- Email: [email protected]
Conceptual Knowledge in Arithmetic: The Core of Calculation Skills
VIEWPOINT CONCEPTUAL KNOWLEDGE IN ARITHMETIC: THE CORE OF CALCULATION SKILLS Carlo Semenza (Department of Psychology, University of Trieste)
A long way has been travelled, in the past twenty years, since cognitive neuropsychology started to tackle the topic of number processing and calculation. Before the early 1980’s, deficits in this cognitive domain had been handled simply by distinguishing between primary acalculia or “anarithmetia”, on the one hand, and impairments of mathematical performance that were demonstrated to be secondary to language or spatial disorders, on the other. By contrast, at the beginning of the 21st century and thanks to the efforts of cognitive neuropsychologists, we can now refer to elaborate models of both number processing and calculation, formulated mainly on the basis of a large corpus of patient data. Characteristically, however, the accumulation of empirical evidence has been rather uneven. In fact, while some aspects of mathematical cognition seem to have enjoyed enduring popularity, other interesting issues that were neglected in early cognitive models continue to be given little attention. Recent reviews (e.g. Cipolotti and van Harskamp, 2001; Noel, 2001) each quote at least 20 studies within sub-domains like number transcoding or the representation of arithmetical facts. On sober reflection, it must be acknowledged that, even on these subjects, consensus and secure conclusions are not widespread. For example, there is still no agreement as to whether number transcoding requires semantic processing or may be mediated by additional asemantic mechanisms (Seron and Noel, 1995). Likewise, there are no conclusive data to inform us about the representational format of arithmetical facts (Noel, 2001). Nevertheless, these issues have stimulated important debates. Little attention, on the other hand, has been paid to other cognitive aspects underlying calculation skills. Pivotal examples of still relatively underdeveloped notions include the “procedural” knowledge that guides the execution of complex calculation (extensively investigated, by the way, in developmental studies) and what has been called mathematical “conceptual” knowledge (Sokol et al., 1991; Hittmair-Delazer et al., 1994). Calculation comprises a very heterogeneous set of procedures, whose common denominator is the schematic, algorithmic nature of the underlying knowledge. Successful application does not require conceptual understanding of each step of the procedure (Van Lehn, 1990); and it can be applied only to familiar tasks. Conceptual knowledge of mathematics, by contrast, implies understanding of arithmetical operations and laws pertaining to these operations. It allows one to make inferences and, unlike procedural knowledge, can be flexibly adapted to new tasks. Thus these two types of knowledge differ considerably: whereas procedural knowledge corresponds to Cortex, (2002) 38, 285-288
286
Carlo Semenza
routine expertise, conceptual knowledge may be labelled adaptive expertise (Hatano, 1988; Delazer, in press). Although cases making reference to these matters have occasionally been reported, they seem to have provoked very little in the way of debate or controversy, perhaps simply because the observations available have been insufficient to stimulate such attention. Conceptual knowledge is not even incorporated in most of the formal models of number processing and calculation currently in use. Yet some important studies have attempted to define these competencies more precisely. For example, deficits in arithmetical procedures have been demonstrated to vary in nature and to be operation-specific (Girelli and Delazer, 1997; Semenza et al., 1997); indeed, these aspects have caught the attention of reviewers (see, for instance, Cipolotti and van Harskamp, 2001). Conceptual knowledge of arithmetic has been shown to be a rather complex but functionally independent domain (Hittmair-Delazer et al. 1994, 1995). As an instance of the critical interplay of procedural and conceptual knowledge, after some early research (Sokol et al., 1991; Sokol and McCloskey, 1991), recent studies (Cacciatori et al., 2000; Pesenti et al., 2000; Granà, 2002) highlight the frequent dissociation between conceptually driven and ‘blind’ use of arithmetical rules like nx0=0, for instance. These examples (reported in greater detail below) reveal the very limited extent to which neuropsychological research has captured the nature of anything beyond overlearned and automatic calculation. Crutch and Warrington’s article (2002, this issue) describes how some complex calculation skills have persisted in a patient, BET, affected by profound semantic dementia. Although she was disturbed in other aspects of numerical processing, BET’s spared capacities were, surprisingly, manifested in a problem solving task requiring very skilful manipulation of conceptual number knowledge (this task was modelled on a popular television gameshow called “Countdown”). She also used arithmetical procedures with considerable success in addition and subtraction, despite being unable to appreciate the meaning of arithmetical signs. One may question whether BET’s expertise at the “Countdown” mathematical game task is so striking, given that she had obviously enjoyed and practised this kind of task much more than the average subject. The important fact, however, is not simply her preserved ability in the face of vast cognitive disruption following brain deterioration. The ability to perform the Countdown tasks implies many distinct sub-abilities: choosing the correct numbers (these are not supplied in the order required), selecting the relevant operations, sorting out a logical sequence for these operations, and so on. An important question then becomes, would the patient be able to employ each of these abilities in other contexts? If these abilities were conceptually based, we should expect them to be used in a flexible way. Because Crutch and Warrington’s report of BET’s strategies provides only a rough idea of their complexity, it leaves some doubt about the correct answer to this question. This notwithstanding, we are given food for thought and for raising other, very interesting, questions. For example: What information are we storing when we learn how to perform complex mathematical tasks? How independent are specific complex calculation skills from the other domains of knowledge? And, as already indicated, how flexible is the conceptual knowledge of mathematics? Let us consider the acquisition of a new complex calculation
Conceptual knowledge in arithmetic
287
skill. It is possible that, as a result of the acquisition process, we store single sub-components in such a way that we are able to apply them easily only within the learned context. As already mentioned, neuropsychological observations have shown us that this may be true for some sets of common procedures or rules. For example, nx0 may be correctly solved within complex multi-digit operations by patients who would systematically answer n instead of 0 when simply asked to solve “n x 0” in isolation. Other patients apparently knowing that “n x 0 = 0” may respond that “0 x n = n”. It may turn out that even what appears to be flexible mathematical conceptual knowledge does not work in some contexts as well as in others. Dissociable processing sub-components might be just part of the story. It may be that, in order to cope successfully with a complex, conceptually based, mathematical task, the cognitive system learns it by combining sets of concepts that are likely to be used together in the task, even if these are logically separable. The use of one component operation would be then much easier in the context of its habitual set than when occasionally required outside this network of pre-established facilitating connections. We do not know yet whether this is the case. But it is plausible, and the accumulation of case descriptions like that of Crutch and Warrington may lead us to an answer. Crutch and Warrington’s article has the usual limitations that are intrinsic to clinical studies. There is no theory or analysis of how the “experimental task” might be performed by normals and little interpretation of what really happens in the patient. While the study encourages us to undertake these difficult analyses, in and of itself it contributes little to our theoretical understanding of calculation processes. The history of neuropsychology has however taught us the heuristic value of such striking findings: we have learned that we cannot afford to neglect these cases merely because we do not fully understand them. Once enough information on further cases has been collected, and proper comparisons among different cases have been made, we can hope to appreciate the meaning of reports like that on BET. We will then be grateful to those who thought it wise to make them known. Acknowledgements: The support is acknowledged of Grant NEUROMATH, from the European Commission, Human Potential Programme, Research Training Networks, to Carlo Semenza. REFERENCES CACCIATORI E, GRANÀ A, GIRELLI L and SEMENZA C. The status of zero in the semantic system: A neuropsychological study. Brain and Language, 74: 414-417, 2000. CIPOLOTTI L and VAN HARSKAMP N. Disturbances of number processing and calculation. In RS Berndt (Ed), Handbook of Neuropsychology, Language and Aphasia. Amsterdam: Elsevier, 2001, Ch 18, pp. 305-334. CRUTCH SJ and WARRINGTON EK. Preserved calculation skills in a case of semantic dementia. Cortex, 38: 389-401, 2002. DELAZER M. Neuropsychological findings on conceptual knowledge in arithmetic. In B Baroody and A Dowker (Eds), The development of arithmetical concepts and skills: Recent research and theory. Hillsdale: Lawrence Erlbaum Associates, in press. GIRELLI L and DELAZER M. Subtraction bugs in an acalculic patient. Cortex, 25: 121-133, 1996. GRANÀ A. L’elaborazione dello zero nel sistema dei numeri e del calcolo. Unpublished doctoral dissertation. The University of Trieste, 2002. HATANO G. Social and motivational bases for mathematical understanding. In GB Saxe and M Gearhart (Eds). Children’s mathematics. S Francisco: Jossey-Bass, 1988, Ch 3, pp. 55-70.
288
Carlo Semenza
HITTMAIR DELAZER M, SEMENZA C and DENES G. Concepts and facts in calculation. Brain, 117: 715-728, 1994. HITTMAIR DELAZER M, SAILER U and BENKE T. Impaired arithmetic facts but intact conceptual knowledge - a single case study of dyscalculia. Cortex, 31: 139-147, 1995. NOEL MP. Numerical cognition. In B Rapp (Ed), The Handbook of Cognitive Neuropsychology. Philadelphia: Psychology Press, 2001, Ch 20, pp. 495-518. PESENTI M, DEPORTER N and SERON X. Noncommutability of the N+0 arithmetical rule: A case of dissociated impairment. Cortex, 36: 445-454, 2000. SEMENZA C, MICELI L and GIRELLI L. A deficit for arithmetical procedures: lack of knowledge or lack of monitoring? Cortex, 33: 483-498, 1997. SERON X and NOEL MP. Transcoding numbers from Arabic code to the verbal one and viceversa: How many routes? Mathematical Cognition, 1: 215-243, 1995. SOKOL SM and MCCLOSKEY M. Cognitive mechanisms in calculation. In R Stembergerg and PA Frensch (Eds), Complex problem solving: Principles and mechanisms. Hillsdale: Lawrence Erlbaum Associates, 1991, Ch 3, pp.85-116. SOKOL SM, MCCLOSKEY M, COHEN NJ and ALIMINOSA D. Cognitive representations and processes in arithmetic: inferences from the performance of brain damaged subjects. Journal of Experimental Psychology: Learning, Memory and Cognition, 17: 355-376, 1991. VAN LEHN K. Mind bugs. The origin of mathematical misconceptions. Cambridge: MIT Press, 1990. Carlo Semenza, Department of Psychology, University of Trieste, via S Anastasio 12, 34123, Trieste, Italy. E-mail: [email protected]
A long way has been travelled, in the past twenty years, since cognitive neuropsychology started to tackle the topic of number processing and calculation. Before the early 1980’s, deficits in this cognitive domain had been handled simply by distinguishing between primary acalculia or “anarithmetia”, on the one hand, and impairments of mathematical performance that were demonstrated to be secondary to language or spatial disorders, on the other. By contrast, at the beginning of the 21st century and thanks to the efforts of cognitive neuropsychologists, we can now refer to elaborate models of both number processing and calculation, formulated mainly on the basis of a large corpus of patient data. Characteristically, however, the accumulation of empirical evidence has been rather uneven. In fact, while some aspects of mathematical cognition seem to have enjoyed enduring popularity, other interesting issues that were neglected in early cognitive models continue to be given little attention. Recent reviews (e.g. Cipolotti and van Harskamp, 2001; Noel, 2001) each quote at least 20 studies within sub-domains like number transcoding or the representation of arithmetical facts. On sober reflection, it must be acknowledged that, even on these subjects, consensus and secure conclusions are not widespread. For example, there is still no agreement as to whether number transcoding requires semantic processing or may be mediated by additional asemantic mechanisms (Seron and Noel, 1995). Likewise, there are no conclusive data to inform us about the representational format of arithmetical facts (Noel, 2001). Nevertheless, these issues have stimulated important debates. Little attention, on the other hand, has been paid to other cognitive aspects underlying calculation skills. Pivotal examples of still relatively underdeveloped notions include the “procedural” knowledge that guides the execution of complex calculation (extensively investigated, by the way, in developmental studies) and what has been called mathematical “conceptual” knowledge (Sokol et al., 1991; Hittmair-Delazer et al., 1994). Calculation comprises a very heterogeneous set of procedures, whose common denominator is the schematic, algorithmic nature of the underlying knowledge. Successful application does not require conceptual understanding of each step of the procedure (Van Lehn, 1990); and it can be applied only to familiar tasks. Conceptual knowledge of mathematics, by contrast, implies understanding of arithmetical operations and laws pertaining to these operations. It allows one to make inferences and, unlike procedural knowledge, can be flexibly adapted to new tasks. Thus these two types of knowledge differ considerably: whereas procedural knowledge corresponds to Cortex, (2002) 38, 285-288
286
Carlo Semenza
routine expertise, conceptual knowledge may be labelled adaptive expertise (Hatano, 1988; Delazer, in press). Although cases making reference to these matters have occasionally been reported, they seem to have provoked very little in the way of debate or controversy, perhaps simply because the observations available have been insufficient to stimulate such attention. Conceptual knowledge is not even incorporated in most of the formal models of number processing and calculation currently in use. Yet some important studies have attempted to define these competencies more precisely. For example, deficits in arithmetical procedures have been demonstrated to vary in nature and to be operation-specific (Girelli and Delazer, 1997; Semenza et al., 1997); indeed, these aspects have caught the attention of reviewers (see, for instance, Cipolotti and van Harskamp, 2001). Conceptual knowledge of arithmetic has been shown to be a rather complex but functionally independent domain (Hittmair-Delazer et al. 1994, 1995). As an instance of the critical interplay of procedural and conceptual knowledge, after some early research (Sokol et al., 1991; Sokol and McCloskey, 1991), recent studies (Cacciatori et al., 2000; Pesenti et al., 2000; Granà, 2002) highlight the frequent dissociation between conceptually driven and ‘blind’ use of arithmetical rules like nx0=0, for instance. These examples (reported in greater detail below) reveal the very limited extent to which neuropsychological research has captured the nature of anything beyond overlearned and automatic calculation. Crutch and Warrington’s article (2002, this issue) describes how some complex calculation skills have persisted in a patient, BET, affected by profound semantic dementia. Although she was disturbed in other aspects of numerical processing, BET’s spared capacities were, surprisingly, manifested in a problem solving task requiring very skilful manipulation of conceptual number knowledge (this task was modelled on a popular television gameshow called “Countdown”). She also used arithmetical procedures with considerable success in addition and subtraction, despite being unable to appreciate the meaning of arithmetical signs. One may question whether BET’s expertise at the “Countdown” mathematical game task is so striking, given that she had obviously enjoyed and practised this kind of task much more than the average subject. The important fact, however, is not simply her preserved ability in the face of vast cognitive disruption following brain deterioration. The ability to perform the Countdown tasks implies many distinct sub-abilities: choosing the correct numbers (these are not supplied in the order required), selecting the relevant operations, sorting out a logical sequence for these operations, and so on. An important question then becomes, would the patient be able to employ each of these abilities in other contexts? If these abilities were conceptually based, we should expect them to be used in a flexible way. Because Crutch and Warrington’s report of BET’s strategies provides only a rough idea of their complexity, it leaves some doubt about the correct answer to this question. This notwithstanding, we are given food for thought and for raising other, very interesting, questions. For example: What information are we storing when we learn how to perform complex mathematical tasks? How independent are specific complex calculation skills from the other domains of knowledge? And, as already indicated, how flexible is the conceptual knowledge of mathematics? Let us consider the acquisition of a new complex calculation
Conceptual knowledge in arithmetic
287
skill. It is possible that, as a result of the acquisition process, we store single sub-components in such a way that we are able to apply them easily only within the learned context. As already mentioned, neuropsychological observations have shown us that this may be true for some sets of common procedures or rules. For example, nx0 may be correctly solved within complex multi-digit operations by patients who would systematically answer n instead of 0 when simply asked to solve “n x 0” in isolation. Other patients apparently knowing that “n x 0 = 0” may respond that “0 x n = n”. It may turn out that even what appears to be flexible mathematical conceptual knowledge does not work in some contexts as well as in others. Dissociable processing sub-components might be just part of the story. It may be that, in order to cope successfully with a complex, conceptually based, mathematical task, the cognitive system learns it by combining sets of concepts that are likely to be used together in the task, even if these are logically separable. The use of one component operation would be then much easier in the context of its habitual set than when occasionally required outside this network of pre-established facilitating connections. We do not know yet whether this is the case. But it is plausible, and the accumulation of case descriptions like that of Crutch and Warrington may lead us to an answer. Crutch and Warrington’s article has the usual limitations that are intrinsic to clinical studies. There is no theory or analysis of how the “experimental task” might be performed by normals and little interpretation of what really happens in the patient. While the study encourages us to undertake these difficult analyses, in and of itself it contributes little to our theoretical understanding of calculation processes. The history of neuropsychology has however taught us the heuristic value of such striking findings: we have learned that we cannot afford to neglect these cases merely because we do not fully understand them. Once enough information on further cases has been collected, and proper comparisons among different cases have been made, we can hope to appreciate the meaning of reports like that on BET. We will then be grateful to those who thought it wise to make them known. Acknowledgements: The support is acknowledged of Grant NEUROMATH, from the European Commission, Human Potential Programme, Research Training Networks, to Carlo Semenza. REFERENCES CACCIATORI E, GRANÀ A, GIRELLI L and SEMENZA C. The status of zero in the semantic system: A neuropsychological study. Brain and Language, 74: 414-417, 2000. CIPOLOTTI L and VAN HARSKAMP N. Disturbances of number processing and calculation. In RS Berndt (Ed), Handbook of Neuropsychology, Language and Aphasia. Amsterdam: Elsevier, 2001, Ch 18, pp. 305-334. CRUTCH SJ and WARRINGTON EK. Preserved calculation skills in a case of semantic dementia. Cortex, 38: 389-401, 2002. DELAZER M. Neuropsychological findings on conceptual knowledge in arithmetic. In B Baroody and A Dowker (Eds), The development of arithmetical concepts and skills: Recent research and theory. Hillsdale: Lawrence Erlbaum Associates, in press. GIRELLI L and DELAZER M. Subtraction bugs in an acalculic patient. Cortex, 25: 121-133, 1996. GRANÀ A. L’elaborazione dello zero nel sistema dei numeri e del calcolo. Unpublished doctoral dissertation. The University of Trieste, 2002. HATANO G. Social and motivational bases for mathematical understanding. In GB Saxe and M Gearhart (Eds). Children’s mathematics. S Francisco: Jossey-Bass, 1988, Ch 3, pp. 55-70.
288
Carlo Semenza
HITTMAIR DELAZER M, SEMENZA C and DENES G. Concepts and facts in calculation. Brain, 117: 715-728, 1994. HITTMAIR DELAZER M, SAILER U and BENKE T. Impaired arithmetic facts but intact conceptual knowledge - a single case study of dyscalculia. Cortex, 31: 139-147, 1995. NOEL MP. Numerical cognition. In B Rapp (Ed), The Handbook of Cognitive Neuropsychology. Philadelphia: Psychology Press, 2001, Ch 20, pp. 495-518. PESENTI M, DEPORTER N and SERON X. Noncommutability of the N+0 arithmetical rule: A case of dissociated impairment. Cortex, 36: 445-454, 2000. SEMENZA C, MICELI L and GIRELLI L. A deficit for arithmetical procedures: lack of knowledge or lack of monitoring? Cortex, 33: 483-498, 1997. SERON X and NOEL MP. Transcoding numbers from Arabic code to the verbal one and viceversa: How many routes? Mathematical Cognition, 1: 215-243, 1995. SOKOL SM and MCCLOSKEY M. Cognitive mechanisms in calculation. In R Stembergerg and PA Frensch (Eds), Complex problem solving: Principles and mechanisms. Hillsdale: Lawrence Erlbaum Associates, 1991, Ch 3, pp.85-116. SOKOL SM, MCCLOSKEY M, COHEN NJ and ALIMINOSA D. Cognitive representations and processes in arithmetic: inferences from the performance of brain damaged subjects. Journal of Experimental Psychology: Learning, Memory and Cognition, 17: 355-376, 1991. VAN LEHN K. Mind bugs. The origin of mathematical misconceptions. Cambridge: MIT Press, 1990. Carlo Semenza, Department of Psychology, University of Trieste, via S Anastasio 12, 34123, Trieste, Italy. E-mail: [email protected]