Commentary on a special issue: Davydov’s approach in the XXI century

The articles in this special issue, collectively, provide overwhelming evidence that a curriculum which has as its primary basis the counting of discrete objects (and hence which introduces numbers as discrete) is not an effective or rational organisation. In this commentary, I discuss the contribution of each article to an understanding of Davydov’s ontology and epistemology, issues around the transposition of a pedagogy or curriculum from one country to another, early algebra and proportional or multiplicative reasoning. From Davydov’s own research and the writing in these articles, we know children are able to understand abstract structures from an early age. The thinking in this special issue provides tools to investigate and question any context in which such understanding is not routinely taking hold.

Releasing the conceptual spring to construct multiplicative reasoning

Constructing multiplicative reasoning is critical for students’ learning of mathematics, particularly throughout the middle grades and beyond. Tzur, Xin, Si, Kenney, and Guebert [American Educational Research Association, ERIC No. ED510991, (2010)] conclude that an assimilatory composite unit is a conceptual spring to multiplicative reasoning. This study examines patterns in the percentages of students who construct multiplicative reasoning across the middle grades based on their fluency in operating with composite units. Multinomial logistic regression models indicate that students’ rate of constructing an assimilatory composite unit but not multiplicative reasoning in sixth and seventh grades is significantly greater than that in eighth and ninth grades. Furthermore, the proportion of students who have constructed multiplicative reasoning in sixth and seventh grades is significantly less than the proportion of those who have constructed multiplicative reasoning in eighth and ninth grades. One implication of this is the quantitative verification of Tzur, Xin, Si, Kenney, and Guebert’s (2010) conceptual spring. That is, students who construct assimilatory composite units early in the middle grades are likely to construct multiplicative reasoning; students who do not construct assimilatory composite units early in the middle grades likely do not construct multiplicative reasoning in the middle grades.

Why Multiply? Area Measurement and Multiplicative Reasoning

Asked to quantify the changes in area of growing rectangles, these students reasoned about multiplicative relationships in interesting new ways.