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Merlyn J. Behr, Ipke Wachsmuth, Thomas R. Post and Richard Lesh

Fourth-grade students' understanding of the order and equivalence of rational numbers was investigated in 11 interviews with each of 12 children during an 18-week teaching experiment. Six children were instructed individually and as a group at each of two sites. The instruction relied heavily on the use of manipulative aids. Children's explanations of their responses to interview tasks were used to identify strategies for comparing fraction pairs of three types: same numerators, same denominators, and different numerators and denominators. After extensive instruction, most children were successful but some continued to demonstrate inadequate understanding. Previous knowledge relating to whole numbers sometimes interfered with learning about rational numbers.

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Richard Lesh and Anthony E. Kelly

This article describes a three-tiered teaching experiment in which teachers were studied over a protracted period of time as they attempted to understand and improve their approaches to one-to-one tutoring. In a three-tiered teaching experiment model, emphasis is placed on (a) establishing a collaborative relationship between the research staff and the teachers, (b) careful choice of tasks for teachers and their students, (c) the development of “learning environments” for the study, and (d) the use of continuous and diverse dependent measures. The study documented the initial and revised strategies of teachers as they tutored children over a period of 10 weeks. Various indices of teacher change are reported, and some of the strengths and limitations of the methodology are discussed.

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Lieven Verschaffel and Erik De Corte

Recent research has convincingly documented elementary school children's tendency to neglect real-world knowledge and realistic considerations during mathematical modeling of word problems in school arithmetic. The present article describes the design and the results of an exploratory teaching experiment carried out to test the hypothesis that it is feasible to develop in pupils a disposition toward (more) realistic mathematical modeling. This goal is achieved by immersing them in a classroom culture in which word problems are conceived as exercises in mathematical modeling, with a focus on the assumptions and the appropriateness of the model underlying any proposed solution. The learning and transfer effects of an experimental class of 10-and 11-year-old pupils—compared to the results in two control classes—provide support for the hypothesis that it is possible to develop in elementary school pupils a disposition toward (more) realistic mathematical modeling.

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Raffaella Borasi

Although teachers and researchers have long recognized the value of analyzing student errors for diagnosis and remediation, students have not been encouraged to take advantage of errors as learning opportunities in mathematics instruction. The study reported here was designed to explore how secondary school students could be enabled to capitalize on the potential of errors to stimulate and support mathematical inquiry. The article provides a case study of the proposed strategy of “using errors as springboards for inquiry” in action, identifies some important variations within the strategy, and discusses its potential contributions to mathematics instruction.

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Maria Blanton, Bárbara M. Brizuela, Angela Murphy Gardiner, Katie Sawrey and Ashley Newman-Owens

The study of functions is a critical route into teaching and learning algebra in the elementary grades, yet important questions remain regarding the nature of young children's understanding of functions. This article reports an empirically developed learning trajectory in first-grade children's (6-year-olds') thinking about generalizing functional relationships. We employed design research and analyzed data qualitatively to characterize the levels of sophistication in children's thinking about functional relationships. Findings suggest that children can learn to think in quite sophisticated and generalized ways about relationships in function data, thus challenging the typical curricular approach in the lower elementary grades in which children consider only variation in a single sequence of values.

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Jinfa Cai, Anne Morris, Charles Hohensee, Stephen Hwang, Victoria Robison, Michelle Cirillo, Steven L. Kramer and James Hiebert

In our March editorial (Cai et al., 2019), we discussed the nature of significant research questions in mathematics education. We asserted that the choice of a suitable theoretical framework is critical to establishing the significance of a research question. In this editorial, we continue our series on high-quality research in mathematics education by elaborating on how a well-constructed theoretical framework strengthens a research study and the reporting of research for publication. In particular, we describe how the theoretical framework provides a connecting thread that ties together all of the parts of a research report into a coherent whole. Specifically, the theoretical framework should help (a) make the case for the purpose of a study and shape the literature review; (b) justify the study design and methods; and (c) focus and guide the reporting, interpretation, and discussion of results and their implications.

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Debra Monson, Kathleen Cramer and Sue Ahrendt

interested in the curriculum we developed on the basis of these teaching experiments, we refer them to http://wayback.archive-it.org/org-121/20190122152857/http://www.cehd.umn.edu/ci/rationalnumberproject/ . UNIT AND PARTITIONING Students need to make sense

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Jessica Hunt and Juanita Silva

number knowledge done through individualized teaching experiments ( Steffe & Thompson, 2000 ) as part of a longer experiment along with two formative clinical interviews ( Ginsburg, 1997 ). The first author was the researcher–teacher. The second author

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J. Peter Fedon

THIS IS A PARTIAL ANALYSIS of an arithmetic teaching experiment conducted in the third grade of an independent Wilmington School during the school year 1957-58.

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Anderson Norton

This article reports on students' learning through conjecturing, by drawing on a semester-long teaching experiment with 6 sixth-grade students. It focuses on 1 of the students, Josh, who developed especially powerful ways of operating over the course of the teaching experiment. Through a fine-grained analysis of Josh's actions, this article integrates Piaget's scheme theory (1950/2001) and Peirce's logic of abduction (1998) into a new theory about conjecturing that explains Josh's learning. Results indicate the power of Josh's operational conjectures in resolving problematic situations and constructing new schemes. Because of the context in which the teaching experiment and Josh's conjecturing occurred, results hold implications for research on fractions and on a particular operation called splitting (Confrey, 1994; Empson, 1999; Sáenz-Ludlow, 1994; Steffe, 2003). The theoretical integration of scheme theory and abduction also holds implications for resolving the learning paradox (Fodor, 1980; Glasersfeld, 2001).