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IES Grant

Title: The Organization of Mathematical Knowledge
Center: NCER Year: 2008
Principal Investigator: Rips, Lance Awardee: Northwestern University
Program: Cognition and Student Learning      [Program Details]
Award Period: 3 years Award Amount: $599,291
Type: Exploration Award Number: R305A080341

Purpose: The purpose of this project is to compare different ways of organizing mathematical information and empirically test which one will lead to better understanding of and memory for the text. Competing theories suggest the benefit of employing either an object-based or deduction-based format. Pilot testing revealed equivalent overall memory for statements with either format but differential benefits when the level of a statement's importance is considered. The deduction-based format led to better memory for text deemed important compared to subsidiary information such as statements that support a main idea than the object-based format. Researchers will explore various cognitive factors that may account for this finding and ultimately offer guidelines for presentation of mathematical content in curricula.

Project Activities: The researchers will create mathematics texts, conceptually organized using either the object-based or deduction-based format. These texts will cover topics unfamiliar to typical high school students (i.e., elementary group theory and non-Euclidian geometry). They will then conduct a series of experiments to examine if and how these different formats affect behaviors (such as reading times and re-reading passages) and performance (such as memory for text, ability to solve problems, and ability to explain text). The goal is to understand which cognitive components (e.g., rehearsal of and memory for important information) might be affected by the format organization and to offer research-based suggestions for curriculum presentation.

Products: The major product from this project will be guidelines for curriculum design based on cognitive theory and empirical findings, along with published reports.

Structured Abstract

Setting: The experiments will be conducted with students living in a suburban area of Illinois.

Population: About 740 eleventh- and twelfth-grade students will participate from a school district that is composed of about 9% Hispanic/Latino, 30% Black/African-American, 52% white, and 3% Asian students, and about 5% of families are below the poverty level.

Intervention: Two organizational formats for presenting math text will be created and tested for their comparable effectiveness in conveying information and promoting student learning. The object-based format organizes the content around the mathematical objects and sub-objects being taught and the properties of those objects. The deduction-based format organizes the content using deductive relations that exist between concepts being taught.

Research Design and Methods: The main comparison will be between the two levels of organizational format: object- and deduction-based. Random assignment to the two levels will be at the student level, with the restriction that student characteristics (e.g., race, pretest math ability) are comparable between conditions. There will be 30 subjects per condition.

Control Condition: There is no control condition.

Key Measures: The key performance measures will be reading latency, amount of re-reading, rating of text importance, recall of presented information, and ability to explain learned text and solve math problems.

Data Analytic Strategy: Comparisons between the two organizational formats on various dependent measures will be made using an independent two-sample t-test. Additionally, regression analyses will be conducted using pretest scores (e.g., math and reading ability) as correlates to performance.


Book chapter

Rips, L.J. (2012). Bootstrapping: How not to Learn. In N.M. Seel (Ed.), Encyclopedia of the Sciences of Learning: Part 2 (pp. 473–477). Berlin: Springer.

Journal article, monograph, or newsletter

Bartels, D.M., and Rips, L.J. (2010). Psychological Connectedness and Intertemporal Choice. Journal of Experimental Psychology: General, 139(1): 49–69.

Hespos, S.J., Dora, B., Rips, L.J., and Christie, S. (2012). Infants Make Quantity Discriminations for Substances. Child Development, 83(2): 554–567.

Rips, L.J. (2010). Two Causal Theories of Counterfactual Conditionals. Cognitive Science, 34(2): 175–221. 

Rips, L.J. (2011). Split Identity: Intransitive Judgments of the Identity of Objects. Cognition, 119(3): 356–373.

Rips, L.J. (2011). Causation From Perception. Perspectives On Psychological Science, 6(1): 77–97.

Rips, L.J. (2013). How Many Is a Zillion? Sources of Number Distortion. Journal of Experimental Psychology: Learning, Memory, and Cognition, 39(4): 1257–1264.

Rips, L.J., and Edwards, B.J. (2013). Inference and Explanation in Counterfactual Reasoning. Cognitive Science, 37(6): 1107–1135.

Rips, L.J., and Hespos, S.J. (2011). Rebooting the Bootstrap Argument: Two Puzzles for Bootstrap Theories of Concept Development. Behavioral and Brain Sciences, 34(3): 145–146.

Rips, L.J., Asmuth, J., and Bloomfield, A. (2013). Can Statistical Learning Bootstrap the Integers?. Cognition, 128(3): 320–330.


Edwards, B., and Rips, L. (2012). Explanations of Counterfactual Inferences. In Proceedings of the Cognitive Science Society.