The Relationship Between Arithmetic Achievement and Magnitude Comparison Skills in Primary School:

Evidence from Canada and Cambodia

Nadia Nosworthy, University of Western Ontario, CanadaMay 8, 2013

Introduction

• It is essential that we advance our approaches to assessing the foundational competencies of numeracy.

• Phonemic awareness

• Magnitude comparison

Symbolic magnitude comparison

Nonsymbolic magnitude comparison

Halberda, Mazzocco & Feigenson (2008)

Nonsymbolic magnitude comparison

Mazzocco, Feigenson & Halberda (2011)

What we know…

• Individual differences in the accuracy and RT measures of symbolic and nonsymbolic magnitude comparison relates to children’s arithmetic skills.

(Durand et al., 2005; Holloway & Ansari, 2009; De Smedt, Verschaffel and Ghesquière, 2009; Halberda, Mazzocco & Feigenson, 2008; Mazzocco, Feigenson & Halberda, 2011)

Correlation of Math Achievement with symbolic magnitude comparison

Holloway & Ansari (2009)

Correlation of Math Achievement with nonsymbolic magnitude comparison

Halberda, Mazzocco & Feigenson (2008)

Challenges

• Current methods used to assess magnitude comparison:

• usually rely on computerized software

• resource intensive

• difficult to implement on a large scale.

Research Goal

• Design a test which directly assesses symbolic and nonsymbolic magnitude processing in children.

• Low-cost paper-and-pencil test

• Quickly and easily administered and scored

• Can be used on a global scale

• Non-symbolic allows for testing across symbol systems

• AND in very young children who may not yet have a fluent understanding of number symbols.

Magnitude comparison task

56 symbolic56 nonsymbolic

Magnitudes 1-9

1-minute time limit per section

Symbolic Nonsymbolic

Project 1 (Canada) – Research Questions

1) Is performance on this assessment related to standardized measures of arithmetic achievement?

2) If significantly related, can it explain significant variance over other factors such as age, working memory, reading skills and IQ?

Methods

n = 160 students (grades 1-3, 83 females, Mage= 8.1 yrs)

Additional Measures

• Woodcock Johnson-III (WJ-III)

Math Fluency

Calculation

Reading Fluency

• Wechsler Abbreviated Scale of Intelligence (WASI)

• Automated Working Memory Assessment (AWMA)

Results

Significant relationship between Math Fluency scores and magnitude comparison scores.

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Math Fluency

r = .43, p < .01(Partial correlation controlling for age)

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-12 -10 -8 -6 -4 -2 0 2 4 6 8

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Calculation

r = .34, p < .01(Partial correlation controlling for age)

Significant relationship between Calculation scores and magnitude comparison scores.

Is this finding math specific?

Results - Regression

• Regression predicting arithmetic skills using age, IQ, WM, Reading Fluency, and magnitude comparison as predictors.

• Result demonstrates that scores on magnitude comparison task account for unique variance in arithmetic scores.

• Relationship not confounded by these other factors

What about reliability?

Reliability

• Sample of 39 first grade students

• Examined correlation between T1 and T2 of testing.

• The test demonstrated test-retest reliability (r = .72, p < .001) between both periods of testing.

Summary

Results suggest that this assessment:

1) is related to standardized measures of arithmetic achievement.

2) demonstrates that performance on magnitude comparison task accounts for unique variance in arithmetic scores.

Next step…

Can these findings from Canadian classrooms be applied to other countries?

Project 2 - Cambodia

Sample

n = 985 students from six schools in Phnom Penh, Cambodia

Mage = 8 years, 1 month (SD = 1.4)

455 males and 530 females in grades 1-3

MethodEach child was given the magnitude comparison task and

Math Fluency subtest in a group setting as follows:

Grade 1: Given magnitude comparison and Math Fluency with Khmer number symbols

Grades 2, 3: Given magnitude comparison and Math Fluency with Arabic number symbols.

Magnitude Comparison - Khmer

Results

Significant relationship between magnitude comparison and Math Fluency

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Math Fluency

r = .338, p < .001(Partial correlation controlling for age)

Cross-national similarities

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Math Fluency

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Math Fluency

Cambodia Canada

Across symbol systems This measure can be adapted and individual differences

can be found across cultures.

Grade one (Khmer) – symbolic comparison correlated with Math Fluency (r = .26, p <.001)

Conclusion and Future Directions

1) This 2 minute assessment of foundational skills is related to measures of arithmetic achievement.

2) Flexible and can be used in a global context to address differences in specific cultural instruction formats.

3) Can be used for early identification of children at risk.

Acknowledgements

Dr. Daniel Ansari Dr. Deepa Srikantaiah

Christian Battista

Stephanie Bugden

Dr. Ian Lyons Cambodian Ministry of

Dr. Lisa Archibald Education, Youth & Sport

Dr. Barrie Evans

Thank you!

nosworthy.n@gmail.com

Numerical Ratio Effect (NRE)

• Ratio effect individuals more quickly and accurately compare two numbers of smaller magnitude vs. two numbers of a larger magnitude, even when the distance between the numbers remains constant (i.e., 3, 4 (0.75) vs. 8, 9 (.89)).

• Ratio effect highly correlates with distance effect but accounts for more variability in response times and accuracy (Moyer and Landauer, 1967).

Linear regression analyses predicting Math Fluency raw scores with chronological age, Reading Fluency, visual spatial working memory,

verbal working memory, IQ, symbolic scores and nonsymbolic scores as predictors.

Linear regression analyses predicting Math Calculation raw scores with chronological age, Reading Fluency, visual spatial working memory,

verbal working memory, IQ, symbolic scores and nonsymbolic scores as predictors.

Summary and future directions

1) This 2 minute assessment of foundational skills is related to measures of arithmetic achievement.

2) Can be used in a global context to assess precursor skills in young children both one-on-one and in large group settings.

Numerical Distance Effect (NDE)

Moyer and Landauer (1967)

Numerical Distance Effect (NDE)

Size of NDE decreases over developmental time (Sekuler & Mierkiewicz, 1977).

Results - ANOVA

* nsns

Results - ANOVA• 2 (format) x 3 (grade) repeated measures ANOVA

• No main effect of format (F(1, 157) = .311, ns)

• Main effect of grade (F(2, 157) = 14.18, p <.001)

• Format x grade interaction (F(2, 157) = 6.61, p <.001)

• Grade one children achieved higher scores on non-symbolic items (t(25) = -3.21, p<.05) compared to symbolic items.

• No significant difference between format within grade 2 (t(55) = 1.38, ns) or grade 3 (t(77) = 1.40, ns) participants.

Results – ANOVA

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Grade 1 Grade 2 Grade 3 Grade 5

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Grade level

Numbers

Dots

Results

• 2 (format) x 3 (grade) repeated measures ANOVA

• Main effect of format (F(1, 1169) = 83.63, p < .001)

• Main effect of grade (F(3, 1169) = 252.62, p < .001)

• Format x grade interaction (F(3, 1169) = 62.90, p < .001)

Results - ANOVA

• Grade 1, 2 and 3 children achieved higher scores on nonsymbolic items compared to symbolic items.

• Grade 1 (t(338) = 5.33, p < .001)

• Grade 2 (t(303) = 8.66, p < .001)

• Grade 3 (t(341) = 14.13, p < .001)

• Grade 5 children achieved higher scores on symbolic items compared to nonsymbolic items (t(187) = 6.56, p < .001)

Linear regression analyses predicting Math Fluency raw scores with chronological age, Reading Fluency, visual spatial working memory,

verbal working memory, IQ, symbolic scores and nonsymbolic scores as predictors.

Linear regression analyses predicting Math Calculation raw scores with chronological age, Reading Fluency, visual spatial working memory,

verbal working memory, IQ, symbolic scores and nonsymbolic scores as predictors.

Measure• 12 page response booklet

• Total of 112 items (56 symbolic and 56 nonsymbolic)

• Time limit – 1 minute (per section)

Measure

• Magnitudes range from 1-9

• The ratio between the two numbers fall between .11 and .89 (i.e., 1:9 = .11 and 8:9 = .89)

• Each number is counterbalanced for the side of presentation (i.e., 2|7, 7|2).

• Non-symbolic stimuli are controlled for area and density.

Results - Correlations

Symbolic scores correlated with: Nonsymbolic scores correlated with:

Odd-One-Out r = .31* Block Design r = .34**

Sentence Recall r = .21* Reading Fluency r = .27*

Listening Recall r = .18*

Vocabulary r = .16*

Block Design r =.20*

Reading Fluency r = .31**

* p < .05, ** p < .01

Results

• 2 (format) x 3 (grade) repeated measures ANOVA

• Main effect of format (F(1, 1169) = 83.63, p < .001)

• Main effect of grade (F(3, 1169) = 252.62, p < .001)

• Format x grade interaction (F(3, 1169) = 62.90, p < .001)

Results - ANOVA

• Grade 1, 2 and 3 children achieved higher scores on nonsymbolic items compared to symbolic items.

• Grade 1 (t(338) = 5.33, p < .001)

• Grade 2 (t(303) = 8.66, p < .001)

• Grade 3 (t(341) = 14.13, p < .001)

• Grade 5 children achieved higher scores on symbolic items compared to nonsymbolic items (t(187) = 6.56, p < .001)

Number sense an analog as important to mathematics learning as phonemic awareness

Introduction• Not enough attention is being paid to foundational

abilities

• We don’t have processing measures to test these skills

• Phonological awareness

• Because math is so important to a child’s overall success it is essential that we advance our approaches to assessing the foundational competencies involved in numeracy development.