21
ANDERSON NORTON and KIRBY DEATER-DECKARD MATHEMATICS IN MIND, BRAIN, AND EDUCATION: A NEO-PIAGETIAN APPROACH Received: 1 June 2013; Accepted: 12 January 2014 ABSTRACT. Because of their focus on psychological structures and operations, neo- Piagetian approaches to learning lend themselves to neurological hypotheses. Recent advances in neural imaging and educational technology now make it possible to test some of these claims. Here, we take a neo-Piagetian approach to mathematical learning in order to frame two studies involving the use of electroencephalography and functional magnetic resonance imaging imaging, as well as the use of iOS-based apps designed to elicit particular ways of operating with mathematics. Results could inform theories of mathematical learning and effective educational game design. KEY WORDS: educational technology, fractions, mathematics education, neuroscience Recent advances in neuroimaging have spurred numerous discussions on the potential value of an educational neuroscience to inform curriculum and instruction. Included among these discussions is the question of whether disparate methods and frameworks across education and neuroscience render educational neuroscience a bridge too far(Bruer, 1997). Optimism continues to grow as new studies emergestudies with clear implications for education. For example, functional magnetic resonance imaging (fMRI) indicates the existence of two distinct, early developing systems in infants for understanding of quantity (Demeyere, Rotshtein, & Humphreys, 2012; Revkin, Piazza, Izard, Cohen, & Dehaene, 2008): subitizing, by which infants can immediately distinguish collections from 1, up to 3 or 4; and magnitude comparisons, by which infants can distinguish collections of any size so long as their relative magnitudes exceed a certain threshold. This finding implies that educators can simultaneously leverage two approaches to early number sense, but successful attempts to bridge education and neuroscience will require careful attention to caveats inherent to the enterprise. Educational commercialization of neuroscience findings presents one important danger to educational neuroscience. By isolating particular findings and applying them to the classroom, with no consideration of educational practice, policy makers and the education industry could justify the use of ineffective, or even harmful, products (Varma, McCandliss, & Schwartz, 2008). For example, fMRI studies indicate that International Journal of Science and Mathematics Education 2014 # National Science Council, Taiwan 2014

MATHEMATICS IN MIND, BRAIN, AND EDUCATION: A NEO-PIAGETIAN APPROACH

  • Upload
    kirby

  • View
    214

  • Download
    0

Embed Size (px)

Citation preview

Page 1: MATHEMATICS IN MIND, BRAIN, AND EDUCATION: A NEO-PIAGETIAN APPROACH

ANDERSON NORTON and KIRBY DEATER-DECKARD

MATHEMATICS IN MIND, BRAIN, AND EDUCATION:A NEO-PIAGETIAN APPROACH

Received: 1 June 2013; Accepted: 12 January 2014

ABSTRACT. Because of their focus on psychological structures and operations, neo-Piagetian approaches to learning lend themselves to neurological hypotheses. Recentadvances in neural imaging and educational technology now make it possible to test someof these claims. Here, we take a neo-Piagetian approach to mathematical learning in orderto frame two studies involving the use of electroencephalography and functional magneticresonance imaging imaging, as well as the use of iOS-based apps designed to elicitparticular ways of operating with mathematics. Results could inform theories ofmathematical learning and effective educational game design.

KEY WORDS: educational technology, fractions, mathematics education, neuroscience

Recent advances in neuroimaging have spurred numerous discussions on thepotential value of an educational neuroscience to inform curriculum andinstruction. Included among these discussions is the question of whetherdisparate methods and frameworks across education and neuroscience rendereducational neuroscience “a bridge too far” (Bruer, 1997). Optimismcontinues to grow as new studies emerge—studies with clear implicationsfor education. For example, functional magnetic resonance imaging (fMRI)indicates the existence of two distinct, early developing systems in infants forunderstanding of quantity (Demeyere, Rotshtein, & Humphreys, 2012;Revkin, Piazza, Izard, Cohen, &Dehaene, 2008): subitizing, by which infantscan immediately distinguish collections from 1, up to 3 or 4; and magnitudecomparisons, by which infants can distinguish collections of any size so longas their relative magnitudes exceed a certain threshold. This finding impliesthat educators can simultaneously leverage two approaches to early numbersense, but successful attempts to bridge education and neuroscience willrequire careful attention to caveats inherent to the enterprise.

Educational commercialization of neuroscience findings presents oneimportant danger to educational neuroscience. By isolating particularfindings and applying them to the classroom, with no consideration ofeducational practice, policy makers and the education industry couldjustify the use of ineffective, or even harmful, products (Varma,McCandliss, & Schwartz, 2008). For example, fMRI studies indicate that

International Journal of Science and Mathematics Education 2014# National Science Council, Taiwan 2014

Page 2: MATHEMATICS IN MIND, BRAIN, AND EDUCATION: A NEO-PIAGETIAN APPROACH

multiplication involves pronounced activity in the angular gyrus, a regionof the brain associated with verbal processing (Delazer et al., 2005). Asimple approach to educational neuroscience might use this finding tojustify instructional tasks designed to stimulate that region as a way toimprove students’ multiplicative reasoning. Such tasks might includememory games or reciting multiplication facts. However, most educatorswould recognize the reason for the association of the angular gyrus andmultiplication, and they would also recognize the major flaw in theprescribed approach to multiplicative reasoning. Namely, whenperforming multiplication, students often rely on a memorized multipli-cation table whose recollection would activate the angular gyrus due toverbal processing (Wang, Lin, Kuhl, & Hirsch, 2007); althoughmemorizing the table can improve fluency, it does little to addressmultiplicative reasoning, which involves the production and coordinationof composite units (e.g., 12 as four units of 3; Steffe, 1994). So if we areto build an educational neuroscience, examples like this accentuate theneed for us to build that bridge as a two-way street (Fischer, Daniel,Immordino-Yang, Stern, Battro, & Koizumi, 2007; Mason, 2009; Szucs& Goswami, 2007; Varma et al., 2008).

In the current paper, we consider a constructivist perspective as anavenue for bridge building, because it has had a substantial impact onmathematics education over the past few decades and because it includeshypotheses about learning that are readily testable using neurosciencemethods. How can neuroscience affirm, inform, and refine constructivisttheories of learning, whereby pruning and plasticity in neural connectionswould correspond with changes in ways of operating? We consider tworelated questions specific to mathematics education that might beaddressed through neuroscience studies—studies that might constitutethe beginning of a mathematics educational neuroscience.

1. What are the neurological correlates for dynamic mathematicaloperations, such as partitioning, iterating, and splitting? Furthermore,is there neurological evidence for splitting as the simultaneouscomposition of partitioning and iterating?

2. Are neural networks for mathematical knowing hierarchical? Inparticular, are various levels of units coordinating represented inhierarchical structures within the intraparietal sulcus? And doesactivity in the intraparietal sulcus function similarly across contexts:whole number, integers, fractions, algebra?

Our questions are framed by theoretical constructs on which wewill elaborate within the discussion of each question. These same

ANDERSON NORTON AND KIRBY DEATER-DECKARD

Page 3: MATHEMATICS IN MIND, BRAIN, AND EDUCATION: A NEO-PIAGETIAN APPROACH

theoretical constructs—chiefly two hypothetical learning trajectories(HLTs; Simon, 1995)—informed the design of two educationalgames. Thus, following the discussion of each question, we describethe corresponding game, as a means of operationalizing the HLTsand as a means of generating tasks that could be used within theneuroimaging studies. The two games, Candy Factory and CandyDepot, are iOS-based apps designed by the Learning TransformationResearch Group at Virginia Tech (LTRG; http://ltrg.centers.vt.edu/),with support from the National Science Foundation. Before elaborat-ing on these apps, the HLTs, our research questions, and theirrelationships, we provide a brief overview of our approach toeducational neuroscience.

NEUROCONSTRUCTIVISM AND THE NEO-PIAGETIAN PERSPECTIVE

In explaining the psychological origins of mathematics, Jean Piaget(1970/1968) posited structures, or “systems closed under transforma-tion” (p. 6), that develop through children’s activity and reflectiveabstractions of that activity. Piaget’s structuralism suggests that neuralsubstrates for mathematical knowledge should be self-regulating,dynamic, and hierarchical. Within neuroconstructivism, such proper-ties are explained in terms of plasticity, which arises from theinteraction of biological components of brain (Westermann et al.,2007). Whereas particular regions of the brain display functionalbiases, those functions are adaptable and change through learning(Karmiloff-Smith, 2009). The brain continually reorganizes itself inresponse to experience and provocation, developing more and morepowerful ways of operating through changes at both the cellular andcortical levels (Epstein, 2001; Fischer & Bidell, 2006; Quartz &Seinowski, 1997).

In line with this neo-Piagetian and neuroconstructivist view, thecurrent paper focuses on several brain regions that are implicated inmathematical knowing and learning, with emphasis on their roles inthe dynamic operations that constitute an individual’s mathematicalknowledge. These regions include the anterior cingular gyrus, theintraparietal sulcus, and the occipital lobe. Figure 1 indicates thelocation of these regions, which we discuss throughout the paper. Tocharacterize how neural imaging of these regions might inform neo-Piagetian theories of learning, we offer a more detailed exampleregarding subitizing.

MATHEMATICS IN MIND, BRAIN, AND EDUCATION

Page 4: MATHEMATICS IN MIND, BRAIN, AND EDUCATION: A NEO-PIAGETIAN APPROACH

NEUROLOGICAL EVIDENCE THAT SUBITIZING IS TOPOLOGICAL

Subitizing describes the innate ability for humans (and some otherspecies) to immediately distinguish between collections of one, two,three, and four objects (Kaufman, Lord, Reese, & Volkmann, 1949). Thespeed and precociousness of apprehension distinguish this ability fromcounting. Recent studies in neuroscience have elaborated on thedistinction (Sophian & Crosby, 2008; Demeyere et al., 2012), while alsodistinguishing subitizing from numerical estimation (Revkin et al., 2008).In particular, numerical estimation follows Weber’s law, which states thatprecision for estimations of quantity diminishes logarithmically as the sizeof the collection increases. If subitizing were a product of numericalestimation for small numbers, then we would expect the degree ofprecision to fit a logarithmic curve representing Weber’s law applied tonumbers from 1 to 4 and beyond. To the contrary, Revkin and colleaguesfound a steep decrease in precision for numbers larger than 4, indicatingthat subitizing is not a product of precise numerical estimation.

For neo-Piagetians, the existence of an innate and precise numericalability, such as subitizing, might come as a surprise. After all, Piaget(1942) characterized early number understanding as a developmentalprocess that merges the algebraic structure of class inclusion (e.g., 4 iscontained within 5) with an ordering relation (e.g., 4 precedes 5). Morerecent research from a Piagetian perspective implicates number sequencesas the result of similar coordinations, developed over several years ofchildhood (Steffe, 1992). However, in addition to algebraic structures andordering relations, Piaget posited a third “mother structure” supportingstudents’ constructions of mathematics, namely topological structures.

Although children might rely on subitizing to support their earlydevelopment of number sense, subitizing might not begin as numerical at

Anterior cingulate gyrus

Occipital lobe

Intraparietal sulcus

Posterior superior parietal lobe

Angular gyrus

Figure 1. Diagram of the brain. Lateral surface of the left hemisphere; dashed link regionon interior/medial surface

ANDERSON NORTON AND KIRBY DEATER-DECKARD

Page 5: MATHEMATICS IN MIND, BRAIN, AND EDUCATION: A NEO-PIAGETIAN APPROACH

all. Rather, an infant’s ability to distinguish between collections of 1, 2, 3,and 4 objects could be the product of topological distinctions betweenthem. These distinctions persist despite rearrangements of the objects: 2 isdistinct from 1 in having a property of separation; 3 is distinct from 1 and2 in having a property of between-ness (Wagner & Walters, 1982).Although not all configurations of 4 have unique properties to distinguishthem from 3, subitizing with 4 is also less reliable: Sophian & Crosby(2008) found that subitizing 4 requires additional attentional processing,possibly because it results from conceptually subitizing a pair of 2s; andPotter & Levy (1968) found that students apprehend 4 with the fewesterrors when the four dots were arranged randomly (versus linearly or asan array), possibly because only random arrangements generate thetopologically distinct property of enclosure (one dot in the interior of thetriangle formed by the other three dots).

If subitizing were a product of topological distinctions, we wouldexpect to find neural substrates common to subitizing and knownapplications of topology. Indeed, brain lesion studies demonstrate thatpatients’ ability to subitize (but not their ability to count) suffers withdamage to the posterior occipital cortex (Demeyere et al., 2012). Thatsame region of the brain (primarily responsible for visual processing andmotion) mediates ability to apprehend Kanizsa figures (Kruggel,Hermann, Wiggins, & von Cramon, 2001), like those shown in Fig. 2.This suggests that subitizing and apprehending Kanizsa figures might berelated to the same mental activity1.

Apprehending the Kanizsa figure in Fig. 2 involves producing a closedand bounded square by continuously extending segments as suggested.Closure, boundedness, and continuity—these are all topological proper-

Figure 2. Kanizsa figure

MATHEMATICS IN MIND, BRAIN, AND EDUCATION

Page 6: MATHEMATICS IN MIND, BRAIN, AND EDUCATION: A NEO-PIAGETIAN APPROACH

ties. Thus, neuroscience studies suggest that subitizing might involve atopological structure. Furthermore, the neurological distinction betweensubitizing and numerical processes (e.g., counting and magnitudeestimations) could correspond with distinctions Piaget made betweenthree mother structures for logico-mathematical development: Countingarises from algebraic and ordering structures; subitizing arises fromtopological structure. Future neuroscience studies could contribute insimilar ways to neo-Piagetian theories of learning, by testing and refiningthem. In the next two sections, we outline two potential studies and theirimplications for mathematics education. These studies correspond toresearch questions 1 and 2.

RESEARCH QUESTION 1

From a neo-Piagetian perspective, the fundamental building blocks formodeling students’ mathematics are mental actions, or operations, thathave been organized within structures for reversing and composing them(Piaget, 1970; von Glasersfeld, 1995). Partitioning and iterating constitutefundamental operations for modeling students’ fractions schemes (Steffe& Olive, 2010). Partitioning involves producing equally sized parts froma continuous whole; iterating involves making connected copies of a part,measuring off a length or area. Researchers have assessed such operationsusing tasks similar to those in Fig. 3 (Wilkins & Norton, 2011).

Initially, these operations arise from physical activity of sharing,folding, and copying objects, and from reflections on such activity.However, students can reorganize mental actions as objects, in the sensethat they can be acted upon. For example, students might learn to partitiona partition (i.e., recursive partitioning; Steffe & Olive, 2010) to producemultiplicities of parts within parts. In other words, when partitioning hasbecome objectified—not merely as figurative partitions drawn out on

Figure 3. Iterating task (top) and partitioning task (bottom)

ANDERSON NORTON AND KIRBY DEATER-DECKARD

Page 7: MATHEMATICS IN MIND, BRAIN, AND EDUCATION: A NEO-PIAGETIAN APPROACH

paper, but as an operative result of mental action—students can act onthat object through further partitioning, keeping track of the unitsproduced.

Students can also learn to coordinate partitioning and iterating asinverse operations so that one operation nullifies the other. For instance, ifa student partitions a line segment into five equal parts and then iteratesone of those parts five times, she will reproduce the original whole. Byreflecting on this way of operating—sequentially coordinatingpartitioning and iterating—students can reorganize the two operations asa single operation, called splitting2 (Steffe, 2002). Splitting is thesimultaneous composition of partitioning and iterating. As such, it canbe thought of as an identity operation, but it contains each of the basicoperations (Wilkins & Norton, 2011). A distinguishing characteristic ofstudents who can split is the ability to solve tasks like the one shown inFig. 4.

The task is iterative in nature, referring to a stick that is “5 times aslong”, but the solution requires the student to partition the given stick intofive equal parts. Thus, the solution requires students to anticipate theinverse relationship between partitioning and iterating (as opposed to task3b, which elicits partitioning alone). Recent research suggests thatstudents who can operate in this way have distinct advantages over otherstudents in developing more advanced conceptions of fractions(Hackenberg, 2007; Norton & Wilkins, 2012) and in making progresstoward algebraic reasoning (Hackenberg, 2010). So, identifying neuralsubstrates for partitioning, iterating, and splitting could have substantialconsequences for mathematics education. On the one hand, it wouldprovide neurological evidence for a theorized way of operating, thusfurther affirming the theory; on the other, it could inform researchers as tohow partitioning and iterating operations become reorganized as asplitting operation at the neurological level, thus suggesting educationalapproaches to supporting this development.

Research question 1 asks, “What are the neurological correlates fordynamic mathematical operations, such as partitioning, iterating, andsplitting?” Neuroimaging research has demonstrated sensitivity todetecting changes in brain activation related to new mathematicalknowledge (Delazer et al., 2005; ,Quinn et al. 2004; Varma &Schwartz, 2008). For example, Qin et al. (2004) observed learning-

Figure 4. Splitting task

MATHEMATICS IN MIND, BRAIN, AND EDUCATION

Page 8: MATHEMATICS IN MIND, BRAIN, AND EDUCATION: A NEO-PIAGETIAN APPROACH

related shifts in brain activity within regions responsible for mathematicalproblem solving involving the manipulation of symbols. Reducedactivation of the prefrontal cortex arising from practice was observed inadolescents who were first learning algebra, as well as in adults who wereexperienced algebra problem solvers. However, the adolescent learnersalso showed reductions in activity with practice in a separate parietalcortex region, whereas adults did not. This suggested a distinct pattern ofchanges in activation in multiple brain regions for adolescents as theyacquired a new knowledge for symbol manipulation.

Turning to our specific question about partitioning and iterating,existing neuroimaging studies indicate that these functions should becorrelated with activity in the intraparietal sulcus and the posteriorsuperior parietal lobule (e.g., Çiçek, Deouell, & Knight, 2009;Rosenberg-Lee, Lovett, & Anderson, 2009; Wilkinson & Halligan,2003). In particular, at least two studies (Çiçek et al., 2009; Wilkinson& Halligan, 2003) found a correspondence between the intraparietalsulcus (IPS) and the mental bisection of a (continuous) line segment; andanother (Rosenberg-Lee et al., 2009) found a correspondence between theposterior superior parietal lobule (PSPL) and spatial aspects of numberprocessing. An existing mathematics education study indicates thatpartitioning and iterating operations are nearly ubiquitous among middleschool students, but that about half of these students lack splittingoperations (Wilkins & Norton, 2011). The collective findings from thesestudies imply that neuroimaging studies of middle school students, usingtasks like the ones illustrated in Figs. 3 and 4, should demonstratedifferentiated activity in the IPS and the PSPL, as students progresstoward the construction of splitting operations.

We propose an accelerated longitudinal design (Tonry, Ohlin, &Farrington, 1991) whereby researchers would work with three cross-sections of middle school students: those who partition and iterate, but notsplit; emerging splitters (students in transition); and students whoregularly split in appropriate situations. Among nonsplitters, we wouldexpect to see activity in two distinct regions within the IPS and/orPSPL—one corresponding to partitioning and one corresponding toiterating. We would expect emerging splitters to experience conflict inresponding to the tasks. For example, their initial response might be toiterate the given stick (ref. Fig. 4), but they might recognize an error inthis way of operating or its associated result. Thus, for these students, wewould expect to see greater activity in the prefrontal cortex and,particularly, the anterior cingulate cortex (involved in resolving conflict).Among splitters, we would expect to see greater functional connectivity

ANDERSON NORTON AND KIRBY DEATER-DECKARD

Page 9: MATHEMATICS IN MIND, BRAIN, AND EDUCATION: A NEO-PIAGETIAN APPROACH

between those regions associated with partitioning and iterating, and lessactivity in the prefrontal cortex. Previous research (e.g., Rivera, Reiss,Eckert, & Menon, 2005) has indicated such neurological differencesbetween more advanced and less advanced students when engaged inmental arithmetic: less advanced students demonstrate greater frontal lobeactivity, presumably due to greater demands on working memory, andmore advanced students demonstrate greater activity in regions of theparietal and occipital lobes, including the IPS.

The longitudinal aspect of the study could involve the use of gameslike Candy Factory (described in the next section) to provoke growthwithin each of the first two cross-sections, so that researchers caninvestigate how neural activity/connectivity evolves as students movefrom one cross-section to the next. In that regard, as students gain moreexperience with the game, we would expect decreased activity in theanterior cingulate cortex and increased activity in the IPS and/or PSPLwhen solving splitting tasks. For electroencephalography (EEG) imaging,the game could be used as a tool for both intervention and assessmentduring the data collection, but localization of associated regions would belimited to the lobular level (frontal and parietal). For fMRI imaging, tasksfrom the game could be projected onto a screen during the scan, butstudents would need to respond imaginatively or through simple buttonpresses; thus, researchers would need to infer the students’ mental activitybased on previous observations of game play.

Findings from such a study could have substantial implications forinstruction, as well as theories of learning. If studies demonstrate thatsubstrates for splitting do not connect with substrates for both partitioningand iterating, the theory for how splitting arises would be called intoquestion. On the other hand, if connections between substrates affirm thetheory, the particular regions involved would be of great interest formathematics educators. For instance, we would want to know whetherregions associated with partitioning and iterating overlap and whethersplitting activity is associated with activation within the overlappingregion or a third region. If a third region is involved, we would want toknow what other activities are associated with that region, which mightsuggest connections between splitting and those activities. For example, ifwe found the neural substrate for splitting to be more strongly associatedwith kinesthetic experience (Martin, Jacobs, & Frey, 2011), this findingmight inform the use of manipulatives for supporting the construction ofsplitting. Moreover, longitudinal data, especially from emerging splitters,could provide clues for how partitioning and iterating become integrated,both behaviorally and neurologically.

MATHEMATICS IN MIND, BRAIN, AND EDUCATION

Page 10: MATHEMATICS IN MIND, BRAIN, AND EDUCATION: A NEO-PIAGETIAN APPROACH

CANDY FACTORY

The CandyFactory Educational Game (Candy Factory) is the first ofthree instructional apps designed by LTRG. The apps are designed toelicit students’ existing ways of operating and provoke new ways ofoperating through game play. In particular, Candy Factory elicitsstudents’ partitioning and iterating operations and provokes students tocoordinate those two operations in ways that lead to the construction ofthe splitting operation (Norton & Wilkins, 2013). The game consists offive levels that progressively challenge students to coordinate partitioningand iterating in service of satisfying customer orders of candy bars offractional sizes.

At level 1, students can assimilate the tasks within a part-wholescheme, which, like partitioning and iterating, is nearly ubiquitous amongmiddle school students (Norton & Wilkins, 2009). The scheme involvesproducing and naming fractions based on the number of equal pieces in aproper fraction out of the number of equal pieces in the whole (e.g., 3/5 isthree equal pieces out of five equal pieces in the whole). The purpose oflevel 1 is to acquaint students with the game in a way that should bemathematically meaningful to most students. Figure 5 illustrates a sampletask at this level. Note that the customer order and the whole candy barare already partitioned for the student.

Beginning at level 2, the customer order and the whole candy bar areno longer partitioned, so that students must produce appropriate partitionswithin the whole to produce the customer order. Because customer orders

Figure 5. Candy Factory, level 1

ANDERSON NORTON AND KIRBY DEATER-DECKARD

Page 11: MATHEMATICS IN MIND, BRAIN, AND EDUCATION: A NEO-PIAGETIAN APPROACH

within level 2 are always unit fractions of the whole, students canappropriately partition the whole by mentally iterating the length of thecustomer order within the whole, thus segmenting the whole into n equalparts. This would involve a sequential coordination of iterating andpartitioning. For example, in the task illustrated in Fig. 6, the studentmight see that the customer order fits into the whole three timesproducing three equal parts, implying that the customer order is 1/3 ofthe whole candy bar.

Customer orders at levels 3 and 4 introduce non-unit proper fractionsand improper fractions, respectively. Corresponding tasks require furthercoordination of partitioning and iterating, as students must iterate a unitfractional part, 1/n, to produce a non-unit fraction, m/n. For example,Fig. 7 illustrates the production of 9/6 (3/2) by iterating a 1/6 part, whichresulted from partitioning the whole into six equal parts. The studentwould continue to iterate (copy) the 1/6 part (top of Fig. 7) by dragging itinto the box (middle of Fig. 7) until producing the length of the customerorder (bottom of Fig. 7). The game keeps track of these iterations byspecifying the resulting fraction; for instance, Fig. 7 displays the result ofdragging two copies of the 1/6 bar into the box: “2/6.” Level 5 requiresstudents to reverse this way of operating, to produce the whole from agiven (proper or improper) fraction of it.

Games like Candy Factory provide special opportunities for re-searchers to conduct neuroscience studies from a neo-Piagetian perspec-tive because, if the mental actions of partitioning and iterating do indeedhave identifiable neural correlates, game play could render their

Figure 6. Candy Factory, level 2

MATHEMATICS IN MIND, BRAIN, AND EDUCATION

Page 12: MATHEMATICS IN MIND, BRAIN, AND EDUCATION: A NEO-PIAGETIAN APPROACH

coordination and reorganization observable, both in terms of the student’sphysical actions with an iOS device, and in terms of corresponding neuralimaging (EEG) during game play. Thus, researchers could begin to relatethe construction of splitting to neurological reorganizations. As notedpreviously, games like Candy Factory could also support acceleratedlongitudinal design studies by promoting growth among three cross-sections of students: those who have not begun coordinating their existingpartitioning and iterating operations, emerging splitters, and splitters.

RESEARCH QUESTION 2

Whole number multiplication, integer addition, fractions concepts, andalgebraic reasoning present notorious challenges for middle schoolstudents in the USA. A growing body of research indicates that thelevels of units students coordinate mediate opportunities for learningacross these domains (Izsák, Jacobsen, de Araujo, & Orrill, 2012; Ellis,2007; Olive & Çağlayan, 2008). With integers, the units include parts of acomposite whole when working through 0. For example, reasoning thevalue of −5+7 requires the student to consider 7 as a unit composed of 5and 2 so that the student can work from −5 through 0, to arrive at thevalue of 2 (Ulrich, 2012). With fractions, the units often include a unitfraction and the whole, as in understanding 7/5 as a unit composed of 7units of 1/5, five of which constitute the whole (Steffe & Olive, 2010).Finally, within algebra, even simple equations involve units of 1 and x.

Figure 7. Candy Factory, level 3

ANDERSON NORTON AND KIRBY DEATER-DECKARD

Page 13: MATHEMATICS IN MIND, BRAIN, AND EDUCATION: A NEO-PIAGETIAN APPROACH

For example, reasoning with the equation 5x= 7 involves understanding 7as a unit of 5 units of x (Hackenberg, 2010).

Units coordinating refer to the structures and operations students canproduce in activity and those students use to assimilate situationsinvolving embedded quantities. Consider the following task: A studentis given a short bar, a medium bar, and a long bar and is told to pretendthat the short bar fits into the medium bar three times and that the mediumbar fits into the long bar four times; then, the student is asked todetermine how many times the small bar would fit into the long bar (seeFig. 8).

Table 1 outlines typical responses to the task, indicating the number oflevels students coordinate. Students at level 1 assimilate only one level ofunits at a time, though they can coordinate two levels of units in activity,

TABLE 1

Levels of units coordination

Students’ unit structuresStudents’ reasoning on the barproblem

Level 1 Students can take one levelof units as given, and maycoordinate two levels ofunits in activity.

Students mentally iterate the short bar,imagining how many times it wouldfit into the longer bar. This activitymight be indicated by head nods orsub-vocal counting.

Level 2 Students can take two levelsof units (a composite unit)as given, and may coordinatethree levels of units in activity.

Students mentally iterate the mediumbar four times, with each iterationrepresenting a 3. This activity mightbe indicated by the student uttering“3, 3, 3, and 3; 12.”

Level 3 Students can take three levelsof units (a composite unit ofcomposite units) as given,and can thus flexibly switchbetween three-level structures.

Students immediately understand thatthere are four threes in the long bar.This assimilation of the task mightbe indicated an immediate responseof “12”, buttressed by an argumentthat 12 is four 3 s.

Figure 8. Units coordination task

MATHEMATICS IN MIND, BRAIN, AND EDUCATION

Page 14: MATHEMATICS IN MIND, BRAIN, AND EDUCATION: A NEO-PIAGETIAN APPROACH

such as when iterating one unit within another unit to determine howmany times it fits. Students at level 2 can assimilate two levels of unitsand can coordinate three levels of units though activity, such as iterating acomposite unit (e.g., the medium bar as three small bars) within a thirdunit (the long bar). Students at level 3 can assimilate three levels of unitsand anticipate moving flexibly between them without having to carry outthe associated activity. We think of these students as having a structurefor simultaneously coordinating all three levels of units: a unit (the longbar) containing four units (medium bars), each of which contains fourunits (small bars).

The levels of units students coordinate are modeled as structures, butdo these hypothetical structures correspond to neurological ones? Becauseexisting neuroscience research implicates the IPS in estimations andcalculations of quantity (e.g., Dehaene, 1997; Ischebeck, Schocke, &Delazer, 2009), we would expect to find that area to be a neural correlateof such structures. Because existing mathematics education researchindicates that units coordinating develops across contexts—wholenumber, integers, fractions, and algebra—we would expect to findfunctional similarities across those contexts. Thus, in approachingresearch question 2, we propose the use of cross-contextual tasks formiddle school students, within neuroimaging studies that focus on activitywithin and around the IPS.

The research by Dehaene (1997) has implicated the role of the IPS innumerical estimation and computation. The accumulator model fornumber (Meck & Church, 1983), which Dehaene draws upon, suggestsa neurological hierarchy within and around the IPS, in which specificneurons for specific numbers fire when specific thresholds are reached(Nieder & Dehaene, 2009; Piazza, Izard, Pinel, Le Bahan, & Dehaene,2004). Likewise, we expect any units coordinating hierarchy to betemporal, as well as spatial, in the sense that activity from one region ofthe brain would shift to a connected region when building up or breakingdown composite units. Furthermore, we expect these regions to center onthe IPS but possibly include neighboring regions, such as the superiorparietal lobule, the inferior parietal lobule, and the sensorimotor cortexregion associated with the hand (Simon, Mangin, Cohen, Le Bihan, &Dehaene, 2002). Finally, we expect greater frontal lobe activity duringunits coordinating tasks among students who have not constructed astructure for coordinating the necessary levels of units, due to thecognitive demand.

Specifically, we expect students at level 1 (ref. Table 1) to demonstrateactivity in and around the IPS when solving tasks that involve only one

ANDERSON NORTON AND KIRBY DEATER-DECKARD

Page 15: MATHEMATICS IN MIND, BRAIN, AND EDUCATION: A NEO-PIAGETIAN APPROACH

level of units, such as counting a collection of objects. When tasksinvolve a second level of units, such as determining how many 3s are in acollection of 18 objects, students might successfully solve the task, butwe would expect to see increased activity in the prefrontal cortex (inadditional to activity in and around the IPS), associated with controlledattention (Rueda, Posner, & Rothbart, 2005) to building a second level ofunits in activity. As students progress the level 2, need for controlledattention in resolving such tasks would be reduced (along with activity inthe prefrontal cortex), and we would expect to see changes in and aroundthe IPS. For example, we might notice activity in a broader region ormore pronounced activity within the same region. Such changes couldindicate spatial or temporal hierarchies in and around the IPS. Likewise,we would expect to see a similar progression in neural activity associatedwith tasks involving three levels of units, as students progress to level 3.

Competing hypotheses on neural substrates for units coordinationmight focus on the role of the angular gyrus, which is associated withremembering multiplication facts. Several studies have demonstratedincreased activity in the angular gyrus among experts performingmultiplication tasks, when compared to novices (e.g., Chochon, Cohen,van de Moortele, & Dehaene, 1999; Delazer et al., 2005; Grabner et al.,2009). Meanwhile, neural activity among novices shows greater activityin the frontal lobe (Rivera et al., 2005). It seems that experts rely onretrieval of number facts as a means for reducing the level of controlledattention (or, more generally, working memory) necessary to complete thetasks (Grabner et al., 2007). It is possible that semantic memory retrievalcould also explain students’ abilities to assimilate higher levels of units. Ifthis were the case, higher levels of units coordination would be associatedwith not only decreased activity in the frontal lobe, but also increasedactivity in the angular gyrus, with no need for a hierarchical structure. Onthe other hand, if our hypothesis were confirmed, it would provideindication of neural activity associated with multiplicative reasoning,rather than retrieval.

CANDY DEPOT

The levels described in Table 1 correspond to levels of game play in asecond app, called Candy Depot. Building on the theme from the CandyFactory, Candy Depot engages students in packaging candy bars intobundles and packaging bundles into boxes. Then, students must fill truckswith bulk orders, specified as either a whole number of bars or a

MATHEMATICS IN MIND, BRAIN, AND EDUCATION

Page 16: MATHEMATICS IN MIND, BRAIN, AND EDUCATION: A NEO-PIAGETIAN APPROACH

fractional number of bundles or boxes. For example, Fig. 9 illustrates asituation in which the student would need to fill an order for 50 bars,using single bars, bundles of four bars each, and boxes of five bundleseach.

Game rewards (money and trophies) are based on the efficiency withwhich students fill the bulk orders, both in terms of time and number ofitems used. For example, a student operating at level 1 might fill the orderby dragging 50 single bars onto the truck. Even if a student does thisquickly, the action involves a relatively large number of items, thusrendering the action inefficient. Students at level 1 might achieve a higherefficiency score by coordinating the number of bars within a bundle, oreven the number of bars in a box. Although students at level 1 wouldhave to coordinate these two levels of units in activity, thus slowing themdown, they are still likely to achieve a higher overall efficiency score, bygreatly reducing the number of items used. Students at level 2 cananticipate how to coordinate these two levels of units (e.g., each bundlecould be assimilated as a unit of four bars, and each box could beassimilated as a unit of 20 bars), which would help them work morequickly. Students at level 3 could work more efficiently still, byanticipating the various ways 50 bars is composed of boxes, bundles,and bars (e.g., 50 bars as two boxes, two bundles, and two bars). As such,

Figure 9. Candy Depot screen shot

ANDERSON NORTON AND KIRBY DEATER-DECKARD

Page 17: MATHEMATICS IN MIND, BRAIN, AND EDUCATION: A NEO-PIAGETIAN APPROACH

Candy Depot is designed to elicit students’ units coordinating activity andprovokes growth, through activity, to higher levels of units coordination.Similar to Candy Factory, these features render Candy Depot usefulwithin a second accelerated longitudinal design study—one that assessesand promotes students’ development of units coordination.

CONCLUDING REMARKS

Hypothetical learning trajectories (Simon, 1995) describe how studentsmove from less advanced to more advanced ways of operating and howteachers can support this progression. Here, we have provided twoexamples: the construction of the splitting operating from partitioning anditerating operations (Steffe & Olive, 2010; Wilkins & Norton, 2011); andthe advancement from n levels of units coordination to n+ 1 levels ofunits coordination (Hackenberg & Lee, 2012; Norton & Boyce, 2013;Steffe & Olive, 2010). The studies we suggest have the potential toadvance our understanding of these HLTs by incorporating two forms oftechnology—neuroimaging and tablet-based apps.

Games like the Candy Factory and Candy Depot, which elicit students’available mental actions and provoke their reorganization, provide specialopportunities for researchers to conduct neuroscience studies from a neo-Piagetian perspective. If the mental actions do indeed have identifiableneural correlates, game play could render their coordination andreorganization observable, both in terms of the student’s physical actionswith an iOS device, and in terms of corresponding neural imaging (EEG)during game play. Thus, researchers could begin to relate hypotheticalreorganizations, such as those specified by an HLT, to neurologicalreorganizations.

Apps based in portable devices, like the iPad, inherit particularaffordances and limitations with regard to neuroimaging studies. On theone hand, iOS devices allow for a wide variety of physical actions thatmight evoke intended mental actions (Olive, 2000). For example, CandyFactory uses fingers swipes to evoke partitioning and multiple drags toevoke iteration. On the other hand, such devices generally cannot be usedduring fMRI imaging (see Tam, Churchill, Strother, & Graham, 2011 foran exception). Even during EEG imaging, we need to be careful thatphysical activity is not disruptive to the electrical activity being recorded.One way around this limitation is to use the devices only during teachingexperiments (Steffe & Thompson, 2000) designed to provoke the develop-ment of targeted ways of operating (e.g., splitting and units coordination),

MATHEMATICS IN MIND, BRAIN, AND EDUCATION

Page 18: MATHEMATICS IN MIND, BRAIN, AND EDUCATION: A NEO-PIAGETIAN APPROACH

and then to conduct EEG and fMRI interviews as pre- and post-tests, withoutthe device. Students’ available ways of operating would be inferred fromtheir activity at the beginning and end of the teaching experiment, and thenstudents could respond to similar tasks during EEG and fMRI interviews,through verbal response or simple button presses. Students could even viewa projection of the iPad game screen during fMRI interviews and be asked toimagine their response to the game tasks.

ACKNOWLEDGMENTS

This work and the development of the two related apps—CandyFactoryand CandyDepot—were supported by a grant from the National ScienceFoundation (DRL-1118571).

NOTES

1 We note that there is a danger of “reverse inference” here (Poldrack, 2006). Theposterior occipital cortex is associated with several activities, so its activation during bothsubitizing and apprehending Kanizsa figures does not necessarily imply that those twoactivities involve the same mental action—but does suggest that they might.

2 We note that the extensive work on splitting by Confrey (e.g., 1994) refers to arelated but distinct operation—one that does not involve iterating.

REFERENCES

Bruer, J. T. (1997). Education and the brain: A bridge too far. Educational Researcher,26(8), 4–16.

Chochon, F., Cohen, L., Van De Moortele, P. F. & Dehaene, S. (1999). Differentialcontributions of the left and right inferior parietal lobules to number processing. Journalof Cognitive Neuroscience, 11(6), 617–630.

Çiçek, M., Deouell, L. Y. & Knight, R. T. (2009). Brain activity during landmark and linebisection tasks. Frontiers in Human Neuroscience, 3.

Confrey, J. (1994). Splitting, similarity, and rate of change: A new approach tomultiplication and exponential functions. In G. Harel & J. Confrey (Eds.), Thedevelopment of multiplicative reasoning in the learning of mathematics (pp. 291–331).Albany: SUNY Press.

Dehaene, S. (1997). The number sense: How the mind creates mathematics. Oxford:Oxford University Press.

Delazer, M., Ischebeck, A., Domahs, F., Zamarian, L., Koppelstaetter, F., Siedentopf, C.M. & Felber, S. (2005). Learning by strategies and learning by drill—Evidence from anfMRI study. NeuroImage, 25(3), 838–849.

ANDERSON NORTON AND KIRBY DEATER-DECKARD

Page 19: MATHEMATICS IN MIND, BRAIN, AND EDUCATION: A NEO-PIAGETIAN APPROACH

Demeyere, N., Rotshtein, P. & Humphreys, G. W. (2012). The neuroanatomy of visualemmuneration: Differentiating necessary neural correlates for subitizing versus countingin a neuropsychological voxel-based morphometry study. Journal of CognitiveNeuroscience, 24(4), 948–964.

Ellis, A. B. (2007). The influence of reasoning with emergent quantities on students’generalizations. Cognition and Instruction, 25(4), 439–478.

Epstein, H. T. (2001). An outline of the role of brain in human cognitive development.Brain and Cognition, 45, 44–51.

Fischer, K.W. & Bidell, T. R. (2006). Dynamic development of action, thought, and emotion.InW. Damon&R.M. Lerner (Eds.), Theoretical models of human development. Handbookof child psychology (6th ed., Vol. 1, pp. 313–399). New York: Wiley.

Fischer, K. W., Daniel, D. D., Immordino-Yang, M. H., Stern, E., Battro, A. & Koizumi,H. (2007). Why mind, brain, and education? Why now? International Mind Brain andEducation Society, 1(1), 1–2.

Grabner, R. H., Ansari, D., Reishofer, G., Stern, E., Ebner, F. & Neuper, C. (2007).Individual differences in mathematical competence predict parietal brain activationduring mental calculation. NeuroImage, 38(2), 346–356.

Grabner, R. H., Ischebeck, A., Reishofer, G., Koschutnig, K., Delazer, M., Ebner, F. &Neuper, C. (2009). Fact learning in complex arithmetic and figural‐spatial tasks: Therole of the angular gyrus and its relation to mathematical competence. Human BrainMapping, 30(9), 2936–2952.

Hackenberg, A. J. (2007). Units coordination and the construction of improperfractions: A revision of the splitting hypothesis. Journal of MathematicalBehavior, 26(1), 27–47.

Hackenberg, A. J. (2010). Students’ reversible multiplicative reasoning with fractions.Cognition and Instruction, 28(4), 383–432.

Hackenberg, A. J., & Lee, M. Y. (2012). Pre-fractional middle school students’ algebraicreasoning. In L. R. Van Zoest, J.-J. Lo, & J. L. Kratky (Eds.), Proceedings of the 34thAnnual Meeting of the North American Chapter of the International Group for thePsychology of Mathematics Education (pp. 943–950). Kalamazoo, MI: WesternMichigan University.

Ischebeck, A., Schocke, M. & Delazer, M. (2009). The processing and representation offractions within the brain: An fMRI investigation. NeuroImage, 47, 403–412.

Izsák, A., Jacobsen, E., de Araujo, Z. & Orrill, C. H. (2012). Measuring mathematicalknowledge for teaching fractions with drawn quantities. Journal for Research inMathematics Education, 43(4), 391–427.

Karmiloff-Smith, A. (2009). Nativism versus neuroconstructivism: Rethinking the studyof developmental disorders. Developmental Psychology, 45, 56–63.

Kaufman, E. L., Lord, M. W., Reese, T. W. & Volkmann, J. (1949). The discrimination ofvisual number. The American Journal of Psychology, 62(4), 498–525.

Kruggel, F., Hermann, C. S., Wiggins, C. J. & von Cramon, D. Y. (2001). Hemodynamicand electroencephalographic responses to illusory figures: Recording of the evokedpotentials during functional MRI. NeuroImage, 14, 1327–1336.

Martin, K., Jacobs, S. & Frey, S. H. (2011). Handedness-dependent and -independentcerebral asymmetries in the anterior intraparietal sulcus and ventral premotor cortexduring grasp planning. NeuroImage, 57(2), 502–512.

Mason, L. (2009). Bridging neuroscience and education: A two-way path is possible.Cortex, 45, 548–549.

MATHEMATICS IN MIND, BRAIN, AND EDUCATION

Page 20: MATHEMATICS IN MIND, BRAIN, AND EDUCATION: A NEO-PIAGETIAN APPROACH

Meck, W. H. & Church, R. M. (1983). A mode control model of counting and timingprocesses. Journal of Experimental Psychology: Animal Behavior Processes, 9(3), 320.

Nieder, A. & Dehaene, S. (2009). Representation of number in the brain. Annual Reviewof Neuroscience, 32, 185–208.

Norton, A. & Boyce, S. (2013). Coordinating n+1 levels of units. Proceedings of the Thirty-Fourth Annual Meeting of the North American Chapter of the International Group for thePsychology of Mathematics Education. Chicago: University of Chicago Press.

Norton, A. & Wilkins, J. (2009). A quantitative analysis of children’s splitting operationsand fractional schemes. Journal of Mathematical Behavior, 28(2/3), 150–161.

Norton, A. & Wilkins, J. L. M. (2012). The splitting group. Journal for Research inMathematics Education, 43(5), 557–583.

Norton, A. & Wilkins, J. (2013). Supporting students’ constructions of the splittingoperation. Cognition & Instruction, 31(1), 2–28.

Olive, J. (2000). Computer tools for interactive mathematical activity in elementaryschool. International Journal of Computers for Mathematical Learning, 5, 241–262.

Olive, J. & Çağlayan, G. (2007). Learner’s difficulties with quantitative units in algebraicword problems and the teacher’s interpretations of those difficulties. InternationalJournal of Science and Mathematics Education, 6, 269–292.

Piaget, J. (1942). The child’s conception of number. London: Routledge & Kegan Paul.Piaget, J. (1970). Structuralism (C. Maschler, Trans.). New York: Basic Books. Originalwork published 1968.

Piazza, M., Izard, V., Pinel, P., Le Bihan, D. & Dehaene, S. (2004). Tuning curves forapproximate numerosity in the human intraparietal sulcus. Neuron, 44(3), 547–555.

Poldrack, R. A. (2006). Can cognitive processes be inferred from neuroimaging data?Trends in Cognitive Sciences, 10(2), 59–63.

Potter, M. C. & Levy, E. I. (1968). Spatial enumeration without counting. ChildDevelopment, 39(1), 265–272.

Qin, Y., Carter, C. S., Silk, E. M., Stenger, V. A., Fissell, K., Goode, A. & Anderson, J.R. (2004). The change of the brain activation patterns as children learn algebra equationsolving. PNAS, 101, 5686–5691.

Quartz, S. R. & Sejnowski, T. J. (1997). The neural basis of cognitive development: Aconstructivist manifesto. Behavioral and Brain Sciences, 20, 537–596.

Revkin, S. K., Piazza, M., Izard, V., Cohen, L. & Dehaene, S. (2008). Does subitizingreflect numerical estimation? Psychological Science, 19(6), 607–614.

Rivera, S. M., Reiss, A. L., Eckert, M. A. & Menon, V. (2005). Developmental changes inmental arithmetic: Evidence for increased functional specialization in the left inferiorparietal cortex. Cerebral Cortex, 15(11), 1779–1790.

Rosenberg-Lee, M., Lovett, M. C. & Anderson, J. R. (2009). Neural correlates of arithmeticcalculation strategies. Cognitive, Affective, & Behavioral Neuroscience, 9(3), 270–285.

Rueda, M. R., Posner, M. I. & Rothbart, M. K. (2005). The development of executiveattention: Contributions to the emergence of self-regulation. Developmental Neuropsy-chology, 28(2), 573–594.

Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivistperspective. Journal for Research in Mathematics Education, 26(2), 114–145.

Simon, O., Mangin, J. F., Cohen, L., Le Bihan, D. & Dehaene, S. (2002). Topographicallayout of hand, eye, calculation, and language-related areas in the human parietal lobe.Neuron, 33(3), 475–487.

ANDERSON NORTON AND KIRBY DEATER-DECKARD

Page 21: MATHEMATICS IN MIND, BRAIN, AND EDUCATION: A NEO-PIAGETIAN APPROACH

Sophian, C. & Crosby, M. E. (2008). What eye fixation patterns tell us about subitizing.Developmental Neuropsychology, 33(3), 394–409.

Steffe, L. P. (1992). Schemes of action and operation involving composite units. Learningand Individual Differences, 4(3), 259–309.

Steffe, L. P. (1994). Children’s multiplying schemes. In G. Harel & J. Confrey (Eds.), Thedevelopment of multiplicative reasoning in the learning of mathematics (pp. 3–39).Albany: State University of New York Press.

Steffe, L. P. (2002). A new hypothesis concerning children's fractional knowledge.Journal of Mathematical Behavior, 20(3), 267–307.

Steffe, L. P. & Olive, J. (Eds.). (2010). Children’s fractional knowledge. New York:Springer.

Steffe, L. P. & Thompson, P. W. (2000). Teaching experiment methodology: Underlyingprinciples and essential elements. In R. Lesh & A. E. Kelly (Eds.), Research design inmathematics and science education (pp. 267–307). Hillsdale: Erlbaum.

Szucs, D. & Goswami, U. (2007). Educational neuroscience: Defining a new discipline forthe study of mental representations. Mind Brain and Education, 1(3), 114–127.

Tam, F., Churchill, N. W., Strother, S. C. & Graham, S. J. (2011). A new tablet forwriting and drawing during functional MRI. Human Brain Mapping, 32, 240–248.doi:10.1002/hbm.21013.

Tonry, M., Ohlin, L. E. & Farrington, D. P. (1991). Accelerated longitudinal design. InHuman Development and Criminal Behavior (pp. 27–34). Springer: New York.

Ulrich, C. L. (2012). Additive relationships and signed quantities (Doctoral dissertation).http://www.libs.uga.edu/etd/.

Varma, S., McCandliss, B. D. & Schwartz, D. L. (2008). Scientific and pragmatic challengesfor bridging education and neuroscience. Educational Research, 37(3), 140–152.

Varma, S. & Schwartz, D. L. (2008). How should educational neuroscience conceptualisethe relation between cognition and brain function? Mathematical reasoning as a networkprocess. Educational Research, 50(2), 149–161.

von Glasersfeld, E. (1995). Radical constructivism: A way of knowing and learning.London: Routledge Falmer.

Wagner, S. H. & Walters, J. (1982). A longitudinal analysis of early number concepts:From numbers to number. In G. E. Forman (Ed.), Action and thought: Fromsensorimotor schemes to symbolic operations (pp. 137–161). New York: Academic.

Wang, Y., Lin, L., Kuhl, P. & Hirsch, J. (2007). Mathematical and linguistic processingdiffers between native and second languages: An fMRI study. Brain Imaging andBehavior, 1, 68–82.

Westermann, G., Mareschal, D., Johnson, M. H., Sirois, S., Spratling, M. W. & Thomas,M. S. C. (2007). Neuroconstructivism. Developmental Science, 10, 75–83.

Wilkins, J. & Norton, A. (2011). The splitting loope. Journal for Research inMathematics Education, 42(4), 386–406.

Wilkinson, D. T. & Halligan, P. W. (2003). Stimulus symmetry affects the bisection offigures but not lines: Evidence from event-related fMRI. NeuroImage, 20, 1756–1764.

Virginia TechBlacksburg, VA, USAE-mail: [email protected]

MATHEMATICS IN MIND, BRAIN, AND EDUCATION