RESEARCH

WHY ARE SOME CHILDREN BETTER AT MATH THAN OTHERS?

There are large individual differences in numeracy. On average, there is a seven-year span in ability within a single elementary classroom (Cockcroft, 1982). Of particular concern is the finding that children who enter school with poor numeracy skills do not catch up (Aunola et al., 2004), likely due to the lack of identification and intervention tools.

As part of our research program, we are developing tools to assess and predict children’s numeracy skills early on, in kindergarten and first grade. We have identified precursor abilities to numeracy and, using patterns of strengths and weaknesses in these abilities, we are able to predict which children will struggle to gain numeracy skills, and even which skills they will struggle with. The ultimate goal of this research is to create individualized interventions.

This line of research was funded by (1) a Social Sciences and Humanities Research Council (SSHRC) grant for the Count Me In Project (www.carleton.ca/cmi). Count Me In was a large-scale, multi-site, longitudinal research study investigating predictors of mathematical development in children from Kindergarten to Grade 5 (N = 500) and (2) a series of three research grants from King’s University College for Improving Numeracy Outcomes in Elementary School Children, a longitudinal research project investigating predictors of mathematical development in children from Kindergarten to Grade 3, and (3) a research grant from King’s University College for Development of an Early-Numeracy Screening Tool.

Key findings:

• There are multiple pathways to math success (LeFevre et al., 2010; Penner, Buckland & Moes, 2019). Children’s linguistic, quantitative, and visuo-spatial working memory skills each contribute independently to the development of mathematical skills. As a result, children may be able to compensate for a weakness in one area with a relative strength in another. This finding of multiple pathways to success has strong implications for math education and the development of math interventions.

• Currently, few evidence-based early screening tools exist to identify children at risk of math difficulties and, once identified, there are even fewer interventions that are shown to be effective (Penner, Buckland & Moes, 2019).

• Researchers use a variety of different performance measures for key numerical tasks including number comparison and enumeration. Our research has shown that these different measures are not indexing the same construct/ability within a given task and, thus, researchers need to be thoughtful about their choices, especially in screening tools (Penner et al., 2019; Penner-Wilger et al., 2017).

• A small set of cognitive predictors can be used to identify kindergarten children who will struggle to learn mathematics: symbolic number comparison (e.g., Which is more? 35), visuo-spatial working memory (e.g., recreating a sequence), phonemic awareness (), and receptive vocabulary (Penner & Waring, 2018). Performance on these four tasks can account for 41% of variability in children’s math scores. Test outcomes can be used for early identification of students in need of math interventions and to more specifically guide the intervention design.

Next steps:

Development of a diagnostic screening tool, MathPath, that teachers can use to identify kindergarten children at risk for math difficulties for intervention. Two of the three pathways (quantitative and working memory) have currently been implemented in our easy to use application. We are working on adding the language pathway tasks and will, in the future, be testing the effectiveness of MathPath in a longitudinal efficacy study following the rigorous criteria outlined in (Penner, Buckland, & Moes, 2019). 

Find out more:

LeFevre, J., Fast, L., Skwarchuk, S. L., Smith-Chant, B. L., Bisanz, J., Kamawar, D., & Penner-Wilger, M. (2010). Pathways to mathematics: Longitudinal predictors of performance. Child Development, 81, 1753-1767.

https://www.ncbi.nlm.nih.gov/pubmed/21077862

Penner, M., *Buckland, C., & *Moes, M. (2019). Early identification of, and interventions for, kindergarten students at risk for mathematics difficulties. In K. M. Robinson, H. Osana & D. Kotsopoulos (Eds.), Mathematical Learning and Cognition in Early Childhood: Integrating Interdisciplinary Research into Practice (pp. 57-78). New York, NY: Springer.

https://doi.org/10.1007/978-3-030-12895-1_5

Penner, M. & *Waring, R. J. (2018, April). Kindergarten predictors of mathematics: Quantitative, working memory and linguistic skills. Presented at the Mathematical Cognition and Learning Society annual meeting, Oxford, UK.

Penner-Wilger, M., Cecala, A. L., & *Elfers, M. (2017, July). Comparing comparison indices: Assessing the construct and predictive validity of different magnitude comparison measures across formats and age groups. Paper presented at the 39th Annual Conference of the Cognitive Science Society, London, UK.

Penner, M., Cecala, A. L., *Moes, M., & *Waring, R. J. (2019, June). How to index subitizing performance: Comparing the concurrent validity of individual subitizing range and subitizing slope measures in predicting kindergarten numeracy. Presented at the Canadian Society for Brain, Behaviour, & Cognitive Science annual meeting, Ottawa, ON.

WHY ARE SOME ADULTS BETTER AT MATH THAN OTHERS?

Poor numeracy is a serious concern, as numeracy skills are at least as important as literacy skills for job outcomes (Bynner & Parsons, 1997). University students’ levels of calculation fluency and conceptual understanding of mathematics have declined significantly. As part of our research program, we are investigating individual differences in university students’ numeracy skills, particularly differences in the mental representations and processes students use to solve arithmetic problems. Our research shows differences in how more- and less-skilled students represent arithmetic operations, and in the solution strategies they use to solve problems. This research has implications for the development of math pedagogy, especially for how operations are conceptually introduced.

This line of research was funded by a Natural Sciences and Engineering Research Council (NSERC) Julie Payette Award and a Canada Graduate Scholarship.

Key findings:

• The calculation fluency of Canadian undergraduates has declined by 20% in the past decade (LeFevre, Penner-Wilger, et al., 2014). Thus, high-school and university instructors cannot expect students to be comfortable solving even simple arithmetic problems ‘in their heads’. 

• There is evidence to suggest that this decline may be the result of the introduction of calculators in elementary schools (LeFevre, Penner-Wilger, et al., 2014). Our findings suggest that calculators are necessary for basic computations, potentially impeding students’ understanding by diverting their attention from the conceptual focus of lessons.

• More-skilled university students tend to represent arithmetic facts in a verbal code (e.g., “six plus two is eight”) and have memorized the basic facts. Less-skilled students tend to represent basic facts spatially, as movement along a mental number line and use a variety of strategies to solve problems, such as counting (e.g., 6+2= six, seven, eight; Penner-Wilger et al., 2011).

Next steps:

Designing and testing a diagnostic tool for university math students. In collaboration with Joseph Turnbull we have developed and are testing the power of a diagnostic math test, along with existing measures of math anxiety and test anxiety, to predict student success in first-year calculus courses. Our preliminary analyses show that 37-40% of variability in final grades can be accounted for by the diagnostics; we are currently completing more rigorous measures of diagnostic accuracy (i.e., sensitivity and specificity; Turnbull et al., in preparation).

Find out more:

LeFevre, J., Penner-Wilger, M., Pyke, A., Shanahan, T., Deslauriers, W. A., Trbovich, P., & Roberts, M. A. (2014). Putting two and two together: Declines in arithmetic fluency among young Canadian adults. Carleton University Cognitive Science Technical Report 2014-01. URL http://www.carleton.ca/ics/TechReports.

Penner-Wilger, M., Landy, D. H., *Zhang, X., & *Weitzer, A. (2011). Going through the motions: Skill differences in the representation of arithmetic operations. Proceedings of the 33rd Annual Cognitive Science Society. Austin, TX: Cognitive Science Society.

Turnbull, J., *Moes, M., Penner, M., Meredith, D., & Lydia, L. (2019). Mathematics learning of undergraduate students. Manuscript in preparation.