Language-responsive support for multiplicative thinking as unitizing: results of an intervention study in the second grade

Multiplicative thinking involves the ability to coordinate bundled units on a more abstract level than additive thinking and implies the identification of the different meanings of the multiplier and the multiplicand. The transition from additive to multiplicative thinking, however, constitutes an obstacle for many children. Specific formulations that are typically used in classroom discourse for talking about multiplicative tasks and situations (e.g., ‘3 times 4’ or ‘3 lots of 4’) might inhibit meaning-making processes because they do not address the idea of unitizing. A language-responsive introduction to multiplication that addresses the core idea of unitizing and that uses phrases such as ‘3 times 4 means you have 3 fours’ may help to overcome these problems. In the study presented in this paper, three second grade primary school teachers joined a teacher program to introduce multiplication in their classes (n = 66) by addressing meaning-making phrases. Another 58 second graders taught by teachers without this teacher program served as the control group. A specially developed multiplication test gave insight into the children’s understanding of multiplication as unitizing immediately after the intervention (posttest) and nearly three months later (follow-up test). We found significant differences between the intervention and control groups in the multiplication posttest. These differences could be underlined in the follow-up test. Our results indicate that a language-responsive teaching intervention that focuses on meaning-making processes can lead to long-term insights and help to develop multiplicative thinking as unitizing.

Atypical Arrays to Stimulate Multiplicative Thinking

Two types of tasks have the potential to help second graders attend to structure and use initialmultiplicative ideas in their strategies to find the total.

Distributivity, partitioning, and the multiplication algorithm

Multiplicative thinking underpins much of the mathematics learned beyond the middle primary years. As such, it needs to be understood conceptually to highlight the connections between its many aspects. This paper focuses on one such connection; that is how the array, place value partitioning and the distributive property of multiplication are related. It is important that students understand how the property informs the written multiplication algorithm. Another component of successful use of the standard multiplication algorithm is extended number facts and the paper also explores students’ ability to understand and generate them. One purpose of the study was to investigate the extent to which students used the standard multiplication algorithm and if their use of it is supported by an understanding of the underpinning components of the array, partitioning, the distributive property, and extended number facts. That is, we seek to learn if students have a conceptual understanding of the multiplication algorithm and its underpinning mathematics that would enable them to transfer their knowledge to a range of contexts, or if they have procedurally learned mathematics. In this qualitative study, data were generated from the administration of a Multiplicative Thinking Quiz with a sample of 36 primary aged students. Data were analyzed manually and reported using descriptive statistics. The main implications of the study are that the connections among the multiplicative array, place value partitioning, base ten property of place value, distributive property of multiplication, and extended number facts need to be made explicit for children in terms of how they inform the use of the written algorithm for multiplication.