The impact of fraction magnitude knowledge on algebra performance and learning

Booth, Julie L.; Newton, Kristie J.; Twiss-Garrity, Laura K.

doi:10.1016/j.jecp.2013.09.001

Cited by 143 publications

(163 citation statements)

References 16 publications

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“…In contrast, in the same data sets, whole number addition, subtraction, and multiplication did not account for significant variance in high school math achievement after the other variables were statistically controlled (though division did). Other studies have also shown both concurrent and predictive relations between knowledge of rational numbers and math achievement (Bailey, Hoard, Nugent & Geary, ; Booth & Newton, ; Booth, Newton & Twiss‐Garrity, ; Schneider, Grabner & Paetsch, ). To omit such strong relations from theories of mathematical development is to commit an unforced error.…”

Section: From Whole To Rational Numbersmentioning

confidence: 88%

Magnitude knowledge: the common core of numerical development

Siegler

2016

Developmental Science

194

167

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The integrated theory of numerical development posits that a central theme of numerical development from infancy to adulthood is progressive broadening of the types and ranges of numbers whose magnitudes are accurately represented. The process includes four overlapping trends: (1) representing increasingly precisely the magnitudes of non-symbolic numbers, (2) connecting small symbolic numbers to their non-symbolic referents, (3) extending understanding from smaller to larger whole numbers, and (4) accurately representing the magnitudes of rational numbers. The present review identifies substantial commonalities, as well as differences, in these four aspects of numerical development. With both whole and rational numbers, numerical magnitude knowledge is concurrently correlated with, longitudinally predictive of, and causally related to multiple aspects of mathematical understanding, including arithmetic and overall math achievement. Moreover, interventions focused on increasing numerical magnitude knowledge often generalize to other aspects of mathematics. The cognitive processes of association and analogy seem to play especially large roles in this development. Thus, acquisition of numerical magnitude knowledge can be seen as the common core of numerical development. Research highlights• The integrated theory of numerical development posits that the unifying theme of numerical development is children's growing understanding of numerical magnitudes.• The theory focuses on four aspects of numerical development: representing increasingly precisely the magnitudes of non-symbolic numbers, connecting small symbolic numbers to their non-symbolic referents, extending understanding from smaller to larger whole numbers, and accurately representing the magnitudes of rational numbers.• For both whole and rational numbers, knowledge of numerical magnitudes is correlated with, predictive of, and causally related to other crucial aspects of mathematics, including arithmetic and overall math achievement.• Interventions that emphasize acquisition of numerical magnitude knowledge have yielded a broad range of positive effects with both whole and rational numbers.

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Section: From Whole To Rational Numbersmentioning

confidence: 88%

Magnitude knowledge: the common core of numerical development

Siegler

2016

Developmental Science

194

167

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“…This is surprising, given that fraction knowledge has been shown to be highly predictive of future math achievement (Siegler et al, 2012), critical for algebra readiness (Booth & Newton, 2012), performance, and learning (Booth, Newton, & Twiss-Garrity, 2014), and is hypothesized to be one of the most important factors in algebra achievement (NMAP, 2008). One reason for the apparent discrepancy may be that scholars have identified foundational knowledge of the magnitudes of fractions, and the understanding of fractions as numbers, as the critical knowledge components impacting algebra (e.g., Booth et al, 2014;Empson & Levi, 2011). In the present study, fraction errors do not represent students' misunderstanding of the values of fractions themselves, but their misunderstanding of the relationships between the numerators and denominators.…”

Section: Discussionmentioning

confidence: 99%

Persistent and Pernicious Errors in Algebraic Problem Solving

Booth¹,

Barbieri²,

Eyer³

et al. 2014

The Journal of Problem Solving

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Students hold many misconceptions as they transition from arithmetic to algebraic thinking, and these misconceptions can hinder their performance and learning in the subject. To identify the errors in Algebra I which are most persistent and pernicious in terms of predicting student difficulty on standardized test items, the present study assessed algebraic misconceptions using an in-depth error analysis on algebra students' problem solving efforts at different points in the school year. Results indicate that different types of errors become more prominent with different content at different points in the year, and that there are certain types of errors that, when made during different levels of content, are indicative of math achievement difficulties. Recommendations for the necessity and timing of intervention on particular errors are discussed.

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“…Future studies might also examine older students' magnitude knowledge of negative numbers and whether their performance on a negative number line task is associated with algebra success and/or later math achievement -something which has been found true for estimates on positive number lines (e.g. Booth & Newton, 2012;Booth, Newton, & Twiss-Garrity, 2014;Booth & Siegler, 2006, 2008. The current study has provided a first stepping stone towards understanding students' magnitude knowledge of negative numbers and has left much room for further efforts to help understand this unique area of mathematical cognition.…”

Section: Discussionmentioning

confidence: 99%

Student Magnitude Knowledge of Negative Numbers

Young

Booth

2015

J. Numer. Cogn.

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Numerous studies have demonstrated the relevance of magnitude estimation skills for mathematical proficiency, but little research has explored magnitude estimation with negative numbers. In two experiments the current study examined middle school students' magnitude knowledge of negative numbers with number line tasks. In Experiment 1, both 6th (n = 132) and 7th grade students (n = 218) produced linear representations on a -10,000 to 0 scale, but the 7th grade students' estimates were more accurate and linear. In Experiment 2, the 7th grade students also completed a -1,000 to 1,000 number line task; these results also indicated that students are linear for both negative and positive estimates. When comparing the estimates of negative and positive numbers, analyses illustrated that estimates of negative numbers are less accurate than those of positive numbers, but using a midpoint strategy improved negative estimates. These findings suggest that negative number magnitude knowledge follows a similar pattern to positive numbers, but the estimation performance of negatives lags behind that of positives.

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The impact of fraction magnitude knowledge on algebra performance and learning

Cited by 143 publications

References 16 publications

Magnitude knowledge: the common core of numerical development

Magnitude knowledge: the common core of numerical development

Persistent and Pernicious Errors in Algebraic Problem Solving

Student Magnitude Knowledge of Negative Numbers

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