Using the Concrete–Representational–Abstract Sequence to Teach Conceptual Understanding of Basic Multiplication and Division

Abstract

To meet increasingly complex mathematics standards in late elementary school, students must conceptually understand and be fluent in the operations of multiplication and division. This includes understanding the operations’ inverse relation. The purpose of the study was to investigate the effects of alternating concrete–representational–abstract (CRA) multiplication and division instruction on students’ mastery of unknown facts and on their conceptual understanding. Fourth through sixth-grade students with learning disabilities who had failed to master all multiplication facts participated in the study. The researchers used a mixed method design, measuring accuracy and fluency of facts with a multiple probe across students design and qualitative methods to capture changes in students’ explanations of their computation. The researchers demonstrated a functional relation between CRA instruction and accuracy and fluency in multiplication and division. Qualitative results indicated differences in students’ understanding of the operations. Implications of the results will be discussed further.

Keywords

mathematics learning strategies concrete–representational–abstract CRA

Get full access to this article

View all access options for this article.

References

Ainsworth

(1999). The functions of multiple representations. Computers & Education, 33, 131–152.

Bicknell

Young-Loveridge

Simpson

(2017). Using quotitive division problems to promote place-value understanding. Australian Primary Mathematics Classroom, 22(2), 28–32.

Braun

Clarke

(2006). Using thematic analysis in psychology. Qualitative Research in Psychology, 3, 77–101. doi:10.1191/1478088706qp063oa

Bruner

J. S.

(1966). Toward a theory of instruction. Cambridge, MA: Harvard University Press.

Clark

Kamii

(1996). Identification of multiplicative thinking in children in grades 1–5. Journal for Research in Mathematics Education, 27, 41–51.

Common Core State Standards Initiative. (2010). Common core state standards for mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers.

Ding

Carlson

M. A.

(2013). Elementary teachers’ learning to construct high-quality mathematics lesson plans: A use of the IES recommendations. The Elementary School Journal, 113, 359–385.

Dubé

A. K.

Robinson

K. M.

(2018). Children’s understanding of multiplication and division: Insights from a pooled analysis of seven studies conducted across seven years. British Journal of Developmental Psychology, 36, 206–219. doi:10.1111/bjdp.12217

Fosnot

Dolk

(2001). Young mathematicians at work: Constructing multiplication and division. Portsmouth, NH: Heinemann.

10.

Greene

Shorter

(2017). Conceptual understanding weighting system: A targeted assessment tool. Teaching Mathematics and Its Applications, 36(1), 1–16.

11.

Harel

Sowder

(2005). Advanced mathematical thinking at any age: Its nature and its development. Mathematical Thinking and Learning: An International Journal, 7, 27–50.

12.

Harris

C. A.

Miller

S. P.

Mercer

C. D.

(1995). Teaching initial multiplication skills to students with disabilities in general education classrooms. Learning Disabilities Research & Practice, 10, 180–195.

13.

Harris

Barmby

(2008). Representing multiplication. Mathematics Teaching Incorporating Micromath, 206, 37–41.

14.

Heinze

Star

J. R.

Verschaffel

(2009). Flexible and adaptive use of strategies and representations in mathematics education. ZDM Mathematics Education, 41, 535–540. doi: 10.1007/s11858-009-0214-4

15.

Matney

G. T.

Daugherty

B. N.

(2013). Seeing spots and developing multiplicative sense making. Mathematics Teaching in the Middle School, 19, 148–155. Retrieved from http://www.nctm.org/publications/article.aspx?id=39544

16.

Mauro

D. G.

LeFevre

J. A.

Morris

(2003). Effects of problem format on division and multiplication performance: Division facts are mediated via multiplication-based representations. Journal of Experimental Psychology: Learning, Memory, and Cognition, 29, 163–170. doi:10.1037/0278-7393.29.2.163

17.

Mercer

C. D.

Miller

S. P.

(1992). Teaching students with learning problems in math to acquire, understand, and apply basic math facts. Remedial and Special Education, 13(3), 19–35.

18.

Mercer

C. D.

Miller

S. P.

(2010a). Strategic math series: Division facts 0-81. Lawrence, KS: Edge.

19.

Mercer

C. D.

Miller

S. P.

(2010b). Strategic math series: Multiplication facts 0-81. Lawrence, KS: Edge.

20.

Miller

S. P.

Harris

C. A.

Strawser

Jones

W. P.

Mercer

C. D.

(1998). Teaching multiplication to second graders in inclusive settings. Focus on Learning Problems in Mathematics, 20(4), 50–70.

21.

Miller

S. P.

Hudson

P. J.

(2006). Using evidence-based practices to build mathematics competence related to conceptual, procedural, and declarative knowledge. Learning Disabilities Research & Practice, 22, 47–57. doi:10.1111/j.1540-5826.2007.00230.x

22.

Miller

S. P.

Mercer

C. D.

(1993). Using a graduated word problem sequence to promote problem solving skills. Learning Disabilities Research & Practice, 8, 169–174.

23.

Miller

S. P.

Stringfellow

J. L.

Kaffar

B. J.

(2011). Developing computation competence among students who struggle with mathematics. Teaching Exceptional Children, 44(2), 38–46. doi:10.1177/004005991104400204

24.

The National Council of Teachers of Mathematics. (2014). Principles to action: Ensuring mathematical success for all. Reston, VA: Author.

25.

Nowell

L. S.

Norris

J. M.

White

D. E.

Moules

N. J.

(2017). Thematic analysis: Striving to meet the trustworthiness criteria. International Journal of Qualitative Analysis, 16(1), 1–13. doi:10.1177/1609406917733847

26.

Pape

S. J.

Tchoshanov

M. A.

(2001). The role of representation(s) in developing mathematical understanding. Theory Into Practice, 40, 118–127. doi:10.1207/s15430421tip4002_6

27.

Park

J.-H.

Nunes

(2001). The development of the concept of multiplication. Cognitive Development, 16, 763–773. doi:10.1016/S0885-2014(01)00058-2

28.

Parker

R. I.

Vannest

K. J.

Davis

J. L.

Sauber

S. B.

(2011). Combining nonoverlap and trend for single-case research: Tau-U. Behavior Therapy, 42, 284–299. doi:10.1016/j.beth.2010.08.006

29.

Ploger

Hecht

(2009). Enhancing children’s conceptual understanding of mathematics through Chartworld software. Journal of Research in Childhood Education, 23, 267–277. doi:10.1080/02568540909594660

30.

Ploger

Hecht

(2012). Meaning, memory, and multiplication: Integrating patterns and properties with basic facts. Childhood Education, 88, 169–176. doi:10.1080/00094056.2012.682550

31.

Pope

Ziebland

Mays

(2000). Qualitative research in health care: Analysing qualitative data. British Medical Journal, 320, 114–116.

32.

Richards

(2014). Working toward a “child friendly” writing progression for division. Mathematics Teaching, 1(240), 15–18.

33.

Scruggs

T. E.

Mastropieri

M. A.

(2013). PND at 25: Past, present, and future trends in summarizing single-subject research. Remedial and Special Education, 34, 9–19. doi:10.1177/0741932512440730

34.

Smith

S. Z.

Smith

M. E.

(2006). Assessing elementary understanding of multiplication concepts. School Science and Mathematics, 106, 140–148. doi:10.1111/j.1949-8594.2006.tb18171.x

35.

Sullivan

Lovin

(2009). Connecting multiplication contexts and language. Teaching Children Mathematics, 15, 502–518.

36.

West

(2014). Divide and conquer: A hands-on exploration of divisibility. Australian Primary Mathematics Classroom, 19(4), 15–19.