Psychology

Previous research by Brown and Burton, and young and Oshea have revealed some systematic nature of problems encountered by children in performing subtractions that more so those that involve multicolumn (Young, 1984153). This paper will focus on studying the systematic nature of these problems that are encountered by children in performing subtraction. In order to achieve this, a model was developed using the Amzi prolog based on examples originating from the young and Oshea model in a previous research.  The model was used to support previous research findings from the production system proposed by Young and Oshea. It its findings, the research established that the errors that the children make are systematic in nature (Young, 1984154). The model developed from the Amzi prolog was used to demonstrate these errors. Through the program it was established that by simply changing or omitting a given procedure, the results obtained would also change in a fashion similar to that exhibited by children in solving problems involving subtraction. The resulted hence demonstrated the earlier assumption made that the problems that the children exhibit are not only systematic but follow a clear procedural nature that can be easily traced.

Young and Oshea in his article acknowledge that childrens errors arise from improper strategies rather than incorrectly recalling number facts. The article resents a production system consistent with earlier nature of cognitive skills. The article clearly elaborates strategies used by children that result to the answers they eventually get whether correct or incorrect. Young and Osheas article, through giving of well articulated examples systematically examined the nature of errors that children make in computing subtraction problems.

Computerized programs are known for their systematic nature. Data input into these programs follow a systematic nature to produce the desired results. Procedural deviations are known to result into undesired outputs. Through use of the Amzi program, this research analyzes the systematic nature of the problems and the resultant effect of omissions or procedural deviations. It establishes the relation of these deviations to the errors already discussed as being exhibited by children.

In Britain, subtraction teaching takes a two way approach namely the decomposition and the equal addition methods (Brown, 1978176). Both methods take the same approach in terms of column arrangement with the operation originating from the right side to the left. Their difference arises from the methods used in borrowing. This research used the Amzi prolog program language to identify the errors that result in applying these methods of solving subtraction errors.

The research put focus on establishing the validity of the theories put forward by Young and Oshea as to the systematic nature errors exhibited by children in solving subtraction problems. Through careful analysis of these errors, the research will be able to recommend possible solutions to these problems.  By establishing the trends that are a result from certain procedural undoing, the research will be able to establish whether it is possible to prescribe similar solutions to childrens errors that display systematic nature of occurrence.

The research aimed to ascertain whether their existence some practical proof that the errors experienced by children in subtraction are actually procedural as proposed by previous research findings of Young and Oshea. Through establishment of this proof, the research would wield the much needed ability in disputing earlier theories that childrens errors are as a result of failures in the process of recalling and determining number facts.

Previous research in the past had proposed that childrens errors basically originate from failure on their part to recall and determine number facts. In Britain, there exist clearly articulated rules used in teaching subtraction amongst children (Young, 1981160). In both rules there exists the minuend, the subtrahend and the difference as illustrated below

The basic and most fundamental rule is that the columns are to be processed from right to left as illustrated below

Whenever the subtrahend digit S presented is greater than the minuend digit M, ten is borrowed from the minuend on the left and subsequently the minuend on the left is decreased by 1 or alternatively the subtrahend increased by one depending on the rue used e.g. in the case mentioned above ten is borrowed from six and added to (3). Subsequently (6) is decreased by one.

Young and Oshea analyzed the various errors that result from omissions or deviations of any of the procedures mentioned. The production system theory involved in his article is supported by analysis of over 1500 subtraction errors by 10 year old children (Young, 1981153). In essence the findings revealed that up to two thirds of the problems originated from systematic errors rather than problems with number recalling (Young, 1981153). Ashlock in his writing stated that the errors fall into clear patterns and can be attributed to erroneous algorithms (Alfred, 197443).

In his research, Young and Oshea expressed the procedure fro correct subtraction as an information processing model which cast in the form of rules with errors accounted for mainly focusing on the omission of one or a combination of the rules (Young, 1981154). The following is a set of examples from the Oshea model.

Modeling errors was deduced from children who had been taught the decomposition method. Their errors were classified according their systematic nature of occurrence. Young and Oshea then concluded that childrens errors were in deed systematic in nature and could be attributed to procedural errors. This formed the basis of this model as it meant to establish the ascertain Young and Osheas assumption about the errors being procedural.

Methodology
The program was developed systematically through careful incorporation of the rules necessary for subtraction. The variables, in this case the numbers involved in subtraction were written as follows
 13  5  8 was represented as minus (13, 5, 8)
8  5  3 was written as minus (8, 5. 3)
The rules were displayed in the program as shown
Example 1
654421subtraction(TopLeft, TopRight, BottomLeft, BottomRight)-
minus(TopRight, BottomRight, AnswerRight),
minus(TopLeft, BottomLeft, AnswerLeft),
writeit(TopLeft, TopRight, BottomLeft, BottomRight, AnswerLeft, AnswerRight).

This case represented a child starting with the right hand side of the column. The child deducts the bottom right from the top right to get the answer right. The child then deducts the bottom left from the top left and gets the answer left. This sin the program appears as (6, 5, 4, 4, 2, 1) represented by the statement writeit(TopLeft, TopRight, BottomLeft, BottomRight, AnswerLeft, AnswerRight).

Example 2
23111922borrow(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).

This is a representation of subtraction involving borrowing. The child adds 10 to the top right to get a new value for top right. The child then deducts the bottom right from the new top right to get the answer right. The child then deducts 1 from the top left value to get a new top left value after which the bottom left value is deducted from the new top left value to give the bottom left answer.
The above mentioned examples are cases where the child followed the procedures required hence the results produced were all right. Talking a look at instances where the rules were deviated from.

Example 3
9816-45234
borrow(TopLeft, TopRight, BottomLeft, BottomRight)-
add(10, TopRight, NewTopRight),
minus(TopRight, BottomRight, AnswerRight),
minus(TopLeft, 1, NewTopLeft),
add(1, bottom left, NewBottom left
            minus(NewTopLeft, NewBottomLeft, AnswerLeft),
writeit(TopLeft, TopRight, BottomLeft, BottomRight, AnswerLeft, AnswerRight).

In this case, the child deviates from the rules. The child unnecessarily borrows and adds ten to the top right figure. However he finds no use fro his borrowed 10 and instead uses the old figures by subtracting the bottom right to the top right to get 4. The child then unnecessarily deducts I from the top left to get 8 and further goes ahead to add one to the bottom left. The child finally subtracts the new bottom left from the new top left to get 5. The deviations from the rules results into a wrong answer.
The program also in analyzing the errors developed the following

Example 4 zero error
70-4737zero_error1(TopLeft, 0, BottomLeft, BottomRight)-
AnswerRightBottomRight,
minus(TopLeft, BottomLeft, AnswerLeft),
writeit(TopLeft, 0, BottomLeft, BottomRight, AnswerLeft, AnswerRight).

The child does not know what to do with the zero and so puts down the bottom right as the answer right. The child goes ahead to correctly subtract the bottom left from the top left to get a correct answer left.

In general the program was made up of a systematic sequence of instructions that were meant to produce results. Variations in instructions was done to establish the effect of missing out on any sequence step with the aim of establishing if the effect took similar approach as that exhibited by children in performing subtraction errors.

Results
The application of the rules already written in the prolog program resulted in the following examples
Case 1 no borrowing required, subtraction rules followed
- subtraction (4, 5, 2, 3 ).

  45
 -23
  __
  22
Yes

The above mentioned reflected the subtraction rule where systematic following of rules was applied. Subtraction originated from the right hand side with each rule adequately followed. The rule was tested numerically with each step followed and thus the Amzi prolog program confirmed the resulting answer as yes

Case 2 borrowing involved, subtraction rules followed
- borrow( 4, 3, 2, 5 ).
  43
 -25
  __
  18
Yes

The above tested the application of borrowing rules. Numerical testing of the subtraction rules involving borrowing was done with each step adequately followed. The Amzi prolog program confirmed the answer as yes
Case 3 error, always subtracting smaller from larger
- smaller_from_larger( 6, 3, 4, 4 ).
  63
 -44
  __
  21
Yes

This case tested the effect of lack of appropriate rule to address which between the subtrahend and the minuend need be deducted from the other. The rule used here represented error of simply subtracting smaller from larger where the child lacks borrowing knowledge and decides to subtract smaller from bigger. Due to the lack of proper step on handling borrowing issue, The Amzi prolog confirmed the answer as yes.
Case 4 always borrowing, mixing up subtraction rules
- always_borrow( 9, 6, 4, 2 ).
  96
 -42
  __
  34
Yes

This rule modeled instances of borrowing unnecessarily. The child always borrows on later to find no use for the borrowed ten. The child then goes ahead to reduce the left value ignoring the ten previously borrowed. The child further mixes up subtraction rules and increases the bottom left by 1. The answer was confirmed by the Amzi prolog as yes
Case 5 add bug error, always adding one to the top left instead of bottom left
- add_bug( 2, 1, 1, 9 ).
  21
 -19
  __
  22
yes

This rule was meant to demonstrate addition bug errors that children exhibit in performing subtraction. After borrowing 10 from the left rather than reduce the top left by 1, the child increases the top left by 1 always resulting to the answer demonstrated.  Upon introduction of this rule the Amzi prolog confirmed the answer as yes.
Case 6 problem with subtracting large from small
- large_from_small( 7, 2, 5, 7 ).
  72
 -57
  __
  20
Yes

This rule was used to demonstrate case errors when subtracting large from small. In instance where the bottom value is greater than the top value, the child does not know how to handle it and instead puts the answer below as zero. The Amzi prolog displayed the resultant answer as yes.


Case 7 zero error, not knowing how to handle zero on top
- zero_errorE( 7, 0, 4, 7 ).
  70
 -47
  __
  37
Yes
This rule was designed to demnostrate the  zero error. It represented a case where teh child lacks knowledge on how to handle the zero and instead decides to take the bottom right value as theright answer. The prolog program confirmed the answer as yes.
Case 8 zero error, not knowing how to handle zero on top
- zero_errorF( 7, 0, 4, 7 ).

  70
 -47
  __
  30
Yes
This rule was for modeling another kind of zero error exhibited by children in doing subtraction. In represented instances where a child lacks the knowledge on how to handle a zero present in the calculation, top right. The child then decides to put the right answer as zero. Testing this numerically resulted in the answer produced by the child being proven by the Amzi prolog as yes Discussion
The ultimate out come was considered a success in confirming the previous assumptions that were derived from the Young and Osheas model. Deletingaltering a part of the procedures in the program rather than leading to a system crash results to a behavior that basically resembles the subtraction errors made by the children.

Examples above, in contrast to earlier claims by Brown and Burton that children errors resulted from poor understanding of number facts (Brown and Burton 1978), clearly demonstrated, that the child had clear knowledge of number facts but instead lacked some procedural knowledge. The errors recorded resulted from procedures not done or poorly done like always borrowing even when unnecessary. Another procedural error recorded was the child always subtracting the small value from the large value rather than follow the procedures required in identifying what to do. Other than these procedural errors, the child had clear knowledge on how to handle number facts and adequately did problems involving number facts to give reasonable outputs. This is in line with Young and OSheas claims that children usually possess necessary number facts but mostly faults on procedural basics.
In the case where there existed zeros on top, the child got confused and ignored the application of multicolumn subtraction procedures. This left the child not knowing what to actually do with zero of which he eventually put as the answer (case 8) the child in this case moved on to the other column displaying good knowledge of number facts but by skipping a procedure was not able to get the correct answer.

Through the Amzi prolog program the model confirmed the theory put forth by Young and Oshea in which he proposed that childrens subtraction errors due to their systematic nature can be modeled using computer languages. The model was able to display mistakes similar to those by children once some procedural steps were either deleted or altered. This displayed the procedural lapses suggested by Young and Oshea being practically obtained. This thus justified Young and Osheas assumption that what children exhibit is a case of erroneous algorithms rather than lack of number facts.

While it may be argued that the program was simply modeled to suit the purpose for which it was intended, this is not true. Its systematic and well sequenced results proved its ability to face the challenge it was up to. Only something that is sequential can be modeled sequentially to produce results similar to those desires. If the program had only been modeled to produce desired results of the researcher, then the results would lack sequence and ability to match what as expected but would instead only give desired result for the sum it was meant for. It therefore is clear that the model adequately served its purpose through providing platform for arguing in support of proposals put forth by young and Oshea.