Inversion in mathematical thinking and learning

Brian Greer

Published online: 2 April 2011# Springer Science+Business Media B.V. 2011

Abstract Inversion is a fundamental relational building block both within mathematics asthe study of structures and within people’s physical and social experience, linked to manyother key elements such as equilibrium, invariance, reversal, compensation, symmetry, andbalance. Within purely formal arithmetic, the inverse relationships between addition andsubtraction, and multiplication and division, have important implications in relation toflexible and efficient computation, and for the assessment of students’ conceptualunderstanding. It is suggested that the extensive research on arithmetic should be extendedto take account of numerical domains beyond the natural numbers and of the difficultiesstudents have in extending the meanings of operations to those of more general domains.When the range of situations modelled by the arithmetical operations is considered, thecomplexity of inverse relationships between operations, and the variability in the forms thatthese relationships take, become much greater. Finally, some comments are offered on thedivergent goals and preoccupations of cognitive psychologists and mathematics educatorsas illuminated by research in this area.

Keywords Inversion . Arithmetic operations . Psychology andmathematics education

1 Introduction

The topic of inversion is of central importance to the arithmetic of the natural numbers andthe four basic operations on them. That is the focus of the experimental studies in thisspecial issue, and that is where I begin, with special emphasis on detecting and exploitingthe structure of arithmetical systems, in particular in relation to inverse operations.However, my main goal in this paper is to broaden the perspective on inversion withinmathematical thinking and learning in a number of respects. Thus, the subsequent sectionsdeal with the pervasive nature of inversion within the architecture of abstract mathematics,with extensions of arithmetical structures beyond the natural numbers, and then with the

Educ Stud Math (2012) 79:429–438DOI 10.1007/s10649-011-9317-2

B. Greer (*)Portland State University, Portland, OR, USAe-mail: brian1060ne@yahoo.com

pervasive nature of inversion within our physical and social experience. Finally, the body ofresearch represented in this special issue is examined as a context in which the tensionsbetween experimental psychologists and mathematics educators are exemplified.

2 Good mathematicians are lazy: exploiting structure in arithmetic

One of my mathematics teachers used to say that good mathematicians are lazy. What hemeant by this was that a good mathematician has an acute sense of structure inmathematical relationships and a disposition to exploit such structure to simplify amathematical task where possible. The prototypical example of this disposition is the story(possibly apocryphal) about the great mathematician Gauss as a child, related, for example,by Wertheimer (1982, originally 1945). In Wertheimer’s (p. 109) version of the story, theteacher of the 6-year-old Gauss asked the class to find the sum of:

1þ 2þ 3þ 4þ 5þ 6þ 7þ 8þ 9þ 10

and Gauss produced the answer extremely quickly. Although there are many ways in whichGauss might have exploited the structure, Wertheimer conjectured that he noticed that thenumbers could be arranged in pairs 1+10, 2+9, etc., of which there are five, all adding to11, hence the total is 5×11=55. Wertheimer’s extensive discussion of this and related taskstouches on many of the key mathematical ideas that are also implicit in the inverserelationships between addition/subtraction and multiplication/division, and in inversemathematical relationships in general.

Wertheimer also discussed (p. 122 et seq.) problems of the form m+a−a and m+a−a+b−b+c−c, and variations. A particularly interesting task that relates to multiplication anddivision as inverses is the following (p. 130):

274þ 274þ 274þ 274þ 274

5

This example illuminates the hard-to-define-in-the-abstract difference between compu-tational fluency and conceptual understanding. A student who computes a repeatedaddition, or multiplication, followed by a division, demonstrates computational fluency butmay well lack conceptual understanding. Wertheimer related his surprise that, while most ofthe bright subjects he asked “enjoyed the joke” (p. 112), “a number of children who wereespecially good at arithmetic … were entirely blind” (p. 113).

These considerations lead to the recommendation that the early teaching of arithmeticand algebra should focus heavily on cultivating a sense of the structure of number systems,the “relational calculus” discussed by Nunes, Bryant, Evans, Bell, and Barros (this issue).With specific reference to the inverse relationships between arithmetic operations, structurecan be explored and exploited in many ways, including the use of inverse operations tocheck calculations and the derivation of alternative computational procedures such assubtraction as complementary addition (Peters, De Smedt, Torbeyns, Ghesquière, &Verschaffel, this issue; Peltenburg, Van den Heuvel-Panhuizen, & Robitzsch, this issue).For teachers and researchers, as illustrated by Wertheimer’s chapter, probes for conceptualunderstanding can be designed, such as the “support problems” illustrated by Vanden Heuvel-Panhuizen (1996, p. 153), in which the presentation of a calculation such as86+57=143 is followed by testing children’s ability to find “smart” ways to get the answersto related calculations such as 143−86 and 85+58. Such probes echo Wertheimer’s strategy

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of testing depth and flexibility of understanding by posing questions requiring increasingdistance of transfer.

3 Inversion as a central structural element within mathematics

Inversion is a basic component in the architecture of mathematical structures, as illustratedbelow. Children’s encounters with inverse relationships in arithmetic accordingly offer anearly chance to sow the first seeds for the formal mathematical treatment of this verypowerful and pervasive idea. Some of the most important manifestations of inversion, inmy judgment, are as follows:

1. The four basic operations and the growth of number systemsBoth the motivation and the mechanisms for expansion of number systems beyond

the natural numbers are logically related to the property of (lack of) closure. Thefollowing is a very rough and schematic sketch of developments that have takenmillennia for humankind but which a child is expected to navigate in a few years ofschool.

Numbers begin with the counting numbers, technically termed the natural numbers.Within the natural numbers, addition is closed in the sense that the sum of any twonatural numbers is also a natural number; the inverse operation, subtraction, is notclosed. This lack of closure creates a kind of disequilibrium, which is resolved byconstructing the negative numbers. Multiplication typically is encountered first inschool mathematics in the form of repeated addition (echoing the definition by Euclid,for example) and is also closed within the natural numbers, but its inverse, division, isnot. This lack of closure is resolved by constructing the rational numbers. The rationalnumbers, both positive and negative, form a number system that is closed under all fouroperations (except that division by zero is not defined).

The process does not stop there. The operation of squaring has an inverse operation,namely taking the (positive) square root, and the Pythagoreans famously discoveredthat this operation is not closed within the rational numbers when it was proved that √2is not a rational number. This lack of closure was resolved by the construction of thereal numbers. There are yet more elaborate number systems; indeed, new kinds ofnumbers are still being constructed (Conway & Guy, 1996). In this process ofsuccessive lacks of closure being resolved by reconceptualizing “number”, there is asuggestive parallel with the Piagetian notion of disequilibrium as a driver of cognitivegrowth.

The number line, extensively referred to in this issue as a teaching and thinking tool,is particularly well suited to representing the successive expansions of the numbersystem up to the real numbers and can be expanded to the representation of complexnumbers in a plane in the form of Argand diagrams (Lakoff & Nunez, 2000).

2. Inverses in the solution of certain algebraic equations and inverse functionsWhen somebody reasons that 6+2=8 → 8−2=6 (or 8−6=2) then that person is—

more or less implicitly/consciously—invoking an algebraic principle that a+b=c→c−b=a (or c−a=b), and similar remarks apply for multiplication/division. This kind of“relational calculus” (Nunes et al., this issue) is central to the articulation betweenarithmetic and algebra (and see Selter, Prediger, Nührenbörger, & Hußmann, thisissue). Closely related are the equations 6+x=8 and x+2=8, the solutions of which canbe derived by “undoing” the operation by its inverse. Certain more complex equations

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(e.g. (x+3)/5+3=7) can be solved by sequences of such undoings. More generally,inverse functions are of great importance within algebra.

3. GroupsA characteristic feature of modern mathematics is the study of abstract structures

defined by certain properties. A group is defined as a set of elements, S, together with abinary operation, *, on the elements of S that has the following properties:

1. Closure. If x and y belong to S, then x * y exists, is unique, and belongs to S2. Associativity. For any x, y, and z belonging to S, x * (y * z)=(x * y) * z3. Identity element. There exists a unique element, e, belonging to S, such that, for

any x belonging to S, x * e=x and e * x=x4. Inverse. For any element x belonging to S, there exists an element y belonging to S

such that x * y=e and y * x=eIf, in addition, it is always true that x * y=y * x, then the group is called

commutative.There are very many groups arising from geometrical and numerical contexts,

as well in the modelling of the physical world (not to mention artefacts likeRubik’s Cube, which, as a Google search will immediately verify, is very widelyused to teach group theory). The integers, under addition, form a commutativegroup, with identity element 0 and the inverse of x being −x. The rationals, reals,and complex numbers also form commutative groups under addition. With theexception of zero, the rationals, reals, and complex numbers form commutativegroups under multiplication, with identity element 1 and inverse of x, 1/x.

A fundamental property of groups, that follows from the axioms, is that, given xand y, the equation x * a=y has a unique solution (which, in the case of additioncan be found by subtraction, and in the case of multiplication, division).

4. Inverse processesA more general idea of inverses in mathematics involves inverse processes. For

example, inverse to the multiplying of n factors x−a1, x−a2, … x−an is the factoring ofa polynomial of degree n. (That a polynomial of degree n, within the complex numbers,can always be factored into n factors, is called the Fundamental Theory of Algebra,proved by Galois—using group theory). As generally stated, the Fundamental Theoryof Calculus states that differentiation and integration are inverse processes.

While the term “inverse” is common to all of these examples, it should be clear that thenature and the meaning of inversion vary among the different contexts. I do not attempthere to clarify this variation, except to comment that it exemplifies the remark by thefamous mathematician Poincare that mathematics is the art of giving the same name todifferent things (the different conceptualizations of “number” provides another example).

4 Beyond the natural numbers

The experimental studies in this special issue, and a great deal of the related literature, arelargely restricted in scope to decontextualised arithmetic operations applied to the naturalnumbers. There are many reasons for going beyond the natural numbers (see abovediscussion of how number systems evolved historically).

At the simplest level, would it not make sense to use “messy” numbers in both researchand teaching, and not just (mostly rather small) natural numbers? For example, what aboutextending strategies such as indirect addition to calculations such as 24.2−11.9? Or, if a

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student has reached the point of immediately answering 3×6÷6 the “smart” way, what ifyou ask her/him to calculate 3.64×8.36÷8.36 (perhaps with a calculator available)? I amnot aware of research using such calculations, but it would seem a good way to test thesolidity of students’ understanding, much as Wertheimer did by posing related challenges.

As sketched above, inversion is embedded within increasingly inclusive number systemsand in algebraic structures. It is thus a key element in the articulation between arithmetic andalgebra. In general, the teaching of mathematics requires a “longitudinal perspective” (Selteret al., this issue), so it is prudent to bear in mind what lies ahead—for example, the extensionof the meaning of multiplication and division beyond the domain of natural numbers (Greer,1994). The advocacy of the “Determine the Difference” conceptualization of subtraction bySelter et al. (this issue) is partly motivated by the relative ease of extending this conceptionto calculations involving negative numbers. Fischbein, Deri, Nello, and Marino (1995)pointed out the consequences for students’ later conceptual change that follow from“implicit models” for the arithmetic operations becoming firmly entrenched. The implicitmodel for subtraction is “take away”. Historically, the dominance of this conception isnicely illustrated by the statement that “3−8 is an impossibility, it requires you to take from3 more than there is in 3, which is absurd”. The source of this statement was an eminentnineteenth century English mathematician (De Morgan, 1910, originally 1831).

5 Inversion in relation to semantic structures

As illustrated below, a given arithmetical equation corresponds to word problems with agreat variety of semantic structures. A considerable body of work (much of it byVerschaffel’s research group) relates to what may be termed as “arithmetical operations asmodels of situations” (e.g. Greer, 1992). Once you move beyond bare calculations, thesituation becomes a great deal more complex.

5.1 Addition and subtraction as models of situations

For addition/subtraction word problems, a standard taxonomy is into combine, change,compare, and equalize problems (e.g. Fuson, 1992). These are not exhaustive, of course,and Freudenthal (1983, Chapter 4) considers much richer phenomenological variety.Arguably, researchers, curriculum planners, textbook authors, and teachers should be moreaware of this variety. Here, I simply want to illustrate the variation in how inversion relatesto uses of the arithmetical operations in modelling distinct classes of situations.

& In the combine problems, putting together and separating are the inverse actions e.g.putting the boys and girls in a class together/separating the class into boys and girls.Here the interpretation of addition and subtraction as inverse operations modelling thenumerical aspects of inverse actions is straightforward.

& In the change problems, there is an initial state, a change, and the resultant final state.The change, an addition or subtraction of n, is inverted (“undone”) by, respectively,subtraction or addition of n—again the relationship between inversion as an action andinversion as an arithmetic process is straightforward. Of course, there are different kindsof problems corresponding to which of the three quantities is unknown and whether thechange is positive or negative (Nunes et al., this issue; Vergnaud, 2009). If the change isunknown, it can be determined as the difference between the initial and final states(corresponding to the “difference between” conception of subtraction).

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& In the compare problems, the inverse relationship relates the difference between a and bto the complementary difference between b and a, quite a different conception.

& An equalize problem is of the form “How much does a need to be increased by to makeb?” This formulation corresponds to subtraction as an indirect addition.

The clear distinctions between the various ways in which inverses play out exemplifywhy Freudenthal argued against taking the decontextualised arithmetic operation as aconceptual bridge without considering the meaning of the operations within each of thedifferent situations. As he stated:

As a calculator one may forget about the origin of one’s numbers and the origin ofone’s arithmetical problem in some word problem. But at the same time, one must beable to return from the algorithmic simplicity to the phenomenal variety in order todiscover the simplicity in the variety. (1983, p. 117)

Relating addition to situations also complicates the arithmetical property of commuta-tivity of addition. In particular, the addends in a combine problem play similar roles,whereas in the other types of situation they play distinct roles, hence giving rise, in eachcase, to two clearly distinct subtraction problems. The implications of this difference weredemonstrated for young children by De Corte and Verschaffel (1987) who showed howsemantic structures interact with computational methods. Related are distinctions betweenaddition and subtraction as unary and binary operations (e.g. Fuson, 1992, p. 244).

5.2 Multiplication and division as models of situations

The variety of situations modelled by multiplication and division is greater, two majorreasons for the added variety being the complexity of the dimensional relationships and theinteraction with the kinds of numbers involved, even when restricting discussion to positivenumbers. This extra complexity applies also to inversion; Freudenthal (1983, p. 114)commented that “The relation of dividing to multiplying is much more involved [i.e.,complex] than that of subtraction to adding”.

A major distinction was suggested in Greer (1992) between “symmetric” situations, inwhich the quantities multiplied involved play equivalent roles (an example beingcomputing the area of a rectangle from its length and breadth) and the more numerous“asymmetric” situations in which the quantities have quite distinct roles, one beingidentifiable as multiplicand and the other as multiplier, the simplest of such case being thatof equal groups where the total number is found by multiplying the number in each group(multiplicand) by the number of groups (multiplier). A consequence of this distinction isthat for each of the asymmetric situations, division takes two distinct forms depending onwhether division is by the multiplier or the multiplicand; it is the former that corresponds tothe inverting of an action. Effects of these distinctions, in relation to the types of numbersinvolved in multiplication and division word problems, were empirically demonstrated by,for example, Greer and Mangan (1986) and by Harel, Behr, Post, and Lesh (1994).

Briefly, (some of) the ways in which division as the inverse of multiplication plays out inrelation to the variety of situations are:

& Equal groups. The combination of n equal groups can be undone by separation (ageneralization of what happens in the combine problems).

& Measure conversion. There is an inverse relationship between changing, say, miles tokilometres, and kilometres to miles.

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& Multiplicative comparison. As with additive comparison, there is an inverse relationship(e.g. 3 times as long as/1/3 times as long as).

& Multiplicative change, e.g. expanded by 25%, the inverse of which is a contraction by20%, 5/4 and

4/5 being reciprocals)

As an example of a rate problem, consider the relationships between quantity ofsomething bought, unit price, and total cost. Finding the total cost by multiplying unit priceby quantity may be considered the inverse of finding the unit cost by dividing the total costby quantity.

The symmetrical situations (area, rectangular arrays, Cartesian product) are characterizedby unidirectionality in the sense that multiplication is the usual operation that will berequired—it is relatively rare that one would want to find the length of a rectangle bydividing the area by the length or breadth, for example.

In short, as with addition and subtraction, a mere facility with arithmetic is insufficientfor dealing with the complexities of modelling situations with multiplication and divisionand understanding the diverse forms of inverse relationship that the operations have withindifferent models.

The fullest phenomenological analysis of situations within children’s (and adults’)experience that can be modelled by arithmetic operations has been provided by Freudenthal(e.g. 1983). Piaget famously constructed a theory of how mathematics is abstracted fromsensorimotor experience, his aim being “to establish better connections between theoperational form of knowledge, which consists in action in the physical and social world,and the predicative form of knowledge, which consists in the linguistic and symbolicexpressions of this knowledge” (Vergnaud, 2009, p. 83).. However, according to Vergnaud(p. 83) Piaget “was slowed down in the analysis of the mathematical contents by hisfascination for logic and his hope to be able to reduce to logical structures the progressivecomplexity gained by children”, and Freudenthal has criticized Piaget on similar grounds,while enthusing about Piaget’s observations of his young children (Freudenthal, 1991,p. 88). Freudenthal (1991, p. 6) also relevantly commented that “Children acquire numberin the stream of their physical and mental activities, which makes it difficult for researchersto find out how this happens in detail”. A very different perspective that makes clearer,through the lens of an anthropological study of a very different culture, how arithmetic issocially embedded is offered by Urton (1997), demonstrating the linguistic complexity andthe relationship of arithmetic to the Quechuan culture.

6 Inverses as models of situations

Inverse actions, movements, and relations are ubiquitous within everyday life. Whentravelling to Leuven for the conference connected with this special issue, for example:

& I flew from Portland to Toronto to Montreal to Brussels, and back from Brussels toMontreal to Toronto to Portland.

& I remembered that when Skyping back to Portland, the time would be 9 h earlier ratherthan 9 h later for Skyping in the other direction.

& I changed dollars into Euros—the exchange rates in opposite directions are reciprocals(but see below).

& The linear function for converting temperature in degrees Centigrade into temperaturein degrees Fahrenheit is the inverse of the function for Fahrenheit to Centigradeconversion.

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In general, situations that can be modelled by inversion (and see Vergnaud, 2009, p. 86)include:

& Physical actions taking place over time, such as moving from A to B, and back from Bto A, reversing or undoing actions, elements of choreography, playing film in reverse

& Social structures, such as lending and getting back, kinship relationships such as A isB’s child/B is A’s parent

& Conventionally defined functional relationships between quantities

However, as with arithmetic operations as models for situations, it is often necessary toadjust the “core model” to take real-world considerations into account. For example,lending and getting back may involve interest; currency exchange typically involves a fee;going from B to A by road, for example, is not the precise opposite of going from A to B.

As with the previous section, this brief account is offered as a preliminary indication ofthe direction in which deeper analyses might proceed.

7 Reflections on research in mathematics education

In this paper, I have argued that the research on inversion in mathematics thinking andlearning—as represented by the papers in this special issue and related literature—is limitedin many ways, and for enlarging its scope in several respects, including:

& Going beyond the natural numbers to more complex number systems and elaboratingon the articulation of arithmetic and algebra

& Considering inversion of arithmetical operations in the context of addition/subtractionand multiplication/division as models of situations, not simply in relation to barecalculations (i.e. dealing with external sense-making as well as internal)

& Considering the pervasive role of inversion in mathematics and in physical and socialexperience—and the links between these (Freudenthal, 1983; Piaget, 1969; Urton, 1997)

Experimental researchers and mathematics educators have divergent aims (De Corte,Greer, & Verschaffel, 1996, p. 492). Mathematics education in classrooms showsdisappointing progress, despite the amount of ingenuity and hard work that has gone intoresearch. Moreover, mathematics educators point to the limited horizons of manyexperimental psychologists; a disproportionate amount of research—as in the case ofresearch relating to inversion—is about the arithmetic of (relatively small) naturalnumbers. In the current context, Robinson and LeFevre (this issue) comment on therelative scarcity of research on the relationships between multiplication and division (andon the principle of associativity). They also show that the SCADS* model, developed bySiegler over a considerable amount of time, cannot be unproblematically extended toprovide a theoretical framework for students’ understanding of the inverse relationshipbetween multiplication and division. So, progress towards a comprehensive architectureof mathematical cognition is slow, both in terms of “making psychological researchcashable in the educational bank” (Greer & Verschaffel, 1990), and in terms of dealingwith more complex mathematical content. It is worth remembering that educationalresearch is “the hardest science of all” (Berliner, 2002, p. 18) for many reasons, including“the power of contexts, the ubiquity of interactions, and the problem of decade byfindings interactions”, and that research on mathematics education is characterized by“reasonable ineffectiveness” (Kilpatrick, 1981), primarily because improving (mathemat-ics) education is a human problem, not a technical problem.

436 B. Greer

Moreover, I am not convinced that experimental psychologists sufficiently recognize thatthere is not an architecture of mathematical cognition that can be studied independently ofschooling and that, when it is complete, will afford a clear guide for mathematics education.Once a relatively low level of complexity is reached, the study of cognition relating toformal mathematics is crucially mediated by instruction (see Greer, 2009). Accordingly, animportant bridge between experimental psychology and mathematics education is the use ofintervention studies that take place in circumstances relatively close to classroom lessons.In relation to such studies, I suggest that several recommendations deserve seriousconsideration:

& Document the instructional history of the students, such as what they have been taught,the teaching approaches adopted, the nature of the curriculum, textbooks, and so on.Such aspects have been amply reported on, or discussed, in several of the studies in thisissue. For example, Robinson and LeFevre (this issue) state that many children asked toevaluate a shortcut strategy considered it a form of cheating and that every calculationshould be performed. Although they do not specifically offer evidence, it seemsplausible that this could be the result of how those children had been taught, andRobinson and LeFevre suggest that North American children are particularly likely toexperience a bias towards an algorithmic solution.

& Pay more attention to the “experimental contract”. How do the students construe whatthey are being asked to do? How do they react, for example, to being asked to carry outlong series of similar calculations? For example, what happens when boredom sets in, aquestion interestingly researched and discussed by Wertheimer (1982)? I suggest thatresearchers conducting intervention studies, particularly short ones, should take intoaccount that students might rationally decide not to abandon well-adapted forms ofbehaviour or invest cognitive effort, knowing that soon they will be back in theirnormal classroom environment.

& Use multiple methods to probe depth and flexibility of understanding, as is wellillustrated in this issue. The performance of a student who has reached the point of veryquickly producing a response to a calculation of the form a+b−b could correspond to awide range of levels of understanding from “surface learning” to “deep learning”(Nunes et al., this issue), including:

(a) Having simply noticed, from a long series of examples, that the answer required isthe first number. The analysis by Nunes et al. (this issue) of results from the studyby Siegler and Stern (1998) makes it clear that for the majority of students in thatstudy, this explanation of their performance is the most parsimonious.

(b) Application of a taught rule(c) Behaving in accordance with a “theorem-in-action” (Vergnaud, 2009, p. 86)(d) Articulation of a principle such as that adding something, then taking the same

thing away, returns one to the starting point (this feeling of logical necessity mightbe termed the Piagetian criterion for understanding)

& Provide support for transfer. If anything is clear by now from research on mathematicseducation, it is that transfer (even when it seems to the experimenter or the teacher thatit should be trivially easy, as the reader will observe more than once in this issue)typically does not happen without considerable nurturing (Greer & Harel, 1998; Maher,Powell, & Uptegrove, 2010). As Nunes et al. (this issue) put it: “it may not be a goodeducational practice to teach primary school children how to calculate and leave thedevelopment of relational calculus skills to the children’s own devices”. Whereaspsychologists may tend to say that what the children learn is “only a matter of what they

Inversion in mathematical thinking and learning 437

are taught” (Wertheimer, 1982, p. 132), mathematics educators aim for conceptualunderstanding that implies the ability to “adapt … cognitive resources and face aproblem never met before” (Vergnaud, 2009, p. 88).

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