. role of the teacher:
Research suggests it is not necessary for n Building on the inquisitive nature
teachers to focus first on skill develop- of children in grades 3 through 5, The quality of the implementation of a
ment and then move on to problem- students should be encouraged to teaching practice also greatly influences
solving. Both can be done together. develop and investigate solutions its impact on student learning. The
Skills can be developed on an as-needed to everyday problems. Instruction value of using manipulative materials
basis, or their development can be should focus on the relationship to investigate a concept, for example,
supplemented through the use of tech- between such processes as addition depends not only on whether manipu-
nology. In fact, there is evidence that if and multiplication, and subtrac- latives are used, but also on how they
students are initially drilled too much tion and division. Students should are used with the students. Similarly,
on isolated skills, they have a harder be introduced to multiplicative small-group instruction will benefit
time making sense of them later. reasoning, equivalence, and a vari- students only if the teacher knows when
ety of methods for computation. and how to use this teaching practice.
52 Principal n September/October 2007 www.naesp.org
3. In addition, to effectively develop Such an approach also helps stu- ematics instruction. By integrating the
students’ mathematical skills, teachers dents develop confidence in their following approaches into classroom
must be effective overall. They must own abilities to do mathematics and instruction, teachers can promote both
exhibit good classroom management gain an even firmer grasp of key student learning and motivation:
skills, especially in classrooms using dif- concepts and processes.
ferentiated instruction; actively engage n Influence learning by posing challeng- n Students are actively engaged in doing
their students; and make efficient use ing and interesting questions. Teachers mathematics. They should not be
of instructional time. A mathematics should present questions that stimu- sitting back watching others students
lesson cannot succeed if the other ele- late students’ curiosity and encour- solve problems.
ments of teaching—classroom manage- age them to investigate further. The n Students are solving challenging prob-
ment, a logical progression of lessons, questions should encourage students lems. Mathematics is a stimulating
an effective use of assessment, and to rely on themselves and their peers and interesting field generating new
time management—are not in place. for ideas about mathematics and knowledge every day, and students
problem-solving. should be exposed to this excite-
An Effective Mathematics n Project a positive attitude about math- ment and challenge, using real-world
Environment ematics and about students’ ability examples when possible.
There are some specific teacher to “do” mathematics. This includes n Interdisciplinary connections and
behaviors that “matter” in the teaching demonstrating enthusiasm for the examples are used to teach mathematics.
of mathematics. In effective classrooms, content as well as a belief that all For example, using literature as a
teachers: students are capable of learning the springboard for mathematical inves-
material, with lessons designed to tigation is a useful way to introduce
n Demonstrate acceptance of students’ encourage curiosity, interest, and authentic problem-solving situations
divergent ideas. They challenge stu- skill-building. that may have “messy” results. This
dents to think more deeply about engages students in connecting the
the problems they are solving and Certain instructional characteristics language of mathematical ideas
ask them to explain the solutions. also are associated with effective math- with numerical representations and
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www.naesp.org Principal n September/October 2007 53
4. develops important skills that sup- In recognition of the importance National Mathematics Advisory Panel.
port students’ abilities to solve word of the topic, a National Mathematics Preliminary Report, 2007. Retrieved from
problems. Advisory Panel was recently estab- www.ed.gov/about/bdscomm/list/
n Students are sharing their mathemati- lished within the U.S. Department Reys, R. E., M. N. Suydam, M. M. Lindquist,
cal ideas while working in pairs and of Education. Its charge is “to foster and N. L. Smith. Helping Children Learn
groups. Research shows that students greater knowledge of and improved Mathematics, 5th ed. New York: John
who work in groups on problems, performance in mathematics among Wiley and Sons, 1999.
assignments, and other mathemati- American students … with respect to Ross, R. and R. Kurtz. “Making
Manipulatives Work: A Strategy for
cal investigations display increased the conduct, evaluation, and effective Success.” Arithmetic Teacher January
achievement. Such opportunities use of the results of research relating (1993), 254-57.
appeal to the social nature of most to proven-effective and evidence- Shellard, E. and P. S. Moyer. What Principals
children, while thinking through based mathematics instruction” (Bush Need to Know about Teaching Math.
problems collaboratively makes 2006). Members of the panel have Alexandria, Va.: National Association
of Elementary School Principals and
it less likely that a student will get been assigned to four task forces Educational Research Service, 2002.
caught in a procedural dead end. focused on critical areas of mathemat-
n Students are provided with a variety ics instruction: learning processes,
of opportunities to communicate math- conceptual knowledge and skills,
ematically. During a lesson, students instructional practices and materials,
should have many opportunities to and teachers and teacher education W eb Resou rc es
communicate their ideas. They may (National Mathematics Advisory Panel Information provided on the NCTM
draw a picture to represent their 2007). Web site includes the council’s focal
ideas or write them in mathematics For teachers and others responsible points by grade as well as questions and
journals. Whole-class discussions for ensuring our students receive the answers about the focal points.
should provide opportunities to hear best possible mathematics education, www.nctm.org/focalpoints.
about and perhaps challenge other ongoing efforts to stay informed about aspx?ekmensel=c580fa7b_10_48_
students’ ideas in an environment of current research findings will be as btnlink
respect and understanding. important as the frameworks provided
n Students are using manipulatives and by state standards. As principal, your “Mathematical Understanding: An
other tools. The long-term use of role in ensuring that happens in your Introduction,” a chapter from the book
mathematics manipulatives is posi- school is critical. P How Students Learn: History, Mathematics,
tively related to student achievement and Science in the Classroom, can be
and attitudes about mathematics. It Nancy Protheroe is director of special accessed at the National Academies
is not enough, however, to simply research projects at the Educational Press Web site.
provide students with manipulatives; Research Service. Her e-mail address is www.nap.edu/catalog.php?record_
they must be taught how to use these [email protected]. id=10126#toc
materials. Several steps can be taken
to ensure students benefit from a References
The K-12 Mathematics Curriculum
Bush, G. W. Executive Order: National
lesson involving manipulatives. First, Mathematics Advisory Panel, 2006. Center, funded by the National Science
the teacher should use manipulatives Retrieved from www.whitehouse.gov/ Foundation, provides information on
that support the lesson’s objectives. news/releases/2006/04/20060418- textbook adoption and a downloadable
Next, before allowing students to 5.html. guide to several mathematics curriculum
handle the materials, the teacher Grouws, D. A. “Chapter 7. Mathematics.”
In Handbook of Research on Improving
should demonstrate how to use the Student Achievement, 3rd ed., edited by www2.edc.org/mcc/default.asp
manipulatives and the procedures G. Cawelti. Arlington, Va.: Educational
for handling them. And finally, the Research Service, 2004.
lesson design should encourage the National Council of Teachers of
active participation of all students Mathematics (NCTM). Principles and
Standards for School Mathematics,
(Ross and Kurtz 1993). 2000. Retrieved from http://standards.
An Ongoing Dialogue National Council of Teachers of
Conversations about math instruc- Mathematics. Curriculum Focal Points
tion will continue. Parents, educators, for Prekindergarten through Grade
8 Mathematics, 2006. Retrieved
policymakers, and future employers are from www.nctm.org/focalpoints.
all concerned about what—and how aspx?ekmensel=c580fa7b_10_48_
well—our students learn mathematics. btnlink.
54 Principal n September/October 2007 www.naesp.org
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