Case Study Errors In Childrens Subtraction Education Essay

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It has been assumed that errors made by children in their subtraction are due to their inability to remember correct number facts, laziness and lack of concentration of the child (Downes and Paling 1958). However, it was later found that it is due to their incorrect strategies rather than their incorrect recall of number facts (Young and O'Shea 1981). A production system model called Prolog is presented in order to model errors made by children in their subtraction. It was expected to be found that children's errors could be modelled by either including or excluding buggy rules from the correct subtraction method which would lead to the production of the same errors made by the child. A total of 6 subtraction errors were taken from Young and O'Shea's (1981) model and entered into the program as buggy rules. The results highlight that children follow step-by-step rules with cognitive skills indicating human behaviour is rule governed. The implications of this on teaching are also discussed.

Introduction

The analysis of cognitive errors made by children has not only been an interest to teachers but also of great interest in the field of cognitive psychology and science for investigating cognitive activities underlying problem solving and reasoning (Greeno and Simon 1984). The work of Anderson (1983) helped establish that human behaviour is rule governed, however, previous studies examining children's arithmetic qualities have stated that errors made by children in their subtraction are due to a number of factors such as their inability to remember correct number facts, laziness and lack of concentration of the child (Downes and Paling 1958). These findings are consistent with the work of Thyne (1954) who highlighted where the failure exists in the process of recalling or determining the number facts. Research has also shown that it is more difficult for children to carry out subtraction than addition (Cox 1974, Ginsburg 1977, Reiss 1943) and that the equation becomes more complex for them to solve when borrowing is required (Cox 1974, Ellis 1972, Graeber and Wallace 1977).

After careful analysis of the errors made by individuals, Lankford (1972) found that these errors result from a failure in the process of executing the subtraction also known as erroneous algorithms or rational errors rather than carelessness or faulty number facts (Ashlock 1976, Brown and VanLehn 1980, Cox 1985). These erroneous algorithms are rule-governed and are applied systematically and deliberately rather than randomly. This means that the child is correctly following a rule however, it is not the correct rule allowing the child to produce an erroneous solution. In line with this VanLehn (1986) introduced the "induction hypothesis" the notion that when a child encounters a problem where they don't know how to solve it, they overspecialize rules from similar examples in order to come up with a result.

Other research shows that inversion errors also known as the smaller-from-larger bug (VanLehn 1983) are the most common types of errors for primary school children, these occur when the child subtracts the smaller number from the larger instead of borrowing (Cox 1974, Smith 1988, Ellis1972). This is due to the child's failure in recalling the correct number fact in subtraction errors (Engelhardt 1977, Morton 1925, Williams and Whitaker 1937). Cebulski (1982) suggested three possible causes for children making inversion errors; the child has no knowledge regarding the borrowing procedure and therefore uses the only available alternative they can think of, the child may not have recognized where borrowing was appropriate and finally the child is not motivated to apply their borrowing skill suggesting that children may apply an inversion strategy because it is easier, allowing them to finish more quickly.

According to Brown and Burton (1978) errors in procedures are caused by faulty algorithms (buggy algorithms), the idea that students are competent rule followers but they often follow the wrong rule due to a faulty version of a component of their skill and invented a diagnostic model known as the "procedural net". They developed a computer program called Buggy to detect bugs in student's subtraction procedures. If the answers of the student were correct then buggy would conclude that the student was applying the correct procedure however, if they made errors then buggy would find one bug that could account for them all or would evaluate all possible combinations that could account for the errors. In response to Brown and Burton, Young and O'shea (1981) produced a production system model to simulate children's subtraction errors. The production system is made up of a set of condition-action rules which they used firstly to model correct subtraction and then added or extracted rules from the correct production system in order to replicate how children produce erroneous algorithms. They also stated that there is no need to include buggy subtraction rules in the PS and found that the errors made in Brown and Burtons study could be explained by including the rules of zero-pattern errors and extracting the correct rules from the PS.

In addition to this Brown and VanLehn (1980) introduced the repair theory which is a theory driven computational model of rational errors that suggests students master a prefix of subtraction algorithm which makes up their core procedure for subtraction, however, when the student reaches a point in the problem solving process where the core procedure lacks the rule to continue an impasse occurs whereby the student will apply various strategies to overcome it. These strategies are called repairs which will modify the core procedure resulting in a rational error.

The ACT-R theory proposed by Anderson (1993) is a theory attempting to understand human cognition and puts forward the idea that production rules are the realisation of cognitive skills and suggests that ACT-R is the correct production system theory. A production system (PS) is a computer program used to provide some form of artificial intelligence, which includes a set of rules about behaviour; "condition-action" rules. ACT-R consists of two forms of long term memory; procedural knowledge and declarative knowledge which compose arithmetic facts that can be later recalled. In this instance to effectively understand the nature of erroneous algorithms several computational models have been developed, originally introduced by Simon and Newell (1972) as psychological models. Previous research on subtraction errors using PS models have demonstrated that this method offers a natural way of presenting how subtraction behaviour includes a number of different strategies (Young 1977).

In consistent with the presented research the present study argues that errors made by children in their subtraction are due to rule-governed behaviour and not carelessness. A PS model called Prolog is presented in order to replicate the subtraction errors made by children. The PS consists of a collection of rules each rule has two parts; a condition part and an action part. It is expected to be found that errors made by children can be replicated by either including or omitting buggy subtraction rules in the PS.

Methodology

Children's subtraction errors were modelled by including buggy subtraction rules or extracting some rules from the PS. Firstly number facts were entered into Prolog and then the rules, to begin with the correct method of subtraction was modelled, the number facts and correct subtraction rules are as follow:

minus(4,2,2).

minus(5,3,2).

subtraction(TopLeft, TopRight, BottomLeft, BottomRight):-

minus(TopRight, BottomRight, AnswerRight),

minus(TopLeft, BottomLeft, AnswerLeft),

writeit(TopLeft, TopRight, BottomLeft, BottomRight, AnswerLeft, AnswerRight).

The correct borrowing procedure was also modelled for which the number facts and correct borrow rules are as follow:

minus(13,5,8).

minus(4,1,3).

minus(3,2,1).

borrow(TopLeft, TopRight, BottomLeft, BottomRight):-

add(10, TopRight, NewTopRight),

minus(NewTopRight, BottomRight, AnswerRight),

minus(TopLeft, 1, NewTopLeft),

minus(NewTopLeft, BottomLeft, AnswerLeft),

writeit(TopLeft, TopRight, BottomLeft, BottomRight, AnswerLeft, AnswerRight).

6 examples of subtraction errors were taken from Young and O'Shea's (1981) paper, these can be found in appendix. The following are a few examples:

Subtraction A:

minus (4, 3, 1).

minus(6, 4, 2 ).

Smaller from Larger (TopLeft, TopRight, BottomLeft, BottomRight):-

Minus (BottomRight, TopRight, AnswerRight),

Minus (TopLeft, BottomLeft, AnswerLeft),

writeit(TopLeft, TopRight, BottomLeft, BottomRight, AnswerLeft, AnswerRight).

In this subtraction, the child incorrectly subtracts the smaller number from the larger one.

Subtraction C:

add( 10, 1, 11 ).

minus( 11, 9, 2 ).

minus( 2, 1, 1 ).

add( 1, 1, 2 ).

add_bug (TopLeft, TopRight, BottomLeft, BottomRight):-

add (10, TopRight, NewTopRight),

minus (NewTopRight, BottomRight, AnswerRight),

minus (TopLeft, 1, NewTopLeft),

add (NewTopLeft, BottomLeft, AnswerLeft),

writeit (TopLeft, TopRight, BottomLeft, BottomRight, AnswerLeft, AnswerRight).

In this subtraction the child correctly borrows from the top left and adds 10 to the 1 which makes 11 however, instead of subtracting the child adds the top and bottom left allowing them to produce an incorrect answer.

Subtraction E:

minus( 7, 4, 3 ).

zero_errorE (TopLeft, 0, BottomLeft, BottomRight):-

AnswerRight=BottomRight,

minus (TopLeft, BottomLeft, AnswerLeft),

writeit (TopLeft, 0, BottomLeft, BottomRight, AnswerLeft, AnswerRight).

In this subtraction instead of subtracting from 7 from the zero, the child writes down the same number that was in the bottom row as the answer.

Results

The subtractions were entered into the program listener window which displayed how the child would write the answer, for example the following was entered as the output query for the correct subtraction rule:

?- subtraction(4, 5, 2, 3 ).

45

-23

__

22

yes

Similarly, the output achieved for the borrow rule query was:

?- subtraction(4, 5, 2, 3 ).

63

-44

__

21

yes

The outputs achieved for subtraction A, C and E was as follow:

Subtraction A:

?- subtraction(4, 5, 2, 3 ).

63

-44

__

21

yes

Subtraction C:

?- add_bug( 2, 1, 1, 9 ).

21

-19

__

22

yes

Subtraction E:

?- zero_errorE( 7, 0, 4, 7 ).

70

-47

__

37

yes

Discussion

The present study aimed to replicate children's subtraction errors by either including or extracting rules from the PS. From the output achieved we can see that the same errors were produced therefore the model was successful in modelling the errors produced by children and in displaying the cognitive processes and the rules involved in subtraction. This study highlights the implications of using cognitive models in order to understand a child's problem solving skills and is also important from an educational standpoint as it portrays knowledge of how children produce errors in their subtraction and this knowledge can be used as basis for guiding them to the correct method of problem-solving. Errors can also be reduced by using techniques of systematic practice and review as well as increasing student's fluency in declarative and procedural knowledge (Hasselbring et al 1988). The study also supports the notion that human behaviour is rule-governed therefore children can be taught when to apply rules.

Although each model is useful in its own way, there are drawbacks to many of them; the theory proposed by Brown and Vanlehn is useful in the sense that it describes the psychological process in which a rational error is produced and gives a clear description of errors and can predict the performance of students over a range of problems rather than just describing their errors. The majority of the bugs presented by Brown and Burton had zeros presented in the top number therefore the bugs were arising by either borrowing or subtracting from a column with a zero in the top number. However, there is nothing surprising about this as borrowing from a zero is the hardest part of the equation. Although Young and O'sheas production system model is useful in describing a step-by-step analysis of a child's problem solving process it does not explain theoretically the motivation for why certain rules are added or deleted from the correct algorithm in order to replicate the errors the child made. They are simply chosen because they best match the student's performance. The model produced in this study is similar to those of Brown and Burton (1978) and Young and O'Shea (1981) as it shares their idea that errors made by children in their subtraction are due to them following an incorrect rule and not because they are careless and unable to perform the subtraction. It could be suggested that our model works best in comparison to earlier models because of the fact that it considered the earlier works of Young and O'shea as well as Brown and Burton's theory.

The use of cognitive modelling helps explain how and why certain cognitive processes cause the behaviour that we observe. The amalgamation of the individual rules makes production system models most suitable for describing the learning of a skill because of the fact that its growth can be measured by the acquirement of new rules. It also provides a natural way of representing the way that subtraction behaviour includes a mixture of different strategies. However, Dutton and Starbuck (1971) have stated that when a model becomes complete, it becomes less understandable. It has been made obvious that with each error produced, we can write a production system which will reproduce that particular error, however to write a separate production system for each error will be of no psychological value because a production system is supposed to have a validity as a representation of a child's subtraction ability and therefore should be able to account for the child's performance over a range of problems. Another limitation to this study is that children in Britain are taught two different methods of doing subtraction (Williams 1971) however; the present study only looks at one. The assumption that children do not make mistakes with number facts was also made regardless of the fact that some may do, this is also a drawback to the study. Not everything was modelled and the possibility of a random error occurring was not considered.

The main drawback of cognitive models is that they do not explain for the gap between the intentions and that of actual behaviour, which consists of social and emotional psychological factors that influence behaviour. Although our model highlights that children do not commit errors due to incapability which is consistent with earlier findings, it fails to give an explanation for why children commit the specific errors.

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