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**Contents:**

As we previously noted, this point sharply contradicts one of the popular—and dangerous—myths about teaching: The uniqueness of the content knowledge and pedagogical knowledge necessary to teach his-. As is the case in history, most people believe that they know what mathematics is about—computation. Most people are familiar with only the computational aspects of mathematics and so are likely to argue for its place in the school curriculum and for traditional methods of instructing children in computation. In contrast, mathematicians see computation as merely a tool in the real stuff of mathematics, which includes problem solving, and characterizing and understanding structure and patterns.

The current debate concerning what students should learn in mathematics seems to set proponents of teaching computational skills against the advocates of fostering conceptual understanding and reflects the wide range of beliefs about what aspects of mathematics are important to know. A growing body of research provides convincing evidence that what teachers know and believe about mathematics is closely linked to their instructional decisions and actions Brown, ; National Council of Teachers of Mathematics, ; Wilson, a, b; Brophy, ; Thompson, Thus, as we examine mathematics instruction, we need to pay attention to the subject-matter knowledge of teachers, their pedagogical knowledge general and content specific , and their knowledge of children as learners of mathematics.

In this section, we examine three cases of mathematics instruction that are viewed as being close to the current vision of exemplary instruction and discuss the knowledge base on which the teacher is drawing, as well as the beliefs and goals which guide his or her instructional decisions. For teaching multidigit multiplication, teacher-researcher Magdelene Lampert created a series of lessons in which she taught a heterogeneous group of 28 fourth-grade students. The students ranged in computational skill from beginning to learn the single-digit multiplication facts to being able to accurately solve n-digit by n-digit multiplications.

The lessons were intended to give children experiences in which the important mathematical principles of additive and multiplicative composition, associativity, commutativity, and the distributive property of multiplication over addition were all evident in the steps of the procedures used to arrive at an answer Lampert, It is clear from her description of her instruction that both her deep understanding of multiplicative structures and her knowledge of a wide range of representations and problem situations related to multiplication were brought to bear as she planned and taught these lessons.

I also taught new information in the form of symbolic structures and emphasized the connection between symbols and operations on quantities, but I made it a classroom requirement that students use their own ways of deciding whether something was mathematically reasonable in doing the work. On the part of the teacher, the principles might be known as a more formal abstract system, whereas on the part of the learners, they are known in relation to familiar experiential contexts. But what seems most important is that teachers and students together are disposed toward a particular way of viewing and doing mathematics in the classroom.

Magdelene Lampert set out to connect what students already knew about multidigit multiplication with principled conceptual knowledge. She did so in three sets of lessons. Another set of lessons used simple stories and drawings to illustrate the ways in which large quantities could be grouped. Finally, the third set of lessons used only numbers and arithmetic symbols to represent problems. Throughout the lessons, students were challenged to explain their answers and to rely on their arguments, rather than to rely on the teacher or book for verification of correctness.

An example serves to highlight this approach; see Box 7. They were able to talk meaningfully about place value and order of operations to give legitimacy to procedures and to reason about their outcomes, even though they did not use technical terms to do so. I took their experimentations and arguments as evidence that they had come to see mathematics as more than a set of procedures for finding answers. Clearly, her own deep understanding of mathematics comes into play as she teaches these lessons.

Helping third-grade students extend their understanding of numbers from the natural numbers to the integers is a challenge undertaken by another teacher-researcher. That is, she not only takes into account what the important mathematical ideas are, but also how children think about the particular area of mathematics on which she is focusing. She draws on both her understanding of the integers as mathematical entities subject-matter knowledge and her extensive pedagogical content knowledge specifically about integers. A wealth of possible models for negative numbers exists and she reviewed a number of them—magic peanuts, money, game scoring, a frog on a number line, buildings with floors above and below ground.

She decided to use the building model first and money later: And if I did this multiplication and found the answer, what would I know about those.

Okay, here are the jars. The stars in them will stand for butterflies. Now, it will be easier for us to count how many butterflies there are altogether, if we think of the jars in groups. Lampert then has the children explore other ways of grouping the jars, for example, into two groups of 6 jars. It is a sign that she needs to do many more activities involving different groupings. Students continue to develop their understanding of the principles that govern multiplication and to invent computational procedures based on those principles.

Students defend the reasonableness of their procedures by using drawings and stories. Eventually, students explore more traditional as well as alternative algorithms for two-digit multiplication, using only written symbols.

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She hoped that the positional aspects of the building model would help children recognize that negative numbers were not equivalent to zero, a common misconception. She was aware that the building model would be difficult to use for modeling subtraction of negative numbers.

Deborah Ball begins her work with the students, using the building model by labeling its floors. Students were presented with increasingly difficult problems. Ball then used a model of money as a second representational context for exploring negative numbers, noting that it, too, has limitations. Like Lampert, Ball wanted her students to accept the responsibility of deciding when a solution is reasonable and likely to be correct, rather than depending on text or teacher for confirmation of correctness. The concept of cognitively guided instruction helps illustrate another important characteristic of effective mathematics instruction: Teachers, it is claimed, will use their knowledge to make appropriate instructional decisions to assist students to construct their mathematical knowledge.

Students make predictions and then see the actual forces between the carts displayed as they collide. The students had the opportunity to seek out information from family members, friends, experts in various fields, on-line computer services, and books, as well as from the teacher. One such system, called Classtalk, consists of both hardware and software that allows up to four students to share an input device e. There were 12 jars, and each had 4 butterflies in it. There's a problem loading this menu right now. During the first week of school Barb Johnson asks her sixth graders two questions:

Cognitively guided instruction is used by Annie Keith, who teaches a combination first- and second-grade class in an elementary school in Madison Wisconsin Hiebert et al. A portrait of Ms. Students spend a great deal of time discussing alternative strategies with each other, in groups, and as a whole class. The teacher often participates in these discussions but almost never demonstrates the solution to problems. Important ideas in mathematics are developed as students explore solutions to problems, rather than being a focus of instruction per se.

For example, place-value concepts are developed as students use base materials, such as base blocks and counting frames, to solve word problems involving multidigit numbers. Everyday first-grade and second-grade activities, such as sharing snacks, lunch count, and attendance, regularly serve as contexts for problem-solving tasks.

Mathematics lessons frequently make use of math centers in which the students do a variety of activities. On any given day, children at one center may solve word problems presented by the teacher while at another center children write word problems to present to the class later or play a math game. She continually challenges her students to think and to try to make sense of what they are doing in math. She uses the activities as opportunities for her to learn what individual students know and understand about mathematics.

As students work in groups to solve problems, she observes the various solutions and mentally makes notes about which students should present their work: Her knowledge of the important ideas in mathematics serves as one framework for the selection process, but her understanding of how children think about the mathematical ideas they are using also affects her decisions about who should present. She might select a solution that is actually incorrect to be presented so that she can initiate a discussion of a common misconception.

Both the presentations of solutions and the class discussions that follow provide her with information about what her students know and what problems she should use with them next.

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She forms hypotheses about what her students understand and selects instructional activities based on these hypotheses. She modifies her instruction as she gathers additional information about her students and compares it with the mathematics she wants them to learn. Her approach is not a free-for-all without teacher guidance: Some attempts to revitalize mathematics instruction have emphasized the importance of modeling phenomena.

Work on modeling can be done from kindergarten through twelth grade K— Modeling involves cycles of model construction, model evaluation, and model revision. It is central to professional practice in many disciplines, such as mathematics and science, but it is largely missing from school instruction.

Modeling practices are ubiquitous and diverse, ranging from the construction of physical models, such as a planetarium or a model of the human vascular system, to the development of abstract symbol systems, exemplified by the mathematics of algebra, geometry, and calculus. The ubiquity and diversity of models in these disciplines suggest that modeling can help students develop understanding about a wide range of important ideas.

Modeling practices can and should be fostered at every age and grade level Clement, ; Hestenes, ; Lehrer and Romberg, a, b; Schauble et al. Taking a model-based approach to a problem entails inventing or selecting a model, exploring the qualities of the model, and then applying the model to answer a question of interest.

For example, the geometry of triangles has an internal logic and also has predictive power for phenomena ranging from optics to wayfinding as in navigational systems to laying floor tile.

Modeling emphasizes a need for forms of mathematics that are typically underrepresented in the standard curriculum, such as spatial visualization and geometry, data structure, measurement, and uncertainty. For example, the scientific study of animal behavior, like bird foraging, is se-. Note, for instance, the rubber bands that mimic the connective function of ligaments and the wooden dowels that are arranged so that their translation in the vertical plane cannot exceed degrees.

Though the search for function is supported by initial resemblance, what counts as resemblance typically changes as children revise their models. For example, attempts to make models exemplify elbow motion often lead to an interest in the way muscles might be arranged from Lehrer and Schauble, a, b. Increasingly, approaches to early mathematics teaching incorporate the premises that all learning involves extending understanding to new situations, that young children come to school with many ideas about mathematics, that knowledge relevant to a new setting is not always accessed spontaneously, and that learning can be enhanced by respecting and encouraging.

Rather than beginning mathematics instruction by focusing solely on computational algorithms, such as addition and subtraction, students are encouraged to invent their own strategies for solving problems and to discuss why those strategies work.

Teachers may also explicitly prompt students to think about aspects of their everyday life that are potentially relevant for further learning. For example, everyday experiences of walking and related ideas about position and direction can serve as a springboard for developing corresponding mathematics about the structure of large-scale space, position, and direction Lehrer and Romberg, b. Two recent examples in physics illustrate how research findings can be used to design instructional strategies that promote the sort of problem-solving behavior observed in experts.

Undergraduates who had finished an introductory physics course were asked to spend a total of 10 hours, spread over several weeks, solving physics problems using a computer-based tool that constrained them to perform a conceptual analysis of the problems based on a hierarchy of principles and procedures that could be applied to solve them Dufresne et al. This approach was motivated by research on expertise discussed in Chapter 2. The reader will recall that, when asked to state an approach to solving a problem, physicists generally discuss principles and procedures.

Novices, in contrast, tend to discuss specific equations that could be used to manipulate variables given in the problem Chi et al. When compared with a group of students who solved the same problems on their own, the students who used the computer to carry out the hierarchical analyses performed noticeably better in subsequent measures of expertise.

For example, in problem solving, those who performed the hierarchical analyses outperformed those who did not, whether measured in terms of overall problem-solving performance, ability to arrive at the correct answer, or ability to apply appropriate principles to solve the problems; see Figure 7. Furthermore, similar differences emerged in problem categorization: See Chapter 6 for an example of the type of item used in the categorization task of Figure 7.

It is also worth noting that both Figures 7. In both cases, the control group made significant improvements simply as a result of practice time on task , but the experimental group showed more improvements for the same amount of training time deliberate practice. Introductory physics courses have also been taught successfully with an approach for problem solving that begins with a qualitative hierarchical analysis of the problems Leonard et al. Undergraduate engineering students were instructed to write qualitative strategies for solving problems before attempting to solve them based on Chi et al.

The strategies consisted of a coherent verbal description of how a problem could be solved and contained three components: That is, the what, why, and how of solving the problem were explicitly delineated; see Box 7. Compared with students who took a traditional course, students in the strategy-based course performed significantly better in their ability to categorize problems according to the relevant principles that could be applied to solve them; see Figure 7.

Hierarchical structures are useful strategies for helping novices both recall knowledge and solve problems. For example, physics novices who had completed and received good grades in an introductory college physics course were trained to generate a problem analysis called a theoretical problem description Heller and Reif, The analysis consists of describing force problems in terms of concepts, principles, and heuristics.

Engaging students in worthwhile learning requires more than a knowledge of underlying principles of good teaching. It demands considerable practice as well . This book provides descriptive cases, accompanied by analytic commentaries, of nine upper-elementary grade mathematics lessons that represent an array of.

With such an approach, novices substantially improved in their ability to solve problems, even though the type of theoretical problem description used in the study was not a natural one for novices. Novices untrained in the theoretical descriptions were generally unable to generate appropriate descriptions on their own—even given fairly routine problems. Skills, such as the ability to describe a problem in detail before attempting a solution, the ability to determine what relevant information should enter the analysis of a problem, and the ability to decide which procedures can be used to generate problem descriptions and analyses, are tacitly used by experts but rarely taught explicitly in physics courses.

Another approach helps students organize knowledge by imposing a hierarchical organization on the performance of different tasks in physics Eylon and Reif, Students who received a particular physics argument that was organized in hierarchical form performed various recall and problem-solving tasks better than subjects who received the same argument.

Similarly, students who received a hierarchical organization of problem-solving strategies performed much better than subjects who received the same strategies organized non-hierarchically. If students had simply been given problems to solve on their own an instructional practice used in all the sciences , it is highly. Students might get stuck for minutes, or even hours, in attempting a solution to a problem and either give up or waste lots of time.

In Chapter 3 , we discussed ways in which learners profit from errors and that making mistakes is not always time wasted. However, it is not efficient if a student spends most of the problem-solving time rehearsing procedures that are not optimal for promoting skilled performance, such as finding and manipulating equations to solve the problem, rather than identifying the underlying principle and procedures that apply to the problem and then constructing the specific equations needed.

In deliberate practice, a student works under a tutor human. Students enrolled in an introductory physics course were asked to write a strategy for an exam problem. Use the conservation of energy since the only nonconservative force in the system is the tension in the rope attached to the mass M and wound around the disk assuming there is no friction between the axle and the disk, and the mass M and the air , and the work done by the tension to the disk and the mass cancel each other out.

First, set up a coordinate system so the potential energy of the system at the start can be determined. There will be no kinetic energy at the start since it starts at rest. Therefore the potential energy is all the initial energy. Now set the initial energy equal to the final energy that is made up of the kinetic energy of the disk plus the mass M and any potential energy left in the system with respect to the chosen coordinate system.

I would use conservation of mechanical energy to solve this problem. The mass M has some potential energy while it is hanging there. When the block starts to accelerate downward the potential energy is transformed into rotational kinetic energy. Through deliberate practice, computer-based tutoring environments have been designed that reduce the time it takes individuals to reach real-world performance criteria from 4 years to 25 hours see Chapter 9!

Before students can really learn new scientific concepts, they often need to re-conceptualize deeply rooted misconceptions that interfere with the learning. As reviewed above see Chapters 3 and 4 , people spend considerable time and effort constructing a view of the physical world through. Mechanical energy is conserved even with the nonconservative tension force because the tension force is internal to the system pulley, mass, rope.

In trying to find the speed of the block I would try to find angular momentum kinetic energy, use gravity. I would also use rotational kinematics and moment of inertia around the center of mass for the disk. There will be a torque about the center of mass due to the weight of the block, M. The force pulling downward is mg. The moment of inertia multiplied by the angular acceleration. By plugging these values into a kinematic expression, the angular speed can be calculated.

Then, the angular speed times the radius gives you the velocity of the block. The first two strategies display an excellent understanding of the principles, justification, and procedures that could be used to solve the problem the what, why, and how for solving the problem. The last two strategies are largely a shopping list of physics terms or equations that were covered in the course, but the students are not able to articulate why or how they apply to the problem under consideration.

Having students write strategies after modeling strategy writing for them and providing suitable scaffolding to ensure progress provides an excellent formative assessment tool for monitoring whether or not students are making the appropriate links between problem contexts, and the principles and procedures that could be applied to solve them see Leonard et al. Starting with the anchoring intuition that a spring exerts an upward force on the book resting on it, the student might be asked if a book resting on the.

The fact that the bent board looks as if it is serving the same function as the spring helps many students agree that both the spring and the board exert upward forces on the book. For a student who may not agree that the bent board exerts an upward force on the book, the instructor may ask a student to place her hand on top of a vertical spring. An educational psychologist with a Ph. Her work includes describing comprehension processes that students use when they read different types of text and developing a system for analyzing the comprehensibility and learnability of textbooks.

She has used cases in her own teaching and conducts research on learning to teach, professional development, culturally-responsive teaching, and education policy. Would you like to tell us about a lower price? If you are a seller for this product, would you like to suggest updates through seller support? Learn more about Amazon Prime. Engaging students in worthwhile learning requires more than a knowledge of underlying principles of good teaching.

It demands considerable practice as well as images of what good teaching in particular situations and for particular purposes might look like.

This volume provides these images. These cases were written from authentic, unrehearsed lessons taught by upper-elementary classroom teachers to diverse groups of real students in intact classrooms. Each lesson contains elements of sound instructional practice from which both preservice and in-service teachers can benefit. Cases are not meant to be ideal, but rather to evoke ways of seeing and thinking about good classroom instruction for all learners. Accompanied by analytic commentaries from experts representing a particular perspective, such as special education and ESOL, these unrehearsed cases are written with the understanding that teaching is complex and multi-dimensional.

The cases are drawn from a four-year study of 4th and 5th grade mathematics instruction of culturally diverse classrooms with relatively high rates of students from low-income families. Read more Read less. Prime Book Box for Kids. Sponsored products related to this item What's this? Page 1 of 1 Start over Page 1 of 1.

Learning The Alphabet Sequence: Teaching A Struggling Reader: This short, fast-read booklet shares some of the basic information that one mom wishes she had earlier to help her dyslexic child learn to read. Attending to Students' Developmental Levels Commentary: A Teacher's Perspective Case 5: Converting Units Within a System of Measurement: Distinguishing Between Area and Perimeter: Exploring the Meanings of "Volume": A Teacher's Perspective Case 8: The Importance of Sample Size: A Teacher's Perspective Case 9: Continuous versus Discrete Data: These math classroom cases are short and to the point — ideal for a focused class discussion about a specific pedagogical principle or teaching dilemma.

I especially appreciated the multiple ways in which the cases are classified — it makes it easier for a teacher educator to decide which ones to use and for what purposes. I also liked the diversity in instructional settings — and especially appreciated having some cases portraying classes with students with disabilities and limited English proficiency. Graeber and Valli provide the community, not only with good cases but also with thoughtful reflections, in the form of short commentaries, by a variety of relevant voices and various perspectives.

I am eager to use these cases in my own teacher education courses. While many works address the theoretical and practical underpinnings needed to deliver exemplary math instruction, few explore how this looks in practice. To meet this need, Graeber emer. Each chapter provides a case study that explores a range of subject matter as delineated by standards, a variety of pedagogical approaches, and a mixture of instructor experience levels and learning settings.