# Concrete–Representational–Abstract (CRA) Instructional Approach in an Algebra I Inclusion Class: Knowledge Retention Versus Students’ Perception

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Perspectives on Manipulatives Use and Need for the Study

- To what extent will students in a high school Algebra I inclusion class retain the knowledge of simplifying square roots after being instructed using the CRA instructional approach?
- How will students in a high school Algebra I inclusion class describe the effectiveness of using a mathematical manipulative to learn about a specific mathematical procedure?

## 3. Materials and Methods

#### 3.1. Context and Participants

#### 3.2. Lesson

#### The Manipulative

^{2}through 10 cm

^{2}.

#### 3.3. Instrumentation

#### 3.4. Procedure

#### 3.4.1. Data Collection

#### 3.4.2. Quantitative Data Analysis

#### 3.4.3. Qualitative Data Analysis

## 4. Results

#### 4.1. Knowledge Acquisition and Retention

#### 4.2. Perception of Manipulative Effectiveness

## 5. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- United States National Commission on Excellence in Education. A Nation at Risk: The Imperative for Educational Reform. 1983. Available online: http://edreform.com/wp-content/uploads/2013/02/A_Nation_At_Risk_1983.pdf (accessed on 8 May 2023).
- U.S. Department of Education. The Condition of Education: Students with Disabilities. 2020. Available online: https://nces.ed.gov/pubs2020/2020144.pdf (accessed on 8 May 2023).
- Association of Mathematics Teacher Educators. Position: Equity in Mathematics Teacher Preparation. 2015. Available online: https://amte.net/sites/default/files/amte_equityposistionstatement_sept2015.pdf (accessed on 6 May 2023).
- National Council of Teachers of Mathematics. Principles and Standards for School Mathematics; National Council of Teachers of Mathematics: Reston, VA, USA, 2000; pp. 1–402. [Google Scholar]
- National Council of Teachers of Mathematics. Principles to Actions: Ensuring Mathematics Success for All; National Council of Teachers of Mathematics: Reston, VA, USA, 2014; pp. 1–139. [Google Scholar]
- Bone, E.K.; Bouck, E.C. Evaluating calculators as accommodations for secondary students with disabilities. Learn. Disabil. Multidiscip. J.
**2018**, 23, 35–49. [Google Scholar] [CrossRef] - Miller, S.P.; Hudson, P.J. Using evidence-based practices to build mathematics competence related to conceptual, procedural, and declarative knowledge. Learn. Disabil. Res. Pract.
**2007**, 22, 47–57. [Google Scholar] [CrossRef] - Jitendra, A.K.; Lein, A.E.; Im, S.H.; Alghamdi, A.A.; Hefte, S.B.; Mouanoutoua, J. Mathematical interventions for secondary students with learning disabilities and mathematics difficulties: A meta-analysis. Except. Child.
**2017**, 64, 21–43. [Google Scholar] [CrossRef] - Marita, S.; Hord, C. Review of mathematics interventions for secondary students with learning disabilities. Learn. Disabil. Q.
**2017**, 40, 29–40. [Google Scholar] [CrossRef] - Moyer, P.S. Are we having fun yet? How teachers use manipulatives to teach mathematics. Educ. Stud. Math.
**2001**, 47, 175–197. [Google Scholar] [CrossRef] - Thompson, P.W. Notations, principles, and constraints: Contributions to the effective use of concrete manipulatives in elementary mathematics. J. Res. Math. Educ.
**1992**, 23, 123–147. [Google Scholar] [CrossRef] - Gagnon, J.C.; Maccini, P. Preparing students with disabilities for algebra. Teach. Except. Child.
**2001**, 34, 8–15. [Google Scholar] [CrossRef] - Strickland, T.K.; Maccini, P. The effects of the concrete-representational-abstract integration strategy on the ability of students with learning disabilities to multiply linear expressions within area problems. Remedial Spec. Educ.
**2012**, 34, 134–153. [Google Scholar] [CrossRef] - Clements, D. ‘Concrete’ manipulatives, concrete ideas. Contemp. Issues Early Child.
**1999**, 1, 45–60. [Google Scholar] [CrossRef] - Burns, B.A.; Hamm, E.M. A comparison of concrete and virtual manipulative use in third- and fourth-grade mathematics. Sch. Sci. Math.
**2011**, 111, 256–261. [Google Scholar] [CrossRef] - Moyer-Packenham, P.; Baker, J.M.; Westenskow, A.; Rodzon, K.; Anderson, K.; Shumway, J.; Jordan, K. A study comparing virtual manipulations with other instructional treatments in third- and fourth-grade classrooms. J. Educ.
**2013**, 193, 25–39. [Google Scholar] [CrossRef] - Westenskow, A.; Moyer-Packenham, P. Using an iceberg intervention model to understand equivalent fraction learning when students with mathematical learning difficulties use different manipulatives. Int. J. Technol. Math. Educ.
**2016**, 23, 45–62. [Google Scholar] [CrossRef] - Ball, D. Magical hopes: Manipulatives and the reform of math education. Am. Educ.
**1992**, 6, 14–47. [Google Scholar] - Corkin, D.S.; Coleman, S.L.; Ekmekci, A. Navigating the challenges of student-centered mathematics teaching in an urban context. Urban. Rev.
**2019**, 51, 370–403. [Google Scholar] [CrossRef] - Anstrom, T. Supporting Students in Mathematics through the Use of Manipulatives (Research Brief); Center for Implementing Technology in Education: Washington, DC, USA, 2006. [Google Scholar]
- Leong, Y.H.; Ho, W.K.; Cheng, L.P. Concrete-pictorial-abstract: Surveying its origins and charting its future. Math. Educ.
**2015**, 16, 1–18. [Google Scholar] - Mullis, I.V.S.; Martin, M.O.; Goh, S.; Cotter, K. (Eds.) TIMSS 2015 Encyclopedia: Education Policy and Curriculum in Mathematics and Science. 2016. Available online: http://timssandpirls.bc.edu/timss2015/encyclopedia/ (accessed on 6 May 2023).
- Witzel, B.S. Using CRA to teach algebra to students with math difficulties in inclusive settings. Learn. Disabil. Contemp. J.
**2005**, 3, 49–60. [Google Scholar] - Swan, P.; Marshall, L. Revisiting mathematics manipulative materials. Aust. Prim. Math. Classr.
**2010**, 15, 13–19. [Google Scholar] - Carbonneau, K.J.; Marley, S.C.; Selig, J.P. A meta-analysis of the efficacy of teaching mathematics with concrete manipulatives. J. Educ. Psychol.
**2013**, 105, 380–400. [Google Scholar] [CrossRef] - Sowell, E.J. Effects of manipulative materials in mathematics instruction. J. Res. Math. Educ.
**1989**, 20, 498–505. [Google Scholar] [CrossRef] - McNeil, N.; Jarvin, L. When theories don’t add up: Disentangling the manipulatives debate. Theory Pract.
**2007**, 46, 309–316. [Google Scholar] [CrossRef] - Bruner, J.S. On learning mathematics. Math. Teach.
**1960**, 53, 610–619. [Google Scholar] [CrossRef] - Bruner, J.S.; Goodnow, J.J.; Austin, G.A. A Study of Thinking, 2nd ed.; Transaction Publishers: New Brunswick, NJ, USA, 1986; pp. 1–350. [Google Scholar] [CrossRef]
- Piaget, J. The Origins of Intelligence in Children, 2nd ed.; International Universities Press: Madison, CT, USA, 1952; pp. 1–419. [Google Scholar] [CrossRef]
- Lawson, A.E.; Wollman, W.T. Encouraging the transition from concrete to formal cognitive functioning: An experiment. J. Res. Sci. Teach.
**2003**, 40, S33–S50. [Google Scholar] [CrossRef] - Ojose, B. Applying Piaget’s theory of cognitive development to mathematics instruction. Math. Educ.
**2008**, 18, 26–30. [Google Scholar] - McKinney, S.; Frazier, W. Embracing the principles and standards for school mathematics: An inquiry into the pedagogical and instructional practices of mathematics teachers in high-poverty middle schools. Clear. House A J. Educ. Strateg. Iss. Ideas
**2008**, 81, 201–210. [Google Scholar] [CrossRef] - Balter, L.; Tamis-LeMonda, C.S. (Eds.) Child Psychology: A Handbook of Contemporary Issues; Psychology Press: London, UK, 2016; pp. 1–522. [Google Scholar]
- Orlich, D.C. Education reform and limits to student achievement. Phi Delta Kappan
**2000**, 81, 468–472. [Google Scholar] - National Governors Association Center for Best Practices, Council of Chief State School Officers. Common Core State Standards for Mathematics. 2022. Available online: https://learning.ccsso.org/wp-content/uploads/2022/11/ADA-Compliant-Math-Standards.pdf (accessed on 15 January 2023).
- Bouck, E.C.; Park, J. A systematic review of the literature on mathematics manipulatives to support students with disabilities. Educ. Treat. Child.
**2018**, 41, 65–106. [Google Scholar] [CrossRef] - Cass, M.; Cates, D.; Smith, M.; Jackson, C. Effects of manipulative instruction on solving area and perimeter problems by students with learning disabilities. Learn. Disabil. Res. Pract.
**2003**, 18, 112–120. [Google Scholar] [CrossRef] - Maccini, P.; Hughes, C.A. Effects of a problem-solving strategy on the introductory algebra performance of secondary students with learning disabilities. Learn. Disabil. Res. Pract.
**2000**, 15, 10–21. [Google Scholar] [CrossRef] - Maccini, P.; Ruhl, K.L. Effects of a graduated sequence on the algebraic subtraction of integers by secondary students with learning disabilities. Educ. Treat. Child.
**2000**, 23, 465–489. [Google Scholar] - Scheuermann, A.M.; Deshler, D.D.; Schumaker, J.B. The effects of the explicit inquiry routine on the performance of students with learning disabilities on one-variable equations. Learn. Disabil. Q.
**2009**, 32, 103–120. [Google Scholar] [CrossRef] - Uttal, D.H.; O’Doherty, K.; Newland, R.; Hand, L.L.; DeLoache, J. Dual representation and the linking of concrete and symbolic representations. Child. Develop Perspect.
**2009**, 3, 156–159. [Google Scholar] [CrossRef] - Gersten, R.; Beckmann, S.; Clarke, B.; Foegen, A.; Marsh, L.; Star, J.R.; Witzel, B. Assisting Students Struggling with Mathematics: Response to Intervention (RTI) for Elementary and Middle Schools. NCEE 2009-4060; National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education: Washington, DC, USA, 2009.
- Kelly, C.A. Using manipulatives in mathematical problem solving: A performance-based analysis. MT Math. Enthus.
**2006**, 3, 184–193. [Google Scholar] [CrossRef] - Larkin, K. Mathematics education and manipulatives: Which, when, how? Aust. Prim. Math. Classr.
**2016**, 21, 12–17. [Google Scholar] - Maccini, P.; Mulcahy, C.A.; Wilson, M.G. A follow-up of mathematics intervention for secondary students with learning disabilities. Learn. Disabil. Res. Pract.
**2007**, 22, 58–74. [Google Scholar] [CrossRef] - Sacks, H. Notes on methodology. In Structures of Social Action: Studies in Conversation Analysis; Heritage, J., Atkinson, J.M., Eds.; Cambridge University Press: Cambridge, UK, 1984; pp. 2–27. [Google Scholar]
- O’Reilly, M.; Kiyimba, N.; Lester, J.N.; Muskett, T. Reflective interventionist conversation analysis. Discourse Comm.
**2020**, 14, 619–634. [Google Scholar] [CrossRef] - Weiner, J.M.; Burton, L.J. The double bind for women: Exploring the gendered nature of turnaround leadership in a principal preparation program. Harv. Educ. Rev.
**2016**, 86, 339–365. [Google Scholar] [CrossRef] - Greene, J.C.; Caracelli, V.J.; Graham, W.F. Toward a conceptual framework for mixed-method evaluation designs. Educ. Eval. Policy Anal.
**1989**, 11, 255–274. [Google Scholar] [CrossRef] - U.S. News and World Report. U.S. News & World Report Releases the 2016 Best High Schools Rankings. 19 August 2016. Available online: https://www.usnews.com/info/blogs/press-room/articles/2016-04-19/us-news-releases-the-2016-best-high-schools-rankings (accessed on 27 August 2022).
- Schultz, K.T.; Bismarck, S.F. Radical thoughts on simplifying square roots. Math. Teach. Middle Sch.
**2013**, 19, 222–228. [Google Scholar] [CrossRef] - Georgia Department of Education. What Do Mathematics Standards Look Like in the Classroom? 2020. Available online: https://www.georgiastandards.org/ (accessed on 27 August 2022).
- New York State Education Department. NYS Next Generation Mathematics Learning Standards Unpacking Documents. 2019. Available online: http://www.nysed.gov/ (accessed on 27 August 2022).
- Ohio Department of Education. Ohio’s Model Curriculum with Instructional Supports: Mathematics. 2018. Available online: http://education.ohio.gov/ (accessed on 27 August 2022).
- Bowling Green State University. NWO/COSMOS: Grant Projects: Middle Grades Resources. n.d. Available online: https://www.bgsu.edu/nwo/current-grant-projects/c2am2p-middle-grades/resources/ (accessed on 27 August 2022).
- Sloutsky, V.M.; Kaminski, J.A.; Heckler, A.F. The advantage of simple symbols for learning and transfer. Psychon. Bull. Rev.
**2005**, 12, 508–513. [Google Scholar] [CrossRef] - Skemp, R. Relational understanding and instrumental understanding. Math. Teach. Middle Sch.
**2006**, 12, 88–95. [Google Scholar] [CrossRef] - Brown, G.T.L.; Irving, E.; Keegan, P. An Introduction to Educational Assessment, Measurement, and Evaluation: Improving the Quality of Teacher-Based Assessment; Dunmore Publishing: Aukland, New Zealand, 2014; pp. 1–194. [Google Scholar]
- Hsieh, H.-F.; Shannon, S.E. Three approaches to qualitative content analysis. Qual. Health Res.
**2005**, 15, 1277–1288. [Google Scholar] [CrossRef] - Kurz, T.L. Insight into Perceptions of Mathematics Using Wordle. In Proceedings of the Society for Information Technology & Teacher Education International Conference, Las Vegas, NV, USA, 2–6 March 2015; Rutledge, D., Slykhuis, D., Eds.; Association for the Advancement of Computing in Education: Waynesville, NC, USA, 2015; pp. 2702–2704. [Google Scholar]
- McNaught, C.; Lam, P. Using Wordle as a supplementary research tool. Qual. Rep.
**2010**, 15, 630–643. [Google Scholar] [CrossRef] - Devlin, K. Finding Your Inner Mathematician. 29 September 2000. Available online: https://www.chronicle.com/article/finding-your-inner-mathematician/ (accessed on 15 January 2023).
- Maccini, P.; Gagnon, J.C. Best practices for teaching mathematics to secondary students with special needs. Focus Except. Child.
**2000**, 32, 1–22. [Google Scholar] [CrossRef] - Wentzel, K.R.; Brophy, J.E. Motivating Students to Learn, 4th ed.; Routledge: Oxfordshire, UK, 2013; pp. 1–312. [Google Scholar]
- D’Mello, S.; Lehman, B.; Pekrun, R.; Graesser, A. Confusion can be beneficial for learning. Learn. Instr.
**2014**, 29, 153–170. [Google Scholar] [CrossRef] - Tanner, K.D. Promoting student metacognition. CBE Life Sci. Educ.
**2012**, 11, 113–120. [Google Scholar] [CrossRef] [PubMed] - Forbringer, L.L.; Fuchs, W.W. RTI in Math: Evidence-Based Interventions for Struggling Students; Routledge: Oxfordshire, UK, 2014; pp. 1–304. [Google Scholar]
- Long, M.J.; Ben-Hur, M. Informing learning through the clinical interview. Arith. Teach.
**1991**, 38, 44–46. [Google Scholar] [CrossRef]

**Figure 1.**A multidigit multiplication problem, 12 × 13, is displayed using each stage of the CRA approach: (

**a**) Panel 1 shows base-ten blocks, which is the concrete stage; (

**b**) Panel 2 shows a diagram, which is the representational stage; and (

**c**) Panel 3 shows symbols that model the problem, which is the abstract stage.

**Figure 2.**These diagrams are representations that show the equivalence of (

**a**) $\sqrt{8}$ to (

**b**) $2\sqrt{2}$.

Class | n | M diff. | SD | t | p | Cohen’s d |
---|---|---|---|---|---|---|

Algebra A | 22 | −1.272 | 1.9623 | −3.04 | 0.0031 | 0.701 |

Algebra B | 22 | −0.9773 | 2.0206 | −2.27 | 0.017 | 0.526 |

Algebra C (Inclusion) | 25 | −1.88 | 1.8044 | −5.21 | <0.0001 | 1.378 |

Geometry A | 10 | −0.8 | 1.2517 | −2.02 | 0.037 | 0.842 |

Geometry B | 18 | −0.5278 | 1.48 | −1.51 | 0.0743 | 0.202 |

Geometry C | 15 | −0.8333 | 1.5079 | −2.14 | 0.025 | 0.744 |

Class | Positive | Neutral | Negative | |||
---|---|---|---|---|---|---|

n | % | n | % | n | % | |

Algebra A | 14 | 87.5 | 2 | 12.5 | 0 | 0 |

Algebra B | 20 | 87 | 2 | 8.7 | 1 | 4.3 |

Algebra C (Inclusion) | 9 | 36 | 8 | 32 | 8 | 32 |

Geometry A | 13 | 100 | 0 | 0 | 0 | 0 |

Geometry B | 17 | 94.4 | 1 | 5.6 | 0 | 0 |

Geometry C | 19 | 95 | 0 | 0 | 1 | 5 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Prosser, S.K.; Bismarck, S.F.
Concrete–Representational–Abstract (CRA) Instructional Approach in an Algebra I Inclusion Class: Knowledge Retention Versus Students’ Perception. *Educ. Sci.* **2023**, *13*, 1061.
https://doi.org/10.3390/educsci13101061

**AMA Style**

Prosser SK, Bismarck SF.
Concrete–Representational–Abstract (CRA) Instructional Approach in an Algebra I Inclusion Class: Knowledge Retention Versus Students’ Perception. *Education Sciences*. 2023; 13(10):1061.
https://doi.org/10.3390/educsci13101061

**Chicago/Turabian Style**

Prosser, Sherri K., and Stephen F. Bismarck.
2023. "Concrete–Representational–Abstract (CRA) Instructional Approach in an Algebra I Inclusion Class: Knowledge Retention Versus Students’ Perception" *Education Sciences* 13, no. 10: 1061.
https://doi.org/10.3390/educsci13101061