Concrete representations and the procedural analogy theory

I I I l l II~ JOURNAL OF MATHEMATICAL BEHAVIOR, 1"/(1), 33-51 ISSN 0364-0213. I¥1 Copyright © 1998 Ablex Publishing Corp. All rights of reproduction in any form reserved.

Concrete Representations and the Procedural Analogy Theory

NElL HALL

John Moores University

This paper describes a procedural analogy theory, which uses an isomorphism index to predict the likely learning outcomes when concrete materials are used in instruction in school mathematics. Literature from mathematics education and cognitive science are reviewed, the role of analogy in learning is considered, and examples of applications of the theory are provided.

Research in mathematics education has been unable to answer significant questions about the use of concrete materials in the teaching and learning of school mathematics. The value of concrete materials, the ways they best represent mathematical systems and the details of how they are supposed to help students learn mathematical concepts and skills, remain unclear. Within typical classrooms, it often seems teachers expect that the mathematical ideas embedded in these materials, and in actions on them, will be absorbed by porous and inanimate students.

For many teachers and teacher educators, the value of concrete materials in teaching and learning mathematics is more-or-less self evident. But it seems that for many teachers the purposes of these manipulative materials are not clear (Lesh, Post, & Behr, 1987a). Fre- quently, teachers tend not to realise that the materials themselves are insufficient, and that mathematical structure must be imposed on them. Lesh et al. argue, too, that teachers see mathematics as "as a collection of isolated rules for manipulating symbols" rather than an interconnected whole. In this way learners' misconceptions, shown up through the use of concrete materials, are taken as a weakness in the materials by teachers, who revert to "superficial computational proficiency" (p. 652).

It may be that this teacher uncertainty, together with the ease of pen-and-paper approaches in classrooms, in comparison to the combination of concrete materials and teacher intervention approaches, explain Scott's (1983) findings. Her survey of 800 teachers indicated that most teachers used few materials other than text books, and that concrete material usage declined as grade level increased. There is, then, a need to investigate how concrete materials actually assist learning, and to devise pedagogies that support these kinds of experiences.

Results from many studies on concrete representations are equivocal, failing to produce the positive outcomes expected. Fennema (1972) lists a number of studies from the 1950s

Direct all correspondence to: Neil Hall, School of Education, Marsh Campus, John Moores University, Liverpool L176BD, England <[email protected]>.

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and 1960s where results achieved through the use of concrete materials in teaching mathematics were inconclusive. Of the 15 studies reported seven showed no significant differences between manipulative and nonmanipulative treatments, four favoured the manipulative groups, three showed mixed results and one favoured the nonmanipulative group. Suydam and Higgins (1976) surveyed 40 studies on the use and effects of manipulative materials in teaching, and reported 24 as showing positive effects on student achievement. Freidman (1978) noted that, in the period 1970-1977, of the 18 doctoral dissertations investigating the use of manipulatives in the teaching of mathematics, only four showed significant differences favouring the manipulative groups over the nonmanipulative groups. So prior to the 1980s there were many studies investigating the impact of concrete materials on student learning, but their findings were inconclusive. This has also been the pattern of results in more recent investigations. Labinowicz (1985) reported little gain in third grade students using multibased arithmetic blocks (MABs) to develop computational skills, whereas Fuson and Briars (1990) reported using MABs to gain high levels of skills in addition and subtraction algorithms. And Wearne and Hiebert (1988) reported fourth, fifth and sixth grade students showed some gain in decimal numeration, addition and subtraction from the use of MABs. Sowell (1989) claimed that individual papers often lacked detail as to who did the teaching, what training they undertook relevant to the study, and what the teaching treatments actually involved. Fennema (1972) had earlier commented on this, when she noted that studies involving concrete materials were often inconclusive, and referred to experimental and control groups "often defined no better than this" (p. 636). And Scott and Neufeld (1976) noted that many studies gave little emphasis to reporting the nature of the concreteness used. Clarifying exactly what the treatment groups did, and how this differed from the comparison groups, remains a problem in the literature. The details of the instruction, and the specificity of what was actually compared between the groups, are frequently not reported.

Sowell (1989) reported a meta-analysis of 60 studies designed to assess the value of manipulative materials in mathematics instruction. The studies ranged from those involving kindergarten children to those in which tertiary students participated, and employed a wide range of manipulatives and mathematics topics. Sowell concluded that treatment last- ing a school year or longer favoured the manipulative groups, but only for concrete materials and not for pictorial representations. Treatments for shorter periods showed no difference between the manipulative and nonmanipulative groups on either post-test or delayed post-test scores. Sowell also reported that those studies in which instruction with concrete materials resulted in greater gains in learning than the alternative instruction, were often taught by university academics or teachers involved in long-term training on the use of such materials for instruction. That is, the more positive findings for the use of concrete materials may partly be explained by built-in bias in the instructor, or by the greater exper- tise of these researchers.

Boulton-Lewis (1992) argued that concrete materials were useful in teaching only if learners recognized "the correspondence between the structure of the materials and the structure of the concept" (p. 10). She explained the failure of concrete materials to improve student learning as a function of the cognitive processing loads required in their use. Unfa- miliarity with the materials or the analogies used, analogies that were inappropriate, and a lack of declarative or procedural knowledge, all increased processing load. This work, together with that of Bobis (1992), and a point made by Lesh, Post, and Behr (1987b),

CONCRETE REPRESENTATIONS 35

about perceptually compelling but misleading cues, suggest that teacher-introduced representations, aimed at assisting learning, may actually make learning more problematic. Con- crete materials have to be used with care in instruction. A line has to be drawn somewhere between concrete aids as a help to teaching and learning, and such materials as yet another barrier between learners and their construction of mathematical knowledge.

Resnick and Omanson (1987) sought to establish the relationship between performing arithmetic and understanding it, especially by illustrating procedural learning with "well-grounded mathematical principles." They developed a mapping instruction which was intended to maintain a step-by-step correspondence between the manipulation of blocks and the writting of symbols. They had 80 fourth, fifth and sixth grade students perform tasks, both written and using MAB materials, where representations of numbers were constructed and decomposed, and where activities involved addition with carrying, and subtraction with decomposition. After a period of instruction, posttest scores showed children taught with the mapping instruction did not differ significantly from children in the comparison group, but in a delayed posttest the mapping instruction group gained higher scores. The researchers expressed disappointment at the children's levels of achievement, and concluded that the mapping instruction was not effective in curing subtraction bugs. These findings suggest at least some of our beliefs about the value of concrete materials are questionable. Mathematics educators need to be concerned with these outcomes, especially since Resnick and Omanson's research was well designed, is frequently cited, and employed what appeared to be a detailed and sensible pedagogy--yet their use of concrete materials did not appear to lead to much in the way of positive learning outcomes.

Fuson and Briars (1990) investigated the use of MAB materials in the addition and subtraction of four-digit numbers by first and second grade pupils. This study was reported in detail, and made explicit many aspects of teacher preparation. The investigation involved teaching approaches, and lesson content, that would normally not be covered until later in elementary school, with older children. The teaching strategies used in this investigation emphasized the link between action on the blocks and written symbols, so that, for example, actions with the blocks were immediately followed by the equivalent written symbol. There was much verbalization about the blocks, in everyday English and in base ten terms. Fuson and Briars reported children performed these calculations at a level well above what was normally expected of their age group, showed little of the small-digit-from-the-large error, labelled digits in their place values, correctly changed word names to numerals and vice versa, and selected the correct digits in trading which they were able to describe in terms of place values. No specific detail is available on the actual classroom experiences of these learners since the lessons were not videotaped, and clearly there would have been differences between groups in their instructional environment. All the same, this research provides strong support for the educational value of concrete materials, and for the need to use them in particular ways to meet specific objectives. According to Fuson and Briars, these teaching approaches emphasised materials that reflected the relative size of the numbers involved, that illustrated the positional nature of place value, and that focused on both understanding and procedural competence.

Both diagrammatic and concrete representation systems have the pedagogical advan- tage of being easier to talk about, to describe and to analyse than language-based solutions. The latter are inherently abstract, and difficult to analyse in learning contexts. This view is consistent with Larkin and Simon (1987) who stressed the importance of perception in

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learning, and who claimed that the only psychological process their analysis referred to was perception. They claimed that "everything that has been learned in the past two decades about expert performance" (p. 97) suggests the importance of focusing attention on a specific part of a diagram, which can later be used to recall relevant components from memory. If learning mathematics effectively means that students develop a richly con- nected cognitive network of concepts and skills, so that their mathematical power is strengthened by one part of the network allowing access to other sections, then the use of a pedagogy involving detailed manipulation, analysis and discussion of concrete materials may provide a useful model. While I accept Kaput's (1985) point that computational solutions developed through some external memory have no logically necessary implications concerning the actual structure of human knowledge, a model of instruction employing concrete materials that explains the cognitive value of such materials, and provides teachers with insights into student learning, is worth investigating.

1. A PROCEDURAL ANALOGY THEORY

Teaching sequences involving concrete materials, whether highly teacher directed or more discovery oriented, generally begin with activities based on exploration of the properties of those materials. In this way students become familiar with the mathematical concepts represented by the materials, and by actions on them. Following these explorations, typically, is a move to some form of systematic use of the materials, and to a symbolic representation where what is written closely reflects the structure of the materials and the learners' actions on them. There may be an intermediary symbolic system, which after some time becomes formalized, before learners move to a final target procedure. It is this target procedure that reflects an expert way of completing an algorithm or problem solving sequence.

The function of a procedural analogy theory, at least in part, must be to explain for a particular teaching instance which teaching approaches, using concrete materials, are likely to be useful and which are not. That is, given that any particular concrete material can be used in teaching and learning in a variety of ways, a procedural analogy theory must explain how to select effective approaches from the range of options available. Since the materials will alter from application to application, and since the mathematical topic being taught will also vary, a procedural analogy theory will have to provide a set of general principles if it is to be useful in instruction. The purpose of the procedural analogy theory, briefly described here, is to guide instruction. Further, it seeks to provide both theoretical principles and practical guidelines as to how teaching and learning with concrete materials may bring about changes to students' cognitive structures.

1.1. Declarative Knowledge

Anderson (1983b) argued that verbal instructions require the student to take what the teacher says, and make a declarative encoding of what is heard. Such a coding is not yet executable, and will become so only when the declarative encoding is transformed into executable, procedural code. Declarative encoding could occur through the discovery of a new relationship while participating in an investigative lesson, or during some problem solving activity where two ideas which had until then been independent are now linked; but


most commonly in school learning declarative encoding takes place on hearing the teacher's description of a new concept or relationship. Further activity, whether it be through discovery, problem solving or teacher direction, allows consolidation of the new coding by making it operational. Knowing what the teacher said, or what you discovered, is not the same as being able to perform what the teacher described or to repeat a newly discovered problem solving technique. This consolidation requires that declarative encoding be followed by proceduralization, the bringing into action of what had previously only been words. The movement from declarative to procedural occurs through teacher instruction, demonstration and example. The cognitive function of teacher talk is to assist declarative encoding, and then to assist the movement from declarative knowledge to procedural knowledge. The use of concrete materials by students requires them to learn a range of declarative and procedural knowledge, knowledge which is then discarded as algorithms are developed, applied and remembered. But it is the declarative and procedural knowledge of concrete materials that provides the vehicle through which learners construct declarative and procedural knowledge about symbol systems. Here, the prior learning about concrete materials makes new learning about symbol systems easier and more meaningful.

Declarative knowledge may include facts, events and generalisations. It is descriptive and constructed of propositions. Procedural knowledge includes strategies, tactics, heuris- tics, plans and the like, and so is prescriptive. Declarative knowledge consists of generalisable facts, but it is limited in that the learner may be able to repeat them to the teacher, but will be unable to put them into practice. Their application requires procedures that are domain or instance-specific (Millward, 1980; Ohlsson, 1991). That is, the learner in a sense "knows" what the teacher said, is aware of the facts, but is unable to operationalise it (Mos- tow, 1983; Neves & Anderson, 1981). The initial declarative encoding is important, since it is inefficient if students have to unlearn incorrect information in a teaching sequence. Yet there are few ways of investigating that the declarative encoding has taken place correctly. Certainly the teacher can ask students to repeat what was said, to write it down, to read it out, to paraphrase it. But all this is just words. We have little control over how students actually encode and store this new data. If declarative encoding does not contain the right information, the proceduralisation process cannot generate the correct procedure (Ohlsson & Hall, 1990), so that errors will quickly develop, and further work is likely to compound these misconceptions. Procedural knowledge is the knowledge needed to put what the student knows declaratively into practice, by following a set of rules. It is procedural knowledge that allows rapid response to everyday situations. These distinctions between declarative and procedural knowledge are important because they illustrate that learning is not a simple one-step process, where knowledge is somehow fed directly into the student by the teacher. They illustrate the need in instruction both to explain clearly and to help operationalise whatever is being taught.

1.2. Procedural Knowledge

Transformation from declarative to procedural encoding can be quite complicated. For example, the teacher must encourage the student to link the declarative content to the relevant goals, when there are many other possible transformations. Indeed one of the consistent findings about novice learners in a domain is that, given the opportunity to read instructions of modest complexity, they are not able to perform the described operations

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without error (Anderson, 1986). Instructions are insufficient to set up procedures to perform skills. Interpretive productions must convert this knowledge into behaviour. Ander- son describes the importance of procedural skills in terms of increasing efficiency of completing tasks, "proceduralisation is a process that builds specialised versions of productions by eliminating retrieval of information from long term memory...the information that would be retrieved from long term memory is encoded directly into the specialised version of the production" (1983a, p. 207). That is, Anderson is claiming that procedural skills are important in that they allow learners to proceduralise solutions to problems and algorithms in an efficient manner, and that this proceduralisation minimises the need to refer to long term memory in order to complete the process. This view is consistent with the role of practice, with the notion of increasing performance speed (Newell & Rosenbloom, 1981), with the importance of chunking steps in procedures (Rosenbloom & Newell, 1982a, 1982b, 1986), and with the importance of automatisation of procedures (Schneider & Fisk, 1983). Proceduralisation allows learners to perform operations, and through a process of simplification and chunking allows sets of procedures to be collapsed into one procedure. This means a sequence of actions, to solve a particular problem, can be reduce to a smaller number of procedures, and in some instances without having to refer to long term memory (Anderson, 1986).

The typical sequence in using concrete materials described previously, where exploration of the materials is followed by a more systematic use leading to the development of a written algorithms, involves many instances of declarative and procedural knowledge. As each step in the sequence is altered, to move toward systematic manipulations and target algorithms, new declarative knowledge has to be established, which then has to be proceduralised. That is, there is a long process of obtaining declarative knowledge and then pro- ceduralising it. If these sequences are to be meaningful to the learner, it is important to simplify and to minimise change as the learner progresses through each step in the sequence. An additional benefit of this sequence with its emphasis on declarative, procedural and simplification, is that as larger numbers are used, or as some learning difficulty is experienced, the learner returns to an earlier stage in the sequence, one that is procedur- ally correct and where the teacher can be confident that understandings exists.

1.3. The Procedural Analogy Theory

The Procedural Analogy Theory discussed here is particularly concerned with the movement from the declarative to the procedural. In the development of whole number algorithms, one of the values of using concrete materials is that they allow the proceduralisation process to begin. The teacher explains the procedure using concrete materials, rather than explaining abstract ideas, and each student constructs a declarative encoding of that procedure which can then be represented through action on concrete materials. This process allows learners to more readily manipulate their thoughts, and makes it easier for the teacher to "see" what the student is thinking. As students construct meanings of mathematical concepts and skills, and as they act on materials and engage in cognitive conflict, this manipulation of materials allows the teacher some access to student cognitive pro- cesses, and so places the teacher in a position to be able to intervene so as to increase learning efficiency. Declarative encodings are converted to executable procedures through practice with materials that are transparent, in the sense that they show both the teacher and


the student what the student is thinking. Such materials have the potential to provide a vis- ible analogy of the student's working memory.

At a time when it seems quicker and easier to tell students a written procedure, give them an example, and then have them practice it (as is common in classrooms), it may not be obvious why a teacher would want to bother with the many declarative and procedural steps required in the use of concrete materials. It seems that the start of a topic is difficult enough without adding what might be seen as extraneous ideas and activities. But these views are simplistic. For example, they ignore the need for students to construct their own meanings, and they overlook the value of reflection on the structure of mathematics by students. These assumptions also ignore a range of curriculum goals, especially those concerning problem solving and the need for experiential learning in school mathematics. The distinction between declarative and procedural knowledge also draws out the point that while information may be given by the teacher, it is the student who has to re-construct that knowledge, to give it meaning by fitting it in with existing cognitive structures.

In the Procedural Analogy Theory, the teacher's initial description of the concrete materials procedure results in a declarative encoding of that procedure, involving a process of language understanding. The teacher's language together with the external representation system allow the learner to construct an internal, cognitive representation system of the external representation. Practice problems allow the declarative encoding of the concrete materials procedure to be proceduralised, converted into an executable cognitive procedure. Once the concrete materials procedure (external representation) has been sufficiently practised, and is efficiently executable, it is moved to the realm of symbols (internal representation) by procedural analogy through a series of transformations and simplifications. The output of this step is a symbolic procedure that is isomorphic to the concrete materials procedure, and in many instances will be the target procedure. The more isomorphic the external representation procedure is to the internal representation procedure, the easier the analogical learning step. Within the concrete materials procedure, within the written procedure and between each of these representational systems, it is the use of analogy, transformation and simplification that provide the means which allow learners to build new meanings from existing knowledge.

The Procedural Analogy Theory emphasises learning through analogy between declarative and procedural representational structures, and through analogy between one set of representational procedures and a second set of procedures. To construct a procedural analogy is to say, in effect, that the solution to this task is like the solution to that task, where the phrase "is like" is to be interpreted with a particular mapping in mind. In this way a new procedure is created by using a previously learned procedure as a model. The purpose then of a procedural analogy theory is to apply analogies so that students are able to move from the concrete representation of a process to a symbolic representation of that process. The closer the isomorphism between the concrete representation and the symbolic representation, the more obvious will be the analogical procedure, the greater the likely level of understanding, and the less the likelihood of errors through misinterpretations.

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2. MEASURING ISOMORPHISMS BETWEEN PROCEDURES

The success of an analogy depends on the ease with which mathematical concepts and skills, involving actions on concrete materials, can be described by the teacher, and the ease with which the learner is able to declaratively encode these descriptions and actions. The analogy depends, too, on the ease with which this declarative encoding can be proce- duralized, the degree of isomorphism between the concrete materials procedure and the symbolic procedure, and the extent to which the transformation from the concrete to the symbolic can be achieved with simplifications.

The Procedural Analogy Theory concentrates on the isomorphism between the concrete materials procedure and the symbolic procedure, which is the key to learning with concrete materials. The task of explaining why an embodiment works well, translates to the task of showing that actions on the materials are isomorphic with the relevant symbolic procedure. Similarly, the task of explaining why given materials and actions on them are unlikely to work well, involves showing that the concrete materials procedure is not isomorphic to the symbolic procedure. But an embodiment is unlikely to be exactly isomorphic with the symbolic procedure. Therefore, we need to be able to talk about degrees of isomorphism between two procedures. That is, we need to be able to quantify the measure of isomorphism (Ohlsson & Hall, 1990).

VanLehn and Brown (1980) described the use of multibased arithmetic blocks (MABs) for the operation of addition. They presented a situation where blocks were not manipu- lated according to their place values: that is, where the blocks were not initially treated as units, tens, and so on. Instead the blocks were piled together, and only then was their place value considered. In such a case, the trading aspect of addition was delayed until after the actual addition of the numbers, as represented by the blocks being joined together. They saw this two-step procedure as unlikely to be as closely analogous to standard addition as the one-pass Dienes Block procedure, where each place value is dealt with in turn. This prompted them to note the need for a theory of analogy to have some formal measure that could predict how close an analogy was, a unit they called the closeness metric. Their closeness metric is a measure of the match between the three parts required for an analogy to occur: the original abstraction or procedure, the new, modified or target abstraction, and the mapping of the first onto the second. In their paper, measures for the closeness metric were arbitrary in the sense that experts were asked to judge the closeness of analogies. They mention the difficulty of quantifying their closeness metric, noting "there are many difficulties and fine points involved in determining such a definition" (p. 120). The Proce- dural Analogy Theory has a more objective, ordered, quantifiable and generalisable approach to measuring degrees of analogy.

What possibilities exist when representing cognitive procedures, so as to make the abstract more accessible? What will this tell us about human learning, what help will it give us in developing instructional strategies, and will it provide insights into possible pedagogies for teachers? Included in the possibilities are flowcharts, planning nets, production rules and procedural sequences. More particularly, which of these representation systems minimise subjectivity, are quantifiable and are likely to help predict pedagogic effectiveness?

Flowcharts can be used to outline the sequence of steps to move through to successfully complete a particular procedure. Flowcharts indicate a line of action, they make decision


5 4 2 -

2 6 3

2 7 9

T A B L E I . A Procedural Sequence for the Subtraction 542 - 263

0.0 542 - 263 (recognise question)

1.0 Process units

1.1 Take away 3 from 2 (cannot)

1.1.1 Trade for more units

1.1.2 R e c a l l 4 - 1 = 3

1.1.3 Cross out 4, write 3 1.1.4 Write 1 next to 2

1.1.5 Recall this is 12

1.2 Take 3 from 12

1.3 Recall 12- 3 = 9

1.4 Record 9 in answer space

2.0 Process tens

2.1 Take 6 from 3 (cannot)

2.1.1 Trade for more tens

2.1.2 Reca l l5 - 1 --4

2.1.3 Cross out 5, write 4

2.1.4 Write 1 next to 3


2.2 Take 6 from 13

2.3 Recall 13- 6-- 7 2.4 Record 7 in answer space

3.0 Process hundreds

3.1 Take 2 from 4

3.2 Reca l l4 - 2 = 2


4.0 Read answer

points clear, and they allow for repetition, iteration and recursion (Hall, 1992). They may also be used to highlight where chunking is possible, where each step listed in the flowchart is a one-step representation of a more complex physical or cognitive action. That is, each step in the flowchart can be enlarged to show more detail of the physical or cognitive actions that are involved. Planning nets are directed graphs, combining nodes that provide choices of action, together with links that direct the next step in the problem's solution. These nets show sequences, reasoning and intent (VanLehn & Brown, 1980). Production rules are if-then statements that, given a particular set of circumstances, specify what to do (Anderson, 1983b). Large numbers of such rules go together in performing quite simple actions, such as adding two numbers.

But these representations raise a series of complex problems. Do these representations reflect what learner's really do? Is there any way of quantifying and comparing these representations? Do these representations distinguish essential steps from the inessential, the sensible from the unnecessary? They are difficult to quantify, and unlikely to have predic- tive capabilities in terms of the quality of learning outcomes. They appear complex and difficult to develop, and are therefore problematic in the context of a helpful tool for teachers. They are incomplete in terms of human learning, and are not especially helpful in terms of pedagogy and classroom practices.

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A more helpful representation may be that of a procedural sequence (Hall, 1993; Ohls- son & Hall, 1990), where both physical and cognitive actions are listed. Two important assumptions are made in using these procedural sequences to represent actions and written algorithms. They are that the teacher has a target procedure in mind that the learners are to re-construct, and that the cognitive aspects of the sequence are logically consistent with completing the algorithm correctly. The procedural sequence for completing the written subtraction algorithm 542-263 is shown in Table 1. The data in Table 1 indicate that for the novice there may be 26 steps in solving this algorithm. Over time chunking of steps prob- ably takes place, as does the development of a more automatic response, so the actual number of steps is likely to decrease. On the other hand, those learners whose recall of number facts requires some calculation beyond immediate recall from long-term memory, will have increased the initial number of steps.

These procedural sequences can be constructed both for written algorithms and for physical actions with concrete materials. In this way, the procedural sequence required to complete a subtraction algorithm in a written form, and the procedural sequence for completing the same subtraction question with MABs, can each be constructed and compared, prior to instruction.

The Procedural Analogy Theory refers to the level of analogy between two sets of procedures as an isomorphism index, which is based on a comparison between expanded traces of the procedures. The trace of a procedure is the sequence of actions that a procedure generates when it is executed. An "expanded trace" is a trace that includes both goals and actions. One way to generate such a trace is to implement the procedure as a computer program, run the program, and list the sequence of goals, subgoals, and actions that the procedure generates. But the procedural analogy theory is not dependent on computer systems. The theory allows teachers to generate the sequence of cognitive and physical actions they want students to perform, by considering expert behavior in completing an algorithm. If this has been done twice for the same procedure, for example, once with concrete materials and once with symbols, it is possible to quantify the isomorphism between them by mapping the traces of the two procedures onto each other. If two goals or actions fulfil similar functions in the two procedures, then we count those goals as corresponding, as analogous.

In the procedural analogy theory the degree of isomorphism (I1,2) between the two procedures is calculated using the formula:

(N 1 + N 2 - 2 ) - (D 1 + 02) I1,2 = N 1 + N 2 - 2

where N 1 and N 2 are the total number of entries in each of the two traces, and D 1 and D 2 are the number of entries in each trace not having a match in the other trace. The "2" is subtracted from the total number of entries because the main goal of each trace must be the same, otherwise an analogy would be nonsense. The index varies between 0 and 1, where the value 0 indicates no entries are in correspondence, and the value 1 indicates all the entries correspond to each other.

In the example shown in Table 2, where 53 is subtracted from 82, N 1 is 16, N 2 is 16, D 1 is 2 (steps 0.1 and 1.1.2) and D 2 is 2 (1.1.2 and 1.1.3). Step 1.1.2 in the MAB procedure involves a trade, a physical movement. This step is not analogous to the trading steps, 1.1.2 and 1.1.3, in the algorithm procedure, which involve subtracting 1 from a number, crossing it out and writing the new digit. But Step 1.1.3 in the MAB procedure involves joining


TABLE 2. Comparing an MAB Procedure with a Written Algorithm Procedure for the Subtraction 82 - 53

MAB procedure 0.0 82 - 53

0.l Subtract 53 from 8T, 2U

1.0 Process units 1.1 Take 3U from 2U (cannot)

1.1.1 Trade for more units 1.1.2 Move lT from 8T to bank,

bring back 10U 1.1.3 Join 10U and 2U 1.1.4 Recall 10U + 2U-- 12U

1.2 Take 3U from 12U 1.3 Recall 12U - 3U = 9U 1.4 Record answer, 9U in answer space

2.0 Process tens 2.1 Take 5T from 7T 2.3 Recall 7T - 5T -- 2T 2.4 Record answer, 2T in answer space

3.0 Read answer (2T 9U)

Target procedure 0.0 82- 53

1.0

2.0

Process units 1.1 Take 3 from 2 (cannot)

1.1.1 Trade for more units 1.1.2 Recall8-1=7 1.1.3 Cross out 8, write 7 1.1.4 Write 1 next to 2 1.1.5 Recall this is 12

1.2 Take 3 from 12 1.3 Recall 12-3=9 1.4 Record 9 in answer space

Process tens 2.1 Take 5 from 7 2.3 Recall7- 5-- 2 2.4 Record 2 in answer space

3.0 Read answer (29)

together ten units and two units, which is analogous to writing a "1" next to the "2" in step 1.1.4 in the algori thm procedure. The values of each of the variables in this case lead to an isomorphism index of 0.87.

In the calculation of the isomorphism index, the final concrete materials procedure is compared with the teacher 's final target procedure. Finding the degree of isomorphism between two procedures requires deciding, again and again, whether each of two entr ies corresponds or not. Certainly procedures can be implemented in different ways, and the different implementat ions generate different traces. But this is part of the theory 's strength. It allows teacher initiative, and provides a measure of l ikely effectiveness of intended

instruction.

Ohlsson and Hall (1990) applied this method to several commonly used embodiments, and found that the Theory appeared to explain some failures to teach arithmetic with embodiments , and to pinpoint the weaknesses of the instruction with these embodiments. In particular, an isomorphism index of 0.46 was calculated for the teaching approach employed by Resnick and Omanson (1987) in their study of MAB materials. This low index is one possible explanation for the lack of increase in mathematics achievement reported in their study. The author (Hall, 1993) investigated three approaches to teaching subtraction using MAB materials. These data suggested that students taught through a high isomorphism index teaching approach, scored more highly on subtraction tests than did students from teaching approaches with a lower isomorphism index.

Table 3 shows another set of traces for MAB materials and for the target subtraction algorithm, in this case for the question 651 - 293. These steps, either with MABs or in the target algorithm, are not unique and must be developed by the teacher. Once the teacher has decided on the target behaviour, a teaching sequence with concrete materials can be devel-

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TABLE 3. Comparing an MAB Procedure with a Writ ten Algori thm Procedure for the Subtraction 651 - 293

MAB procedure

0.0 651-293 0.1 Subtract 293 from 6H, 5T, 1U

1.0 Process units

1.1 Take 3U from 1U (cannot) 1.1.1 Trade for more units

I. 1.2 Move 1T from 5T to bank,

bring back 10U

1.1.3 Join 10U and IU

1.1.4 Recall 10U + 1U = 11U

1.2 Take3Ufrom l l U

1.3 Recall 11U - 3U = 8U

1.4 Record answer, 8U in answer space

2.0 Process tens

2.1 Take 9T from 4T (cannot)


2.1.2 Move 1H from 6H to bank,

bring back 10T

2.1.3 Join 10T and 4T

2.1.4 Recall 10T+4T- - 14T

2.2 Take 9T from 14T

2.3 Recall 14T - 9T = 5T

2.4 Record answer, 5T in answer space


3.1 Take 2H from 5H

3.2 Recall 5H - 2H = 3H 3.3 Record answer, 3H in answer space

4.0 Read answer (3H 5T 8U)

Target procedure

0.0 651-293

1.0 Process units

1.1 Take 3 from 1 (cannot)

1.1.1 Trade for more units

1.1.2 R e c a l l 5 - 1 = l

1.1.3 Cross out 5, write 4

1.1.4 Write 1 next to 1 1.1.5 Recall this is 11

1.2 Take 3 from 11

1.3 R e c a l l l l - 3 - - 8


2.0 Process tens 2.1 Take 9 from 4 (cannot)


2.1.2 Recall 6 - 1 = 5 2.1.3 Cross out 6, write 5

2.1.4 Write 1 next to 4


2.2 Take 9 from 14

2.3 Recall 14 - 9 = 5



3.1 Take 2 from 5

3.2 Recall 5 - 2 -- 3 3.3 Record 3 in answer space


oped, so as to give a high isomorphism index. Applying the isomorphism index formula, from the procedural analogy theory, to the MAB and target procedures described in Table 3, gives N 1 = 25, N 2 = 26, D 1 = 3 (steps 0.1, 1.1.2 and 2.1.2) and D 2 -- 4 (steps 1.1.2, 1.1.3, 2.1.2 and 2.1.3), and a high isomorphism index of 0.86.

Slight variations to the procedures in Table 3 will lead to a lower isomorphism index. For example, if the teacher ignores the "recall" steps in the MAB procedure, the value of D 2 will increase to 9 (the additional steps 1.1.4, 1.3, 2.1.4, 2.3 and 3.2). From a teaching perspective omission of such a small detail may seem inconsequential, but in completing algorithms students have to recall such facts. This omission of the "recall" steps in the MAB procedure, reduces the isomorphism index to 0.73. These and other variations are shown in Table 4. Another alteration that would seem of little importance from a pedagogical perspective, would involve deleting the "join" statements in the MAB procedure (steps 1.1.3, and 2.1.3), giving an index of 0.67.

As a final alteration to the initial procedures in Table 3, consider those students who use finger counting or tally marks to perform simple subtractions. For them the target proce-


TABLE 4. Comparing an MAB Procedure with a Written Algorithm Procedure for the Subtraction 651-293

MAB procedure 0.0 651 -293

0.1 Subtract 293 from 6H, 5T, IU

1.0 Process units 1.1 Take 3U from 1U (cannot)

1.1.1 Trade for more units 1.1.2 Move 1T from 5T to bank,

bring back 10U 1.1.3* 1.1.4*

1.2 Take3Ufrom l lU 1.3"

1.4 Record answer, 8U in answer space

2.0 Process tens 2.1 Take 9T from 4T (cannot)

2.1.1 Trade for more tens 2.1.2 Move 1H from 6H to bank,

bring back 10T 2.1.3" 2.1.4*

2.2 Take 9T from 14T 2.3*

2.4 Record answer, 5T in answer space

3.0 Process hundreds 3.1 Take 2H from 5H 3.2* 3.3 Record answer, 3H in answer space

4.0 Read answer (3H 5T 8U)

Notes: * changes to Table 3.

Target procedure 0.0 651-293

1.0 Process units 1.1 Take 3 from 1 (cannot)

1.2 1.3

1.4

1.1.1 Trade for more units 1.1.2 R e c a l l 5 - 1 = l 1.1.3 Cross out 5, write 4 1.1.4 Write l n e x t t o l 1.1.5 Recall this is 11

Take 3 from 11 Recall 1 1 - 3 = 8 1.3.1" Using fingers or tally Record 8 in answer space

2.0 Process tens 2.1 Take 9 from 4 (cannot)

2.1.1 Trade for more tens 2.1.2 Recall6- 1--5 2.1.3 Cross out 6, write 5 2.1.4 Write 1 next to 4 2.1.5 Recall this is 14

2.2 Take 9 from 14 2.3 Recall 14- 9 = 5

2.3.1" Using fingers or tally 2.4 Record 5 in answer space

3.0 Process hundreds 3.1 Take 2 from 5 3.2 Recall 5- 2 = 3 3.3 Record 3 in answer space


dure r equ i res add i t iona l s teps (s teps 1.3.1 and 2.3.1), l ead ing to N l = 18, N 2 = 28, D 1 = 3

and D 2 = 13, and g iv ing an i s o m o r p h i s m index o f 0.64.

T h e " reca l l " and " j o i n " o m i s s i o n s m a y be u n i m p o r t a n t w h e n the l ea rne r has c h u n k e d

s teps and a u t o m a t i s e d the sequence . B u t those l ea rne r s sti l l d e v e l o p i n g m e a n i n g s f rom this

conc re t e to s y m b o l i c m o v e m e n t , are l ike ly to bene f i t f r o m as de ta i l ed an a n a l o g y as poss i -

ble. I f the t e ache r dec ides no t to e m p h a s i s e such a n a l o g o u s steps, or i f the l ea rne r does no t

u n d e r s t a n d the c o r r e s p o n d e n c e o f t he se steps, M A B s m a y b e c o m e li t t le more than ca lcu la -

tors. In such c i r c um s t ances , M A B s a p p e a r to be u sed for the i r o w n sake, as an a l t e rna t ive

to f inge r s or a tally, r a the r than as ma te r i a l s to ass i s t the d e v e l o p m e n t o f m a t h e m a t i c a l pr in-

c iples , cogn i t i ve n e t w o r k s and m e a n i n g s . T h e P rocedura l A n a l o g y T h e o r y sugges t s tha t

the s e q u e n c e ou t l i ned in T a b l e 3 w o u l d p rove to b e a m o r e e f fec t ive t e ach ing a p p r o a c h than

that ou t l i ned in T a b l e 4. T h e a u t h o r ' s 1993 da ta suppor t this.

46 HALL

TABLE 5. Applying the Procedural Analogy Theory to the addition of fractions.

Manipulative procedure 1.0 i + i

2 3

2.0 Process denominators 1.1 What is a common

denominator of 2 and 3 1.2 Recall LCD is 6

3.0 Draw diagram

4.0 Term 1 4.1 Represent 1/2 4.2 Recall there are 6 sections 4.3 Calculate 1/2 of 6 4.4 Recall 1/2 of 6 is 3 4.5 Draw it

5.0 Term 2 5.1 Represent 1/3 5.2 Recall there are 6 sections 5.3 Calculate 1/3 of 6 5.4 Recall 1/3 of 6 is 2 5.5 Draw it

6.0 Add 1/2 and 1/3

7.0 Recognise answer is coloured sections divided by all sections 6.1 Count coloured sections (5) 6.2 Count all sections (6)

8.0 Record answer (5/6)

Target procedure

1.0 l + i 2 3

2.0 Process denominators 1.1 What is a common

denominator of 2 and 3 1.2 Recall LCD is 6

3.0 Represent addition as

i + 1 = __ + __

2 3 6 6

4.0 Numerator of term 1 4.1 What do I multiply 2 by to get 6 (3) 4.2 Multiply numerator by 3 4.3 Recall 3 X 1 = 3 4.4 Write 3 (3/6)

5.0 Numerator of term 2 5.1 What do I multiply 3 by to get 6 (3) 5.2 Multiply numerator by 2 5.3 Reca l l2X 1 - -2 5.4 Write 2 (2/6)

1 + 1 = 3 . 2 2 3 6 6

6.0 Recognise answer is sum of numerators 6.1 Add numerators 6.2 Recall 3 + 2 = 5 6.3 Recall denominator

7.0 Record answer (5/6)


TABLE 6. Applying the Procedural Analogy Theory to the addition of fractions, modifying Table 5.

Manipulative procedure

4.0 Term 1 4.1 How do I represent 1/2 4.2 By colouring 1/2 the sections 4.3 How many will be coloured 4.4 Colour them 4.5 What fraction is that (3/6)

5.0

I I I I Term 2 5.1 How do I represent 1/3 5.2 By colouring 1/3 the sections 5.3 How many will be coloured 5.4 Colour them 5.5 What fraction is that (2/6)

Target procedure

4.0 Numerator of term 1 4.1 How do I represent 1/2 4.2 What is 1/2 of 6 4.3 Recall 1/2 of 6 is 3 4.4 Write 3 4.5 What fraction is that (3/6)

5.0 Numerator of term 2 5.1 How do I represent 1/3 5.2 What is 1/3 of 6 5.3 Recall 1/3 of 6 is 2 5.4 Write 2 5.5 What fraction is that (3/6)

In summary, the procedural analogy theory predicts that the degree of isomorphism between an embodiment procedure and a symbolic procedure is a major determinant of pedagogical effectiveness. To measure this variable for a particular use of an embodiment involves six steps: (a) write a procedural sequence for the concrete material, (b) generate an expanded trace by running the sequence on a problem, (c) write a sequence for the symbolic procedure, (d) generate an expanded trace, (e) map the traces onto each other, and (f) calculate the isomorphism index (Ohlsson & Hall, 1990).

Tables 3 and 4 describe the use of one form of concrete materials in one mathematical operation: MABs in subtraction. However, the Procedural Analogy Theory is likely to be applicable to a range of teaching materials, and to a range of school mathematics topics. If teaching materials are intended to be acted on by students, so as to increase their mathematical knowledge and improve their cognitive structure, then the Procedural Analogy Theory is likely to have a role. The theory may be useful in whole number, rational number and decimal number work, as a means for teachers and curriculum developers to contrast alternative teaching approaches. Of course, there is a need for empirical evidence to support such claims. The possibility of using rectangular shapes to teach the concepts associated with the addition of fractions, is explored in Tables 5 and 6.

48 HALL

Table 5 provides a trace for one way of teaching the addition of fractions with rectangular shapes, together with a trace for the final target procedure. The symbolic procedure is a typical approach. Each term is represented as an equivalent fraction, with both terms altered so as to have a common denominator. The numerators are then added. The rectangular shapes procedure is formulated to maximise the analogies between it and the symbolic procedure. Initially, in this procedure, the need for a lowest common denominator of 6 is explored. The denominators 2 and 3 provide the dimensions of the rectangle, 2 by 3. This is intentionally recalled as 6, since this is what has to happen in the symbolic procedure. The two terms of the addition are then represented in the rectangle, with an emphasis on changing each term to an equivalent fraction. Finally, the two representations are added. These data in Table 5 indicate N 1 = 24, N 2 = 20, D 1 = 8 (steps 3.0, 4.2, 4.3, 4.4, 5.2, 5.3, 5.4 and 6.0) and D 2 = 8 (steps 3.0, 4.2, 4.3, 4.4, 5.2, 5.3, 5.4 and 6.1), and so/(1, 2)becomes 26/ 42 or 0.62. This is not a high isomorphism index.

If steps 4 and 5 from Table 5 are modified so as to make the concrete representation procedure and the target procedure more alike, as is shown in Table 6, the isomorphism index will be higher. With this new version of Table 5, represented in part in Table 6, N 1 = 22, N 2 = 22, D 1 = 4 (steps 3.0, 4.4, 5.4, 6.0) and D 2 = 2 (steps 3.0, 4.4, 5.4, 6.1), and so I(1, 2) becomes 34/42 or 0.81. This is considerably higher than the value of 0.62 obtained from Table 5.

The procedures and traces in Tables 5 and 6 show clearly that different teaching approaches will lead to different isomorphisms indices. Of course further empirical study is needed, but if the changes made in Table 6 (in comparison to Table 5) lead to improved learning outcomes, then the Procedural Analogy Theory appears to be a useful instructional tool.

3. CONCLUSION

This paper began by addressing the equivocal nature of research findings concerning the impact of concrete materials on mathematical learning. Research by Sowell (1989), by Resnick and Omanson (1987), and by Fuson and Briars (1990) were used to illustrate the range of findings, and were used too, to suggest that the teaching approaches used in this field of research are reported in varying degrees of detail. Generally speaking, the majority of research reports give too little detail for the teaching approach investigated to be repli- cated.

The Procedural Analogy Theory (Ohlsson & Hall, 1990) is suggested as a detailed and replicable instructional technique, applicable in those situations where a teacher intends to use concrete materials to help learners develop a written algorithm. In particular, the theory provides guidelines for the comparison of various teaching approaches before instruction commenced. The theory claims to be able to quantify the comparison of the concrete mate- dais procedure with the symbolic procedure, and represented this as an isomorphism index. Further, the theory indicates that the higher the level of analogy between the concrete mate- dais procedure and the written procedure, the higher the isomorphism index, and so the greater the likelihood of higher student achievement scores.

Concrete materials may be useful because it is easier for the teacher to describe actions on physical objects than to describe operations on symbols, and because it is easier for stu-


dents to correctly proceduralise such a description. These materials may also be useful because, through student action on these materials, teachers gain insights into learners' cognitive structures. The Procedural Analogy Theory illustrates how these procedures with concrete materials can be transferred to create a written algorithm. The theory emphasises that this transfer involves analogy, substitution and simplification, rather than the creation of a symbol system from nothing.

Reference was made to data that gave support to the theory, based on an investigation where children learned subtraction algorithms. And suggestions were made for its application beyond whole number arithmetic. Clearly, there is need for further investigation. For example, is the theory applicable to all whole number operations, to fractions, and to other mathematics topics? And what impact will the Procedural Analogy Theory have on students' understandings, on their cognitive structures, and on their problem solving abilities? While acknowledging the need for further research, the theory reported here appears to have potential in facilitating the design of teaching approaches involving the use of concrete materials, and in helping teachers guide learners in their construction of mathematical knowledge.

Acknowledgment: The procedural analogy theory was developed by Ohlsson and Hall (1990), during a period when the present author was on sabbatical leave at the Learning Research and Development Center, University of Pittsburgh. The author acknowledges the importance of this Ohlsson and Hall publication, and the continued support by Stellan Ohlsson, to the present article.

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Concrete representations and the procedural analogy theory

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