# Mathematics Learning

# MATHEMATICS LEARNING

algebra

mitchell j. nathan

complex problem solving

alan h. schoenfeld

geometry

vera kemeny

learning tools

susanne p. lajoie

nancy c. lavigne

myths, mysteries, and realities

michael t. battista

number sense

chris lowber

teruni lamberg

numeracy and culture

yukari okamoto

mary e. brenner

reagan curtis

word-problem solving

lieven verschaffel

brian greer

erik de corte

## ALGEBRA

Algebraic reasoning is a major development (circa 800), both culturally and individually. Culturally, the invention of algebraic representations in graphical and symbolic form is viewed as central for the advancement of mathematics and science. Algebra provides a succinct notation for recording mathematical relationships and describing computational algorithms and scientific laws. Algebraic reasoning is viewed as a major conceptual advancement beyond arithmetic thinking. For individuals, algebra can serve as a *thinking tool,* or, more aptly, a toolkit, for describing relationships in terms of unknown quantities and modeling complex and dynamic situations. For many, algebra instruction is the sole source for the formal study of abstract representations and problem solving.

Algebra has also been identified as a societal gatekeeper for further development of mathematical and scientific instruction, and for wide-ranging economic opportunities. Consequently, at the beginning of the twenty-first century, there is a major shift in the United States to move algebra education into the middle and primary grades, and to reconceptualize instruction appropriately for these age groups. The utility of algebra is boundless. However, learning and teaching algebra, particularly algebra word problems, is often viewed as the bane of mathematics education.

### What Is Algebra?

Abstract algebra refers to the use of formal mathematical structures and symbols, such as F (*X* ), to represent relations between terms or objects. It includes the operations that operate on those structures, such as the inverse, F-1 (*X* ), and the identity, I (*X* ). An especially important structure is the *function,* which specifies a one-to-many relationship (or mapping) between an independent variable (the *input* ) and a set of dependent variables (the *output* ).

School-based algebra is most commonly viewed as the generalization of arithmetic to include the use of literal symbols (such as the letters *X* and *Y* ) and arbitrarily complex symbolic expressions. School algebra also includes the study of functions as well as the construction of abstract formalisms that inductively describe a pattern of instances, predict future instances, and characterize the general form of the pattern (e.g., linear).

### Challenges of Learning Algebra

To encompass the multifaceted nature of school-based algebra, new concepts arise that contribute to its learning difficulties. The *variable* is expanded from being a place-holder (or box) in arithmetic, to representing an unknown value or set of values that stand in relation to (and may covary with) other values and expressions. Symbols that represent variables in algebra must denote the same thing everywhere in a problem, but generally take on new meanings with each new problem.

Detailed analyses of problem solving show that result-unknown problems, such as 25x4+8=?, are solvable by direct application of the arithmetic operators, or by using counting objects to physically model the number sentence; whereas start-unknown problems, such as 25Y+8=108, defy modeling and are considered to be algebraic. Both children and adults exhibit lower levels of performance with start-unknown than with result-unknown problems.

In algebra, the equal sign (=) takes on a relational or structural role, as when two sides of an equation are compared, such as 25Y=108. This is in addition to its operational role in arithmetic where the equal sign signals one to perform a computation, as with 25x4=?. Facility with and among multiple representations, including symbolic, tabular, graphical, and verbal formats, is an important aspect of algebraic reasoning, as is an understanding of the relative utility of each. Each representational format has unique advantages. However, in practice, equations receive far more attention that other representations, such as graphs.

The prototypical activities for school-based algebra are solving symbolic equations and word problems. Equation solving most typically involves applying legal rules of symbolic manipulation to isolate an unknown value (see Figure 1, part a). Legal rules typically entail performing the same symbolic calculations to both sides (such as subtracting 10 from both sides of Equation 1 in step 1) to maintain the relations specified in the original problem.

Novices often perform actions that violate the syntactic, hierarchically nested relationships contained in equations, and so inappropriately change their original meanings. For example, in Figure 1 it is algebraically illegal to combine 10 and 5 in step 1 because the original relation is no longer preserved. However, preconceptions from reading and arithmetic, where processing is done from left to right, can lead students to misstep.

Word-problem solving is the next most common activity in algebra education. Word problems can be in the form of a story (Figure 1, part b), or a word equation (Figure 1, part c). Although based on the same quantitative relations as Equation 1, students perform very differently on these three tasks. A misconception generally held by high school mathematics teachers is that high school students solve Equation 1 more easily than a matched story problem or word equation. Teachers justify this prediction by noting that a student must first write a symbolic equation that models the verbal statement, and that this invites other types of errors. While translation from words to mathematical expressions is error-ridden for novices, high school students typically circumvent this step when permitted. Instead, they use highly reliable informal methods, such as guess-and-test and working backwards, which produce higher levels of performance than equation solving.

**FIGURE 1**

### Curricular and Technological Advances in Algebra Education

Like many algebra teachers, traditional algebra textbooks take a largely *symbol precedence* view of the development of algebraic reasoning, introducing algebraic concepts through symbolic problem solving, and later applying them to verbal reasoning activities. In contrast, several alternative curricula have recently emerged that begin by eliciting students' invented strategies and representations for describing patterns and data, and developing from these inventions algebraic equations and graphs through a process called *progressive formalization.* These reform-based curricula typically draw on problem-based learning (PBL), which emphasizes complex, multi-day, collaborative problem solving. Three of these approaches, Mathematics in Context, Connected Mathematics, and The Adventures of Jasper Woodbury, have produced commercially available curricula that cover the major topics in middle grade mathematics, such as geometry and algebra.

Technology has also been effectively wedded to innovative curriculum designs. Graphing calculators have had a profound effect on the teaching of algebra using graphical, tabular, and programming forms. The Algebra Sketchbook supports the relationship between verbal descriptions and graphics. The Animate system helps students to construct situation-based meaning for equations. Jasper uses multimedia to present rich problem contexts and encourage production of Smart Tools, representations that support modeling, analysis, and comparison. The Pump Algebra Tutor provides individualized computer-based instruction by relying on adaptive cognitive models of individual students.

** See also: ** Mathematics Education, Teacher Preparation; Mathematics Learning,

**Complex Problem Solving, Geometry, Learning Tools, Word-Problem Solving.**

*subentries on*## bibliography

The Cognition and Technology Group at Vanderbilt. 1997. *The Jasper Project: Lessons in Curriculum, Instruction, Assessment, and Professional Development.* Mahwah, NJ: Erlbaum.

English, Lyn, ed. 2002. *Handbook of International Research in Mathematics Education: Directions for the 21st Century.* Mahwah, NJ: Erlbaum.

Kaput, James J., 1999. "Teaching and Learning a New Algebra." In *Mathematics Classrooms that Promote Understanding,* ed. Elizabeth Fennema and Thomas A. Romberg. Mahwah, NJ: Erlbaum.

Kieran, Carolyn. 1992. "The Learning and Teaching of School Algebra." In *Handbook of Research on Mathematics Teaching and Learning,* ed. Douglas A. Grouws. New York: Macmillan.

Koedinger, Kenneth R.; Anderson, John R.; Hadley, William H.; and Mark, Mary A.1997. "Intelligent Tutoring Goes to School in the Big City." *International Journal of Artificial Intelligence in Education* 8:30–43.

Ladson-Billings, Gloria. 1997. "It Doesn't Add Up: African-American Students' Mathematics Achievement." *Journal for Research in Mathematics Education* 28 (6):697–708.

Lappan Glenda; Fey, James T.; Fitzgerald, William M.; Friel, Susan N.; and Phillips, Elizabeth D. 1998. *Connected Mathematics.* Palo Alto, CA: Dale Seymour.

Lehrer, Richard, and Chazan, Daniel, eds. 1998. *Designing Learning Environments for Developing Understanding of Geometry and Space.* Mahwah, NJ: Erlbaum.

Mayer, Richard E. 1982. "Different Problem-Solving Strategies for Algebra Word and Equation Problems." *Journal of Experimental Psychology: Learning, Memory, and Cognition* 8:448–462.

Nathan, Mitchell J.; Kintsch, Walter; and Young, Emilie. 1992. "A Theory of Algebra Word Problem Comprehension and Its Implications for the Design of Computer Learning Environments." *Cognition and Instruction* 9 (4):329–389.

Nathan, Mitchell J.; Long, Scott D.; and Alibali, Martha W. 2002. "The Symbol Precedence View of Mathematical Development: An Analysis of the Rhetorical Structure of Algebra Textbooks." *Discourse Processes* 33 (1):1–21.

National Center for Research in Mathematical Sciences Education, and Freudenthalinstitute, eds. 1997. *Mathematics in Context: A Connected Curriculum for Grades 5–8.* Chicago: Encyclopaedia Britannica Educational Corporation.

Owens, S.; Biswas, G.; Nathan, Mitchell J.; Zech, L.; Bransford, J. D.; and Goldman,S. R. 1995. "Smart Tools: A Multi-Representational Approach to Teaching Function Relations." In *Proceedings of the Seventh World Conference on Artificial Intelligence in Education, AI-ED'95* (Washington, D.C.). Charlottesville, VA: Association for the Advancement of Computing in Education.

Usiskin, Zalmon. 1997. "Doing Algebra in Grades K–4." *Teaching Children Mathematics* 3:346–349.

Mitchell J. Nathan

## COMPLEX PROBLEM SOLVING

In April 2000, the National Council of Teachers of Mathematics (NCTM) published *Principles and Standards for School Mathematics,* a document intended to serve as "a resource and a guide for all who make decisions that affect the mathematics education of students in prekindergarten through grade 12," and that represented the best understandings regarding mathematical thinking, learning, and problem solving of the mathematics education community at the dawn of the twenty-first century. It also reflected a radically different view from the perspective that dominated through much of the twentieth century.

*Principles and Standards* specifies five mathematical content domains as core aspects of the curriculum: number and operations, algebra, geometry, measurement, and data analysis and probability. These content areas reflect an evolution of the curriculum over the course of the twentieth century. The first four were present, to various degrees, in 1900. Almost all children studied number and measurement, which comprised the bulk of the elementary curriculum in 1900. Algebra and geometry were mainstays of the secondary curriculum, which was studied only by the elite; approximately 10 percent of the nation's fourteen-year-olds attended high school. Data analysis and probability were nowhere to be seen. Over the course of the twentieth century, the democratization of American education resulted in increasing numbers of students attending, and graduating from, high school.

Curriculum content evolved slowly, with once-advanced topics such as algebra and geometry becoming required of increasing numbers of students. The study of statistics and probability entered the curriculum in the 1980s, and by 2000 it was a central component of most mathematics curricula. This reflected an emphasis on the study of school mathematics for "real world" applications, as well as in preparation for mathematics at the collegiate level.

While content changes can thus be seen as evolutionary, perspectives on mathematical *processes* must be seen as representing a much more fundamental shift in perspective and curricular goals. Given equal weight with the five content areas in *Principles and Standards* are five process standards: problem solving, reasoning and proof, communication, connections, and representation. All of these are deeply intertwined, representing an integrated view of complex mathematical thinking and problem solving. Problem solving might be viewed as a "first among equals," in the sense that the ultimate goal of mathematics instruction can be seen as enabling students to confront and solve problems–not only problems that they have been taught to solve, but unfamiliar problems as well. However, as will be elaborated below, the ability to solve problems and to use one's mathematical knowledge effectively depends not only on content knowledge, but also on the process standards listed above.

Solving difficult problems has always been the concern of professional mathematicians. Early in the twentieth century, *problem books* were viewed as ways for advanced students to develop their mathematical understandings. Perhaps the best exemplar is George Pólya and Gabor Szegö's *Problems and Theorems in Analysis,* first published in 1924. The book offered a graded series of exercises. Readers who managed to solve all the problems would have learned a significant amount of mathematical content, and (although implicitly) a number of problem-solving strategies.

The idea that one could isolate and teach strategies for problem solving remained tacit until the publication of Pólya's *How to Solve It* in 1945. Pólya introduced the notion of *heuristic strategy* –a strategy that, while not guaranteed to work, might help one to better understand or solve a problem. Pólya illustrated the use of certain strategies, such as drawing diagrams; "working backwards" from the goal one wants to achieve; and decomposing a problem into parts, solving the parts, and recombining them to obtain a solution to the original problem. Pólya's ideas resonated within the mathematical community, but they were exceptionally difficult to implement in practice. For example, while it was clear that one should draw diagrams, it was not at all clear which diagrams should be drawn, or what properties those diagrams should have. A problem could be decomposed in many ways, but it was not certain which ways would turn out to be productive.

Means of addressing such issues became available in the 1970s and 1980s, as the field of artificial intelligence (AI) flourished. Researchers in AI wrote computer programs to solve problems, basing the programs on fine-grained observations of human problem solvers. Allen Newell and Herbert Simon's classic 1972 book *Human Problem Solving* showed how one could abstract regularities in the behavior of people playing chess or solving problems in symbolic logic–and codify that regularity in computer programs. Their work suggested that one might do the same for much more complex human problem-solving strategies, if one attended to fine matters of detail. Alan Schoenfeld's 1985 book *Mathematical Problem Solving* (and his subsequent work) showed that such work could be done successfully. Schoenfeld provided evidence that Pólya's heuristic strategies were too broadly defined to be teachable, but that when one specified them more narrowly, students could learn to use them. His book provided evidence that students could indeed learn to use problem-solving strategies–and use them to solve problems unlike the ones they had been taught to solve. It also indicated, however, along with other contemporary research, that problem solving involved more than the mastery of relevant knowledge and powerful problem-solving strategies.

One issue, which came to be known as *metacognition* or *self-regulation,* concerns the effectiveness with which problem solvers use the resources (including knowledge and time) potentially at their disposal. Research indicated that students often fail to solve problems that they might have solved because they waste a great deal of time and effort pursuing inappropriate directions. Schoenfeld's work indicated that students could learn to reflect on the state of their problem solving and become more effective at curtailing inappropriate pursuits. This, however, was still only one component of complex mathematical behavior.

Research at a variety of grade levels indicated that much student behavior in mathematics was shaped by students' beliefs about the mathematical enterprise. For example, having been assigned literally thousands of "problems" that could be solved in a few minutes each, students tended to believe that all mathematical problems could be solved in just a few minutes. Moreover, they believed that if they failed to solve a problem in short order, it was because they didn't understand the relevant method. This led them to give up working on problems that might well have yielded to further efforts. As Magdalene Lampert observed, "Commonly, mathematics is associated with certainty; knowing it, with being able to get the right answer, quickly. These cultural assumptions are shaped by school experience, in which *doing* mathematics means following the rules laid down by the teacher; *knowing* mathematics means remembering and applying the correct rule when the teacher asks a question; and mathematical *truth is determined* when the answer is ratified by the teacher. Beliefs about how to do mathematics and what it means to know it in school are acquired through years of watching, listening, and practicing"(p. 31).

Lampert argued that the very practices of schooling resulted in the development of inappropriate beliefs about the nature of mathematics, and that those beliefs resulted in students' poor mathematical performance. Given the link between students' experiences and their beliefs, the necessary remedy was to revise instructional practices–to create instructional contexts in which students could engage in mathematics as an act of sense-making, and thereby develop a more appropriate set of knowledge, beliefs, and understandings.

Research on mathematical thinking and problem solving conducted in the 1970s and 1980s established the underpinnings for the first major "reform" document, *Curriculum and Evaluation Standards for School Mathematics* (1989), also published by NCTM. The climate was right for change, for the nation was concerned about its students' mathematical performance. Reports such as *A Nation at Risk* (1983) had documented American students' weak mathematical performance in comparison to that of students from other nations, and there was a sense of national crisis regarding the nation's mathematical and scientific capacities. In the early 1990s the U. S. National Science Foundation began to support the development of curricular materials consistent with emerging research on mathematical thinking and learning. The first wave of curricula developed along these lines began to be adopted in the late 1990s.

Many of the new curricula call for students to work on complex problems over extended periods of time. In some cases, important mathematical ideas are introduced and developed through working on problems, rather than taught first and "applied" later. Either way, the fundamental idea is that students will need to have opportunities to develop both the content and process understandings described in *Principles and Standards.* As indicated above, this calls for changes in classroom practices. The best way for students to develop productive mathematical dispositions and knowledge is for them to be supported, in the classroom, in activities that involve meaningful mathematical problem solving. Given a complex problem, students can work together, under the guidance of a knowledgeable teacher, to begin to understand the task and the resources necessary to solve it. This can help them develop productive mathematical dispositions (i.e., the understanding that complex problems will yield to sustained, systematic efforts) and analytic skills. Complex problems may span mathematical areas or be drawn from real-world applications, thus helping students make mathematical connections.

Understanding and working through such problems calls for learning various representational tools–the symbolic and pictorial languages of mathematics. Tasks that call for explaining one's reasoning (i.e., asking students to make a choice between two options and to write a memo that justifies their choice on mathematical grounds) can help students develop their skills at mathematical argument. They also reinforce the idea that obtaining an answer is not enough; one must also be able to convince others of its correctness. Teachers can help students understand that there are standards for communicating mathematical ideas. The arguments students present should be coherent and logical, and ultimately, as students develop, formalizable as mathematical proofs. In these ways, complex problem solving becomes a curricular vehicle as well as a curricular goal.

** See also: ** Mathematics Education, Teacher Preparation; Mathematics Learning,

**Learning Tools, Myths, Mysteries, AND Realities, Number Sense, Word-Problem Solving.**

*subentries on*## bibliography

Brown, Ann L. 1978. "Knowing When, Where, and How to Remember: A Problem of Metacognition." In *Advances in Instructional Psychology,* Vol. 1, ed. Robert Glaser. Hillsdale, NJ: Erlbaum.

Henry, Nelson B., ed. 1951. *The Teaching of Arithmetic.* Chicago: University of Chicago Press.

Lampert, Magdalene. 1990. "When the Problem Is Not the Question and the Solution Is Not the Answer: Mathematical Knowing and Teaching." *American Educational Research Journal* 17:29–64.

Lester, Frank. 1994. "Musings about Mathematical Problem-Solving Research: 1970–1994." *Journal for Research in Mathematics Education* 25 (6):660–675.

National Commission on Excellence in Education. 1983. *A Nation at Risk: The Imperative for Educational Reform.* Washington, DC: U.S. Government Printing Office.

National Council of Teachers of Mathematics. 1989. *Curriculum and Evaluation Standards for School Mathematics.* Reston, VA: National Council of Teachers of Mathematics.

National Council of Teachers of Mathematics. 2000. *Principles and Standards for School Mathematics.* Reston, VA: National Council of Teachers of Mathematics.

Newell, Allen, and Simon, Herbert A. 1972. *Human Problem Solving.* Englewood Cliffs, NJ: Prentice-Hall.

Pólya, George. 1945. *How to Solve It.* Princeton, NJ: Princeton University Press.

Pólya, George, and SzegÖ, Gabor. 1972. *Problems and Theorems in Analysis.* New York: Springer-Verlag.

Schoenfeld, Alan H. 1985. *Mathematical Problem Solving.* Orlando, FL: Academic Press.

Schoenfeld, Alan H. 1992. "Learning to Think Mathematically: Problem Solving, Metacognition, and Sense-Making in Mathematics." In *Handbook of Research on Mathematics Teaching and Learning,* ed. Douglas A. Grouws. New York: Macmillan.

Whipple, Guy M., ed. 1930. *Report of the Society's Committee on Arithmetic.* (The twenty-ninth yearbook of the National Society for the Study of Education.) Bloomington, IL: Public School Publishing Company.

Alan H. Schoenfeld

## GEOMETRY

Geometry originated in the ancient practice of earth measurement used in agriculture, the building of pyramids, and the observation of the patterns in the movement of the stars applied in navigation. In spite of the very practical origin of geometry in the investigation of the world, geometry is also the subject area where the development of abstract reasoning began, culminating in the first systematic organization of mathematical knowledge by Euclid around 300 b. c. e. Euclid's deductive system, built on definitions, postulates, theorems, and proofs, has served as the blueprint for representing mathematical knowledge since its inception.

Tension between the experiential, empirical origins of geometry and its abstract deductive representation characterizes contemporary instructional practice and research. Critics of the traditional Euclidean approach to the teaching and learning of geometry argue that the severance of geometric knowledge from its foundation in an inherently geometric world is a pedagogical error.

Educators in the United States have been reluctant to introduce geometry in the primary grades. The traditional view of geometry as an exemplification of abstract reasoning and a fear of exposing students prematurely to formal thinking may be among the reasons for this reluctance. When primary grade teachers choose to spend a short instructional period on geometry, it is usually limited to having students recognize and recall the names of prototypical two-dimensional shapes like triangles, squares, and rectangles. This practice fails to take advantage of the host of informal geometric knowledge children bring to school.

Even before entering school, children develop intuitions about geometric shapes and their characteristics during their early explorations with their environment. For example, in exploring the objects around them, children experience that surfaces can be bumpy or smooth. Building with blocks or stacking other objects, children learn about differences in forms and sizes. Using boxes and other containers, they form intuitive ideas of space-filling or volume. As children walk around in their neighborhood they develop informal notions of spatial arrangements, distance, and directionality. The learning of geometry can be built on this naturally acquired spatial sense. Guiding children to reflect on the characteristics and regularities of their spatial experience can easily lead to the development of the basic concepts (abstractions) of geometry, such as straight and curved lines, points as intersections, planes, and planar and three-dimensional shapes. Uncultivated or ignored, however, children's natural spatial sense fades away, and it is difficult to retrieve it for use when students enroll in their first official geometry course in high school.

A programmatic document, the 1989 *Curriculum and Evaluation Standards for School Mathematics,* produced by the National Council of Teachers of Mathematics (NCTM) to guide reform in mathematics education, recommends that geometric topics be introduced and applied to real-world situations whenever possible. However, this does not imply that immersing children in real-world situations automatically leads to mathematical or geometrical understanding. Hands-on activities are a popular way to establish a connection between instruction and real life, but as instructional means they are only as good as the meanings derived from them. The challenge of geometry instruction is to elevate children's experience with real-world objects to the level of mathematics.

This happens in well-designed instructional tasks that promote reflection on the geometric features of real-life situations, leading to the development of geometric concepts and spatial reasoning. Children learn to generate geometric arguments by participating in carefully orchestrated conversations where they articulate, share, and discuss their ideas regarding spatial problems. Children develop skills of modeling spatial situations when they are invited to publicly display and discuss their visualizations in drawings. These drawings can then be turned into mathematical representations during revision cycles, in the course of which the geometrical features are accentuated while the mathematically irrelevant features (e.g., material, color, and other decorative elements) gradually fade away.

At the secondary level, the traditional Euclidean geometry curriculum that revolves around deductive proof procedures has been criticized because it separates geometry from its empirical, inductive foundation. Critics refer to the typical lack of student appreciation for the subject–often accompanied by low achievement. The deductive organization of the geometry course has been seen as a viable model to help introduce students to mathematical reasoning. However, this is a misrepresentation of the actual reasoning that goes on among expert mathematicians. The deductive logic applied in proofs constitutes only a subset of the rules, and it seldom accounts for the actual thought processes that contributed to the discovery communicated in a proof. Actual discovery usually follows an inductive line of reasoning that begins with empirical investigation and the observation of regularities. It continues with making conjectures based on the observed regularities, and then testing them on multiple examples. Attempts at explaining and generalizing the observed relationship with the help of proof come only after the long process of empirical exploration.

Alternatives to a traditional Euclidean secondary geometry curriculum have been offered based on this more grounded view of mathematical reasoning that incorporates exploration and induction. In the process of exploration, students learn to deconstruct geometric objects into their constitutive elements, and to rely on properties–such as the number and relative size of the sides of the objects, the measure of angles, and their relationships–rather than a prototypical or customary presentation of an image when they identify shapes. Ideally, students will learn to go beyond the appearance of an actual drawing of a shape and argue about generalized concepts of shapes as defined by their properties (for example, a rectangle is a quadrilateral with four right angles and with opposite sides equal and parallel). These skills serve as the foundation of geometrical understanding and need to be acquired–ideally in the primary grades–before students are exposed to proofs.

Some of the new secondary geometry curricula have been organized around technology tools, including geometry construction programs such as the Geometer's Sketchpad and the Geometric Super-Supposer. These programs provide an electronic environment for geometric explorations and allow the learner to generate multiple solutions of geometric construction problems, thus facilitating the generation and testing of hypotheses. Proofs gain a different meaning in this context, becoming the means of explaining why the conjectures developed by the students themselves hold beyond the examples created by the program. This has a motivating effect on the learner. Without such an inductive foundation, students see proofs as an unnecessary procedure to arrive at a simple truth that they already know and accept.

** See also: ** Mathematics Education, Teacher Preparation; Mathematics Learning,

**Algebra, Complex Problem Solving, Learning Tools, Word-Problem Solving.**

*subentries on*## bibliography

Chazan, Daniel, and Yerushalmy, Michal. 1998. "Charting a Course for Secondary Geometry." In *Designing Learning Environments for Developing Understanding of Geometry and Space,* ed. Richard Lehrer and Daniel Chazan. Mahwah, NJ: Erlbaum.

Lakatos, Imre. 1976. *Proofs and Refutations: The Logic of Mathematical Discovery.* Cambridge, Eng.: Cambridge University Press.

Lehrer, Richard, and Chazan, Daniel, eds. 1998. *Designing Learning Environments for Developing Understanding of Geometry and Space.* Mahwah, NJ: Erlbaum.

Lehrer, Richard; Jacobson, Cathy; Kemeny, Vera; and Strom, Dolores. 1999. "Building on Children's Intuitions to Develop Mathematical Understanding of Space." In *Mathematics Classrooms that Promote Understanding,* ed. Elizabeth Fennema and Thomas A. Romberg. Mahwah, NJ: Erlbaum.

National Council of Teachers of Mathematics. 1989. *Curriculum and Evaluation Standards for School Mathematics.* Reston, VA: National Council of Teachers of Mathematics.

National Council of Teachers of Mathematics. 2000. *Principles and Standards for School* *Mathematics.* Reston, VA: National Council of Teachers of Mathematics.

Schwartz, Judah L.; Yerushalmy, Michal; and Wilson, Beth, eds. 1993. *The Geometric Supposer: What Is It a Case Of?* Hillsdale, NJ: Erlbaum.

Serra, Michael. 1989. *Discovering Geometry: An Inductive Approach.* Berkeley, CA: Key Curriculum Press.

Vera Kemeny

## LEARNING TOOLS

The manner in which students learn mathematics influences how well they understand its concepts, principles, and practices. Many researchers have argued that to promote learning with understanding, mathematics educators must consider the tasks, problem-solving situations, and tools used to represent mathematical ideas. Mathematical tools foster learning at many levels–namely, the learning of facts, procedures, and concepts. Tools can also provide concrete models of abstract ideas, or, when dealing with complex problems, they can enable students to manipulate and think about ideas, thereby making mathematics accessible and more deeply understood.

Mathematical learning tools can be traditional, technological, or social. The most frequently employed tools are traditional, which include physical objects or manipulatives (e.g., cubes), visualization tools (e.g., function diagrams), and paper-and-pencil tasks (e.g., producing a table of values). Technological tools, such as calculators (i.e., algebraic and graphic) and computers (e.g., computation and multiple-representation software), have gained attention because they can extend learning in different ways. Social tools, such as small-group discussions where students interact with one another to share and challenge ideas, can be considered a third type of learning tool. These three tools can be used independently or conjointly, depending on the type of learning that is intended.

### Learning Tools in Mathematics

A learning tool can be as simple as an image or as complex as a computer-based environment designed to improve mathematical understanding. The key characteristic of a learning tool is that it supports learners in some manner. For example, a tool can aid memory, help students to review their problem-solving processes, or allow students to compare their performance with that of others, thereby supporting self-assessment. Learning tools can represent mathematical ideas in multiple ways, providing flexible alternatives for individuals who differ in terms of learner characteristics. For example, learners who have difficulty understanding the statistical ideas of arithmetic mean (center) and variance (spread) may be assisted through interactive displays that change as data points are manipulated by the learner. A mathematical learning tool can scaffold the learner by performing computations, providing more time for students to test mathematical hypotheses that require reasoning. In the statistics example, learners can focus on why changes to certain parameters affect data–and in what ways, rather than spending all their time calculating measures.

**Traditional Tools.** Traditional tools are best suited for facilitating students' learning of basic knowledge and skills. Objects that can be manipulated, such as cubes, reduce the abstract nature of concepts, such as numbers, thereby making them real and tangible, particularly for younger children. Such tools support the development of children's understanding of arithmetic by serving as a foundation for learning more complex concepts. Visualization tools, such as graphs, can support data interpretation, while paper-and-pencil tools that provide practice of computational skills can support memory for procedures and an ability to manipulate symbols. Combining physical tools with visualization tools can substantively increase students' conceptual knowledge. Dice and spinners, for example, can be used to support elementary school students in creating graphs of probability distributions, helping them develop an understanding of central tendency.

**Technological Tools.** Technological tools are most effective in facilitating students' understanding of complex concepts and principles. Computations and graphs can be produced quickly, giving students more time to consider why a particular result was obtained. This support allows students to think more deeply about the mathematics they are learning. Electronic tools are necessary in mathematics because they support the following processes: (a) conjectures–which provide access to more examples and representational formats than is possible by hand; (b) visual reasoning–which provides access to powerful visual models that students often do not create for themselves; (c) conceptualization and modeling–which provide quick and efficient execution of procedures; and (d) flexible thinking–which support the presentation of multiple perspectives.

Spreadsheets, calculators, and dynamic environments are sophisticated learning tools. These tools support interpretation and the rapid testing of conjectures. Technology enables students to focus on the structure of the data and to think about what the data mean, thereby facilitating an overall understanding of a concept (e.g., function). The graphics calculator supports procedures involving functions and students' ability to translate and understand the relationship between numeric, algebraic, and graphical representations. Transforming graphical information in different ways focuses attention on scale changes and can help students see relationships if the appropriate viewing dimensions are used. Computers may remove the need for overlearning routine procedures since they can perform the task of computing the procedures. It is still debatable whether overlearning of facts helps or hinders deeper understanding and use of mathematics. Technology tools can also be designed to help students link critical steps in procedures with abstract symbols to representations that give them meaning.

Video is a dynamic and interactive learning tool. One advantage of video is that complex problems can be presented to students in a richer and more realistic way, compared to standard word problems. An example is The Adventures of Jasper Woodbury, developed by the Learning Technology Center at Vanderbilt University. Students are required to solve problems encountered by characters in the Woodbury video by taking many steps to find a solution. This tool supports students' ability to solve problems, specifically their ability to identify and formulate a problem, to generate subgoals that lead to the solution, and to find the solution. However, the information presented in a video cannot be directly manipulated in the same way that data can be changed in spreadsheets and calculators.

Learning tools that present the same information in several ways (e.g., verbal equation, tabular, graphic) are referred to as multiple-representation tools. The ability to interpret multiple representations is critical to mathematical learning. There is evidence to suggest that multiple representations can facilitate students' ability to understand and solve word problems in functions, and to translate words into tables and graphs. However, interpretation is not easy without some kind of support. One type of support involves highlighting common elements between the different representations to make the relationship between each explicit, thereby facilitating interpretation in both contexts. In some cases, this type of support is insufficient and students need to be explicitly taught to make the connections. Multiple representations can be a powerful learning tool for difficult problems–when students have acquired a strong knowledge base.

Additional research is needed to determine the exact benefits of multiple representational tools. It is important to emphasize that, as with any educational innovation, mathematical learning tools must be designed with a consideration of the teacher, curriculum, and student in mind. For example, with the help of curricular teams and teachers, complex computer environments that present students with multiple representation tools for learning algebra and geometry were successfully adopted in several school systems in the United States.

**Social tools.** Social tools are a fairly recent consideration. In the 1990s, small-group work where students share strategies for solving problems began to be used as a powerful learning tool. This tool facilitates students' ability to solve word problems and to understand arithmetic. Group collaboration while learning with technology can help students develop the perspectives and practices of mathematics, such as what constitutes acceptable mathematical evidence. Peers and computers can provide feedback that makes students aware of contradictions in their thinking. In this way, social tools can assist learning and transform understanding.

### Issues for Further Consideration

Mathematical learning tools should be an important part of students' educational experience. However, a few issues must be addressed before their potential is fully realized. First, use of technological tools is fairly limited in classrooms, despite their potential in changing the nature of mathematical learning. Moreover, software used in schools is often geared towards the practice of computational skills. For example, there may be a potential misuse of the graphing calculator if it is not utilized in the context of sense-making activities. There is a fine line between using a tool for understanding and using it because problems cannot be solved without its use.

Second, learning tools should be an integral part of instructional activities and assessment tasks. Learning tools should be a regular part of the mathematics experience at every educational level, and different tools should be used for various purposes. The question of ethics and equity is raised when technological tools that are used in instruction are not accessible in assessment situations.

Third, learning tools will only meet their promise through professional development. Teachers who understand the strengths and weaknesses of tools can have a strong impact on how they are used. Support is needed at all levels of education to ensure that sophisticated learning tools are available for use in every mathematics classroom. Learning tools are only as good as the activities that provide the mathematical experiences. The effectiveness of such tools is thus highly dependent on the purpose of the activity and the learning that is intended.

** See also: ** Mathematics Learning,

**Complex Problem Solving; Science Learning,**

*subentry on***Tools; Technology in Education,**

*subentry on***Current Trends.**

*subentry on*## bibliography

Burill, Gail. 1997. "Graphing Calculators and Their Potential for Teaching and Learning Statistics." In *Research on the Role of Technology in Teaching and Learning Statistics,* ed. Joan B. Garfield and Gail Burrill. Voorburg, Netherlands: International Statistical Institute.

Cobb, Paul. 1999. "Individual and Collective Mathematical Development: The Case of Statistical Data Analysis." *Mathematical Thinking and Learning* 1 (1):5–43.

Cognition and Technology Group at Vanderbilt. 1992. "The Jasper Series as an Example of Anchored Instruction: Theory, Program Description, and Assessment Data." *Educational Psychologist* 27 (3):291–315.

Fennema, Elizabeth, and Romberg, Thomas, eds. 1999. *Mathematics Classrooms That Promote Understanding.* Mahwah, NJ: Erlbaum.

Horvath, Jeffrey K., and Lehrer, Richard. 1998. "A Model-Based Perspective on the Development of Children's Understanding of Chance and Uncertainty." In *Reflections on Statistics: Learning, Teaching, and Assessment in Grades K–12,* ed. Susanne P. Lajoie. Mahwah, NJ: Erlbaum.

Kaput, James. 2000. "Teaching and Learning a New Algebra." In *Mathematics Classrooms That Promote Understanding,* ed. Elizabeth Fennema and Thomas A. Romberg. Mahwah, NJ: Erlbaum.

Koedinger, Kenneth, R.; Anderson, John, R.; Hadley, William, H.; and Mark, Mary, A.1997. "Intelligent Tutoring Goes to School in the Big City." *International Journal of Artificial Intelligence in Education* 8:30–43.

Lajoie, Susanne P., ed. 2000. *Computers as Cognitive Tools: No More Walls,* Vol. 2. Mahwah, NJ: Erlbaum.

Lesgold, Alan. 2000. "What Are the Tools For? Revolutionary Change Does Not Follow the Usual Norms." In *Computers as Cognitive Tools,* Vol. 2, ed. Susanne P. Lajoie. Mahwah: Erlbaum.

National Council of Teachers of Mathematics. 2000. *Principles and Standards for School Mathematics.* Reston, VA: National Council of Teachers of Mathematics.

Susanne P. Lajoie

Nancy C. Lavigne

## MYTHS, MYSTERIES, AND REALITIES

According to the National Research Council, "Much of the failure in school mathematics is due to a tradition of teaching that is inappropriate to the way most students learn" (p. 6). Yet, despite the fact that numerous scientific studies have shown that traditional methods of teaching mathematics are ineffective, and despite professional recommendations for fundamental changes in mathematics curricula and teaching, traditional methods of teaching continue. Indeed, mathematics teaching in the United States has changed little since the mid–twenieth century–essentially, teachers demonstrate, while students memorize and imitate.

### Realities

Although research indicates that learning that emphasizes sense-making and understanding produces a better transfer of learning to new situations, traditional classroom instruction emphasizes imitation and memorization. Even when traditional instruction attempts to promote understanding, most students fail to make sense of the ideas because classroom derivations and justifications are too formal and abstract. Though research indicates that mathematical knowledge is truly understood and usable only when it is organized around and interconnected with important core concepts, traditional mathematics curricula make it difficult for students to meaningfully organize knowledge. This is because such curricula provide little time for, or attention to, the type of sense-making activities that enable students to genuinely understand and organize mathematical knowledge. Indeed, the major finding that caused the authors of the Third International Mathematics and Science Study (TIMSS) to characterize the U.S. mathematics curricula as "a mile wide and inch deep" is that traditional curricula cover far too many topics, almost all superficially. As a result, though the same topics are retaught yearly, many are never learned, and few are truly understood.

Furthermore, because traditional instruction focuses so much on symbolic computation procedures, many students come to believe that mathematics is mainly a matter of following fixed and rigid procedures that have no connection to their thinking about realistic and meaningful situations. Instead of seeing mathematics as thoughtful, reflective reasoning, students see it as a matter of parroting procedures, as an academic ritual that has no genuine usefulness. Such ritualistic mathematics, stripped of its power to explain anything that matters and devoid of the interconnections that arise from sense-making, becomes a hodgepodge of memorized–and easily forgotten–rules. The National Research Council dubbed such knowledge "mindless mimicry mathematics."

**The modern scientific view of mathematics learning.** Almost all current major scientific theories describing how students learn mathematics with genuine understanding (instead of by rote) agree that: (a) mathematical ideas must be mentally constructed by students as they intentionally try to make personal sense of situations; (b) how students construct new ideas is heavily dependent on the cognitive structures students have previously developed; and (c) to be effective, mathematics teaching must carefully guide and support the processes by which students construct mathematical ideas. According to these *constructivist-based* theories, the way a student interprets, thinks about, and makes sense of newly encountered mathematical ideas is determined by the elements and the organization of the relevant mental structures that the student is currently using to process his or her mathematical world. Consequently, instruction that promotes understanding cannot ignore students' current ideas and ways of reasoning, including their many informal, and even incorrect, ideas.

However, despite the value of the general notion that students must actively construct their own mathematical knowledge, a careful reading of research in mathematics education reveals that the power and usefulness of the these *constructivist* theories arise from: (a) their delineation of specific learning mechanisms, and (b) the detailed research they have spawned on students' mental construction of meaning for particular mathematical topics such as whole-number operations, fractions, and geometric shapes. It is this elaboration and particularization of the general constructivist theory to specific mathematical topics and classroom situations that make the theory and research genuinely relevant to teaching mathematics.

**The modern view of mathematics teaching.** Both research and professional recommendations suggest a type of mathematics instruction very different from that found in traditional classrooms. In the spirit of inquiry, problem solving, and sense-making, such instruction encourages students to invent, test, and refine their own ideas, rather than unquestioningly follow procedures given to them by others. This type of instruction guides and supports students' construction of personally meaningful ideas that are increasingly complex, abstract, and powerful, and that evolve into the important formal mathematical ideas of modern culture.

However, unlike instruction that focuses only on classroom inquiry, this type of instruction is based on detailed knowledge of students' construction of mathematical knowledge and reasoning. That is, this teaching is based on a deep understanding of: (a) the general stages that students pass through in acquiring the concepts and procedures for particular mathematical topics; (b) the strategies that students use to solve different problems at each stage; and (c) the mental processes and the nature of the knowledge that underlies these strategies. This teaching uses carefully selected sequences of problematic tasks to provoke appropriate perturbations and reformulations in students' thinking.

An abundance of research has shown that mathematics instruction that focuses on student inquiry, problem solving, and personal sense-making–especially that guided by research on students' construction of meaning for particular topics–produces powerful mathematical thinkers who not only can compute, but have strong mathematical conceptualizations and are skilled problem solvers.

### Myths and Misunderstandings

**Misunderstanding the nature of mathematics.** One of the most critical aspects of effective mathematics learning is developing a proper understanding of the nature of mathematics. The chairperson of the commission that wrote the National Council of Teachers of Mathematics (NCTM) *Standards* stated, "The single most compelling issue in improving school mathematics is to change the epistemology of mathematics in schools, the sense on the part of teachers and students of what the mathematical enterprise is all about" (Romberg, p. 433).

Mathematics is first and foremost a form of reasoning. In the context of analytically reasoning about particular types of quantitative and spatial phenomena, mathematics consists of thinking in a logical manner, making sense of ideas, formulating and testing conjectures, and justifying claims. One does mathematics when one recognizes and describes patterns; constructs physical or conceptual models of phenomena; creates and uses symbol systems to represent, manipulate, and reflect on ideas; and invents procedures to solve problems. Unfortunately, most students see mathematics as memorizing and following little-understood rules for manipulating symbols.

To illustrate the difference between mathematics as reasoning and mathematics as rule-following, consider the question: "What is 2-1/2 divided by 1/4?" Traditionally taught students are trained to solve such problems by using the "invert and multiply" method: 2-1/2 ÷ 1/4 = 5/2 × 4/1. Students who are lucky enough to recall how to compute an answer can rarely explain or demonstrate why the answer is correct. Worse, most students do not know when the computation should be applied in real-world contexts.

In contrast, students who have made genuine sense of mathematics do not need a symbolic algorithm to compute an answer to this problem. They quickly reason that, since there are 4 fourths in each unit and 2 fourths in a half, there are 10 fourths in 2-1/2. Furthermore, such students quickly recognize when to apply such thinking in real-world situations.

Obviously, not all problems can be easily solved using such intuitive strategies. Students must also develop an understanding of, and facility with, symbolic manipulations. Nevertheless, students' use of symbols must never become disconnected from their powerful intuitive reasoning about actual quantities. For when it does, students become over-whelmed with trying to memorize countless rules.

**The myth of coverage.** One of the major components of traditional mathematics teaching is the almost universal belief in the myth of *coverage.* According to this myth, if mathematics is "covered" by instruction, students will learn it. This myth is so deeply embedded in traditional mathematics instruction that, at each grade level, teachers feel tremendous pressure to teach huge amounts of material at breakneck speeds. The myth has fostered a curriculum that is superficially broad, and it has encouraged acceleration rather than deep understanding. Belief in this myth causes teachers to criticize as inefficient curricula that emphasize depth of understanding because students in such curricula study far fewer topics at each grade level.

But research on learning debunks this myth. Based on scientific evidence, researchers John Bransford, Ann Brown, and Rodney Cocking explain that covering too many topics too quickly hinders learning because students acquire disorganized and disconnected facts and organizing principles that they cannot make meaningful. Indeed, in his article "Teaching for the Test," Alan Bell, from the Shell Centre for Mathematical Education at the University of Nottingham, presents research evidence showing the superiority of sense-making curricula. Consistent with Bell's claim, TIMSS data suggest that Japanese teachers, whose students significantly outperform U.S. students in mathematics, spend much more time than U.S. teachers having students delve deeply into mathematical ideas.

In summary, because students in traditional curricula learn ideas and procedures rotely, rather than meaningfully, they quickly forget them, so the ideas must be repeatedly retaught. In contrast, in curricula that focus on deep understanding and personal sense-making, because students naturally develop and interrelate new and rich conceptualizations, they accumulate an ever-increasing network of well-integrated and long-lasting mathematical knowledge. Thus, curricula that emphasize deep understanding may cover fewer topics at particular grade levels, but overall they enable students to learn more material because topics do not need to be repeatedly taught.

**Putting skill before understanding.** Many people, including teachers, believe that students, especially those in lower-level classes, should master mathematical procedures first, then later try to understand them. However, research indicates that if students have already rotely memorized procedures through extensive practice, it is very difficult for later instruction to get them to conceptually understand the procedures. For example, it has been found that fifth and sixth graders who had practiced rules for adding and subtracting decimals by lining up the decimal points were less likely than fourth graders with no such experience to acquire conceptual knowledge from meaning-based instruction.

**Believing that bright students are doing fine.** Although there is general agreement that most students have difficulty becoming genuinely competent with mathematics, many people take solace in the belief that bright students are doing fine. However, a closer look reveals that even the brightest American students are being detrimentally affected by traditional teaching. For instance, a bright eighth grader who was three weeks from completing a standard course in high school geometry applied the volume formula in a situation in which it was inappropriate, getting an incorrect answer:

Observer: How do you know that is the right answer?

Student: Because the equation for the volume of a box is length times width times height.

Observer: Do you know why that equation works?

Student: Because you are covering all three dimensions, I think. I'm not really sure. I just know the equation. (Battista, 1999)

This student did not understand that the mathematical formula she applied assumed a particular mathematical model of a real-world situation, one that was inappropriate for the problem she was presented. Although this bright student had learned many routine mathematical procedures, much of the learning she accomplished in her accelerated mathematics program was superficial, a finding that is all too common among bright students. Indeed, only 38 percent of the students in her geometry class answered the item correctly, despite the fact that all of them had scored at or above the ninety-fifth percentile in mathematics on a widely used standardized mathematics test in fifth grade. Similarly, in the suburbs of one major American city in which the median family income is 30 percent higher than the national average, and in which three-quarters of the students were found to be at or above the international standard for computation, only between one-fifth and one-third met the international standard for problem solving.

**Misunderstanding inquiry-based teaching.** Many educators and laypersons incorrectly conceive of the inquiry-based instruction suggested by modern research as a pedagogical paradigm entailing nonrigorous, intellectual anarchy that lets students pursue whatever interests them and invent and use any mathematical methods they wish, whether these methods are correct or not. Others see such instruction as equivalent to cooperative learning, teaching with manipulatives, or *discovery* teaching in which a teacher asks a series of questions in an effort to get students to discover a specific, formal mathematical concept. Although elements of the latter three conceptions are, in altered form, similar to components of the type of instruction recommended by research in mathematics education, none of these conceptions is equivalent to the modern view. What separates the new, research-based view of teaching from past views is: (a) the strong focus on, and carefully guided support of, students' construction of personal mathematical meaning, and (b) the use of research on students' learning of particular mathematical topics to guide the selection of instructional tasks, teaching strategies, and learning assessments.

To illustrate, consider the topic of finding the volume of a rectangular box. In traditional didactic teaching, students are simply shown the procedure of multiplying the length, width, and height. In classic discovery teaching, students might be given several boxes and asked to determine the boxes' dimensions and volumes using rulers and small cubes. The teacher would ask students to determine the relationship between the dimensions and the volumes, with the goal being for students to discover the "length times width times height" procedure. In contrast, research-based inquiry teaching might give students a sequence of problems in which students examine a picture of a rectangular array of cubes that fills a box, predict how many cubes are in the array, then make the box and fill it with cubes to check their prediction. The goal would be for each individual student to develop a prediction strategy that not only is correct but also makes sense to the student. Research shows that the formula rarely makes sense to students, and that, if given appropriate opportunities, students generally develop some type of layering strategy, for instance, counting the cubes showing on the front face of an array and multiplying by the number of layers going back. Because the layering strategy is a natural curtailment of the concrete counting strategies students initially employ on these problems, it is far easier for students to make personal sense of layering than using the formula.

Modern research further guides inquiry teaching by describing the cognitive obstacles students face in learning and the cognitive processes needed to overcome these obstacles. For instance, research indicates that before being exposed to appropriate instruction, most students have an incorrect model of the array of cubes that fills a rectangular box. Because of a lack of coordination and synthesis of spatial information, students can neither picture where all the cubes are nor appropriately mentally organize the cubes. Instruction can support the development of personal meaning for procedures for finding volume only if it ensures that (a) students develop proper mental models of the cube arrays, and (b) students base their enumeration strategies on these mental models.

**Forgetting the need for fluency.** Because of mistaken beliefs about the type of instruction suggested by research and professional recommendations, low-fidelity implementations of reform curricula often focus so much on promoting class discussions and reasoning that they lose sight of the critical need to properly crystallize students' thinking into a sophisticated and fluent use of mathematics. Although modern approaches to instruction have rightly shifted the instructional focus from imitating procedures to understanding and personal sense-making, it is clearly insufficient to involve students only in sense-making, reasoning, and the construction of mathematical knowledge. Sound curricula must also assure that students become fluent in utilizing particularly useful mathematical concepts, ways of reasoning, and procedures. Students should be able to readily and correctly apply important mathematical strategies, procedures, and lines of reasoning in various situations, and they should possess knowledge that supports mathematical reasoning. For instance, students should know the *basic number facts,* because such knowledge is essential for mental computation, estimation, performance of computational procedures, and problem solving.

### Mysteries and Challenges

**To inquire or not to inquire.** Scientific research and professional standards recommend inquiry-based instruction because such instruction elicits classroom cultures that support students' genuine sense-making, and because such classrooms focus on the development of students' reasoning, not the disconnected rote acquisition of formal, ready-made ideas contained in textbooks. However, the critical ingredient in research-based teaching is the focus on fostering students' construction of personal mathematical meaning. This focus suggests that inquiry-based teaching that does not focus on students' construction of personally meaningful ideas is not completely consistent with research-based suggestions for teaching. It also suggests that demonstrations, and even lectures, might create meaningful learning if students are capable of, and intentionally focus on, personal sense-making and understanding. However, the question of whether, and when, lecture/demonstration–the most common mode of teaching found in American schools–can produce meaningful mathematics learning has not received much research attention. Research is needed that thoroughly investigates the role that this cherished traditional instructional tool can play in meaningful mathematics learning.

**Scientific practice versus tradition.** One of the major reasons that school mathematics programs in the U.S. are so ineffective is because they ignore modern scientific research on mathematics learning and teaching. For instance, many popular approaches to improving mathematics learning focus on getting students to "try harder" or take more rigorous courses. Or, in attempts to increase students' motivation, educators use gimmicks to try to make mathematics classes–but not mathematics itself–more interesting. But almost all of these approaches are rooted in a traditional perspective on mathematics learning; they ignore the cognitive processes that undergird mathematical sense-making. So even when these approaches are "successful," they produce only mimicry-based procedural knowledge of mathematics.

It is not that increasing motivation and effort are bad ideas. If students are unwilling to engage in intellectual activity in the mathematics classroom, there is little chance that mathematics instruction of any kind, no matter how sound, will induce or support their mathematics learning. However, students' motivation and effort to learn mathematics are strongly dependent on their beliefs about the value that mathematics, and school in general, has for their lives. The nature of these beliefs is determined partly by students' interaction with family, peers, schools, and community, but also by the quality of their mathematics instruction. Instruction that does not properly support students' mathematical sense-making builds counterproductive beliefs about mathematics learning.

Thus, because instructional approaches that are not based on modern scientific research on the learning process ignore the workings of the very process they are attempting to affect, they cannot support genuine mathematical sense-making or produce productive beliefs about learning mathematics. One of the greatest challenges is to determine how to get teachers, administrators, and policymakers to base their instructional practices and decisions on modern scientific research.

**Assessment.** Because commonly used assessments inadequately measure students' mathematics learning, there is a critical need for the creation and adoption of new assessment methods that more accurately portray student learning. Assessments are needed that not only determine *if* students have acquired particular mathematical knowledge, skills, and types of reasoning, but also determine precisely *what* students have learned. Such assessments must be firmly and explicitly linked to scientific research on students' mathematics learning, something that is sorely missing in traditional assessment paradigms. To be consistent with such research, assessment must focus on students' mathematical cognitions, not their overt behaviors.

** See also: ** Instructional Strategies; Mathematics Education, Teacher Preparation.

## bibliography

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Boaler, Jo. 1998. "Open and Closed Mathematics: Student Experiences and Understandings." *Journal for Research in Mathematics Education* 29 (1):41–62.

Bransford, John D.; Brown, Ann L.; and Cocking, Rodney R. 1999. *How People Learn: Brain, Mind, Experience, and School.* Washington, DC: National Research Council.

Carpenter, Thomas P.; Franke, Megan L.; Jacobs, Victoria R.; Fennema, Elizabeth; and Empson, Susan B. 1998. "A Longitudinal Study of Invention and Understanding in Children's Multidigit Addition and Subtraction." *Journal for Research in Mathematics Education* 29 (1):3–20.

Cobb, Paul; Wood, Terry; Yackel, Erna; Nicholls, John; Wheatley, Grayson; Trigatti, Beatriz; and Perlwitz, Marcella.1991. "Assessment of a Problem-Centered Second-Grade Mathematics Project." *Journal for Research in Mathematics Education* 22 (1):3–29.

De Corte, Erik; Greer, Brian; and Verschaffel, Lieven. 1996. "Mathematics Teaching and Learning." In *Handbook of Educational Psychology,* ed. David C. Berliner and Robert C. Calfee. London: Simon and Schuster; New York: Macmillan.

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Michael T. Battista

## NUMBER SENSE

What does it mean to suggest that an individual possesses good *number sense* ? The ability to see patterns and relationships between numbers, to work flexibly with operations and procedures, to recognize order and relative quantities, and to utilize estimation and mental computation are all components of what is termed *number sense.* Individuals who quickly calculate a 15 percent gratuity at a restaurant, know that the seven-digit display 0.498732 is approximately 1/2, or recognize that calculating 48 × 12 will be less problematic than calculating 48 × 13 are said to manifest qualities associated with good number sense.

Most mathematics educators agree that developing number sense is important, yet there is no single definition that is unanimously accepted. Number sense is highly personalized and thought to develop gradually. It includes self-regulation, an ability to make connections in number patterns, and an intuition regarding numbers. Number sense "refers to a person's general understanding of number and operations along with the ability and inclination to use this understanding in flexible ways to make mathematical judgments and to develop useful strategies for handling numbers and operations" (McIntosh et al., p. 3).

### Historical Background

Before the term *number sense* came into use, the word *numeracy* was coined in 1959 to denote those within the realm of mathematics who had a propensity to comprehend higher-level mathematical concepts. Yet the general public took numeracy to be the mathematical analogue of literacy, and therefore reduced its meaning to connote the propensity to comprehend basic arithmetic. A book by John Allen Paulos, *Innumeracy: Mathematical Illiteracy and Its Consequences* (1988), demonstrated the dangers of a population that lacks a basic understanding of mathematics and that views the subject as enigmatic due to poor education or psychological anxiety. Many of those involved with mathematics education felt that math pedagogy was in need of serious reform due to a superficial acquisition of knowledge based merely on a procedural understanding of mathematics (e.g., "just follow this algorithm").

In the late 1980s and the 1990s researchers and educators saw a heightened need to examine the role of computation as it related to elementary mathematics, reflecting on both the process and result of employing algorithmic strategies. It was during this period that the term *number sense* gained wide acceptance, epitomizing the desired outcome for the teaching and learning of mathematics. Yet due to its implicit nature, succinctly describing how number sense is revealed can be problematic. The mathematician Stanislas Dehaene, in his 1997 book *The Number Sense: How the Mind Creates Mathematics,* states, "Our number sense cannot be reduced to the formal definition provided by rules or axioms" (p. 240). In addition, James Greeno relates, "We recognize examples of number sense, even though we have no satisfactory definition that distinguishes its features"(p. 171).

Similar to the ambiguous implications of common sense, number sense is open to a variety of interpretations. The National Council of Teachers of Mathematics, in *Curriculum and Evaluation of Standards for School Mathematics* (1989), defines number sense as "an intuition about numbers that is drawn from all the varied meanings of number. It has five components: (1) having well-understood number meanings, (2) developing multiple relationships among numbers, (3) understanding the relative magnitudes of numbers, (4) developing intuitions about the relative effect of operating on numbers,(5) developing referents for measures of common objects" (pp. 39–40). However, others would argue that such descriptors and boundaries for the nature of number sense do not characterize it in forms that guide instruction. Lauren Resnick and Judith Sowder categorize number sense as an open-ended form of reasoning that is nonalgorithmic, complex, and involves uncertainty. These multiple views are highlighted merely to show the somewhat amorphous nature of number sense and qualities ascribed to it.

### Examples of Number Sense

Most often, number sense is recognized through example. One ascribed attribute is the ability to use numbers flexibly when mentally computing an abstract numerical operation. This flexibility evolves through infixing connections and relationships between numbers and their representations. By augmenting the number of connections to analogous situations, more flexibility and utility ensues. For example, a simple computation involving subtraction is the problem 7-4. The ability to place this abstraction of symbols into multiple situations signifies a certain number sense, such as: (1) a set or group–seven cookies take away four cookies; (2) a distance–in order to move from space 4 to space 7 in a board game, 3 moves are required; (3) a temperature reading–to change from 7° C to 4° C, the temperature must drop 3° C. These mental models seem natural to most adults and children who have been guided to think with such models. With the simple transition of this problem, reversing the minuend and subtrahend mandates an ability to move into negative numbers: 4 - 7 equals what? For a child who has only the group mental model (4 cookies take away 7 cookies), this operation seems problematic or impossible. A child who has multiple models can utilize the one that gives a more intuitive representation of the abstract operation–if the temperature is 4° C and then falls 7°C, then the new temperature would be negative (or minus) 3° C.

In addition, being able to compare the relative size of numbers would be a sign of number sense. Students should recognize that 4,562 is large compared to 400 but small compared to 400,000. There should also be emphasis placed on providing context to compare large numbers. For example, a million and a billion are ubiquitous quantities in many economies. Therefore, to recognize that it takes roughly eleven-and-a-half days for a million seconds to pass and nearly thirty-two years for a billion seconds to pass connotes a deeper appreciation for the relative magnitude of quantities.

Number sense extends beyond the set of whole numbers and integers. Consider a more frequent area of concern for many school children, fractions. Consider the following example: ^{2} §3 + 1/4. For conceptual understanding, fractions and ratios necessitate the skill of proportional reasoning in order to make sense of this abstract representation. Considering a part-to-a-whole relationship, the adroit student can recognize the necessity to compare equal size parts (and therefore find a common denominator) before total parts can be computed:

In contrast, a child who has no intuitive grasp for fractions will most likely commit the error of adding the numerators and adding the denominators. This algorithmic error might also be attributed to those who rely on a strictly procedural understanding, because this procedure is correct when it relates to multiplying fractions,

and students often confuse these two rules. Furthermore, this nonconventional result for addition can be justified with concrete examples. If Barry Bonds plays in both games of a double-header, and he bats 2 for 3 in the first game and 1 for 4 in the second game, then his correct batting average for the day is 3 for 7, which, in terms of the traditional procedure for adding fractions, is not conventionally correct:

Therefore, number sense involves knowing *when* a specific model is applicable.

Sometimes, number sense can be grasped intuitively through visual clues as well. Some people have an affinity for understanding visual models, which they might then internalize and incorporate into their personal number sense. Figure 1 contains no symbolic representation of numerals; rather, actual quantities are depicted as the objects themselves. The question at hand is to compare the available cake for girls and for boys and determine in which of the two groups does an individual receive more cake. Students who are versed in strictly procedural understandings might set up ratios that symbolize the situation, then try to rely on memorized algorithms to simplify the symbols:

Someone with a more flexible understanding might simply notice that for the boys there is one cake for a group of three; therefore, an equal ratio based on three cakes would be a group of nine girls. From this equivalency, they would deduce that since there are less than nine girls, then each girl must receive more cake than each boy.

### Developing Number Sense

The acquisition of number sense is often considered to develop as stages along a continuum, rather than as a static object that is either possessed or not. Dehaene reports that most children enter preschool with a well-developed understanding of approximation and counting. Dehaene presents research from cognitive psychologists, such as Jean Piaget, Prentice Starkey, and Karen Wynn, suggesting contradictory results about what skills are innate, when skills are developed, and how they are acquired. Part of the complexity to succinctly describe a development of number sense stems both from the subtlety of multiple factors it encompasses and the lack of explicit demonstrability. For example, with the problem 18 × 5, someone demonstrating number sense might recognize the relationship of the quantity *5* comparedto *10* is simply half, and knowing that, taking half of this result would give the desired result, 90. This sophisticated innovation may be entirely internal, with only the final solution given and no account of the process. Although we can recognize number sense when we see it, the question as to how one's cognitive process completes individual tasks is less certain. It is similar to mathematicians' demands for valid proofs to be *rigorous,* though they are unable to adequately describe what is meant by *rigor.*

There are several factors regarding the development of number sense that mathematics educators have come to agree upon from empirical research during the 1990s. Results from Paul Cobb et al., Judith Sowder, Sharon Griffin and Robbie Case, and

**FIGURE 1**

Eddie Gray and David Tall have provided a more clearly agreed upon framework regarding advantageous skills for building number sense. Sowder notes that computational estimation and mental computation are important links to building number sense. Both Cobb et al. and Greeno state that both the use of mental models and creating a conceptual environment are necessary facilitators to make these links. Educators see a necessity to incorporate rich examples that guide students toward conceptual understandings, instead of superficial procedures that are not considered malleable. Developing mental models and utilizing mental computation are increasingly considered vital skills in mathematics; however research about reasoning with mental models is in a preliminary state.

### Current Trends and Their Effects on Mathematics Education

The twentieth century saw its share of reforms in mathematics pedagogy–from the algorithmic framework of connectionist theory attributed to Edward L. Thorndike to the axiomatic formalization of modern mathematics pursued by Bourbaki (a pseudonym taken by a group of French mathematicians) and Piaget's constructivist theory, which dominated the second half of the century and emphasized individuals as constructing their own knowledge through a process of abstraction, generalization, and concept formation. The concern in the 1990s surrounding a superficial (or merely procedural) understanding of mathematics with a lack of conceptual understanding was the catalyst that galvanized a push toward interpreting mathematics not as rote and memorization, but as problem solving, intuitive reasoning, and pattern recognition. The concept of number sense sprang forth from these shifts in philosophy regarding mathematics education. With this shift, a question arises: What significance does number sense have on mathematics education and pedagogy?

Another difficulty in encapsulating pedagogy that develops number sense stems from the fact that most mathematicians fail to recognize their own number sense and how they employ it. Their ability to move beyond procedures and definitions into the realm of concepts is rarely a conscious process. To a mathematician, the act is incorporated into their thinking process such that its nature becomes an involuntary action, like blinking or breathing. The mathematical paradox of striving for efficiency, both in notation and procedures, can oftentimes add to a lack of understanding for the student. To communicate efficiently, all those involved must be fluent in the language of mathematics.

Obviously, some students are successful in mathematics regardless of the pedagogical approach used. If this were not the case, the explosion within the new fields of mathematics that occurred after 1950 would not have occurred. Philip Davis and Reuben Hersh attest that more than half of all of mathematics was discovered after World War II. The question, then, is what percentage of those who completed traditional education were finding this success. Gray and Tall speculate that only 30 percent of students were able to develop an intuitive grasp of mathematics and higher-order thinking with previous pedagogies. So what about the other 70 percent? The research by Paul Cobb et al., Sharon Griffin and Robbie Case, and others consider this a central focus of current pedagogical issues.

This search for conceptual understanding seems to be the focus of research and pedagogy in the beginning of the twenty-first century. Empirical evidence supports a curriculum that stresses practical, intuitive, and rich real-world examples within mathematics. The Rightstart project, developed by Case and Griffin in 1997, is one such example. Their research focused on children living in urban, low-income communities who were lagging behind their peers in terms of age-level mathematics abilities. After participating in forty twenty-minute sessions that incorporated numerical games and concrete materials (using thermometers, board games, and number lines) these children were propelled to the top of their class, and they maintained this placement over a longitudinal study lasting several years. This success was achieved by focusing on two main goals: (1) to help students to develop a set of symbolic states and operations that are intimately tied to real-world quantities, and (2) to develop students' explicit knowledge of notational systems in conjunction with their implicit and intuitive knowledge, thus ensuring that these two types of knowledge act as natural companions to each other. Both of these goals coincide with the parameters of developing number sense.

The mathematician Warren McCulloch (1965) once observed, "What is a number, that a man may know it, and a man, that he may know a number?" The answer to this question, which has been posed in various forms since antiquity, changes with the understanding of mathematics. Since the 1990s, mathematics educators have been researching how number sense ameliorates students' understanding of mathematics. Mathematics educators have embraced this shift toward a pedagogy that strives to merge intuition, formal notation, and conceptual understanding. Number sense helps students eschew the notion that mathematics is merely a collection of rules to memorize. Number sense fosters students' ability to make judgments about the reasonableness of solutions and to build on their intuitions and insights. Number sense helps convince students that mathematics makes sense.

** See also: ** Mathematics Education, Teacher Preparation; Mathematics Learning,

**Myths, Mysteries, AND Realities, Numeracy and Culture.**

*subentries on*## bibliography

Anghileri, Julia. 2000. *Teaching Number Sense.* London: Continuum.

Cobb, Paul; Wood, Terry; Yackel, Erna; Nicholls, John; Wheatley, Grayson; Trigatti, Beatriz; and Perlwitz, Marcella.1991. "Assessment of a Problem-Centered Second Grade Mathematics Project." *Journal for Research in Mathematics Education* 22 (1):3–29.

Davis, Philip, and Hersh, Reuben. 1981. *The Mathematical Experience.* Boston: Mariner Books.

DeHaene, Stanislas. 1997. *The Number Sense: How the Mind Creates Mathematics.* New York: Oxford University Press.

Gray, Eddie, and Tall, David. 1994. "Duality, Ambiguity, and Flexibility: A 'Proceptual' View of Simple Arithmetic." *Journal for Research in Mathematics Education* 25 (2):116–140.

Greeno, James G. 1991. "Number Sense as Situated Knowing in a Conceptual Domain. "*Journal for Research in Mathematics Education* 22 (3):170–218.

Griffin, Sharon, and Case, Robbie. 1997. "Rethinking the Primary School Math Curriculum: An Approach Based on Cognitive Science." *Issues in Education: Contributions from Educational Psychology* 3 (1):1–50.

Hiebert, James; and Lefevre, Patricia. 1986. "Conceptual and Procedural Knowledge in Mathematics: An Introductory Analysis." In *Conceptual and Procedural Knowledge: The Case of Mathematics,* ed. James Hiebert. Hillsdale, NJ:Erlbaum.

Markovits, Zvia, and Sowder, Judith. 1994. "Developing Number Sense: An Intervention Study in Grade 7." *Journal for Research in Mathematics Education* 25 (1):4–30.

McCulloch, Warren. 1965. *Embodiments of Mind.* Cambridge, MA: MIT Press.

McIntosh, Alistair; Reys, Barbara J.; and Reys, Robert E. 1992. "A Proposed Framework for Examining Basic Number Sense." *For the Learning of Mathematics* 12 (3): 2–8.

National Council of Teachers of Mathematics. 1989. *Curriculum and Evaluation of Standards for School Mathematics.* Reston, VA: National Council of Teachers of Mathematics.

Paulos, John Allen. 1988. *Innumeracy: Mathematical Illiteracy and Its Consequences.* New York: Vintage.

Piaget, Jean. 1965. *The Child's Conception of Number.* New York: Norton.

Resnick, Lauren B. 1989. "Defining, Assessing, and Teaching Number Sense." In *Establishing Foundations for Research on Number Sense and Related Topics: Report of a Conference,* ed. Judith T. Sowder and Bonnie P. Schappelle. San Diego, CA: San Diego State University Center for Research in Mathematics and Science Education.

Sowder, Judith T. 1992. "Estimation and Number Sense." In *Handbook of Research on Mathematics Teaching and Learning: A Project of the National Council of Teachers of Mathematics,* ed. Douglas A. Grouws. New York: Macmillan.

Sowder, Judith T. 1992. "Making Sense of Numbers in School Mathematics." In *Analysis of Arithmetic for Mathematics Education,* ed. Gaea Leinhardt, Ralph Putnam, and Rosemary Hattrup. Hillsdale, NJ: Erlbaum.

Chris Lowber

Teruni Lamberg

## NUMERACY AND CULTURE

In simple terms, *numeracy* can be defined as the ability to understand basic mathematical concepts and operations. Numeracy thus encompasses a wide range of topics, including formal symbolic mathematics, cultural practices, children's intuitions about mathematics, and everyday behaviors mediated by mathematics. There are different forms of numeracy, and their realization in various cultural contexts has commonly been called *ethnomathematics.* Researchers have explored how pedagogy can be changed to incorporate cultural practices related to numeracy.

### Numeracy in Cultural Context

All cultures have developed various representational systems that provide ways of thinking about quantitative information. Different systems highlight different aspects of knowing. For example, the Oksapmin of Papua New Guinea have a counting system that uses body parts to express numbers from one to twenty-seven. Though no base is used in this system, it is adequate when trading goods using a one-to-one correspondence. It is inadequate, however, for computing or counting objects beyond twenty-seven. A contrasting example is the enumeration system of many Asian languages that is congruent to the structure of base ten. Asian children using this system tend to recite more number names in correct sequence and show earlier mastery of place-value concepts (i.e., relations among number words, multi-digit numerals, and quantities) than children using less regular base-ten systems.

In addition to enumeration systems, cultures have developed representational systems for locating (geometry, navigation), measuring, designing (form, shape, pattern), playing (rules, strategies), and explaining (abstraction). These representational systems entail beliefs and values associated with numeracy, and they support numeracy activities using tools such as abaci, clocks, and digital computers. A broad array of human activities to which mathematical thinking is applied is thus interwoven with cultural artifacts, social conventions, and social interactions.

Cultural variations in mathematical behavior are also seen in the ways people use mathematical representations in their everyday activities. For example, children working as street vendors in Brazil were found to use different computational strategies when selling than when doing school-like problems. While selling they used such strategies as oral computation, decomposition, and repeated groupings, whereas when given school-like problems they used standard algorithms. These children were found to be much more accurate in the context of selling than in the school setting. Research in other domains, such as measurement and proportional reasoning, further confirms that informal mathematics can be effective and does not depend upon schooling for its development.

Mathematics is used in everyday life in pursuit of goals that differ from the goals of academic mathematics found in schools and universities. This type of mathematics is often referred to as *informal,* or *naive,* mathematics. Representational systems and practices deriving from everyday activities are modified as new goals emerge. Thus, everyday mathematics is an adaptive system that can be used to creatively meet new challenges. For example, as the Oksapmin became more involved with the currency system, their body counting system described earlier began to change toward a base system. Although knowledge acquired through informal experiences are often distinguished from school mathematics, skills developed in the informal domain can be used to address goals and practices in the school setting. For example, the most successful elementary students in Liberian schools combine the strategies from their indigenous mathematics with school algorithms. Use of informal mathematics in school settings, therefore, may be an effective way to help children learn school mathematics. Several authors have argued for building bridges between informal and formal mathematics.

Although most research on numeracy and culture has been done outside of the United States, understanding informal and everyday mathematics is important for educators in the United States for a number of reasons. According to the census conducted in October 1999, about 2.5 million foreign-born children came to U.S. schools, bringing with them different mathematics representational systems and associated computational skills. In addition, everyday mathematical activities and language repertoires for American children of different ethnic groups have been shown to differ both across groups and when compared to the school curriculum. In the case of some groups, such as Native Americans, mathematical reasoning derived from cultural traditions is distinct from that of the schools, posing major conceptual problems for these children in the regular school curriculum.

### Research on Curricular Change

A number of projects have attempted to make mathematics instruction more culturally relevant for groups of children who have traditionally underachieved in the U.S. school system. Mary Brenner has worked with teachers to improve mathematics teaching for Native Hawaiian children. She interviewed parents and children and observed children in everyday settings to determine what kinds of numerical skills children brought with them to school. At the kindergarten level, adapting the existing curriculum consisted of reordering topics to begin with counting and computation (areas of student strength), more use of the students' nonstandard dialect in mathematics lessons, and more emphasis upon hands-on and game-like activities. At the higher grade levels, adaptations focused more on including activities, such as a school store, that enabled students to move from informal mathematical activities to more standard mathematical practices.

Ethnographic research has revealed many mathematically rich activities in everyday adult life. Luis Moll and James Greenberg have developed a culturally relevant pedagogy for Latino students by building classroom activities from the "funds of knowledge" that are present in their family networks. Teachers and researchers worked together to plan lessons based upon ethnographic data, and they also invited parents to teach. New mathematics units, such as one involving candy making, were developed as the contexts for teaching specific mathematical ideas.

In a different approach, Jerry Lipka and Ester Ilutsik, who work with the Yup'ik people in Alaska, advocate giving the community control over the process of curriculum development. The goal is to make the schools a local institution, rather than having schools act as representatives of the dominant society. Researchers, Yup'ik teachers, and tribal elders have worked together to translate Yup'ik mathematical knowledge into a form that can be utilized in classrooms. Like the Funds of Knowledge project, this group has analyzed everyday adult activities, such as fish camps, to understand the culturally relevant mathematics. In addition, they have worked to better understand the Yup'ik number system and how it can be used in the classroom. The goal is to create an entire mathematics curriculum based upon Yup'ik culture, rather than adapting existing curricula.

Gloria Ladson-Billings has conducted research on culturally relevant mathematics instruction for African-American children. This research has highlighted a variety of attitudinal changes that teachers must make in their teaching, including expecting higher academic standards for students, emphasizing cultural competence, and instilling critical consciousness in students.

### Teaching and Cultural Context

Teaching is an inherently cultural activity; it is situated in a bed of routines, traditions, beliefs, expectations, and values of students, teachers, administrators, parents, and the public. Thus, the inclusion of cultural and everyday mathematical knowledge in school mathematics must take into account the school-based assumptions about the appropriate way to teach mathematics.

For example, cultural assumptions about effective ways to improve teaching in Japan include lifelong professional development activities carried out by ordinary teachers. Typically, a few teachers with similar goals and interests form a study group. They select a few lessons that need improvement and analyze what is and is not working in the current practice in terms of learning goals for students, students' misunderstandings, and use of activity. They gather information on the topics by reading about other teachers' ideas, as well as other sources of recommended practices. A revised lesson is then planned, and one of the teachers from the group implements it while the others observe and evaluate what is and is not working. This process of evaluation, planning, and implementation is repeated until a satisfactory lesson is crafted, and may consume an entire school year. This is in sharp contrast to the type of model lesson developed by expert teachers and handed down to ordinary teachers in the United States. This is also different from the "one-day, make-it-and-take-it" type professional development workshops often implemented in the United States–a practice whose long-term effectiveness is questionable.

### Ideas for Teachers

The importance of raising awareness of cultural diversity among teachers has been extended to the teaching of mathematics in the United States. For example, in the mid-1990s a task force for the National Council of Teachers of Mathematics recommended the publication of a series, *Changing the Faces of Mathematics,* in order to help make the slogans of *Mathematics for All* and *Everybody Counts* real. Particular efforts were made to focus on education of ethnic and cultural minority students. Included in this series are volumes on African-American perspectives, Latino perspectives, and Asian-American and Pacific Islander perspectives. Each of these volumes includes articles that discuss successful pedagogical strategies for culturally diverse groups of students. The volumes also feature articles that may help educators develop a deeper understanding of the cultural differences that influence classroom dynamics, behavior, and environment.

** See also: ** Mathematics Education, Teacher Preparation; Mathematics Learning,

**Learning Tools, Myths, Mysteries, AND Realities, Number Sense.**

*subentries on*## bibliography

Bishop, Alan. 1991. *Mathematical Enculturation: A Cultural Perspective on Mathematics Education.* Dordrecht, Netherlands: Kluwer.

Brenner, Mary E. 1998. "Adding Cognition to the Formula for Culturally Relevant Instruction in Mathematics." *Anthropology and Education Quarterly* 29:214–244.

Brenner, Mary E. 1998. "Meaning and Money." *Educational Studies in Mathematics* 36:123–155.

Edwards, Carol A., ed. 1999. "Perspectives on Asian Americans and Pacific Islanders." In *Changing the Faces of Mathematics,* ed. Walter G. Secada. Reston, VA: National Council of Teachers of Mathematics.

Gallimore, Ronald. 1996. "Classrooms Are Just Another Cultural Activity." In *Research on Classroom Ecologies: Implications for Inclusion of Children with Learning Disabilities,* ed. Deborah L. Speece and Barbara K. Keogh. Mahwah, NJ: Lawrence Erlbaum.

Gay, John, and Cole, Michael. 1967. *The New Mathematics and an Old Culture.* New York: Holt, Rinehart and Winston.

Hiebert, James, and Stigler, James W. 2000. "A Proposal for Improving Classroom Teaching: Lessons from the TIMSS Video Study." *Elementary School Journal* 101:3–20.

Ladson-Billings, Gloria. 1995. "Making Mathematics Meaningful in Multicultural Contexts." In *New Directions for Equity in Mathematics Education,* ed. Walter G. Secada, Elizabeth Fennema, and Lisa Byrd Adajian. Cambridge, Eng.: Cambridge University Press.

Lipka, Jerry, and Ilutsik, Esther. 1995. "Negotiated Change: Yup'ik Perspectives on Indigenous Schooling." *Bilingual Research Journal* 19:195–207.

Miller, Kevin F.; Smith, Catherine M.; Zhu, Jianjun; and Zhang, Houcan. 1995. "Pre-school Origins of Cross-National Differences in Mathematical Competence." *Psychological Science* 6:56–60.

Miura, Irene T.; Okamoto, Yukari; Kim, Chung-soon C.; Steere, Marcia; and Fayol, Michel.1993. "First Graders' Cognitive Representation of Number and Understanding of Place Value: Cross-National Comparisons: France, Japan, Korea, Sweden, and the United States." *Journal of Educational Psychology* 85:24–30.

Moll, Luis C., and Greenberg, James B. 1990. "Creating Zones of Possibilities: Combining Social Contexts." In *Vygotsky and Education: Instructional Implications and Applications of Sociohistorical Psychology,* ed. Luis C. Moll. Cambridge, Eng.: Cambridge University Press.

Nunes, Terezinha; Schliemann, Analucia D.; and Carraher, David. W. 1993. *Street Mathematics and School Mathematics.* Cambridge, Eng.: Cambridge University Press.

Ortiz-Franco, Luis; Hernandez, Norma G.; and De La Cruz, Yolanda, eds. 1999. "Perspectives on Latinos." In *Changing the Faces of Mathematics,* ed. Walter G. Secada. Reston, VA: National Council of Teachers of Mathematics.

Pinxten, Rik. 1997. "Applications in the Teaching of Mathematics and the Sciences." In *Ethnomathematics,* ed. Arthur B. Powell and Marilyn Frankenstein. Albany, NY: State University of New York.

Saxe, Geoffrey B. 1982. "Developing Forms of Arithmetic Operations among the Oksapmin of Papua New Guinea." *Developmental Psychology* 18:583–594.

Secada, Walter G. 1992. "Race, Ethnicity, Social Class, Language, and Achievement in Mathematics." In *Handbook of Research on Mathematics Teaching and Learning,* ed. Douglas A. Grouws. New York: Macmillan.

Strutchens, Marilyn E.; Johnson, Martin L.; and Tate, William F., eds. 2000. "Perspectives on African Americans." In *Changing the Faces of Mathematics,* ed. Walter G. Secada. Reston, VA: National Council of Teachers of Mathematics.

### internet resource

U.S. Census Bureau. 2001. "School Enrollment in the United States–Social and Economic Characteristics of Students." <www.census.gov/prod/2001pubs/p20-533.pdf>

Yukari Okamoto

Mary E. Brenner

Reagan Curtis

## WORD-PROBLEM SOLVING

A *word problem* is a verbal description of a problem situation wherein one or more questions are posed, the answers to which can be obtained by the application of mathematical operations to information (usually numerical data) available in the text. In its most typical form, a word problem describes the essentials of some situation assumed to be familiar to the solver. Within the text, certain quantities are explicitly given, while others are not. The student is required to give a numerical answer to a stated question by making exclusive use of the quantities given–and of the mathematical relationships between these quantities. Simple examples include: "Pete wins 3 marbles in a game and now has 8 marbles. How many marbles did he have before the game?" and "One kilogram of coffee costs 12 euros. Susan buys 0.75 kilogram of coffee. How much does she have to pay?"

Despite its label, a word problem need not constitute a *problem* in the cognitive-psychological sense of the word–higher-order thinking going beyond the application of a familiar routine procedure is not necessarily required. Indeed, in typical elementary mathematics instruction, many word problems provide thinly disguised practice in adding, subtracting, multiplying, or dividing.

### Structural Dimensions of Word Problems

Several structural dimensions can be distinguished in word problems that affect their difficulty and how they are solved:

*Mathematical structure,*which includes the nature of the given and unknown quantities of the problem, and the mathematical operations by which the unknowns can be derived from the givens.*Semantic structure,*which includes the ways in which an interpretation of the text points to particular mathematical relationships. For example, addition or subtraction is indicated when the text implies a combination of disjoint subsets into a superset, a change from an initial quantity to a subsequent quantity by addition or subtraction, or the additive comparison between two collections.*Context,*meaning the nature of the situation described. For example, an additive problem involving combination of disjoint sets might deal with physically combining collections of objects or with conceptually combining collections of people in two locations.*The format,*meaning how the problem is formulated and presented. Format involves such factors as the placement of the question, the complexity of the lexical and grammatical structures, the presence of superfluous information, and so on.

Over several decades, numerous studies have analyzed the role of these task variables on the difficulty of problems, on the kind of strategies students use to solve these problems, and on the nature of their errors, particularly for simple word problems involving addition and subtraction or multiplication and division.

### Roles of Word Problems

Why does school mathematics include word problems? Perhaps simply because they are there, and have been for many centuries. Indeed, their role in mathematics education dates back to antiquity; the oldest known being in Egyptian papyri dating from 2000 B. c.e., with strikingly similar examples in ancient Chinese and Indian manuscripts. The following example is from the first printed mathematical textbook, a Treviso arithmetic of 1478: "If 17 men build 2 houses in 9 days, how many days will it take 20 men to build 5 houses?"

Despite this striking continuity across time and cultures, until recently there was little explicit discussion of why word problems should be such a prominent part of the curriculum, or of the variety of purposes behind their inclusion. Some have a puzzle-like nature and act as "mental manipulatives" (Toom, p. 36) to guide thinking within mathematical structures. Such problems are intended to train students to think creatively and develop problem-solving abilities. By contrast, the type mainly used educationally consists of a text representing (at least putatively) a real-world situation in which the derived answer would "work." Ostensible goals for the use of this type include offering practice for the situations of everyday life in which the mathematics learned will be needed, thereby showing students that the mathematics they are learning will be useful.

### Apparent Suspension of Sense-Making

In recent years, the characteristics, use, and rationale of word problems have been critically analyzed from multiple perspectives, including linguistic, cultural, and sociological perspectives. In particular, it has been argued by many mathematics educators that the stereotyped and artificial nature of word problems typically represented in mathematics textbooks, and the discourse and activity around these problems in traditional mathematics lessons, have detrimental effects. Many observations have led to the conclusion that children answer word problems without taking into account realistic considerations about the situations described in the text, or even whether the question and the answer make sense. The most dramatic example comes from French researchers who posed children nonsensical questions such as: "There are 26 sheep and 10 goats on a ship. How old is the captain?" It was found that the majority of students were prepared to offer an answer to such questions. In another study, thirteen-year-old students in the United States were asked the following question: "An army bus holds 36 soldiers. If 1,128 soldiers are being bussed to their training site, how many buses are needed?" The division was correctly computed by 70 percent of the students to get a quotient of 31 and remainder 12–but only 23 percent gave the appropriate answer, "32 buses." Nineteen percent gave the answer as "31 buses" and 29 percent gave the answer as "31, remainder 12."

To explain the abundant observations of this "suspension of sense-making" when doing word problems, it has been suggested by Erik De Corte and Lieven Verschaffel that the practice surrounding word problems is controlled by a set of (largely implicit) rules that constitute the "word-problem game." These rules including the following assumptions: (1) every problem presented by the teacher or in a textbook is solvable and makes sense; (2) there is only one exact numerical correct answer to every word problem; and (3) the answer must be obtained by performing basic arithmetical operations on all numbers stated in the problem.

### Reconceptualizing Word Problems as Modeling Exercises

One reaction to criticisms of traditional practice surrounding word problems in schools is to undermine the approach that allows students to succeed using superficial strategies based on the "rules." This is done by breaking up the stereotypical nature of the problems posed. For example, by including problems that do not make sense or contain superfluous or insufficient data, students can be guided to interpret word problems critically.

A more radical suggestion is to treat word problems as exercises in mathematical modeling. The application of mathematics to solve problem situations in the real world, termed *mathematical modeling,* is a complex process involving several phases, including understanding the situation described; constructing a mathematical model that describes the essence of the relevant elements embedded in the situation; working through the mathematical model to identify what follows from it; interpreting the computational work to arrive at a solution to the problem; evaluating that interpreted outcome in relation to the original situation; and communicating the interpreted results.

This schema can be used to describe the process of solving mathematical word problems as application problems. In the simplest cases, situations may be directly modeled by addition, subtraction, multiplication, or division, and children need to learn the variety of prototypical situations that fit unproblematically onto these operations. In other cases, the modeling is not so straightforward if serious attention is given to the reality of the situation described. In the example from the Treviso arithmetic, attention would be drawn to the assumptions that under-pin an answer based on direct proportionality–and to the fact that the answer thus derived would at best provide a rough approximation in the real situation. In the bus problem, the "raw" result of the computation has to be appropriately refined in the context of the situation described.

### Reforming the Teaching of Word Problems

In line with the above criticisms and recommendations with respect to the traditional practice surrounding word problems in schools, researchers have set up design studies to develop, implement, and evaluate experimental programs aimed at the enhancement of strategies and attitudes about solving mathematical word problems. In these studies, positive outcomes have been obtained in terms of both outcomes (test scores) and underlying processes (beliefs, strategies, attitudes). Characteristics common to such experimental programs include:

- The use of more realistic and challenging tasks than traditional textbook problems.
- A variety of teaching methods and learner activities, including expert modeling of the strategic aspects of the competent solution process, small-group work, and whole-class discussions.
- The creation of a classroom climate that is conducive to the development in pupils of an elaborated view of mathematical modeling, and of the accompanying beliefs and attitudes.

To some extent, these characteristics of a new approach to word-problem solving are beginning to be implemented in mathematical frameworks, curricula, textbooks, and tests in many countries. Much remains to be done, however, to align the teaching of word problems with widely accepted principles that children should make connections between mathematics and their lived experience–and that mathematics should make sense to them.

** See also: ** Mathematics Education, Teacher Preparation; Mathematics Learning,

**Complex Problem Solving, Learning Tools, Myths, Mysteries, and Realities, Number Sense.**

*subentries on*## bibliography

Carpenter, Thomas P.; Lindquist, Mary M.; Matthews, Walter; and Silver, Edward A.1983. "Results of the Third NAEP Mathematics Assessment: Secondary School." *Mathematics Teacher* 76:652–659.

Cognition and Technology Group at Vanderbilt. 1997. *The Jasper Project: Lessons in Curriculum, Instruction, Assessment, and Professional Development.* Mahwah, NJ: Erlbaum.

De Corte, Erik, and Verschaffel, Lieven. 1985. "Beginning First Graders' Initial Representation of Arithmetic Word Problems." *Journal of Mathematical Behavior* 4:3–21.

Fuson, Karen. 1992. "Research on Whole-Number Addition and Subtraction." In *Handbook of Research on Mathematics Teaching and Learning,* ed. Douglas A. Grouws. New York: Macmillan.

Greer, Brian. 1992. "Multiplication and Division as Models of Situations." In *Handbook of Research on Mathematics Teaching and Learning,* ed. Douglas A. Grouws. New York: Macmillan.

Lave, Jean. 1992. "Word Problems: A Microcosm of Theories of Learning." In *Context and Cognition: Ways of Learning and Knowing,* ed. Paul Light and George Butterworth. New York: Harvester Wheatsheaf.

Nesher, Pearla. 1980. "The Stereotyped Nature of School Word Problems." *For the Learning of Mathematics* 1:41–48.

Reusser, Kurt. 1988. "Problem Solving Beyond the Logic of Things: Contextual Effects on Understanding and Solving Word Problems." *Instructional Science* 17:309–338.

Schoenfeld, Allen H. 1991. "On Mathematics as Sense-Making: An Informal Attack on the Unfortunate Divorce of Formal and Informal Mathematics." In *Informal Reasoning and Education,* ed. James F. Voss, David N. Perkins, and Judith W. Segal. Hillsdale, NJ: Lawrence Erlbaum.

Toom, AndrÉ. 1999. "Word Problems: Applications or Mental Manipulatives." *For the Learning of Mathematics* 19 (1):36–38.

Verschaffel, Lieven; De Corte, Erik; Lasure, Sabine; Van Vaerenbergh, Griet; Bogaerts, Hedwig; and Ratinckx, Elie. 1999. "Design and Evaluation of a Learning Environment for Mathematical Modeling and Problem Solving in Upper Elementary School Children." *Mathematical Thinking and Learning* 1:195–229.

Verschaffel, Lieven; Greer, Brian; and De Corte, Erik. 2000. *Making Sense of Word Problems.* Lisse, Netherlands: Swets and Zeitlinger.

Lieven Verschaffel

Brian Greer

Erik De Corte

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mathematics. For the original article on Bourbaki see DSB, vol. 2.
Bourbaki was a pseudonym adopted in 1934 by a group of young Fre… David Hilbert , Hilbert, David
Hilbert, David
(b. Königsberg, Germany [now Kaliningrad, R.S.F.S.R.], 23 January 1862; d. Göttingen, Germany, 14 February 1943)
mathem…

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