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A computer based system for developing expertise in the diagnosis and remediation of common error patterns in the domain of fractions

Rod Nason and Christopher Martin
Queensland University of Technology - Carseldine

In our mathematics education courses at Queensland University of Technology, a significant amount of time is spent in trying to develop high level diagnostic and remediation skills. One method which is used to achieve this aim is to have the student teachers analyse error patterns taken from children's work. This analysis usually proceeds in three phases. First, the student teachers are required to identify the error pattern. Then they are required to identify the underlying causes of the error pattern and state possible reasons for the children's use of the error pattern. Finally, the student teachers are required to plan a sequence of remediation activities which first would help the children understand why their answers were wrong and second would help them learn how to generate correct answers for the problems.

Most student teachers experience few problems with the first phase of the analysis; the identification of the error pattern. They however experience great difficulties with the second and third phases of the analysis; identifying the underlying causes and planning an appropriate sequence of remediation activities. This is not surprising because these two phases of the analysis require the student teachers to do more than just recall information which they have acquired from their mathematics education courses. These two phases in fact require the student teachers to utilise higher level application thinking skills based on accurate observation of working children. Because of this, most of our students tend to utilise a shotgun approach in these two phases and thus produce rather inexact and impractical diagnostic and remediation reports. For example, given an error pattern such as that in Figure 1 below, it is not uncommon for student teachers to recommend a remediation program which consists of all the activities which they have recently done in a series of workshops on fraction concepts. That this sequence of activities may cover work to be done over eight years within a school curriculum seems to be overlooked.

1/3 + 2/3 = 3/6 1/4 + 3/4 = 4/8 3/5 + 5/8 = 8/13

Figure 1: An error pattern for addition of fractions

Unfortunately, at present, the student teachers rarely have an opportunity to implement and evaluate their planned sequences of remediation activities. Many of them thus graduate without realising the inappropriateness of the shotgun approach. This is one reason why most of our graduates exhibit low levels of expertise in diagnosing and remediating mathematical learning problems.

It is in this context that the System for Diagnosis and Remediation of Mathematical Error Patterns is being developed. Unlike many other computer based systems whose major purpose is to replace or supplement human experts, the major aim of this system is to provide the users with experiences which will help them to develop such high levels of expertise, that they soon will no longer be dependent on human and/or computer experts.

In the section that follows, the architecture and the functioning of the system in its present form will be described. Following this, a description of how the system will be used by teacher education students at our college will be presented. In the final section of this paper, future directions for this research and development project will be discussed.

The present system

During the preparation of the system, a number of assumptions based on close classroom observations and laboratory research were made about how children learn and do mathematics and a number of criteria for an effective system were specified. The assumptions made about how children learn and do mathematics were:
  1. that many of the errors that children make in mathematics are systematic, and

  2. that the underlying causes of these error patterns are related to the manner in which mathematical knowledge is stored and structured within the children's long term memory.
The criteria for an effective system were:
  1. that the system's actions should parallel the actions of real children as closely as possible;

  2. that the system should give students easy access to its knowledge base so that they can better understand the underlying causes of why children use error patterns in mathematics;

  3. that the system should have psychological validity; and

  4. that the domains of knowledge explored by the system should be those in which most children experience difficulties.
One major consequence of these sets of assumptions and criteria has been that the system presently being developed is a network of production systems which focuses on the addition and subtraction of vulgar fractions with unlike denominators. Before proceeding on to a description of the system, it first would be desirable to briefly describe what a production system is and how it operates.

Production systems

Production systems were first employed by Newell & Simon (1972) in order to help in the analysis of detailed protocols of human problem solving behaviour. According to Young, the technique attempts to capture regularities in a subject's behaviour by writing a set of independent rules - known as a production system (PS) - that expresses what the subject does under what conditions (Newell, 1968; Newell & Simon, 1972). Each individual rule (or PR for 'production rule') is a condition action statement of the form C ===> A, and means simply that in the circumstances specified by C the subject performs action(s) A. (Young, 1971, p3)

The condition side of a production refers to the symbols in the Short Term Memory (STM) that represent the goals and the elements of knowledge existing in the system's momentary knowledge state. The action consists of transformations such as the generation, interruption and satisfaction of goals, modifications of existing elements and the addition of new elements to the STM.

The architecture of a classical production system (Klahr, Langley and Neches, 1987) is presented in Figure 2 below.


T1 (Temp < 70) (Furnace off) ===> Turn on Furnace
T2 (Temp > 72) (Furnace on) ===> Turn off Furnace.

(Temp = 71 ) (Furnace on) AA BB CC

Figure 2: Overall architecture of a production system

The STM holds an ordered set of symbolic expressions or chunks. New expressions always enter the STM at the front or left hand end. Thus, when (Temp > 72) enters the STM in Figure 2, the contents of the STM will be:

(Temp > 72) (Temp = 71) (Furnace on) AA BB CC.
The conditions of the production rules examine the expressions in the STM in order starting from the front. Front expressions in the STM thus may pre-empt later expressions. The LTM consists entirely of an ordered set of production rules. Each production rule is written with the condition on the left separated from the action(s) on the right by an arrow. In Figure 2, only two production rules are shown: T1 and T2.

This particular production system operates a thermostatic control device whose purpose is to keep the temperature of a room between 70-72 degrees Fahrenheit. As the system stands initially, none of the production rules in the LTM is satisfied by the contents of STM and nothing happens. However, as is shown in Figure 2, (Temp > 72) is about to enter the system from the external world. When it does, the contents of the STM will contain the two conditions necessary for the selection and the execution of Rule T2 by the system:

(Temp > 72) and (Furnace on).
The action of Rule T2 will be to turn the furnace off.

In contrast, if the contents of the STM had indicated that the furnace was off and if the incoming information from the external world was:

room temperature less than 70 (Temp < 70), then Rule T1 would have been executed. Its action would be to turn the furnace on and heat up the room.

The prototype

Although production systems have predominantly been used to analyse adult protocols, production systems have a number of properties which make them most appropriate for the present project of producing a system for the diagnosis and remediation of error patterns associated with the addition and subtraction of fractions with unlike denominators:
  1. Production Systems can serve as a miniature Artificial Subject whose reactions to different experimental manipulations can be empirically explored. The production system thus can be used to predict a child's cognitive behaviour under certain conditions much more precisely than with most other techniques.

  2. Production Systems analyse problem solving into components which specify not only what a child does but also when he (or she) does it. Most techniques, such as Gagne's technique of task analysis (Gagne, 1962, 1968) and Pascual-Leone's technique of M-Space analysis, only indicate what the child does. The solution of a problem, however, involves more than the simple linking together of a list of subtasks. Information about when and how these subtasks are linked together is just as important. This capability of production systems to specify what the child does, when s/he does it and how the child does it is therefore most useful when one is trying to develop a complete processing model.

  3. The explicit control structure of a Production System makes clear exactly which decisions the child has control over and which decisions are forced on him or her by the environment. Production Systems thus allow one to clearly indicate how the child's goals affect performance on a task. With many other techniques, this is well nigh impossible especially if one tries to simultaneously indicate how environmental factors may have affected performance.

  4. Production Systems can cope with minor variations to a Task. This enables construction of a single processing system to handle all variations of the addition and subtraction of fractions tasks rather than the construction of separate systems for each variation of the tasks as has to be done with most other techniques such as flow chart and M -Space analyses (Pascual-Leone, 1970).

  5. Production Systems are the most homogeneous form of programming organisation known to man (Newell & Simon, 1972). Each production rule represents a fragment of potential activity that is a meaningful component of the total problem solving process. Individual production rules therefore frequently possess a kind of local plausibility which enables them to be discussed sensibly in isolation. With most other techniques, the parts cannot be isolated out of the system to be discussed and modified; the system has to be discussed and modified in toto.
This structural independence of the production rules has two important consequences. Various sets of production rules can be combined freely to form working production systems. This makes it possible to show how certain production rules or schemes perform critical functions in more than one number related task. Secondly, it is possible to incrementally add new production rules or modify previous rules to extend a production system so that it can handle more complex tasks or the same tasks in a variety of new contexts. This makes it possible to identify consistencies and differences between successful and unsuccessful performances on the complex number tasks very precisely.

The use of production rules in the system being produced thus enables the authors to produce a powerful and flexible system which meets all of the criteria required of the system. As is illustrated in Figure 3 below, the prototype system consists of a network of three interrelated production subsystems: an executive subsystem (called MAINLIST), an equivalent fraction generating subsystem (called EQUIVFRACT), and a lowest common multiple generating subsystem (called LCM).

MAINLIST < - - - - - > EQUIVFRACT < - - - - > LCM

Figure 3: The prototype system

MAINLIST affects the overall operation of the system by selecting which of the other subsystems will be operational at any point in time. EQUIVFRACT generates equivalent fractions when asked to do so by MAINLIST; LCM is activated by EQUIVFRACT to generate the lowest common denominators.

For example, if the system is asked to calculate

1/2 + 1/3
MAINLIST first peruses the problem. When it finds that the two fractions have different denominators, MAINLIST activates EQUIVFRACT. Because EQUIVFRACT has not got any rules which enables it to generate the lowest common denominator, it in turn activates LCM. When LCM has determined that the lowest common denominator is 6, EQUIVFRACT is reactivated. EQUIVFRACT then converts 1/2 to 3/6 and 1/3 to 2/6. MAINLIST is then reactivated. The rules in MAINLIST then generate the answer 5/6.

The power of the system as a means to produce expert teacher diagnosticians however only really becomes apparent if one of the three following modifications are made to the system:

  1. one or two critical production rules are removed from one or more of the subsystems;

  2. one or more "buggy rules" (Van Lehn and Brown, 1980) are included in one or more of the subsystems; or

  3. a combination of the first two types of modifications occur.

PR2B If you are required to add two fractions, then
add the numerators and add the denominators.

Figure 4: "Buggy Rule" PR2B

For example, if Buggy Rule PR2B is added to MAINLIST, the following scenario could occur if the system is asked to add 1/2 and 1/3.

The system peruses the problem, notes the addition sign and immediately operationalises the addition operation rules. Rule PR2B then is executed. The system thus adds the two numerators (1 and 1) and the two denominators (2 and 3) to produce an incorrect answer of 2/5.

1/2 + 1/3 = 2/5
This, by the way, is one of the most common errors that children make when adding fractions with unlike denominators. By experimenting with modifications to the four production systems, the experimenters have been able to reproduce most of the common errors that children make when adding and subtracting fractions.

How the system will be used

As was stated earlier, by making modifications to the lists of production rules within the sub systems, it is possible to reproduce most of the error patterns that occur when children attempt to add or subtract fractions with unlike denominators.

This characteristic of the system enables the production of a very dynamic type of interactive, computer based tutorial. To maximise the effectiveness of the system, it is recommended that each tutorial should be attempted by a group of two to four student teachers. At the beginning of each tutorial, the student teachers will be presented with a worksheet of fraction problems done by an imaginary child. The student teachers first will be asked to identify the error pattern being used by the child. This is done by asking them to show on the computer screen how the child would do a problem similar to those on the worksheet. Following their correct identification of the type of error pattern, the student teachers will be asked to identify why the child is making the error and to suggest what the child needs to be offered in order to overcome the error pattern. In order to test their hypothesis about why the child is making the errors, the students will be asked to teach the relevant knowledge to the system.

The students teach the system by adding and/or removing production rules from the system. This process is done by the means of a special menu. Once the student teachers have made their changes to the system, the system reworks the original worksheet of problems. If the student teachers diagnosis and remediation plan is sound, then the system will get all the problems on the worksheet correct. If their diagnosis and proposed remediation program is unsound, then two possible scenarios can occur; the system either makes the same errors as before or a new set of error patterns appears. In either case, the student teachers will be required to revise their diagnosis and remediation program and rerun the simulation. To help the student teachers gain deeper insights into what is occurring, they will be encouraged to hack into the production system and explore how the information within the system is structured.

Because this system's ability to reproduce most of the common error patterns children use when adding and subtracting fractions with unlike denominators, it will be possible to produce a graded sequence of tutorials so that student teachers may become more adept at diagnosing and remediating most of the common errors that occur within this domain of mathematics. The first tutorials thus will simulate the simpler types of error patterns while the later tutorials will simulate the compound and more complex types of error patterns. This will be an improvement on the one off types of error pattern activities which student teachers presently do in their mathematics education courses.

Future directions

The present version of the system is written in muLisp and is still being modified and improved to make it quicker and more user friendly. Once it is fully operational, a Parser will be added to the system so that the users will be able to use almost Natural language to interact with and modify the system. The student teachers thus may be able to teach the system and explore how changes to the amount and the structure of knowledge stored in a child's Long Term Memory affects mathematical performance. This has a number of important possible consequences. First, it may facilitate the development of teachers as not just transmitters of knowledge but also as researchers. Second, it may help overcome many teachers' resistance to technology by helping them to realise that there are more educational uses for computers than just word processing.

Once the natural language version of this system has been produced and field tested, the shell produced for this system will be modified for use with other domains of knowledge such as:

  1. addition and subtraction of whole numbers;

  2. multiplication and division of whole numbers;

  3. addition and subtraction of decimal fractions;

  4. multiplication and division of decimal fractions;

  5. operations with percentages; and

  6. operations with ratios.
A set of computer simulation tutorials for most of the content domain areas in the K-10 mathematics curriculum thus will be generated from the prototype system which has been described in this paper.


Gagne, R. M. (1962). The acquisition of knowledge. Psychological Review, 69, 355-365.

Gagne, R. M. (1968). Learning hierarchies. Educational Psychologist, 6,1.

Klahr, D., Langley, P. and Neches, R. (1987). Production system models of learning and development. Cambridge, MA.: MIT Press.

Newell, A. and Simon, H. A. (1972). Human problem solving. Englewood Cliffs, NJ: Prentice-Hall.

Pascual-Leone, J. (1970). A mathematical model for transition in Piaget's developmental stages. Acta Psychologica, 32, 301-345.

Young, R. M. (1971). Production systems for children's seriation behaviour: An introduction and an example. CIP Paper No. 202. Dept of Psychology, Carnegie-Mellon University.

Authors: Rod Nason and Christopher Martin, Centre for Information Technology in Education (CITIE), Queensland University of Technology - Carseldine

Please cite as: Nason, R. and Martin, C. (1990). A computer based system for developing expertise in the diagnosis and remediation of common error patterns in the domain of fractions. In J. G. Hedberg, J. Steele and M. Mooney (Eds), Converging Technologies: Selected papers from EdTech'90, 205-211. Canberra: AJET Publications.

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