# Mathematics Learning

## 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.

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MICHAEL T. BATTISTA

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