In an 8-month teaching experiment, I investigated how 4 sixth-grade students reasoned with reversible multiplicative relationships. One type of problem involved a known quantity that was a whole number multiple of an unknown quantity, and students were asked to determine the value of the unknown quantity. To solve these problems, students needed to produce a fraction of the known quantity that could be repeated some number of times to make the known, rather than repeat the known quantity to make the unknown quantity. This aspect of the problems involved reversibility because students who do not make a fraction of the known quantity tend to repeat the known quantity (Norton, 2008; Steffe, 2002). All four students constructed schemes to solve such problems and more complex versions where the relationship between known and unknown quantities was a fraction. Two students could not foresee the results of their schemes in thought—they had to carry out some activity, review its results, and then carry out more a...

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Students’ Reasoning With Reversible Multiplicative Relationships

This collective case study examines the influence of 4 third-grade teachers’ beliefs and knowledge on their error-handling practices during class discussion of mathematics. Across cases, 3 dimensions of teachers’ error-handling practices are identified and discussed in relation to teacher beliefs and knowledge: (a) intentional focus on flawed solutions in class discussion, (b) promotion of conceptual understanding through discussion of errors, and (c) mobilization of a community of learners to address errors. Study findings suggest that, although teachers’ ways of handling student errors during class discussion of mathematics are clearly linked to both teacher beliefs and teacher knowledge, some aspects of teacher response are more strongly linked to knowledge and others are influenced more by beliefs.

https://www.jstor.org/stable/10.5951/jresematheduc.42.1.0002

A collective case study of the influence of teachers' beliefs and knowledge on error-handling practices during class discussion of Mathematics

This study investigated discursive positioning moves that facilitated Latino/a English learners’ (ELs) opportunities to take on agentive problem-solving roles in group mathematical discussion. A focus on mechanisms that support students’ agentive participation is consistent with our view that recurrent experiences participating and being positioned in particular ways contribute to identity development. Findings suggest several ways that discursive positioning facilitated ELs’ agentive participation, including via: (a) explicit statements that validated ELs’ reasoning, (b) invitations to share, justify, or clarify thinking that positioned ELs as competent problem solvers, and (c) inviting peers to respond to an EL’s idea in ways that positioned the idea as important and/or mathematically justified.

https://www.jstor.org/stable/10.5951/jresematheduc.44.1.0199

English Learners' Participation in Mathematical Discussion: Shifting Positionings and Dynamic Identities

Students' Whole Number Multiplicative Concepts: A Critical Constructive Resource for Fraction Composition Schemes.

Child proportional scaling: Is 1/3 = 2/6 = 3/9 = 4/12?

Although equal sharing problems appear to support the development of fractions as multiplicative structures, very little work has examined how children's informal solutions reflect this possibility. The primary goal of this study was to analyze children's coordination of two quantities (number of people sharing and number of things being shared) in their solutions to equal sharing problems and to see to what extent this coordination was multiplicative. A secondary goal was to document children's solutions for equal sharing problems in which the quantities had a common factor (other than 1). Data consisted of problem-solving interviews with students in 1st, 3rd, 4th, and 5th grades (n=112). We found two major categories of strategies: (a) Parts Quantities strategies and (b) Ratio Quantities strategies. Parts quantities involved children's partitions of continuous units expressed in terms of the number of pieces that would be created. Ratio quantities involved children's creation of associated sets of discrete quantities. Within these strategies, children drew upon a range of relationships among fractions, ratio, multiplication, and division to mentally or physically manipulate quantities of sharers and things to produce exhaustive and equal partitions of the items. Additionally, we observed that problems that included number combinations with common factors elicited a wider range of whole-number knowledge and operations in children's strategies and therefore appeared to support richer interconnections than problems with relatively prime or more basic number combinations.

Fractions as the Coordination of Multiplicatively Related Quantities: A Cross-Sectional Study of Children's Thinking

Our study investigated the knowledge 13elementary teachers gained implementing astudent-centered curriculum in the context ofdistrict-wide reform. Participants comprisedall the teachers in grades three, four and fiveat a single elementary school. We believed thatinvestigating teachers' responses to fictionalpedagogical scenarios involving nonstandardalgorithms would yield insights into criticalcomponents of their knowledge base. We lookedin particular at teachers' knowledge of children's mathematics. We found that teacherswere in the midst of creating a knowledge basefocused on children's mathematics and groundedin knowledge about alternative conceptualtrajectories through the elementary curriculum.Teachers' knowledge of nonstandard strategiessupported by the curriculum materials wasstronger and more coherent than their knowledgeof students' novel nonstandard strategies.Strong mathematical knowledge was notnecessarily associated with strong knowledge ofchildren's mathematics. Teachers' thinkingvaried by a topic's treatment in the writtencurriculum materials used, suggestingimplementation of the curriculum as a source oflearning.

Teachers' Knowledge of Children's Mathematics after Implementing a Student-Centered Curriculum

Seven university professors participated in lesson study of their own mathematics methods teaching. They presented a research lesson four times in elementary mathematics methods classes and a fifth time as a public research lesson at a national mathematics education conference. The lesson focused on the use of invented strategies in elementary mathematics instruction. The project demonstrated that lesson study is a powerful tool for mathematics teacher educator professional development, supporting the premise that mathematics instruction is a problem to solve rather than a set of teaching techniques to implement.

Walking the Talk: Lessons Learned by University Mathematics Methods Instructors Implementing Lesson Study for Their Own Professional Development

Building understanding of multiplicative relationships is a key goal of mathematics instruction in the upper elementary and middle grades. Multiplicative thinking includes comparing numbers through many processes: multiplication and division (rather than addition and subtraction), ratio, proportions, stretching and shrinking, magnification, scaling, and splitting. Research has shown that multiplicative thinking develops slowly in children, over long periods of time (Clark and Kamii 1996; Vergnaud 1988). Initially, children tend to reason additively about multiplicative situations, and this additive thinking is often resistant to change (Hart 1984). Students need practice with tasks that help develop multiplicative thinking—in particular, tasks that help them recognize and reason about multiplicative relationships. Solving and discussing mathematical problems is also an essential part of doing and learning mathematics. Many teachers agree that such discussion is important, but helping students articulate, justify, and debate ideas can be challenging (O’Connor 2001). Teachers need to structure opportunities for students to share their thinking and consider the ideas of others. Although research has shown that all students, including lower-performing and English Language Learner (ELL) students, can and do participate in discussions (see, e.g., Empson 2003; Moschkovich 1999), some students need additional support (see, e.g., Baxter, Woodward, and Olson 2001). Also, not all tasks generate rich mathematical discussions. Students need to reason and communicate about interesting tasks that can be solved in various ways and that lead to important mathematical ideas (Hiebert et al. 1997). In this article, we focus on a particular task—paper folding—and describe its potential to address both these challenges: supporting the development of multiplicative thinking and facilitating discussion of mathematical problems for all students.

http://www.4j.lane.edu/wp-content/uploads/2014/09/folding-Multiplicitive-thinking.pdf

Debra L. Junk

Papers

Fractions as the Coordination of Multiplicatively Related Quantities: A Cross-Sectional Study of Children's Thinking

Teachers' Knowledge of Children's Mathematics after Implementing a Student-Centered Curriculum

Walking the Talk: Lessons Learned by University Mathematics Methods Instructors Implementing Lesson Study for Their Own Professional Development

The Power of Paper- Folding Tasks: Supporting Multiplicative Thinking and Rich Mathematical Discussion