MENTAL CALCULATION STRATEGIES FOR ADDITION AND SUBTRACTION IN THE SET OF RATIONAL NUMBERS

Transcription

1 MENTAL CALCULATION STRATEGIES FOR ADDITION AND SUBTRACTION IN THE SET OF RATIONAL NUMBERS Sebastian Rezat Justus-Liebig-University, Giessen Studies on mental calculation strategies usually focus on elementary school students and accordingly on problems in the set of natural numbers. But, mental calculation is also an issue at secondary school in the context of other number sets. In the paper an exploratory study of eight eighth grade students mental calculation strategies for addition and subtraction problems in the set of rational numbers is presented. The study focuses on the analysis of the used strategies and on students adaptive expertise in the choice of the strategies. RATIONALE A considerable body of research investigated issues related to mental calculation of elementary school students in the set of natural numbers. This body of research mainly focuses on three aspects: Firstly, strategies students are using are identified (Reys, Reys, Nohda, & Emori, 1995; Selter, 2001). Secondly, the issue of adaptive expertise, i.e. flexible and adaptive strategy use, is addressed (Blöte, Klein, & Beishuizen, 2000; Threlfall, 2002; Torbeyns, De Smedt, Ghesquière, & Verschaffel, 2009; Verschaffel, Luwel, Torbeyns, & Van Dooren, 2009). Finally, factors affecting different aspects of mental calculation, e.g. performance or adaptive expertise are analysed (Heinze, Marschick, & Lipowsky, 2009; Heirdsfield & Cooper, 2004). The development of mental calculation strategies and their flexible and adaptive use by students in secondary school related to other number sets than the natural numbers is rarely investigated so far. This is surprising, because on the one hand students in secondary school have had the opportunity to gain more experience in the number system. In Germany the set of natural numbers is extended to the integers, the rationals, and the real numbers in secondary school. Getting familiar with new numbers and their computation might possibly have positive effects on the development of number sense. Besides metacognitive, and affective factors number sense is regarded as one of the main influential factors on mental calculation performance and adaptive expertise (Heirdsfield & Cooper, 2004). On the other hand, mental calculation is a subject that is addressed in secondary school standards and national curricula. The NCTM-Standards expect that In grades 6 8 all students should [...] select appropriate methods and tools for computing with fractions and decimals from among mental computation. The call for mental calculation is continued in the upper grades: In grades 9 12 all students should [ ] develop fluency in operations with real numbers, vectors, and matrices, using mental computation or paper-and-pencil calculations (National Council of Teachers of Mathematics, 2000). Similar expectations can be found in the English National

2 Curriculum (Qualifications and Curriculum Development Agency, 1999) and the German Bildungsstandards (Ständige Konferenz der Kultusminister der Länder in der Bundesrepublik Deutschland, 2005). In this paper an exploratory study of eight eighth grade students mental calculation strategies in the set of rational numbers is presented. THEORETICAL FRAMEWORK Mental calculation strategies Research on mental calculation for addition and subtraction problems in the set of natural numbers has identified numerous strategies that can be divided into different groups. Threlfall (2009) firstly distinguishes approach strategies and numbertransformation strategies. He defines an approach strategy in mental calculation as the general form of mathematical cognition used for the problem for example counting, or recall, or application of a learned method, or visualisation of a procedure, or exploiting known number relations (Threlfall, 2009, p. 541). A number-transformation strategy in mental calculation is the detailed way in which the numbers have been transformed to arrive at a solution (Threlfall, 2009, p. 542). In the literature varying conceptualizations of these strategies can be found (see e.g. Threlfall, 2002 for an overview). In this paper Threlfall s terminology is adopted. Mental calculation strategy is used as an overarching term whenever it is referred to both, an approach and a number-transformation strategy. In this section an a priori analysis of possible approach- and number-transformation strategies in the set of rational numbers is presented. From a mathematical point of view an analysis of mental calculation strategies for addition and subtraction in the set of rational numbers might focus on different aspects: 1. Addition and subtraction with integers 2. Addition and subtraction with fractions 3. Addition and subtraction with decimals Addition and subtraction with fractions (2) is carried out by treating the nominator and the denominator separately. Therefore it is likely that the number-transformation strategies students use for solving addition and subtractions problems with fractions are the same as with natural numbers and integers. Furthermore, in everyday life mental calculation with fractions is hardly needed in contrast to mental calculation with integers and decimals (Profke, 1991). Therefore, the study reported in this paper focuses on mental calculation strategies related to integers (1) and decimals (3). Besides the occurrence of negative numbers a major novelty in calculating with integers is a zero-transition. In order to avoid the zero-transition, addition and

3 subtraction of integers can be reduced to addition and subtraction of natural numbers either by definition or by proving the following rules: For any integers n, m (-n) + n = 0 (-n) + m = m n, if m > n (-n) + m = -(n m), if n > m (-n) + 0 = -n (-n) + (-m) = -(n + m) 0 n = -n (-n) 0 = -n 0 (-n) = n (-n) (-m) = (-n) + m (-n) m = -(n+m) m (-n) = n + m Therefore, mental addition and subtraction of integers can be approached by applying these definitions / rules combined with number-transformation strategies for natural numbers. Another way of approaching mental addition and subtraction problems with integers is by referring to the mental image of the number line. In Germany a common way of introducing negative numbers is the extension of the number line. This geometrical model is sometimes derived from temperature, altitudinal, or monetary (bankaccount-balance-model) contexts (Vollrath & Weigand, 2007). Table 1 summarizes the 5 different approach strategies that were identified in the a priori analysis: transformation number line bank-accountbalance model (b-a-b-m) refers to an approach strategy where the original problem is transformed into an equivalent problem in the set of natural numbers according to the above stated rules refers to an approach strategy where students solve the problem with reference to an internal image of the number line. refers to an approach strategy where students solve the problem with reference to monetary contexts of deposit, withdrawal and depts. temperature scale model refers to an approach strategy where students solve the problem with reference to a temperature context. altitude model refers to an approach strategy where students solve the problem with reference to a altitude context. Table 1: Idealized approach strategies for addition and subtraction problems in the set of rational numbers. Decimals are introduced in Germany according to two major approaches: They are either regarded as a special kind of common fractions, e.g. 23,45 = 2345/100, or they are introduced via an extension of the place-value-system (Padberg, 2009). In the former case, addition and subtraction of decimals is reduced to addition and

4 subtraction of fractions, which is as we have seen earlier just a special case of addition and subtraction with integers. In the latter case, the number-transformation rules for decimals are traced back to the number-transformation rules of natural numbers. In summary, all addition and subtraction problems with rational numbers can be approached by reducing them to equivalent problems with natural numbers. Therefore, it is likely that the actual transformation of numbers in order to arrive at a solution is carried out with number-transformation strategies for natural numbers. In this paper I will refer to the conceptualization of idealized number-transformation strategies for addition and subtraction in the set of natural numbers that is put forward by Heinze et al. (2009), because it is appropriate for the German situation in the way that it comprises the strategies that are well known in German arithmetic literature. An overview of the different strategies is given in table 2. Stepwise strategy = = = 57 Split strategy = = = = 57 Compensation strategy = = = 54 Simplifying strategy = = 70 Indirect addition (ind. add.) = = 53 Table 2: Idealized number-transformation strategies for addition and subtraction according to Heinze et al. (2009). Whereas the strategy, the strategy, the compensation strategy, and indirect addition can be applied to integers and to decimals without any modification a variation of the simplifying strategy is specific to the set of integers: Problems like can be solved by simplifying into Flexible and adaptive strategy use The use of the terms flexibility and adaptivity is not consistent in the literature (Selter, 2009). Based on a literature review Verschaffel et al. suggest that flexibility refers to switching (smoothly) between different strategies and adaptivity puts more emphasis on selecting the most appropriate strategy (Verschaffel, et al., 2009, p. 337). As a working definition they suggest that an adaptive choice of strategy is characterized by the conscious or unconscious selection and use of the most appropriate solution strategy on a given mathematical item or problem, for a given individual, in a given sociocultural context (Verschaffel, et al., 2009, p. 343). This definition implies that strategies can be adaptive related to different aspects: they can adapt to task characteristics as well as to individual or sociocultural conditions. Furthermore, adaptivity seems to imply flexibility. This is even more evident in Selter s slight variation of Verschaffel et al. s definition: Adaptivity is

5 the ability to creatively develop or to flexibly select and use an appropriate solution strategy in a (un)conscious way on a given mathematical item or problem, for a given individual, in a given sociocultural context (Selter, 2009, p. 624). Whereas flexibility might be operationalised by showing at least a certain number of different strategies, the question when a strategy is considered to be appropriate and which criteria are relevant to this are critical and challenging [ ] fundamental theoretical questions in this context (Heinze, Star, & Verschaffel, 2009, p. 536). Therefore, a normative perspective is usually taken in order to decide whether a strategy adapts to a given problem or not. The findings of studies on students mental number-transformation strategies for addition and subtraction problems in the set of natural numbers indicate that elementary school students hardly choose among different strategies with respect to task characteristics, but favor one or two strategies that they apply to every problem (Selter, 2001; Torbeyns, et al., 2009; Torbeyns, Verschaffel, & Ghesquière, 2006). Based on the discussion of mental calculation strategies and adaptive strategy use the following research questions are addressed in the study presented in this paper: 1. Which approach and number-transformation strategies do students use for mental calculation in the set of rational numbers? Do students use strategies that are specific to the rational numbers or do they refer to strategies that they are familiar with from the set of natural numbers? 2. Do students in secondary grade use mental calculation strategies adaptively according to tasks characteristics? STUDY DESIGN AND METHODOLOGY Data on students mental calculation strategies were collected in video recorded interviews. The tasks were read out loud to the students and the students were asked to solve the tasks mentally without using any notes. Afterwards the students were asked to explain the way they solved the task [1]. Eight eighth grade students from a German comprehensive school took part in the study. The school is located in a rural area in Germany. According to their grades in mathematics the students are considered to be medium-achieving students. The problems posed in this study can be grouped into three different categories: 1. Addition and subtraction problems with natural numbers (tab. 3: P1 P4) 2. Addition and subtraction problems with integers (tab. 3: P5 P8) 3. Addition and subtraction problems with positive and negative decimals (tab. 3: P9 P14) Whereas addition and subtraction problems with natural numbers (category 1) were included in order to get an idea which mental computation strategies students use in

6 the set of natural numbers, problems of categories 2 and 3 relate directly to the main aims of the study. Since a zero-transition is a major novelty when students are introduced to calculating with rational numbers two problems containing a zero-transition (P7, P14) were included. In this study adaptivity is investigated related to task characteristics (research question 2). A normative perspective is taken in order to decide if a strategy adapts to a task or not. Therefore, problems suggesting the application of different numbertransformation strategies were included. From a normative perspective the or the strategy are most appropriate for problems P1, P2, P3, P9, P10, P11, P12. The compensation strategy is appropriate for problems P3, P7, P8, P13, P14. P13 is also suitable for applying the indirect addition strategy. P4 and P5 might be solved by a variation of the simplifying strategy, since e.g =(50+3)-(30-3)= FINDINGS Table 3 provides an overview of used approach and number-transformation strategies related to problems P1-P14. The analysis of the eight students approach strategies reveals that the main strategy applied to problems containing negative numbers was the transformation into equivalent problems with natural numbers using laws for addition and subtraction of integers, e.g = Only the problems containing a zero-transition (P7 and P14) were approached by referring to a mental image of the number line or to the bank-account-balance-model. Student 1 approaches problem P4 through the number line model: Interviewer: Minus 11 plus 28? [Minus 11 plus 28?] 1 Student 1: Plus 17! [Plus 17!] Interviewer: Richtig! Student 1: [Correct!] Hab ich erstmal von den 28 die 11 abgezogen bis ich bei 0 bin, dann plus den Rest. [I subtracted 11 from 28 so that I will be at 0 and then I added the rest.] Interviewer: Und wie genau hast du das abgezogen? Student 1: [And how exactly did you subtract?] Ich hab 28 erstmal minus 11, dass ich wieder bei null bin, sind 17, dann null plus 17 sind 17. [I took 28 minus 11 so that I am at 0, is 17, then 0 plus 17 is 17.] 1 All translations that are provided in brackets were carried out by the author.

7 problem (P1) (P2) (P3) (P4) (P5) (P6) (P7) (P8) (P9) 3,5+1,2 (P10) 7,2-2,1 (P11) -7,6+2,3 (P12) 3,8+2,6 (P13) 6,4-3,8 (P14) 0,8-2,9 S1 -(47-23) 2, -(12+25), number line 28-11,? -(33+19), -(7,6-2,3), -(2,9-0,8), S2 -(47-23), -(12+25), -11+1=-10; -(10-7)=-3; -3+20=17 -(33+19), -(7,6-2,3), 0,8-0,9=-0,1; -(0,1+2)=- 2,1; S3 -(47-23),? -(12+25); number line -11 up to 0 is 11; 28-11,? -(33+19); -(7,6-2,3); -(2,9-0,8);? S4 -(47-23); -(12+25) number line 28-11;? -(33+19); -(7,6-2,3); b-a-b-m -(2,9-0,8) 2-0,8=1,2; 1,2+0,9=2,1 S5 -(47-23); -(12+25) 28-11;? -(33+19) -(7,6-2,3); -(2,9-0,8) S6 ind. add.? -(12+25); number line ind. add. -(33-19);? -(7,6-2,3); ind. add. number line -(2,9-0,8) S7 -(47-23); -(12+25); 28-11; -(33+19) ; 64-38; -(29-0,8) S8 -(47-23); -(12+25); (33+19); -(7,6-2,3); b-a-b-m 2,9-0,8; Table 3: Approach and number-transformation strategies used by students S1-S8. 2 Whenever students use the approach strategy transformation the actual transformation will be specified in the table.

8 Student 4 refers to the bank-account-balance-model in order to calculate the algebraic sign of the result of problem P14: Interviewer: 0,8 minus 2,9? [0,8 minus 2,9?] Student 4: das sind minus 2,1, stimmt s? [is minus 2,1, is it?] Interviewer: Ja! [Yes!] [...] Student 4: Ich hab von 2 die 0,8 abgezogen, dann hab ich 1,2, dann rechne ich die 9 dazu, dann fehlen noch 8 bis zur 2, weil ich hab ja 9 dann sind es 2,1, ist es minus, weil ich bin ja plus 0,8, wenn ich 8 Cent aufm Konto hab und mach 2,9 Euro Schulden bin ich ja im Minus. [I subtracted 0.8 from 2, makes 1.2, then I add 9, 8 are missing upto 2, and I have got 9, makes 2.1, it s minus, because I am plus 0.8, when I have got 8 Cent deposit on my account and I make a withdrawal of 2.9 Euro, it s negative.] Regarding number-transformation strategies, the analysis reveals that all the problems were solved using familiar number-transformation strategies from the set of natural numbers. No strategies specifically related to rational numbers were observed. Six of the eight students solved problems with decimal numbers by applying either the strategy or the strategy. Two of them (S1, S2) used the strategy for every problem, no matter if it was a problem with natural numbers, integers or decimals. S8 almost exclusively used the strategy. Three students (S3, S4, S5) used the and the strategy. Only two students (S7, S8) apply other strategies than the or the strategy: S7 transformed some of the problems to problems with natural numbers by leaving the decimal point out and setting it correctly after the calculation. S6 also applied the indirect addition strategy once to a subtraction problem with decimals. In terms of adaptive strategy use, the analysis of the used number-transformation strategies reveals that students do not apply a variety of strategies to the posed problems: Three of the eight students (S1, S2, and S8) use the same numbertransformation strategy for almost all problems. The rest of the students use mainly two strategies. Although the compensation strategy would have been appropriate for problems P3, P7, P8, P12, P13, P14 none of the students used this strategy. However, the three students using the and the strategy (S3, S4, and S5) show the tendency to solve problems with integers using the strategy and problems with decimals according to the strategy. This might be an indication for adaptive strategy use. But, the reasons for the strategy choice cannot be derived from the data.

9 DISCUSSION AND CONCLUSION In the present study all problems from the set of rational numbers were solved according to number-transformation strategies that students are familiar with from the set of natural numbers. No strategies specific to the set of rational numbers were observed. The students in the present study hardly choose adaptively among different computation strategies with respect to task characteristics. In none of the cases where a compensation strategy would have been appropriate the students applied this strategy. Furthermore, students in the study favour the and the strategy. Regarding the small scope of the study these results cannot be generalized. Further research is needed to support the findings from this study. In terms of approach strategies, it is remarkable that students only refer to didactic models of negative numbers (number line, bank-account-balance-model) associated with problems that contain a zero-transition. This might indicate that problems with zero-transition present particular difficulties to students. This hypothesis could be approached in further research. The findings from the present study reflect results from previous studies on mental calculation in the set of natural numbers: students apply only one or two strategies to almost all problems without choosing strategies adaptively to task characteristics. To investigate factors affecting adaptive expertise in mental calculation is therefore not only an issue for further studies related to the set of natural numbers, but also related to the set of rational numbers. NOTES 1. Data was collected by Jens Hubert in the context of his final thesis for earning a teaching degree in secondary school. REFERENCES Blöte, A. W., Klein, A. S., & Beishuizen, M. (2000). Mental computation and conceptual understanding. Learning and Instruction, 10(3), Heinze, A., Marschick, F., & Lipowsky, F. (2009). Addition and subtraction of threedigit numbers: adaptive strategy use and the influence of instruction in German third grade. ZDM, 41(5), Heinze, A., Star, J., & Verschaffel, L. (2009). Flexible and adaptive use of strategies and representations in mathematics education. ZDM, 41(5), Heirdsfield, A. M., & Cooper, T. J. (2004). Factors affecting the process of proficient mental addition and subtraction: case studies of flexible and inflexible computers. The Journal of Mathematical Behavior, 23(4), National Council of Teachers of Mathematics (Ed.). (2000). Principles and standards for school mathematics. Reston, Va.: National Council of Teachers of Mathematics.

10 Padberg, F. (2009). Didaktik der Bruchrechnung (4., erw., stark überarb. Aufl. ed.). Heidelberg: Spektrum. Profke, L. (1991). Bruchrechnung im Mathematikunterricht. In H. Postel & u. a. (Eds.), Mathematik lehren und lernen. Festschrift für Heinz Giesel (pp ). Hannover: Schroedel. Qualifications and Curriculum Development Agency. (1999). Mathematics: the National Curriculum for England. Key stages 1-4: Qualifications and Curriculum Authority. Reys, R. E., Reys, B. J., Nohda, N., & Emori, H. (1995). Mental Computation Performance and Strategy Use of Japanese Students in Grades 2, 4, 6, and 8. Journal for Research in Mathematics Education, 26(4), Selter, C. (2001). Addition and Subtraction of Three-digit Numbers: German Elementary Children's Success, Methods and Strategies. Educational Studies in Mathematics, 47(2), Selter, C. (2009). Creativity, flexibility, adaptivity, and strategy use in mathematics. ZDM, 41(5), Ständige Konferenz der Kultusminister der Länder in der Bundesrepublik Deutschland. (2005). Bildungsstandards im Fach Mathematik für den Hauptschulabschluss (Jahrgangsstufe 9). Beschluss vom München: Luchterhand. Threlfall, J. (2002). Flexible Mental Calculation. Educational Studies in Mathematics, 50(1), Threlfall, J. (2009). Strategies and flexibility in mental calculation. ZDM, 41(5), Torbeyns, J., De Smedt, B., Ghesquière, P., & Verschaffel, L. (2009). Jump or compensate? Strategy flexibility in the number domain up to 100. ZDM, 41(5), Torbeyns, J., Verschaffel, L., & Ghesquière, P. (2006). The development of children's adaptive expertise in the number domain 20 to 100. Cognition and Instruction, 24, Verschaffel, L., Luwel, K., Torbeyns, J., & Van Dooren, W. (2009). Conceptualizing, investigating, and enhancing adaptive expertise in elementary mathematics education. European Journal of Psychology of Education, 24(3), Vollrath, H.-J., & Weigand, H.-G. (2007). Algebra in der Sekundarstufe (2 ed.). München: Elsevier, Spektrum.