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.

The academic search for principles of high-quality subject-matter teaching has been informed by different perspectives, in particular normative, epistemological, empirical, and pragmatic perspectives. While these perspectives have sometimes been treated as competing, we emphasize the need of their integration to identify sets of principles that can inform professional development programs for quality development. This paper starts from characterizing the four perspectives, and then shows how they are iteratively intertwined in providing a research base for specifying high-quality principles for teaching, in our case exemplified for the school subject mathematics. For this goal, we present the set of five principles that we have deemed as core principles for a new nationwide, ten-year professional development program in Germany: Conceptual Focus, Cognitive Demand, Student Focus and Adaptivity, Longitudinal Coherence, and Enhanced Communication. We will discuss these five principles against their backgrounds stemming from combing normative, epistemological, empirical, and pragmatic perspectives. This set of principles serves as an exemplary case to substantiate our general argumentation that contemporary educational research and professional development activities should not choose between perspectives but strive for combining them.

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