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An understanding of how students learn and teachers teach is at the core of education policy.

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2015 SSHRC Knowledge Synthesis Grants - Faculty of Education

Problem solving as a method of teaching may be used to accomplish the instructional goals of learning basic facts, concepts, and procedures, as well as goals for problem solving within problem contexts. For example, if students investigate the areas of all triangles having a fixed perimeter of 60 units, the problem solving activities should provide ample practice in computational skills and use of formulas and procedures, as well as opportunities for the conceptual development of the relationships between area and perimeter. The "problem" might be to find the triangle with the most area, the areas of triangles with integer sides, or a triangle with area numerically equal to the perimeter. Thus problem solving as a method of teaching can be used to introduce concepts through lessons involving exploration and discovery. The creation of an algorithm, and its refinement, is also a complex problem solving task which can be accomplished through the problem approach to teaching. Open ended problem solving often uses problem contexts, where a sequence of related problems might be explored. For example, the problems in the investigations in the insert evolved from considering gardens of different shapes that could be enclosed with 100 yards of fencing:

Study Design and Setting: We discuss how to complete a broad range of knowledge syntheses.


Constructivist theories have received considerable acceptance in mathematics education in recent years. In the constructivist perspective, the learner must be actively involved in the construction of one's own knowledge rather than passively receiving knowledge. The teacher's responsibility is to arrange situations and contexts within which the learner constructs appropriate knowledge (45,48). Even though the constructivist view of mathematics learning is appealing and the theory has formed the basis for many studies at the elementary level, research at the secondary level is lacking. Our review has not uncovered problem solving research at the secondary level that has its basis in a constructivist perspective. However, constructivism is consistent with current cognitive theories of problem solving and mathematical views of problem solving involving exploration, pattern finding, and mathematical thinking (36,15,20); thus we urge that teachers and teacher educators become familiar with constructivist views and evaluate these views for restructuring their approaches to teaching, learning, and research dealing with problem solving.

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Results: Many groups worldwide conduct knowledge syntheses, and some Methods, are applicable to most reviews.

Educational research is conducted within a variety of constraints -- isolation of variables, availability of subjects, limitations of research procedures, availability of resources, and balancing of priorities. Various research methodologies are used in mathematics education research including a clinical approach that is frequently used to study problem solving. Typically, mathematical tasks or problem situations are devised, and students are studied as they perform the tasks. Often they are asked to talk aloud while working or they are interviewed and asked to reflect on their experience and especially their thinking processes. Waters (48) discusses the advantages and disadvantages of four different methods of measuring strategy use involving a clinical approach. Schoenfeld (32) describes how a clinical approach may be used with pairs of students in an interview. He indicates that "dialog between students often serves to make managerial decisions overt, whereas such decisions are rarely overt in single student protocols."

The basis for most mathematics problem solving research for secondary school students in the past 31 years can be found in the writings of Polya (26,27,28), the field of cognitive psychology, and specifically in cognitive science. Cognitive psychologists and cognitive scientists seek to develop or validate theories of human learning (9) whereas mathematics educators seek to understand how their students interact with mathematics (33,40). The area of cognitive science has particularly relied on computer simulations of problem solving (25,50). If a computer program generates a sequence of behaviors similar to the sequence for human subjects, then that program is a model or theory of the behavior. Newell and Simon (25), Larkin (18), and Bobrow (2) have provided simulations of mathematical problem solving. These simulations may be used to better understand mathematics problem solving.

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It is useful to develop a framework to think about the processes involved in mathematics problem solving. Most formulations of a problem solving framework in U. S. textbooks attribute some relationship to Polya's (26) problem solving stages. However, it is important to note that Polya's "stages" were more flexible than the "steps" often delineated in textbooks. These stages were described as understanding the problem, making a plan, carrying out the plan, and looking back. To Polya (28), problem solving was a major theme of doing mathematics and "teaching students to think" was of primary importance. "How to think" is a theme that underlies much of genuine inquiry and problem solving in mathematics. However, care must be taken so that efforts to teach students "how to think" in mathematics problem solving do not get transformed into teaching "what to think" or "what to do." This is, in particular, a byproduct of an emphasis on procedural knowledge about problem solving as seen in the linear frameworks of U. S. mathematics textbooks (Figure 1) and the very limited problems/exercises included in lessons.

Countries from Low & Middle-Income have been involved in the production of systematic reviews and other forms of evidence synthesis. Some have even been able to translate the knowledge and use it in decision-making. GESI aims to enhance these efforts by building Evidence Synthesis Centres in these countries, increasing the capacity to synthesise evidence and using evidence in local and regional practice and policy decision-making.

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Another aspect of problem solving that is seldom included in textbooks is problem posing, or problem formulation. Although there has been little research in this area, this activity has been gaining considerable attention in U. S. mathematics education in recent years. Brown and Walter (3) have provided the major work on problem posing. Indeed, the examples and strategies they illustrate show a powerful and dynamic side to problem posing activities. Polya (26) did not talk specifically about problem posing, but much of the spirit and format of problem posing is included in his illustrations of looking back.

The Educational Theory of Herbert Spencer - …

... science as the most important subject matter; curriculum to be a synthesis of thought based on science (especially evolution), and including philosophies of education, biology, psychology, sociology, ethics, and politics (Magill); sciences are superior in all respects to languages as educational material; history -- nature and action of government, intellectual condition of the nation, description of people's food, shelters, and amusements, importance of drawing in education (Spencer, p. xiv); individual to teach self, aided by teachers, books, observation, laboratory work

Herbert Spencer's theory of education analyzed into eight factors.

Knowledge Synthesis Grants are not intended to support original research. Rather, they are intended to support the synthesis of existing research knowledge and the identification of knowledge gaps. This call is particularly focused on the state of research knowledge emerging over the past 10 years.

Knowledge | Internet Encyclopedia of Philosophy

Proposals should address one or more of the specific questions listed under the themes. However, applicants may choose to identify and/or develop specific aspects or elements within the questions to frame the knowledge synthesis to be produced.

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