- Email: [email protected]
From origins to destinations on the right track
Journal of Applied Developmental Psychology 30 (2009) 61–64
Contents lists available at ScienceDirect
Journal of Applied Developmental Psychology
Book review From origins to destinations on the right track Catherine Sophian, The origins of mathematical knowledge in childhood. Erlbaum (Taylor & Francis Group), New York, 2007, ISBN: 978-0-8058-5758-0 (cloth), 196 pp., $59.95 From its vantage point in developmental psychology and mathematics education, Origins of Mathematical Knowledge in Childhood is a bold under-taking in addressing two dominant positions that diverge over the origins of number, one in counting, the other in comparison. These two positions are analysed in six chapters covering numeracy in infancy, quantity comparison without numbers, understanding units, additive/multiplicative reasoning, and fractions. Two final chapters cover their implications. If you are familiar with this area, you may be thinking “Why was this book written right now?” Both positions continue to develop with the jury still out, attested by the honest admission that none of the research in Origins is “decisive in choosing between the positions” [p.150]. A compounding factor is the time lag between journal papers and book publication. However, the answer is to Catherine Sophian's credit. Her own work is multiple and influential. Keeping track of where we are now is necessary both for staying on track or for changing direction. Most work in this area has been problem- rather than position-centered. The main aim in Origins is the analysis of both positions with the overall verdict in favour of comparisons, admittedly a provisional verdict. Since the standoff between psychology and education has a long history, and research with a dual focus is rare, Origins is welcome, and should be read. 1. Counting and comparisons The starting-point in Chapter 1 is Piaget's account from which both positions are taken to part company in different ways. In one, the origin of number is in counting [CON]. In this domain-specific account, counting—determining the numerical values of discrete quantities—is taken to be something not investigated by Piaget in depth. A typical mechanism is proposed to be an accumulator open to impulses but whose gate is closed by default with content remaining constant, opening each time a to-becounted item is encountered, thereby increasing its content in a principled way: — the gate opens just once for each item to be counted, i.e. one-to-one — the states occur in a fixed order, from less to more full, each time a series is counted — the final state indicates the number of items encountered, i.e. the extent to which the initial steady state has been augmented. This mechanism is operative nonverbally in infancy containing implicit knowledge of how counting works. Verbal counting is built up by children on the same principles in becoming explicit knowledge eventuating in reasoning about number [pp. 2, 4–6]. In the other, the origin of number is in comparisons of equalities/inequalities, interpreted as non-numerical comparisons [COP]. Quantities are defined as “physical properties of things that we can measure” [p.4]. Measures can be compared along several dimensions, such as length [a shelf in relation to an alcove] or area [a door in relation to a hole in a wall]. Number is another such dimension. These comparisons occur “developmentally and conceptually antecedent to numerical ones” [p.42]. Secondly, despite its initial overlap with Piaget's account, COP parts company in regarding its major problem to be about units of comparison that are developed during childhood—what counts as “one” and its iteration. As well, there are crossover issues that straddle the gap between teaching and learning in mathematics [pp.3, 7, 42]. 2. Analysis CON is addressed in chapter 2 with its review of number knowledge in infancy based on visual preference studies. Infants familiarized with an array are later presented with an alternative differing solo numero, e.g. familiar and novel squares identical in area, color and length but non-identical in number [2-v-3]. An innumerate infant is predicted to view both indiscriminately, a numerate infant preferentially looking longer at the novel array. Most evidence is in line with infant numeracy, i.e. CON. But there are qualifications. Some studies have reported the non-differentiation of number/equality in early infancy, thus these precocious discriminations may be due to comparing, not to counting [p.23]. Infants' discriminations are confined to small numbers, though in excess of single digits [8-v-16]. Competing interpretations include expected/non-expected events, original/changed values, doi:10.1016/j.appdev.2008.10.006
62
Book review
familiar/novel preferences—each is intriguing, but none entails numeracy [pp.27, 30]. Also reviewed is number cardinality with research attesting that by 3–4 years children base cardinality on the last number in a series, rather than by re-counting. However, also attested is number recitation in incorrect counting with divergent reports about children's capacity to detect errors and the age of its onset. In short, CON has been scientifically productive but without a consensus emerging [p.39]. COP is presented in chapter 3. Two alternatives are presented—Piaget's in which one dimension [length] is taken by children as the indicator of another [number], and COP in which different dimensions are non-differentiated [pp.44, 61]. Well known experimental studies are interpreted through COP with the implication that children only gradually learn that a quantity can be greater in one dimension but not so in another by differentiating dimensions [p.50]. This learning is not easy for children since it requires the resolution of an apparent contradiction “X is the same and X is not the same”—something that instruction is taken typically to by-pass [p.63]. Chapter 4 deals with how the same thing can have different numerical properties depending on the unit, e.g. 4 pairs/8 shoes. Typical findings are that, presented with a collection of whole and broken objects, 3 year-olds can correctly count them as discrete objects, but only 5-6 year-olds as kinds [p.67]. In turn, children have difficulties in developing number units with higher power—here CON and COP should part company [p.70]. Implicated problems are the composition/decomposition of large numbers and place value\e.g., .80 N .8 = .08\with neither error detected by youngsters [p.81]. Key instructional recommendations are that a number unit is not an everyday object; units are constituted differently in different contexts; coordination of multiple units is complex [p.83]. Additive and multiplicative reasoning are separated in chapter 5. Relations in the former generate an independent unit [7 is neither 3 nor 4], whereas relations in the latter are determined through the others [12 is 3 times 4]. The proposal that addition has its origin in part-whole and other schemas is taken to be attractive, apart from its non-explanation of why the early availability of a schema leads to difficulty in school arithmetic. Actually, this is a general issue facing CON; but explanatory factors—pictures, language, familiarity—due to COP fare no better in securing their explanatory purchase [p.94]. Under CON, the seeds of multiplication are contained in initial enumeration; but without the formation and systematization of higher units, such as one packet containing five things, these seeds are barren. Yet under COP, instruction may be counter-productive if multiplication is taught as further addition [p.104]. In chapter 6, the complexity of fractions is attributed to how two numbers generate a single unit and how members of this triple are inter-related. A compounding factor is said to be the naive use of pictorial aids such as pie charts and the ambiguity of equal fractions without the formation of a higher unit. The latter is regarded as a serious problem for CON [p.108]. Particular flashpoints are improper fractions\ 6/5\and systematic ambiguities\ 1/2 b 2/3 N 5/8\in the absence of higher-order units. Encouraging ongoing work is reported, notably the Rational Number Project with a focus on multiple models for representing fractions along with inter-model translations. In a controlled study of fifth graders [n = 1600] from 66 classrooms, no difference was found between experimentals and controls where the RNP curriculum was frugal in contrast to a 30% gain by the former over the latter where the RNP curriculum was rich [p.125]. However, neither durability nor transfer was tested. And the adverse finding from comparative assessments—US fourth graders perform as well, but eighth graders far less well, than their international peers—is a well known cause for concern. Asserted to be a problem here is the ethics of instructional interventions [p.130]. Main implications for future work are presented in the two final chapters. For developmental psychology, these include: cognitive sequences in the serial formation of units, something not transparent to young children; identifying the contributions of canalization and instruction in ontogenesis; identifying specialized experiences, including the role of number representations; identifying general processes, such as language and working memory; exploiting cultural factors in instructional interventions; and securing a consensus as to the relative standings of CON and COP. For mathematics education, main implications are: gauging the fit between instructional practices and different psychological positions; tutor/learner task-matching; bridging the gap between number-solutions and number-arguments; design/delivery of effective instruction by grade with attention to the constraints of cultural norms. 3. Commentary I really do stand by my opening claim that Origins is welcome due to its resolute focus on positions, not merely issues. I encourage you to read and think through its analysis of its two leading positions about mathematical knowledge. All the same, I see three omissions, all major I would say. 3.1. Education “A major goal of this book is to examine issues concerning the teaching and learning of mathematics (traditionally the purview of mathematics education) from a developmental perspective” [p.2]. This sets up false expectations. The difference between psychology, a theoretical science, and education, an applied science, has long been attested (Bruner, 1966; De Corte, 2000; James, 1899). At least five relations run from-psychology-to-education (Smith, 2005): [i] [ii] [iii] [iv] [v]
independence: theory alone relevance: global claims implication: testable implications applications: tested implications inter-dependence: two-way testing.
Book review
63
Its final chapters place Origins at [iii]. That is quite an achievement when many of the studies reported are at [i] with most of the others at [ii], a seductive road to nowhere. The case for [v]—a science that is jointly theoretical and applied—has been well stated both in general using Pasteur's work (Stokes, 1997) and in the specific context of developmental psychology-and-education by Ann Brown with colleagues (Bransford, Brown & Cocking, 2001). Whilst this joint science is still a long way off, developmentalists could take a lead in the advance from [iii] to [iv]. In fact, they already have done so. One omission in Origins is manifest in its non-review of studies en route to [iv]. These studies lie at the interface, dealing with class size and the quality of classroom learning (Bennett, 1998; Biggs, 1998; Whitburn, 2000), or learning and pedagogy (Baroody & Dowker, 2003; Smith, 2004; Stigler & Hiebert, 1999). Other studies are already at [iv], notably interventions in mathematics and science learning driven by developmental models—this master-class work is on-going, repeatedly demonstrating huge gains in which experimentals outperform controls by up to 20% in national tests and examinations during childhood or adolescence (Adey, 2007; Shayer, 2008; Shayer & Adhami, 2003). The model behind these successful inventions is both Piagetian and Vygotskian (Shayer & Adey, 1981, 2002). It is astonishing that Origins did not cover this educationally significant evidence under a model due to the “two giants” of developmental psychology. 3.2. Reasoning Mathematical knowledge is interpreted in Origins to include reasoning, notably in chapter 5 where a theoretical problem is by-passed. Addition [qone and oneq] in arithmetic and in logic are not the same thing: specifically, 1 + 1 = 1 and 1 + 1 = 2 are both sound (Russell, 1964, §114). The same distinction was drawn by Piaget (1966, p.265): a tautology (A + A = A) in logic differs from an iteration (A + A = 2A) in arithmetic. Recognising “a” star and recognising “a” star is not thereby to recognise two stars—recall Frege's (1950) famous example of the Morning Star and the Evening Star and differences in “cognitive value”. Origins is silent about the origins of this distinction. Under CON and COP, why is a number-judgment made about equality rather than a classification through identity? A developmental problem is how the advance is made from one to the other. Piaget (1942; reviewed in Smith, 2002) gave an answer in terms of reasoning by mathematical induction [MI], incidentally a type of reasoning that is interesting in its own right, though by-passed in Origins. MI is distinctive, uniquely combining the necessity of deduction and the universality of induction. MI is recognized to be important in philosophy [its reducibility to the other two types of reasoning—Russell Yes, Poincaré No]; MI is used by mathematicians [Wiles' recent proof of Fermat's Last Theorem]; and MI is required in the curriculum for higher mathematics education. I know of no studies of its development during childhood other than a 1963 Genevan study, recently replicated (Smith, 2002; for commentary, see Rips, Bloomfield, & Asmuth, in press; Smith, in press). Two main conclusions were drawn in this replication: direct support for the Genevan conclusion that children aged 5– 7 years can reason by MI and indirect support for their conclusion that these children understand the necessity of this inference. Since current educational practice has confined MI to the advanced curriculum, a good opportunity to teach some form of MI during childhood is being lost. Two studies, and so much remains to be done. If MI is used by adults, and if adults were formerly children, what are the sequences and mechanisms of this advance, including interactions with other forms of reasoning? 3.3. Reasons The stance taken in Origins about the nature/nurture problem is standard with mathematical origins interpreted primarily through nativism [CON] or social experience [COP]. In the case of mathematical knowledge, both are seriously incomplete, explanatory failures with regard to two properties of mathematical knowledge, objectivity and necessity. Any mathematical truth is objective. Consider 4 + 3 = 7. If you count four children in his family and three in her family as “so six altogether,” the conclusion to draw is “you've made a mistake,” not “the laws of arithmetic have been falsified.” Further, any mathematical truth is necessarily true. Consider (4 + 3 = 7) = N (4 × 3 = 12). Three necessities are nested here: the antecedent (4 + 3 = 7) is an equality, the consequent (4 × 3 = 12) is an equality, and the relation (=N) linking them is entailment. All equalities and all entailments are necessitations, normatively speaking. What, then, is the origin of necessary knowledge, how is it acquired by contingent beings such as ourselves? Exactly that was Piaget's principal problem over 60 years (quoted in Smith, 2002, p.110). Piaget's (2006) answer was in terms of reasons whose role is “to introduce new necessities into systems where they were merely implicit or remained unacknowledged” [p.8]. Reasons straddle the gap between subjective-objective, contingentnecessary. For example, presented with a line of glasses and a line of bottles and asked whether there is the same in each, you might say “Yes—six in this line, six in that”. What matters is the normative framework behind this response (Piaget, 2006, Note 16): six is my favorite number six is what I observed six is the self-same number six is necessarily the self-same number. Adult-speak, of course; but I was young once, and Piaget's argument is that children's knowledge becomes adults' knowledge in virtue of the normativities in reasons. Although norms are not facts, normative facts are facts open to empirical investigation (Smith, 2006). Significantly, it is these normativities that enable human development to be on the right tracks from the origins of knowledge to its eventual destinations.
64
Book review
References Adey, P. (2007). The CASE for a general factor in intelligence. In M. J. Roberts (Ed.), Integrating the mind: Domain general versus domain specific processes in higher cognition (pp. 369−386). Hove, UK: Psychology Press. Baroody, A. J., & Dowker, A. (2003). The development of arithmetic concepts and skills: Constructing adaptive expertise. Mahwah. NJ: Erlbaum. Bennett, N. (1998). Annotation: Class size and the quality of educational outcomes. Journal of Child Psychology and Psychiatry, 39, 797−804. Biggs, J. (1998). Learning from the Confucian heritage: So size doesn't matter? International Journal of Educational Research, 29, 723−738. Bransford, J., Brown, A., & Cocking, R. (2001). How people learn. Washington, DC: National Academy Press. Bruner, J. (1966). Towards a theory of education. Cambridge, MA: Harvard University Press. De Corte, E. (2000). Marrying theory building and the improvement of school practice. Learning and Instruction, 10, 249−266. Frege, G. (1950). Translations from the philosophical writings of Gottlob Frege. Oxford: Blackwell. James, W. (1899). Talks to teachers on psychology. London: Longman. Piaget, J. (1942). Classes, relations, nombres: Essai sur les groupements de la logistique et sur la réversibilité de la pensée. Paris: Vrin. Piaget, J. (1966). Part II. In E. Beth & J. Piaget (Eds.), Mathematical epistemology and psychology (pp. 131−304). Dordrecht: Reidel. Piaget, J. (2006). Reason (translation and commentary by Leslie Smith). New Ideas in Psychology, 24, 1−29. Rips, L., Bloomfield, A., & Asmuth, J. (in press). From numerical concepts to concepts of number. Behavioral and Brain Sciences. Russell, B. (1964). The principles of mathematics, 2nd edition New York: W.W. Norton & Co, Inc. Shayer, M. (2008). Intelligence for education: As described by Piaget and measured by psychometrics. British Journal of Educational Psychology, 78, 1−29. Shayer, M., & Adey, P. (1981). Towards a science of science teaching. London: Heinemann. Shayer, M., & Adey, P. (2002). Learning intelligence. Buckingham: Open University Press. Shayer, M., & Adhami, M. (2003). Realising the cognitive potential of children 5-7 with a mathematical focus. International Journal of Educational Research, 39, 743−775. Smith, L. (2002). Reasoning by mathematical induction in children's arithmetic. Oxford: Elsevier Science Pergamon Press. Smith, L. (2004). Developmental epistemology and education. In J. Carpendale & U. Müller (Eds.), Social interaction and the development of knowledge (pp. 175−194). Mahwah, NJ: Erlbaum. Smith, L. (2005). Developmental psychology and education. In B. Hopkins (Ed.), The Cambridge encyclopedia of child development (pp. 487−490). Cambridge: Cambridge University Press. Smith, L. (2006). Norms and normative facts in human development. In L. Smith & J. Vonèche (Eds.), Norms in human development (pp. 103−137). Cambridge: Cambridge University Press. Smith, L. (in press). Mathematical induction and its formation during childhood. Behavioral and Brain Sciences. Stigler, J., & Hiebert, J. (1999). The teaching gap. New York: Free Press. Stokes, D. (1997). Pasteur's quadrant: Basic science and technological innovation. Washington, DC: Brookings Institute. Whitburn, J. (2000). Strength in numbers. London: NIESR.
Leslie Smith E-mail address: [email protected]. URL: http://www2.clikpic.com/ls99/.
Contents lists available at ScienceDirect
Journal of Applied Developmental Psychology
Book review From origins to destinations on the right track Catherine Sophian, The origins of mathematical knowledge in childhood. Erlbaum (Taylor & Francis Group), New York, 2007, ISBN: 978-0-8058-5758-0 (cloth), 196 pp., $59.95 From its vantage point in developmental psychology and mathematics education, Origins of Mathematical Knowledge in Childhood is a bold under-taking in addressing two dominant positions that diverge over the origins of number, one in counting, the other in comparison. These two positions are analysed in six chapters covering numeracy in infancy, quantity comparison without numbers, understanding units, additive/multiplicative reasoning, and fractions. Two final chapters cover their implications. If you are familiar with this area, you may be thinking “Why was this book written right now?” Both positions continue to develop with the jury still out, attested by the honest admission that none of the research in Origins is “decisive in choosing between the positions” [p.150]. A compounding factor is the time lag between journal papers and book publication. However, the answer is to Catherine Sophian's credit. Her own work is multiple and influential. Keeping track of where we are now is necessary both for staying on track or for changing direction. Most work in this area has been problem- rather than position-centered. The main aim in Origins is the analysis of both positions with the overall verdict in favour of comparisons, admittedly a provisional verdict. Since the standoff between psychology and education has a long history, and research with a dual focus is rare, Origins is welcome, and should be read. 1. Counting and comparisons The starting-point in Chapter 1 is Piaget's account from which both positions are taken to part company in different ways. In one, the origin of number is in counting [CON]. In this domain-specific account, counting—determining the numerical values of discrete quantities—is taken to be something not investigated by Piaget in depth. A typical mechanism is proposed to be an accumulator open to impulses but whose gate is closed by default with content remaining constant, opening each time a to-becounted item is encountered, thereby increasing its content in a principled way: — the gate opens just once for each item to be counted, i.e. one-to-one — the states occur in a fixed order, from less to more full, each time a series is counted — the final state indicates the number of items encountered, i.e. the extent to which the initial steady state has been augmented. This mechanism is operative nonverbally in infancy containing implicit knowledge of how counting works. Verbal counting is built up by children on the same principles in becoming explicit knowledge eventuating in reasoning about number [pp. 2, 4–6]. In the other, the origin of number is in comparisons of equalities/inequalities, interpreted as non-numerical comparisons [COP]. Quantities are defined as “physical properties of things that we can measure” [p.4]. Measures can be compared along several dimensions, such as length [a shelf in relation to an alcove] or area [a door in relation to a hole in a wall]. Number is another such dimension. These comparisons occur “developmentally and conceptually antecedent to numerical ones” [p.42]. Secondly, despite its initial overlap with Piaget's account, COP parts company in regarding its major problem to be about units of comparison that are developed during childhood—what counts as “one” and its iteration. As well, there are crossover issues that straddle the gap between teaching and learning in mathematics [pp.3, 7, 42]. 2. Analysis CON is addressed in chapter 2 with its review of number knowledge in infancy based on visual preference studies. Infants familiarized with an array are later presented with an alternative differing solo numero, e.g. familiar and novel squares identical in area, color and length but non-identical in number [2-v-3]. An innumerate infant is predicted to view both indiscriminately, a numerate infant preferentially looking longer at the novel array. Most evidence is in line with infant numeracy, i.e. CON. But there are qualifications. Some studies have reported the non-differentiation of number/equality in early infancy, thus these precocious discriminations may be due to comparing, not to counting [p.23]. Infants' discriminations are confined to small numbers, though in excess of single digits [8-v-16]. Competing interpretations include expected/non-expected events, original/changed values, doi:10.1016/j.appdev.2008.10.006
62
Book review
familiar/novel preferences—each is intriguing, but none entails numeracy [pp.27, 30]. Also reviewed is number cardinality with research attesting that by 3–4 years children base cardinality on the last number in a series, rather than by re-counting. However, also attested is number recitation in incorrect counting with divergent reports about children's capacity to detect errors and the age of its onset. In short, CON has been scientifically productive but without a consensus emerging [p.39]. COP is presented in chapter 3. Two alternatives are presented—Piaget's in which one dimension [length] is taken by children as the indicator of another [number], and COP in which different dimensions are non-differentiated [pp.44, 61]. Well known experimental studies are interpreted through COP with the implication that children only gradually learn that a quantity can be greater in one dimension but not so in another by differentiating dimensions [p.50]. This learning is not easy for children since it requires the resolution of an apparent contradiction “X is the same and X is not the same”—something that instruction is taken typically to by-pass [p.63]. Chapter 4 deals with how the same thing can have different numerical properties depending on the unit, e.g. 4 pairs/8 shoes. Typical findings are that, presented with a collection of whole and broken objects, 3 year-olds can correctly count them as discrete objects, but only 5-6 year-olds as kinds [p.67]. In turn, children have difficulties in developing number units with higher power—here CON and COP should part company [p.70]. Implicated problems are the composition/decomposition of large numbers and place value\e.g., .80 N .8 = .08\with neither error detected by youngsters [p.81]. Key instructional recommendations are that a number unit is not an everyday object; units are constituted differently in different contexts; coordination of multiple units is complex [p.83]. Additive and multiplicative reasoning are separated in chapter 5. Relations in the former generate an independent unit [7 is neither 3 nor 4], whereas relations in the latter are determined through the others [12 is 3 times 4]. The proposal that addition has its origin in part-whole and other schemas is taken to be attractive, apart from its non-explanation of why the early availability of a schema leads to difficulty in school arithmetic. Actually, this is a general issue facing CON; but explanatory factors—pictures, language, familiarity—due to COP fare no better in securing their explanatory purchase [p.94]. Under CON, the seeds of multiplication are contained in initial enumeration; but without the formation and systematization of higher units, such as one packet containing five things, these seeds are barren. Yet under COP, instruction may be counter-productive if multiplication is taught as further addition [p.104]. In chapter 6, the complexity of fractions is attributed to how two numbers generate a single unit and how members of this triple are inter-related. A compounding factor is said to be the naive use of pictorial aids such as pie charts and the ambiguity of equal fractions without the formation of a higher unit. The latter is regarded as a serious problem for CON [p.108]. Particular flashpoints are improper fractions\ 6/5\and systematic ambiguities\ 1/2 b 2/3 N 5/8\in the absence of higher-order units. Encouraging ongoing work is reported, notably the Rational Number Project with a focus on multiple models for representing fractions along with inter-model translations. In a controlled study of fifth graders [n = 1600] from 66 classrooms, no difference was found between experimentals and controls where the RNP curriculum was frugal in contrast to a 30% gain by the former over the latter where the RNP curriculum was rich [p.125]. However, neither durability nor transfer was tested. And the adverse finding from comparative assessments—US fourth graders perform as well, but eighth graders far less well, than their international peers—is a well known cause for concern. Asserted to be a problem here is the ethics of instructional interventions [p.130]. Main implications for future work are presented in the two final chapters. For developmental psychology, these include: cognitive sequences in the serial formation of units, something not transparent to young children; identifying the contributions of canalization and instruction in ontogenesis; identifying specialized experiences, including the role of number representations; identifying general processes, such as language and working memory; exploiting cultural factors in instructional interventions; and securing a consensus as to the relative standings of CON and COP. For mathematics education, main implications are: gauging the fit between instructional practices and different psychological positions; tutor/learner task-matching; bridging the gap between number-solutions and number-arguments; design/delivery of effective instruction by grade with attention to the constraints of cultural norms. 3. Commentary I really do stand by my opening claim that Origins is welcome due to its resolute focus on positions, not merely issues. I encourage you to read and think through its analysis of its two leading positions about mathematical knowledge. All the same, I see three omissions, all major I would say. 3.1. Education “A major goal of this book is to examine issues concerning the teaching and learning of mathematics (traditionally the purview of mathematics education) from a developmental perspective” [p.2]. This sets up false expectations. The difference between psychology, a theoretical science, and education, an applied science, has long been attested (Bruner, 1966; De Corte, 2000; James, 1899). At least five relations run from-psychology-to-education (Smith, 2005): [i] [ii] [iii] [iv] [v]
independence: theory alone relevance: global claims implication: testable implications applications: tested implications inter-dependence: two-way testing.
Book review
63
Its final chapters place Origins at [iii]. That is quite an achievement when many of the studies reported are at [i] with most of the others at [ii], a seductive road to nowhere. The case for [v]—a science that is jointly theoretical and applied—has been well stated both in general using Pasteur's work (Stokes, 1997) and in the specific context of developmental psychology-and-education by Ann Brown with colleagues (Bransford, Brown & Cocking, 2001). Whilst this joint science is still a long way off, developmentalists could take a lead in the advance from [iii] to [iv]. In fact, they already have done so. One omission in Origins is manifest in its non-review of studies en route to [iv]. These studies lie at the interface, dealing with class size and the quality of classroom learning (Bennett, 1998; Biggs, 1998; Whitburn, 2000), or learning and pedagogy (Baroody & Dowker, 2003; Smith, 2004; Stigler & Hiebert, 1999). Other studies are already at [iv], notably interventions in mathematics and science learning driven by developmental models—this master-class work is on-going, repeatedly demonstrating huge gains in which experimentals outperform controls by up to 20% in national tests and examinations during childhood or adolescence (Adey, 2007; Shayer, 2008; Shayer & Adhami, 2003). The model behind these successful inventions is both Piagetian and Vygotskian (Shayer & Adey, 1981, 2002). It is astonishing that Origins did not cover this educationally significant evidence under a model due to the “two giants” of developmental psychology. 3.2. Reasoning Mathematical knowledge is interpreted in Origins to include reasoning, notably in chapter 5 where a theoretical problem is by-passed. Addition [qone and oneq] in arithmetic and in logic are not the same thing: specifically, 1 + 1 = 1 and 1 + 1 = 2 are both sound (Russell, 1964, §114). The same distinction was drawn by Piaget (1966, p.265): a tautology (A + A = A) in logic differs from an iteration (A + A = 2A) in arithmetic. Recognising “a” star and recognising “a” star is not thereby to recognise two stars—recall Frege's (1950) famous example of the Morning Star and the Evening Star and differences in “cognitive value”. Origins is silent about the origins of this distinction. Under CON and COP, why is a number-judgment made about equality rather than a classification through identity? A developmental problem is how the advance is made from one to the other. Piaget (1942; reviewed in Smith, 2002) gave an answer in terms of reasoning by mathematical induction [MI], incidentally a type of reasoning that is interesting in its own right, though by-passed in Origins. MI is distinctive, uniquely combining the necessity of deduction and the universality of induction. MI is recognized to be important in philosophy [its reducibility to the other two types of reasoning—Russell Yes, Poincaré No]; MI is used by mathematicians [Wiles' recent proof of Fermat's Last Theorem]; and MI is required in the curriculum for higher mathematics education. I know of no studies of its development during childhood other than a 1963 Genevan study, recently replicated (Smith, 2002; for commentary, see Rips, Bloomfield, & Asmuth, in press; Smith, in press). Two main conclusions were drawn in this replication: direct support for the Genevan conclusion that children aged 5– 7 years can reason by MI and indirect support for their conclusion that these children understand the necessity of this inference. Since current educational practice has confined MI to the advanced curriculum, a good opportunity to teach some form of MI during childhood is being lost. Two studies, and so much remains to be done. If MI is used by adults, and if adults were formerly children, what are the sequences and mechanisms of this advance, including interactions with other forms of reasoning? 3.3. Reasons The stance taken in Origins about the nature/nurture problem is standard with mathematical origins interpreted primarily through nativism [CON] or social experience [COP]. In the case of mathematical knowledge, both are seriously incomplete, explanatory failures with regard to two properties of mathematical knowledge, objectivity and necessity. Any mathematical truth is objective. Consider 4 + 3 = 7. If you count four children in his family and three in her family as “so six altogether,” the conclusion to draw is “you've made a mistake,” not “the laws of arithmetic have been falsified.” Further, any mathematical truth is necessarily true. Consider (4 + 3 = 7) = N (4 × 3 = 12). Three necessities are nested here: the antecedent (4 + 3 = 7) is an equality, the consequent (4 × 3 = 12) is an equality, and the relation (=N) linking them is entailment. All equalities and all entailments are necessitations, normatively speaking. What, then, is the origin of necessary knowledge, how is it acquired by contingent beings such as ourselves? Exactly that was Piaget's principal problem over 60 years (quoted in Smith, 2002, p.110). Piaget's (2006) answer was in terms of reasons whose role is “to introduce new necessities into systems where they were merely implicit or remained unacknowledged” [p.8]. Reasons straddle the gap between subjective-objective, contingentnecessary. For example, presented with a line of glasses and a line of bottles and asked whether there is the same in each, you might say “Yes—six in this line, six in that”. What matters is the normative framework behind this response (Piaget, 2006, Note 16): six is my favorite number six is what I observed six is the self-same number six is necessarily the self-same number. Adult-speak, of course; but I was young once, and Piaget's argument is that children's knowledge becomes adults' knowledge in virtue of the normativities in reasons. Although norms are not facts, normative facts are facts open to empirical investigation (Smith, 2006). Significantly, it is these normativities that enable human development to be on the right tracks from the origins of knowledge to its eventual destinations.
64
Book review
References Adey, P. (2007). The CASE for a general factor in intelligence. In M. J. Roberts (Ed.), Integrating the mind: Domain general versus domain specific processes in higher cognition (pp. 369−386). Hove, UK: Psychology Press. Baroody, A. J., & Dowker, A. (2003). The development of arithmetic concepts and skills: Constructing adaptive expertise. Mahwah. NJ: Erlbaum. Bennett, N. (1998). Annotation: Class size and the quality of educational outcomes. Journal of Child Psychology and Psychiatry, 39, 797−804. Biggs, J. (1998). Learning from the Confucian heritage: So size doesn't matter? International Journal of Educational Research, 29, 723−738. Bransford, J., Brown, A., & Cocking, R. (2001). How people learn. Washington, DC: National Academy Press. Bruner, J. (1966). Towards a theory of education. Cambridge, MA: Harvard University Press. De Corte, E. (2000). Marrying theory building and the improvement of school practice. Learning and Instruction, 10, 249−266. Frege, G. (1950). Translations from the philosophical writings of Gottlob Frege. Oxford: Blackwell. James, W. (1899). Talks to teachers on psychology. London: Longman. Piaget, J. (1942). Classes, relations, nombres: Essai sur les groupements de la logistique et sur la réversibilité de la pensée. Paris: Vrin. Piaget, J. (1966). Part II. In E. Beth & J. Piaget (Eds.), Mathematical epistemology and psychology (pp. 131−304). Dordrecht: Reidel. Piaget, J. (2006). Reason (translation and commentary by Leslie Smith). New Ideas in Psychology, 24, 1−29. Rips, L., Bloomfield, A., & Asmuth, J. (in press). From numerical concepts to concepts of number. Behavioral and Brain Sciences. Russell, B. (1964). The principles of mathematics, 2nd edition New York: W.W. Norton & Co, Inc. Shayer, M. (2008). Intelligence for education: As described by Piaget and measured by psychometrics. British Journal of Educational Psychology, 78, 1−29. Shayer, M., & Adey, P. (1981). Towards a science of science teaching. London: Heinemann. Shayer, M., & Adey, P. (2002). Learning intelligence. Buckingham: Open University Press. Shayer, M., & Adhami, M. (2003). Realising the cognitive potential of children 5-7 with a mathematical focus. International Journal of Educational Research, 39, 743−775. Smith, L. (2002). Reasoning by mathematical induction in children's arithmetic. Oxford: Elsevier Science Pergamon Press. Smith, L. (2004). Developmental epistemology and education. In J. Carpendale & U. Müller (Eds.), Social interaction and the development of knowledge (pp. 175−194). Mahwah, NJ: Erlbaum. Smith, L. (2005). Developmental psychology and education. In B. Hopkins (Ed.), The Cambridge encyclopedia of child development (pp. 487−490). Cambridge: Cambridge University Press. Smith, L. (2006). Norms and normative facts in human development. In L. Smith & J. Vonèche (Eds.), Norms in human development (pp. 103−137). Cambridge: Cambridge University Press. Smith, L. (in press). Mathematical induction and its formation during childhood. Behavioral and Brain Sciences. Stigler, J., & Hiebert, J. (1999). The teaching gap. New York: Free Press. Stokes, D. (1997). Pasteur's quadrant: Basic science and technological innovation. Washington, DC: Brookings Institute. Whitburn, J. (2000). Strength in numbers. London: NIESR.
Leslie Smith E-mail address: [email protected]. URL: http://www2.clikpic.com/ls99/.