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Equity Within Mathematics Education Research as a Political Act: Moving From Choice to Intentional Collective Professional Responsibility

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In 2005, the NCTM Research Committee devoted its commentary to exploring how mathematics education research might contribute to a better understanding of equity in school mathematics education (Gutstein et al., 2005). In that commentary, the concept of equity included both conditions and outcomes of learning. Although multiple definitions of equity exist, the authors of that commentary expressed it this way: “The main issue for us is how mathematics education research can contribute to understanding the causes and effects of inequity, as well as the strategies that effectively reduce undesirable inequities of experience and achievement in mathematics education” (p. 94). That research commentary brought to the foreground important questions one might ask about equity in school mathematics and some of the complexities associated with doing that work. It also addressed how mathematics education researchers (MERs) could bring a “critical equity lens” (p. 95, hereafter referred to as an “equity lens”) to the research they do. Fast forward 10 years to now: Where is the mathematics education researcher (MER) community in terms of including an equity lens in mathematics education research? Gutiérrez (2010/2013) argued that a sociopolitical turn in mathematics education enables us to ask and answer harder, more complex questions that include issues of identity, agency, power, and sociocultural and political contexts of mathematics, learning, and teaching. A sociopolitical approach allows us to see the historical legacy of mathematics as a tool of oppression as well as a product of our humanity.
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178 Teaching Children Mathematics / October 2007
Four-year-old Carina is intrigued by the mate-
rials set up in the manipulatives area of the
classroom. The table has child-sized cups and
saucers, notebooks and pencils, and watercolor
paints and brushes arranged in corresponding
sets (e.g., a cup and a saucer make up one set),
an arrangement that encourages one-to-one cor-
respondence. Carina goes over to the cups and
saucers and, after briefly looking into the bowl that
contains the cups, takes one out and puts it on top
of a saucer. She continues to take out one cup at
time and matches each with a saucer. After she has
matched one cup from the bowl to every saucer on
the table, Carina notices that there are still some
cups in the bowl. She takes one cup out and looks
around the table, apparently searching for another
saucer to match the cup. After briefly surveying the
table, she puts the cup back in the bowl and moves
on to the next set of objects.
Carmen Brown was Carina’s teacher in this
Head Start program. The following are her reflec-
tions on her experience.
Thinking about
Children’s Thinking
In the past, I might have noticed that Carina was
successful in placing a set of objects in one-to-
one correspondence with another set of objects. I
would have been satisfied if she appeared to have
mastered one-to-one correspondence or completed
the activity without any adult assistance. However,
I would not have asked myself what other activi-
ties could reinforce or enhance her thinking at this
developmental level. I would not have asked any
of the following questions: What other mathemati-
cal skills could be developed or enhanced with this
activity or a modification of it? Are there other
types of one-to-one correspondence that Carina
may still need to develop? How might I assist her
Carmen S. Brown, Mrs.Carmen@att.net, is a doctoral student at the University at Buffalo, State
University of New York, Buffalo, NY 14260-1660. She is interested in mathematics professional
development in preschool settings. Julie Sarama, jsarama@buffalo.edu, is an associate pro-
fessor of mathematics education at the University of Buffalo. Her research interests include
the implementation and effects of software environments in mathematics classrooms and the
development of research-based learning trajectories for young children. Douglas H. Clements,
clements@buffalo.edu, previously a preschool and kindergarten teacher, is now a professor
of early childhood, mathematics, and computer education at the University of Buffalo. He
conducts research in computer applications in education, early development of mathemati-
cal ideas, and the learning and teaching of geometry. Together, Sarama and Clements have
created curriculum materials for young children and are directing several large-scale research
projects in early mathematics.
By Carmen S. Brown, Julie Sarama, and Douglas H. Clements
Thinking about
Learning
Trajectories
in Preschool
Copyright © 2007 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.
This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
Teaching Children Mathematics / October 2007 179
to progress to the next developmental level?
Now, however, I consistently ask myself such
questions after observing children in my classes.
Knowing about learning trajectories has helped me
develop goals and objectives for my students’ math-
ematical knowledge, identify their developmental
levels in specific mathematical topics, and chart
paths for instruction that will guide their learning of
mathematics. How did I come to learn about these
learning trajectories?
Joining a Research Team
My interest in research on children’s mathematical
thinking began with my supervisor suggesting that
I participate in studies conducted by Julie Sarama
and Doug Clements, researchers at the University
at Buffalo, SUNY (and co-authors of this article).
These studies evaluated the impact of innovative
preschool mathematics curricula and interven-
tions on the mathematics knowledge of at-risk
preschoolers (Clements and Sarama 2007b; see
also the TRIAD Web site for the positive results of
these studies). These studies intrigued me because
my preschool classroom consisted primarily of
children who came from low-income families. As
a participant in Sarama and Clements’s research,
I was happy to be randomly assigned to the group
using the Building Blocks curriculum, the first cur-
riculum to be based fully on learning trajectories, a
concept proposed by Simon (1995).
Thinking about Thinking and
about Teaching: Learning
Trajectories
Learning trajectories have three parts: a math-
ematical goal, a developmental path through which
children progress to reach that goal, and a set of
activities matched to each of those levels (Clements
and Sarama 2004). Because my prekindergarten
classroom was multiage and children come to
school at different developmental levels, I had to
individualize instruction. The Building Blocks cur-
riculum helped me in three ways: It helped me clar-
ify the mathematics goal. It allowed me to assess
the children’s developmental level by observing
and recording their mathematical thinking and their
progress in achieving that goal. Finally, it helped
me determine what activities, and what modifica-
tion of activities, would be effective in helping my
students’ progress.
Understanding learning trajectories helped me
teach children such as Carina more effectively.
First, I understand the mathematics goals at a
deeper level. In matching cups with saucers, Carina
was working on an object correspondence task. I
knew this understanding was important, but I also
knew being able to complete this task did not mean
that, for example, she understood one-to-one corre-
spondence when counting objects. Instead, this task
might help her learn comparing numbers, a distinct
learning trajectory.
Second, I could determine Carina’s develop-
mental level in that learning trajectory. Because she
Photograph by Carmen S. Brown; all rights reserved
180 Teaching Children Mathematics / October 2007
was able to match the cups to
the saucers and made certain
that each saucer received only
one cup, she was capable
of thinking at the nonverbal
comparer of dissimilar objects
level. (Note: These levels of
learning trajectories (itali-
cized) are set forth in Clements
and Sarama [2007a].) To deter-
mine whether Carina is at the
subsequent matching comparer
level, I would ask her to compare
the number of saucers and cups and
would note whether she confidently
stated that both groups had the same
number. If so, I would give her another
set of saucers to match and ask her to first count
the number of saucers and then tell me how many
cups she would need to match the saucers. I would
encourage her to begin by counting the number of
each type of object (five cups and then five saucers)
instead of using one-to-one correspondence (pair-
ing one cup with one saucer). This process would
help her develop the skills that will eventually bring
her to the counting comparer level. Understanding
the learning trajectories helped me provide devel-
opmentally appropriate activities that would guide
Carina, and other children, to increasingly sophisti-
cated levels of thinking.
Steps to Mathematical
Understanding
Perhaps the most important learning trajectory is
in counting. Picture Joseph exploring the Unifix
cubes in the table toys area of the classroom. From
the container holding the cubes, he takes out one
cube and places it on the table. He then takes out
three cubes. As he holds the three cubes in his
hand, Joseph connects them and places the three-
cube tower directly next to the single cube. From
the container, he takes out a handful of cubes,
without appearing to count them. He places the
cubes on the table and, one by one, connects them.
After connecting all of them (he has taken seven
cubes from the container), he places this tower
next to the three-cube tower. Joseph promptly
goes to another container, and takes out a teddy
bear counter. He places the teddy bear on the
single cube, then moves it to the three-cube tower,
and then moves it to the seven-cube tower. Excit-
edly, he exclaims, “I made it to the top!”
Joseph is building stairs
for the teddy bear counter to
climb. “Build Stairs” is a
Building Blocks activity
that reinforces counting by
producing sets of objects
and sequencing (e.g.,
reinforcing the notion of
“plus one” in the count-
ing sequence). At circle
time, in preparation for a
small-group activity, I had
introduced the concept of
building stairs. During this
introduction, I took one cube
from the container and placed
it on the rug to make the first stair. I then asked the
children to use the counting sequence as a clue and
tell me how many cubes should be in the next stair.
My objective was for the children to eventually
create cube buildings having from one to at least
five cubes and then order these, so I intentionally
emphasized the “plus one” idea. The children were
successful in determining that the next stair should
have two cubes in it. I then took two cubes out of
the container, counted them out loud, connected
them, and placed them next to the single cube. I
continued in this manner until the children and I
had built a five-cube stairs.
On the table toys shelf, I had placed an addi-
tional container of Unifix cubes for the children
to explore independently. Working at this center,
Joseph appeared to model the strategies that I had
demonstrated during the whole-group activity. He
was not successful in creating a “plus one” pattern
but seemed to understand that the stairs needed to
increase in length. The learning trajectories helped
me determine specific objectives for Joseph. He
needed more experience producing sets up to
five, a level describing counting a set of the same
size. Where was his mathematical thinking on
the development path? I have seen Joseph count
and produce sets of small numbers (up to two
and sometimes three). He seemed to understand
the idea of one and then another one, but did not
appear to completely understand the concept of
“plus one.” For example, he did not appear to rec-
ognize that the stairs that I had built had a “plus
one” pattern.
What instructional activities could I provide
to allow Joseph more practice and experience in
acquiring the skills and competence necessary to
move to the next developmental level? The Build-
Teaching Children Mathematics / October 2007 181
ing Blocks curriculum presented me with several
activities and strategies. We had been playing
games with dot cards, or counting cards, that illus-
trated the numbers 1, 2, 3, 4, and so forth, so we
arranged the cards in sequence. Joseph used the
cards to figure out that after making a one-cube
step he needed to make a two-cube step. He briefly
referred to the counting cards and recited, “One,
two, three.” Before I could ask him how many
cubes he should take out of the container, he asked
if he could take out three cubes for the next tower.
Joseph counted out three cubes, connected them,
and placed them next to the two-cube tower. Using
the cards for reinforcement, Joseph was successful
at making a four-cube tower and a five-cube tower.
His confidence was evident when he used a count-
ing bear to climb the steps. As he made the bear
jump from step to step, I encouraged him to count.
He was successful at counting and noting the cor-
rect number of cubes in each tower. This activity
also allowed Joseph to begin numeral recognition,
sequencing, and use of the “plus one” concept.
Learning Trajectories
and My Teaching
By incorporating the three parts of the learning
trajectories—setting a mathematical goal, creating
a sequential developmental path, and developing
an activity to match the level I wanted my students
to attain—I was able to assist them in acquiring
the ability to distinguish between patterns and
nonpatterns.
When I first engaged the children in pattern-
ing activities that involved concrete objects, they
were unable to differentiate between a pattern and
a nonpattern. To assist them in identifying pat-
terns, I set an initial mathematical goal: to help the
children transition from the pre-pattern level to
the pattern recognizer level. My next step was to
determine the teaching strategies and correspond-
ing activities I would use to assist the children. I
collected and created various examples of patterns
and nonpatterns, put these examples on tagboard,
and displayed them in the area where we held our
large-group meetings. I began each lesson with a
discussion of what a pattern is and gave the chil-
dren a developmentally appropriate definition: “A
pattern is a unit that repeats.” After a brief discus-
sion on what a unit is, the children comprehended
that any unit that repeats over and over is a pat-
tern. Each day we looked at the examples I had
collected, discussed whether each had a repeating
unit, and determined whether the repeating unit
was a pattern or not. At the end of the lesson, most
of the children had reached the pattern recognizer
level.
The Building Blocks learning trajectories per-
mitted me to be flexible, adaptable, and responsive
to the children’s changing needs. I developed new
knowledge about their mathematical thinking and
the skills that correspond to their mathematical
development. The instructional activities were
easily adaptable to children who were struggling
to acquire new concepts as well as children who
needed extra challenges to expand on their curios-
ity. The main objective of the Building Blocks cur-
riculum is to benefit students, not simply convert
teachers to a new way of teaching mathematics
to preschool children. The learning trajectories
allowed me to realize
that my previous teach-
ing strategies, which I
considered appropriate
and satisfactory, were
narrow in scope and
did not grow as the
children’s understand-
ing did. My perception
of my role as a teacher,
assessor, implementer,
and modifier has
changed significantly.
Participating in the
research and learning
about research-based learning trajectories have
encouraged me to develop new instructional tech-
niques, refine my current practice, and broaden
my thinking as a preschool mathematics teacher.
References
Clements, D. H., and J. Sarama. “Learning Trajectories in
Mathematics Education.” Mathematical Thinking and
Learning 6 (2004): 81–89.
——. Building Blocks: SRA Real Math Teacher’s Edi-
tion, Grade PreK. Columbus, OH: SRA/McGraw-
Hill, 2007a.
——. “Effects of a Preschool Mathematics Curriculum:
Summative Research on the Building Blocks Project.
Journal for Research in Mathematics Education 38
(2007b): 136–63.
Simon, M. A. “Reconstructing Mathematics Pedagogy
from a Constructivist Perspective.” Journal for Re-
search in Mathematics Education 26, no. 2 (1995):
114–45.
TRIAD. http://www.gse.buffalo.edu/org/triad/tbb/index
.asp?local=parent. s
Understanding the
learning trajectories
helped me provide
developmentally
appropriate activities
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Building Blocks: SRA Real Math Teacher's Edition, Grade PreK
  • D H Clements
  • J Sarama
Clements, D. H., and J. Sarama. "Learning Trajectories in Mathematics Education." Mathematical Thinking and Learning 6 (2004): 81-89. --. Building Blocks: SRA Real Math Teacher's Edition, Grade PreK. Columbus, OH: SRA/McGraw-Hill, 2007a. --. "Effects of a Preschool Mathematics Curriculum: Summative Research on the Building Blocks Project." Journal for Research in Mathematics Education 38 (2007b): 136-63.