AAAS Conference
on Developing Textbooks That
Promote Science Literacy
February
27-March
2, 2001
American Association for the Advancement of Science
Washington, D.C.
Developing
Coherent, High Quality Curricula:
The Case of the Connected Mathematics Project
A background
paper commissioned for the
AAAS Project
2061 Science Textbook Conference
Washington, D.C.
February 27-March
2, 2001
Elizabeth Difanis Phillips
Michigan State University
Glenda Lappan
Michigan State University
Susan N. Friel
University of North
Carolina
James
T. Fey
University of Maryland
Table of
Contents
Developing Coherent, High
Quality Curricula:
The Case of the Connected Mathematics Project (CMP)
The Connected Mathematics
Project is one of the
13 curriculum development projects
that were funded by NSF between
1990 and 1993 to help realize
the National Council of Teachers
of Mathematics’ Curriculum
and Evaluation Standards for
School Mathematics in classrooms. Connected
Mathematics, the project’s
middle grades curriculum, was
awarded an “Exemplary” designation
by the Mathematics Panel of
the U.S. Department of Education
and was the most highly ranked
middle grades curriculum material
in an evaluation of mathematics
textbooks conducted by the American
Association for the Advancement
of Science.
Connected Mathematics is
a complete curriculum for teachers
and students in grades six,
seven, and eight. The curriculum
is published in eight modules
for each grade level accompanied
by a comprehensive teacher’s
guide for each module. While
the modules might appear to
be interchangeable within a
grade level, they are carefully
sequenced to provide the most
powerful mathematics learning
experience for students. Within
a module, the mathematics is
developed through a series of
investigations, each focusing
on key mathematical concepts,
skills, processes, and ways
of thinking. Investigations
involve work on two to five
mathematically rich situations
that engage students in extended
problem-solving activity, often
in collaboration with their
classmates.
About
the Project
In the Connected Mathematics
Project (CMP), the curriculum
materials for students and teachers
reflect a commitment to the
philosophy that teaching and
learning are not distinct—“what
to teach” and “how
to teach it” are inextricably
linked. We believe that
The circumstances in which
students learn affect what
is learned. We have worked
to produce a curriculum that
helps teachers and those who
work to support teachers to
examine their expectations
for students and analyze the
extent to which classroom mathematics
tasks and teaching practices
are aligned with their goals
and expectations. (Lappan & Phillips,
1998)
Prior
Experiences
A critical factor
in the development of Connected
Mathematics was the collective
experience of the authors. Each
of us has had extensive prior
experiences with research and
with innovative approaches to
mathematics curriculum and teaching.
In the Middle Grades Mathematics
Project (MGMP), Glenda
Lappan, Elizabeth Phillips,
and William Fitzgerald did extensive
research on creative new teaching
strategies and professional
development of teachers. They
developed five MGMP curriculum
units (Mouse and Elephant, Spatial
Visualization, Factors
and Multiples, Probability, and Similarity)
that challenge students to become
active learners, both intellectually
and physically. These units
were built on a teaching model
that employed mathematical investigations
as the primary locus for mathematics
learning.
The MGMP materials were used
in a variety of research studies.
One such study explored teacher
professional development through
expert coaching (Lappan et al.,
1988). Several studies were
conducted on student learning,
in proportional reasoning (Lappan & Friedlander,
1987) and spatial visualization
(Ben-Chaim et al., 1986, 1988,
1989; Lappan et al., 1985).
A study of the learning and
teaching of growth relationships
was conducted by Fitzgerald
and Shroyer in 1979. From 1975
to –1990, Fitzgerald,
Lappan, and Phillips spent many
hours in the classroom doing
teaching experiments around
drafts of units, conducting
research, and working with teachers
and students in a variety of
settings.
In 1988, Glenda Lappan and
Perry Lanier were funded through
the National Center for Research
on Teacher Learning to conduct
a multi-year study of the development
of a cadre of 24 elementary
teacher education students in
both mathematics and the teaching
of mathematics. We used the
MGMP curriculum as a core set
of ideas from which to develop
a sequence of university mathematics
courses for these students.
This research project gave us
another opportunity to examine
carefully the mathematical development
of students and then to follow
them into their classrooms as
elementary teacher over the
first two years of teaching.
We learned a great deal about
curriculum design and effective
teaching that has been of use
to us in our current work.
In the Used Numbers Project,
Susan Friel had done similar
curriculum development work
focusing on making statistics
ideas accessible to upper elementary
grades students. She followed
that curriculum work with the Teach-Stat
Project that focused on
the professional development
of teachers. Jim Fey’s
work on the Computer Intensive
Algebra Project (CIA) explored
strategies for teaching algebraic
ideas by embedding them in authentic
problem-solving activities and
by employing calculators and
computers as learning and problem-solving
tools. Fey is also an author
of Core-Plus, a high
school curriculum project funded
by the National Science Foundation
(NSF).
Each of the research and development
projects in which the authors
participated prior to CMP was
related to curriculum, teaching,
and/or student learning. Each
led to publications in refereed
journals, presentations at meetings,
and to the development of curriculum
materials that were implemented
in many schools. The CMP authors
also had a deep understanding
of the National Council of Teachers
of Mathematics (NCTM) standards. Lappan
was Grades 5–8 Chair of
the NCTM Curriculum and
Evaluation Standards for School
Mathematics and Chair of
the NCTM Professional Standards
for Teaching Mathematics (1989).
Friel was Chair of the Professional
Development Writing Group for
the NCTM Professional Standards
for Teaching Mathematics (1991).
Phillips was an author of Patterns
and Functions for Middle Grades—the
NCTM Addenda to the Standards (1991).
Fey wrote a chapter, “Quantity,” for On
the Shoulders of Giants (National
Research Council, 1990) and
was a member of the NRC committee
that produced curriculum recommendations
for Reconstructing School
Mathematics (NRC, 1990).
Lappan was also NCTM president
during the development of NCTM
Principles and Standards 2000.
Emerging
Philosophy: A Problem-Centered
Curriculum
Formal mathematics
begins with undefined terms,
axioms, and definitions and
deduces important conclusions
logically from those starting
points. However, mathematics
is produced and used in a much
more complex combination of
exploration, experience-based
intuition, and reflection. The
CMP team began its work with
some shared assumptions about
mathematics, learning, and teaching.
Over time and with experience,
many of these assumptions have
evolved into a philosophical
framework to guide our design
of curriculum materials. Among
these guiding philosophies are
those that relate to:
Curriculum materials:
-
An effective curriculum
has coherence—it builds
and connects from investigation
to investigation, unit to
unit, and grade to grade.
-
The “big,” or
key, mathematical ideas around
which the curriculum will
be built should be identified.
-
Each key idea may be related
to a number of smaller concepts,
skills, or procedures. These
need to be identified, elaborated,
exemplified, and connected.
-
Ideas must be explored in
sufficient depth to allow
students to make sense of
them. Superficial treatment
of an area produces shallow
and short-lived understanding
and does not support the making
of connections among ideas.
-
Mathematical tasks that
students work on inside and
outside the classroom are
the primary vehicle for student
engagement with the mathematical
concepts to be learned.
-
Posing mathematical tasks
in context (i.e., a problem-centered
curriculum) provides support
both for making sense of the
ideas and for processing them
so that they can be recalled.
Field testing and evaluation:
-
Effective curriculum materials
development requires careful
field-testing, evaluation,
and rewriting over several
trials.
-
Research—including
formative and summative student
evaluation as well as other
relevant studies—is
essential in determining what
seems to be working and where
revision is needed.
-
Research is needed on student
and teacher learning from
the curriculum to provide
evidence for school districts
that may consider adopting
the materials.
Teacher support:
-
Successful implementation
of materials depends on teachers
enacting the curriculum in
a way that supports the philosophy
of learning and teaching underlying
the materials’ development.
In order for this to be the
case, teachers need effective,
ongoing professional development
that focuses on the materials
themselves rather than generic
teacher enhancement.
-
Teachers need opportunity
and support to collaborate
with each other in study and
planning for teaching the
curriculum.
-
Teacher support materials
for the curriculum are needed
to provide help with mathematics,
assessment, and pedagogy.
Administrative and community
support:
-
Superintendents, principals,
and other administrators as
well as school boards and
parents must have access to
clear information about the
project materials. The development
staff needs a well-developed
strategy for providing the
mechanisms through which such
information is made available
and kept updated.
The validity and importance
of these guiding principles
were not completely obvious
to us at the outset of our work.
However, we can see from experience
over the past decade that each
has been significant, particularly
those that relate to focusing
on key ideas and providing the
right context within which students
can engage mathematical concepts
and skills.
Key ideas. Our
development team made a commitment
to studying a small and select
set of important ideas deeply
rather than skimming a larger
set of ideas in a shallow manner.
This means that time is allocated
to developing understanding
of key ideas in contrast to “covering” the
book. The concept of “spiraling” is
philosophically appealing; but,
too often, not enough time is
spent initially with a new concept
to build on it at the next stage
of the spiral. Without this
deeper understanding of concepts
and how they are connected,
students come to view mathematics
as 1,001 different techniques
to be memorized. They cannot
apply or communicate about mathematical
ideas in any way that makes
sense.
This commitment to developing
key ideas in depth is illustrated
in the way that Connected
Mathematics treats proportional
reasoning, an important topic
for middle school mathematics.
Conventional treatments of this
central topic are often limited
to a brief expository presentation
of the ideas of ratio and proportion,
followed by training in techniques
for solving proportions. In
contrast, the CMP curriculum
materials develop core elements
of proportional reasoning in
at least four major units:
-
Stretching and Shrinking introduces
proportionality concepts in
the context of geometric problems
involving similarity. Students
connect visual ideas of enlarging
and reducing figures, numerical
ideas of scale factors and
ratios, and applications of
similarity through work on
problems focused around the
question: “What would
it mean to say two figures
are similar?”
-
Comparing and Scaling connects
fractions, percents, and ratios
through investigation of various
situations in which the central
question is: “What strategies
make sense in describing how
much greater is one quantity
than another?” Through
a series of problem-based
investigations, students explore
the meaning of ratio comparison
and develop, in a progression
from intuition to articulate
procedures, a variety of techniques
for dealing with such questions.
-
Moving Straight Ahead is
a unit on linear relationships
and equations. Proportional
thinking is connected and
extended to the core ideas
of linearity—constant
rate of change and slope.
-
Data Around Us extends
the theme of finding sensible
ways to compare numbers by
giving students tasks that
call for developing and using
number sense to analyze quantitative
data. Differences, rates,
and ratios again occur as
possible strategies. The meaning
and power of these mathematical
tools are compared and contrasted.
Providing a context. In
trying to help students make
sense of mathematics we found
that embedding the concepts
and skills within a context
or problem not only helped students
to make sense of the mathematics,
it also helped them to process
the mathematics in a retrievable
way. The following passage from Getting
To Know CMP (Lappan et.
al, 2001) provides a short summary
of why we chose to develop a
problem-centered curriculum.
Over the past three to four
decades, a growing body of
knowledge from the cognitive
sciences has supported the
notion that students develop
understanding as they construct
and evaluate solutions to problems.
This is quite different from
the previous assumption that
students learn by observing
a teacher as she demonstrates
how to solve a problem and
then practicing that method
on similar problems.
Students’ perceptions
about a discipline come from
the tasks or problems they
are asked to engage in. For
example, if students in a geometry
course are asked to memorize
definitions, they think geometry
is about memorizing definitions.
If students spend a majority
of their mathematics time practicing
paper-and-pencil computations,
they come to believe that mathematics
is about calculating answers
to arithmetic problems as quickly
as possible. They may become
faster at performing specific
types of computations, but
they may not be able to apply
these skills to other situations
or to recognize problems that
call for these skills.
On the other hand, if the
purpose of studying mathematics
is to be able to solve a variety
of problems, then students
should spend most of their
mathematics time solving problems.
If time is spent solving problems,
reflecting on solution methods,
examining why the methods work,
comparing methods, and relating
methods to those used in previous
situations, then students are
likely to build more robust
understandings and strategies.
(pp. xx)
In our curriculum work important
mathematical ideas are embedded
in the context of interesting
problems. As students explore
a series of connected problems,
they develop understanding
of the embedded ideas and with
the aid of the teacher, abstract
powerful mathematical ideas,
problem-solving strategies,
and ways of thinking. (pp.
xx)
To help students develop mathematical
understanding and skill through
work on substantial and interesting
problems, a key curriculum development
task was constructing problems
that were engaging for students
as well as carriers of important
mathematical ideas. We found
that problems could take a variety
of forms, from analysis of mathematical
games to practical tasks like
interpreting product advertisements
or news reports based on data,
organizing and modeling data
from science experiments, and
optimizing design of geometric
shapes. This variety is illustrated
in the following sample of problems
taken from several points in
the CMP curriculum.
In the Roller Derby game,
student teams place 12 markers
on a board that looks like
the following table:
1 |
|
2 |
X |
3 |
X |
4 |
X |
5 |
XX |
6 |
X |
7 |
XX |
8 |
X |
9 |
X |
10 |
X |
11 |
|
12 |
X |
Two dice are rolled in turns,
and when the sum of faces matches
a number on which students
have a marker, they can remove
one such marker. The goal is
to remove one’s markers
first. This game leads students
naturally to consideration
of empirical probability concepts
and also to the role that finite
sample spaces can play in modeling
probability experiments. (p.
xx)
The Comparing and Scaling unit
begins with a task that asks
students to think about comparisons
often made in advertising:
Advertisements often refer
to surveys showing that people
prefer one product over another.
For example, an ad for Bolda
Cola starts like this:
Which Soft Drink
Do You Like Better?
Bolda Cola or Cola
Nola
Take the Cola Taste
Test Yourself!
To complete the ad, Bolda
Cola wants to report the results
of a taste test. A copywriter
for the advertising department
has proposed four possible
concluding statements.
In a taste test, people who
preferred Bolda Cola out-numbered
those who preferred Cola
Nola by a ratio of
17,139 to 11,426.
In a taste test, 5,713 more
people preferred Bolda
Cola.
In a taste test, 60% of the
people preferred Bolda
Cola.
In a taste test, people who
preferred Bolda Cola outnumbered
those who preferred Cola
Nola by a ratio of
3 to 2.
Problem 1.1
A. Which of the proposed
statements do you think would
be most effective in advertising Bolda
Cola?
B. Is it possible that all
four advertising claims are
based on the same survey data?
C. What other concluding
statements could express the
same survey data?
D. Based on the survey results
used to write the Bolda
Cola ad, what data
would you expect in a new
survey of 1000 cola drinkers?
(pp. xx)
In the Mathematical Modeling unit,
students explore direct and
inverse variation by analyzing
data from classroom experiments
in which they build paper bridges
and compare carrying load (in
pennies) with length and thickness
of the bridges. They are asked
to make some predictions about
the relationships involved before
experimenting. Then they collect
data, display it graphically,
and describe the patterns revealed
in those displays. (p. xx)
Field trials of the materials
in which these investigations
were used revealed remarkable
instructional power in such
context-based teaching. Students
with many different prior experiences
and interests became engaged
in the tasks. When the same
underlying mathematical ideas
were encountered in later work,
students often connected the
new problems to the settings
in which those mathematical
structures were first studied.
However, we also learned that
without careful questioning
and reflection, it is quite
possible for students to leave
the classroom with nothing more
than a pleasant and lively experience.
Furthermore, we found that if
a particular context for mathematics
is carried on for too long,
students lose interest. As a
result, we have limited the
use of any single problem storyline
and introduced a variety of
features in student and teacher
materials that prompt reflection
on problem-solving experiences
and encourage students to articulate
the underlying mathematical
relationships and procedures
used to solve the problems.
Each investigation includes
Application/Connection/Extension
exercises that push students
to use their new ideas in different
but related contexts and to
connect new ideas to related
familiar ideas. Each investigation
also closes with a set of Mathematical
Reflection questions that ask
students to articulate (orally
and in journal writing) their
understanding about the key
mathematical ideas of the preceding
material. Finally, each unit
closes with a series of overall
reflection questions and additional
problems that test student understanding
of the key ideas.
When mathematical ideas are
embedded in problem-based investigations
of rich context, the teacher
has a critical responsibility
for ensuring that students get
the mathematical messages. In
a problem-centered classroom,
teachers take on new roles—moving
from always being the source
of knowledge to one of guiding
and facilitating the learner
in making sense of the mathematics.
Teachers become an even more
integral part of the learning
process. To teach a problem-centered
curriculum requires a deep knowledge
of mathematics; a broad and
coherent view of the subject
matter; and an understanding
of effective ways to conduct
a class based on inquiry. The
teacher support materials must
provide these kinds of help
for the teacher. Our goal from
the beginning of the project
was to develop a curriculum
from which both teachers and
students could learn.
Development
of Connected Mathematics
After the release of the National
Council of Teachers of Mathematics
(NCTM) Standards for School
Mathematics in 1989, the
National Science Foundation
put forth a call for proposals
to develop mathematics curricula
for grades K-5, 6-8, and 9-12.
It was clear from our past experiences,
that curriculum materials were
an essential part of improving
school mathematics. Even though
we had already written five
mathematics modules that were
widely in use, teachers wanted
a more comprehensive curriculum
that reflected the NCTM standards.
In 1991 CMP submitted a proposal
to NSF to write a complete mathematics
curriculum for teachers and
students in the middle grades,
building on and expanding our
earlier work. Our original research
and development team of Fitzgerald,
Lappan, and Phillips now included
James Fey, who brought high
school experience to the task,
and Susan Friel, who provided
elementary school experience.
Together, we would have the
expertise to build on what good
elementary curricula were doing,
while also preparing students
for what was to come in high
school. An advisory board consisting
of mathematicians, scientists,
mathematics educators, teachers,
parents, and business representatives
was created. They played a critical
role in deciding what content
to emphasize, in the format
of materials, and in planning
the research and evaluation
conducted throughout the development
process.
Five professional development
centers (PDC) were established,
each with a well-known mathematics
educator at its head. These
centers were established in
San Diego, CA; Portland, OR;
Pittsburgh, PA; Queens, NY;
and in a consortium representing
Michigan (Portland, Flint, Shepherd,
Traverse City, Bloomfield Hills,
Sturgis, and Waverly School
Districts). The PDCs served
as the primary sites for field-testing.
Additional sites in Tallahassee,
FL; Toledo, OH; Evanston, IL;
and Durham, NC were added during
the trial phases of the project.
Collectively, the field test
sites represented diversity
of geographic location, academic
ability, ethnicity, and socioeconomic
condition.
Collaborators
A research and evaluation
team was formed under the direction
of Diana Lambdin at the University
of Indiana, along with Judith
Zawojewski at Purdue University
and Sandra Wilcox at Michigan
State University. Their role
was also critical to the development
of materials. They trained observers
at each of the professional
development centers; and they
collected information on classroom
environment, teacher needs and
practices, student attitudes
and achievement, and management
and related school issues. In
addition, during the second
and third years of the field
trials, they conducted a large-scale
comparative study of CMP students
and students in classrooms using
more traditional texts. The
Iowa Test of Basic Skills and
a problem-solving test developed
by the Balanced Assessment Project
were used to assess student
performance. This information
was critical to the author team
as we prepared the final versions
of the modules for students
and the supporting teacher guides.
Classroom teachers also helped
to shape Connected Mathematics.
A number of collaborating teachers
worked with us on all phases
of the development and, most
important, served as our eyes
in the classroom. A full-time
middle school teacher headed
up the development of the assessment
package that was included in
each module’s teacher
guide. Graduate students in
mathematics and mathematics
education contributed fresh
ideas to the project and learned
a great deal from their experiences.
As the next generation of curriculum
development professionals, these
graduate students will have
a better understanding of what
it takes to develop effective
materials.
Setting
Goals
We began with some assumptions
about what students would know
when they entered grade six.
Then, working with our Advisory
Board, our next task was to
define the knowledge and skills
in each content strand that
students should have by the
time they leave grade eight.
To aid in the process and help
us think more broadly about
the curriculum, we wrote a set
of papers that outlined the
exit goals for grade eight in
the four mathematical strands
that we would develop—number,
algebra, probability and statistics,
and geometry and measurement.
These papers were our touchstone
for the development of each
strand. Key ideas or clusters
of related ideas were identified
in each strand and incorporated
into units. Initial sequencing
of units and appropriate grade
level placements were initially
identified. Subsequent revisions
were based on extensive field
tests.
The overarching goal of Connected
Mathematics is to help
students and teachers develop
mathematical knowledge, understanding,
and skill along with an awareness
and appreciation of the rich
connections among mathematical
strands and between mathematics
and other disciplines. As the
CMP materials were developed,
the authors synthesized multiple
mathematical goals into a single
standard that has been a guide
for all of our curriculum development:
All students should be able
to reason and communicate proficiently
in mathematics. They should
have knowledge of and skill
in the use of the vocabulary,
forms of representation, materials,
tools, techniques, and intellectual
methods of the discipline of
mathematics, including the
ability to define and solve
problems with reason, insight,
inventiveness, and technical
proficiency. (Getting to
Know CMP, p. xx)
Another goal for CMP was to
align teaching, learning, and
assessment with each other as
integral parts of the material.
To accomplish that broad program
goal, development of student
and teacher materials was guided
by the following five fundamental
mathematical and instructional
themes:
-
CMP is organized around
a selected number of important
mathematical content and
process goals, each of which
is studied in depth.
-
CMP emphasizes significant
connections among various
mathematical topics and between
mathematics and problems
in other disciplines that
are meaningful to students.
-
The instruction in CMP
emphasizes inquiry and discovery
of mathematical ideas through
the investigation of rich
problem situations.
-
CMP helps students grow
in their ability to reason
effectively with information
represented in graphic, numeric,
symbolic, and verbal forms
and to move flexibly among
these representations.
-
The goals and teaching
of CMP approaches reflect
the information-processing
capabilities of calculators
and computers and the fundamental
changes such tools are making
in the way people learn mathematics
and apply their knowledge
of problem-solving tasks.
(Getting To Know CMP,
p. xx)
As materials for students
and teachers began to take shape,
difficult problems and decisions
surfaced at every stage of the
process. The major development
issues fall into three categories—student
materials, teacher support,
and project management.
Design
of Student Materials
As we set out to
write a complete connected curriculum
for grades six, seven, and eight,
certain basic questions about
the mathematical content and
format of student materials
quickly surfaced:
-
What are the overarching
mathematical goals in each
content strand?
-
What size problem is feasible
for the teacher and students
to explore?
-
What kind of sequencing
or scaffolding is needed to
support investigations?
-
How much help is needed
to move from a contextual
setting to mathematical understanding
that transcends specific situational
factors?
-
What computational skills
should be developed and how?
-
What combination of whole
class, small group, and individual
work is most effective?
-
What kinds of practice or
homework and reflection are
needed to ensure solid understanding
of key ideas and automaticity
of critical skills?
-
What forms of assessment
will best inform teachers
and students of progress and
needed next steps in teaching
and learning?
-
How much help do teachers
need in understanding and
implementing the content and
pedagogy of the curriculum?
It took classroom trials and
observations, discussions, revisions,
more trials, research, and reflections
to resolve such issues into
a coherent problem-centered
curriculum. Although our initial
analysis of overarching goals
remained sound, our notions
of what could be taught in one
year changed. To foster students’ deep
understanding of the key ideas
would require more time.
Characteristics of
good problems. To
be effective, problems must
embody critical concepts and
skills and have the potential
to engage students in making
sense of mathematics. And,
since students build understanding
by reflecting and communicating,
the problems need to encourage
them to use these processes.
Each problem in Connected
Mathematics satisfies
some or all of the following
criteria:
-
The problem must have
important, useful mathematics
embedded in it.
-
Students must be able
to approach the problem in
multiple ways, using different
solution strategies.
-
The problem should allow
various solution strategies
or lead to alternative decisions
that can be taken and defended.
-
The problem should engage
students and encourage classroom
discourse.
-
Solution of the problem
should require higher-level
thinking and problem solving.
-
Investigation of the problem
should contribute to students’ conceptual
development.
-
The mathematical content
of the problem should connect
to other important mathematical
ideas.
-
Work on the problem should
promote skillful use of mathematics
and opportunities to practice
important skills.
-
The problem should create
opportunities for the teacher
to assess what students are
learning and where they are
experiencing difficulty.
(Lappan & Phillips, 1998,
pp. 87-88)
Undoubtedly the most common
challenge we faced was deciding
how big a problem can be. While
rich, open problems encourage
students to generate many interesting
patterns and conjectures and
to solve problems in a variety
of ways, it can be difficult
to decide which conjectures
and problem-solving strategies
to follow up on. We struggled
to select and sequence problems
in a way that (a) would allow
both students and teachers a
chance to make sense of the
mathematics, (b) wouldn’t
lead to activity that was irrelevant
to the mathematical goals (e.g.,
mathematics projects that become
art projects), and (c) would
be manageable for teachers,
both mathematically and pedagogically.
Our decisions about the grain-size
of a problem were generally
based on classroom experience.
We considered whether the time
required to develop an idea
fully was time well spent; whether
students could grasp the mathematical
subtlety of the ideas; and whether
students were reaching useful
closure on important concepts,
strategies, or skills.
For example, as we formulated
plans for a unit on elementary
data analysis and informal inference,
we experimented with a student
investigation comparing the
length of first and last names.
In its most open form, students
could be given the simple task
of collecting and organizing
data to decide whether there
seemed to be any significant
patterns in such data. We found
it useful to allow some initial
exploration of this open question
(to get a reading of students’ prior
knowledge and experiences).
However, this sort of open task
rarely leads students to utilize
useful standard techniques for
data display (line plots, box
plots, or scatter plots) or
standard concepts for data summary
(median, mean, and measures
of variation). Pilot teachers
almost always recommend that
such materials need to provide “more
structure” for students.
Another challenge was to find
engaging contextual problems
that would embody the mathematical
concepts we wanted to target.
Ideally, the context helps make
concepts meaningful to students
and enables them to retrieve
that meaning more readily later
on. We often found that when
students were eventually presented
with mathematical ideas symbolically—without
a contextualizing storyline--they
were able to go back to the
original context in which they
had learned those ideas to help
them make sense of the symbols.
It is important to note, however,
that students’ ability
to generalize knowledge in this
way does not occur without being
attended to specifically. (Lappan & Phillips
1998, p. 90).
Organization of the
units. Each
CMP unit is centered on an
important mathematical idea
or cluster of related ideas.
A unit consists of four to
seven investigations, each
containing two to five problems
for the class to explore. Each
investigation ends with a set
of problems called Applications,
Connections, and Extensions
(ACE). The ACE problems provide
homework practice with applications
of key ideas. We found a relatively
effective mix of skills and
applications through our extensive
pilots and field-testing (each
unit went through a three-year
development and testing cycle).
Finally, the Mathematical Reflections
at the end of each investigation
and the assessment package
provide opportunities for teachers
and students to assess their
understanding throughout the
unit (Lappan & Phillips,
1998, p. 90).
Providing
Support for Teachers
We developed the
CMP curriculum materials from
the perspective that students
develop deep understanding of
mathematical concepts, skills,
procedures, and processes through:
-
solving problems;
-
observing patterns and
relationships among variables
in a situation;
-
conjecturing, testing,
discussing, verbalizing, and
generalizing these patterns
and relationships;
-
discovering salient mathematical
features of patterns and relationships
and abstracting from these
underlying mathematical concepts,
processes, and relationships;
-
developing a language for
talking about problems and
for representing and communicating
their ideas; and
-
striving to make sense
of and to connect the mathematical
concepts they abstract from
their experiences.
These kinds of student learning
activities require a specific
set of teaching practices to
support them. (Lappan & Phillips,
1998, p. 85)
The teaching model. As
we developed the CMP student
materials and supporting teacher
materials, we took into account
the demands of problem-centered
teaching, formulating an instructional
model that provides a lesson-planning
template. This model looks at
instruction in three phases—launching,
exploring, and summarizing.
During the first phase, the
teacher launches the
investigation with the whole
class by setting the context
for the problem. This involves
making sure the students understand
the setting or situation in
which the problem is posed.
More important, the problem
must be launched in such a way
that the mathematical context
and challenge are clear. In
planning a lesson launch, teachers
need to consider questions like
these:
-
What mathematical questions
are or can be asked in this
situation?
-
What are students expected
to do?
-
How are they expected to
record and report their work?
-
Will they be working individually,
in pairs, or in groups?
-
What tools are available
that might be helpful?
This is also the time when,
if necessary, the teacher introduces
new ideas, clarifies definitions,
reviews old concepts, and connects
the problem to students’ prior
knowledge. Teachers need to
make sure they are launching
tasks in such a way that the
challenge of the task is left
intact even though students
are given a clear picture of
what is expected. It is easy
to tell too much, thus lowering
the challenge of the task to
something fairly routine.
After the task is launched,
students explore the
task as the teacher circulates,
asking focusing questions when
a student or group is struggling
and extending questions when
students have solved the problem
but have not taken it as far
as possible. The teacher takes
stock of who is understanding
and who needs help; who has
a strategy, a good idea, a generalization,
or an interesting way of explaining
the problem solution that needs
to be shared in the summary.
The final phase of the instructional
model is the summary.
This is the most important and,
perhaps, the hardest phase to
do well. Here the students and
the teacher work together to
make the mathematics of the
problem more explicit; to generalize
certain situations; to abstract
useful mathematical ideas, processes,
and concepts; to make connections;
and to foreshadow mathematics
that is yet to be studied.
As an example of how this launch/explore/summarize lesson
is implemented, consider one
of the tasks in a CMP algebra
unit:
When a sports team wins a
big game, they often celebrate
by saluting each other with
high fives all around. How
many high fives will there
be for winning teams with different
