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Anne K. Morris and James Hiebert

Two studies were conducted to identify the conditions under which instructors teaching the same mathematics teacher preparation course would continuously improve their shared instructional products (lesson plans for class sessions) using small amounts of data on preservice teacher performance. Findings indicated that when lesson-level student performance data were simply collected, by course section, the instructors could make important changes to the lessons but did not often do so. However, when the instructors were encouraged to compare data across semesters, they generated hypotheses that guided instructional improvements, which then were tested through multiple cycles. The cycles of hypothesis testing helped instructors clarify the goals for improvement, use the performance data to test whether changes were actually improvements, and reduce their tolerance for marginal student performance.

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James Hiebert and Diana Wearne

Conceptually based instruction on place value and two-digit addition and subtraction without regrouping was provided in four first-grade classrooms, and more conventional textbook-based instruction was provided in two first-grade classrooms. An observer compiled extensive notes of 20 lessons in each kind of classroom. Students who received conceptually based instruction performed significantly better on items measuring understanding of place value and two-digit addition and subtraction with regrouping and used strategies more often that exploited the tens and ones structure of the number system. Content and pedagogical differences between the instruction lessons are linked to the learning differences and are used to explain between-group differences in levels of performance and understanding. Observations are offered on the complex interactions between instruction, understanding, and performance.

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Diana Wearne and James Hiebert

A theory is proposed for how students develop competence with the written symbols of decimal fractions. The theory views competence as the cumulative and sequential mastery of four separate cognitive processes in working with written symbols. The first two processes develop for the user the semantics of the symbol system, and the second two elaborate its syntax. Initial predictions from the theory were tested by instructing small groups of students for about 2 weeks. As predicted, (a) most students were able to acquire the first two processes, (b) these processes generated correct performance and transferred to novel tasks, and (c) it was more difficult for students to acquire the semantic-based processes once they had routinized syntactic processes. The theory and data are interpreted in the context of some persistent issues in mathematics learning.

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James Hiebert

Edited by Leroy G. Callahan

This report summarizes what research says about the learning and teaching of decimal fractions and suggests some instructional strategies and activities to improve learning. Decimal fractions ordinarily are introduced in grade 4 or 5 and are treated intensively in grades 6 and 7. Research has focused on students at these grade levels, although some studies have included high school students. The following presents some of the significant findings obtained primarily from studies of students in conventional instructional programs in the United States and other countries.

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James Hiebert and Lowell H. Tonnessen

Efforts to define the fraction concept have frequently led to the identification of several different interpretations, such as part-whole, ratio, quotient, and multiplicative operator (Kieren, 1976). The part-whole interpretation is particu larly important, since it is often used by elementary school mathematics programs to introduce the fraction concept.

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Diana Wearne and James Hiebert

Edited by Patricia F. Campbell

Students who get the correct answer may differ in their understanding .

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Thomas P. Carpenter, James M. Moser and James Hiebert

Forty-three first-grade children who had received no formal instruction in addition and subtraction were individually administered 10 verbal problems. These problems were selected to represent the following semantic types: Joining, Separating, Part-Part-Whole, Comparison, and Equalizing. In spite of the lack of formal instruction, most children successfully solved the problems. In general, children's solution processes were consistent with a predictive model proposing that solution processes would directly represent the action or relationship described in individual problems.

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James Hiebert, Thomas P. Carpenter and James M. Moser

This study investigated the relationships between first-grade children's performance on Piagetian logical reasoning tasks and an information processing task, and the processes they used to solve verbal addition and subtraction problems. The problems varied systematically in semantic structure, number size, and the availability of objects to aid the solution process. Statistically significant differences in arithmetic performance between the developmental groups were found for some of the cognitive variables. The information processing variable was the cognitive ability most consistently related to accuracy of solution and use of advanced solution strategies. However, none of the cognitive abilities was required to solve the arithmetic problems or to use a given solution strategy. This calls into question the use of these cognitive tasks as readiness variables for arithmetic instruction.

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James Hiebert, Thomas P. Carpenter and James M. Moser

Investigating relationships between major domains of knowledge is a complex task. A number of fundamental questions often accompany such attempts, and our study on the relationship between cognitive skills and arithmetic performance is no exception. Steffe and Cobb (1983) identified some of these questions. This interchange, we hope, will help to clarify the issues and the alternative views.

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Anne K. Morris, James Hiebert and Sandy M. Spitzer

The goal of this study is to uncover the successes and challenges that preservice teachers are likely to experience as they unpack lesson-level mathematical learning goals (i.e., identify the subconcepts and subskills that feed into target learning goals). Unpacking learning goals is a form of specialized mathematical knowledge for teaching, an essential starting point for studying and improving one's teaching. Thirty K–8 preservice teachers completed 4 written tasks. Each task specified a learning goal and then asked the preservice teachers to complete a teaching activity with this goal in mind. For example, preservice teachers were asked to evaluate whether a student's responses to a series of mathematics problems showed understanding of decimal number addition. The results indicate that preservice teachers can identify mathematical subconcepts of learning goals in supportive contexts but do not spontaneously apply a strategy of unpacking learning goals to plan for, or evaluate, teaching and learning. Implications for preservice education are discussed.