The variation theory of learning emphasizes variation as a necessary condition for learners to be able to discern new aspects of an object of learning. In a substantial number of studies, the theory has been used to analyze teaching and students’ learning in classrooms. In mathematics education, variation theory has also been used to explore variation in sets of instructional examples. For example, it has been reported how teachers, by using variation and invariance within and between examples, can help learners to engage with mathematical structure. In this paper, we describe the variation theory of learning, its underlying principles, and how it might be appropriated by teachers. We illustrate this by an analysis of one teacher’s teaching before and after he participated in three lesson studies based on variation theory. Both the theory and the empirical illustration focus on ‘what is made possible to learn’ in different learning situations. We show that in the two analyzed lessons, different things were made possible to learn.

Using illustrative teaching case studies, this book demonstrates how teaching informed by a learning theory, specifically Variation theory, can equip teachers to facilitate possibilities for students’ learning in effective and powerful ways. For a long period of time teaching has been “black-boxed”, in favour of other explanations of why students learn or not, such as motivation and social interaction. A large amount of research on teaching and learning, not the least made using Variation theory, has shown that students often need to experience the same aspects of the focused content or capability in order to learn, indicating that relationships between teaching and learning are not unique or even qualitatively different for every individual and every situation. This perspective on the relationship between teaching and learning emphasizes content-specific aspects and in that sense structural components of teaching, while other aspects of schooling such as social interaction and general well-being recede into the background. The authors argue for the importance of this in the direct development of teachers' independent collective professional knowledge about teaching, and the leverage this gives for developing student learning. They introduce theoretical tools to help teachers to increase the probability that teaching focusing a specific content or capability is predictive of students learning of that specific content or capability, while decreasing contextual dependency without assuming that teaching and learning have a one-to-one relationship. Intended for teachers, graduate students in education, teacher educators, student teachers, and researchers, this book shows that while there is no simple equation between teaching and learning, there are general, though content specific, aspects of teaching that can be systematically planned and analyzed and used to improve the quality of student learning. The Open Access version of this book, available at www.taylorfrancis.com, has been made available under a Creative Commons Attribution-Non Commercial-No Derivatives (CC-BY-NC-ND) 4.0 license. <!----> <!----> <!----> <!---->

The idea of using fingers as a key component in arithmetic development has received a great deal of support, much of which is based on neuroscientific evidence. However, this body of work pays limited attention to how fingers are used and possible different outcomes in arithmetic problem solving. The aim of our paper, based on an analysis of 126 observations of 4–5-year-olds solving a simple subtraction task, is to discuss different ways of using fingers, with some of the ways appearing more, and others less, powerful. The analysis suggests there is much more complexity to children’s finger-related strategies than prior research has indicated. Empirical findings in our study point to the decisive effects of different ways of using fingers, and in particular for either keeping track of counted units or for presenting a structured awareness of number. Three ways of using fingers emerge in the analysis, which are discussed in relation to their rate of success in solving the subtraction task and with attention to why the differences matter for the success rate. Through this discussion we suggest that the complexity of how fingers are used must be considered.

In this paper, we report how 5-year-olds’ arithmetic skills developed through participation in an 8-month-long intervention. The intervention program aimed to enhance the children’s ways of experiencing numbers’ part-part-whole relations as a basis for arithmetic skills and was built on principles from the variation theory of learning. The report is based on an analysis of assessments with 103 children (intervention groupn = 65 and control groupn = 38) before and after the intervention and a follow-up assessment 1 year after the intervention. Our findings show that the learning outcomes of the intervention group were significantly higher compared to those of the control group after the intervention and that differences between the groups remained even 1 year after the intervention. In particular, the results show that children participating in the intervention group learned to recognize and use part-part-whole relations in novel arithmetic tasks.

In this paper, we present a way of describing variation in young children’s learning of elementary arithmetic within the number range 1–10. Our aim is to reveal what is to be learnt and how it might be learnt by means of discerning particular aspects of numbers. The Variation theory of learning informs the analysis of 2184 observations of 4- to 7-year-olds solving arithmetic tasks, placing the focus on what constitutes the ways of experiencing numbers that were observed among these children. The aspects found to be necessary to discern in order to develop powerful arithmetic skills were as follows: modes of number representations, ordinality, cardinality, and part-whole relation (the latter has four subcategories: differentiating parts and whole, decomposing numbers, commutativity, and inverse relationship between addition and subtraction). In the paper, we discuss particularly how the discernment of the aspects opens up for more powerful ways of perceiving numbers. Our way of describing arithmetic skills, in terms of discerned aspects of numbers, makes it possible to explain why children cannot use certain strategies and how they learn to solve tasks they could not previously solve, which has significant implications for the teaching of elementary arithmetic.

In this paper we report on findings from a study of 5-to-6-year-old children’s ways of structuring part-part-whole relations using finger patterns. We focused our analysis on data from interviews with 28 children who during their last year of preschool learned to enact a structural approach. We used this data set to analyze their different ways of structuring a task with one part unknown, and what constitutes the ability to structure the task in a conceptually powerful way. The way children structure number relations was interpreted as being related to how they experience the task at hand. We identified some ways of structuring as being more powerful for future learning, particularly those that facilitate the child in experiencing parts and the whole simultaneously. We suggest that there are three aspects that children need to discern in order to structure the task successfully in both the short and the long term: what constitutes the whole, the parts within the whole, and finger patterns as a representation of the cardinality of a set. The pedagogical implications are that attention to children’s ways of experiencing the number relations in arithmetic tasks gives clues to why some children develop powerful strategies, and how to support children in their learning to solve arithmetic tasks.