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The contents of this zip file can be downloaded from the attachments area to the right. Guidance on organisation and planning to cater for more able pupils in mathematics, with advice on other sources of help and material. This book addresses class organisation, planning and teaching through answers to commonly asked questions. The book provides 98 pages of puzzles and problems for Key Stages 1 and 2.
They are intended to challenge pupils and extend their thinking. While some of them may be solved fairly quickly, others will need perseverance and may extend beyond a single lesson. Pupils may need to draw on a range of skills to solve the problems. Register for an account, or log in if you are already a member. Show all recent publications. Skip to page content Skip to main navigation Accessibility help Text size: Site search Search this site: Primary Mathematics subject area Lesson resources remove.
Mathematical challenges for able pupils in Key Stages 1 and 2. Page 1 of 11 Next. Key Stage 1 5 How can I adapt my termly planning? Key Stage 2 6 How can I use the 'extra' week each term? Lead intervention teacher, Subject leader, Teacher Function: Knowing mathematics for teaching also entails more than knowing mathematics for oneself. Teachers certainly need to be able to understand concepts correctly and perform procedures accurately, but they also must be able to understand the conceptual foundations of that knowledge.
In the course of their work as teachers, they must understand mathematics in ways that allow them to explain and unpack ideas in ways not needed in ordinary adult life. Knowledge of students and how they learn mathematics includes general knowledge of how various mathematical ideas develop in children over time as well as specific knowledge of how to determine where in a developmental trajectory a child might be.
Buy Teaching Mathematics to Able Children 1 by Valsa Koshy (ISBN: ) from Amazon's Book Store. Everyday low prices and free delivery on. Mathematics to Able. Children. Valsa Koshy. David Fulton Publishers. London. Teaching. Mathematics to Able. Children. Valsa Koshy. David Fulton Publishers.
It includes familiarity with the common difficul-. Knowledge of instructional practice includes knowledge of curriculum, knowledge of tasks and tools for teaching important mathematical ideas, knowledge of how to design and manage classroom discourse, and knowledge of classroom norms that support the development of mathematical proficiency. Teaching entails more than knowledge, however. Teachers need to do as well as to know. For example, knowledge of what makes a good instructional task is one thing; being able to use a task effectively in class with a group of sixth graders is another.
Understanding norms that support productive classroom activity is different from being able to develop and use such norms with a diverse class. Because knowledge of the content to be taught is the cornerstone of teaching for proficiency, we begin with it. Many recent studies have revealed that U. The mathematical education they received, both as K students and in teacher preparation, has not provided them with appropriate or sufficient opportunities to learn mathematics. As a result of that education, teachers may know the facts and procedures that they teach but often have a relatively weak understanding of the conceptual basis for that knowledge.
Many have difficulty clarifying mathematical ideas or solving problems that involve more than routine calculations. Teachers frequently regard mathematics as a fixed body of facts and procedures that are learned by memorization, and that view carries over into their instruction.
Many have little appreciation of the ways in which mathematical knowledge is generated or justified. Preservice teachers, for example, have repeatedly been shown to be quite willing to accept a series of instances as proving a mathematical generalization. Although teachers may understand the mathematics they teach in only a superficial way, simply taking more of the standard college mathematics courses does not appear to help matters. The evidence on this score has been consistent, although the reasons have not been adequately explored.
For example, a study of prospective secondary mathematics teachers at three major institutions showed that, although they had completed the upper-division college mathematics courses required for the mathematics major, they had only a cursory understanding of the concepts underlying elementary mathematics. For the most part, the results have been disappointing: Most studies have failed to find a strong relationship between the two. Many studies, however, have relied on crude measures of these variables.
The measure of teacher knowledge, for example, has often been the number of mathematics courses taken or other easily documented data from college.
Such measures do not provide an accurate index of the specific mathematics that teachers know or of how they hold that knowledge. Teachers may have completed their courses successfully without achieving mathematical proficiency. Or they may have learned the mathematics but not know how to use it in their teaching to help students learn.
They may have learned mathematics that is not well connected to what they teach or may not know how to connect it. It is widely believed that the more a teacher knows about his subject matter, the more effective he will be as a teacher. 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. 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.
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.
He wants to put all the cookies on 6 plates. National Center for Education Statistics. 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. School policies and practices affecting instruction in mathematics NCES 98— New practices for the new millennium. Journal for Research in Mathematics Education , 20 , 41—
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-. 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. 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.
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.