IT IS MENTAL COMPUTATION

Literature at national and international levels argues the importance of including mental computation in a mathematics curriculum that promotes number sense. However, mental computation does not feature in importance in the current Queensland mathematics syllabus documents. Hopefully, with the writing of a new mathematics syllabus, mental computation will feature with more prominence. 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, and develop number sense. (Heirdsfield, 2000)

Introduction

Three studies were reported the architecture of mental and substraction at the the Australian Association for Research in Education Conference in 1997 (Heirdsfield & Cooper, 1997). In particular, the third study found a complex interaction among factors that appeared to be connected with proficient mental computation; it (reported in Heirdsfield & Cooper, 1997) constituted a small part of a study, whose main purpose was to develop an explanation of why some children are better at addition and subtraction mental computation than others. The idea is mental computation that is defined as “the process of carrying out arithmetic calculations without the aid of external devices” (Sowder, 1988, p. 182).

According to Heirdsfield, the purpose of the research that was developed is to identify factors and the relationships between factors which influence children’s proficiency in addition and subtraction mental computation. As a result, the literature has shown that mental computation may be viewed as a subset of number sense, as students who exhibit proficiency in mental computation also display number sense (e.g., McIntosh, 1996; McIntosh, Reys, & Reys, 1992; Sowder, 1990, 1992).

In order to know what factors that related with mental computation, I try to make summary in which envision and enlighten some related factors to mental strategies. So, this article was aimed to identify factors and the relationships between factors which influence children’s proficiency in addition and subtraction mental computation.

Mental Strategies for Addition and Subtraction

Based on Heirdsfield’s study, we may know some strategies of mental computation such as counting, separation, aggregation, wholistic. In her research, she used instruments that address mental computation strategies, number facts, computational estimation, numeration, number and operations, and investigated metacognition, affect, beliefs and evidence of mental representations.

Examples of some strategies are described below.

By counting, students would do in two ways such as

Count on by 1 (28+35: 28, 29, 30, …)

Count back by 1 (52-24: 52, 51, 50, …)

By separation, students would do in three ways such as

Right to left (u-1010)

Example, 28+35: 8+5=13, 20+30=50, 63; 52-24: 12-4=8, 40-20=20, 28), (subtractive)

Left to right (1010)

Example, 4+8=12, 20+20=40, 28) (additive)

Cumulative sum or difference

Example, 28+35: 20+30=50, 8+5=13, 63; 52-24: 40-20=20, 12-4=8, 28(subtractive),

20+20=40, 4+8=12, 28 (additive)

28+35: 20+30=50, 50+8=58, 58+5=63

52-24: 50-20=30, 30+2=32, 32-4=28

By aggregation, students would do in two ways such as

Right to left (u-N10)

Example, 28+35: 28+5=33, 33+30=63

52-24: 52-4=48, 48-20=28 (subtractive)

: 24+8=32, 32+ 20=52, 28 (additive)

Left to right (N10)

Example, 28+35: 28+30=58, 58+5=63

52-24: 52-20=32, 32-4=28 (subtractive)

: 24+20=44, 44+8=52, 28 (additive)

By wholistic, students would do in two ways such as

Compensation

Example, 28+35: 30+35=65, 65-2=63

52-24: 52-30=22, 22+6=28(subtractive)

24+26=50, 50+2=52, 26+2=28 (additive)

Levelling

Example, 28+35: 30+33=63

52-24: 58-30=28 (subtractive)

22+28=50, 28 (additive)

Reflection as Literature Review

Heirdsfield reported an overview of the findings of the pilot study. The four students in the pilot study were Clare (accurate and flexible), Mandy (accurate but inflexible), Emma (inaccurate and flexible), and Rosie (inaccurate and inflexible). Results for Clare and Mandy have been reported elsewhere (Heirdsfield, 1998; Heirdsfield & Cooper, 1997). Moreover, from the result asserted a picture of a proficient mental computer was starting to emerge. It appeared that a well-connected network of knowledge of the effects of operations on number, numeration, number facts, and computational estimation contributed towards flexibility and accuracy in mental computation (Heirdsfield, 2000)

Mental computation requires concurrent processing and temporary storage of information (holding interim calculations in memory), and retrieval of facts and strategies; that is, mental computation is cognitively demanding (Heirdsfield, 2000). This assertion supports Heirdsfield’s (2000) statement below

“In the case of Mandy, who was accurate but not flexible, few links were made among the factors that were investigated; yet she was capable of holding many interim calculations in memory, resulting in overall accuracy. Mandy’s number facts were fast and accurate (although it could be argued, not very efficient). Her number facts might have contributed to accuracy in mental computation. However, it is argued that her mental strategies would have taxed working memory. In contrast, Clare’s mental strategies did not require such a load on working memory. Rather, memory was involved in making connections, for instance, remembering previously calculated number facts. On the other hand, many of Emma’s errors were attributed to memory problems. Thus, memory seemed to impact on mental computation. To date, there is a paucity of studies investigating memory and mental computation, when mental strategies are not confined to mental images of pen and paper algorithms”.

Two aspects of memory seemed to be significant: load on working memory while calculating, and retrieval from long-term memory of facts and strategies.

Mental Strategies from the seventh grader

Problem of International Competition and Assessment for Schools Mathematics 2009 (ICAS, 2009) in number 13 described below.

“An automatic teller machine (ATM) has only $20 and $50 notes. When possible, $50 notes are given out instead of $20 notes. For example, $100 is given out as two $50 notes instead of five $20 notes.

Six people use the automatic teller machine.

Who received exactly one $50 note?

As you know, this problem needs student’s experience to understand an automatic teller machine (ATM). How does it works become real question to know the students are really understand about the machine. Students’ understanding of ATM is as well as students’ experience, so they will know withdrawal as if they have been. Now, there is one of student in SMP Negeri 1 Palembang who I gave this problem, then he solved like showed below.

I know the students try to understand the question “Who received exactly one $50 note?”. As a result, you see students do mental computation. From 4 strategies that be described above, The student may choose separation as the way to find out the question. Firstly, he tried to understand for six people who use ATM and much of money that they withdraw. Secondly, he understood what for “one $50” related with amount withdrawn. Then, making analysis for each of amount withdrawn toward “one $50” as well as grouping money into some types of group such as 50, or 20. Finally, make conclusion from the analysis of grouping $50 or $20. As the picture above, the students concluded that Rene, Tom, Vance, and Will received exactly one $50 note.

Although it seems different way, the student does not realize to show his way of thinking when division can be one of strategies. For example, $60 isn’t divisible number by 50; it means that the user of ATM wouldn’t withdraw $50 as part of money. But, the way of thinking by developing concept of grouping can encourage the student to enhance level of division. On the other words, separation is one of mental strategies, it can be developed into another strategy that students are learning by doing with their mathematical knowledge, for instance, division.

Conclusion

The purpose of this article is to identify another mental strategy that we may find in students’ mathematical learning. From literature you may know separation as one of mental strategies, but can be developed based on the problem. Like what student of SMPN 1 Palembang did, separation can be improvised to another strategy that is division.

Reference

Heirdsfield, A. M., & Cooper, T. J. (1997). The architecture of mental addition and subtraction. Paper presented at the annual conference of the Australian Association of Research in Education, Brisbane, Australia. (http://www.swin.edu.au/aare97pap/HEIRA97395.htm) 15 January 1998

Heirdsfield, Ann (2000). Mental Computation: Is it more than mental architecture? Presented at the annual meeting of the Australian Association for Research in Education, Sidney, 4 – 7 December 2000. Retrieved March 28, 2010.

McIntosh, A. (1996). Mental computation and number sense of Western Australian students. In J. Mulligan & M. Mitchelmore (Eds.), Children’s number learning. (pp. 259-276). Adelaide: Australian Association of Mathematics Teachers, Inc.

McIntosh, A., Reys, B., & Reys, R. (1992). A proposed framework for examining basic number sense. For the Learning of Mathematics, 12, 2-8.

Sowder, J. (1988). Mental computation and number comparisons: Their roles in the development of number sense and computational estimation. In J. Hiebert & M. Behr (Eds.), Number concepts and operations in the middle grades. Hillsdale: NJ: Lawrence Erlbaum Associates.

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.

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