Fuson, K. C., Murata, A., & Abrahamson, D. (2015). Using learning path research to balance mathematics education: Teaching/learning for understanding and fluency. 

In R. Cohen Kadosh & A. Dowker (Eds.), The Oxford handbook of numerical cognition (pp. 1020-1038). Oxford, UK: Oxford University Press.

This chapter is an overview of central research-based perspectives that support teaching-learning for understanding and for fluency. We summarize the Class Learning Path Model that integrates two theoretical foci – a Piagetian focus on learning and a Vygotskiian focus on teaching – and specifies phases in learning that reflect Vygotsky’s assertion about the move from spontaneous to scientific concepts. Major aspects of the model were drawn from national research-based reports. This model connects understanding and fluency with a focus on mathematically important but also accessible methods in the middle and on maths drawings and other supports for understanding these methods. Such methods can be generated by students and can bridge from less-advanced student methods to formal methods that are unnecessarily complex. For three maths domains in Grades Kindergarten through Grade 6, we illustrate and discuss methods in the middle and drawings (diagrams) that support these methods: problem solving and especially the full range of word problem situations with each quantity the unknown; multidigit addition, subtraction, multiplication, and division; and ratio and proportion. Central features of the Common Core State Standards Mathematical Practices (CCSSO/NGA 2010) in these domains are identified, and how these can support understanding and fluency are briefly discussed. Further aspects of how the pedagogical supports help students move through the Class Learning Path in their own individual ways, and implications for research and for designing maths programmes are then discussed.