מחקרים חדשים בחינוך מתמטי

PME28 דיווח מ

2004

מלאה הרצאות

Klette, KirstiClassroom Business Same as Usual? (What) Do Reseachers and Policy Makers Learn from Classroom Studies? Powell, Arthur BClass, Diversity, and Race in Research on the Psychology of Mathematics LearningJaworski, BarbaraGrappling with Complexity: Co-Learning in Inquiry Communities in Mathematics Teaching DevelopmentHershkowitz, RinaFrom Diversity to Inclusion and Back: A Lens on Learning

EMPOWERING ANDREA TO HELP YEAR 5 STUDENTSCONSTRUCT FRACTION UNDERSTANDING.Annette R Baturo, Centre for Mathematics and science Education

Queensland University of Technology, Brisbane, Australia

The collaboration. The first activity was the development of the fraction test to determine Andrea’s students’ Entry knowledge. As soon as Andrea saw the developed instrument, she knew immediately the importance of the reversing and nonprototypic representations, exclaiming: That’s what I’ve been missing! Thus, both the instrument and the students responses (which were poor in significant areas became excellent springboards for discussion of the teaching sequences for developing fraction understanding. than before.

As a consequence, Andrea spent most of Term 2 almost exclusively on re-teaching the fraction concepts, focusing first on partitioning a variety of prototypic and nonprototypic wholes, and then reunitising area, set, and linear models including nonprototypic representations of the whole and the parts, as well as on reversing activities (i.e., whole.part; part.whole). However, she found that her students were unable to process set models so these were delayed until the end of the year. Her excellent pedagogy skills combined with the more sophisticated techniques and richer representations that emerged from the joint planning meant that she was able to challenge her students’ understanding at a greater depth than before.

GENERALIZED DIAGRAMS AS A TOOL FOR YOUNG CHILDREN’S PROBLEM SOLVINGBarbara J. Dougherty and Hannah SlovinUniversity of Hawai‘i

Measure Up is a research and development project that uses findings from Davydov (1975) and others to introduce mathematics through measurement and algebra in grades 1–3. This paper illustrates the use of generalized diagrams and symbols in solving word problems for a group of 10 children selected from a grade 3 Measure Up classroom. Students use the diagrams to help solve word problems by focusing on the broader structure rather than seeing each problem as an entity in and of itself.The consistent use of the diagrams is related to students’ experience with simultaneous presentations of physical, diagrammatic, andsymbolic representations used in Measure Up.

To begin mathematics learning with abstractions

Davydov (1975a) believed that very young children should begin their mathematics learning with abstractions so that they could use formal abstractions in later school years and their thinking would develop in a way that could support and tolerate the capacity to deal with more complex mathematics. He (1975b) and others (Minskaya,1975) felt that beginning with specific numbers (natural and counting) led to misconceptions and difficulties later on when students worked with rational and real numbers or algebra.

Spontaneous and Scientific Concepts

Vygotsky’s distincts between spontaneous and scientific concepts (1978). Spontaneous or empirical concepts are developed when children can abstract properties from concrete experiences or instances. Spontaneous concepts progress from natural numbers to whole, rational, irrational, and finally real numbers, in a very specific sequence. Scientific concepts, on the other hand, develop from formal experiences with properties themselves, progressing to identifying those properties in concrete instances. Scientific concepts focus on real numbers in the larger sense first, with specific cases found in natural, whole, rational, and irrational numbers at the same time.Davydov (1966) conjectured that a general-to-specific approach in the case of the scientific concept was much more conducive to student understanding than using the spontaneous concept approach.

Starting children’s mathematical experiences with basic conceptual ideas about mathematics

Davydov (1975a) proposed starting children’s mathematical experiences with basic conceptual ideas about mathematics and its structure, and then build number from there. Thus young children begin their mathematics program in grade 1 by describing and defining physical attributes of objects that can be compared. Davydov (1975a) advocated children begin in this way as a means of providing a context to explore relationships, both equal and unequal. Six-year-olds physically compare objects’ attributes (length, area, volume, and mass), and describe those comparisons withrelational statements like H < B, where H and B represent unspecified quantities being compared, not objects. The physical context of these explorations and means by which they are recorded, link measurement and algebra so that children develop meaning for statements they write and do not see them as abstract.

Sample 1

Ama caught k fishes and Chris caught e fishes less. How many fishes did Chris catch?

Dusty had b fishes in a bucket. When Anthony added his fish, there were g fishes in the bucket. How many fishes did Anthony catch?

Students’ representations

Sample 3

THE DOTS PROBLEM: THIRD GRADERS WORKING WITHFUNCTIONSEarnest , Schliemann TERC Tufts University

We highlighted the shift from thinking about relations between particular numbers and measures towards thinking about relations among sets of numbers and measures, from computing numerical answers to describing and representing relations between variables, and aimed at the understanding of arithmetic operations as functions.

Students’ work across multiple representational systems provides evidence that, when given the opportunity to work with algebraic concepts and representations, third graders can develop a rich understanding of functions and can represent and solve problems usually taught to be accessible only to older children. Our results support proposals that algebra should become a central part of the elementary school mathematics curriculum.

TRACKING PRIMARY STUDENTS’ UNDERSTANDING OF THE EQUALITY SIGNFreiman Lesley LeeUniversitי de Moncton Universitי du Quיbec א Montrיal

In the spring of 2003, we began exploring a number of “early algebra” themes in

kindergarten (35 students, ages 5-6), two grade three classes (20 and 11 students,

ages 8-9) and a grade six class (23 students, ages 11-12) in a French Montreal area

private primary school with a math enrichment program for all students.

Thus, by order of importance in a short list of problems to include in a written test instrument across the grades, we would suggest:

a + b = c + _

a + b = _ + d

c = a + _

a + b = _ + d.