Meta-Analysis of Mathematics Teaching with Manipulatives

Introduction

  • Topic: Efficacy of Teaching Mathematics with Concrete Manipulatives
  • Authors: Kira J. Carbonneau, Scott C. Marley, James P. Selig, University of New Mexico.
  • Purpose: Examine empirical evidence regarding the use of manipulatives in mathematics instruction.

Literature Review

Foundational Concepts

  • Manipulatives are physical objects used to teach mathematical concepts.
  • Common instructional techniques utilizing manipulatives range from elementary (e.g., play money) to high school (e.g., algebra tiles).
  • NCTM (2000) recommends access to manipulatives for mathematical understanding.

Current State of Mathematics Achievement in the U.S.

  • National Assessment of Educational Progress (2011) shows 60% of fourth-grade and 57% of eighth-grade students in the U.S. failed to meet proficiency standards in mathematics.
    • Only 10% of fourth graders and 6% of eighth graders met international standards for advanced proficiency.
  • Initiative: President Obama launched "Educate to Innovate" to enhance student achievement in science, technology, engineering, and math (STEM).

Meta-Analysis Overview

Research Goals

  1. Determine the average effect of using concrete manipulatives in mathematics instruction.
  2. Examine the relationship between manipulatives and different student learning outcomes.
  3. Investigate instructional and methodological characteristics that may moderate this relationship.

Methodology

  • Study Selection: Systematic literature search identified 55 studies comparing instruction with manipulatives to instruction using only abstract math symbols, involving a total of 7,237 students.
  • Effect Size Measurement: Statistically significant results indicated a small to moderate effect size (Cohen’s d) favoring manipulatives.
    • Retention: k = 53, N = 7,140.
    • Problem Solving: k = 9, N = 477.
    • Transfer: k = 13, N = 3,453.
    • Justification: k = 2, N = 109.
  • Effect Sizes: Moderate to large effects on retention; small effects on problem solving, transfer, and justification.

Analysis of Moderator Variables

Instructional Characteristics

  1. Abstract Reasoning: Young children benefit from manipulatives as they support cognitive development and the emergence of abstract reasoning (Bruner, 1964; Piaget, 1962).
  2. Real-World Knowledge: Manipulatives can help link abstract concepts with real-world experiences (Brown, McNeil, & Glenberg, 2009).
  3. Learner-Driven Exploration: Opportunities for students to discover concepts through manipulatives may enhance learning outcomes, although unstructured approaches might not always be more effective than guided instruction (Mayer, 2004).
  4. Instructional Guidance: High instructional guidance generally leads to better performance outcomes, whereas too much guidance may restrict learner interpretation.

Methodological Characteristics

  • Differences in research designs affect the credibility of findings: peer-reviewed studies and those using within-subjects designs tend to report larger effect sizes.
  • Statistical independence in analyses is crucial; violations may inflate effect sizes.

Study Coding and Characteristics

Inclusion Criteria for Studies

  1. Studies must have compared manipulatives with abstract symbols.
  2. Studies must involve direct instruction and adequate reporting of effect sizes.
  3. Types of manipulative definitions were operationalized (excluding tools like calculators).

Summary of Studies

  • Tables provide comprehensive summaries of included studies, highlighting characteristics such as sample size, duration, design, and means for effect sizes.

Effect Sizes and Findings

Aggregated Results

  • Mean effect size across studies was 0.37 (p < .001).
  • Variability in effect sizes primarily influenced by instructional guidance, topic of mathematics (fractions had the highest effect size at 0.69), age of learners, and instructional time.

Disaggregated Outcomes

  • Retention: Mean effect size = 0.59.
  • Problem Solving: Mean effect size = 0.46.
  • Transfer: Mean effect size = 0.13.
  • Justification: Mean effect size = 0.38.

Detailed Findings by Learning Outcomes

  • Retention: Larger effect sizes for students receiving high instructional guidance and using non-perceptually rich manipulatives.
  • Problem Solving: High guidance yields larger effects, particularly for fractions.
  • Transfer: Low guidance showed higher effects; perceptually rich materials produced robust effects contradicting previous findings.

Publication Bias Check

  • Rosenthal's fail-safe N analysis suggests approximately 9,501 studies would be required to nullify the overall significance of manipulatives in improving learning outcomes.

Discussion and Conclusion

  • Findings show a small to moderate effect of manipulatives compared to purely abstract instruction. However, various contexts and instructional characteristics critically shape these outcomes.
  • Manipulatives should be integrated thoughtfully within mathematics instruction to enhance their efficacy.
  • Further empirical investigations are recommended to clarify the complexities surrounding the use of manipulatives.

References

  • Detailed list of references used in the investigation, categorized by works cited throughout the analysis.