Jurnal Mental komputasi

17

Munirah Ghazali, Rohana Alias, Noor Asrul Anuar Ariffin and Ayminsyadora Ayub Journal of Science and Mathematics Education in Southeast Asia 2010, Vol. 33 No. 1, 17-38

Identification of Students’ Intuitive Mental Computational Strategies for 1, 2 and 3 Digits Addition and Subtraction:

Pedagogical and Curricular Implications

Munirah Ghazali Universiti Sains Malaysia

Rohana Alias Universiti Teknologi MARA Perlis

Noor Asrul Anuar Ariffin Universiti Sains Malaysia

Ayminsyadora Ayub Universiti Sains Malaysia

This paper reports on a study to examine mental computation strategies used by Year 1, Year 2, and Year 3 students to solve addition and subtraction problems. The participants in this study were twenty five 7 to 9 year-old students identified as excellent, good and satisfactory in their mathematics performance from a school in Penang, Malaysia. The students were interviewed individually and asked to solve 6 questions consisting of 1, 2 and 3 digits addition and subtraction problems mentally without the use of pen and paper. The students were also required to explain how they arrived at their answer. Findings from this study showed a range of strategies employed by the students even though addition and subtraction mental strategies are not formally taught to Year 1, Year 2 and Year 3 students. The different types of addition and subtraction strategies used were highlighted. The strategies for 1-digit, 2-digit and 3-digit mental computation of both addition and subtraction problems were compared. Further, the students employed intuitive mental computation strategies and there are qualitative differences in these strategies which may prove invaluable to help teachers in strengthening students’ understanding of the number concepts. Pedagogical and curricular implications arising from the findings were discussed.

Key Words: Mental computation; Number sense; Addition; Subtraction; Primary mathematics

18

Identification of Students’ Intuitive

Introduction There is an emerging trend to focus on mental computation in the mathematics education curriculum of many countries (Harnett, 2007). Curricular reform documents in the United States, Australia, the United Kingdom, New Zealand and the Netherlands have included mental computation as an important aspect of primary mathematics (National Council of Teachers of Mathematics, 2000, DfES, 2007, Treffers & De Moor, 1990 and Australian Education Council, 1991). In Malaysia, mental computation is documented in the primary school Mathematics curriculum as evidenced in the curriculum specification documents (Ministry of Education Malaysia, 2002; 2003a; 2003b). For example, the Year One curriculum specification actually specifies, ‘Emphasise mental calculation’ under ‘points to note’ for the curriculum specification on the topic of whole numbers under the learning area addition with the highest total of 10, 18, and subtraction within the range of 18. The curriculum specification documents recommend and encourage teachers to emphasise mental calculation for all the operations addition, subtraction, multiplication and division from Years One to Six. An analysis of the Mathematics Special Guide Book published by the Curriculum Development Centre of the Ministry of Education, Malaysia, showed that the number sense initiative was already embedded in the Integrated Curriculum for Primary Schools, Ministry of Education since 1988. As an example, the Mathematics Special Guide Book 1 published by the Ministry of Education, 1988 stated that: “The aims and objectives of the subject stress that knowledge in numbers and operations using numbers is the foundation for the said subject” ( pp vii). Mastery in those said skills however necessitates that it be done congruent with principles of child psychology and current educational principles.

Even though, no specific mental computation strategies was mentioned in the document, it is encouraging to see that mental computation is being positioned in the Malaysia Primary Mathematics Curriculum. This research intends to investigate students’ intuitive mental computational strategies when solving addition and subtraction problems given that the students might or might not be formally introduced to mental computation. Research shows that many of the mental computation strategies were spontaneous and self-developed by the students even in classes where mental computation is formally introduced (Caney & Watson, 2003). Therefore, it would be of

19

Munirah Ghazali, Rohana Alias, Noor Asrul Anuar Ariffin and Ayminsyadora Ayub

interest to document mental computation strategies to help mediate mental computation acquisition and use by students.

Mental Computations and Number Sense Mental computation and computational estimation are two important aspects of number sense (McIntosh, De Nardi & Swan, 1994; Sowder, 1992; Reys, 2006). Moreover, number sense is seen as an indicator of mathematical thinking (McIntosh et al., 1994). Number sense is seen as having an intuition about numbers (Howden, 1989). Most characteristics of number sense identified by researchers concentrate on its intuitive nature, its gradual and evolutionary development, and the multiple ways in which it is revealed. These characteristics include: making sense of numbers, operations, and their relationships; using numbers flexibly when doing mental computation and computational estimation; comparing and ordering numbers; judging the reasonableness of results and calculations; decomposing and recomposing numbers; and creating a variety of strategies to solve problems involving numbers, all of these coming from an inclination to make sense of numerical situations (Reys & Yang, 1998; Yang, 1995). Apart from being difficult to define, number sense is difficult to measure (Sowder & Schappelle, 1989). Nonetheless, despite being difficult to define and measure, number sense is an important trait for students to develop (Hope, 1989; NCTM, 2000). All definitions of number sense discussed so far have consistently seen number sense as a requirement towards understanding operations and numeracy exercises that children carry out. In conclusion, number sense refers to a person’s general understanding of number and operations along with the ability and inclination to use this understanding in flexible ways to make mathematical judgments and to develop useful strategies for handling numbers and operations’ (McIntosh, Reys & Reys, 1993; Carpenter, Fennema & Franke, 1996; Fosnot & Dolk, 2001).

For many people, primary school mathematics is synonymous with written computations (Reys & Nohda, 1994). Teachers allocate a lot of time teaching pupils written computations (McIntosh, A., Reys, B., Reys, R., Bana, J., & Farrell, B., 1997). There is a belief among teachers that if pupils hold understanding regarding number concepts, then this understanding can be measured by means of written computations (Jones, Kershaw & Sparrow, 1994). Nonetheless, skills in solving written computations will not differentiate whether the pupil possesses number sense or whether he/she

20


is able to find the solution based on the use of algorithms and certain procedures.

Research on students’ number sense in Malaysia showed that there were students who could perform the arithmetic calculations well, but lacked number sense (Munirah, 2000; Munirah Ghazali, Siti Aishah Sheikh Abdullah, Zurida Ismail, Mohd Irwan Idris, 2005; Clarke, 2001). Moreover, analyses from the study showed that while students were able to do calculations for certain computation questions, ironically they faced difficulty doing the same questions in number sense format. Research on number sense so far has established that the unchallenged belief that ‘if students can compute then they have understood’ is now being questioned (McIntosh et al., 1997). It is therefore important for students to understand numbers in order to make sense of the way numbers are used in students’ everyday world (NCTM, 2000). The same mathematics scenario is apparent in Malaysian schools. Research in Malaysia (Munirah et al., 2005; Munirah & Noor Azlan, 1999) have strongly indicated that most school children do not display good number sense and predictably have an average standing in mathematics achievement compared to other nations. An analysis of the Third International Mathematics and Science Study – Repeat (TIMSS-R) showed that Malaysian students performed well above the international average for questions that require computation but face difficulty and thus perform below the international average for questions that require understanding of basic concepts (Ministry of Education, Malaysia, 1999).

Research on mental computation and number has proposed connections between mental computation and number sense, particularly number facts, computational estimation, numeration, and properties of number and operation; social and affective issues including attributions, self efficacy, and social context (e.g., classroom and home); and metacognitive processes (Heirdsfield, 2000). It has been posited that when children are encouraged to formulate their own mental computation strategies, they learn how numbers work, gain a richer experience in dealing with numbers, develop number sense, and develop confidence in their ability to make sense of number operations (Kamii & Dominick, 1998; Reys & Barger, 1994; Sowder, 1990). Mental computation’s significance is seen in terms of its contribution to number sense as a whole; for example, as a “vehicle for promoting thinking, conjecturing, and generalising based on conceptual understanding rather than as a set of skills that serve as an end of instruction” (Reys & Barger,

21


1994, p.31). Therefore, it appears necessary for students to develop proficiency in mental computation through the acquisition of self-developed or spontaneous strategies rather than memorisation of procedures (Kamii, Lewis, & Livingston, 1993; Reys & Barger, 1994). Mental computation in this form features in various models of computation (e.g., National Council of Teachers of Mathematics, 1989; Trafton, 1994), although usually in combination with written, and calculator methods. Studies show that students who benefit from effective mental computation skills can use a variety of strategies in different situations. Moreover, the students were able to display flexible use of the structure of the number systems because they were disposed to making sense of mathematics (Cooper et al., 1996; Sowder, 1992, 1994; Heirdsfield & Cooper, 2004a, 2004b).

Mental Computations in the Year 1, Year 2 and Year 3 Curriculum Written arithmetic procedures for addition and subtraction are introduced at the early stage in Malaysian schools. The children are expected to be able to calculate three digit addition and subtraction standard written method by the end of Year 2. The strategies for written computation taught to students require them to follow fixed sequential steps to solve arithmetic addition and subtraction, usually using the place value procedure.

Addition and subtraction mental computation is mentioned in the curriculum document as mental calculation. The curriculum indicates that in Year 1, mental computation should be emphasised in addition with the highest total of 18, and subtraction within the range of 18 where they are expected to recall rapidly basic facts of addition, basic facts of subtraction, and the total of two numbers and the difference of two numbers. Similarly, in Year 2 mental computation is covered in addition with the highest total of 1000 and subtraction within the range of 1000 with different combinations of one digit, two digit and three digits numbers. In Year 3, emphasis on mental computation are made in addition with the highest total of 10000 and subtraction within the range of 10000 where students are expected to be able to add and subtract two numbers up to four digit. However, no specific mental computation strategies are mentioned in any of these curriculum specifications. An important observation made is that the emphasis on mental computation is taught concurrently with the standard written method for both addition and subtraction.

22


Interviews with teachers indicate that mental computation in the curriculum is confined to basic facts and to the mental computation considered necessary so that one could perform written calculations where speed and accuracy were emphasised. Though mental computations are addressed in the curriculum and in the classrooms, they are not assessed in any of the achievement test. The students are tested in their ability to perform the addition and subtraction standard written method only. Thus, much emphasis is placed on these procedures.

Methodology Twenty seven students from Year 1, Year 2 and Year 3, ranging from seven to nine years in age from a school in Penang, were selected in this study. They were chosen from three different classes with different levels of achievement – excellent, good and satisfactory. One student each from both Year 1 and Year 2 were reluctant to answer the majority of the questions given and thus had to be omitted from the study. The actual participants for this study were 25 students; eight from Year 1, eight from Year 2 and nine from Year 3.

The students were interviewed individually and asked to solve six questions consisting of one, two and three-digit addition and subtraction problems mentally without the use of pen and paper. The problems were: (i) 5 + 9; (ii) 37 + 19; (iii) 106 + 228; (iv) 14 -7; (v) 63 – 15; and (vi) 241 – 167. These problems were given in an interview room where the sessions were videotaped together with all the nonverbal behaviour. The students were also required to explain how they arrived at their answer.

The responses were analysed for the strategy choice and also accuracy. The strategies in mental computation used were identified using the categorisation scheme given in Table 1 with reference to Beishuizen, 1993; Cooper, Heirdsfield, & Irons, 1996; Reys, Reys, Nohda, & Emori, 1995; and Thompson, 1999.

23


Table 1 Mental Strategies for Addition and Subtraction

Strategy Example

Counting all 5+9: ‘1, 2, 3, 4, … 14’ Counting on from first 5+9: ‘4, 5, 6, 7, ... 14’ number Counting on from larger 5+9: ‘9, 10, 11, … 14’ Counting back from 14–7: ‘14, 13, 12, 11, 10, 9, 8, 7’ Counting up from 14–7: ‘7, 8, 9, 10, 11, 12, 13, 14’ (complementary addition) The answer is 7 Doubles fact (subtraction) 14–7: ‘The answer is 7 because 7 and 7

is 14’ Subtraction as the inverse 14–7: ‘I knew 7 and 7 is 14 … and I just of addition took away 7’ Bridging through ten 5+9: ‘9 and 1 is 10 ... take 1 off the 5 … (addition) and add all the leftover from before … so

I get 14’ Separation 37+19: 7+9=16, 30 + 10 = 40, 56

37+19: 30+10=40, 7+9=16, 56 Wholistic 37+19: 37+20=57,57–1 =56

37+19: 36+20=56 Mental image of pen Student reports placing numbers under and paper algorithm each other as on paper and doing

the operation right to left Mental recall Student reports that they just know

the answer Standard written Student use pen and paper to get their method (algorithm) answers using the standard written

method Inappropriate strategies Strategies where the wrong operation

was used in the calculation, for example 14+7 instead of 14-7

Unidentified Student was unable to explain the method used or insufficient probing on the part of the interviewer

Right to left Left to right Compensation Levelling

24


Results and Discussion Findings from this study showed a range of strategies employed by the students even though addition and subtraction mental strategies are not formally taught to Year 1, Year 2 and Year 3 students. Analysis of the responses suggests that the students use from an inefficient strategy of counting on their fingers to more sophisticated and simpler strategies such as: Using doubles, Bridging through tens, Separation and Wholistic. The mental computation addition strategies used were Counting all, Counting on from first number, Counting on from larger, Bridging through ten, Separation, Wholistic, Mental recall and Mental image of the pen and paper algorithm. The mental computation subtraction strategies used were Counting back from, Doubles fact, Separation, Mental recall, Mental image of pen and paper algorithm, and Direct modelling. Moreover students tend to use Counting strategies for one and two digits and prefer to use Mental image of pen and paper algorithm as their mental computation addition strategies for three-digit problems. Findings from this study suggest that students do have intuitive mental computation strategies and that there are qualitative differences in these strategies which may prove invaluable to help teachers in strengthening students’ understanding of the number concepts.

Categorisation of Students’ Responses The first stage of the analysis categorises students’ responses into three categories: correct response, wrong response and no response. The no response category was excluded from further analysis. The categorisation is shown in Table 2. The mental calculation problems include addition and subtraction with one digit, two-digit and three-digit problems. It can be seen that there are altogether 150 responses, where 63 (42%) were correct responses, 63 (42%) wrong responses and 24 (16%) no responses. The percentage of correct responses and wrong responses were similar. Among the general trends that can be observed are (a) there is a higher percentage of correct response for addition than subtraction problems, (b) the percentage of correct response decreases with higher digits addition and subtraction problems, and (c) the percentage of questions not attempted (no response) increases with higher digits addition and subtraction problems.

25


Table 2 Response Category

Response Addition Subtraction Total Category One Two Three One Two Three

digit digit digit digit digit digit

Correct 23 11 8 14 6 1 63 Wrong 2 12 10 10 13 16 63 No 0 2 7 1 6 8 24 response Total 25 25 25 25 25 25 150

Correct Responses Table 3 show the students’ responses categorised according to the types of problems and year in school. Analysis of the data indicates that the percentage of correct response increased with the year in school, while the wrong response and no response decreased with the year in school. The percentages are 17%, 46%, 61% correct response; 54%, 37%, 35% wrong response; and 29%, 17%, 4% no response for Year 1, Year 2 and Year 3 respectively.

The addition problems were attempted with higher frequency and accuracy than subtraction problems. Among Year 1 students the number of responses were 18 (75%) addition and 16 (66%) subtraction; for Year 2 students 21 (88%) addition and 19 (79%) subtraction; and for Year 3 students 27 (100%) addition and 25 (93%) subtraction problems. The number of correct responses for Year 1 students were seven (29%) addition and one (4%) subtraction; for Year 2 students 14 (58%) addition and eight (33%) subtraction; and for Year 3 students 21 (77%) addition and 12 (44%) subtraction problems. Thus, it can be concluded that students were more able to do addition than subtraction problems.

26


Table 3 Response Category by Year of Schooling

Addition Subtraction

Year Response One Two Three One Two Three Total (%) Category digit digit digit digit digit digit

Year 1 Correct 6 1 0 1 0 0

8 (17) Wrong 2 6 3 6 5 4 26 (54) Excluded 0 1 5 1 3 4 14 (29)

Year 2 Correct 8 3 3 6 2 0 22 (46) Wrong 0 4 3 2 4 5 18 (37) Excluded 0 1 2 0 2 3 8 (17)

Year 3 Correct 9 7 5 7 4 1 33 (61) Wrong 0 2 4 2 4 7 19 (35) Excluded 0 0 0 0 1 1 2 (4)

Further analysis of the data also indicates that the percentage of correct responses decreased as the number of digits increased in the problems given to them. This is true for both addition and subtraction problems and also for Year 1, Year 2 and Year 3 students. For example, the number of correct responses for Year 3 students were nine (33%) for one digit addition problem, seven (26%) for two-digit addition problem and five (19%) for three-digit addition problem. Similarly for the same year, there were seven (26%) correct responses for one digit subtraction problem, four (15%) for two-digit problem and one (4%) for three-digit subtraction problem.

In addition, the data in this study also showed that none of the Year 1 students were able to solve the three-digit addition and subtraction problems. This is not beyond expectation because although the Year 1 curriculum contains the topic of whole numbers up to 100, it includes only addition and subtraction within the range of 18. Furthermore, none of the Year 2 students and only one Year 3 student was able to solve the three-digit subtraction problems.

Mental Computation Strategies Among Correct Responses This section discusses the strategies employed by students who obtained correct answers in the addition and subtraction problems. There were 42 correct responses for addition problems and 21 for subtraction problems. Analysis of the responses indicates that the students used a variety of

27


strategies from an inefficient strategy of counting on their fingers to more sophisticated and simpler strategies such as Using doubles, Bridging through ten, Separation and Wholistic strategy.

Mental Computation Addition Strategies Overall, a variety of strategies were used to solve the addition problems. These strategies were, Counting all, Counting on from first number, Counting on from larger, Bridging through ten, Separation (left to right and right to left), Wholistic-compensation, Mental recall, and Mental image of the pen and paper algorithm. It should be noted here that there were three responses (7%) from the students who requested to use the Standard written method where the student uses pen and paper to get their answers. The strategy for one of the responses was categorised as Unidentified even though the answer was correct.

Examples of the strategies used identified from the sample of this study are as follows:

Wholistic-compensation 37+19: 10+30=40; 9+1=10; 40+10=50; 7-1=6; 50+6=56

Separation-left to right 37+19: 3+1=4; 9+7=16; 4+1=5; 56

Separation-right to left 106+228: 6+8=14; 2+1=3; 334

The researchers felt that it was important to highlight these examples for there might be some difference in the strategies as outlined by Beishuizen (1993), Cooper, Heirdsfield, and Irons (1996), Reys, Reys, Nohda, and Emori (1995) and Thompson (1999). For example, 106+228 using the Separation right to left strategy, the sum would be solved as 6+8=14; 20+10=30; 200+100=300; 334. While 37+19 with the Separation left to right strategy would be solved as 30+10=40; 7+9=16; 56. Despite the fact that the students in this study presented the 30 and 10 as 3 and 1 which resembles the Mental image of pen and paper algorithm strategy, the difference lies in that they did not report using the method where they mentally place the numbers under each other and carrying out the ‘carrying the one’ procedure. Similarly, under the same categorisation scheme 37+19 with Wholistic-compensation strategy would be solved as 37+20=57; 57-1=56; 56. The data in our study however showed one respondent using a more naïve step when he did the 9+1 and 7-1 step which resembled the compensation part.

28


The data in Table 4 indicates that students preferred to use Counting strategies for one and two digit addition problems and some students were able to use the Mental recall strategy for one digit problems. The most common Counting strategies used was Counting on from first number. However, the students used more efficient strategies for two digits and three digits addition problems such as Separation left to right strategy for the two digit problem (one response from Year 2); Separation right to left for the two digit problem (one response from Year 2); Separation right to left strategy for the three digit problem (one response from Year 3); and the Wholistic-Compensation strategy for the two digit problem (one response from Year 1). Another observation made of the data was that there were students who tended to use either the Standard written method or Mental image of the pen and paper algorithm for the two and three digit addition problems. They were from Year 2 and Year 3.

Table 4 Frequency of Mental Computation Addition Strategies

Year in school One digit Two digit Three digit 1 2 3 Total 1 2 3 Total 1 2 3 Total

Counting all 1 1 2 1 1 2 Counting on 3 4 7 1 2 3 from first number Counting on 2 5 7 from larger Bridging 1 1 2 through ten Separation left 1 1 to right Separation 1 1 1 1 right to left Wholistic – 1 1 compensation Mental recall 1 1 3 5 Mental image 1 2 3 3 3 of algorithm Standard 3 3 written method Unidentified 1 1

Total 6 8 9 23 1 3 7 11 3 5 8

29


Mental Computation Subtraction Strategies Table 5 demonstrates the different strategies used to solve one digit, two digit and three digit subtraction problems. The total number of correct responses for one digit, two digit and three digit subtraction problems were 14, 6 and 1 responses respectively. The range of mental computation subtraction strategies used were from Direct modelling (five responses) to Counting strategies such as Counting back from (two responses) and Calculating strategies (five responses) and Doubles fact (three responses) and Bridging through ten (two responses), to a more efficient strategy such as Separation (one response). Note that the Direct modelling strategy (either using counters such as marbles or fingers) was not utilised by any of the students for addition problems. There were also students who preferred the Mental image of the pen and paper algorithm (five responses) and one student used the Standard written method. Thus, among the mental computation subtraction strategies mentioned, the preferred strategies for one digit problem were the Counting strategies and the Calculation strategies and for two and three digit problems the Mental image of the pen and paper algorithm strategy was preferred.

Examples of the mental computation subtraction strategies employed by the students are as follows:

Bridging through ten 14-7: 10-7= 3; 4+3=7; 7

Doubles fact (think addition) 14-7: 7+7 = 14; therefore 14-7 =7

Doubles fact (think multiplication) 14-7: 7x2 =14; therefore 14-7 = 7

Separation left to right 63-15: 6-1=5; 10+3=13; 13-5=8; 5-1=4; 48

The Bridging through ten strategy that was used by the two students to solve 14 - 7 was by partitioning the 14 into a ten and a four. This action might be influenced by the fact that their fingers can accommodate only the ten and not the four and thus forced them to keep the four mentally. The researcher categorised the strategy as such in order to differentiate between the Direct modelling strategy used by another three students who used their ten fingers to first subtract the seven and then replacing four fingers before counting again to get the answer 7.

30


Table 5 Frequency of Mental Computation Subtraction Strategies


Counting back from 1 1 2 Doubles fact 1 1 2 (think addition) Doubles fact 1 1 (think multiplication) Bridging through 1 1 2 ten Separation left 1 1 to right Mental recall 1 1 2 Mental image 1 3 4 1 1 of algorithm Standard written 1 1 method Direct modeling 3 2 5

Total 1 6 7 14 0 2 4 6 0 0 1 1

Mental Computation Strategies with Wrong Responses This section discusses the strategies exhibited by students who obtained wrong answers in the addition and subtraction problems. There were 24 wrong responses for addition problems and 39 for subtraction problems. The data in the study further indicates that the mental computation strategies used by those who obtained wrong responses (refer to Table 6) were almost similar to those with correct responses given in Table 4 with the exception of Direct modelling and Guessing strategy. It should be highlighted here that for the two digit addition problem (37+19) no distinction can be made with first number and larger number because it refers to the same number, thus, the study categorised this strategy as Counting on from first number.

31


Table 6 Frequency of Mental Computation Addition Strategies with Wrong Answers


Counting on 2 1 3 from first number Separation right 1 1 2 1 1 2 to left Separation 1 1 2 1 3 left to right Mental image 1 1 2 2 of algorithm Direct modeling 1 1 2 Guessing 2 2 1 1 1 1 Unidentified 1 0 1 2 1 1 2

Total 2 2 6 4 2 12 3 3 4 10

Observations made on the mental computation addition strategies used by those who obtained wrong answers when compared to those who obtained correct answers are: usage of Direct modelling (two responses); more on account of students Guessing (four responses); higher frequency of Separation strategy (eight responses); and lower frequency of using the Mental image of pen and paper algorithm strategy (three responses).

The students made attempts at answering the problems given but made errors during the process of solving. Examples of mental computation addition strategies used and the errors made are as follows:

Separation right to left 37+19: 7+9=16; 3+1=4; 416 (place value error)

Separation right to left 37+19: 7+9=15; 3+1=4; 4+1=6; 65 (addition error)

Separation left to right 37+19: 3+1=4; 7+9=16; 40+10=50; 50+6=55; 55 (addition error)

Counting on from first 37+19: 37, 38, 39…55 (counting error)

Direct modelling 37+19: 57 (counting error)

32


Table 7 shows the mental computation subtraction strategies used by students who obtained wrong responses. In comparison with the information provided in Table 5, this indicates that the strategies and their frequencies were almost identical with the following differences: higher frequency of Separation (left to right and right to left) strategy (three responses); lower frequency usage of Direct modeling (two responses); occurrence of Inappropriate strategy (two responses); higher frequency of Standard written method (three responses); and none on Doubles fact.

Similar comparison in strategies used for wrong responses between the different numbers of digit in the problems was omitted since there was a higher percentage of wrong responses for the two digit and three digit problems for both addition and subtraction.

Examples of mental computation subtraction strategies used and the errors made are as follows:

Counting back from 14-7: 14,13, …8; 8 (error include the number they were counting back from)

Bridging through ten 14-7: 10-7=3; 4 (addition error)

Inappropriate 14-7: 14, 15 … 21; 21 (addition instead of subtraction)

Separation left to right 246-167: 2-1=1; 4-6=2; 6-7 =1; 127 (subtraction error)

33


Table 7 Frequency of Mental Computation Subtraction Strategies with Wrong Answers


Counting back from 1 1 1 3 1 1 2 Bridging through 1 1 ten Separation right 1 1 1 1 to left Separation left 2 2 to right Mental recall 1 1 Mental image of 1 1 1 1 2 3 3 algorithm Standard written 2 1 3 method Direct modeling 1 1 1 1 Guessing 1 1 2 1 1 1 1 Inappropriate 1 1 1 1 1 1 strategy Unidentified 0 0 3 0 2 5 3 2 5

Total 6 2 2 10 5 4 4 13 4 5 7 16

Conclusions This research revealed that students invent or use their own intuitive strategies when asked to solve problems using mental computation even when mental computation may or may not have been formally taught to them. Secondly, while some students did invent their own intuitive strategies, there were other students who did not display their ability for mental computation. While this study did not connect actual teachers’ teaching strategies with students’ strategies, the findings from the study raised questions whether students’ do invent their own strategies or whether their use of intuitive strategies were indirectly encouraged by modelling teachers’ own mental computation strategies. Moreover, for students who did not display the ability for mental computation, the question raised was whether they could be encouraged to invent their own strategies if the classroom

34


and the teachers encouraged them to do so. These questions give some suggestions for further research into teachers’ pedagogical practice when teaching mathematics. Another finding from this research is that there was a tendency for Year 2 and 3 students to resort to algorithmic computation rather than mental computation especially when the number magnitude for the questions increases. While this paper argued that students’ ability to carry out mental computation reflect their understanding of numbers. The question that arises is whether the trend for older children to resort to algorithmic computation be seen as a concern, and in what ways could the curriculum address such situation.

References Australian Education Council (AEC). (1991). A national statement on

mathematics for Australian schools. Carlton, Vic: Curriculum Corporation. Beishuizen, M. (1993). Mental strategies and materials or models for addition

and subtraction up to 100 in Dutch second class. Journal for Research in Mathematics Education, 24(4), 194-323.

Caney, A. and Watson, J. M. (2003). Mental computation strategies for part- whole numbers, Australian Association for Research in Education AARE 2003. Conference Papers: International Education Research Conference, 30 November - 3 December 2003, Auckland, New Zealand EJ ISBN 1324- 9320.

Carpenter, T. P., E. Fennema, and M. L. Franke. (1996). “Cognitively Guided Instruction: A Knowledge Base for Reform in Primary Mathematics Instruction.” The Elementary School Journal 97 (1): 3-20.

Clarke, D. (2001). Understanding, assessing and developing young children’s mathematical thinking: Research as powerful tool for professional growth. In J. Bobis, B. Perry & M. Mitchelmore (Eds.), Numeracy and beyond: Proceedings of the 24th Annual Conference of the Mathematics Education Research Group of Australasia (Vol. 1, pp. 9-26). Sydney: MERGA.

Cooper, T. J. , Heirdsfield, A , & Irons, C. J. (1996). Children’s mental strategies for addition and subtraction word problems. In J. Mulligan & M. Mitchelmore (Eds.), Children’s number learning (pp 147-162) Adelaide: Australian Association of Mathematics Teachers.

35


DfES (Department for Education and Studies). (2007). Primary framework for literacy and mathematics: Guidance paper: calculation.[online] Retrieved on 21 July 2009 from http://www.standards.dfes.gov.uk/ primaryframeworks/mathematics/Papers/Calculation/

Fosnot, C. T. & Dolk, M. (2001). Young mathematicians at work: Constructing number sense, addition, and subtraction. Portsmouth, NH: Heinemann Press.

Hartnett, Judy E. (2007). Categorisation of mental computation strategies to support teaching and to encourage classroom dialogue. In Watson, Jane and Beswick, Kim, Eds. Proceedings 30th Annual Conference of the Mathematics Education Research Group of Australasia - Mathematics: Essential Research,Essential Practice. (pp. 345-352). Hobart, Tasmania. MERGA Inc.

Heirdsfield, Ann M. (2000). Mental computation: Is it more than mental architecture? In: Australian Association for Research in Education, December, 2000, Sydney.

Heirdsfield, A. N., & Cooper, T. J. (2004a). Factors affecting the process of proficient mental addition and subtraction: Case studies of flexible and inflexible computers. Journal of Mathematical Behavior, 23, 443-463.

Heirdsfield, A. N., & Cooper, T. J. (2004b). Inaccurate mental addition and subtraction: Causes and compensation. Focus on Learning Problems in Mathematics, 26(3), 43-66.

Hope, J. A. (1989). Fifth and seventh grade preference for doing computation: Percentage results. Unpublished raw data.

Howden, H. (1989). Teaching number sense. Arithmetic Teacher, 36(6), 6-11. Jones,K.; Kershaw, L; Sparrow, L. (1994). Number sense and computation in the

classroom.(monograph) Mathematics, Science and Technology Education Centre, Edith Cowan University, Perth, Western Australia.

Kamii, C., & Dominick, A. (1998). The harmful effects of algorithms in grades 1-4. In L. J. Morrow & M. J. Kenney (Eds.), The teaching and learning of algorithms in school mathematics, 1998 yearbook. (pp. 130-140). Reston, VA: NCTM

Kamii, C., Lewis, B. A., & Livingston, S. L. (1993). Primary arithmetic: Children inventing their own procedures. Arithmetic Teacher, 41, 200-203.

Ministry of Education Malaysia. (1988). Mathematics Special Guide Book 1 . Kuala Lumpur: Pusat Perkembangan Kurikulum.

36


Ministry of Education Malaysia, (1999). Kajian TIMSS-R: Laporan Analisis Item Matematik dan Sains. Kuala Lumpur: Bahagian Perancangan dan Penyelidikan Dasar Pendidikan. Academic engagement and integrated science process skills achievement.

Ministry of Education Malaysia (2002). Huraian sukatan pelajaran matematik tahun 1. Kurikulum Bersepadu Sekolah Rendah. Kuala Lumpur: Pusat Perkembangan Kurikulum.

Ministry of Education Malaysia (2003a). Huraian sukatan pelajaran matematik tahun 2. Kurikulum Bersepadu Sekolah Rendah. Kuala Lumpur: Pusat Perkembangan Kurikulum.

Ministry of Education Malaysia (2003b). Huraian sukatan pelajaran matematik tahun 3. Kurikulum Bersepadu Sekolah Rendah. Kuala Lumpur: Pusat Perkembangan Kurikulum.

McIntosh, A. J., De Nardi, E., & Swan, P. (1994). Think mathematically. Melbourne:Longman.

McIntosh, A. J., Reys, B. J., & Reys, R. E. (1993). A proposed framework for examining number sense. For The Learning of Mathematics, 12(3), 2-8.

McIntosh, A., Reys, B. J., Reys, R. E., Bana, J. & Farrell, B. (1997). Number sense in school mathematics: Student performance in four countries, Perth, Australia: Edith Cowan University.

Munirah Ghazali & Noor Azlan Ahmad Zanzali (1999). Assessment of school children’s number sense. Proceedings of The International Conference on Mathematics Education into the 21 Century: Societal Challenges: Issues and Approaches, pp.30-36. Cairo, Egypt.[online]. Retrieve on 17th June 2010. http://math.unipa.it/~grim/Evol3.PDF

Munirah Ghazali (2000). Kajian kepekaan nombor murid tahun lima. (Unpublished Doctor of Philosophy Thesis, Universiti Teknologi Malaysia).

Munirah Ghazali, Siti Aishah Sheikh Abdullah, Zurida Ismail, Mohd Irwan Idris. (2005). Dominant Representation in the Understanding of Basic Integrals Among Post Secondary Students. The Mathematics Education into the 21st Century Project Universiti Teknologi Malaysia Reform, Revolution and Paradigm Shifts in Mathematics Education, Johor Bahru, Malaysia, Nov 25th - Dec 1st 2005, 97.

37


National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, Virginia: NCTM.

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.

Reys, B. J., & Barger, R. H. (1994). Mental computation: Issues from the United States perspective. In R. E. Reys & N. Nohda (Eds.), Computational alternatives for the twenty-first century. Reston, Virginia: The National Council of Teachers of Mathematics.

Reys, R. E. & Nohda, N. (1994). Computation and the need for change. In R. E. Reys & N. Nohda (Eds.), Computational alternatives for the twenty-first Century: Cross cultural perspectives from Japan and the United States (pp. 1- 11). Reston, VA: The National Council of Teachers of Mathematics.

Reys, B. J. (Ed.) (2006). The intended mathematics curriculum as represented in state-level curriculum standards: Consensus or confusion? Charlotte, NC: Information Age Publishing.

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 of Research in Mathematics Education, 26(4), 304-326.

Reys, R.E. & Yang, D.C. (1998). Relationship between computational performance and number sense among sixth and eighth grade students in Taiwan. Journal for Research in Mathematics Education, 29, 225237.

Sowder, J. T., & Schappelle, B. P. (Eds.). (1989). Establishing foundations for research on number sense and related topics: Report of a Conference. San Diego, CA: San Diego State University, Center for Research in Mathematics and Science Education.

Sowder, J. (1990). Mental computation and number sense. Arithmetic Teacher, 37(7), 18-20.

Sowder, J. (1992). Making sense of numbers in school mathematics. In G. Leinhardt, R. Putman & R. Hattrup (Eds.), Analysis of arithmetic for mathematics teaching. Hillsdale, New Jersey: Lawrence Erlbaum Associates.

Sowder, J. T. (1994). Cognitive and metacognitive processes in mental computation and computational estimation. In R.Reys, & N. Nohda (Eds.), Computational alternatives for the twenty-first century (pp. 139–146). Reston, VA: NCTM.

38


Thompson, I. (1999). Mental calculation strategies for addition and subtraction Part 1. Mathematics in School. Vol. 28, No.5, November 1999, pp 2-4. London: The Mathematical Association.

Trafton, P. (1994). Computational estimation: Curriculum development efforts and instructional issues. In R. Reys & N. Nohda (Eds.), Computational alternatives for the twenty-first century: Crosscultural perspectives from Japan and the United States. (76-86). Reston, Virginia: NCTM.

Treffers, A., & De Moor, E (1990). Towards a national mathematics curriculum for the elementary school. Part 2. Basic Skills and Writing Computation, Zwijssen, Tilburg, The Netherlands.

Yang, D.C. (1995). Number sense performance and strategies possessed by sixth- and eighth grade students in Taiwan. Unpublished Doctoral dissertation, University of Missouri, Columbia.

Authors:

Munirah Ghazali, School of Educational Sciences, Universiti Sains Malaysia, Penang; e-mail: [email protected]

Rohana Alias, Universiti Teknologi MARA Perlis; e-mail: [email protected]

Noor Asrul Anuar Ariffin, School of Educational Sciences, Universiti Sains Malaysia, Penang; e-mail: [email protected]

Ayminsyadora Ayub, School of Educational Sciences, Universiti Sains Malaysia, Penang; e-mail: [email protected]

Documents

Jurnal Mental komputasi