
Publications
Abstracts
Algebra in Elementary School
Schliemann, A.D., et al. (2003). Algebra in
Elementary School. Proceedings of the 27th International Conference for the
Psychology of Mathematics Education. Honolulu, HI, July, 2003.
Increasing numbers of mathematics educators, policy makers,
and researchers believe that algebra should become part of the elementary
curriculum. Such endorsements require careful research. This paper presents the
general results of a longitudinal classroom investigation of children's thinking
and representations over two and a half years, as they participate in Early
Algebra activities. Results show that 3rd and 4th grade students are capable of
learning and understanding elementary algebraic ideas and representations as an
integral part of the early mathematics curriculum.
Guess my rule revisited
Schliemann, A.D., et al. (2003). Algebra in
Elementary School. Proceedings of the 27th International Conference for the
Psychology of Mathematics Education. Honolulu, HI, July, 2003.
We present classroom research on a variant of the
guessmyrule game, in which nineyearlold students make up linear functions
and challenge classmates to determine their secret rule. We focus on issues
students and their teacher confronted in inferring underlying rules and in
deciding whether the conjectured rule matched the rule of the creators. We
relate the findings to the tension between semantically and syntactically driven
algebraic reasoning.
Fourth graders solving equations
Schliemann, A.D., et al. (2003). Algebra in
Elementary School. Proceedings of the 27th International Conference for the
Psychology of Mathematics Education. Honolulu, HI, July, 2003.
We explore how fourth grader (9 to 10 year olds) students
can come to understand and use the syntactic rules of algebra on the basis of
their understanding about how quantities are interrelated. our classroom data
comes from a longitudiansl study with students who participated in weekly Early
Algebra activities from grades 2 through 4. We describe the results of our work
with the students during the second semester of their fourth grade academic
year, during which equations became the focus of our instruction.
Linear function graphs and multiplicative reasoning in
elementary school
Schliemann, A.D., et al. (2003). Algebra in
Elementary School. Proceedings of the 27th International Conference for the
Psychology of Mathematics Education. Honolulu, HI, July, 2003.
The introduction of function graphs in elementary school
may support the understanding of graphs and of multiplicative reasoning. Drawing
on a longitudinal study in which we developed six to eight 90minute Early
Algebra activities per term, with the same 2nd to 4th grade students, we
describe their discussions on function graphs and their use of multiplication
across two lessons that took place in third grade.
Bringing Out the Algebraic Character of Arithmetic
Schliemann, A.D., Carraher, D.W., & Brizuela, B. (2003,
in preparation). Bringing Out the Algebraic Character of Arithmetic: From
Children's Ideas to Classroom Practice. Studies in Mathematical Thinking and
Learning Series. Mahwah, NJ: Lawrence Erlbaum Associates.
This book describes and discusses the results of our first
studies of Early Algebra. Chapter 1, an introductory chapter, looks at the
findings of research studies on algebraic reasoning and descibes the main tenets
of our approch to early algebra. Chapters 2 to 6 are divided into two amin
parts. Part 1 focuses on individual children's interview studies. Chapter 2
reports on interviews conducted in Brazil and in the USA on young children's
understanding of principles underlying algebraic manipulation across different
contexts. Chapter 3 deals with children's development of notations to solve
algebraic problems.
Chapters in Part 2 covers a one year teaching experiment we
carried out in a third grade public school in Boston. Chapter 4 focuses on how
students reason about addition and subtraction as functions. Chapter 5 focuses
on reasoning about multiplication as a function. Chapter 6 looks at students
working with fractions from an algebraic perspective and examines children's use
of their own notations to solve problems. To the extent that students' reasoning
took place in instructional settings we designed, chapters 4 to 6 will also look
at instructional issues in treating arithmetical operations as functions. In
each of chapters 4 to 6 we will discuss some of the main challenges we
identified in children's development as they deal with number relations and
contextual constraints that are inherent to problem solving activities. A final
discussion chapter (Chapter 7) will summarize the findings of our studies and
will point to the theoretical and practical implications of our results.
As a complement to chapters 4 to 6 the reader will find a
CDROM with short videopapers containing (a) detailed descriptions of the tasks
and materials we used in the classroom study, (b) examples of children's
participation and of their ways to solve and represent the problems, (c) the
challenges children faced and how they progressed, (d) children's written work,
and (e) comments and questions hyperlinked to specific video clips.
Additive relations and function tables
Brizuela, B. M. & LaraRoth, S. (2002). Additive
relations and function tables. Journal of Mathematical Behavior, 20 (3),
309319.
We present work with a second grade classroom where we
carried out a teaching experiment that attempted to bring out the algebraic
character of arithmetic. In this paper, we specifically illustrate our work with
the second graders on additive relations, through the children's work with
function tables.
We explore the different ways in which the children
represented the information of a problem in the form of a selfdesigned function
table.
We argue that the choices children make about the kind of
information to represent or not, as well as the way in which they constructed
their tables, highlight some of the issues that children may find relevant in
their construction of function tables. This openended format pointed to how
they were understanding and appropriating tables into their thinking about
additive relations.
Empirical and logical truth in Early Algebra activities:
From guessing amounts to representing variables
Carraher, D. & Schliemann, A.D. (2002). Empirical and
Logical truth in Early Algebra activities: From guessing amounts to representing
variables. Symposium paper NCTM 2002 Research Presession. Las Vegas, Nevada,
April 1921.
We describe classroom activities and discussions in the
"Early Algebra, Early Arithmetic Project" that attempted to take into account
the widely accepted principle that teachers should carefully listen to what
students say and try to build on their initial understandings. The lesson was
also guided by the ideas that (a) arithmetical operations can be viewed as
functions; (b) generalizing lies at the heart of algebraic reasoning; and (d) we
should provide students with opportunities to use letters to stand for unknown
quantities and for variables. We illustrate how these considerations were
reflected in the design of algebraicarithmetic tasks related to the operations
of addition and subtraction.
The evolution of mathematical understanding: Everyday
versus idealized reasoning
Schliemann, A.D. & Carraher, D.W. (2002). The Evolution
of Mathematical Understanding: Everyday Versus Idealized Reasoning.
Developmental Review, 22(2), 242266.
Developmental psychology lacks a theory of mathematical
reasoning that accounts for how learners appropriate conventional symbol systems
into their thinking. In this essay we attempt to consider how students'
mathematical thinking evolves not only as a result of their actions and everyday
experiences but also from their increasing reliance on introduced mathematical
principles and representations. First we contrast how certain mathematical ideas
are represented diversely in school and out of school. Then we exemplify, from
our own research, how 8 to 10yearold children's personal representations come
to face with (what for them are novel and for us are conventional)
representations involving algebraic concepts. Finally we explore some
implications for theories of instruction and longterm development of
mathematical reasoning.
Functions and graphs in third grade
Schliemann, A.D., Goodrow, A. & LaraRoth, S. (2001).
Functions and Graphs in Third Grade. Symposium Paper. NCTM 2001 Research
Presession, Orlando, FL.
Teaching about multiplicative functions is traditionally
postponed until the middle or high school years. It seems, however, that
children are able to deal with functional relationships at an earlier age. In
this paper we analyze how second graders complete function tables and how
instructional activities involving ratios and graphs may encourage third graders
to focus on functional relationships.
The reification of additive differences in early algebra
Carraher, D., Brizuela, B. M., & Earnest, D. (2001). The
reification of additive differences in early algebra. In H. Chick, K. Stacey, J.
Vincent, & J. Vincent (Eds.), The future of the teaching and learning of
algebra: Proceedings of the 12th ICMI Study Conference (vol. 1). The University
of Melbourne, Australia.
We look at the emergence of 9year old student's concept of
additive difference. The concept entails a tension between process and object.
But even more strikingly, reifying the concept requires that children adopt
analogies across diverse representational contexts. We will look at examples of
students' reasoning about children's heights in contexts associated with number
lines, counting, line segment diagrams, and arithmeticalgebraic notation. The
examples show that subtraction comprises a small yet essential part of the
concept of difference. We consider implications for research and curriculum
development in early algebra and early arithmetic education.
When tables become function tables
Schliemann, A.D., Carraher, D.W. & Brizuela, B.M.
(2001). When tables become function tables. Proceedings of the XXV Conference of
the International Group for the Psychology of Mathematics Education, Utrecht,
The Netherlands, Vol. 4, 145152.
This study explores thirdgrade students' strategies for
dealing with function tables and linear functions as they participate in
activities aimed at bringing out the algebraic character of arithmetic.
We found that the students typically did not focus upon the
invariant relationship across columns when completing tables. We introduced
several changes in the table structure to encourage them to focus on the
functional relationship implicit in the tables.
With a guessmyrule game and functionmapping notation we
brought functions explicitly into discussion. Under such conditions
nineyearold students meaningfully used algebraic notation to describe
functions.
Can young students operate on unknowns?
Carraher, D., Schliemann, A., & Brizuela, B. (2001). Can
Young Students Operate on Unknowns? Proceedings of the XXV Conference of the
International Group for the Psychology of Mathematics Education, Utrecht, The
Netherlands (invited research forum paper), Vol. 1, 130140.
Algebra instruction has traditionally been delayed until
adolescence because of mistaken assumptions about the nature of arithmetic and
about young students' capabilities. Arithmetic is algebraic to the extent that
it provides opportunities for making and expressing generalizations. We provide
examples of nineyearold children using algebraic notation to represent a
problem of additive relations. They not only operate on unknowns; they can
understand the unknown to stand for all of the possible values that an entity
can take on. When they do so, they are reasoning about variables.
Mathematical notation to support and further reasoning
('to help me think of something')
Brizuela, B., Carraher, D., & Schliemann, A. (2000).
Mathematical notation to support and further reasoning ('to help me think of
something'). Symposium paper, 2000 NCTM Research Presession Meeting (18 pp.).
Many researchers and educators now believe that elementary
algebraic ideas and notation should be an integral part of young students'
understanding of early mathematics. To support this change in thinking and
practice, the field needs research on young learners' algebraic reasoning. In
this study we take children's "algebraic reasoning" to refer to cases in which
they express general properties of numbers or quantities. We believe that they
can also express these properties and relations through written representation
or notation, without having to treat conventional notation as a mere appendage
to reasoning. The specific examples we focus upon refer to Sara, a student in
the third grade classroom we taught in once every two weeks. Sara exemplifies,
through her actions and her words, how notations can represent not only what was
done while solving a problem and what happened in the context of the problem,
but also how notations can become tools for thinking and reflecting about the
relationships between quantities in the problem. In this way, we can begin to
think about children's notations not only as tools for learners to represent
their understanding and thinking about algebraic relations or as precursors of
conventional algebra representation, but also as tools to further those
understandings and that thinking.
Bringing out the algebraic character of arithmetic
Carraher, D., Brizuela, B. & Schliemann, A., (2000).
Bringing out the algebraic character of arithmetic. Instantiating variables in
addition and subtraction. In T. Nakahara & M. Koyama (Eds.) Proceedings of the
XXIV Conference of the International group for the Psychology of Mathematics
Education. Hiroshima, Japan, Vol. 2, 145152.
We report findings from a oneyear teaching experiment
designed to document and help nurture the early algebraic development of third
grade students. We focus on an arithmetic problem that fixes some measures but
allows more than one solution set. We highlight how children dealt with the fact
that the quantitative relations referred to particular measures on one hand (and
in that sense were arithmetical), and were meant to express general properties
not bound to particular values, on the other (and in this sense were algebraic).
We look at the role of instantiated variables in this tension and transition
between the particular and the general.
Children's early algebraic concepts
Carraher, D., Schliemann, A.D., & Brizuela, B. (2000).
Children's Early Algebraic Concepts. Plenary address. XXII Meeting of the
Psychology of Mathematics Education, North American Chapter, Tucson, AZ,
October, 2000.
We believe there are good reasons for treating arithmetical
operations as functions in early mathematics instruction. In this paper we
present partial results of an exploratory, year long, third grade classroom
intervention study. The first section refers to addition as functions and the
second to multiplication as functions. The children's participation in the
activities involving additive functions convinced us that third graders can
begin to think about addition and subtraction as additive functions and to
understand and use algebraic notation, such as n > n +3. In the case of
multiplicative functions, we found that although the children could correctly
fill in function tables, they seemed to do so with a minimal of thought about
the invariant relationship between the values in the first and second columns.
Several didactical maneuvers were introduced in an attempt to break the students
habit of building up in the tables. Within the context of a guess my rule game
students were finally able to break away from the building up strategies they
had been using. The children in the study had to deal with the fact that the
quantitative relations referred to particular numbers and measures on one hand
(and in that sense were arithmetical), and were meant to express general
properties not bound to particular values, on the other (and in this sense were
algebraic). Surprisingly, they did not need concrete materials to support their
reasoning about numerical relations and could even deal with notations of an
algebraic nature. In fact, the introduction of algebraic notation helped them to
move from specific computation results to generalizations about how two series
of numbers are interrelated.
Solving algebra problems before algebra instruction
Schliemann, A.D., Carraher, D.W., Brizuela, B., & Pendexter, W. (1998).
Second Early Algebra Meeting. University of
Massachusetts at Dartmouth/Tufts University.
If equivalent operations are
performed on the left and right terms of an equation, new equation results. This
principle allows one to produce equations with a variable isolated on one side
and its value(s) on the other. It also underlies problemsolving in situations
where equations are not explicitly used, but the problem calls for recognizing
that two quantities are equal in value and for using that information to derive
conclusions about values of unknown quantities.
The present paper focuses on how thirdgrade
children recognize and use this logical principle in solving problems. It also
looks at issues children face as the try to represent unknowns through written
notation and use their written symbols to draw inferences about unknown values.
The result showed that children comfortably recognized that equal additive
operations upon equal quantities produce equal results (Study 1). Further, they
easily produced written representations of known (numerically quantified or
measured) quantities. However, the children showed considerable hesitation about
producing written representations for unknown quantities (Study 2). Their
Hesitation seems to stem from the challenge of finding a symbol to represent a
quantity without constraining or making incorrect presumptions about values it
may stand for.
