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This article discusses algebraic thinking regarding positive integers and rational numbers when students, 6 to 9 years old in multilingual classrooms, are engaged in an algebraic learning activity proposed by the El’konin and Davydov curriculum. The main results of this study indicate that young, newly arrived students, through tool-mediated joint reflective actions as suggested in the ED curriculum, succeeded in analysing arithmetical structures of positive integers and rational numbers. When the students participated in this type of learning activity, they were able to reflect on the general structures of numbers established as additive relationships (addition and subtraction) as well as multiplicative relationships (multiplication and division) and mixtures thereof, thus a core foundation of algebraic thinking. The students then used algebraic symbols, line segments, verbal, written, and gesture language to elaborate and construct models related to these relationships. This is in spite of the fact that most of the students were second language learners. Elaborated in common experiences staged in the lessons, the learning models appeared to bridge the lack of common verbal language as the models visualized aspects of the relationships among numbers in a public manner on the whiteboard. These learning actions created rich opportunities for bridging tensions in relation to language demands in the multilingual classroom.

This study examines the collective mathematical reasoning when students and teachers in grades 3, 4, and 5 explore fractions derived from length comparisons, in a task inspired by the El´konin and Davydov curriculum. The analysis showed that the mathematical reasoning was mainly anchored in mathematical properties related to fractional or algebraic thinking. Further analysis showed that these arguments were characterised by interplay between fractional and algebraic thinking except in the conclusion stage. In the conclusion and the evaluative arguments, these two types of thinking appeared to be intertwined. Another result is the discovery of a new type of argument, identifying arguments, which deals with the first step in task solving. Here, the different types of arguments, including the identifying arguments, were not initiated only by the teachers but also by the students. This in a multilingual classroom with a large proportion of students newly arrived. Compared to earlier research, this study offers a more detailed analysis of algebraic and fractional thinking including possible patterns within the collective mathematical reasoning. An implication of this is that algebraic and fractional thinking appear to be more intertwined than previous suggested.

We will present cultural transposition as a particular perspective to frame the use of foreign mathematics education practices as an opportunity for questioning the didactic practices of one’s own cultural context. This requires a process activated by researchers, who deconstruct the cultural layers underpinning the foreign education practice before proposing it to teachers. We discuss the theoretical premises of this approach and, in accordance with them, we propose a transposition process of the El’konin-Davydov curriculum. In particular, we will show how our deconstruction process has affected the design and implementations of particular Professional Development courses (PDs) in Italy. Finally, we will present a case study of a teacher involved in one of these PDs to observe her new educational awareness.

The goal of this article is to investigate Davydov’s concept of the concept against the backdrop of its philosophical system, namely, dialectical materialism. In the first part, after briefly sketching the context of Davydov’s work, I consider some ontological and epistemological ideas on which Davydov bases his concept of the concept. I pay particular attention to Hegel’s and Marx’s contributions. Then, I discuss Davydov’s concept of the concept and the relationship between the logical and the historical—a relationship that proved to be crucial in the making of the educational curricular program he and El’konin launched in the 1960s in Russia. I argue that, in tune with the dominant epistemology of the twentieth century, Davydov’s concept of the concept is based on a scientific outlook of the world, one in which theoretical scientific thought is considered the pinnacle of human cognition. I conclude with a critique that intends to place the notion of the concept in a broader dialectical materialist perspective.

This study is focused on the relational thinking of first-grade students following their first 3 months of instruction from the Measure Up (MU) curriculum, an adaptation of the El’konin-Davydov curriculum. Following Davydov’s outline of instructional material, the MU firstgrade materials were designed to have students identify the quantitative attributes of objects, learn to designate the properties using certain symbols, and carry out elementary analyses of the relationships. Additionally, MU emphasized the use of specific, multiple concurrent representations so students could use concrete, diagrammatic, and symbolic ways to analyze comparisons and convey their findings. Our research focused on the characteristics of MU first-grade students’ thinking about relations without numbers. Additionally, we were interested in the role symbols had in their ability to communicate their thinking. We analyzed video recordings and transcriptions of six semi-structured student interviews and then reanalyzed the data for specific evidence of symbolic understandings. Recognizing MU as a symbolically structured environment, we connected our data to this paradigm. Our findings show that students were able to make direct and indirect comparisons and that they relied on symbolizing to explain their thinking. These results show further support for Davydov’s hypothesis––a non-numeric introduction to relational symbols can develop children’s theoretical thinking abilities by going beyond empirical ways of knowing.

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.

Recent scholarship around teaching elementary mathematics supports the learning of early algebra with 5- to 12-year olds. However, in spite of the recognition of the affordances of early algebra, issues about how to introduce it remain open. Within this context, Davydov’s work is often cited as a source of impressive demonstration of young learners’ capacity for algebraic thinking. This work requires further exploration in order to yield a clearer picture of a very particular teaching approach, which focuses on early abstractions and symbolic language. We argue that in order to fully understand how Davydov’s work contributes to current conversations and what Davydov was trying to do, we need to shed light on the context-and time-specific discourse of the 1960 Soviet educational reforms that made it possible for Davydov to develop his vision about algebraic thinking and to set in motion appropriate teaching approaches for young learners. In this paper, we look back to the Soviet debates that unfolded in Russia on the integration of early algebra in elementary school word-problem solving. Drawing on these debates and the results of Davydov’s school experiments, we lay out the developmental axes of capacity building. This can be done by emphasizing ascent from the abstract to the concrete using a variety of representational modeling tools to support the emergence of algebraic thinking while targeting particular habits of mind within carefully designed learning activities. We conclude with some insights about current arithmetic-algebra debates, and how these could be enriched and deepened by Davydov’s work, which yet remains open to future discussion and reflection.

Early experiences with theoretical thinking and generalization in measurement are hypothesized to develop constructs we name here as logical reasoning and preparedness for algebra. Based on work of V. V. Davydov (1975), the Measure Up (MU) elementary grades experimental mathematics curriculum uses quantities of area, length, volume, and mass to contextualize the relationships among the quantities in, for example, R + C = T. This quasi-experimental study, conducted with 129 fifth- and sixth-grade students, examines MU effects on students' preparedness for algebra. Structural equation modeling is used to identify a system of relationships among the variables in our proposed model. Findings show significant direct standardized effects from MU to preparedness (0.28, p<.05) and from logical reasoning to preparedness (0.89, p<.05). Although positive, the effect of MU mediated by logical reasoning was not statistically significant. This suggests that the development of logical reasoning abilities, attributed to theoretical thinking and generalization, lag preparedness for algebra. It also suggests that MU can potentially contribute to algebra preparedness for students who may not have developed strong logical reasoning abilities. The findings are discussed in terms of their theoretical and practical implications for the successful study of algebra.