Using Stories to Teach Maths Ages 9 to 11 (Using Stories to Teach...)

Maths Stories

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The number names used in a language provide children with a readymade representation for number. Linguistic structure of number names. Names for numbers have been generated according to a bewildering variety of systems. These may be freely combined, with the place of a digit indicating the power of 10 that it represents. First, it is a widely used system for writing numbers.

Second, it is as consistent and concise as a base system could be. Box 5—1 shows how spoken names for numbers are formed in three languages: English, Spanish, and Chinese. All of these languages use a base system, but the languages differ in the clarity and consistency with which the base structure is reflected in the number names. As the first section of the figure shows, representations for numbers from 1 to 9 consist of an unsystematically organized list. There is no way to predict that 5 or five or wu come after 4, four , and si , respectively, in the Arabic numeral, English, and Chinese systems.

Names for numbers above 10 diverge in interesting ways among these different languages, as the second part of Box 5—1 demonstrates. The Chinese number-naming system maps directly onto the Hindu-Arabic number system used to write numerals. For example, a word-for-word translation of shi qi 17 into English produces ten-seven. English has unpredictable names for 11 and 12 that bear only a historical relation to one and two. The English names for numbers in the teens beyond 12 do have an internal structure, but it is obscured by phonetic modifications of many of the elements used in the first 10 numbers e.

Furthermore, the order of word formation reverses the place value, unlike the Hindu-Arabic and Chinese systems and the English system above 20 , naming the smaller value before the larger value. Spanish follows the same basic pattern for English to begin the teens, although there may be a clearer parallel between uno, dos, tres and once, doce, trece than between one, two, three and eleven, twelve, thirteen. The biggest difference between Spanish and English is that after 15 the number names in Spanish abruptly take on a different structure.

Thus the name for 16 in Spanish, diez y seis literally ten and six , follows the same basic structure as Arabic numerals and Chinese number names starting with the tens value and then naming the ones value , rather than the structures of the number names in English from 13 to 19 and the names in Spanish from 11 to 15 starting with the ones value and then naming the tens value. Above 20, all these number-naming systems converge on the Chinese structure of naming the larger value before the smaller one.

Despite this convergence, the systems continue to differ in the clarity of the connection between the decade names and the corresponding unit values. Chinese numbers are consistent in forming decade names by combining a unit value and the base ten. Decade names in English and Spanish generally can be derived from the name for the corresponding unit value, with varying degrees of phonetic modification e. Psychological consequences of number names. Although all the number-naming systems being reviewed are essentially base systems, they differ in the consistency and transparency with which that structure is reflected in the number names.

Several studies comparing English-. The relative complexity of English number names has other cognitive consequences. Speakers of English and other European languages face a complex task in learning to write Arabic numerals, one that is more difficult than that faced by speakers of Chinese. Speakers of languages whose number names are patterned after Chinese including Korean and Japanese are better able than speakers of English and other European languages to represent numbers using base blocks and to perform other place-value tasks.

When learning to count, children must acquire a combination of conventional knowledge of number names, conceptual understanding of the mathematical principles that underlie counting, and ability to apply that knowledge in solving mathematical problems. Language differences during preschool. In one study, for example, Chinese and American preschoolers did not differ in the extent to which they violated the previously discussed counting principles or in their ability to use counting to produce sets of a given size in the course of a game.

Nevertheless, these effects have implications for learning Arabic numerals and thus for understanding the principal symbol system used in school mathematics. As with other aspects of mathematics, counting requires combining a conceptual understanding of the nature of number with a fluent mastery of procedures that allow one to determine how many objects there are. When children can count consistently to figure out how many objects there are, they are ready to use counting to solve problems.

It also helps support their learning of conventional arithmetic procedures, such as those involved in computation with whole numbers. Preschool children bring a variety of procedures to the task of learning simple arithmetic. Most of these procedures begin with strategic application of counting to arithmetic situations, and they are described in the next section. As with the distinction between conceptual understanding and procedural fluency, this categorization is somewhat arbitrary, but it provides a good example of how children can build on procedures such as counting in extending their mathematical competence to include new concepts and procedures.

Strategic competence refers to the ability to formulate mathematical problems, represent them, and solve them. An important feature of mathematical development is the way in which situations that involve extended problem solving at one point can later be handled fluently with known procedures. Simple arithmetic tasks provide a good example. Most preschoolers show that they can understand and perform simple addition and subtraction by at least 3 years of age, often by modeling with real objects or thinking about sets of objects.

In one study, children were presented with a set of objects of a given size that were then hidden in a box, followed by another set of objects that were also placed in the box. The majority of children around age 3 were able to solve such problems when they involved adding and subtracting a single item, although their performance decreased quickly as the size of the second set increased. A wealth of strategies.

Much research has described the diversity of strategies that children show in performing simple arithmetic, from preschool well into elementary school. Some children will model the problem using available object or fingers; others will do it verbally. These strategies are discussed in detail in Chapter 6. Kindergartners use all of these strategies, and second graders use all of them except for counting all.

When 5-year-olds were given four individual sessions over 11 weeks in which they solved more than addition problems, most of them discovered the counting-on-from-larger strategy, which saves effort by requiring them to do less counting. They then were most likely to apply it to problems e. The diversity of strategies that children show in early arithmetic is a feature of their later mathematical development as well. In some circumstances the number of different strategies children show predicts their later learning.

Young children are able to make sense of the relationships between quantities and to come up with appropriate counting strategies when asked to solve simple word, or story, problems. Word problems are often thought to be more difficult than simple number sentences or equations. Young children, however, find them easier. If the problems pose simple relationships and are phrased clearly, preschool and kindergarten children can solve word problems involving addition, subtraction, multiplication, or division.

For example, if a picture of five birds and four worms is shown to preschoolers, most of them can answer the following: Will every bird get a worm? In addition to using counting to solve simple arithmetic problems, preschool children show understanding at an early age that written marks on paper can preserve and communicate information about quantity.

But they are less able to represent changes in sets or relationships between sets, in part because they fail to realize that the order of their actions is not automatically preserved on paper. Adaptive reasoning refers to the capacity to think logically about the relationships among concepts and situations and to justify and ultimately prove the correctness of a mathematical procedure or assertion.

Adaptive reasoning also includes reasoning based on pattern, analogy, or metaphor. Research suggests that young children are able to display reasoning ability if they have a sufficient knowledge base, if the task is understandable and motivating, and if the context is familiar and comfortable. Situations that require preschoolers to use their mathematical concepts and procedures in unconventional ways often cause them difficulty.

For example, when preschool children are asked to count features of objects e. A major challenge of formal education is to build on the initial and often fragile understanding that children bring to school and to make it more reliable, flexible, and general. Most preschool children enter school with an initial understanding of procedures e. In addition to the concepts and skills that underlie mathematical proficiency, children who are successful in mathematics have a set of attitudes and beliefs that support their learning. They see mathematics as a meaningful, interesting, and worthwhile activity; believe that they are capable of learning it; and are motivated to put in the effort required to learn.

Reports on the attitudes of preschoolers toward learning in general and learning mathematics in particular suggest that most children enter school eager to become competent at mathematics. In a survey that examined a number of personality and motivational features relevant to success in mathematics, teachers and parents reported that kindergarteners have high levels of persistence and eagerness to learn although teachers differed in their perceptions of children from different ethnic groups, as we discuss below.

In one study, first graders rated their interest in mathematics on average at approximately 6 on a scale from 1 to 7 with 7 being the highest. One important factor in attaining a productive disposition toward mathematics and maintaining the motivation required to learn it is the extent to which children perceive achievement as the product of effort as opposed to fixed ability.

Extensive research in the learning of mathematics and other domains has shown that children who attribute success to a relatively fixed ability are likely to approach new tasks with a performance rather than a learning orientation, which causes them to show less interest in putting themselves in challenging situations that result in them at least initially performing poorly. Most preschoolers enter school interested in mathematics and motivated to learn it. The challenge to parents and educators is to help them maintain a productive disposition toward mathematics as they develop the other strands of their mathematical proficiency.

In some circumstances, preschool children show impressive mathematical abilities that can provide the basis for their later learning of school mathematics. These abilities are, however, limited in a number of important ways. Because the algorithms that preschoolers develop are based on counting and on their experience with sets of objects, they do not generalize to larger numbers.

For example, preschool children can show a mastery of the concepts of addition and subtraction for very small numbers. This limitation is an important feature of preschool mathematical thinking and is an important way in which preschool mathematical proficiency differs from adult proficiency. As stated above, the way in which a word problem is phrased can be the difference between success and failure.

Furthermore, if children succeed, the strategy they use is a direct model of the story; they, in effect, act out the story to find the answer. These abilities include understanding the magnitudes of small numbers, being able to count and to use counting to solve simple mathematical problems, and understanding many of the basic concepts underlying measurement.

For example, a large survey of U. A number of children, however, particularly those from low socioeconomic groups, enter school with specific gaps in their mathematical proficiency. Several promising approaches have been developed to deal with this developmental immaturity in mathematical knowledge. For example, the Rightstart program consists of a set of games and number-line activities aimed at providing children needing remedial assistance with an understanding of the relative magnitudes of numbers.

Another intervention is aimed at ensuring that Latino children understand the base structure of number names, something that, as noted above, U. Taken together, these results suggest that relatively simple interventions may yield substantial payoffs in ensuring that all children enter or leave first grade ready to profit from mathematics instruction. The kindergarten survey cited above reported smaller ethnic differences in factors related to productive disposition persistence, eagerness to learn, and ability to pay attention than in mathematical knowledge.

There were, however, some noteworthy differences between the reports of teachers and parents for different ethnic groups. Parents reported high levels of eagerness to learn e. Teachers and parents are, of course, judging children against different comparison groups, but the data at least raise the possibility that kindergarten teachers may be underestimating the eagerness of their students to learn mathematics.

For preschool children, the strands of mathematical proficiency are particularly closely intertwined. Although their conceptual understanding is limited, as their understanding of number emerges they become able to count and solve simple problems. It is only when they move beyond what they informally understand—to the base system for teens and larger numbers, for example—that their fluency and strategic competencies falter.

Young children also show a remarkable ability to formulate, represent, and solve simple mathematical problems and to reason and explain their mathematical activities. The desire to quantify the world around them seems to be a natural one for young children. They are positively disposed to do and understand mathematics when they first encounter it.

Preschool children generally show a much more sophisticated understanding of small numbers than they do of larger numbers. They also have a great deal of difficulty in moving from the number names in languages such as English and Spanish to understanding the base structure of number names and mastering the Arabic numerals used in school mathematics. Furthermore, not all children enter school with the intuitive understanding of number described above and assumed by the elementary school curriculum.

Recent research suggests that effective methods exist for providing this basic understanding of number. Similar suggestions have been made by Baroody, a, b; Fuson, , ; and Siegler, The so-called Hindu-Arabic numeration system is in some sense a misnomer because the Chinese numeration system has been a decimal one from the time of the earliest historical records.

Because of the frequent contact between the Chinese and the Indians since the time of antiquity, there has always been some question of whether the Indians got their decimal system from the Chinese. Language has to be the product of its culture. So the fact that the names for numbers in Chinese, especially for the teens, reflect a base system indicates that the decimal system has been in place in China all along. By contrast, the Hindu-Arabic system did not take root in the West until the sixteenth century, long after the names for numbers in the various Western languages had been set.

The irregularities in the English and Spanish number names may perhaps be understood better in this light. Alibali and Goldin-Meadow, , showed that in learning to solve problems involving mathematical equivalence, students were most successful when they had passed through a stage of considering multiple solution strategies. Alexander, White, and Daugherty, , propose these three conditions for reasoning in young children. There is reason to believe that the conditions apply more generally. Analogical reasoning and early mathematics learning.

Analogies, metaphors, and images pp. The development of written representations for some mathematical concepts. Gesture-speech mismatch and mechanisms of learning: Cognitive Psychology , 25 , — Perception of numerical invariance in neonates. Child Development , 54 , — A developmental framework for preschool, primary, and special education teachers. The development of counting strategies for single-digit addition.

Journal for Research in Mathematics Education , 18 , — Journal for Research in Mathematics Education , 20 , — Remedying common counting difficulties. The relationship between initial meaningful and mechanical knowledge of arithmetic. The case of mathematics pp. Developmental Psychology , 20 , — Models of problem solving: Journal for Research in Mathematics Education , 24 , — The acquisition of addition and subtraction concepts in grades one through three.

Journal for Research in Mathematics Education , 15 3 , — How children learn mathematics 4th ed. Their role in motivation, personality, and development. Child Development , 60 , — Research on whole number addition and subtraction. Relationships children construct among English number words, multiunit base-ten blocks, and written multidigit addition.

The nature and origins of mathematical skills pp. The acquisition of early number work meanings. The acquisition and elaboration of the number work sequence. Matching, counting, and conservation of numerical equivalence. Child Development , 54 , 91— Journal for Research in Mathematics Education , 28 , — First principles organize attention to and learning about relevant data: Number and the animate-inanimate distinction as examples.

Cognitive Science , 14 , 79— A rational-constructivist account of early learning about numbers and objects. Advances in research and theory pp. Cognition , 13 , — Cognitive Development , 1 , 1— Connecting research with practice. Child psychology and practice 5th ed. Providing the central conceptual prerequisites for first formal learning of arithmetic to students at risk for school failure. Integrating cognitive theory and classroom practice pp.

Implications for judgments of the self and others. Child Development , 69 , — Relationship to beliefs about goodness. Child Development , 63 , — A mental model for early arithmetic. The empirical literature suggests that this belief needs drastic modification and in fact suggests that once a teacher reaches a certain level of understanding of the subject matter, then further understanding contributes nothing to student achievement. The notion that there is a threshold of necessary content knowledge for teaching is supported by the findings of another study in that used data from the Longitudinal Study of American Youth LSAY.

The NAEP data revealed that eighth graders taught by teachers who majored in mathematics outperformed those whose teachers. Fourth graders taught by teachers who majored in mathematics education or in education tended to outperform those whose teachers majored in a field other than education. That crude measures of teacher knowledge, such as the number of mathematics courses taken, do not correlate positively with student performance data, supports the need to study more closely the nature of the mathematical knowledge needed to teach and to measure it more sensitively.

The research, however, does suggest that proposals to improve mathematics instruction by simply increasing the number of mathematics courses required of teachers are not likely to be successful. As we discuss in the sections that follow, courses that reflect a serious examination of the nature of the mathematics that teachers use in the practice of teaching do have some promise of improving student performance.

Teachers need to know mathematics in ways that enable them to help students learn. The specialized knowledge of mathematics that they need is different from the mathematical content contained in most college mathematics courses, which are principally designed for those whose professional uses of mathematics will be in mathematics, science, and other technical fields. Why does this difference matter in considering the mathematical education of teachers? First, the topics taught in upper-level mathematics courses are often remote from the core content of the K curriculum.

Although the abstract mathematical ideas are connected, of course, basic algebraic concepts or elementary geometry are not what prospective teachers study in a course in advanced calculus or linear algebra.

Second, college mathematics courses do not provide students with opportunities to learn either multiple representations of mathematical ideas or the ways in which different representations relate to one another. Advanced courses do not emphasize the conceptual underpinnings of ideas needed by teachers whose uses of mathematics are to help others learn mathematics.

While this approach is important for the education of mathematicians and scientists, it is at odds with the kind of mathematical study needed by teachers. Consider the proficiency teachers need with algorithms. The power of computational algorithms is that they allow learners to calculate without having to think deeply about the steps in the calculation or why the calculations work. Over time, people tend to forget the reasons a procedure works or what is entailed in understanding or justifying a particular algorithm. Because the algorithm has become so automatic, it is difficult to step back and consider what is needed to explain it to someone who does not understand.

Most advanced mathematics classes engage students in taking ideas they have already learned and using them to construct increasingly powerful and abstract concepts and methods. Once theorems have been proved, they can be used to prove other theorems. It is not necessary to go back to foundational concepts to learn more advanced ideas.

Teaching, however, entails reversing the direction followed in learning advanced mathematics. In helping students learn, teachers must take abstract ideas and unpack them in ways that make the basic underlying concepts visible. For adults, division is an operation on numbers. Jane has 24 cookies. She wants to put 6 cookies on each plate. How many plates will she need? Jeremy has 24 cookies. He wants to put all the cookies on 6 plates. If he puts the same number of cookies on each plate, how many cookies will he put on each plate?

These two problems correspond to the measurement and sharing models of division, respectively, that were discussed in chapter 3. Young children using counters solve the first problem by putting 24 counters in piles of 6 counters each. They solve the second by partitioning the 24 counters into 6 groups.

In the first case the answer is the number of groups; in the second, it is the number in each group. Until the children are much older, they are not aware that, abstractly, the two solutions are equivalent. Teachers need to see that equivalence so that they can understand and anticipate the difficulties children may have with division.

To understand the sense that children are making of arithmetic problems, teachers must understand the distinctions children are making among those problems and how the distinctions might be reflected in how the children think about the problems. The different semantic contexts for each of the operations of arithmetic is not a common topic in college mathematics courses, yet it is essential for teachers to know those contexts and be able to use their knowledge in instruction.

The division example illustrates a different way of thinking about the content of courses for teachers—a way that can make those courses more relevant to the teaching of school mathematics. Teachers are unlikely to be able to provide an adequate explanation of concepts they do not understand, and they can hardly engage their students in productive conversations about multiple ways to solve a problem if they themselves can only solve it in a single way. Most of the investigations have been case studies, almost all involving fewer than 10 teachers, and most only one to three teachers.

In general, the researchers found that teachers. Not surprisingly, these teachers gave the students little assistance in developing an understanding of what they were doing. Some of the same studies contrasted the teaching practices of teachers with low levels of mathematical knowledge with the teaching practices of teachers who had a better command of mathematics. The teacher also needs to be sensitive to the unique ways of learning, thinking about, and doing mathematics that the student has developed. Each student can be seen as located on a path through school mathematics, equipped with strengths and weaknesses, having developed his or her own approaches to mathematical tasks, and capable of contributing to and profiting from each lesson in a distinctive way.

Teachers also need a general knowledge of how students think—the approaches that are typical for students of a given age and background, their common conceptions and misconceptions, and the likely sources of those ideas. We have described some of those progressions in chapters 6 through 8. Using that body of evidence, researchers have also. From the many examples of misconceptions to which teachers need to be sensitive, we have chosen one: Children can develop this impression because that is how the notation is often described in the elementary school curriculum and most of their practice exercises fit that pattern.

Knowing classroom practice means knowing what is to be taught and how to plan, conduct, and assess effective lessons on that mathematical content. We have discussed these matters in chapter 9. In the sections that follow, we consider how to develop an integrated corpus of knowledge of the types discussed in this section. First, however, we need to clarify our stance on the relation between knowledge and practice. We have discussed the kinds of knowledge teachers need if they are to teach for mathematical proficiency.

Although we have used the term knowledge throughout, we do not mean it exclusively in the sense of knowing about. Teachers must also know how to use their knowledge in practice.

Improve primary maths skills through storytelling, researcher says

Effective programs of teacher preparation and professional development cannot stop at simply engaging teachers in acquiring knowledge; they must challenge teachers to develop, apply, and analyze that knowledge in the context of their own classrooms so that knowledge and practice are integrated. In chapter 4 we identified five components or strands of mathematical proficiency. Teaching is a complex activity and, like other complex activities, can be conceived in terms of similar components.

Just as mathematical proficiency itself involves interwoven strands, teaching for mathematical proficiency requires similarly interrelated components. In the context of teaching, proficiency requires:. Like the strands of mathematical proficiency, these components of mathematical teaching proficiency are interrelated.

In this chapter we discuss the problems entailed in developing a proficient command of teaching. In the previous section we discussed issues relative to the knowledge base needed to develop proficiency across all components. Now we turn to specific issues that arise in the context of the components. It is not sufficient that teachers possess the kinds of core knowledge delineated in the previous section.

One of the defining features of conceptual understanding is that knowledge must be connected so that it can be used intelligently. Teachers need to make connections within and among their knowledge of mathematics, students, and pedagogy. The implications for teacher preparation and professional development are that teachers need to acquire these forms of knowledge in ways that forge connections between them. For teachers who have already achieved some mathematical proficiency, separate courses or professional development programs that focus exclusively on mathematics, on the psychology of learning, or on methods of teaching provide limited opportunities to make these connections.

Unfortunately, most university teacher preparation programs offer separate courses in mathematics, psychology, and methods of teaching that are taught in different departments. The difficulty of integrating such courses is compounded when they are located in different administrative units. It is not enough, however, for mathematical knowledge and knowledge of students to be connected; both need to be connected to classroom practice.

Teachers may know mathematics, and they may know their students and how they learn. But they also have to know how to use both kinds of knowledge effectively in the context of their work if they are to help their students develop mathematical proficiency. Similarly, many inservice workshops, presentations at professional meetings, publications for teachers, and other opportunities for teacher learning focus almost exclusively on activities or methods of teaching and seldom attempt to help teachers develop their own conceptual understanding of the underlying mathematical ideas, what students understand about those ideas, or how they learn them.

Alternative forms of teacher education and professional development that attempt to teach mathematical content, psychology. The second basic component of teaching proficiency is the development of instructional routines. Just as students who have acquired procedural fluency can perform calculations with numbers efficiently, accurately, and flexibly with minimal effort, teachers who have acquired a repertoire of instructional routines can readily draw upon them as they interact with students in teaching mathematics.

Some routines concern classroom management, such as how to get the class started each day and procedures for correcting and collecting homework. Other routines are more grounded in mathematical activity. For example, teachers need to know how to respond to a student who gives an answer the teacher does not understand or who demonstrates a serious misconception.

Teachers need businesslike ways of dealing with situations like these that occur on a regular basis so that they can devote more of their attention to the more serious issues facing them. When teachers have several ways of approaching teaching problems, they can try a different approach if one does not work. Researchers have shown that expert teachers have a large repertoire of routines at their disposal.

Novice teachers, in contrast, have a limited range of routines and often cannot respond appropriately to situations. Expert teachers not only have access to a range of routines, they also can apply them flexibly, know when they are appropriate, and can adapt them to fit different situations. The third component of teaching proficiency is strategic competence.

Although teachers need a range of routines, teaching is very much a problem-solving activity. These are problems that every teacher faces every day, and most do not have readymade solutions. Conceptual understanding of the knowledge required to teach for proficiency can help equip teachers to deal intelligently with these problems. It is misleading to claim that teachers actually solve such problems in the sense of solving a mathematical problem. There is never an ideal solution to the more difficult problems of teaching, but teachers can learn to contend with these problems in reasonable ways that take into account the mathematics that students are to learn; what their students understand and how they may best learn it; and representations, activities, and teaching practices that have proven most effective in teaching the mathematics in question or that have been effective in teaching related topics.

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Teacher education and professional development programs that take into account the strategic decision making in teaching can help prepare teachers to be more effective in solving instructional problems. Teachers can learn to recognize that teaching involves solving problems and that they can address these problems in reasonable and intelligent ways. The fourth component of teaching proficiency is adaptive reasoning.

Teachers can learn from their teaching by analyzing it: Many successful programs of teacher education and professional development engage teachers in reflection, but the reflection, or perhaps more appropriately the analysis, is grounded in specific examples. In those programs, teachers engage in analyses in which they are asked to provide evidence to justify claims and assertions. As with other complex activities, teacher learning can be enhanced by making more visible the goals, assumptions, and decisions involved in the practice of.

Teachers are often asked to pose a particular mathematical problem to their classes and to discuss the mathematical thinking that they observe. Just as students must develop a productive disposition toward mathematics such that they believe that mathematics makes sense and that they can figure it out, so too must teachers develop a similar productive disposition.

Teachers whose learning becomes generative perceive themselves as in control of their own learning. The teachers become more comfortable with mathematical ideas and ripe for a more systematic view of the subject. Teachers whose learning becomes generative see themselves as lifelong learners who can learn from studying curriculum materials 35 and from analyzing their practice and their interactions with students. Programs of teacher education and professional development that portray to the participants that they are in control of their own learning help teachers develop a productive dispo-.

Programs that provide readymade, worked-out solutions to teaching problems should not expect that teachers will see themselves as in control of their own learning. In a teacher preparation program, teachers clearly cannot learn all they need to know about the mathematics they will teach, how students learn that mathematics, and how to teach it effectively.

Consequently, some authorities have recommended that teacher education be seen as a professional continuum, a career-long process. They need to be able to adapt to new curriculum frameworks, new materials, advances in technology, and advances in research on student thinking and teaching practice. They have to learn how to learn, whether they are learning about mathematics, students, or teaching. Teachers can continue to learn by participating in various forms of professional development.

But formal professional development programs represent only one source for continued learning. We consider below examples of four such program types that represent an array of alternative approaches to developing integrated proficiency in teaching mathematics. For example, prospective elementary school teachers may take a mathematics course that focuses, in part, on rational numbers or proportionality rather than the usual college algebra or calculus.

Such courses are offered in many universities, but they are seldom linked to instructional practice. The lesson depicted in Box 10—1 comes from a course in which connections to practice are being made. The prospective teachers stare at the board, trying to figure out what the instructor is asking them to do. After calculating the answer to a simple problem in the division of fractions and recalling the old algorithm—invert and multiply—most of them have come up with the answer, It is familiar content, and although they have not had occasion to divide fractions recently, they feel comfortable, remembering their own experiences in school mathematics and what they learned.

But now, what are they being asked? The instructor has challenged them to consider why they are getting what seems to be an answer that is larger than either of the numbers in the original problem and. Confused, they are suddenly stuck. None of them noticed this fact before. The instructor proposes a new task: Can you come up with an example or a model that shows what is going on with dividing one and three fourths by one half?

The prospective teachers set to work, some in pairs, some alone. The instructor walks around, watching them work, and occasionally asking a question. Most have drawn pictures like those below:. I have two pizzas. My little brother eats one quarter of one of them and then I have one and three quarters pizzas left.

My sister is very hungry, so we decide to split the remaining pizza between us. We each get pieces of pizza. I have cups of sugar. Each batch of sesame crackers takes cup of sugar. How many batches of crackers can I make? And another pair has envisioned filling -liter containers, starting with liters of water. After about 10 minutes, the instructor invites students to share their problems with the rest of the class.

One student presents the pizza situation above. Most students nod appreciatively. When a second student offers the sesame cracker problem, most nod again, not noticing the difference. The instructor poses a question: How does each problem we heard connect with the original computation? Are these two problems similar or different, and does it matter? Through discussion the students gradually come to recognize that, in the pizza problem, the pizza has been divided in half and that the answer is in terms of fourths —that is, that the pieces are fourths of pizzas.

In the case of the sesame cracker problem, the answer of batches is in terms of half cups of sugar. In the first instance, they have represented division in half, which is actually division by two; in the second they have represented division by one half. The instructor moves into a discussion of different interpretations of division: After the students observe that the successful problems— involving the sesame crackers and the liters of water—are measurement problems, she asks them to try to develop a problem situation for that represents a sharing division.

In other words, could they make a sensible problem in which the is not the unit by which the whole is being measured, but instead is the number of units into which the whole has been divided? For homework, the instructor asks the students to try making representations for several other division situations, which she chooses strategically, and finally asks them to select two numbers to divide that they think are particularly good choices and to say why.

In this excerpt from a university mathematics course, the prospective teachers are being asked to unpack familiar arithmetic content, to make explicit the ideas underlying the procedures they remember and can perform. Repeatedly throughout the course, the instructor poses problems that have been strategically designed to expose concepts on which familiar procedures rest. A second principle is to link that work with larger mathematical ideas and structures. For example, the lesson on the division of fractions is part of a larger agenda that includes understanding division, its relationship to fractions and to multiplication, and the meaning and representation of operations.

Moreover, throughout the development of these ideas and connections, the prospective teachers work with whole and rational numbers, considering how the mathematical world looks inside these nested systems. The overriding purpose of a course like this is to provide prospective teachers with ample opportunities to learn fundamental ideas of school mathematics, how they are related, and how students come to learn them.

But the course is not about how to teach, nor about how children learn. It is explicitly and deliberately a sustained opportunity for prospective teachers to learn mathematical ideas in ways that will equip them with mathematical resources needed in teaching.

Teachers do not learn abstract concepts about mathematics and children. In the programs, teachers look at problem-solving strategies of real students, artifacts of student work, cases of real classrooms, and the like. Furthermore, the teachers in these programs are challenged to relate what they learn to their own students and their own instructional practices. They learn about mathematics and students both in workshops and by interacting with their own students. Specific opportunity is provided for the teachers to discuss with one another how the ideas they are encountering influence their practice and how their practice influences what they are learning.

The workshop described in Box 10—2 forms part of a professional development program designed to help teachers develop a deeper understanding of some critical mathematical ideas, including the equality sign. The program, modeled after Cognitively Guided Instruction CGI , which has proven to be a highly effective approach, 41 assists teachers in understanding how to help their students reason about number operations and relations in ways that enhance the learning of arithmetic and promote a smoother transition from arithmetic to algebra.

Several features of this example of professional development are worth noting. Although they begin by considering how children think, the teachers. Before attending the workshop, participating teachers ask their students to find the number that they could put in the box to make the following open-number sentence a true number sentence: At the workshop, the teachers share their findings with the other participants.

These findings, which surprised most teachers, have led them to begin to listen to their students, and a number of teachers have engaged their students in a discussion of the reasons for their responses. The discussion generates insights about how children are thinking and what teachers can learn by listening to their students.

The task is to decide whether the sentence is true or false. Sometimes the decision requires calculation e. The teachers work in small groups to construct true and false number sentences they might use to elicit various views of equality. Using these sentences, their students could engage in explorations that might lead to understanding equality as a relation.

The sentences could also provide opportunities for discussions about how to resolve disagreement and develop a mathematical argument. The teachers work together to consider how their students might respond to different number sentences and which number sentences might produce the most fruitful discussion. Falkner, Levi, and Carpenter, Used by permission of the authors. The teachers also begin to ponder how notation is used and how ideas are justified in mathematics.

A central feature of their discussion is that math-.

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The mathematical ideas and how children think about them are seen in classroom interactions. The programs do not deal with general theories of learning. For example, to understand the different strategies that children use to solve different problems, teachers must understand the semantic differences between problems represented by the same operation, as illustrated by the sharing and measurement examples of dividing cookies described above in Box 10—1.

Gains in student achievement generally have been in the areas of understanding and problem solving, but none of the programs has led to a decline in computational skills, despite their greater emphasis on higher levels of thinking. Case examples are yet another way to build the connections between knowledge of mathematics, knowledge of students, and knowledge of practice. Although the cases focus on classroom episodes, the discussions the teachers engage in as they reflect on the cases emphasize mathematics content and student thinking.

The cases involve instruction in specific mathematical topics, and teachers analyze the cases in terms of the mathematics content being taught and the mathematical thinking reflected in the work the children produce and the interactions they engage in. Cases can be presented in writing or using multiple media such as videotapes and transcriptions of lessons. These teachers are probing the concept of functions from several overlapping perspectives. They dig into the mathematics through close work on and analysis of the task that the teacher posed.

And they revisit the mathematical ideas by looking carefully at how the teacher deals with the mathematics during the lesson.

Read & Respond

A dozen teachers are gathered around a table. They have read a case of a teacher teaching a lesson on functions. The written case includes the task the teacher used and a detailed narrative account of what happened in the class as students worked on the problem. The teacher used the following task:. Sara has made several purchases from a mail-order company. Sara decides that the company must be using a simple rule to determine how much to charge for shipping.

Help her figure out how much it would most likely cost to ship a 1-kg package and how much each additional kilogram would cost. Before the teachers studied the case and the accompanying materials, they solved the mathematical problem themselves. To begin the discussion, the workshop leader asks the teachers to look closely at one segment of the lesson in which two students are presenting solutions to the problem.

She asks them to interpret what each student did and to compare the two solutions.

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The teachers launch into a discussion of the mathematics for several minutes. They note that if the given values weight, cost are graphed, the points lie on the same straight line. Reading the graph provides a solution. Also, by asking how much each additional kilogram would cost, the problem suggests there is a constant difference that can be used in solving it. She asks them to analyze the text closely and try to categorize what the teacher is doing.

This discussion yields surprises for most of the teachers. Suddenly the intricate work that the teacher is doing becomes visible. They begin to describe and name the different moves she makes. One teacher becomes intrigued with how the teacher helps students express their ideas by asking questions to support their explanations before she asks other students to comment. It is quite clear that this is no generic skill, for the mathematical sensitivity and knowledge entailed are quite visible throughout.

The teachers become fascinated with what looks like an important missed opportunity to unpack a common misconception about function. Speculating about why that happened leads them to a productive conversation about what one might do to seize and capitalize on the opportunity.

The session ends with the teachers agreeing to bring back one mathematical task from their own work on functions and compare it with the task used in the case. Several are overheard to be discussing features of this problem that seem particularly fruitful and that have them thinking about how they frame problems for their students. The group briefly discusses some ways to vary the problem to make it either simpler or more complex. The leader then closes by summarizing some of the mathematical issues embedded in the task.

She points out that it is not obvious what the value of 2. It is the cost of sending a package of zero weight, an idea that does not appear anywhere in the problem itself or in real life. She also says that it is important to understand that x refers to whole numbers only. Finally, she notes that with a different function, the differences might not be constant. The assumption of constant differences is one suggested by the problem and common in situations like those involving shipping costs, but it is not necessarily always warranted. They gained a greater repertoire of ways to represent mathematical ideas, were able to articulate connections among mathematical ideas, and developed a deeper understanding of mathematical structures.

At least some of the teachers continued the process of learning mathematics by examining the mathematical work of their own students in their own classrooms. The case-based programs that focus on classroom instruction treat the cases as problematic situations that serve as a basis for discussion and inquiry rather than as models of instruction for the teachers to emulate. A somewhat different approach to professional development is represented by so-called lesson study groups, which are used in Japan see Box 10—4. They design a lesson, and one member of the group teaches it while the others watch.

Afterwards they discuss what happened in light of their anticipations and goals. Based on this experience, the group revises the lesson and someone else teaches it. The cycle continues of trying the lesson, discussing and analyzing how it worked, and revising it. Their knowledge becomes a basis for further learning through the study of a lesson. Lesson study groups might follow somewhat different formats and schedules than the one described above, but most meet regularly during the year and focus on improving a very few lessons with clear learning goals.

Box 10—4 The Japanese Lesson Study. Small groups of teachers form within the school around areas of common teaching interests or responsibilities e. Each group begins by formulating a goal for the year. Sometimes the goal is adapted from national-level recommendations e. The more specific goal might focus on a curriculum topic that has been problematic for students in their classrooms. A few lessons then are identified that ordinarily deal with that topic, and the group begins its yearlong task to improve those lessons. Lesson study groups meet regularly, often once a week after school e.

Some groups divide their work into three major phases, each taking about one third of the school year. During the first phase, teachers do research on the topic, reading and sharing relevant research reports and collecting information from other teachers on effective approaches for teaching the topic.

During the second phase, teachers design the targeted lessons often just one, two, or perhaps three lessons. During the third phase, the lessons are tested and refined. The first test often involves one of the group members teaching a lesson to his or her class while the other group members observe and take notes. After the group refines the lesson, it might be tested with another class in front of all the teachers in the school.

In this case, a follow-up session is scheduled, and the lesson study group engages their colleagues in a discussion about the lesson, receiving feedback about its effectiveness.

Working directly on improving teaching is their means of becoming better teachers. Learning in ways that continue to be generative over time is best done in a community of fellow practitioners and learners, as illustrated by the Japanese lesson study groups. Studies of school reform efforts suggest that professional development is most effective when it extends beyond the individual teacher. Professional development can create contexts for teacher collaboration, provide a focus for the collaboration, and provide a common frame for interacting with other teachers around common problems.

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