The Curriculum: One Program’s
Alignment with the NCTM
Standards
Dr. Lee Stiff
Program Author of Houghton Mifflin Math,
past President of the National Council of
Teachers of Mathematics (NCTM), and
Professor of Mathematics Education at
North Carolina State University.
To align with the standards in the National Council
While many programs tout that they are in
of Teachers of Mathematics (NCTM) Principles and
alignment, not all programs give the rigorous nature
Standards for School Mathematics, an instructional
of mathematics the careful attention that this
program must fully engage all students in learning,
program does. Moreover, Houghton Mifflin Math
understanding, and applying the content and
helps students understand the underlying meaning
process standards that describe the mathematics
of mathematics and provides opportunities for them
that students should know and be able to do. The
to develop the skills and master the procedures
first five standards describe the mathematical
needed to be successful in school mathematics. Its
content goals; they are Numbers and Operations,
sequence of lessons and organization encourage
Algebra, Geometry, Measurement, Data Analysis,
students to make the connections necessary for
and Probability. The next five standards address the
mathematics understanding and success.
processes of mathematics; they are Problem Solving,
Reasoning and Proof, Communication, Connections,
CONNECTIONS WITH PRIOR UNDERSTANDING
and Representation. The content and process
Houghton Mifflin Math provides systematic instruc-
standards are inextricably linked (NCTM, 2000).
tion on the concepts and ideas of mathematics,
Effective, aligned programs reflect this vital link in
being careful to draw upon what has been taught
their teaching and learning of mathematics.
and learned previously. The balance between conceptual understanding and computational proficien-
Houghton Mifflin Math, a new, coherent and com-
cy is achieved when the curriculum enables and
prehensive K–6 mathematics instructional
encourages students to build on the mathematics
program, fully aligns with the NCTM standards.
they already know. Quality instructional materials
T H E C U R R I C U LU M : O N E P R O G R A M ’ S A L I G N M E N T W I T H T H E N CT M S TA N D A R D S
demonstrate computational algorithms and explain
clearly the mathematics involved in each step of the
Student Book, grade 5, page 68
LINKING THE CONTENT STANDARDS WITH THE
PROCESS STANDARDS
procedure. This
Principles and Standards for School Mathematics
helps students
emphasizes the value and use of modeling in
make the integral
several of its standards, including algebra, geometry,
connections that
and representation. Houghton Mifflin Math
enable them to
uses many
understand the
powerful models
underlying
to represent
concepts exemplified in procedures as well as
mathematical
improve their computational proficiency.
ideas and
procedures,
Consider, for example, the program’s grade 4
which is a strong
measurement lessons wherein students first
feature of the
participate in hands-on activities and games that
program. In this
expose relationships about perimeter and area.
Later, when the
formulas for
perimeter and
area are introduced and
developed,
students are
able to connect
their previous
Student Book, grade 4, page 456
grade 4 example,
Student Book, grade 4, page 262
the average of
whole numbers is represented with counters. The
importance of this model is that it enables students
to experience the effect of the algorithm for finding
the average of a group of numbers. That is, the
effect of the addition and division used in the
algorithm is demonstrated simply and naturally by
using counters.
concrete experi-
Throughout all grade levels, other process standards
ences with the
are emphasized in the context of content standards.
new, symbolic
Reasoning and problem-solving skills are routinely
representation.
developed.
This offers an
Students are
ideal way for students to utilize what they know
asked to
and understand, and then recognize and understand
communicate
the elements (variables) of a formula and why
about mathe-
they are what they are. Students have had many
opportunities to investigate and learn about
matics by
Student Book, grade 6, page 201
talking and
perimeter before they learn the formula for the
writing about what they have learned or what they
perimeter of a rectangle in this lesson.
think about the possibilities of a situation.
With Houghton Mifflin Math, students participate in
activities, games, and explorations that challenge
their understanding of mathematical ideas and
Student Book, grade 3, page 508
relationships. In
In the lesson below, students manipulate figures to
these more infor-
compare them and evaluate whether they are con-
mal settings,
gruent. As new
students explain
types of figures
the reasons for
are considered, it
the particular
is necessary for
actions they take,
students to make
and plan and
conjectures about
strategize in order
the qualities of
to be successful
congruent figures.
at games. Class-
Finding contradic-
mates question
tions is a valuable
decisions, making
way to identify
it necessary for
relevant attributes
of concepts and
students’ explanations to be clear and convincing.
Hands-on activities that are part of formal lesson
instruction can contribute to students’ insights about
Student Book, grade 3, page 442
relationships.
With Houghton
how mathematical ideas are related. For example,
Mifflin Math, students learn the need to make and
in grade 3 students construct fraction models from
test conjectures, knowing that contradictions provide
paper and manipulate them in order to make the
useful information in that process.
connection between the symbols for equivalent
fractions and their physical counterparts.
Just as making conjectures allows students to
experience mathematical situations and wonder how
Opportunities for students to reason about mathe-
objects or ideas are related, formulating questions
matics and reflect on their thinking are plentiful in
about the world around us promotes mathematics
this program. Each lesson requires students to
understanding. Often events and situations in our
focus on and think about the mathematics they
world can be quantified by counting things in which
are studying when they answer “Think” or “Ask
students have an
Yourself ” problem-solving and reasoning questions.
interest. For
example, how
Student Book, grade 3, page 521
It is important for
many people like
students to experience
various flavors of
what it feels like to be
ice cream? What
a mathematician. This
is the most popu-
is achieved whenever
lar type of video
students encounter
rented during the
situations in which
summer months?
mathematical relationships may exist that they can
Which sports do
identify. Learning to recognize the commonalities in
girls prefer to
situations is an important mathematical skill that
play; which do
Houghton Mifflin Math promotes.
Student Book, grade 3, page 148
boys? Such
questions are common and require the use of data
also includes
analysis to analyze the outcomes. It is important for
problem solving
students to recognize that questions must be framed
and reasoning.
in special ways so that the information obtained is
Many activities
useful and accurate. Hence, it is necessary that
and explorations
students practice both formulating questions and
engage students’
determining the best ways to evaluate the results.
problem-solving
skills. As a result,
To ensure and deepen mathematics understanding
one of this
students need to ask themselves questions, explain
program’s
their thinking, and discuss their reasoning with
strengths is its
contextualized
development of
Student Book, grade 6, page 278
problem solving.
Student Book, grade 1, page 129
As the NCTM Curriculum Principle states, a curricu-
others. In Houghton Mifflin Math students respond
lum is more than a collection of activities; it must be
to Why? and How? questions in each
coherent, focused on important mathematics, and
lesson’s Explain Your Thinking feature. When
well articulated across the grades (NCTM, 2000).
students think mathematically, they consider
The organized sequence of mathematical topics in
multiple approaches to problems and discuss them
Houghton Mifflin Math ensures that students form
with others. They write about their solutions and
and build upon a strong foundation that is made
reflect on what they did. When students share
stronger and expanded by connections with prior
their thinking with others, they gain a deeper
understanding and among mathematical
understanding of the mathematics. This program
topics. Its instruction focuses on the key skills and
involves all of these scenarios in highlighted
important mathematical concepts that comprise the
problem-solving and skills practice.
content standards. More important, the instruction
Copyright © 2003 by Houghton Mifflin Company. All rights reserved. Litho in U.S.A. G-23511 SHA15M703
and applications involve the process standards.
AN EMPHASIS ON PROBLEM SOLVING
This deeper integration of the content and process
One of the strongest features of Houghton Mifflin
standards, combined with the extensive connections
Math is the way in which problem solving is woven
among standards and with prior understanding,
into the fabric of the instruction. Students learn the
are what bring Houghton Mifflin Math into full
value of planning to solve problems and the impor-
alignment with the NCTM standards.
tance of looking back on what they have done.
Effective problem solvers constantly monitor and
REFERENCE
adjust what they are doing (NCTM, 2000). The
National Council of Teachers of Mathematics
problem-solving model of Understand, Plan, Solve,
(NCTM). Principles and Standards for School
Look Back is revisited throughout the program and
Mathematics. Reston, VA: NCTM, 2000.
exemplified in each Problem-Solving Strategy,
Application, and Skills lesson. Each regular lesson
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