This article addresses the contradiction between the assumption that the goal of mathematics instruction is to transmit knowledge to students and the view that students construct mathematical knowledge by reorganizing their cognitive structures. Transmission and constructivist analyses of communication are contrasted. It is concluded that the task of accounting for successful instruction is not one of explaining how students take in and process information transmitted by the teacher. Instead, it is to explain how students actively construct knowledge in ways that satisfy constraints inherent in instruction. A brief case study is presented to support this contention, and general implications that take account of both cognitive and noncognitive instructional goals are discussed.

The Tension Between Theories of Learning and Instruction in Mathematics Education

Representational systems, learning, and problem solving in mathematics

Fractions and multiplicative reasoning

Every word problem has a solution—The social rationality of mathematical modeling in schools

We present an empirical study that investigated seventh-, ninth-, and eleventh-grade students’ understanding of the infinity of numbers in an interval. The participants (n = 549) were asked how many (i.e., a finite or infinite number of numbers) and what type of numbers (i.e., decimals, fractions, or any type) lie between two rational numbers. The results showed that the idea of discreteness (i.e., that fractions and decimals had “successors” like natural numbers) was robust in all age groups; that students tended to believe that the intermediate numbers must be of the same type as the interval endpoints (i.e., only decimals between decimals and fractions between fractions); and that the type of interval endpoints (natural numbers, decimals, or fractions) influenced students’ judgments of the number of intermediate numbers in those intervals. We interpret these findings within the framework theory approach to conceptual change.

How Many Decimals Are There Between Two Fractions? Aspects of Secondary School Students’ Understanding of Rational Numbers and Their Notation

Modelling Reality in Mathematics Classrooms: The Case of Word Problems.

It has been firmly established that a pupil's choice of operation for the solution of a single-operation word problem is often affected by the types of numbers in the problem (Bell, Fischbein, & Greer, 1984; Bell, Swan, & Taylor, 1981; Fischbein, Deri, Nello, & Marino, 1985; Hart, 1981). A more specific observation is that a choice of different operations occurs even when two problems differing only in the numbers involved are juxtaposed and attention is drawn to the similarity between them (Ekenstam & Greger, 1983). Although several examples have been cited in the literature, such nonconservation of operations remains to be explored in more depth. Existing research affords four overlapping perspectives on the phenomenon.

Nonconservation of Multiplication and Division Involving Decimals.

Three hundred seventy-seven subjects in six age groups, ranging from 10 to 20 years, were asked to identify the appropriate operation for 24 multiplication and division problems involving price, speed, and measure conversion. For multiplication problems, difficulty was strongly dependent on the type of number in the preferred-multiplier role. Important differences from the general tendency were noted in those change-of-size and mixture problems in which both quantities were measured in the same units. Choice of operation also was affected by the misconception that multiplication makes bigger and division smaller. In division problems, choice was dominated by the numerical preference for dividing by an integer or the smaller of the two numbers; decimal points were sometimes ignored in determining this preference. MMBDS also operated, but not in all problems. In a second experiment with 42 fifteen-yearold students, half were asked to estimate answers and half to choose the correct operation for change-of-size and mixture problems. Estimating the outcome was easier than both choosing the operation for division, and multiplication by numbers less than 1. For multiplication by numbers substantially greater than 1, choosing the operation was easier.

https://www.jstor.org/stable/pdfplus/749419.pdf

Children's Performance on Multiplicative Word Problems: Elements of a Descriptive Theory

The Role of Isomorphisms in Mathematical Cognition

The growth of mathematics through conceptual restructuring

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