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IES Grant

Title: Making Sense of Concrete Models for Mathematics
Center: NCER Year: 2008
Principal Investigator: Mix, Kelly Awardee: Michigan State University
Program: Cognition and Student Learning      [Program Details]
Award Period: 3 years Award Amount: $1,319,945
Type: Development and Innovation Award Number: R305A080287

Co-Principal Investigator: Linda B. Smith

Purpose: To help children connect their intuitive understanding of mathematics to related symbolic procedures, some educational theorists have advocated for the use of concrete models during instruction. Current theories identify several mechanisms that might be engaged by concrete models for mathematics. These models are meant to embody mathematical relationships more transparently than everyday objects. For example, to teach place value concepts, a set of beads may be used to illustrate the relations among ones, tens, hundreds, and thousands. These objects are different from the objects children are likely to encounter in their day-to-day activities and in that sense, may be less grounded in everyday experience. Yet, the objects are hypothesized to provide a way to directly experience place value relations that is lacking in written symbols. Concrete models are implemented in many different ways based on a range of variation along several dimensions. The goal of this project is to develop and document the feasibility of an intervention based on the use of concrete models to teach mathematics. Specifically, the research team is investigating the conditions under which concrete models support learning based on the cognitive mechanisms they might engage. The project focuses on the acquisition of place value concepts in the early elementary grades because this is a well-known challenge for young learners and one that interferes with the subsequent acquisition of more complex algorithms and concepts.

Project Activities: In this project, the research team will conduct six experiments designed to examine the effects of various instructional approaches of using math manipulatives on the cognitive mechanisms these approaches might engage. In each experiment, rising first graders will be randomly assigned to either the control condition or to one of multiple experimental conditions. By manipulating factors such as amount of direct contact, length of exposure, teacher-directed structure, exposure to written algorithms, numbers of models, and models that elicit different actions, the project seeks to determine not only whether manipulatives work but under what conditions they do and why, at a process level. Children's learning of multi-digit calculation problems, as well as place value, will be measured in both conditions.

Products: The products of this project will be published reports detailing a set of experimental data that identifies not only whether concrete math manipulatives work but under what conditions these models support learning based on the cognitive mechanisms they might engage. Other products include coded videotapes of child-controlled variables that demonstrate the way children use and act upon materials during training.

Structured Abstract

Setting: Researchers will organize and offer summer camps and after-school programs for recruiting and testing participants in two ethnically diverse mid-size Midwestern cities (Michigan and Indiana).

Population: Study participants include approximately 400 rising first graders who reflect the demographic diversity of the research setting, both in terms of race and socioeconomic status. Twenty-four children, with approximately equal numbers of boys and girls, will participate in each of the proposed conditions of each experiment.

Intervention: Guided by current cognitive theory related to symbol grounding, the research team will identify the instructional practices for teaching place value and multi-digit calculation with base-10 models, such as place value blocks, Montessori beads, and trading chips. Participating children will attend the instructional program for 2 hours every day over a period of 3 weeks. Although the proposed intervention is not aimed at creating new teaching materials or activities, researchers will generate a solid theoretical and empirical basis for implementing concrete models that already exist.

Research Design and Methods: Using the same general method across six experiments, researchers will examine under what conditions base-10 models support learning. Experiments use a pre-test—intervention—post-test design, in which the pre-test and post-tests are equivalent (though not identical) and the type of instruction that children receive is manipulated. Researchers will recruit children by organizing and offering summer math camps and after-school programs. Factors to be manipulated include amount of direct contact, length of exposure, teacher-directed structure, exposure to written algorithms, numbers of models, and models that elicit different actions. In Experiment 1, for example, children will be randomly assigned to either an experimental condition (concrete place value blocks; virtual place value blocks; or, teacher demonstration only with no direct contact for students) or to a control condition (in which children are taught and will practice written algorithms only). Children will be pulled out in small groups (5–10 students) for brief (30 minutes), daily training sessions across 12 weeks. Using an iterative design, subsequent experiments will be based on these outcomes; and, this process of iteration will be repeated as subsequent experiments build on results from previous experiments.

Control Condition: Conditions for each of the six experiments depend on the research question posed, as follows: In Experiment 1, researchers compare the performance of children taught with place value blocks to those taught only the written algorithms. In Experiment 2, children either solve multi-digit calculation problems with full access to the blocks during practice or solve written algorithms only. In Experiment 3, students are provided unlimited access to materials while others only use the materials for teacher-sanctioned activities. In Experiment 4, children either receive the blocks only, written algorithms only, one before the other, or simultaneously. In Experiment 5 students either receive one model, multiple models in random order, or multiple models that decrease in concreteness. Finally, in Experiment 6, effects of different types of concrete models on students—place value blocks, Montessori beads, and interlocking blocks—are examined.

Key Measures: The same outcome measure will be used across all six experiments and conditions and will consist of randomly generated multi-digit calculation problems, as well as place value items drawn from standardized tests, textbook tests, and previous research. Researchers will also collect and code videotapes of student behaviors observed during each experiment.

Data Analytic Strategy: Standard analysis of variance techniques will be used to compare student outcomes across experimental conditions.



Mix, K. S. (2010). Spatial tools for mathematical thought. In K. S. Mix, L.B. Smith & M. Gasser (Eds.) The Spatial Foundations of Language and Cognition, New York: Oxford University Press.

Byrge, L. Smith, L.B., & Mix, K.S. (2014). Beginnings of place value: How preschoolers write three-digit numbers. Child Development, 85 (2), 437-443. doi:10.1111/cdev.12162.

Mix, K. S., Prather, R. W., Smith, L. B., & Stockton, J. D. (2014). Young children’s interpretations of multi-digit number names: From emerging competence to mastery. Child Development, 85v (3), 1306-1319. doi: 10.1111/cdev.12197

Mix, K. S., Smith, L. B., Stockton, J. D., Cheng, Y.L., & Barterian, J. A. (2016). Grounding the symbols for place value: Evidence from training and long-term exposure to base-10 models. Journal of Cognition and Development, 18(1), 129-151. doi: 10.1080/15248372.2016.1180296

Yuan, L., Prather, R., Mix, K. S., & Smith, L.B. (2019) Preschoolers and multi-digit numbers: A path to mathematics through symbols themselves. Cognition, 189 (1), 89-104. doi:10.1016/j.cognition.2019.03.013

Yuan, L., Prather, R., Mix, K. S., & Smith, L. B. (2020). Number representations drive number-line estimations. Child Development, 91 (4), e952-e967.

Yuan, L., Byrge, L., Mix, K. S., & Smith, L. B. (under review). Learning before school: Individual differences in what preschoolers know about multi-digit numbers. Developmental Science.

Yuan, Mix, Prather & Smith (invited submission in prep)