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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