# Analogies in Mathematics

When teachers draw on children’s prior knowledge during a new lesson, it’s like making an analogy across new and old concepts. One primary goal of my research is to explore the consequences of analogies in instruction. Recently, Clarissa and I wrote an overview of this work and argued for its importance! Here is the citation for our Current Directions in Psychological Science paper, and highlights in this line are below!

Sidney, P. G. & Thompson, C. A. (2019). Implicit analogies in learning: Supporting transfer by “warming up”. Current Directions in Psychological Sciences, 28(6). DOI: 10.1177/0963721419870801

###### Which Analogies?

What are the consequences of making analogies between prior knowledge that might help or hinder new fraction division concepts either explicitly or implicitly? We found that implicitly linking to a conceptually-similar domain (whole number division) was better for new learning than linking to perceptually-similar domains (fraction addition and subtraction) or explicitly linking to whole number division without providing support for the link.

Sidney, P.G. & Alibali, M.W. (2015). Making connections in math: Activating a prior knowledge analogue matters for learning. Journal of Cognition and Development, 16(1) 160-185. DOI:10.1080/15248372.2013.79209.

###### Setting up “Implicit” Analogies

To further explore the boundaries of implicit analogies in understanding fraction division concepts, I examined children’s understanding of fraction division concepts after merely activating their knowledge of relevant whole number division concepts in contrast to activating their knowledge of fraction multiplication. In this study, we found that children whose whole number division concepts were very recently activated were better at modeling what it means to divide by a fraction.

Sidney, P. G., & Alibali, M. W. (2017). Creating a context for learning: Activating children’s whole number knowledge prepares them to understand fraction division. Journal of Numerical Cognition, 3(1), 31-57. DOI: 10.5964/jnc.v3i1.71

Given that implicit analogies appear to work under certain conditions, I compared an implicit instructional analogy between whole number and fraction division to a highly-cued, explicit instructional analogy (or no analogy at all). Indeed, it turns out that implicit analogies to children’s prior knowledge may have different consequences for learning than explicit analogies!

Sidney, P. G. (2020). Children’s learning from implicit analogies during instruction: Evidence from fraction division. Cognitive Development, 56. DOI: 10.1016/j.cogdev.2020.100956

###### How to Support “Implicit” Analogies

In my ongoing work, I am examining other instructional features that might moderate the effectiveness of implicit analogies. We have found that diagrams may be critical for eliciting students’ prior knowledge. Currently, we are examining the effects of self-explanation on analogical learning.

Sidney, P.G., Shirah, J. +, Zahrn, L.*, & Thompson, C.A. (2022). Diagrams support spontaneous transfer across whole number and fraction concepts. Contemporary Educational Psychology, Advanced online publication: 102066.