Award Abstract # 1844140
Creating a Theory of Decimal Arithmetic Learning

NSF Org: BCS
Division of Behavioral and Cognitive Sciences
Recipient: FLORIDA STATE UNIVERSITY
Initial Amendment Date: August 23, 2019
Latest Amendment Date: August 23, 2019
Award Number: 1844140
Award Instrument: Standard Grant
Program Manager: Soo-Siang Lim
slim@nsf.gov
 (703)292-7878
BCS
 Division of Behavioral and Cognitive Sciences
SBE
 Directorate for Social, Behavioral and Economic Sciences
Start Date: September 1, 2019
End Date: August 31, 2024 (Estimated)
Total Intended Award Amount: $550,000.00
Total Awarded Amount to Date: $550,000.00
Funds Obligated to Date: FY 2019 = $550,000.00
History of Investigator:
  • David Braithwaite (Principal Investigator)
    dbraithwaite@fsu.edu
  • Robert Siegler (Co-Principal Investigator)
Recipient Sponsored Research Office: Florida State University
874 TRADITIONS WAY
TALLAHASSEE
FL  US  32306-0001
(850)644-5260
Sponsor Congressional District: 02
Primary Place of Performance: Florida State University
FL  US  32306-4166
Primary Place of Performance
Congressional District:
02
Unique Entity Identifier (UEI): JF2BLNN4PJC3
Parent UEI:
NSF Program(s): Science of Learning,
Discovery Research K-12
Primary Program Source: 01001920DB NSF RESEARCH & RELATED ACTIVIT
04001920DB NSF Education & Human Resource
Program Reference Code(s): 059Z, 7645, 8089
Program Element Code(s): 004Y00, 764500
Award Agency Code: 4900
Fund Agency Code: 4900
Assistance Listing Number(s): 47.075

ABSTRACT

Proficiency with rational numbers-fractions, decimals, and percentages-is essential for success in more advanced mathematics such as algebra. It is also important for occupational success; majorities of both white- and blue-collar workers report using rational numbers in their jobs. Yet, many children struggle with rational numbers even after years of instruction. The goal of this project is to create a theory of children's learning in one area of rational numbers: decimal arithmetic. The project will identify types of knowledge that help children to learn decimal arithmetic more easily, clarify the mechanisms by which this facilitation occurs, and develop a computational model that simulates the process of learning decimal arithmetic. Based on the results, recommendations will be generated for improving children's learning of decimal arithmetic including recommendations (1) to focus classroom and practice time on conceptual approaches that are used by successful learners, (2) to place special emphasis on types of problem that pose difficulty for children, (3) to devote classroom time to illustrating common errors and explaining why they are incorrect, and (4) to use discussion of common errors as an opportunity to illustrate general concepts. These recommendations are anticipated to have implications for improving mathematics instruction in general.

Learning mathematics involves learning both concepts and procedures. Concepts include principles and relations; procedures are step-by-step action sequences for solving problems. Understanding of concepts is believed to help children learn procedures, but how this occurs is not known. This project aims to create a theory of how conceptual understanding - when present - facilitates learning of procedures within a particularly difficult and important area of math: decimal arithmetic. To accomplish this goal, the project will adopt a three-pronged approach including longitudinal, microgenetic, and computational modeling methods. Longitudinal methods will identify specific types of conceptual knowledge that predict success in learning decimal arithmetic procedures; microgenetic methods will provide evidence for specific mechanisms by which these types of conceptual knowledge facilitate learning; computational modeling will be used to describe these mechanisms precisely and to simulate the empirical phenomena observed using the previous two methods. The computational model will build on and extend a modeling architecture previously employed in a model of fraction arithmetic learning, FARRA; its success will be assessed based on its ability to generate levels of accuracy, patterns of errors, and correlations between conceptual and procedural knowledge similar to those observed among children. The proposed research will advance scientific knowledge in three ways: by connecting individual differences in learning outcomes with a theory of learning processes, by advancing understanding of the relations between conceptual and procedural knowledge, and by extending theories of numerical development into a new domain, decimal arithmetic.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH

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(Showing: 1 - 10 of 11)
Braithwaite, David W "Domain effects on interpretations of general conditionals: The case of mathematics" Thinking & Reasoning , 2024 https://doi.org/10.1080/13546783.2024.2406197 Citation Details
Braithwaite, David W. and Liu, Qiushan "Affordances of Fractions and Decimals for Arithmetic" Journal of experimental psychology , 2022 Citation Details
Braithwaite, David W. and McMullen, Jake and Hurst, Michelle A. "Cross-notation knowledge of fractions and decimals" Journal of Experimental Child Psychology , v.213 , 2022 https://doi.org/10.1016/j.jecp.2021.105210 Citation Details
Braithwaite, David W. and Siegler, Robert S. "A unified model of arithmetic with whole numbers, fractions, and decimals." Psychological Review , 2023 https://doi.org/10.1037/rev0000440 Citation Details
Braithwaite, David W. and Siegler, Robert S. "Testing a unified model of arithmetic" Proceedings of the 44th Annual Conference of the Cognitive Science Society , 2022 Citation Details
Braithwaite, David W. and Siegler, Robert S. "Testing a Unified Model of Arithmetic" Proceedings of the Annual Conference of the Cognitive Science Society , v.44 , 2022 Citation Details
Braithwaite, David W. and Sprague, Lauren "Conceptual Knowledge, Procedural Knowledge, and Metacognition in Routine and Nonroutine Problem Solving" Cognitive Science , v.45 , 2021 https://doi.org/10.1111/cogs.13048 Citation Details
Braithwaite, David W. and Sprague, Lauren and Siegler, Robert S. "Toward a unified theory of rational number arithmetic." Journal of Experimental Psychology: Learning, Memory, and Cognition , 2021 https://doi.org/10.1037/xlm0001073 Citation Details
Siegler, Robert S. and Im, Soo-hyun and Schiller, Lauren K. and Tian, Jing and Braithwaite, David W. "The Sleep of Reason Produces Monsters: How and When Biased Input Shapes Mathematics Learning" Annual Review of Developmental Psychology , v.2 , 2020 https://doi.org/10.1146/annurev-devpsych-041620-031544 Citation Details
Tian, Jing and Braithwaite, David W. and Siegler, Robert S. "Distributions of textbook problems predict student learning: Data from decimal arithmetic." Journal of Educational Psychology , v.113 , 2021 https://doi.org/10.1037/edu0000618 Citation Details
Tian, Jing and Braithwaite, David W. and Siegler, Robert S. "How do people choose among rational number notations?" Cognitive Psychology , v.123 , 2020 https://doi.org/10.1016/j.cogpsych.2020.101333 Citation Details
(Showing: 1 - 10 of 11)

PROJECT OUTCOMES REPORT

Disclaimer

This Project Outcomes Report for the General Public is displayed verbatim as submitted by the Principal Investigator (PI) for this award. Any opinions, findings, and conclusions or recommendations expressed in this Report are those of the PI and do not necessarily reflect the views of the National Science Foundation; NSF has not approved or endorsed its content.

    Fractions, decimals, and percentages are among the most difficult topics that children encounter in early math education. These topics are covered extensively in fourth to sixth grade, but many children have not mastered them as late as eighth grade. Difficulties with fractions and decimals interfere with learning algebra and other more advanced topics. This research has yielded five key findings about how individuals learn fraction and decimal arithmetic, each of which has implications for educational practice in this area.
    (1) Most errors involve overgeneralization: using procedures that would be correct for one arithmetic operation to solve problems involving a different operation. Common errors include adding or subtracting fractions by performing the arithmetic operation separately on the numerators and denominators (e.g., 3/5+1/4 = 4/9), which is correct for multiplication but not addition or subtraction; multiplying fractions with equal denominators by multiplying only the numerators and passing the common denominator into the answer (e.g., 3/5*1/5 = 3/5), which is correct for addition and subtraction but not multiplication; adding or subtracting decimals by aligning the operands at their rightmost digits rather than at their decimal points (e.g., 2.46+4.1 = 2.87), which is correct for multiplication but not for addition or subtraction; and when multiplying decimals, placing the decimal point at the same location in the answer as where it appears in the operands (e.g., 2.4*1.2 = 28.8), which is correct for addition and subtraction but not for multiplication. Learning activities should emphasize not only how to execute each procedure, but also when each procedure should be used and why.
    (2) Different children display qualitatively different patterns of errors. Some children persistently use procedures that would be correct for addition and subtraction on most or all problems, resulting in high accuracy on addition and subtraction problems, but low accuracy on other arithmetic operations. Others persistently use procedures that would be correct for multiplication on most or all problems, resulting in high accuracy on multiplication problems, but low accuracy on other arithmetic operations. Still other children use a variety of procedures and choose among them somewhat randomly. If a child is struggling with fraction or decimal arithmetic, it is important to diagnose which of these error patterns the child displays in order to provide intervention that is matched to the child’s specific difficulties.
    (3) Most children (and even many adults) cannot explain why fraction and decimal arithmetic procedures make sense. Rather than learning procedures by rote, children should understand their conceptual rationales. For example, children should learn that a common denominator is needed when adding fractions because addition requires a common unit and the denominator of a fraction functions like a unit. Analogously, children should learn that one aligns the decimal points of decimals before adding them because doing so ensures that one will add digits that have the same place value.
    (3) Certain types of problems are challenging for children because children almost never encounter them. Such problems include division of a fraction by a fraction (e.g., 2/3÷3/5), multiplication of fractions with equal denominators (e.g., 3/5*1/5), and addition of a decimal and a whole number (e.g., 0.415+52). These types of problems are surprisingly difficult for children because they are rarely appear in children's math textbooks and homework. Children should be given more opportunities to practice solving these rare types of problems.
    (4) Addition is intrinsically harder with fractions than with decimals, while multiplication is intrinsically harder with decimals than fractions. When children or adults are presented equivalent fraction and decimal addition problems (e.g., 3/5+1/4 or 0.6+0.25), they prefer to solve, and are more accurate with, the decimal problems. The opposite is true for equivalent fraction and decimal multiplication problems (e.g., 3/5*1/4 or 0.6*0.25). Children might benefit by being taught rational number addition (and probably subtraction) with decimals before fractions, contrary to the instructional sequence that is currently typical. However, the findings support the current practice of teaching multiplication (and probably division) with fractions before decimals.
    (5) Children who understand connections between fractions and decimals tend to be more proficient with fraction and decimal arithmetic, even when controlling for other differences among children. This finding is related to the previous one, in that children who understand connections between notations may use their understanding of an arithmetic operation in one notation to help understand the same arithmetic operation in a different notation (e.g., using understanding of fraction multiplication to help understand decimal multiplication), and when presented a problem that is difficult to solve in one notation, they may translate it into an easier notation (e.g., translate a decimal multiplication problem into fraction form). We observed that adults use such cross-notation conversions spontaneously when doing fraction and decimal arithmetic. Children may profit from being taught to do so as a legitimate problem solving strategy, and being given opportunities to practice this strategy.


Last Modified: 12/16/2024
Modified by: David W Braithwaite

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