Multiplicative Thinking

Multiplicative thinking has been widely accepted as a critically important 'big idea' of mathematics and one which underpins much mathematical understanding beyond the primary years of schooling. It is therefore of importance to consider the capacity of children to think multiplicatively but also to consider the capacity of their teachers to teach multiplicative thinking in a conceptual manner. This article focusses specifically on the conceptual links between the multiplicative array, the notion of numbers of equal groups in the multiplicative situation, factors and multiples, the commutative property of multiplication, and the inverse relationship between multiplication and division. A study involving a large sample of primary school students found that whilst most students demonstrated an understanding of some of the aforementioned elements, hardly any of the students were able to connect the ideas or to explain them in terms of each other. As a consequence of the findings, the impact of teacher knowledge on children's capacity to think multiplicatively was considered.

http://www.lectitopublishing.nl/Article/Detail/children-have-the-capacity-to-think-multiplicatively-as-long-as

Multiplicative thinking is a 'big idea' of mathematics that underpins much of the mathematics learned beyond the early primary school years. This article reports on a recent study that utilised an interview tool and a written quiz to gather data about children's multiplicative thinking. Our research has so far revealed that many primary aged children have a procedural view of multiplicative thinking which we believe inhibits their progress. There are two aspects to this article. First, we present some aspects of the interview tool and written quiz, along with some of findings, and we consider the implications of those findings. Second, we present a key teaching idea and an associated task that has been developed from our research. The main purpose of the article is to promote the development of conceptual understanding of the multiplicative situation as opposed to the teaching of procedures. In doing so, we encourage the explicit teaching of the many connections within the multiplicative situation and between it and other 'big ideas' such as proportional reasoning and algebraic thinking.

Background: Given the context of low attainment in primary mathematics in South Africa, improving learners’ understanding of multiplicative reasoning is important as it underpins much of later mathematics.

Aim: Within a broader research programme aiming to improve Foundation Phase (Grades 1–3, 7–9-year-olds) learners’ mathematical performance, the aim of the particular research reported
on here was to improve learners’ understanding of and attainment in multiplicative reasoning when solving context-based problems.

Setting: The research was conducted in a suburban school serving a predominantly historically disadvantaged learner population, and involved teachers and learners from three classes in each of Grades 1–3.

Methods: A 4-week intervention piloted the use of context-based problems and array images to encourage learners to model (through pictures and diagrams) the problem situations, with the models produced used both to support problem solving and to support understanding of the multiplicative structures of the contexts.

Results: Cleaning the data to include those learners participating at all three data points – pre-, post- and delayed post-test – provided findings based on 233 matched learners. These findings show that, on average, Grade 1 learners had a mean score average increase of 22 percentage
points between the pre-test and the delayed post-test, with Grades 2 and 3 having mean increases of 10 and 9 percentage points, respectively.

Conclusion: The findings of this study demonstrate that young learners can be helped to better understand and improve their attainment in multiplicative reasoning, and suggest the usefulness of trialling the intervention model more broadly across schools.

Keywords: Foundation Phase; multiplicative reasoning; assessment; raising standards.

The purpose of this paper is to suggest how to modify and apply multiplication models in order to link to students’ conceptual understanding on multiplicative relationship based on a measurement perspective. A research group in Russia proposed the measurement perspectives based quantitative reasoning. To this end, we analyze multiplication models commonly used in basal texts such as repeated addition, equal size group, and an array. In addition, we apply the measurement approach in order to explain how each model or diagram demonstrates the mathematical meaning of multiplication and multiplicative reasoning in whole numbers and rational numbers.

As technologies that put the body at the center of mathematics learning enter formal and informal learning spaces, we still know little about the teaching methods educators can use to support students' learning with these specialized systems. Drawing on ethnomethodology and conversation analysis (EMCA) and the CoOperative Action framework, we present three multimodal ways that educators can be responsive to learners' embodied ideas and help them transform sensorimotor patterns into mathematically significant perceptions. These techniques include (1) encouraging learners to use gesture to express and reflect on their ideas, (2) presenting multimodal candidate understandings to check comprehension of learners' embodied ideas, and (3) co-constructing multimodally expressed embodied ideas with learners. We demonstrate how these techniques create opportunities for learning and discuss implications for a multimodal, embodied practice of responsive teaching.