Decimal multiplication and division in mathematics textbooks for prospective elementary teachers

Jia He

doi:10.29333/iejme/15898

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Jia He ¹ ^*

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¹ Department of Mathematics, Augusta University, Augusta, GA, USA^* Corresponding Author

Abstract

This study examined 11 U.S. textbooks written for prospective teachers to investigate how standard decimal multiplication and division algorithms are presented, especially the rationale of both algorithms. Analytical frameworks of various methods used in different textbooks were developed. The findings suggest that half of the textbooks do not fully present the standard algorithms. The results indicate variations in textbooks’ explanations for the procedures involved in the standard algorithms. Some procedures are explained more frequently than others. The reason why moving the decimal point in both dividend and divisor keeps the quotient unchanged is provided in most of the textbooks. Half of the textbooks explain why adding the decimal places in the multiplication algorithm. However, only two books explain the rationale for placing the decimal point in the quotient. The findings suggest that textbook writers pay more attention to explaining the rationales of the standard algorithms for decimal multiplication and division.

Keywords

textbook analysis
teacher education
decimal multiplication
decimal division

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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Article Type: Research Article

INT ELECT J MATH ED, Volume 20, Issue 2, May 2025, Article No: em0815

https://doi.org/10.29333/iejme/15898

Publication date: 01 Apr 2025

Online publication date: 29 Jan 2025

Article Views: 36

Article Downloads: 15

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References

Aricha-Metzer, I., & Zaslavsky, O. (2017). The nature of students’ productive and non-productive example-use for proving. Journal of Mathematical Behavior, 53, 304-322. https://doi.org/10.1016/j.jmathb.2017.09.002
Ashlock, R. B. (2010). Error patterns in computation: Using error patterns to help each student learn (10th ed.). Allyn & Bacon.
Association of Mathematics Teacher Educators. (2017). Standards for preparing teachers of mathematics. Information Age Publishing.
Berk, D., & Cai, J. (2019). Mathematics teacher beliefs. In M. A. Peters (Ed.), Encyclopedia of teacher education (pp. 1-6). Springer. https://doi.org/10.1007/978-981-13-1179-6_236-1
Conference Board of the Mathematical Sciences. (2012). The mathematical education of teachers II. American Mathematical Society and Mathematical Association of America.
Fischbein, E., Deri, M., Nello, M. S., & Marino, M. S. (1985). The role of implicit models in solving verbal problems in multiplication and division. Journal for Research in Mathematics Education, 16(1), 3-17. https://doi.org/10.5951/jresematheduc.16.1.0003
Graeber, A. O., & Tirosh, D. (1990). Insight fourth and fifth grades brings to multiplication and division with decimals. Educational Studies in Mathematics, 21(6), 565-588. https://doi.org/10.1007/BF00315945
Graeber, A. O., Tirosh, D., & Glover, R. (1989). Pre-service teachers misconceptions in solving verbal problems in multiplication and division. Journal for Research in Mathematics Education, 20(1), 95-102. https://doi.org/10.5951/jresematheduc.20.1.0095
Greenberg, J., & Walsh, K. (2008). No common denominator: The preparation of elementary teacher in mathematics by American education schools. National Council on Teacher Quality.
Hiebert, J., & Wearne, D. (1985). A model of students’ decimal computation procedures. Cognition and Instruction, 2(3-4), 175-205. https://doi.org/10.1080/07370008.1985.9648916
Hiebert, J., & Wearne, D. (1986). Procedures over concepts: The acquisition of decimal number knowledge. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 199-223). Lawrence Erlbaum Associates.
Jones, D., Hollas, V., & Klepsis, M. (2017). The presentation of technology for teaching and learning mathematics in textbooks: Content courses for elementary teachers. Contemporary Issues in Technology & Teacher Education, 17(1), 53-79.
Joung, E., Lin, C. Y., & Kim, Y. (2021). Pre-service teachers’ understanding of decimals using standard algorithm and alternative strategies. International Journal for Mathematics Teaching and Learning, 22(1), 66-85. https://doi.org/10.4256/ijmtl.v22i1.329
Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding it up: Helping children learn mathematics. The National Academies Press. https://doi.org/10.17226/9822
Knuth. E. J. (2002). Secondary school mathematics teachers’ conceptions of proof. Journal for Research in Mathematics Education, 33(5), 379-405. https://doi.org/10.2307/4149959
Lai, M. Y., & Murray, S. (2014). What do error patterns tell us about Hong Kong Chinese and Australian students’ understanding of decimal numbers? International Journal for Mathematics Teaching & Learning.
Laursen, S. L., Hassi, M., & Hough, S. (2016). Implementation and outcomes of inquiry-based learning in mathematics content courses for pre-service teachers. International Journal of Mathematical Education in Science and Technology, 47(2), 256-275. https://doi.org/10.1080/0020739X.2015.1068390
Li, S. (2006). Practice makes perfect: A key belief in China. In F. K. S. Leung, K. Graf, & F. J. Lopez-Real (Eds.), Mathematics education in different cultural traditions: A comparative study of East Asia and the West (pp.129-138). Springer. https://doi.org/10.1007/0-387-29723-5_8
Liu, R. D., Ding, Y., Zong, M., & Zhang, D. (2014). Concept development of decimals in Chinese elementary students: A conceptual change approach. School Science and Mathematics, 114(7), 326-338. https://doi.org/10.1111/ssm.12085
Lortie-Forgues, H., Tian, J., & Siegler, R. S. (2015). Why is learning fraction and decimal arithmetic so difficult? Developmental Review, 38, 201-221. https://doi.org/10.1016/j.dr.2015.07.008
Maher, E. M., & He, J. (2023). Supporting spatial reasoning: Identifying aspects of length, area, and volume in textbook definitions. International Journal on Studies in Education, 5(1), 64-87. https://doi.org/10.46328/ijonse.100
Martin, W. G., & Harel, G. (1989). Proof frames of pre-service elementary teachers. Journal for Research in Mathematics Education, 20(1), 41-51. https://doi.org/10.5951/jresematheduc.20.1.0041
Matthews, M., Rech, J., & Grandgenett, N. (2010). The impact of content courses on pre-service elementary teachers’ mathematical content knowledge. Teacher Education Faculty Publications, 22.
McCrory, R. (2006). Mathematicians and mathematics textbooks for prospective elementary teachers. Notices of the AMS, 53(1), 20- 29.
McCrory, R., & Stylianides, A. J. (2014). Reasoning-and-proving in mathematics textbooks for prospective elementary teachers. International Journal of Educational Research, 64, 119-131. https://doi.org/10.1016/j.ijer.2013.09.003
McCrory, R., Francis, A., & Young, S. (2008). Resource use by instructors of mathematics classes for future elementary teachers: Results of a survey [Paper presentation]. The meeting of the International Committee on Mathematics Instruction.
Muir, T., & Livy, S. (2012). What do they know? A comparison of pre-service teachers’ and in-service teachers’ decimal mathematical content knowledge. International Journal for Mathematics Teaching and Learning.
National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common core state standards for mathematics. http://www.corestandards.org/wp-content/uploads/Math_Standards1.pdf
NRC. (2002). Curriculum as a channel of influence: What shapes what is taught to whom? In I. R. Weiss, M. S. Knapp, K. S. Hollweg, & G. Burrill (Eds.), Investigating the influence of standards: A framework for research in mathematics, science, and technology education (pp. 39-47). National Academy Press.
OECD. (2014). PISA 2012 results: What students know and can do–Student performance in mathematics, reading and science. OEDC Publishing. https://doi.org/10.1787/9789264201118-en
Okazaki, M., & Koyama, M. (2005). Characteristics of 5^th graders’ logical development through learning division with decimals. Educational Studies in Mathematics, 60(2), 217-251. https://doi.org/10.1007/s10649-005-8123-0
Philipp, R. A. (2007). Mathematics teachers’ beliefs and affect. In F. Lester (Ed.), Secondhand book of research on mathematics teaching and learning (pp. 257-315). National Council of Teachers of Mathematics.
Remillard, J. T. (2005). Examining key concepts in research on teachers’ use of mathematics curricula. Review of Educational Research, 75(2), 211-246. https://doi.org/10.3102/00346543075002211
Rittle-Johnson, B., Siegler, R. S., & Alibali, M. W. (2001). Developing conceptual understanding and procedural skill in mathematics: An iterative process. Journal of Educational Psychology, 93(2), 346-362. https://doi.org/10.1037//0022-0663.93.2.346
Siegler, R. S., & Lortie-Forgues, H. (2017). Hard lessons: Why rational number arithmetic is so difficult for so many people. Current Directions in Psychological Science, 26(4), 346-351. https://doi.org/10.1177/0963721417700129
Skemp, R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77, 20-26.
Star, J. R. (2005). Reconceptualizing procedural knowledge. Journal for Research in Mathematics Education, 36(5), 404-411. https://doi.org/10.2307/30034943
Steffe, L. (1990). On the knowledge of mathematics teachers. In R. B. Davis, C. A. Maher, & N. Noddings (Eds.), Constructivist views on teaching and learning mathematics (pp. 167-186). National Council of Teachers of Mathematics. https://doi.org/10.2307/749919
Steffe, L. P. (1994). Children’s multiplying schemes. In G. Harel, & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 3-39). State University of New York Press.
Stylianides, G. J., Stylianides, A. J., & Philippou, G. N. (2007). Preservice teachers’ knowledge of proof by mathematical induction. Journal of Mathematics Teacher Education, 10, 145-166. https://doi.org/10.1007/s10857-007-9034-z
Tian, J., Braithwaite, D. W., & Siegler, R. S. (2021). Distributions of textbook problems predict student learning: Data from decimal arithmetic. Journal of Educational Psychology, 113(3), 516-529. https://doi.org/10.1037/edu0000618
Tirosh, D. (1990). Improving prospective early childhood teachers’ content knowledge and attitudes toward mathematics. In L. P. Steffe, & T. Wood (Eds.), Transforming children’s mathematics education: International perspectives. Lawrence Erlbaum Associates.
Tirosh, D., Fischbein, E., Graeber, A. O., & Wilson, J. W. (1998). Prospective elementary teachers’ conceptions of rational numbers. http://jwilson.coe.uga.edu/Texts.Folder/Tirosh/Pros.El.Tchrs.html
Wearne, D., & Hiebert, J. (1988). Conceptual knowledge and procedural knowledge in mathematics: An analysis of the relationship between students' understanding of mathematics and their ability to use mathematical procedures. Educational Studies in Mathematics, 19(4), 361-381.

How to cite this article

APA

He, J. (2025). Decimal multiplication and division in mathematics textbooks for prospective elementary teachers. International Electronic Journal of Mathematics Education, 20(2), em0815. https://doi.org/10.29333/iejme/15898

Vancouver

He J. Decimal multiplication and division in mathematics textbooks for prospective elementary teachers. INT ELECT J MATH ED. 2025;20(2):em0815. https://doi.org/10.29333/iejme/15898

AMA

He J. Decimal multiplication and division in mathematics textbooks for prospective elementary teachers. INT ELECT J MATH ED. 2025;20(2), em0815. https://doi.org/10.29333/iejme/15898

Chicago

He, Jia. "Decimal multiplication and division in mathematics textbooks for prospective elementary teachers". International Electronic Journal of Mathematics Education 2025 20 no. 2 (2025): em0815. https://doi.org/10.29333/iejme/15898

Harvard

He, J. (2025). Decimal multiplication and division in mathematics textbooks for prospective elementary teachers. International Electronic Journal of Mathematics Education, 20(2), em0815. https://doi.org/10.29333/iejme/15898

MLA

He, Jia "Decimal multiplication and division in mathematics textbooks for prospective elementary teachers". International Electronic Journal of Mathematics Education, vol. 20, no. 2, 2025, em0815. https://doi.org/10.29333/iejme/15898