gravatunlebfi.tk/dont-be-that-man-how.php The concept of ratio is much more difficult than many people realize. Proportional reasoning is the term given to reasoning that involves the equality and manipulation of ratios.
Children often have difficulty comparing ratios and using them to solve problems. Research tracing the development of proportional reasoning shows that proficiency grows as students develop and connect different aspects of proportional reasoning. Further, the development of proportional reasoning can be supported by having students explore proportional situations in a variety of problem contexts using concrete materials or through data collection activities.
We see ratio and proportion as underdeveloped components of grades pre-K-8 mathematics:.
The curriculum should provide extensive opportunities over time for students to explore proportional situations concretely, and these situa tions should be linked to formal procedures for solving proportion problems whenever such procedures are introduced. Students often view the study of whole numbers, decimal fractions, common fractions, and integers as disconnected topics. One tool that we believe may be useful in developing numerical understanding and in making connections across number systems is the number line, a geometric representation of numbers that gives each number a unique point on the line and an oriented distance from the origin, depicting its magnitude and direction.
Although it may be difficult to learn, the number line gives a unified geometric representation of integers and rational numbers within the real number system, later to be encountered in geometry, algebra, and calculus. The geometric models of operations afforded by the number line apply uniformly to all real numbers, thus presenting one unified number system.
The number line may become particularly useful as students are learning about integers and rational numbers, for it may help students develop a sense of the magnitudes and relationships of those numbers in a way that is less clear in other representations:. Because it can serve as a tool for simultaneously representing whole numbers, integers, and rational numbers, teachers and researchers should explore effective uses of the number line representation when students learn about operations with numbers, relations among number systems, and more formal symbolic representations of numbers.
Students currently encounter the expansion of the number domain by starting with whole numbers, gradually incorporating fractions, and only much later expanding the domain to include negative integers and irrational numbers. That sequence has a long history, but there are arguments for an alternative. For example, expanding the whole numbers to take in the negative integers in the early grades would allow students to do more with addition and subtraction before venturing into the rational number system, which requires multiplication and division.
Systematic study of this alternative is needed:. Teachers, curriculum developers, and researchers should explore the possibility of introducing integers before rational numbers.
Methods of Solving Problems in Elementary, Middle, and High School Mathematics: Mihai Rosu: edutiwytil.tk: The Book Depository UK. METHODS OF SOLVING PROBLEMS IN Elementary, Middle, and High School MATHEMATICS. By Mihai Rosu. Also available as: E-Book, Perfect Bound.
Ways to engage younger children in meaningful uses of negative integers should be devel oped and tested. The formal study of algebra is both the gateway into advanced mathematics and a stumbling block for many students. The transition from arithmetic to algebra is often not an easy one. The difficulties associated with the transition from the activities typically associated with school arithmetic to those typically associated with school algebra representational activities, transformational activities, and generalizing and justifying activities have been extensively studied.
Research has documented that the visual and numerical supports provided for symbolic expressions by computers and graphing calculators help students create meaning for expressions and equations. The research, however, has shed less light on the long-term acquisition and retention of transformational fluency.
Although through generalizing and justifying, students can learn to use and appreciate algebraic expressions as general statements, more research is need on how students develop such awareness. The study of algebra, however, does not have to begin with a formal course in the subject. New lines of research and development are focusing on ways that the elementary and middle school curriculum can be used to support the development of algebraic reasoning. These efforts attempt to avoid the difficulties many students now experience and to lay a better foundation for secondary school mathematics.
We believe that from the earliest grades of elementary school, students can be acquiring the rudiments of algebra, particularly its representational aspects and the notion of variable and function. By emphasizing both the relationships among quantities and ways of representing these relationships, instruction can introduce students to the basic ideas of algebra as a generalization of arithmetic. They can come to value the roles of definitions and see how the laws of arithmetic can be expressed algebraically and be used to support their reasoning. We recommend that algebra be explicitly connected to number in grades pre-K The basic ideas of algebra as generalized arithmetic should be anticipated by activities in the early elementary grades and learned by the end of middle school.
Teachers and researchers should investigate the effectiveness of instructional strategies in grades pre-K-8 that would help students move from arithmetic to algebraic ways of thinking. In some countries by the end of eighth grade, all students have been studying algebra for several years, although not ordinarily in a separate course.
In the United States, however, some efforts to promote algebra for all have involved simply offering a standard first-year algebra course algebra through quadratics to everyone. We believe such efforts are virtually guaranteed to result in many students failing to develop proficiency in algebra, in part because the transition to algebra is so abrupt. Instead, a different curriculum is needed for algebra in middle school:. Teachers, researchers, and curriculum developers should explore ways to offer a middle school curriculum in which algebraic ideas are devel oped in a robust way and connected to the rest of mathematics.
Research has shown that instruction that makes productive use of computer and calculator technology has beneficial effects on understanding and learning algebraic representation. It is not clear, however, what role the newer symbol manipulation technologies might play in developing proficiency with the transformational aspects of algebra. We recommend the following:. An important part of our conception of mathematical proficiency involves the ability to formulate and solve problems coming from daily life or other domains, including mathematics itself.
That ability is not being developed well in U. Studies in almost every domain of mathematics have demonstrated that problem solving provides an important context in which students can learn about number and other mathematical topics. Problem-solving ability is enhanced when students have opportunities to solve problems themselves and to see problems being solved. Further, problem solving can provide the site for learning new concepts and for prac-. We believe problem solving is vital because it calls on all strands of proficiency, thus increasing the chances of students integrating them.
Other activities, such as listening to an explanation or practicing solution methods, can help develop specific strands of proficiency, but too much emphasis on them, to the exclusion of solving problems, may give a one-sided character to learning and inhibit the formation of connections among the strands. We see problem solving as central to school mathematics:. Problem solving should be the site in which all of the strands of math ematics proficiency converge. Analyses of the U. How teachers might understand and use instructional materials to help students develop mathematical proficiency is not well understood.
On the basis of our reasoned judgment, we offer the following recommendations for improving instructional materials in school mathematics:. Textbooks and other instructional materials should develop the core content of school mathematics in a focused way, in depth, and with continu ity in and across grades, supporting all strands of mathematical proficiency. Textbooks and other instructional materials should support teacher understanding of mathematical concepts, of student thinking and student errors, and of effective pedagogical supports and techniques.
Activities and strategies should be developed and incorporated into instructional materials to assist teachers in helping all students become proficient in mathematics, including students low in socio-economic status, English language learners, special education students, and students with a special interest or talent in mathematics.
Efforts to develop textbooks and other instructional materials should include research into how teachers can understand and use those materials effectively.
A government agency or research foundation should fund an inde pendent group to analyze textbooks and other instructional materials for the extent to which they promote mathematical proficiency. The group should recommend how these materials might be modified to promote greater math ematical proficiency. Research indicates that a key requirement for developing proficiency is the opportunity to learn.
In many U. On some days in some classes they are spending little or no time at all on the subject. Mathematical proficiency as we have defined it cannot be developed unless regular time say, one hour each school day is allocated to and used for mathematics instruction in every grade of elementary and middle school. Further, we believe the strands of proficiency will not develop in a coordinated fashion unless continual attention is given to every strand. The following recommendation expresses our concern that mathematics be given its rightful place in the curriculum:.
Mathematical proficiency as we have defined it cannot be developed unless regular time is allocated to and used for mathematics instruction in every grade of elementary and middle school. Substantial time should be devoted to mathematics instruction each school day, with enough time devoted to each unit and topic to enable stu dents to develop understanding of the concepts and procedures involved.
Time should be apportioned so that all strands of mathematical proficiency together receive adequate attention. Practice is important in the development of mathematical proficiency. When students have multiple opportunities to use the computational procedures, reasoning processes, and problem-solving strategies they are learning, the methods they are using become smoother, more reliable, and better understood. Practice alone does not suffice; it needs to be built on understanding and accompanied by feedback. In fact, premature practice has been shown to be harmful.
The following recommendation reflects our view of the role of practice:.
Practice should be used with feedback to support all strands of math ematical proficiency and not just procedural fluency. In particular, practice on computational procedures should be designed to build on and extend under standing. At present, substantial time every year is taken away from mathematics instruction in U. Often, those tests are not well articulated with the mathematics curriculum, testing content that has not been taught during the year or that is not central to the development of mathematical proficiency.
Preparation for such tests, moreover, does not ordinarily focus on the development of proficiency. Instead, much time is given to practicing calculation procedures and reviewing a multitude of topics.
Teachers and students often waste valuable learning time because they are not informed about the content to be tested or the form that test items will take. It needs to provide opportunities for students to learn rather than taking time away from their learning. Assessments in which students are learning as well as showing what they have already learned can provide valuable information to teachers, schools, districts, and states, as well as the students themselves.
Such assessments help teachers modify their instruction to support better learning at each grade level. Time and money spent on assessment need to be used more effectively so that students have the opportunity to show what they know and can do. In that way, students and teachers alike can learn from assessments instead of having assessments used only to rank students, teachers, or schools.
The following recommendations will help make assessment more effective in developing mathematical proficiency:. Students and teachers alike can learn from assessments instead of having assessments used only to rank students, teachers, or schools. Assessment, whether internal or external, should be focused on the development and achievement of mathematical proficiency.
In particular, assessments used to determine qualification for state and federal funding should reflect the definition of mathematics proficiency presented in this report. Information about the content and form of each external assessment should be provided so that teachers and students can prepare appropriately and efficiently. The results of each external assessment should be reported so as to provide feedback useful for teachers and learners rather than simply a set of rankings.
A government agency or research foundation should fund an inde pendent group to analyze external assessment programs for the extent to which they promote mathematical proficiency. The group should recommend how programs might be modified to promote greater mathematical proficiency. Effective teaching—teaching that fosters the development of mathematical proficiency over time—can take a variety of forms. Consequently, we endorse no single approach.
All forms of instruction configure relations among teachers, students, and content. The development of mathematical proficiency requires thoughtful planning, careful execution, and continual improvement of instruction. It depends critically on teachers who understand mathematics, how students learn, and the classroom practices that support that learning. They also need to know their students: who they are, what their backgrounds are, and what they know.
Planning, whether for one lesson or a year, is often viewed as routine and straightforward. However, plans seldom elaborate the content that the students are to learn or develop good maps of paths to take to reach learning goals. Instructional materials need to support teachers in their planning, and teachers need to have time to plan. Instruction needs to be planned with the development of mathematical proficiency in mind:. Content, representations, tasks, and materials should be chosen so as to develop all five strands of proficiency toward the big ideas of math ematics and the goals for instruction.
Planning for instruction should take into account what students know, and instruction should provide ways of ascertaining what students know and think as well as their interests and needs. Rather than simply listing problems and exercises, teachers should plan for instruction by focusing on the learning goals for their students, keep ing in mind how the goals for each lesson fit with those of past and future lessons.
Their planning should anticipate the events in the lesson, the ways in which the students will respond, and how those responses can be used to further the lesson goals. Mathematics classrooms are more likely to be places in which mathematical proficiency develops when they are communities of learners and not collections of isolated individuals. Research on creating classrooms that function as communities of learners has identified several important features of these classrooms: ideas and methods are valued, students have autonomy in choosing and sharing solution methods, mistakes are valued as sites of learning for everyone, and the authority for correctness lies in logic and the structure of the subject, not in the teacher.
In such classrooms the teacher plays a key role as the orchestrator of the discourse students engage in about mathematical ideas. Teachers are responsible for moving the mathematics along while affording students opportunities to offer solutions, make claims, answer questions, and provide explanations to their peers. Teachers need to help bring a mathematical discussion to a close, making sure that gaps have been filled and errors addressed. To develop mathematical proficiency, we believe that students require more than just the demonstration of procedures. They need experience in investigating mathematical properties, justifying solution methods, and analyzing problem situations.
A significant amount of class time should be spent in developing math ematical ideas and methods rather than only practicing skills. Discourse should not be confined to answers only but should include discussion of connections to other problems, alternative representations and solution methods, the nature of justification and argumentation, and the like.
Students acquire higher levels of mathematical proficiency when they have opportunities to use mathematics to solve significant problems as well as to learn the key concepts and procedures of that mathematics. Although mathematics gains power and generality through abstraction, it finds both its sources and applications in concrete settings, where it is made meaningful to the learner. There is an inevitable dialectic between concrete and abstract in which each helps shape the other.
Research reveals that various kinds of physical materials commonly used to help children learn mathematics are often no more concrete to them than symbols on paper might be. Concrete is not the same as physical. To ensure that progress is made toward mathematical abstraction, we recommend the following:. Part of becoming proficient in mathematics is becoming an independent learner. For that purpose, many teachers give homework. Neither the quality nor the function of homework has been studied. Homework can have different purposes.
For example, it might be used to practice skills or to prepare the student for the next lesson. We believe that independent work serves several useful purposes. Regarding independence and homework, we make the following recommendations:. Students should be provided opportunities to work independently of the teacher both individually and in pairs or groups.
When homework is assigned for the purpose of developing skill, stu dents should be sufficiently familiar with the skill and the tasks so that they are not practicing incorrect procedures. In the discussion above, we mention the special role that calculators and computers can play in learning algebra. But they have many other roles to play throughout instruction in grades pre-K Using calculators and computers does not replace the need for fluency with other methods.
Confronted with a complex arithmetic problem, students can use calculators and computers to see beyond tedious calculations to the strategies needed to solve the problem. Technology can relieve the computational burden and free working memory for higher-level thinking so that there can be a sharper focus on an important idea. Further, skillfully planned calculator investigations may reveal subtle or interesting mathematical ideas, such as the rules for order of operations. A large number of empirical studies of calculator use, including long-term studies, have generally shown that the use of calculators does not threaten the development of basic skills and that it can enhance conceptual understanding, strategic competence, and disposition toward mathematics.
For example, students who use calculators tend to show improved conceptual understanding, greater ability to choose the correct operation, and greater skill in estimation and mental arithmetic without a loss of basic computational skills. They are also familiar with a wider range of numbers than students who do not use calculators and are better able to tackle realistic mathematics problems. Timothy Gowers. How to Study for a Mathematics Degree. Lara Alcock. Fermat's Last Theorem. Simon Singh. The Signal and the Noise. Nate Silver. How to Think Like a Mathematician. Kevin Houston.
Mental Arithmetic 4. Power of 2. David Joseph Sharp. Game Theory. Morton D. James Gleick. Cryptography: A Very Short Introduction. Fred C. Ken Binmore. For another thing, students are more active, more at the center of the classroom, than in the traditional model; their choices count and their voices are heard. Even parents who are open to being convinced of the value of Whole Language may seem skeptical about conceptual approaches to math.
The idea of reading for understanding is clear enough few adults, after all, spend their time underlining topic sentences or circling vowels , but how many of us have any experience with math instruction that emphasizes understanding? Indeed, it asks a lot for people to support, or even permit, a move from something they know to something quite unfamiliar. As with any of the issues discussed in this book, there are basically three ways to convince skeptics.
Finally, there are examples, ideally gleaned from first-hand observation of extraordinary classrooms — or, next best, descriptions that give a flavor of what these ideas look like in practice and how they compare to the usual fare. Now imagine a teacher who has first graders figure out how many plastic links placed on one side of a balance are equivalent to one metal washer on the other side.
Then, after discovering that the same number of links must be added again to balance an additional washer, the children come to make sense of the concept of ratio for themselves. Which approach do you suppose will lead to a deeper understanding? In which classroom are they more likely to see math as relevant, appealing, and something at which they can be successful?
Finally, consider — which is to say, remember — a conventional middle-school or high school math curriculum. Which is more rigorous? Which do you want for your child? The key question is whether understanding is passively absorbed or actively constructed. In the latter case, math actually becomes a creative activity. She can be excused for ignorance of this detail since it was omitted from virtually all discussions of those results in the popular press. More interesting is her belief that it would obviously be ludicrous to have mathematical laws reinvented by students.
Coincidentally, the very same example was offered by Piaget several decades earlier to argue in favor of this kind of learning. By thinking through the possibilities, students come up with their own ways of finding solutions. They have to invent their own procedures. He has to bite his tongue a lot, and also refrain from having children put their answers down on paper too early, since that can get in the way of really thinking through the problems. I believe this whole approach makes sense for four reasons.
It has to be constructed. As Kamii comments,.
Wrong ideas have to be modified by the child. They cannot be eliminated by the teacher. An open-ended invitation to tackle a new kind of problem lets the teacher see how they think, whether they can integrate earlier concepts, and exactly where they get stuck — as opposed to judging only whether they got the right answer. The third argument for this approach is that it really works.
She wanted to know how many were left. I watched the children as they struggled to solve the problem mentally. They were extremely quiet. Some of them were staring into space intently, as if they were solving the problem on an invisible chalkboard. Others sat nodding their heads as if in rhythm to the numbers. A few manipulated their fingers, and one child was biting his lip and looking quite puzzled.
After several minutes of thinking time had elapsed, most of the children had raised a hand to let me know that they had an answer to share with the group. Their proposed answers were 29, 22, 18, 28, and The children immediately got involved in thinking about the answers that had been proposed. Steve, on the other hand, was not so tentative. He simultaneously raised his hand, stood up, and began to speak. So 29 is too high. Ben barely waited for Steve to pause and take a breath.
So, 40 minus 20 is Take away 2 more. I was not disappointed. Over the next few weeks, they wrestled with other problems. In fact, though, not only is it inaccurate to say that a constructivist math classroom is based on that relativistic premise, but exactly the opposite is true. By now we understand that the teacher is vitally active, integrally involved. She sets things up so students can play with possibilities, think through problems, converse and revise. But we can say more than that this approach is effective.
The final justification for teaching math this way is that the conventional transmission approach can be positively harmful. A teacher or parent for whom the right answer means everything is one who will naturally want to tell the child the most efficient way of getting that right answer. This creates mindlessness.
Such a student, armed with algorithms, gets in the habit of looking to the adult, or the book, instead of thinking it through herself. She feels less autonomous, more dependent. But these kids might as well be learning nonsense syllables.