Inaccurate mental addition and subtraction: causes and compensation.

Abstract

This paper reports on a study of seven Year 3 students' diminished performance in mental computation, and compares their mental architecture. Although all students were identified as being inaccurate, three students used some variety of mental strategies, while the other students used only one strategy that reflected the written procedure for each of the addition and subtraction algorithms taught in the classroom. Interviews were used to identify students' knowledge and ability with respect to number sense (including number facts, estimation, numeration, and effect of operation on number), metacognition and affects. Two conceptual frameworks were developed, one representing the "flexible" mental computers, and the other representing the inflexible mental computers. These frameworks identified factors and relationships between factors that influence mental computation. The frameworks were compared with an ideal framework that had been developed from a study of proficient mental computers. These frameworks showed that inaccuracy resulted from disconnected and deficient cognitive, metacognitive, and affective factors; and in some cases might have been affected by deficient short-term memory. It appeared that students' choices of mental strategies resulted from different forms of compensation for varying levels of deficiencies.

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Inaccurate Mental Addition and Subtraction: Two Case Studies

Researchers and educators have stressed the importance of including mental computation in number strands of mathematics curricula (e.g., Cobb & Merkel, 1989; McIntosh, 1996; Reys & Barger, 1994; Sowder, 1990; Treffers & de Moor, 1990; Willis, 1990). Reasons for its inclusion are that mental computation: (1) enables children to learn how numbers work, make decisions about procedures, and create strategies (e.g., Reys, 1985; Sowder, 1990); (2) promotes greater understanding of the structure of number and its properties (Reys, 1984); and (3) can be used as a "vehicle for promoting thinking, conjecturing, and generalizing based on conceptual understanding" (Reys & Barger, 1994, p. 31). In effect, mental computation promotes number sense (National Council of Teachers of Mathematics, 1989; Sowder, 1990). In fact, Willis (1992) suggested that mental computation should be the main form of computation, with written computation to serve as memory support.

Mental computation involves a wider range of strategies than traditional written procedures. A wide variety of mental addition and subtraction strategies has been identified in the literature (e.g., Beishuizen, 1993; Blote, Klein, & Beishuizen, 2000; Cooper, Heirdsfield, & Irons, 1996; Reys, Reys, Nohda, & Emori, 1995; Thompson & Smith, 1999). These strategies are summarized in Table 1.

The terms 1010 and u-1010 are used for separation strategies in the Dutch literature, N10 and u-N10 are used for the aggregation strategies, and N10C is used for the compensation strategy which is described here as wholistic (e.g., Blote, Klein, & Beishuizen, 2000). The strategy mental image of pen and paper algorithm is included in the table because of its presence in the literature (Reys, Reys, Nohda, & Emori, 1995). However, most literature considers mental image of pen and paper algorithm to be an inefficient strategy (Carraher, Carraher, & Schliemann, 1987; Ginsberg, Posner, & Russell, 1981; Hope, 1985; Kamii, 1989; Maier, 1977; Plunkett, 1979; Reys, Reys, Nohda, & Emori, 1995).

In terms of efficiency, Thompson and Smith (1999) classified the strategies so that aggregation and wholistic were the most sophisticated. Similarly, Heirdsfield and Cooper (1997) argued that separation right to left, separation left to right, aggregation and wholistic represented increasing levels of strategy sophistication.

While it has been posited in the literature that different strategy choice is effected by the semantic structure of word problems (e.g., Riley, Greeno, & Heller, 1983; Verschaffel & DeCorte, 1990), Blote, Klein, and Beishuizen (2000) also found that the number characteristics of problems can affect which strategy is chosen. However, some students do not consider either semantic structure of the word problem or the number characteristics; they employ a single strategy continuously. As mentioned before, this strategy is usually mental image of pen and paper algorithm.

Proficiency in mental computation has been the focus of several research projects (e.g., Beishuizen, 1993; Heirdsfield, 1996; Hope & Sherrill, 1987; McIntosh & Dole, 2000; Reys, Reys, Nohda, & Emori, 1995). In The Netherlands, where mental computation is taught before written computation, mathematics programs emphasize the use of aggregation (N10) as a more efficient mental strategy. However, weaker students tended to use less efficient separation strategies (Beishuizen, 1993). Hope and Sherrill (1987) reported that unskilled mental computers used strategies that reflected pen and paper algorithms. In contrast, skilled mental computers employed a variety of strategies that reflected understanding of number and operations. Reys, Reys, Nohda, and Emori (1995) also found that accuracy in mental computation was associated with strategies other than mental image of pen and paper algorithm. In contrast to these findings, McIntosh and Dole (2000) reported higher accuracy when students employed mental image of pen and paper algorithm than when they employed alternative mental strategies (although these alternative strategies revealed number sense). Heirdsfield (1996) also found that accuracy in mental computation did not need to be accompanied by employment of a variety of efficient mental strategies. Therefore, while some research appears to indicate that accuracy in mental computation is a result of efficient mental strategies; other research has reported accuracy as a result of employment of strategies that reflect pen and paper algorithms.

Research reported by the authors investigated mental computers and the factors that supported accuracy (Heirdsfield, 1998, 2001a, Heirdsfield & Cooper, 2002). This study investigated the part played by number sense knowledge (e.g., number facts, estimation, numeration, and effect of operation on number), metacognition (metacognitive knowledge, strategies and beliefs), affects (e.g., beliefs, attitudes), and memory (working memory and long-term memory) in mental computation. Flexibility in mental computation was defined as employment of efficient mental strategies, taking into account the number combinations to inform the mental strategy choice. The research showed that students proficient in mental computation (accurate and flexible) possessed integrated understandings of number facts (speed, accuracy, and efficient number facts strategies), numeration, and effect of operation on number. These proficient students also exhibited some metacognitive strategies and beliefs, and affects (e.g., beliefs about self and teaching) that supported their mental computation. Further, proficient mental computers had reasonable short-term recall to hold interim calculations and recall number facts (phonological loop--see Baddeley, 1986), and well developed central executive (Baddeley, 1986) to attend to the demanding task of mental computation and retrieve strategies and facts from a well-connected knowledge base in long-term memory. Proficient mental computers chose alternative and efficient strategies, as they possessed extensive and connected knowledge bases to support these strategies. Thus, there was evidence of the importance of connected knowledge, including domain specific knowledge, and metacognitive strategies, affects and memory for proficient mental computation. As a result of this study, a conceptual framework identifying associated factors involved in proficient mental computation (see Figure 1).

[FIGURE 1 OMITTED]

This leads to the question as to what are the effects on mental computation of less knowledge and fewer connections? It would be expected that one effect would be less accuracy. The purpose of this paper is to report on seven students who were inaccurate in mental computation. These students were part of a study that investigated addition and subtraction mental computation in seven and eight year old students. Conceptual frameworks for these students are developed and compared with a framework for the "ideal" mental computer (flexible and accurate) that was developed in the large study from which these students were drawn (Heirdsfield, 2001b).

Method

Participants

The participants were seven students, selected from a population of sixty Year 3 children from three classrooms (see Figure 2). The students were selected on the basis of an interview that probed for accuracy and flexibility (employment of variety of strategies) in mental computation. Emma, Jane and Sarah were inaccurate, yet they employed some variety of strategies; therefore they were categorized as inaccurate and "flexible." Sarah employed the full hierarchy of strategies, but Jane and Emma employed separation only (see Table 1 for explanation of strategies). While Rosie, Vicki, Jane and Angela were also inaccurate, they employed a single strategy consistently. This strategy was mental image of pen and paper algorithm (although Rosie calculated left to right). Therefore, these four students were categorized as inaccurate and inflexible.

Instruments

The students were presented with a series of tests and in-depth interviews. These were number fact knowledge, mental computation (one-, two-, and three-digit addition and subtraction), computational estimation, numeration, effects of operation on number, and memory. Examples of tasks from the respective interviews are presented in Figure 3. In order to choose neuropsychological tests relevant to these aspects, Lezak (1995) was consulted. The neuropsychological tests, which were used with the Year 3 students, were aimed at investigating short-term recall and executive functioning. The tests were modifications of a Digit Span Test (short-term recall, addressing the phonological loop) and a maze test (addressing central executive, e.g., planning and attention).

[FIGURE 3 OMITTED]

Further, questions were asked addressing self-efficacy, beliefs, and metacognition. The students were also required to complete the Student Preference Survey (SPS) (McIntosh, 1996), to identify whether they would and could solve computational tasks mentally. In order to get a feel for classroom and home contexts, the children were encouraged to indulge in general conversation, and the teacher was invited to respond to initial and general inferences.

Procedure

The students were withdrawn from their classroom on a one to one basis, and interviewed in a quiet room. The interviews were videotaped, and each interview session lasted for no more than 30 minutes at a time. Because of the variety of aspects covered, each child received four interview sessions. The order of the interviews is presented in Figure 4.

[FIGURE 4 OMITTED]

Analysis

The analysis of the interviews incorporated three stages. Firstly, each interview for each student was analyzed separately. Secondly, relationships across interviews for each student were considered (e.g., whether understanding of the effect of operation on number was used for mental computation, whether the same number facts strategies were employed in both the number facts test and in the mental computation interview). Thirdly, analysis compared commonalities and differences across students.

Mental computation responses were analyzed for strategy choice (Table 1), flexibility, accuracy, understanding of number facts, computational estimation, numeration, and the effect of operation on number. It was also noted whether the students could access alternative mental strategies, when encouraged to do so (specific scaffolding questions were presented to the students). Number facts were analyzed for accuracy, speed and strategy choice (number fact strategies are summarized in Table 2). Estimation strategies were identified and proficiency and flexibility were noted. Analysis of students' responses to numeration tasks was based on Ross's five levels (1986), which included canonical (e.g., 54=5 tens and 4 ones) and noncanonical (e.g., 54=4 tens and 14 ones) understanding of number. Also, evidence of multiplicative understanding (e.g., ten tens are the same as one hundred) was investigated. The tasks addressing the effect of operation on number were analyzed for understanding of arithmetic properties (e.g., associativity, inverse, the effect of changing the addend and subtrahend) as they apply to computational relationships (e.g., 70-43=27, [therefore] 70-44=26).

For the analysis of the memory tests, Lezak (1995) was consulted. As so few students were interviewed, it was decided to compare individuals' raw scores for the Digit Span Test, and note any trends with memory problems evident in mental computation tasks. Evidence of any memory strategy as reported by the student and observable from review of the videotapes was also noted. The maze completion times and the number of errors, such as retracing lines or entering blind alleys, were recorded. Furthermore, strategies used and verbalizations elicited were documented. Although there might be a tenuous link between executive functioning in completing mazes and executive functioning in completing mental computation tasks, the fact that a student could attend to a task and plan would affect mental functioning in any domain. Evidence for planning and decision-making was compared with the same in the mental computation interviews.

Results

Results of inaccurate and "flexible" students

The results for the inaccurate and "flexible" students are summarized in Table 3. Emma, Jane and Sarah used some variety of strategies, mostly separation (left to right, right to left, and cumulative sum/difference) but few high order strategies; although, Sarah also attempted to employ aggregation left to right and wholistic strategies for addition in the in-depth interview.

Cognitive

The mental computation strategies that Emma, Jane and Sarah used revealed some flexibility; however, there was no consistent understanding of numeration, or effect of operation on number, and little knowledge of number facts. All three students used "buggy algorithms" for subtraction. However, Jane did not persist with the "buggy algorithm" of "take smaller from larger," when the examples became more difficult (i.e., three-digit). It appeared that she had habituated the "bug" for two-digit subtraction, but, when faced with unfamiliar three digit examples, she worked from first principles, and, as a result, used a correct strategy, separation right to left or separation left to right, sometimes resulting in success, although at other times errors occurred because of memory overload. Sarah's "buggy algorithm" resulted from incorrectly applying an addition strategy (cumulative sum) to subtraction (cumulative difference e.g., 58-36: 50-30=20, 20+6=26, 26+8=34). However, with scaffolding, she successfully employed wholistic strategies to solve some subtraction examples (e.g., 234-99: 234-100=134, 134+1=135).

During the selection interviews, all three students experienced memory problems (they reported forgetting interim calculations). However, only Emma continued to experience memory problems in the in-depth interview. Jane and Sarah did not, possibly because they used more efficient mental strategies (e.g., aggregation and wholistic) in the in-depth interview than in the selection interview. Another reason Jane might not have experienced these problems is that she used reasonably efficient number facts strategies (recall and through 10), although there were times she counted on her fingers, but this would have served as an external memory aid. Further, Jane employed cumulative sum (76+43: 7+4=11, 110+6=106, 106+3=109), where only one interim calculation need be remembered, and so, again reducing memory load.

All three students accessed alternative strategies, but with varying degrees of success. Emma could not access any alternative strategies for addition, but Jane and Sarah accessed wholistic (e.g., 45+19: 45+20-1), sometimes successfully. Sarah even went further and employed wholistic for 246+99, as an initial strategy, and without prompting. It appeared that learning had taken place. All three students attempted to use wholistic (e.g., 234-99: 234-100+1) for subtraction as an alternative strategy to their original choice of strategy, again with varying levels of success. Neither Jane nor Emma was successful in completing the subtraction examples, when employing wholistic, because of lack of understanding of the effect of changing the subtrahend. In contrast, Sarah was successful, with scaffolding.

Jane's number facts supported her mental computation, as she employed reasonably efficient derived facts strategies for interim calculations. In contrast, although Emma and Sarah employed some derived facts strategies in the number facts test, they did not use them for interim calculations. In fact, the count strategy, employed by Emma (for interim calculations in mental computation) contributed to memory problems.

Jane and Sarah employed some appropriate estimation strategies during the interview addressing estimation. However, Emma was unsuccessful and generally attempted to calculate the estimation tasks. None of the three students used estimation during mental computation to predict or check solutions. However, Jane did appear to check her solutions, but by recalculating. Neither Emma nor Sarah seemed to check their mental computation solutions. In summary, estimation did not support mental computation for Emma, Jane, and Sarah.

Emma, Jane, and Sarah exhibited some understanding of numeration, which was reflected in various aspects of mental computation. One aspect of numeration that was not mentioned in the literature, but is used in this study is labeled proximity of number. In order to access wholistic compensation, students had to first recognize that, for example, 99 is close to 100, and that 100 is an appropriate number to use, rather than, say, 98. While all three flexible students recognized "close numbers," only Sarah was successful in completing examples when applying the principle. However, Sarah demonstrated least numeration understanding of the three students, as she possessed only face value understanding in the numeration tasks (when asked about the "1" in "16," she picked up a single block to indicate that it meant merely one). However, some of her mental computation strategies reflected numeration understanding; for instance, Sarah successfully calculated 76+43 (aggregation left to right: 76+21=96, 96+20=130 [incorrect], 130+3), 246+199 (wholistic compensation: 200+100=300, 300+100=400, 400+45=445), 107-15 (separation left to right: 100-10=90, 7-5=2, 92), and 234-99 (wholistic: 234-100=134, 134+1=135). In contrast, neither Emma nor Jane exhibited understanding of the multiplicative principle, but they did have positional understanding of number. All three students showed some understanding of regrouping/renaming, mostly with the use of concrete material (MAB [Multibase Arithmetic Blocks--base 10] were provided for the numeration tasks if the students wanted to use material), particularly for noncanonical representations. In general, there was canonical and noncanonical understanding, which was necessary for some of the low-level mental computation strategies the students used.

Some understanding of the effect of operation on number supported mental computation for Sarah (e.g., the effect of changing the addend for 246+199), and to less extent for Jane, but lack of understanding for Emma was reflected in her inability to apply the principles in mental computation. Emma was unable to use the principles of the effect of changing the addend or the subtrahend in any of the mental computation tasks.

Affect and metacognition

Both Jane and Sarah stated that it was "important to be able to work things out in your head," because "you don't always have a piece of paper" (Jane) and "because they stick in my head for a long time" (Sarah), although Sarah had a great deal of difficulty having things "stick" in her head. Jane maintained that she worked out change in her head in class, further support for her belief of the value of mental computation.

Neither Jane nor Sarah held accurate perceptions of their ability in mental computation, as they stated they would calculate several examples mentally, yet they could not. On the other hand, Emma realized that she experienced difficulties, and stated that she would only complete easier addition and subtraction examples mentally.

All three students showed some metacognitive strategies. Emma assessed the level of difficulty of tasks, before calculating, and, once, evaluated a strategy (although her judgment relied only on previous success with the strategy). Jane appeared to check her solutions to some mental computation tasks, for instance, "no, that's not right." However, there was no other evidence of metacognitive strategies. Although Sarah did not appear to exhibit metacognition in mental computation, she seemed to make conscious choices of number facts strategies. In general, although there was some evidence of metacognitive strategies, they did not appear to support mental computation.

Memory

Digit span scores of 6 and above indicated good short-term recall. It is posited that any memory problems the students experienced in mental computation resulted from inefficient mental computation strategies and inefficient number facts strategies.

Students completed the mazes, although not always efficiently. Thus, there was some evidence of planning, and this may have supported flexibility in mental computation, as strategy choice would depend on some planning.

Summary

In general, Emma, Jane and Sarah lacked sufficient understanding of number facts, numeration, and effect of operation on number to support advanced mental computation strategies. As well, their estimation did not support their mental computation. Being unsuccessful with the taught written procedures, they compensated by inventing strategies, although most (but not all) of these strategies were not high-order strategies. Some numeration understanding and metacognitive strategies assisted mental calculation using alternative strategies. Emma was the only inaccurate and flexible student who held accurate perceptions of her ability (or rather, inability) to perform the tasks. Jane and Sarah believed they would be successful with the tasks, but were unable to complete many of the examples successfully.

Thus, lack of procedural understanding of the pen and paper procedures resulted in the students' inventing mental strategies. However, as they did not possess sufficient understanding of number facts and effect of operation on number, they were rarely successful. Some numeration understanding and metacognitive strategies assisted the invention of mental strategies; however, there was insufficient understanding to support high-level mental strategies.

Thus, they attempted to compensate for their lack of procedural understanding, but their knowledge was disconnected. A conceptual framework for the inaccurate and "flexible" students is presented in Figure 5. This frame-work was developed from that of the proficient mental computer (Figure 1).

[FIGURE 5 OMITTED]

Results of inaccurate and inflexible students.

The results for the inaccurate and inflexible students are summarized in Table 4. As the table shows, Rosie, Vicki, Georgia, and Angela were inaccurate and predominantly employed mental image of pen and paper algorithm. However, Rosie calculated left to right (tens first), although she imagined the vertical format of the written algorithms.

Cognitive

The mental strategies that Rosie, Vicki, Georgia, and Angela used revealed poor number facts knowledge, lack of procedural understanding of the subtraction algorithm, poor numeration understanding, poor understanding of the effect of operation on number, and little inclination to make sense of their answers. Further, Rosie and Georgia lacked procedural understanding of the addition algorithm. Most errors could be attributed to lack of conceptual and procedural understanding, and memory problems. These four students tended to maintain their poor accuracy levels or score lower in the in-depth interviews than in the selection interviews.

All four students were unsuccessful with regrouping examples for subtraction, as they used the "buggy algorithm" of "take smaller from larger." Memory problems contributed to other errors in subtraction. Rosie was unable to complete any regrouping examples in either addition or subtraction, so her accuracy levels remained less than fifty percent. Vicki also experienced difficulties with regrouping addition examples; mostly because of memory problems (she forgot the interim calculations).

Unlike Rosie, the other three students were able to access (although not always successfully) the more efficient mental strategy, wholistic, for addition and subtraction, with scaffolding. This required understanding of numeration and the effect of operation on number. However, none of these students exhibited consistent understanding in the mental computation interviews or the related interviews with regard to numeration or effect of operation on number. Further, the students still experienced memory problems when using wholistic strategies, possibly because they persisted in employing mental image of pen and paper algorithm for the interim calculation (e.g., 246+100 was calculated using mental image of pen and paper algorithm).

Rosie, Georgia, and Angela used count to complete most interim calculations, and also used count predominantly in the number facts test. In contrast, Vicki used a strategy based around ten, although she also used some count for subtraction in the number facts test. Further, Vicki generally used a strategy based on ten for interim calculations in the mental computation interview. However, she still experienced memory problems, because the process was lengthy (e.g., 75+28: 8+5? 10+5=15, 9+514, therefore, 8+5=13 ...). In Georgia's case, if she recalled the number fact, she found the calculation easier to complete; for instance, she experienced no problems remembering interim calculations for 246+99, for which she employed mental image of pen and paper algorithm, as she knew her "plus nines." However, when she did not know the number fact, she resorted to count. Thus, when number facts could be easily retrieved there was more success in mental computation. Therefore, these students' number facts did not support mental computation. Even when what would appear to be more efficient number facts strategies were used, these were lengthy and resulted in memory overload.

None of the four students used estimation to check solutions or to get a feel for the reasonableness of the answer. Computational estimation did not support mental computation in any of the students. However, Vicki, Georgia and Angela were able to solve one estimation task each.

All four students exhibited poor numeration understanding. Even when they were provided with concrete material (MAB--Multibase Arithmetic Blocks [base 10]), they were able to represent numbers in canonical form only. In general, noncanonical understanding was not evident in the four students, and it was not utilized in mental computation. Noncanonical understanding is not required for the "buggy algorithm" of "take smaller from larger" for subtraction, which all the students used. Rosie did not even recognize positional property of number. No meaning was given to individual digits (e.g., 26 was "two" and "six"). Only Vicki and Georgia exhibited multiplicative understanding. Overall, inconsistent understanding in numeration did not support mental computation.

Understanding of the effects of operation on number ranged from no understanding to some understanding in addition, and inconsistent understanding in subtraction. When the students attempted to apply some of these principles in mental computation (generally when asked to think of alternative strategies), there was some success, but performance was erratic.

Affect and Metacognition

Only Angela held a reasonably accurate perception of her ability to perform mental computation. The other three students held inaccurate perceptions, as evidenced by their responses on Student Preference Survey (SPS) and when asked if they thought they would be able to complete the in-depth interview. They stated that they would be able to solve all the mental computation tasks and yet they could not do so. Rosie and Vicki were particularly confident, although this confidence was ill-founded.

Rosie, Vicki, Georgia, and Angela agreed that it might be useful to "work things out in your head," but they seemed to equate this with number facts, rather than mental computation. Their discussions focused on saving time, and avoiding the use of pen and paper and fingers (for external memory). Yet, they depended on these forms of external memory, more than any other group of students.

None of the four students evaluated their mental computation strategies. Rosie and Georgia did not exhibit any metacognitive strategies. In contrast, Georgia checked the reasonableness of some of her subtraction answers, but only after having to explain her solutions. Some of Angela's verbalizations indicated that she evaluated tasks before calculating, for instance, before commencing the calculation for 246+199, Angela stated, "Mmm, big numbers." Then, while calculating, she was aware it was difficult, because there was "too much to remember." Although Angela exhibited some metacognitive strategies, these did not contribute to success in mental computation.

Memory

Digit Span Test scores were 5 or lower, the lowest of all the students in the study. It is posited that diminished short-term memory resulted in the students' being unable to recall interim calculations in mental computation.

To alleviate memory problems, the students resorted to using the pen and paper strategy as a form of visual memory. Georgia and Angela also used their fingers as a form of external memory to aid in counting for interim calculations.

The students could not complete the last two mazes, indicating reduced executive functioning. Although they tended to aerial trace or follow a path with their eyes before putting pen to paper, the extra care taken did not guarantee success, as they tended to retrace incorrect paths. It is posited that deficiencies in executive functioning (efficient planning, decision-making, and allocation of attention) compounded with other shortcomings in the students' knowledge base to contribute to diminished performance in mental computation.

Diminished short-term memory was reflected partially in memory load in mental computation. However, it is posited that there are other factors (poor number facts, low-level mental computation and number facts strategies) that also contributed to memory overload. Poor executive functioning resulted in lack of planning and retrieval from long-term memory of facts and strategies. Poor performance was compounded by the absence of such factors as numeration, number facts, and efficient mental strategies.

Summary

In general, poor number facts knowledge, and poor understanding of estimation, numeration, and effect of operation on number contributed to inaccuracies in mental computation, and the inability to access alternative strategies. Although Vicki, Georgia, and Angela did not exhibit such poor knowledge as Rosie, they made insufficient connections to compensate for lack of conceptual and procedural understanding. Also, poor short-term recall and diminished executive functioning compounded these deficiencies. To compensate for a poor knowledge base and memory overload, they attempted to employ a teacher-taught strategy, which required little conceptual understanding (and also might have provided a mental image to support memory). However, because of a lack of procedural understanding, errors still resulted. None of the students held accurate perceptions of their mental computation abilities. Finally, the students who were categorized as inaccurate and inflexible exhibited little or no understanding of any of the factors investigated in relation to mental computation, resulting in deficient and disconnected knowledge.

In summary, the students exhibited deficient and disconnected knowledge. To compensate, the students resorted to an automatic strategy, but lack of procedural understanding and other deficiencies resulted in inaccurate application of this strategy. A conceptual framework for the inaccurate and inflexible students is presented in Figure 6. This framework was developed from that of the proficient mental computer (Figure 1).

[FIGURE 6 OMITTED]

Comparison of "flexible" and inflexible inaccurate mental computers

Although all students were inaccurate, there were differences in the mental computation strategies they employed. These strategies appeared to be used to compensate for limited and disconnected knowledge. A comparison between the "flexible" and inflexible students is made in Figure 7.

In general, all the students lacked sufficient understanding of number facts, estimation, numeration, and effect of operation on number to support advanced mental computation strategies. Although the students were unsuccessful with the taught procedures, the "flexible" students attempted to compensate by inventing strategies, although most (but not all) of these strategies were not high-order strategies. Some numeration understanding and metacognitive strategies assisted mental calculation using alternative strategies. Thus, the flexible students attempted to compensate for their lack of procedural understanding, but their knowledge was diminished and disconnected.

On the other hand, the inflexible students attempted to compensate by employing the teacher-taught strategy, which required little conceptual understanding (and also provided a mental image to support memory). However, because of a lack of procedural understanding, errors still resulted. None of these students held accurate perceptions of their mental computation abilities. Also, poor short-term recall and diminished executive functioning compounded these deficiencies. In other words, their knowledge was so deficient and disconnected, even strategies that had been taught (but not learnt) could not be followed.

Discussion and conclusions

This study has four important findings. First, students are unsuccessful for as complex a set of reasons as they are successful, and understanding that complexity may provide direction for remediation. Each of the seven students was inaccurate with mental computation; yet, each student failed in a different way from a different structure of knowledge, using different strategies and from different perspectives and perceptions. It is evident that to remediate these students' inability to mentally compute would necessitate taking account of the particular cognition, affect, metacognition, and memory of the students.

Second, although error patterns are a useful way to study mathematical difficulties, insight can also be gained from students' use of strategies (here discussed in terms of flexible and inflexible) and from studying the knowledge they have that is associated with the topic, here, mental computation. By studying their use of strategies, the study was able to categorize students as using a variety of strategies (flexible) or only using one strategy, mental image of pen and paper algorithm (inflexible). By comparing flexible and inflexible (see Figure 7), the study was able to identify particular traits of each type of student.

Third, failure in mathematical activities is a result of active attempts to compensate for deficiencies. As was described earlier, the flexible students (Emma, Jane, and Sarah) lacked sufficient understanding of number facts, numeration, effect of operation on number, and estimation to support advanced mental computations strategies. They compensated by inventing strategies, supported by some, but minimal understanding. Similarly, compensation was also evident for the inflexible students. Poor number facts knowledge, short-term recall, and understanding of estimation, numeration and effect of operation on number contributed to inaccuracies in mental computation and the inability to access alternative strategies. Vicki, Georgia, Angela, and Rosie employed a teacher-taught strategy which required little conceptual understanding (and possibly provided a mental image to support memory) to compensate for a poor knowledge base and memory overload.

Fourth, the different forms of compensation resulted from differences in knowledge and can lead to different ways of failing. For the students in this study, the key differences between the two groups of inaccurate computers were number facts strategies, numeration understanding, metacognitive strategies, and memory.

Knowledge and compensation

It would seem obvious that knowing number facts by immediate recall would aid in mental computation, as there would be less memory load. Many of the flexible students (and to less extent the inflexible students) used efficient derived facts strategies (c.f., count) when they could not recall number facts in the number facts test. However, they resorted to count for calculating interim calculations during mental computation. The reason might lie in the extra load placed on working memory when derived facts strategies are employed. Count seems to be a more primitive strategy that the flexible and inflexible students resorted to for interim calculations. Possibly, permitting students to use pen and paper would alleviate working memory load due to having to hold all aspects of the computation in the mind. Still, students should be encouraged to develop efficient derived facts strategies, where understanding rather than speed is a focus.

Although numeration understanding was lacking in both flexible and inflexible students, some numeration understanding supported alternative low-level mental computation strategies. There were differences in the little numeration understanding the students held; flexible students had more noncanonical knowledge of numeration and more knowledge of proximity of a number than the inflexible students. This provides some explanation for the differences in mental computation behaviour; non-canonical numeration is one basis of flexibility in accurate mental computers, while proximity of number is obviously needed for higher order mental computation strategies. This appears to indicate some evidence for numeration teaching affecting mental computation. The success of teaching experiments (e.g., Buzeika, 1999; Kamii, 1989) where students are encouraged to formulate and discuss self-developed computational strategies seems to indicate that the opposite relationship, the development of efficient computational strategies, has a positive effect on the development of numeration.

It appears that some children do not (and cannot) learn the taught computational procedures, nor do they use these procedures mentally. There is evidence that these children can develop their own computational strategies, but not necessarily accurately. However, if they are encouraged to use and explain these strategies, these students might experience more success.

Implications for teaching and learning

The implications for teachers, classrooms, and curriculum materials are at two levels. At the more general level, all three have to take into account the differences between flexible and inflexible mental computers, and have identification of these two types of students connected to particular teaching and learning activities. They also have to take into account the role of basic facts, numeration operation, metacognition, and affect on mental computation.

At present, a teaching experiment is investigating the use of representations (empty number line and 100 board) to support Year 3 children's learning of mental computation strategies (Heirdsfield, ongoing). While all ability levels of Year 3 children are involved, it will be interesting to compare the pre- and post-mental computation interview data to check whether there has been improvement in the mental computation strategies used by those students who experienced greatest difficulty in the pre-interviews. Data from the pre-interviews indicated that students who were experiencing difficulty with mental computation had poor number fact knowledge and little numeration understanding. While the emphasis in the teaching experiment is on children discussing their strategies and using representations to support mental computation, it is hoped that the students also develop associated understandings (numeration and number facts) while mental strategies develop. Initial findings seem to indicate that numeration understanding has improved, but to develop number facts strategies, the students need constant encouragement to think about efficient number facts strategies, rather than resort to counting.

At a more particular level, implications for teachers, classrooms, and curriculum materials that emerge from this study are that diagnosis and remediation of student difficulties with mental computation (and therefore, the teaching of mental computation in the first place) is a more complex and idiosyncratic activity than might be expected at first glance. Even within the two categories of flexibility and inflexibility, there were subtle variations of knowledge and affect that appeared to make each student's experience with failure unique to them, and indicated that the experience was caused by a mix of knowledge, affect, and memory particular to that student. This indicates that powerful forms of remediation and teaching require more detailed diagnosis of students' performance than might be expected.

TABLE 1 Mental Strategies for Addition and Subtraction

Strategy                       Example

Counting                       28+35: 28, 29, 30, ... (count on by 1)
                               52-24: 52, 51, 50, ... (count back by 1)
Separation   Right to left     28+35: 8+5=13,
             (u-1010)                 20+30=50, 63
                               52-24: 12-4=8,
                                      40-20=20, 28 (subtractive)
                                    : 4+8=12,
                                      20+20=40, 28 (additive)
             Left to right     28+35: 20+30=50,
             (1010)                   8+5=13, 63
                               52-24: 40-20=20,
                                      12-4=8, 28 (subtractive)
                                    : 20+20=40,
                                      4+8=12, 28 (additive)
             Cumulative sum    28+35: 20+30=50,
             or difference            50+8=58,
                                      58+5=63
                               52-24: 50-20=30,
                                      30+2=32,
                                      32-4=28
Aggregation  Right to left     28+35: 28+5=33,
             (u-N10)                  33+30=63
                               52-24: 52-4=48,
                                      48-20=28 (subtractive)
                                    : 24+8=32,
                                      32+20=52, 28 (additive)
             Left to right     28+35: 28+30=58,
             (N10)                    58+5=63
                               52-24: 52-20=32,
                                      32-4=28 (subtractive)
                                    : 24+20=44,
                                      44+8=52, 28 (additive)
Wholistic    Compensation      28+35: 30+35=65,
             (N10C)                   65-2=63
                               52-24: 52-30=22,
                                      22+6=28 (subtractive)
                                    : 24+26=50,
                                      50+2=52, 26+2=28 (additive)
             Levelling         28+35: 30+33=63,
                               52-24: 58-30=28 (subtractive)
                                    : 22+28=50, 28 (additive)
Mental image of pen and        Child reports using the method taught in
paper algorithm                class, placing numbers under each other,
                               as on paper, and carrying out the
                               operation, right to left.

Flexible            Inflexible

Emma                Rosie
Jane                Vicki
Sarah               Jane
                    Angela

Figure 2. Inaccurate students selected for study.

TABLE 2. Number Fact Strategies

Strategy                                Example

Count              count all            3+2: 1, 2, 3, 4, 5.
                   count on from
                     smaller            3+2: 3, 4, 5.
                   count on from
                     larger             3+2: 4, 5.
                   count back by
                     subtrahend         8-3: 7, 6, 5.
                   count back to
                     subtrahend         8-5: 7, 6, 5; answer 3.
                   count on (for
                     subtraction)       8-5: 6, 7, 8; answer 3.
Derived facts
  strategies       use doubles          8+7: 7+7=14, 14+1=15.
                   through 10           8+5: (8+2)+3=13.
                   use another
                     fact               9+3: 9+2=11, [therefore] 9+3=12.
                   use addition (for)
                     subtraction)      15-8: 8+7=15, [therefore] 15-8=7.
                   pattern with 9       9+6: 1 less than 6 is 5, 15.
Immediate fact
  recall                                8+4: answer 12.

TABLE 3. Summary of Results for "Flexible" and Inaccurate Students

Factors                   Emma                     Jane

Mental computation:
*  Accuracy (%)           addition: 70%;           addition: 100%
                          subtraction: 60%.        subtraction: 60%.
*  Flexibility            Used some variety,       Used some variety,
                          but no higher order      but no higher order
                          strategies (all          strategies (all
                          separation).             separation).
*  Access to alternative  Attempted to apply       Attempted to apply
   strategies             wholistic for            wholistic, but
                          subtraction (only        rarely successful
                          partial solution).       (partial solution).
Number facts
*  Speed & accuracy (no   addition: 0.18/s.        addition: 0.20/s.
   correct/sec)           subtraction: 0.07/s.     subtraction: 0.14/s.
*  strategies             Recall, count, use       addition: through
                          another fact             10; subtraction: use
                                                   addition & use
                                                   another fact.
Computational estimation  Poor. Calculated         Some proficiency
                          most examples
Numeration                Some understanding       Canonical and
                          with concrete material.  noncanonical
                          Interpreted digits by    understanding,
                          face value. No evidence  mostly without
                          of understanding of      blocks. No
                          multiplicative           multiplicative
structure                 understanding.
Effect of operation on    Made no connections.     Made no connections.
number
Metacognition             Evaluated tasks before   Some evidence of
                          calculating. Generally,  checking solutions in
                          no other metacognitive   mental computation,
                          strategies. Accurate     by recalculating.
                          perception of inability  Inaccurate perception
                          to perform tasks.        of ability to perform
                                                   tasks.
Affects:                  Some evidence of belief  Believed it was
*  Beliefs                that mathematics makes   important to "work
                          sense. Not confident in  things out in your
                          ability to solve tasks,  head "Believed she
                          and could not.           was good it maths.
                                                   Believed she could
                                                   solve tasks.
Memory
*  Digit span score       Not completed.           6
*  Executive functioning                           Completed all mazes,
   maze test                                       appeared to plan.

Factors                        Sarah

Mental computation:
*  Accuracy (%)                addition: 90%; subtraction: 40%.
*  Flexibility                 Used full hierarchy of strategies for
                               addition, only.
*  Access to alternative       Accessed some alternative strategies
   strategies                  (including wholistic) with
                               scaffolding--some success
Number facts
*  Speed & accuracy (no        addition: 0.19/s. subtraction: 0.06/s.
   correct/sec)
*  strategies                  derived facts strategies
Computational estimation       Mostly calculated, but some understanding
Numeration                     Canonical (without concrete material) and
                               noncanonical (with concrete material)
                               understanding. Only positional
                               understanding. Some multiplicative
                               understanding.
Effect of operation on         Not consistent--some understanding for
number                         addition and subtraction (where minuend
                               changes).
Metacognition                  Appeared to make conscious choices of
                               number facts strategies. Inaccurate
                               perception of ability to perform tasks.
Affects:                       Believed it was important to do things
*  Beliefs                     in head. Didn't like maths. Believed she
                               could solve tasks on SPS.
Memory
*  Digit span score            6
*  Executive functioning       Completed all mazes, appeared to plan.
   maze test

TABLE 4 Summary of Results for Inflexible and Inaccurate Students

Factors                 Rosie                  Vicki

Mental computation
* Accuracy (%)          addition: 40%;         addition: 60%.
                        subtraction: 40%       subtraction: 40%
* Flexibility           Used mental image of   Used mental image of pen
                        pen and paper          and paper algorithm
                        algorithm, but left
                        to right
* Access to             None                   Attempted to apply
  alternative                                  wholistic, but rarely
  strategies                                   successful.
Number facts
* speed & accuracy      addition: 0.07/s;      addition: 0.12/s.
  (no. correct/sec)     subtraction: 0.12/s.   subtraction: 0.07/s. I
                        Slow, but accurate.    error in each of addition
                                               and subtraction
* strategies            Nearly all solved by   addition: use another
                        using count.           fact (based on 10).
                                               subtraction: use addition
                                               & count.
Computational           Poor.                  Some proficiency
estimation
Numeration              Poor understanding     No noncanonical
                        even with concrete     understanding Appeared
                        materials. Positional  to use multiplicative
                        property only.         structure.
Effect of operation on  No connections.        Poor, Few connections
number                                         made
Metacognition           No evidence of         Checked reasonableness of
                        strategies Inaccurate  some solutions Inaccurate
                        ability.               perception of ability.
Affects:
* Beliefs               Any answer will do     Liked maths. Believed she
                        Mathematics need not   could solve all SPS
                        make sense. Number     tasks.
                        facts are not
                        important. Confident
                        she could solve
                        tasks. Became
                        agitated when could
                        not solve tasks.
Memory
* Digit span score      Not completed          5
* Executive                                    Could not complete last 2
  functioning                                  mazes. Retraced wrong
  maze test                                    paths several times.

Factors                 Georgia                      Angela

Mental computation
* Accuracy (%)          addition: 70%;               addition: 70%;
                        subtraction: 30%.            subtraction: 20%.
* Flexibility           Used mental image of pen     Used mental image
                        and paper algorithm.          of pen and paper
                                                     algorithm.
* Access to             Encouraged to use            Encouraged to use
  alternative           wholistic, but completely    wholistic, but
  strategies            unsuccessful.                generally
                                                     unsuccessful.
Number facts
* speed & accuracy      addition: 0.15/s;            addition: 0.12/s;
  (no. correct/sec)     subtraction: 0.06/s. 3.      subtraction: 0.05/s
                        errors in subtraction.       3 errors in
                                                     subtraction.
* strategies            addition: count, use         addition: recall &
                        another fact, recall,        count subtraction:
                        subtraction: count, use      count, use another
                        another fact.                fact.
Computational           Poor, but one estimation     Poor, but solved
estimation              strategy used successfully,  one example.
                        once.
Numeration              No noncanonical              No noncanonical
                        understanding Appeared to    understanding
                        have multiplicative          trades all or none.
                        structure                    Multiplicative
                                                     principle possibly
                                                     understood.
Effect of operation on  No connections for           Successful when
number                  subtraction. Was able to     addend, subtrahend
                        use effect of changing       or minuend changed!
                        addend.
Metacognition           No evidence of strategies.   Evaluated some
                        Inaccurate perception of     tasks and some
                        ability.                     solutions.
                                                     Reasonably accurate
                                                     perception of
                                                     ability.
Affects:
* Beliefs               Believed she was good at     Liked maths.
                        maths, because she "usually
                        gets the right answer"
                        Thought it was important to
                        be able to work things out
                        in head.
Memory
* Digit span score      4                            4
* Executive             Could not complete last 2    Could not complete
  functioning           mazes Drew paths before      last 2 mazes Drew
  maze lest             putting pencil to paper.     paths before
                                                     putting pencil to
                                                     paper Very hesitant

Not accurate and flexible (n=3)    Not accurate and inflexible (n=4)

Mental computation:                Mental computation:
* Strategies: Separation           * Strategies: Used mental image of
  strategies only.                   pen and paper algorithm.
* Alternative strategies: yes,     * Alternative strategies: yes, but
  but not always successful.         rarely successful
Number facts:                      Number facts:
* Accuracy: accurate for           * Accuracy: inaccurate and slow
  addition, generally slow and
  inaccurate for subtraction
* Strategies: derived facts        * Strategies: derived facts
  strategies, recall, and count.     strategies, recall, and count. Used
  Used count mostly in mental        count mostly in mental
                                     calculations.
Numeration: Varied, canonical,     Numeration: Generally poor, mostly
  noncanonical, proximity of         only canonical understanding with
  numbers.                           material.
Effect of operation on number:     Number and operation: Mostly poor.
  Mostly poor.
Metacognition: some strategies,    Metacognition: mostly no strategies,
  mostly inaccurate beliefs.         inaccurate beliefs.
Affects: Varied beliefs and no     Affects: varied beliefs.
  strong beliefs evident.
Memory: Evidence of phonological   Memory: Evidence of diminished
  loop and central executive.        phonological loop and central
                                     executive. Appeared to use visual
                                     image, but did not support mental
                                     computation.
Digit Span Test scores:            Digit Span Test scores:
  [greater than or equal to] 6       [less than or equal to] 5

Figure 7. Comparison of "flexible" and inflexible inaccurate mental
computers.

REFERENCES

Baddeley, A. D. (1986). Working memory. Oxford: Oxford University Press.

Beishuizen, M. (1993). Mental strategies and materials or models for addition and subtraction up to 100 in Dutch second grades. Journal for Research in Mathematics Education, 24(4), 294-323.

Blote, A. W., Klein, A. S., & Beishuizen, M. (2000). Mental computation and conceptual understanding. Learning and Instruction, 10, 221-247.

Carraher, T. N., Carraher, D. W., & Schliemann, A. D. (1987). Written and oral mathematics. Journal for Research in Mathematics Education, 18(2), 83-87.

Cobb, P., & Merkel, G. (1989). Thinking strategies: Teaching arithmetic through problem solving. In P. R. Trafton & A. P. Schulte (Eds.), New directions for elementary school mathematics. 1989 yearbook (pp. 70-84). Reston: National Council of Teachers of Mathematics.

Cooper, T. J., Heirdsfield, A. M., & 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, Inc.

Ginsburg, H. P., Posner, J. K., & Russell, R. L. (1981). The development of mental addition as a function of schooling and culture. Journal of Cross Cultural Psychology, 12(2), 163-178.

Heirdsfield, A. M. (1996). Mental computation, computational estimation, and number fact knowledge for addition and subtraction in year 4 children. Unpublished master's thesis, Queensland University of Technology, Brisbane.

Heirdsfield, A. M. (1998, July). Flexible/inflexible: Clare and Mandy's story. In C. Kanes, M. Goos, & E. Warren. (Eds.), Teaching mathematics in new times (pp. 241-249). Gold Coast, Australia: Mathematics Education Research Group of Australasia (MERGA).

Heirdsfield, A. M. (2001a, July). Integration and compensation in accurate mental computation. In J. Bobis, B. Perry, & M. Mitchelmore (Eds.), Numeracy and beyond (Vol 2, pp. 273-280). Sydney: MERGA.

Heirdsfield, A. M. (2001b). Mental computation: The identification of associated cognitive, metacognitive and affective factors. Unpublished doctoral thesis, Queensland University of Technology, Brisbane.

Heirdsfield, A. M., & Cooper, T. J. (1997, November). The architecture of mental addition and subtraction. Paper presented at the annual conference of the Australian Association for Research in Education, Brisbane, Australia. Retrieved September 25, 2003, from http://www.aare.edu.au/97pap/heira395.htm

Heirdsfield, A. M., & Cooper, T. J. (2002). Flexibility and inflexibility in accurate mental addition and subtraction: Two case studies. Journal of Mathematical Behavior, 21, 57-74.

Hope, J. A. (1985). Unravelling the mysteries of expert mental calculation. Educational Studies in Mathematics, 16, 355-374.

Hope, J. A., & Sherrill, J. M. (1987). Characteristics of unskilled and skilled mental calculators. Journal for Research in Mathematics Education, 18(2), 98-111.

Kamii, C. (1989). Young children continue to reinvent arithmetic--2nd grade: Implications of Piaget's theory. New York: Teachers College Press.

Lezak, M. D. (1995). Neuropsychological assessment. Third edition. New York: Oxford University Press.

Maier, B. (1977). Folk math. Instructor, 86(6), 84-89, 92.

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., & Dole, S. (2000). Mental computation, number sense and general mathematics ability: Are they linked? In J. Bana & A. Chapman (Eds.), Mathematics education beyond 2000 (pp. 40 1-408). Perth: MERGA.

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

Plunkett, S. (1979). Decomposition and all that rot. Mathematics in Schools, 8(3), 2-5.

Reys, B. J. (1985). Mental computation. Arithmetic Teacher, 32(6), 43-46.

Reys, B. J., & Barger, R. H. (1994). Mental computation: Issues from the United States perspective. In R. B. Reys & N. Nohda (Eds.), Computational alternatives for the twenty-first century (pp. 31-47). Reston, Virginia: The National Council of Teachers of Mathematics.

Reys, R. E. (1984). Mental computation and estimation: Past, present and future. Elementary School Journal, 84(5), 546-557.

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), 304-326.

Riley, M. S., Greeno, J. G., & Heller, J. I. (1983). Development of children's problem solving ability in arithmetic. In H. P. Ginsburg (Ed.), The development of mathematical thinking (pp. 153-196). New York: Academic Press.

Ross, S. H. (1986). The development of children's place value numeration concepts in grades two through five. Paper presented at the annual meeting of the American Educational Research Association. San Francisco, April.

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

Thompson, I., & Smith, F. (1999). Mental calculation strategies for the addition and subtraction of 2-digit numbers. Final report. University of Newcastle, Newcastle upon Tyne.

Treffers, A., & de Moor, E. (1990). Towards a national mathematics curriculum for the elementary school. Part 2. Basic skills and written computation. Zwijssen, Tilburg, The Netherlands.

Verschaffel, L., & DeCorte, B. (1990). Do non-semantic factors also influence the solution process of addition and subtraction word problems? In H. Mandl, E. DeCorte, N. Bennett, & H. E. Friedrich (Eds.), Learning and instruction: European research in an international context (pp. 415-429). Oxford: Pergamon.

Willis, S. (1990). Numeracy and society: The shifting ground. In S. Willis (Ed.), Being numerate: What counts? (pp. 1-23). Melbourne: ACER.

Willis, S. (1992). The national statement on mathematics for Australian schools: A perspective on computation. In C. J. Irons (Ed.), Challenging children to think when they compute (pp. 1-13). Brisbane: Centre for Mathematics and Science Education, Queensland University of Technology.

Ann M. Heirdsfield and Tom J. Cooper

Queensland University of Technology