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Teaching in today's mixed-ability classroom can be a challenge. These days, it's not uncommon to find a wide range of abilities in the one classroom—from students struggling to grasp new concepts, to those who are way ahead of their peers from day one. This factor has contributed to a range of problems for early math learners, including a large achievement gap between students.
Not a MyNAP member yet? Register for a free account to start saving and receiving special member only perks. In the previous chapter, we examined teaching for mathematical proficiency. We now turn our attention to what it takes to develop proficiency in teaching mathematics. Proficiency in teaching is related to effectiveness: consistently helping students learn worthwhile mathematical content.
Proficiency also entails versatility: being able to work effectively with a wide variety of students in different environments and across a range of mathematical content. Teaching in the ways portrayed in chapter 9 is a complex practice that draws on a broad range of resources.
Despite the common myth that teaching is little more than common sense or that some people are just born teachers, effective teaching practice can be learned. In this chapter, we consider what teachers need to learn and how they can learn it. First, what does it take to be proficient at mathematics teaching?
If their students are to develop mathematical proficiency, teachers must have a clear vision of the goals of instruction and what proficiency means for the specific mathematical content they are teaching.
They need to know the mathematics they teach as well as the horizons of that mathematics—where it can lead and where their students are headed with it. They need to be able to use their knowledge flexibly in practice to appraise and adapt instructional materials, to represent the content in honest and accessible ways, to plan and conduct instruction, and to assess what students are learning. A Chinese teacher on how a profound understanding of fundamental mathematics is attained.
One thing is to study whom you are teaching, the other thing is to study the knowledge you are teaching. If you can interweave the two things together nicely, you will succeed…. Believe me, it seems to be simple when I talk about it, but when you really do it, it is very complicated, subtle, and takes a lot of time. It is easy to be an elementary school teacher, but it is difficult to be a good elementary school teacher. Used by permission from Lawrence Erlbaum Associates.
Teaching requires the ability to see the mathematical possibilities in a task, sizing it up and adapting it for a specific group of students. In short, teachers need to muster and deploy a wide range of resources to support the acquisition of mathematical proficiency. In the next two sections, we first discuss the knowledge base needed for teaching mathematics and then offer a framework for looking at proficient teaching of mathematics.
In the last two sections, we discuss four programs for developing proficient teaching and then consider how teachers might develop communities of practice. Three kinds of knowledge are crucial for teaching school mathematics: knowledge of mathematics, knowledge of students, and knowledge of instructional practices.
Mathematical knowledge includes knowledge of mathematical facts, concepts, procedures, and the relationships among them; knowledge of the ways that mathematical ideas can be represented; and knowledge of mathematics as a discipline—in particular, how mathematical knowledge is produced, the nature of discourse in mathematics, and the norms and standards of evidence that guide argument and proof. In our use of the term, knowledge of mathematics includes consideration of the goals of mathematics instruction and provides a basis for discriminating and prioritizing those goals.
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.
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. Third, advanced mathematical.
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 a MyNAP member yet? Register for a free account to start saving and receiving special member only perks. In the previous chapter, we examined teaching for mathematical proficiency. We now turn our attention to what it takes to develop proficiency in teaching mathematics. Proficiency in teaching is related to effectiveness: consistently helping students learn worthwhile mathematical content. Proficiency also entails versatility: being able to work effectively with a wide variety of students in different environments and across a range of mathematical content. Teaching in the ways portrayed in chapter 9 is a complex practice that draws on a broad range of resources.
Students in the United States are struggling with math. Although average scores have increased over longer periods of time, they are not where they should be today. Math scores among students in the U. Within the school system, teachers are the most important factor contributing to student achievement. This one-hour webinar from the Consortium on Reaching Excellence in Education CORE discusses four best practices in math instruction and related research based instructional strategies math teachers can use in their classrooms to improve student achievement.
We all want our kids to succeed in math. In most districts, standardized tests are the way understanding is measured, yet nobody wants to teach to the test. Being intentional and using creative approaches to your instruction can get students excited about math. As early as second grade, girls have internalized the idea that math is not for them. Rather than being born with or without math talent, kids need to hear from teachers that anyone who works hard can succeed. Psst… you can snag our growth mindset posters for your math classroom here.
Keywords: Instructional Strategies, Teacher, stpetersnt.org, Mathematics, Learning a method of teaching must be selected by the teachers which should be interesting.
A thirty students sample was taken from six Government elementary schools and divided them into two groups, one was experimental and the second was control group. This article offers a brief introduction to some inductive teaching strategies, and how to implement them in class. It is a method of verification.
In contemporary education , mathematics education is the practice of teaching and learning mathematics , along with the associated scholarly research. Researchers in mathematics education are primarily concerned with the tools, methods and approaches that facilitate practice or the study of practice; however, mathematics education research , known on the continent of Europe as the didactics or pedagogy of mathematics, has developed into an extensive field of study, with its concepts, theories, methods, national and international organisations, conferences and literature. This article describes some of the history, influences and recent controversies. Elementary mathematics was part of the education system in most ancient civilisations, including Ancient Greece , the Roman Empire , Vedic society and ancient Egypt.
Пожалуйста, ваше удостоверение. Сьюзан протянула карточку и приготовилась ждать обычные полминуты. Офицер пропустил удостоверение через подключенный к компьютеру сканер, потом наконец взглянул на. - Спасибо, мисс Флетчер. - Он подал едва заметный знак, и ворота распахнулись.
Все смешалось в ее голове - лица бесчисленных мужчин, склонявшиеся над ней, потолки гостиничных номеров, в которые она смотрела, мечты о том, что когда-нибудь все это кончится и она заведет детей… Внезапно, без всякого предупреждения, тело немца выгнулось, замерло и тут же рухнуло на. Это. - подумала она удивленно и с облегчением и попыталась выскользнуть из-под. - Милый, - глухо прошептала. - Позволь, я переберусь наверх.
Ничего похожего. У Халохота был компьютер Монокль, мы и его проверили. Похоже, он не передал ничего хотя бы отдаленно похожего на набор букв и цифр - только список тех, кого ликвидировал. - Черт возьми! - не сдержался Фонтейн, теряя самообладание. - Он должен там. Ищите. Джабба окончательно убедился: директор рискнул и проиграл.
Вот. Он печально на нее посмотрел.
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