The multiplication principle (MP) is a fundamental aspect of combinatorial enumeration, serving as an effective tool for solving counting problems and underlying many key combinatorial formulas. In this study, we used guided reinvention to investigate 2 undergraduate students' reasoning about the MP, and we sought to answer the following research…

Counting is fundamental to early mathematics. Most studies of teaching counting focus on teachers observing children count. The present study compares mathematical ideas that 12 PK, transitional kindergarten (TK), and kindergarten teachers noticed from observing their own students count during a classroom session of Counting Collections with ideas…

We investigated the extent to which one elementary school child with working-memory differences made sense of number as a composite unit and advanced her reasoning. Through ongoing and retrospective analysis of eight teaching-experiment sessions, we uncovered four shifts in the child's real-time negotiation of number over time: (a) initial…

Although approximately 5-8% of students have a mathematical learning disability (MLD), researchers have yet to develop a consensus operational definition. To examine how MLD has been identified and what mathematics topics have been explored, the authors conducted a systematic review of 165 studies on MLD published between 1974 and 2013. To move…

This study examines how collective activity related to multiplication evolved over several class sessions in an elementary mathematics content course that was designed to foster prospective elementary teachers' number-sense development. We document how the class drew on as-if-shared ideas to make sense of multidigit multiplication in terms of…

This case study examined the performance of a 13-year-old highly skilled mental calculator on mental multiplication tasks. She solved difficult tasks by using various ingenious calculation methods, including distributing, factoring, and recalling the product directly. Implications for instruction are reported. (Author/RH)

Four methods of solutions and 12 calculative strategies were found from introspective reports of 15 skilled and 15 unskilled students in grades 11 and 12 doing mental multiplication. Unskilled students used strategies more suited to written than mental computation, while skilled students used strategies based on number properties. (MNS)

Assessed attitude, computational preferences, and mental computational performance of 176, 187, 186, and 206 Japanese students in grades 2, 4, 6, and 8, respectively. Found wide ranges of performance on mental computation, that visual modes of presentation increased performance levels, and a narrow range of strategies. Contains assessments. (37…

Examined the effects of intervention in the instruction of 12 male seventh-grade students for the purpose of developing number sense. Results indicate that after instruction students were more likely to use strategies to solve number magnitude, mental computation, and computational estimation problems that reflected number sense. (Contains 50…

Use of the commutativity, addition-subtraction complement, and N+1 progression principles was studied by interviewing 54 capable pupils in grades 1-3. Commutativity was used extensively at each grade, while the addition-subtraction principle to solve subtraction varied across grades, and the N+1 pattern was seldom used. (MNS)

Results of this study indicated calculator algorithms can serve as an effective alternative instructional strategy where computational skill is an instructional goal. (MP)

A nine-year-old's conception of fractions was compared with his knowledge of units. He had effective schemes for solving some partition problems but did not consistently use units of different sizes in interpreting fractions. His solutions to equivalence problems showed no coherent method of verification. (MNS)

This study: (1) documented the spontaneous derived facts strategies (DGSs) that second-grade children (N=23) used; (2) investigated how training in use of DFSs influenced the solution strategies children used to solve addition and subtraction problems; and (3) examined the role of DFSs in the transition to recall of number facts. (JN)

This critique discusses Gagne's position that students should understand how to mathematize a concrete situation and validate a solution but need not understand how a solution is derived. Reconciling his views with those of mathematics educators and raising questions are both included. (MNS)

Gagne's reply to critiques by Wachsmuth and by Steffe and Blake notes that their approaches are from different points of view. He urges that mathematics educators examine critically the view that understanding involves some aspects of the structure of mathematics. (MNS)