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When students begin work with fraction division in fifth grade or sixth grade, they bring with them experiences from whole-number division. Many students think that a division problem should lead to a quotient that is smaller than the dividend. It is also common for students to believe that the dividend should be larger than the divisor. Many, if not most, of the whole-number division problems that students have worked with are of the form 15 ÷ 5 = 3 or 666 ÷ 18 = 37, where "division makes smaller." When students begin to study fraction division, it is important that their understanding of division is expanded to address why it is possible to divide two numbers and find a quotient that is larger than either dividend or divisor. Supporting students to develop efficient procedures for fraction division, although important, should not be the only goal. Instruction should also support the development of number and operation sense. Fraction magnitude is a prerequisite for developing such number sense and operation sense (Behr et al. 1984). With fractions, number sense includes understanding magnitude or the size of an individual fraction in relation to a whole and in relation to other fractions. With fraction division, understanding what a reasonable quotient might be involves being able to compare the magnitude of the dividend with the magnitude of the divisor when the operation of division is carried out. This article explores both modeling and equivalence as reasoning tools to support students' developing number sense and understanding of the measurement interpretation of fraction division.

Estimation is more than a skill or an isolated topic. It is a thinking tool that needs to be emphasized during instruction so that students will learn to develop algorithmic procedures and meaning for fraction operations. For students to realize when fractions should be added, subtracted, multiplied, or divided, they need to develop a sense of size for various fraction quantities. They also need number sense, which will enable them to use estimation as a tool with fractions. When wrestling with quantities like three-fifths or nine-tenths, as individual fractions and in relation to each other, students will develop a sense of size and quantity for individual fractions. This experience is necessary if students are to develop meaning for fraction operations. This article explores the ways that estimation is used when students are learning to add, subtract, multiply, and divide fractions. (Contains 1 table and 2 figures.)

This article describes one prevalent practice that a group of 6th- and 7th-grade students engaged in when they used fractions in the context of area and perimeter, decimal operations, similarity, and ratios and proportions. The study's results revealed that students did not simply take the concepts and skills learned in formal fractions units and use them in these other mathematical content areas. Their understanding of how to use fractions was tied to their understanding of situations in which they could be used. Students had to take into account both mathematical and situational contexts when making choices about how to use fractions. This led students to raise questions regarding what was appropriate when using fractions in these new contexts and how fractions and the new contexts were related. (Contains 3 tables, 6 figures, and 3 footnotes.)

This article will share results from research that investigated how sixth-, seventh-, and eighth-grade students who had not been exposed to formal algebraic methods approached word problems of an algebraic nature. Student use of systematic guess and check, the predominate approach taken by these students, is the focus. The goal is to consider the students' use of systematic guess and check reasoning in terms of the broadening perspective of algebra and algebraic thinking by highlighting ways in which this reasoning can provide a basis for developing some of the thinking patterns and discourse of formal algebra. Two perspectives will be highlighted "relationships among quantities and function-based reasoning." (Contains 1 table and 3 figures.)

This study explored the problem-solving schemas developed by 7th-grade pre-algebra students as they participated in a teaching experiment that was designed to help students develop effective schemas for solving algebraic problem situations involving contexts of (1) growth and change and (2) size and shape. This article describes the qualities and types of schemas that students developed through examples of problems used to develop schemas, examples of students' reasoning and writing, and excerpts from student interviews. Findings from the study indicated that there is a link between the type of generalizations students construct and the schemas they are forming. By using these schemas students recognize, extend, and generalize patterns and quantitative relationships both verbally and symbolically.

Uses qualitative analysis to understand how middle school students thought about and approached problems while they wrote descriptions of why and how they solve problems. Indicates that rich learning experiences are possible when writing is used as a way to prepare for small group discussion in mathematics. (Contains 13 references.) (Author/ASK)

The article presents a lesson plan for primary education which teaches the development of algorithms through the addition and subtraction of fraction.

In the course of a school year teachers are faced with a full calendar of mathematics to cover. Every minute matters. Engaging students in mathematical activity as soon as they enter the door increases the number of minutes they spend studying mathematics in a year. Warm-ups can be used to review and practice skills. They can be used to build new ideas and forms of mathematical thinking. This paper shares three different sets of warm-ups that can be used with middle school students to encourage different types of mathematical thinking and reasoning. This article was supported by the Middle Grades Mathematics Initiative project funded by The Ohio Board of Regents Improving Teacher Quality Grant No. 08-39. [ABSTRACT FROM AUTHOR]

The article provides mathematical concepts to help students develop their learning capacity for transfer. It highlights the different concepts in learning transfer including transfer via connections, transfer via contrasting ideas, and transfer via error analysis. It offers guidelines for aiding the students ability to transfer including the importance of building on student's experiences, and the use of instructions that deepens the student's mathematical knowledge.

The article offers examples of instructional task of how middle-grades students can pace from the foreground and background on the study of mathematics. It notes a coherent curriculum that was developed based the study of fractions taken from a curriculum unit on decimal operations, similarity. Units of fraction and similarity can lead a way wherein a curriculum can be organized to provide continuing, non-repetitive experiences for students in understanding mathematics.

Explores the processes employed by a sixth-grade teacher in establishing a classroom environment that promoted the development of autonomous learners. Need for students to assume autonomy for their own learning; Efforts of the teacher to cultivate her student's role as learners of mathematics; Ways in creating a classroom learning community.

The current reform effort in mathematics education emphasizes the importance of students' being active in their own development and understanding of mathematical ideas. The reform recommends that students read, write, and discuss mathematics as a way of developing mathematical thinking. This study emphasizes writing as a way to help students learn to think mathematically. Typically, writing has been used as a dialogue between the teacher and the student or as a product for the teacher to examine. However, this study is significant because students wrote as a preparation for discourse with the teacher and other students. Qualitative analysis was used to understand how middle school students think about and approach problems while they wrote descriptions of why and how they solve problems. For this study, students wrote detailed descriptions of their approaches to nonroutine problems and then met in groups to discuss and evaluate their approaches. In order to understand their mathematical thinking, the researcher examined students' writing, participated in and audiotaped group discussions, and interviewed students. The results of the analysis showed that rich learning experiences are possible when writing is used as a way to prepare for small group discussion in mathematics. [ABSTRACT FROM AUTHOR]