Prospective Middle School Mathematics Teachers’ Solution Strategies Regarding Multiplication with Decimals

Author(s):

Semanur Kandil(presenting / submitting)Seçil Yemen-Karpuzcu Mine Işıksal-Bostan

Conference:

ECER 2017

Network:

Emerging Researchers' Group (for presentation at Emerging Researchers' Conference)

Format:

Paper

Session Information

ERG SES D 04, Mathematics and Education

Paper Session

Time:

2017-08-21

13:30-15:00

Room:

W2.05

Chair:

Meinert Arnd Meyer

Contribution

Decimals are alternative representations of rational numbers where each rational number can be written as a decimal. The importance of decimals in the mathematics curriculum is emphasized while using calculators and understanding metric measures (Thipkong & Davis, 1991). Besides, decimals are widely used in daily life and are critical for students’ pre-algebraic readiness for learning further mathematic skills (NCTM, 2000). In Turkish national middle school mathematics curriculum (MoNE, 2013), decimals have an important place through different grade levels. While students are expected to learn to order decimals, place decimals on number line and perform operations (addition and subtraction) with decimals in the 5^th grade, they are expected to decompose decimals and perform operations (multiplication and division) in 6^th grade.

While curricula put significant emphasis on decimals, research studies showed that students have difficulties with decimals based on their overgeneralization of whole number properties (Brown, 1981; Roche & Clarke, 2004; Wearne & Kouba, 2000). Misconceptions and difficulties with decimals are not only observed in students, but also in samples of prospective teachers (Widjaja, Stacey, Steinle, 2008). The research studies present that prospective teachers have difficulties with decimal numeration (Putt, 1995; Stacey, Helme, Steinle, Baturo, Irwin, & Bana, 2001; Thipkong & Davis, 1991), in identifying decimals between two decimals (Widjaja, Stacey & Steinle, 2008), with operations in decimals word problems (Graeber & Tirosh, 1988). However, there were limited number of studies that focus on prospective teachers’ solution strategies while working with decimal. Investigation of solution strategies would give valuable information on prospective teachers’ understanding on decimals and highlight the points that they have difficulty with.

Considering this importance, we focused on the subject matter knowledge (SMK) of prospective middle school mathematics teachers in the domain of multiplication with decimals. SMK is the knowledge that involves categories related with content knowledge that is vital in teachers’ mathematical knowledge for teaching (Hill, Ball & Schilling, 2008; Shulman, 1986). The category that is specific to teaching task is named as specialized content knowledge (SCK) and SCK includes “providing explanation for mathematical explanations for common rues and explanations” (Hill et al., 2008, p. 378). That is, teachers’ SCK shows their profound and connected understanding of school mathematics. We considered the mathematical knowledge for teaching framework (Hill et al., 2008) while performing the items in a written test of decimals and analyzing the outcome. Specifically, it was decided to investigate prospective teachers’ strategies of placing the decimal point and characteristics of their strategies about the placing the decimal point. In this regard, the following research questions were determined:

What are the prospective middle school mathematics teachers’ strategies of placing the decimal point in the multiplication of two decimals?

What are the characteristics of prospective middle school mathematics teachers’ strategies of placing the decimal point in the multiplication of two decimals?

Method

This study is a part of a larger study that investigated prospective middle school mathematics teachers’ mathematical knowledge for teaching multiplication of decimals. In this paper, we presented prospective teachers’ SMK for multiplication of decimals. Our case was middle school mathematics teacher candidates who completed general mathematics courses and did not have teaching experience. The participants were 49 third grade students in a middle grades mathematics education majors in a public university in Ankara, Turkey. Participants were in the course of methods of teaching mathematics I in 2015-2016 fall semester. The textbook of the course was Van De Walle, Karp, & Bay-Williams’s (2013) book ‘Elementary and middle school mathematics: Teaching developmentally’. At the time of the data collection (fifth semester), they did not have any class on decimals and operations with decimals in the course. Besides, they had classes on problem structures of multiplication and division in in natural numbers and objectives on number learning area in middle school mathematics curriculum in Turkey. Until taking the course, they had taken content courses such as mathematics, physics, statistics and general education courses. In this study, the reported data came from the items in a written test of multiplication of two decimals. The items involved three computation questions that asked the result of the multiplication of two decimals using the information of the product of two numbers with same digits and without decimal point. In addition, in the items, the rationale for placing the decimal point were required for each result of the multiplication. For example, one of the items asked “It is known that 1268x155=196540. Using the result of this computation and not multiplying the decimals, where should the decimal point go for the result of the operation 12.68x1.55=? Explain your solution way and reasoning.” In the analysis, we used coding and clustering as basic qualitative analysis methods (Miles & Huberman, 1994). First, we coded the participants’ written explanations of placing the decimal point in multiplication through the big ideas in decimals topic. Then we utilized the concepts related with decimals and multiple representations mentioned in the literature to characterize the codes.

Expected Outcomes

The emerging outcomes of this study showed that prospective middle school mathematics teachers’ strategies were clustered in four categories. 6 of the responses (12.2 %) were based on estimation, 26 of the responses (53.1 %) were based on knowledge of fraction, 9 of the responses (18.4 %) were based on knowledge of division and 14 of the responses (28.6 %) included shortcut. These categories represented the strategies of understanding of multiplication with decimals and some teachers’ had explanations involving two of the categories. It was seen that characteristics of the estimation strategy involves four dimensions that were named as ‘multiplying by less than 1’, ‘front-end’, ‘rounding’ and ‘front-end and checking’. For example, in the first item, two teachers used both rounding and front-end methods. That is, they covered up 12.68 as 12 and 1.55 as 1 in front-end method and they round 1.55 to 2 in rounding method. Then, they considered the product value was between 12 and 24. As another example, in the second item, four teachers used multiplying by less than 1 conception that requires an understanding of relation between product and multiplier being less than 1. They knew that multiplying by 0.155 makes the number smaller. Therefore, teachers used both their knowledge of computational estimation and multiplication of whole numbers while placing the decimal point. In addition, in the categories of fraction, division and shortcut, four characteristics emerged that were named as fractional part, moving the place, as verbalization, symbolic representation, and decimal point. These outcomes showed that teachers used both their knowledge of fraction and place value relating with decimals.

References

Brown, M. (1981). Place value and decimals. In K. M. Hart (Ed.), Children’s understanding of mathematics: 11–16 (pp. 48-65). London: Alden Press. Graeber, A., & Tirosh, D. (1988). Multiplication and division involving decimals: Preservice teachers' performance and beliefs. Journal of Mathematical Behavior, 7, 263-280. Hill, H. C, Ball, D. L., & Schilling, S. G. (2008). Unpacking pedagogical content knowledge. Journal for Research in Mathematics Education, 39, 372-400. Miles, MB. & Huberman, AM. (1994). Qualitative Data Analysis (2nd edition). Thousand Oaks, CA: Sage Publications. Milli Eitim Bakanlığı (MoNE) (2013). Ortaokul matematik dersi (5, 6, 7 ve 8. sınıflar) öğretim programı, Ankara. National Council of Teachers of Mathematics (2000). Principles and Standards for School Mathematics. Reston, VA: NCTM. Putt, I. J. (1995). Preservice teacher ordering of decimal numbers: When more is smaller and less is larger! Focus on Learning Problems in Mathematics, 17(3), 1-15. Roche, A., & Clarke, D., (2004). When does successful comparison of decimals reflect conceptual understanding? In I. Putt, R. Faragher, & M. McLean (Eds), Mathematics education for the third millennium: Towards 2010 (Proceedings of the 27th Annual Conference of the Mathematics Education Research Group of Australasia, pp. 486–493). Townsville: MERGA. Shulman, L.S. (1986). Paradigms and research programs for the study of teaching. In M.C. Wittrock (Ed.), Handbook of research on teaching (3rd ed.). New York: Macmillan. Stacey, K., Helme, S., Steinle, V., Baturo, A., Irwin, K., & Bana, J. (2001). Preservice teachers’ knowledge of difficulties in decimal numeration. Journal of Mathematics Teacher Education, 4(3), 205-225. Thipkong, S., & Davis, E. J. (1991). Preservice elementary teachers' misconceptions in interpreting and applying decimals. School Science and Mathematics, 91(3), 93-99. Van de Walle, J. A., Karp, K. S., & Williams, J. M. B. (2013). Elementary and middle school mathematics. Teaching developmentally (8th ed.). Boston, MA: Pearson Education, Inc. Wearne, D., & Kouba, V. L., (2000). Rational numbers. In E. A. Silver & P. A. Kenny (Eds), Results from the seventh mathematics assessment of the national assessment of educational progress (pp. 163–191). Reston VA: National Council of Teachers of Mathematics. Widjaja, W., Stacey, K., & Steinle, V. (2008). Misconceptions about density of decimals: Insights from Indonesian pre-‐service teachers. Journal of Science and Mathematics Education in Southeast Asia, 31(2), 117–131.

Author Information

Semanur Kandil (presenting / submitting)

Middle East Technical University

Elementary Education

Ankara

Seçil Yemen-Karpuzcu

Middle East Technical University, Turkey

Mine Işıksal-Bostan