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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 guess-my-rule game, in which nine-yearl-old 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 90-minute 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 CD-ROM 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. & Lara-Roth, S. (2002). Additive relations and function tables. Journal of Mathematical Behavior, 20 (3), 309-319.

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 self-designed 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 open-ended 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 19-21.

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 algebraic-arithmetic 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), 242-266.

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 10-year-old 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 long-term development of mathematical reasoning.

Functions and graphs in third grade
Schliemann, A.D., Goodrow, A. & Lara-Roth, 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 9-year 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 arithmetic-algebraic 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, 145-152.

This study explores third-grade 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 guess-my-rule game and function-mapping notation we brought functions explicitly into discussion. Under such conditions nine-year-old 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, 130-140.

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 nine-year-old 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 Pre-session 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, 145-152.

We report findings from a one-year 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 problem-solving 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 third-grade 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.

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