Fractions speak louder than words

In the first of a series exploring how Neuroscience research impacts teaching, Hampton Tutors coach and PhD in Neuroscience, Tim Barnes, explains how research is shining light on the way students understand Math. 

 
Einstein
 

Answer as quickly as you can: which of the following numbers is closest to the correct answer to the question below?

13/11 + 7/8

A) 1  B) 2  C) 19  D) 20

[The correct answer is given at the bottom of this post]

In a recent study, the 4th to 8th graders who were recently asked this question tended to give an answer that was midway between the two fractions; they were, in effect, taking the average of the fractions rather than adding them [1]. The authors of the study concluded that this was a conceptual problem, as students instinctively know that adding two numbers should result in a higher number.

Research conducted by Professor Bob Siegler at Carnegie Mellon has attempted to use fractions to answer the fundamental question of why some people learn mathematical concepts more easily than others [2]. To answer this question, scientists need to determine what stage of the process causes differences between students. Common theories include differences in how students mentally model the size of numbers; how they relate those sizes to written digits; their ability to hold and manipulate concepts in working memory; their ability to hear and understand math-related words; and socioeconomic factors [3,4]. The 'fraction experiment' checked that the subjects’ errors were more likely to be conceptual than to be related to their mental estimates of fraction magnitudes, their ability to do whole number arithmetic, or their ability to do the same estimation problem for whole numbers.

Practically, this means that tutors and coaches should remember that students rely on informal thinking to check their math work, something that may actually increase, rather than diminish, as they age [5]. Students can benefit, therefore, from improving their 'informal learning' along with their formal arithmetic methods. At its simplest, working on mental math (i.e. math without a calculator) reinforces math learning even at the highest levels. In some ways, there is an elegance that even the highest mathematical concepts are held together by the simplest mental techniques.  

[The answer was B) 2. The actual answer is 2 5/88, or 2.056]

[1] Braithwaite DW, Tian J, and Siegler RS (2017). Do children understand fraction addition? Developmental Science:e12601. https://doi.org/10.1111/desc.12601

[2] Siegler RS, Fazio LK, Bailey DH, and Zhou X (2013). Fractions: The new frontier for theories of numerical development. Trends in Cognitive Sciences 17(1):13–99. https://doi.org/10.1016/j.tics.2012.11.004

[3] Vanbinst K and De Smedt B (2016). Individual differences in children's mathematics achievement: The roles of symbolic numerical magnitude processing and domain-general cognitive functions. Progress in Brain Research 227:105–130. https://doi.org/10.1016/j.tics.2012.11.004

[4] Siegler RS and Lortie-Forgues H (2017). Hard lessons: Why rational number arithmetic is so difficult for so many people. Current Directions in Psychological Science 26(4):346–351. https://doi.org/10.1177/0963721417700129

[5] Braithwaite DW, Goldstone RL, van der Maas HLJ, and Landy DH (2016). Non-formal mechanisms in mathematical cognitive development: The case of arithmetic. Cognition 149:40–55. https://doi.org/10.1016/j.cognition.2016.01.004


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