Learning and Teaching Mathematics
Major theoretical approaches to learning and teaching mathematics have been developed from three psychological perspectives of human learning. The social construc-tivist approach to mathematical learning emphasizes classroom learning as a process of both individual and
social construction. Critical to this framework is the need for mathematics teachers to construct a form of practice that fits with their students' ways of learning mathematics (Wood, Cobb, & Yackel, 1995). “The most basic responsibility of constructivist teachers is to learn the mathematical knowledge of their students and how to harmonize their teaching methods with the nature of that mathematical knowledge” (Steffe & Wiegel, 1992, p. 17). This is the planning whereby teachers plant powerful mathematical ideas in a personally meaningful context for students to investigate. Cobb, Wood, and Yackel (1993) further elaborate on teachers' responsibility in the mathematics classroom as playing the dual role of fostering the development of conceptual knowledge among students and facilitating the constitution of what is often referred to as taken-as-shared knowledge in the classroom community. This is the teaching in which, without direct access to one another's understanding, members of the classroom community achieve through social interaction a sense of some aspects of knowledge being shared, promoted by classroom social norms understood by members as constituting effective participation in the mathematics classroom community.
The underlying metaphor of social constructivism is persons in conversation, which highlights the critical role of language (see Ernest, 1996). For example, when teaching the theorem that the sum of the three inner angles of a triangle is 180 degrees, a teacher practicing social constructivism would present the idea of moving the three angles together to calculate the sum and would then have students talk about or share their ideas about moving angles. Manipulatives would be provided to students so that they could take a triangle apart and experiment with different ways of making three angles come together. Students would find that many of their ideas about moving angles would produce three angles forming a straight line indicating 180 degrees. This physical construction would then be followed by another “talk point” for students promoted by the teacher about how to move angles mathematically for geometric proof. Students, now with papers and pencils, would engage in continuing conversation about moving angles mathematically with the main idea being creating equal angles in different places in order to pull three angles together (the mathematical construction would refer frequently to the physical construction). The language elements are always critical during the whole process from moving angles physically to moving angles mathematically.
The cognitive science approach to mathematical learning emphasizes the nature of knowledge representation (this separates cognitive psychologists from constructivist theorists). Representation in mathematical learning can take on different forms: cognitive (the internal representation of a learner's knowledge), mathematical (the mathematical representation of a mathematical structure), symbolic (the external representation of a mathematical notation), explanatory (the model and theory developed to account for cognitive structures and processes), and computational (the hypothetical mental representation in computer programs based on observed human behaviors) (see English & Halford, 1995). These representations provide a way to construct cognitive models of mental structures of mathematical knowledge and cognitive processes of mathematical learning. Cognitive processes operate on cognitive representations (mental structures) that are viewed as a network of interrelated mathematical ideas.
The quality of mathematical learning is measured through the number of connections that learners can make among mental representations and the strength of each connection (Hiebert & Carpenter, 1992). The notion of connected representation “provides an effective link between theoretical cognitive issues and practical classroom issues” (English & Halford, 1995, p. 13). Cognitive science has proven to be the most effective way to investigate problem-solving behaviors in mathematics. In addition to solid mathematical knowledge, successful problem solving in mathematics is found to require “a repertoire of general problem-solving heuristics” that remind mathematics educators of the classic work of Polya (1957) (English & Halford, 1995, p. 14).
For example, English and Halford (1995) argued that “a mental model of the relations between numbers themselves” is needed to truly understand numbers (p. 60). The mental model starts with a correspondence between a number and a set of objects (e.g., 5 corresponds to five apples). Relations between numbers can then correspond to relations between sets. Because the set of five apples has more objects (members) than the set of three apples, 5 is larger than 3. Students establish the mental model of number by experimenting with sets of objects to reinforce the correspondence between number relations and set relations. Without this mental model, most students would have to rely on the succession relation to understand numbers, with rote learning emphasizing the counting order of the numbers. Obviously, the notion that 5 is larger than 3 because 5 comes after 3 is at best vague and fuzzy to most students.
The sociocultural approach to mathematical learning emphasizes the effort to situate mathematical ideas within culturally organized activities. Because education is a process of enculturation (or socialization), it is important for learners to engage in social interaction with more knowledgeable experts in what Lev Vygotsky called the zone of proximal development and for teachers to use culturally developed sign systems and culturally appropriate artifacts as psychological tools for instruction (Vygotsky, 1978). “The qualities of thinking [and learning] are actually generated by the organizational features of the social interaction” (van Oers, 1996, p. 93). Internalization is another central notion in Vygotsky's cultural-historical theory of human development which “appears first between people as an intermental category and then within the child as an intramental category” (Vygotsky, 1960, p. 197–198). Particularly relevant to mathematics education is the notion that “learning is the initiation into a social tradition [of mathematical inquiry, mathematical discovery, mathematical argument, and so on]” (Solomon, 1989, p. 150).
With enculturation as the emphasis, a teacher who employs the sociocultural approach to teach mathematics would design a learning task through which students can interact with experts. Simply put, it is a process of guided participation and interaction of learning by solving problems just beyond a student's current capability with the help of a more expert other via scaffolding. Experts do not necessarily imply mathematicians. Teachers are most often the experts in real classrooms, and continuing professional development strengthens their expertise in mathematics. In addition, teaching assistants, advanced peers, and even parents can all be trained to become mathematics experts in real classrooms.
For the theorem that the sum of the three inner angles of a triangle is 180 degrees, experts would ask well-thought questions around the main idea of creating equal angles in different places so that the three angles can be pulled together to examine the sum. These questions prepare students for the demonstration of experts who would show students how to move one angle (both physically and mathematically), would have students comment on the move, and would ask students to move the other angles following a similar strategy. Observing students closely, experts would alert students of wrong moves, would point out reasons for wrong moves, and would demonstrate correct moves repeatedly. This modeling and coaching process between experts and students, often considered apprenticeship in nature, is the very foundation of the sociocultural approach that requires social participation and interaction of the whole classroom community. Finally, experts would evaluate with students on different strategies of moving angles and would reason with them on the best way to move angles. The whole interaction between experts and students would end up with the establishment (or enculturation) of an important mathematical tradition (or practice) on the use of auxiliary lines for geometric proof. As a result, experts would have trained students on geometric proof in a way very similar to how masters would have trained apprentices in, say, furniture building.
“Students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge” (National Council of Teachers of Mathematics [NCTM], 2000, p. 11). The mathematics education research community has been fairly united in identifying understanding as the major learning goal for mathematics education—teaching mathematics for understanding. Simon (2006) stated:
Recent discourse in mathematics education has coalesced around the importance of focusing on and fostering students' mathematical understanding. This agreement among mathematics educators has led to a commitment to generate new learning goals for students that are less skewed in favor of skill and facts learning and more focused on student thinking. (p. 359)
Although mathematics education has a long history, teaching mathematics for understanding is a fairly new notion.
Believing that two effectively different subjects are taught under the same name called mathematics, Richard Skemp published his classic paper on mathematical understanding in 1976. He distinguished between instrumental understanding (knowing how) and relational understanding (knowing both how and why). According to Skemp (1976), there are three advantages in instrumental understanding of mathematics: (a) easier to understand, (b) more immediate and more apparent rewards, and (c) quicker in getting correct answers. But the metaphor for instrumental understanding is getting an instruction to go to a certain place (i.e., still not knowing how to go to any new place). There are four advantages in relational understanding of mathematics: (a) more adaptable to new tasks, (b) easier to remember, (c) effective as a goal in itself, and (d) organic in quality. The metaphor for relational understanding is getting a map to go to a certain place (i.e., knowing how to go to any new place).
It is common among teachers to use rhymes to help students understand mathematical properties. Such an effort in many cases results in instrumental understanding. For example, many teachers use the following rhyme to help students understand (or memorize) the order of number operation: Please (parenthesis) Excuse (exponential) My (multiplication) Dear (division) Aunt (addition) Sally (subtraction). Many students who have gained instrumental understanding through this teaching practice would mistakenly believe that, for example, they should do addition before subtraction because of the order shown in the rhyme. This rhyme does not illustrate the fact that there are equivalent levels of operation. The order of operation does not matter between multiplication and division. Neither does it between addition and subtraction.
In contrast to instrumental understanding, relational understanding aims to make students know the very reason behind every mathematical action (or manipulation). For example, when solving the equation 2X–4 = 8, for X, a teacher adopting instrumental understanding would procedurally have students move the 4 from the left side of the equation to the right side. Their instrumental understanding also reminds them to change the sign of the number (in this case from negative to positive): 2X = 8 + 4 or 2X = 12. However, many students have no understanding of why this rule governs this mathematical action. Instead of having students learn this procedural rule, a teacher adopting relational understanding would emphasize properties of equation operation, such as the one that adding the same number to both sides of an equation keeps equality. Therefore, students with relational understanding would perform: 2X–4 + 4 = 8 + 4 resulting in 2X = 12.
Instrumental understanding produces procedural knowledge, whereas relational understanding produces conceptual knowledge. Hiebert and Lefevre (1986) defined conceptual knowledge as “knowledge that is rich in relationships … a network in which the linking relationships are as prominent as the discrete pieces of information” (pp. 3–4, cited in Simon, 2006). This is the reason why effective classroom teaching for mathematical understanding often emphasizes one key aspect of understanding: connection.
Researchers have increasingly recognized the importance of affect-related factors in mathematical learning and teaching. Stuart (2000) stated that “mathematics is like a sport: [performance is] 90 percent mental—one's mathematics confidence—and 10 percent physical— one's mathematics competence in performing mathematical skills” (p. 330–331). After a review of studies, Eynde, de Corte, and Verschaffel (2002) noted that both affective and cognitive factors play a key role as the constituting elements of the learning process:
Motivation and volition (i.e., the conative factors) are no longer seen as just the fuel or the engine of the
learning process, but are perceived as fundamentally determining the quality of the learning. In a similar way, self-confidence and positive emotions (affective factors) are no longer considered as just positive side effects of learning, but become important constituent elements of learning and problem solving. (p. 14)
NCTM has been highlighting five new goals for mathematical learning and teaching since 1989 when it published the first-ever curriculum standards. Two of the five new goals are affective in nature: (a) students learn to value mathematics, and (b) students become confident in their ability to do mathematics.
As early as 1985, Slavin and Karweit stated that “one of the most troublesome and enduring problems of mathematics instruction is accommodating heterogeneity in student preparation and learning rate” (p. 351). The problem of classroom heterogeneity is much more pronounced in mathematics than in any other school subject because mathematics is structured in the most highly sequential fashion among all school subjects—access to and performance in mathematics courses are determined by prior success in particular courses often referred to as prerequisites that systematically regulate student performance and progress in mathematics (see Oakes, 1990). This means that mathematical problems and difficulties accumulate, and the degree of classroom heterogeneity enlarges over time.
Traditionally, there are two ways to deal with the problem of classroom heterogeneity. One is ability grouping at various levels: (a) grouping students within a class (e.g., regular mathematics groups and advanced mathematics groups), (b) grouping classes within a school (e.g., curriculum tracking and curriculum placement), and (c) grouping schools within a district (e.g., special education programs and magnet programs for gifted students). The other prevalent way to accommodate classroom heterogeneity is individualized instruction in which students work on learning materials at their own level and rate with frequent assistance from teachers.
NCTM (1989) introduced another means to accommodate diverse learning needs: content differentiation. This concept emphasizes that the depth to which a topic is explored should depend on the level of abstraction at which students are capable or operating. To apply this concept, classroom teachers make concrete examples and applications of a topic open to all students to learn but make higher levels of abstraction and generalization available to, but not required of, all students. Classroom heterogeneity remains a major challenge in mathematics instruction today because the effectiveness of ability group and individualized instruction has been challenged, and the concept of content differentiation is difficult to implement even for experienced teachers.
A related, but largely isolated, challenge in mathematics instruction is the presence, in some cases substantial, of English as a second language (ESL) students. ESL students face enormous difficulties in mathematical learning because both mathematics and the language that carries it are foreign to them. NCTM (1994) clarifies that all students, regardless of their language or cultural background, must study a core curriculum in mathematics based on its curriculum standards. In 1991 NCTM recommended five major changes in mathematics classroom to help ESL students succeed in mathematics: (a) selecting mathematics tasks that engage students' interests and intellect, (b) orchestrating classroom discourse in ways that promote the investigation and growth of mathematical ideas, (c) using, and helping students use, technology and other tools to pursue mathematical investigations; (d) seeking, and helping students seek, connections to previous and developing knowledge; and (e) guiding individual, small-group, and whole-class work (students benefit from a variety of instructional settings in the classroom).
Various types of instructional technology (IT) have been developed to enhance mathematical learning and teaching. Lou, Abrami, and d'Apollonia (2001) classified IT into five major categories according to its educational functions: (a) tutorial, (b) communication media, (c) exploratory environment, (d) tools, and (e) programming language (with limited application so far). Tutorial refers to programs that can directly teach mathematics by creating a stimulating environment in which information, demonstration, and practice are shared with students. Computer-assisted instruction (CAI) and various mathematics games (e.g. Math Blaster) are typical examples. Research points to CAI as a potentially effective mechanism for teaching students (including those with special needs), with increased mathematics performance as positive outcomes (e.g., Xin, 1999).
Communication media refers to communication tools such as email, computer-supported-collaborative learning (CSCL) systems, videoconferencing, and the Internet. These tools promote effective communication and information sharing. Multimedia program and videoconferencing have begun to emerge as potentially effective tools for mathematical learning and teaching (e.g., Irish, 2002; SBC Knowledge Ventures, 2005). For example, videoconferencing is found to help meet state and national curriculum standards, reach new heights in staff professional development, and help students take classes not offered at their school. In particular, students involved in the operation of the videoconferencing equipment benefit the most by having learned both subject contents and technical skills.
Exploratory environment seek to encourage active learning through discovery and exploration. Logo, simulations, and hypermedia-based learning are typical examples. One of the hypermedia-based learning programs in mathematics is the Adventures of Jasper Woodbury, a mathematics program developed at the Vanderbilt University and widely used around the world. Based on the theory of anchored instruction, the program uses video and multimedia computing technology to provide problem-scenarios to help students develop skills and knowledge for problem solving and critical thinking. Implementation of this program has yielded positive findings (e.g., Mushi, 2000; Shyu, 1999).
Tools serve the technological purpose of making learning and teaching attractive, effective, and efficient. Word processors, PowerPoint, spreadsheet, Geometer's Sketchpad, data-analysis software, and various virtual manipulatives are typical examples. In mathematics classrooms, particularly at the elementary level, manipulatives are used extensively to help students build a foundation for understanding abstract mathematical concepts. Accessed via the Internet, virtual manipulatives are replicas of real manipu-latives and capable of connecting dynamic visual images with abstract symbols (a limitation of regular manipulatives). A variety of studies have examined the use of virtual manipulative tools in mathematics classrooms and have found positive effects on student mathematics achievement and attitude toward mathematics (e.g., Moyer & Bolyard, 2002; Reimer & Moyer, 2005; Suh, Moyer, & Heo, 2005).
As an emerging concept, classroom assessment is different from student assessment in that the goal of classroom assessment is to understand learners' learning in order to improve teachers' teaching. Based on results of a comprehensive review of empirical research on whether classroom assessment can benefit learning (in mathematics, science, and English) by Black and Wiliam (1998), Har-len and Winter (2004) offered this summary:
[There is] convincing evidence that classroom assessment raises students' attainment when it has these key characteristics: that information is gathered about the processes and products of learning and is used to adapt teaching and learning; that learners receive feedback that enables them to know how to improve their work and take forward their learning; that teachers and learners share an understanding of the goals of particular pieces of work; that learners are involved in assessing their work (both self- and peer-assessment); that pupils are actively involved in learning rather than being passive recipients of information. (p. 390)
Some developments have also occurred in student assessment. There have been calls for traditional large-scale assessments not only to measure (or assess) but also improve (or support) student mathematical learning. Chudowsky and Pellegrino (2003) argued that large-scale assessments can and should do a much better job of “gauging student learning, holding education systems accountable, signaling worthy goals for students and teachers to work toward, and providing useful feedback for instructional decision making” (p. 75). As one of the efforts, large-scale tests are moving toward a closer alignment with mathematics curriculum and instruction.
Portfolios are emerging as a popular alternative student assessment. A portfolio is a purposeful collection of a students' work as indicators of their effort, progress, and achievement over time to gauge their cognitive and affective development. For mathematical learning, the purpose of a portfolio is to understand “student thinking, student's growth over time, mathematical connections, student views of themselves as mathematicians, and the problem solving process” (Stenmark, 1991, p.37). However, there ought to be a note of caution on alternative student assessments. Herman and Winters (1994) discussed the lack of empirical evidence on claims of advantages of portfolios over traditional student assessments. Carney (2001) noted that the research literature on portfolios did not change much from 1994 to 2001. An inspection of the leading Journal for Research in Mathematics Education shows again the lack of empirical research on all forms of alternative assessments since 2001.
After an evaluation of current mathematics curriculum projects based on his Curriculum Research Framework, Clements (2004) ranked the Realistic Mathematics Education (RME), the Investigations in Number, Data, and Space (Investigations), Everyday Mathematics, and the Connected Mathematics Project (CMP) as (large-scale) research-based curriculum projects in mathematics education. The RME actually is a learning and teaching theory about mathematics education, developed by the Freudenthal Institute in the Netherlands. Freudenthal (1991) views mathematics as connected to reality and as a human activity. “Realistic” refers not only to the connection of mathematics with the real world and everyday life but also to problems real in mind of students. The RME organizes mathematics education into a process of guided reinvention to allow students to experience the similarity between the process by which mathematics is learned and the process by which mathematics is invented. “Invention” emphasizes the need to develop steps in the learning process, and “guided” emphasizes the need to create an instructional environment for the learning process. The RME can be characterized as making good use of (a) contexts (phenomenological exploration), (b) models (instrument bridging), (c) productions and constructions from students (student contribution), (d) educational interactions (interactivity), and (e) intertwining (of various learning strands).
The Investigations is a complete K-5 reform-based mathematics curriculum developed under the leadership of Dr. Susan Russell at TERC in Cambridge, Massachusetts. It designs activity-based investigations to encourage students to think creatively, develop their own problem-solving strategies, and work cooperatively. In addition to talking, writing, and drawing about mathematics, students use manipulatives, calculators, and computers to explore mathematical concepts and procedures. Classroom assessment is embedded within each investigation for improvement in teaching. Studies are in favor of Investigations students in (a) straight calculation problems (basic facts and whole number operations), (b) understanding of number concepts and number relationships, (c) word problems and complex calculation problems, and (d) performance in a high-stakes standardized test (administered in Massachusetts). Studies also indicate that Investigations works well with students across all mathematics achievement levels.
Research and development of Dr. Max Bill and his colleagues during the 1980s and 1990s resulted in the University of Chicago School Mathematics Project. The major component, Everyday Mathematics, is a research-based K-6 curriculum, with the goal to significantly improve both curriculum and instruction for all students to learn mathematics. The project began with a research phase in which a thorough review was conducted of existing research on mathematics curriculum and instruction with an emphasis on mathematical thinking. This review study was accompanied by interviews of hundreds of K-3 children and surveys of curricular and instructional practices in other countries. The research phase eventually established several basic principles to guide the curriculum development, including that (a) students learn mathematics from their own experiences obtained in an environment in which mathematics is rooted in real life, learners are actively involved in learning, and teachers provide learners with rich and meaningful mathematical contexts; (b) K-6 mathematics curriculum should build on an intuitive and concrete foundation, gradually shifting children to an abstract and symbolic foundation; and (c) teachers are the key factor in the success of any curriculum project, requiring adequate attention to their working lives.
Based on these principles, curriculum at each grade level went through a 3-year development cycle (1 year of writing, 1 year of extensive field testing, and 1 year of revising) with collaboration between mathematicians, education specialists, and classroom teachers. This unique development process has resulted in a comprehensive K-6 curriculum with a sequence of instruction carefully building upon and extending knowledge and skills obtained in the previous learning experience.
Led by Dr. Glenda Lappan at Michigan State University, the Connected Mathematics Project (CMP) is a research-based and field-tested middle school mathematics curriculum, different from many existing mathematics curricula because it takes a problem-centered curricular approach to develop powerful mathematical concepts, skills, and procedures with a focus on the ways of mathematical thinking and reasoning. CMP emphasizes the following principles reflecting both research and policy stances in mathematics education regarding what truly supports mathematical learning and teaching: (a) curriculum should be built around a number of “big” mathematical ideas, (b) underlying concepts, skills, or procedures should be in an appropriate developmental sequence; (c) an effective curriculum should be coherent connecting investigation to investigation, unit to unit, and grade to grade; (d) focus of classroom instruction should be inquiry and investigation of mathematical ideas embedded in rich problem situations, (e) mathematical tasks should be the primary vehicle for student engagement, (f) mathematical ideas should be explored deeply enough to enable students to make sense of them, (g) curriculum should develop student ability to reason effectively with information represented in multiple formats (graphic, numeric, symbolic, and verbal) and to interchange flexibly among these representations, and (h) curriculum should reflect the information-processing capability of technology and appreciate how technology is fundamentally changing the way that people learn and apply mathematics.
Most reforms in mathematics instruction can be reasonably characterized as standards-based (referring to the 1989 NCTM Standards) and inquiry-oriented (referring to the hands-on and minds-on discovery approach). Different instructional reforms employ different strategies to promote standards and discoveries in mathematics. Some of them are research-grounded reforms, such as QUASAR (Quantitative Understanding: Amplifying Student Achievement and Reasoning), a national project particularly aimed at improving mathematics instruction for socially disadvantaged middle school students (see Silver & Stein, 1996). The most influential of the research-grounded instructional reforms in mathematics may well be the Cognitively Guided Instruction (CGI). Based on their research, Dr. Thomas Carpenter and Dr. Elizabeth Fennema at the University of Wisconsin at Madison led the development of the CGI approach to teaching mathematics (e.g., Carpenter, Fennema, Franke, & Empson, 1999).
Essentially, CGI encourages mathematics teachers to utilize what they know about the mathematical understanding of their students to select or create mathematical problems, promote students to think about the problems, stimulate students with a well-designed sequence of questions to guide them gradually to approach or discover the solutions, and facilitate discussion and idea-sharing throughout the entire instruction. The CGI instruction basics for teachers can be roughly summarized as: (a) giving sufficient waiting or thinking time for students to respond, (b) allowing free use of all materials to assist problem solving, (c) creating context of problem solving relevant to students, (d) reading a problem multiple times as necessary, (e) focusing on thinking process rather than product, (f) asking for and probing into students' explanations to learn about their mathematical thinking, (g) analyzing wrong answers from students with the whole class, (h) sharing solutions and strategies with the whole class, (i) promoting students to compare solutions to identify best problem-solving strategies, and (j) delaying symbol manipulations.
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