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COGNITIVE NEUROSCIENCE ANDMATHEMATICS EDUCATION

FRANK QUINN

Abstract. This article is primarily addressed to cognitive scientists and neu-

roscientists interested in education.For quite some time mathematics education has seemed an area in which

cognitive neuroscience might make important contributions. This has not hap-

pened: studies have been large in number but small in impact, and educationhas been influenced more by misunderstandings and over-simplifications than

actual science. Are these ‘important contributions’ an illusion? If not, why

have they not been realized?The first part of the article describes difficulties and obstacles to effective

use of neuroscience in education. There are a great many: some in the science

itself, many in the education community, and lack of subject sophistication isa particular problem. These obstacles could explain lack of impact, but do not

demonstrate that it is possible.The second part describes four neuroscience experiments that could have

significant impact in mathematics education. The goals are to show this really

is possible, and to see how the concerns of part one play out in examples.

Contents

Introduction 21. Background, and outline of part one 22. Technical difficulty, and consequences 32.1. Technical difficulties 32.2. Needs of neuroscience 52.3. Implications for application 63. Ineffective neuroscience 63.1. Multiplication 63.2. Solving equations 83.3. Errors 83.4. Summary 124. Pitfalls in education 124.1. Lack of scientific skills 124.2. Lack of content sophistication 134.3. Hidden cognitive errors 155. Mathematics and learning 165.1. Teaching vs. diagnosis 165.2. Modern mathematics 175.3. Educational hostility 175.4. Theoretical incoherence 18

Date: Draft, November 2010.

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5.5. Summary 186. Sample collaborations 186.1. Structure 186.2. Cognitive interference, outline 196.3. Subliminal learning and reenforcement, outline 207. Cognitive interference 227.1. Cognitive interference in multiplication 227.2. Cognitive interference in word problems 268. Subliminal learning and reenforcement 298.1. Subliminal algebra in integer multiplication 298.2. Kinetic reenforcement of geometric structure 31References 33

Introduction

The original goal of this article was to explore potential applications of neuro-science to mathematics education, and the kinds of collaborations that would makethese possible. The conclusions, however, are discouraging. Good applications willbe subtle and difficult, there are many pitfalls that can render neuroscience in-vestigations ineffective, and misused neuroscience could cause damage. Further,good collaborators are not likely to be found in the psychological or educationalcommunities. These conclusions are explained in §§1–5.

The second part of the article (§§6–8) was developed as an antidote of sorts tothe first part. As evidence that high-impact educational neuroscience is possible inspite of all the pitfalls, we give four detailed proposals for neuroscience experimentsthat address issues described elsewhere [41, 40]. A detailed outline is given in §6.

1. Background, and outline of part one

Cognitive neuroscience is concerned with the neural mechanisms underlying hu-man behaviour and cognition. The area has roots in medicine, psychology, soci-ology, and philosophy, but it was largely advances in brain imaging that led todevelopment of a distinct discipline in the 1990s.

Mathematics education was an early theme in cognitive neuroscience. Elemen-tary mathematical activity is more well-defined and consistently localized thanmost cognitive activities, and in the late 1990s Stanislas Dehaene [12, 13] exploitedthis in a pioneering exploration of mechanisms behind number sense. Applicationsto education seemed a natural and valuable next step. At the same time Bruer[8] warned that direct meddling by neuroscientists in education is a “bridge toofar”. Instead, he recommended a two-stage approach with educational applica-tions undertaken by cognitive psychologists, and neuroscience providing input topsychology. Bruer’s views prevailed for almost a decade.

By 2005 there were calls for direct education-neuroscience interactions as a “two-way street” [5], [22], and the term “educational neuroscience” (with “cognitive”removed) began to be used. For later accounts see [35], [10]. The reason offered wasthat Bruer’s two-stage approach was not working: educators were using distortedpopular accounts rather than solid science [23] [19], and psychologist intermediariesseemed to be ineffective. There is something to this: [51] for instance gives a

COGNITIVE NEUROSCIENCE AND MATHEMATICS EDUCATION 3

generic, uncritical summary of neuroscience findings that could be used in all sortsof ways. On the other hand, by then many of the people doing education-relatedcognitive neuroscience were themselves in departments of psychology or education,and they were being ineffective directly. The main benefit of a ‘direct connection’is that it would enable them to identify themselves as neuroscientists rather thanas psychologists or educators, and they had found that this brought more respectand influence.

The ‘two-way-street’ approach is still the main theme. For instance, a majorconference intended to plot a course for neuroscience and mathematics educationwas held in 2009; see the program [16] and position paper [15]. However ‘brain-based’, ‘brain-friendly’, etc. educational methods are multiplying, and still basedon ‘neuro-myths’ [3]. Something isn’t working.

Section 2.1 explores what neuroscience needs from a partner. In a nutshell, theexpense and technical difficulty of doing anything at all means genuinely useful re-search requires targeted and insightful guidance about what to look for, and what itmeans. §3 illustrates how ‘edu-myths’ and lack of subject sophistication can renderneuroscience studies ineffective. On the other side of the collaboration, complexityand ambiguity of outcomes means care and sophistication are needed to properlyapply them. More than that, sophistication is needed to avoid misapplication thatcould discredit the approach. §4 explains why the education community cannot pro-vide such care and sophistication. Finally, §5 describes more precisely the expertisethat—in light of the examples—seems to be necessary.

Background and an outline of the second part of the paper are given in §6.

2. Technical difficulty, and consequences

The section begins with a brief review of the technical difficulties of brain imag-ing. We see that these difficulties impose strong constraints on how the work isconducted and on the feedback needed to make progress. It also makes good useof the outcomes a challenge.

2.1. Technical difficulties. The first problem is that the brain is encased in thedensest bone in the body. In principle, sensors could be implanted inside the skull,but this is invasive, permanent, expensive, and current sensors are unsuitable forall but the most urgent human applications. Education-oriented imaging must bedone from outside. As a result all techniques must deal with signal attenuationand distortion by the skull, and the inversion problem (reconstruction of internalactivity from external data) requires difficult blends of mathematics, physics, andanatomical knowledge. No method has completely satisfactory inversion: see [52]for an illustration of how better anatomical knowledge could improve interpretationof fMRI data, for example.

Next, brains are busy places, and signals relevant to the question at hand mustbe extracted from the general hustle and bustle. This requires relatively strong,topic-specific signals, and some imaging methods require relatively long duration.Further, in most cases multiple trials must be statistically combined before a sig-nificant signal can be seen.

A great deal remains inaccessible. None of these methods give information aboutneurotransmitter activity, for instance. Serious imbalances in neurotransmitteractivity are well known to effect cognition, and can cause differences in activitythat show up in imaging. It is probably common for individual differences to effect

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both cognition and imaging, even when not severe enough to be diagnosed as a“serious imbalance”. Currently there is no way to anticipate or compensate forsuch effects. They must be treated as noise and this, no doubt, is one reasonstatistical aggregation is necessary for clear outcomes.

Different methods have their own specific difficulties:

• Positron emission tomography (PET) gives good images but requires in-gestion of radioactive substances. Total exposure is low compared to X-raytomography, but the radiation is higher-energy and the target is an organthat we don’t want to damage. It is also expensive.• The usual explanation of PET is that it tracks glucose uptake associated

with increased neural activity. This is not literally true and exactly what itdoes track, and whether it really correlates with glucose delivery, is a matterof debate. For practical purposes it may be more important to understandhow it correlates with other imaging data.• Functional magnetic resonance imaging (fMRI) requires high magnetic fields.

The machines are large, noisy, and expensive to operate. The data shouldbe similar to PET data, and nicely complementary to EEG or MEG. Un-fortunately the high field strength makes it difficult or impossible to usethese methods at the same time, so data from a single trial usually can-not be correlated. There has been some progress with simultaneous EEGand high-field fMRI [26]. Low-field fMRI that might address some of theseproblems is in development [9], but has a long way to go.• fMRI primarily images the so-called Blood Oxygen Level Dependent (BOLD)

response. The presumption is that deoxygenated hemoglobin indicates en-ergy consumption, therefore neural activity, and therefore cognitive use ofthe area. The connection may not always be so direct: this signal may betriggered by anticipated use [46], for example.• Diffusion Spectrum Imaging (DSI) is a variation on MRI that detects the

movement of water [24, 25]. DIS can be used to image fiber structure andconnectivity in white matter because water is constrained to move withinfibers. Neural activity can be detected because it causes net displacementalong the fiber. This method produces much more information than othersbut is not often used because the technical requirements are high: very highfields, high gradients, long acquisition times, and complex mathematicalanalysis. There are less-demanding and less-informative variants (diffusiontensor imaging, etc., [24]).• Near infrared spectroscopy (NIRS) exploits the fact that tissues contain-

ing relatively little blood—in particular the skull—are translucent in thenear infrared (650–950 nm). This window includes absorption lines foroxygenated and deoxygenated hemoglobin, and these can be used to detectenergy-intensive activity in the outermost few mm of the brain [37], [1].The thick diffusive layer (skull) and multitude of artifacts (cardiovascularand respiratory activity, head movement, scalp effects, etc.) limit imagingto very coarse resolution. On the other hand the equipment is modest andunobtrusive compared to other methods, so NIRS is attractive for large-scale studies or individual assessment when high-resolution pilot studieshave shown what to look for [17].


• Magnetoencephalagraphy (MEG) uses magnetic fields resulting from cur-rent loops in the brain. These fields are weak and detection requires elabo-rate shielding, and liquid-helium-cooled sensors a few millimeters from thescalp.• Inversion techniques for MEG are still primitive, and it is a stretch to call

the output “images”. Fields detectible outside the skull must be generatedby relatively large currents, but the anatomical structures that supportthese currents are not fully understood and cannot be inferred from thefield data. DSI, which does image these structures, should eventually givea ‘wiring diagram’ that should enable better inversion.• There is a lot of individual variation not related to the questions at hand.

People with tinnitus, for example, have significantly different MEG profiles[50]. Again, at present these differences are treated as noise and degradethe signal.• Electroencephalagraphy (EEG) makes use of electric fields. These are easier

to detect than magnetic fields but the data is more complex: the fields aredistorted by the skull and scalp; fluctuating reference levels; and artifactsfrom pulse and cardiovascular electrical activity. As with MEG, genuineimages cannot yet be extracted.• Both MEG and EEG have response fast enough to enable real-time track-

ing of neural activity, while the slower image rate of PET and fMIR givetime-averaged results. Fast response offers opportunities but also addi-tional challenges. Making sense of data that involves vision, for instance,requires tracking and compensating for eye movement. Eyes move a lot, andthe brain pre-processes optic nerve signals to produce stable perceptions.This compensates for limitations of light receptors and gives better visualperception, but the pre-processing produces complex and mostly irrelevantsignals.• Tracking eye movement can give information on attention focus, and pupil

dilation sometimes correlates with cognitive resource allocation [32].

2.2. Needs of neuroscience. Neuroscience studies are expensive, with costs or-ders of magnitude higher than traditional educational studies of comparable scope.To attract funding, and to avoid wasting it, education-oriented experiments mustbe designed so there is a good chance there will be a signal above the noise level, andthat this signal will be meaningful. Moreover the technical difficulties mean thatexperimental design must be quite insightful and targeted to have a good chanceof success.

As De Smedt et.al. [15] put it (p. 102):

“few attempts have been made to study more complex and higherorder mathematical skills. . . . A particular challenge of this re-search is that it requires educational and psychological theories,which specify cognitive processes that are detailed enough to beexamined by neuroimaging.”

More precisely, insights into cognitive processes are needed from somewhere. Recallthat these are mathematical processes. Educational and psychological expertise areimportant but it seems likely that mathematical sophistication will play a key role.This is illustrated through analysis of recent neuroscience studies in §3.

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2.3. Implications for application. The big problem is that misapplication ofpowerful methodologies will cause damage, not just fail to be beneficial. Currentlythere is public interest and hope that hard science might provide a way to untanglethe educational mess. However modern culture has strong anti-intellectual andanti-science currents. Mishandling applications of neuroscience to education couldeasily lead to a backlash, and hostility like that now directed at nuclear energy andgenetic modification of food plants. Cognitive neuroscience is fast approaching thepoint where it will be more dangerous than ignorance. It is consequently at risk,and care is essential.

One component of good application is that the science must be appropriate. Inparticular it must incorporate subject sophistication and address long-term educa-tional goals, not just ‘learning difficulties’. Getting the science right is a responsi-bility of neuroscientists.

The other component of good application is the use of science by educators. Thedifficulty here is that findings in neuroscience are complicated, and often ambigu-ous or unclear. They may eventually become unambiguous, but they will never beuncomplicated. In short, this is science, and good application requires scientificsophistication and intellectual discipline. Unfortunately these are in short supplyin the educational community; see [3] [31], and §4. This means it will be a re-sponsibility of educationally-oriented collaborators to try to manage educationalapplications, and in particular to try to head off the more egregious abuses.

3. Ineffective neuroscience

The need for outside collaboration, and content sophistication in particular, isillustrated by analysis of neuroscience studies related to mathematics education.Section 3.1 describes two studies of multiplication, §3.2 concerns solving simpleequations, and §3.3 discusses the role of error analysis.

3.1. Multiplication. In this section we review two fMRI studies of integer multi-plication, [29] [45]. These are relatively clear and are the sort of study one couldimagine educators trying to use in some way. However both omit subtle but crucialissues, particularly concerning errors, and one cannot imagine educators compen-sating for this.

Krueger, Landgraf, et al. [29] find activity in five main areas. Lacking “theo-ries which specify cognitive processes that are detailed enough to be examined byneuroimaging” (quote from [15], above), they simply note which areas are activein each of a sequence of time blocks, and from the coincidences infer correlatedactivity.

The first concern is that the study uses three tasks identified as having increasingdifficulty: multiplication of two 1-digit integers; one 1-digit and one 2-digit integer;and two 2-digit integers. In fact these tasks have qualitatively different cognitiveand mathematical structures:

• multiplication of two 1-digit integers is simple fact recall and input-output.• multiplication of a 1-digit and a 2-digit number requires two multiplication

facts, and short-term storage and addition of outcomes. The addition re-quires shifting one output by one place, usually a single 1-digit addition,and occasionally dealing with an overflow.


• multiplication of two 2-digit numbers engages the standard algorithm. Hereit is used to organize and combine two 1× 2-digit products, each generatedas above.

Difficulty increases because successive tasks have qualitatively different subtasks,not because they are more of the same. Moreover the algorithm used for the 2× 2-digit multiplication is more representative of later mathematics than is single-digitproduct memorization. An important question potentially addressed by this studyis: how does 2× 2-digit multiplication differ from the simpler cases?

More generally, there is an urgent need to understand how components of math-ematical algorithms interact cognitively. It seems likely that some could be re-designed to reduce cognitive difficulty, and this could have profound educationalconsequences. Substantial exploration of the multiplication algorithm would re-quire products with three or more digits in each factor, but this would be pointlesswithout sophisticated mathematical input (see §6).

Another concern is that [29] was envisioned as a test of mental arithmetic, soparticipants were unable to do the scratch work usually employed in multiplication.In particular in the 2× 2 digit task, the output from the first 2× 1 multiplicationhad to be held in internal working memory rather than written. This may haveintroduced an artifact: instead of being written it may have been held in internalmemory associated with writing, and this may partially account for the observedrecruitment of the left precentral gyrus (subjects were right-handed). Their inter-pretation of this as being connected with use of fingers in counting is doubtful, andcould be misused if wrong.

A general context for this concern is that most mathematical procedures usewritten intermediate results as external working memory. External memory hasa tradeoff: it is more accurate and durable, but requires input-output processingand shifts in attention focus. A key part of algorithm design is to optimize writtencomponents for this use (and other things; see §3.3). Neuroscience studies couldcertainly contribute to this, but the main point is that excluding scratch workrenders studies of all but the very simplest tasks useless.

A third concern is more neurological than mathematical. There seem to be semi-autonomous facilities, for instance in the anterior cingulate cortex, that check forconflicts and inconsistencies (see §3.3). This means that generating an outcome,and recognizing whether or not a proposed outcome is correct, can be substantiallydifferent neural activities. A lack of coordination between these activities seems tounderlie some learning difficulties [18]. However many mathematical neuroscienceexperiments (including this one) tacitly assume that correct alternatives are iden-tified by generating an outcome and comparing. For that matter, this assumptionis used to justify wide use of multiple-choice tests in mathematics education. Thisassumption is problematic and urgently needs to be tested.

In another direction, Rosenberg-Lee, Lovett, and Anderson [45] describe an fMRIstudy of 3 × 1- and 5 × 1-digit multiplication, comparing two different strategiesto predictions of ACT-R computer models (see [2], [4], and the Wikipedia entry).Differences from the study above:

• The models give predictions that enable detailed imaging (Granger analysisnot needed), and the imaging essentially supports the model.• Subjects gave answers rather than identified them among choices.

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• The use of a single-digit factor avoids the cognitive complexity of the fullmultiplication algorithm, so this should have been described as a study of acomponent of multiplication, not the full activity. This is a concern aboutpossible misinterpretation, not the science.• The single-digit factor restriction, and comparison of two strategies, pro-

vides a clearer picture of this component of the algorithm.The main shortcoming of [45] is that only correct responses were analyzed. There isno information on how errors occur or how to avoid them, and the design preventederror-checking, see §3.3.

3.2. Solving equations. Most studies of equation-solving have been neurologi-cal explorations without significant educational goals. Lack of a coherent overallcontext for this activity1 makes large-scale goal formation difficult, but small-scalegoals were available.

[4] and [49] seem primarily designed to show that ACT-R programs [2] can effec-tively model the activities. This suggests that ACT-R is ready to be challenged byquestions with important consequences, but that not much will happen without suchchallenges. In detail, [49] compares solving equations with symbolic and numericalcoefficients, and finds relatively little difference. This supports the idea that a lotof arithmetic is more symbolic than numerical, but the task was too simple to testthis effectively. [4] compares solving numerical equations to a symbol-manipulationtask that is less relevant than one might have liked. In both of these studies errorswould probably have been more revealing than correct work, but the studies werelimited to correct solutions because, so far, the models are.

A study of elementary calculus [30] showed activation patterns similar to algebra.The routines used are essentially algebraic so mathematically these are just more-complicated algebra problems, and apparently the brain sees them the same way.They are significantly more complicated than the tasks used in other trials, butmore subject sophistication seems to be needed to draw useful conclusions.

[42] studies errors in solving simple numerical equations. However the conclusionsare weak in a number of ways:

• Coefficient arithmetic has a significant error rate. How much of the observederror was due to arithmetic, how much to the solving component, and howmuch (if any) to interference between these tasks?• The task was mental (i.e. did not allow scratch work). See the next section

for an explanation why this is problematic.• It would have been useful to know if there were patterns in the errors

that got noticed and triggered correction. However the protocol inhibitedcorrections.

3.3. Errors. In principle, mathematical methods can give results that are 100%reliable. Complex methods depends on this: if a calculation requires twenty op-erations, for instance, and each is 100% reliable, then the result is 100% reliable.Elementary educators seem to find 75% success acceptable, but twenty 75%-reliableoperations gives a final outcome 0.3% reliable. Twenty 90% reliable operations stillonly gives a 12% success rate. These error rates will block most students from math-ematics requiring more than two or three operations, which is to say essentially allof it. Better accuracy is thus crucial for better outcomes.

1Equation-solving should probably be seen as symbolic pattern recognition and manipulation.


Procedures designed to minimize error are a vital component of accuracy. Veryhigh accuracy, however, comes from noticing and correcting errors that happeneven when good procedures are used. This is a cognitive issue, and probably themost important opportunity for cognitive neuroscience. In more detail:

Developmental and procedural errors: These result from failure to fol-low procedures that minimize errors. The remedy is better internalizationand use of procedure, e.g. through practice. Neuroscience experiments thatdo not include use of appropriate procedures simply provoke errors we al-ready know how to address. This may have value but experimenters shouldbe realistic about it. The best opportunities are in understanding how theprocedures minimize errors; see ‘Minimizing error’ below

Random errors: These happen. They should be noticed and corrected, sothe key neuroscience questions concern characterization and developmentof “noticing”; see ‘Conflict alerts’ below.

Systematic errors: These do not result from failure to follow procedures,but are too frequent or too hard to correct to consider random. These mayindicate cognitive shortcomings in the procedure. Neuroscience can clarifythis and help design more effective procedures. Examples are given in thesecond part of this paper.

Different types of errors will almost certainly have different neural signatures. Thisis a serious problem when statistical amalgamation is necessary to get reliable orcomprehensible outcomes: error types must be identified (insightfully) and sortedbefore analysis. Clearly, this will depend heavily on sophisticated subject expertise.

3.3.1. Minimizing error. Standard mathematical procedure is to maintain a written(or graphic) transcript that enables reconstruction of reasoning and can be checkedfor errors, see [41](a). Random scratch work does not qualify. Following thisprocedure substantially reduces errors even if the transcript is not actually checked.Presumably this is a cognitive effect and understanding it would enable sharperdesign of transcript templates.

The benefits of transcripts seem to have several components. First, writing anintermediate step seems to fix it more accurately in working memory, even if it isused immediately. In some cases the kinetic aspect (writing or drawing) seems tobe essential. In others, visual feedback may be sufficient, though this is probablyage-dependent. This must be understood before we can design learning-effectivecomputational environments, for instance; see [41](b). Second, the transcript servesas durable working memory for material not used immediately. Assessing formateffectiveness for identification and retrieval of ‘working memory’ material is animportant potential role for neuroscience.

The main point here, however, is that written auxiliary work is part of mathe-matical activity and should be included in all but the most primitive neurosciencestudies. Errors that result from failure to use proper procedures are not deeplyinteresting.

3.3.2. Conflict alerts. A “conflict alert” is an awareness of a potential conflict ineither input or cognition. Humans have fast and powerful primitive mechanismsthat generate conflict alerts. For instance, it is common practice to check spellingby writing a guess and seeing if it “looks right” (doesn’t provoke a conflict alert).

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The goal is to use conflict alerts to trigger deliberate and reliable error check-ing. They should not be thought of as directly detecting errors. Feedback fromdeliberate checking trains the alert mechanisms, and experts eventually developimpressively reliable alerts. These are fast and seemingly effortless (apparently dueto implementation in the cingulate cortex) and give the appearance of ‘instinctive’access to the subject. Some educators see these as a goal, or perhaps confuse deeply-learned and innate, and encourage students to develop ‘instinctive understanding’.This is a mistake because it undercuts the training aspect of deliberate checking.Instead of being refined, unreliable alerts are reenforced by repetition. Erroneousalerts are hard to fix or inhibit [6] so this locks in high error rates. Finally, evenexperts do not have fully reliable alerts. Good alerts make checking efficient, notunnecessary.

We tentatively identify two types of conflict alerts:

Internal: Awareness that the outcome of a cognitive action is doubtful.Input: Awareness of potential inconsistency in input.

Certain motor activities qualify as “input”. Reading written material aloud, orrepeating auditory material often gives much better cognitive access. Similarly,hand-copying written material, or writing down auditory material often gives muchbetter cognitive access. Motor activity of the tongue, lips, hands etc. acts as inputchannels, and reading aloud, copying, etc. essentially reformats input into thesechannels. In particular it a mistake to think of it as output, and damage hasresulted from failing to understand this. This is discussed further in §8.2, as partof ‘reenforcement’.

3.3.3. Conflict alert mechanisms. There have been many studies of conflict alertsand the patterns so far are summarized in [11]. Most of these studies concernperceptual tasks irrelevant to mathematics. They are useful guides, but suggestionsthat they all apply directly to mathematics should trigger a conflict alert. Withthis in mind we give more detail.

Internal conflict alerts originate in the region where the questionable cognitiontakes place. For instance the fMRI study [42] shows a correlation between errors insolving simple numerical equations, and reduced preliminary activity in a region inthe prefrontal cortex associated with procedural fact retrieval. Presumably insuffi-cient activity increases the likelihood of some sort of loading error. However theirdata also shows significantly increased activity during and after an erroneous out-come, and ‘Error-Related Negativity’ (ERN) is seen at this time in analogous EEGstudies. Confusion or conflict due to the loading error seems to cause formulationof a problem report and the ERN results from dispatch of this report to the DorsalAnterior Cingulate Cortex (DACC). The DACC apparently determines what sortof conflict might result from the problem, identifies the relevant control region inthe DorsoLateral PreFrontal Cortex (DLPFC), and forwards an amplified report tothat region for consideration. It seems to be the DLPFC that actually issues thealert, determines the level of awareness, and perhaps provides preliminary plans forresponse.

Input conflict alerts are similar. Somebody (more about this below) notices apossible discrepancy between input and something internal, and sends a problemreport to the DACC. The DACC then notifies the appropriate region in the DLPFC,which may or may not do anything about it.


It is unclear where input discrepancies are first detected. This is often attributedto the DACC, but for very simple tasks the first detection may be so fast andhappen so deep in the brain that the first significant activity occurs in the DACC.Mathematical tasks that do not engage the DACC (i.e. seem unproblematic) seemto engage the precuneus [?], so it may be that first detection occurs there. Thisis an important issue. Advanced mathematics depends on development of deeplyembedded “automatic” models of complex mathematical objects. Discrepancies arepresumably first noticed in the regions where these models are located, and it seemsunlikely this is the DACC.

Input alerts concern discrepancies between input and either internal representa-tions, or active processes. Some examples of the latter:

• A cognitive process can produce output, and an alert can result from adiscrepancy between the intended output and the actual output when itcomes back as input. This is a ‘slip of action’ and usually triggers a fastresponse.• A cognitive act can produce output, and an alert can result from a dis-

crepancy between output when it comes back as input and the state of thecognitive process when the input is received. This is common in speededforced response trials, or deliberately confusing tasks like the Stroop test,when the first output occurs before processing is complete, is frequentlywrong, and is recognized as doubtful when received as input.

Changes in state of a process do not seem to generate alerts unless output beforethe change comes back as input after it.

3.3.4. Error checking. The model described above has three stages: a process orstructure notices a problem and sends a report to the DACC; the DACC processesthis and sends a conflict notice to the appropriate control region in the DLPFC;and the DLPFC decides what, if anything, to do about it. The DACC activityseems to be fast, automatic and reliable, so we focus on the other two steps. Inthese, highly accurate work seems to require:

• Development of cognitive processes or internalizations that either get thingsright or reliably submit problem reports, and• Training the DLPFC to care enough to issue conflict alerts that actually

trigger error checking.[20] focuses on internal conflict alerts. Consistently effective alerts, referred to as“introspective awareness”, was found to correlate with gray matter volume in aregion in the prefrontal cortex. Presumably this is the control region relevant tothe task in question. In any case it seems to be the bottleneck and (as observedin [20]) effective error correction may require enough training that neural plasticityleads to an increase in volume. A similar finding for mathematical error correctionwould have profound educational significance because error correction is largelyabsent from current curricula.

As mentioned above, [42] shows a correlation between errors in solving simplenumerical equations, and reduced preliminary activity in a region in the prefrontalcortex associated with procedural fact retrieval. This suggests that additional train-ing (leading to increased activity) would lower the error rate. The data also showsincreased activity during and after an error, interpreted here as preparation andtransmission of a problem report. However the data does not address reliability of

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these error reports. Did the reduced activity that made errors more likely also in-crease the likelihood that an error would fail to generate an error message? Again,evidence for this would have significant educational implications.

3.4. Summary. Neuroscience experiments genuinely effective for education will re-quire appropriate educational and psychological expertise and, crucially, cognitively-informed mathematical sophistication. Lack of these has prevented cognitive neu-roscience from having much impact.

4. Pitfalls in education

The first subsection expands on the reasons the education community cannotinteract effectively with neuroscience. The second shows this is just as well: withoutsubject sophistication, neuroscience compatible with standard educational theorieswould not lead to better long-term outcomes. The third section illustrates thedangers of hidden assumptions.

4.1. Lack of scientific skills. The core problem is that education is not a science.This is not for lack of trying: systematic efforts to develop ‘education science’ goback more than a century, and progress of a sort has been made. Research went froma top-down view dominated by social theory and forceful personalities, thoroughbehavioral models of children essentially as small animals, and later tried to cometo grips with the complexity of actual classroom practice and human learning. See[31] for a detailed recounting of the general story and [28] for mathematics in theUS. Unfortunately the obstacles are immense and the complexity overwhelming:

• Children are surely the most complicated and difficult experimental sub-jects possible;• There are strong constraints on how the subjects can be controlled or ma-

nipulated, and on how much control investigators have over actual practice;• There are many important variables that cannot be measured, let alone

controlled;• The situation is so complex and the theories so vague that scientific preci-

sion in terminology is impossible; and• theories and ‘facts’ are not precise or strong enough to support logical

analysis or deductive reasoning.

Recent education research in the US has attempted to be more scientific by use oflarge-scale trials, statistical analysis, etc. more-or-less modeled on medical studies.The resemblance is rather superficial, however, and many of the practices would beconsidered unethical in medicine. For instance:

• The double-blind protocol found to be necessary for reliable medical con-clusions is impossible in education. But rather than accept the medicalconclusion that findings will be biased and unreliable, educators concludethat it somehow doesn’t matter.• Small studies often have sample sizes two orders of magnitude too small to

justify the statistical analysis.• Large-scale trials often have little more than the name in common with the

pilot studies, but outcomes are still interpreted as validations of the pilotdesign.


• Significant variables such as per-student cost and teacher expertise areknown but not reported, and others such as student disruption are some-times controlled without mention.• Complexity is controlled by severely restricting—in advance—possible in-

terpretations of the data.One can sympathize with the difficulty and admire the effort, but the consequence isstill that educational researchers do not develop scientific sensibilities or discipline.

Educational research outside the US tends to be more qualitative and thought-ful, but not more scientific. Influences include dubious and conflicting psychologicaltheories, abstract models of “the student”, and deeply-held convictions derived fromclassroom practice, particularly at elementary levels. A complicating factor is thathumans have powerful instinctive responses to children. Some educational theoryseems as much informed by the emotional response of teachers as by dispassion-ate facts about children. In any case the effort is still not data-driven or logic-constrained, and educators do not develop the facilities to deal with data-drivenand logic-constrained scientific material.

4.1.1. Summary. Educators, as embodied in colleges of education, teachers associ-ations, etc., do not develop scientific sensibilities or discipline. Their theories arebound together by the strength of their convictions, not the strength of their data orlogic. As a result they are rarely able to make use of precision when it is available,and are rarely able to draw logical consequences when logic does apply. Scienceincompatible with their beliefs is usually misinterpreted or dismissed as irrelevant,and efforts to point out incompatibilities are usually seen as attacks to be defendedagainst. In short, the education community is not capable of making good use ofcomplicated science.

In 2006 Goswami [23] described educational abuse of cognitive neuroscience andobserved the inability to make use of complex or nuanced information, but expressedhope that broad-brush or big-picture messages might lead to better outcomes. The2010 article of Alferinka and Farmer-Dougana [3] describes more extensive misuse,with less optimism. So far neuroscience does not have broad-brush or big-pictureformulations that avoid the need for scientific precision (more about this below).Again, this means the mainstream educational community cannot be expected touse it carefully or correctly.

4.2. Lack of content sophistication. The point here is that apparent good align-ment of neuroscience and educational theory does not ensure good outcomes. Thisis illustrated with material from a profile of Stanislas Dehaene, by Jim Holt in theNew Yorker magazine [27].

First, a general principle:“The fundamental problem with learning mathematics is that whilethe number sense may be genetic, exact calculation requires cul-tural tools—symbols and algorithms—that [. . . ] must be absorbedby areas of the brain that evolved for other purposes. The processis made easier when what we are learning harmonizes with built-incircuitry. If we can’t change the architecture of our brains, we canat least adapt our teaching methods to the constraints it imposes.”

This is certainly true as stated, but most educators would interpret it as “adaptour teaching goals to the constraints imposed by brain architecture”. They would

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want to build on innate number senses and connection to spacial sense, as mappedout by Dehaene and others, and avoid some of the obviously unnatural materialnow taught.

A content-aware interpretation would be: understand skills needed in the longterm; find out how they are implemented in brains of successful users; and de-sign teaching methods to develop this implementation as quickly and painlessly aspossible.

The educational and content-aware interpretations differ: there are unnaturalthings that really are needed for long-term success. Returning to [27]:

Our inbuilt ineptness when it comes to more complex mathemati-cal processes has led Dehaene to question why we insist on drillingprocedures like long division into our children at all. There is, afterall, an alternative: the electronic calculator. ‘Give a calculator toa five-year-old, and you will teach him how to make friends withnumbers instead of despising them,’ he has written. By removingthe need to spend hundreds of hours memorizing boring proce-dures, he says, calculators can free children to concentrate on themeaning of these procedures . . .

Boring memorization is indeed a problem, and this solution would be welcomed bymost people in the education community. In fact calculators are already widelyused in the US.

In the long term, however, this is the wrong solution. The reasons are describednext, and some are expanded in the experimental section.

First, calculators make numbers friendly in a very superficial way: they all seemthe same and have no valuable structure. A basic feature of the place-value nota-tion, for instance, is that multiplication or division by ten can be accomplished bymoving the decimal point. Hand-arithmetic students know this because it is a bigtime-saver when it can be used, and it is an integral part of multi-digit multipli-cation and division algorithms. Calculator students do not, because it is useless:on a calculator moving the decimal point would be accomplished by multiplyingor dividing by ten. Similarly when a calculation calls for both multiplying anddividing by the same number, hand-arithmetic students will cancel them to avoidboth operations. Calculator students almost never do this, partly because they donot have written intermediates that they could scan for opportunities, and partlybecause it has so little payoff that such scanning is not worthwhile. The result isthat calculator students frequently have much weaker number sense.

The second problem is more subtle. For long-term purposes, internalizing thealgebraic structure of numbers in a way that extends to symbols is more importantthan fast or perfect numerical multiplication. In the past, much of this internal-ization seems to have been a subliminal consequence of the fact that much of themanipulation in the boring and unnatural algorithms is essentially symbolic, see §8.Symbols may even seem friendlier than numbers because one doesn’t have to recallnumerical multiplication facts. Calculator students never see these quasi-symbolicmanipulations, and so do not get this subliminal exposure. Further, encoding op-erations as keystrokes seems to make them inaccessible to abstraction, and becausesymbols cannot be manipulated the same way, symbols seem completely differ-ent from numbers. The result is that calculator students frequently have weakersymbolic skills.


Weaker number sense and symbolic skill, however, are long-term problems. Bothcause serious difficulty in college courses that are supposed to prepare students forscience and engineering careers. Pre-college education does not have external goalsso is more flexible. Tests in calculator-oriented curricula, for instance, heavilyemphasize numerical work to enable students to “display their calculator skills”,and coincidently downplay their symbolic weakness. As a result these problems areignored in educational theories whose horizon includes only pre-college education.

4.2.1. Summary. Dehaene offers “broad brush messages” of the sort Goswami [23]observes are needed by educators. They lack the nuance and precision of the originalscience, but surely Dehaene should be qualified to formulate broad-brush messagesthat are true to the science. They are also quite compatible with dominant con-ventional wisdom in education, so they are instances of “successful” collaborationbetween neuroscience and education. They are nonetheless counterproductive inthe long term because they are insensitive to the needs of mathematics.

This does not solve the original difficulty, however. The easy, neuroscience-endorsed calculator alternative may be counterproductive, but memorization is stillboring and the antique approach still leaves a lot to be desired. Is there a way touse calculators that does not undercut subliminal learning of algebraic structure?Exploring this will require a neuroscience collaboration with much more contentsophistication, c.f. §8.1.

4.3. Hidden cognitive errors. Many educational practices are based on hiddenassumptions, and many of these are wrong, at least on the cognitive level. Whenthey are accepted as valid and incorporated into neuroscience experimental design,the neuroscience will be ineffective.

A relatively minor example appears in §3.1 above: multi-digit integer multipli-cation is assumed in [29] to be a numerical issue, thereby missing opportunitiesto clarify symbolic and algorithmic aspects. A more problematic example, con-cerning the relationship of mathematics to the real world, is discussed next. Theeducational reasoning is:

(1) Mathematics is an abstraction of systematic structure in the physical world;(2) We have a great deal of intuitive understanding of the physical world, either

innate or learned;(3) Therefore, our intuitive understanding can and should serve as a basis for

developing mathematical knowledge and skills;(4) Further, word problems play an essential role in keeping mathematics tied

to its roots, and providing concrete instances of mathematical structure.However this seems to be wrong in many ways. We return to the philosophical

assertion (1) later. (2) is true: it is now well-known to cognitive psychologiststhat we do have an innate version of physics, but it is non-Newtonian and must beovercome to learn the real thing [18], [6]. In other words (3) is wrong for physics.It should not be a big surprise that it is wrong for mathematics too.

We do have some innate number sense but, as discussed above, it is quite insuffi-cient for mathematics and again must be avoided or overcome. We have some senseof space and geometry, and in antiquity it was found that with a little proddingthis could be used to do Euclidean plane geometry. This is still about all we can dowith it two thousand years later, and it is not a good foundation for the real thing.In short, conclusion (3) is wrong. Once scientific or mathematical understandings

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are established, then intuitive ideas can be retrained and recruited to enrich thisunderstanding, but trying to do this too soon inhibits rather than advances reliablelearning. This belief was rejected in professional practice about a century ago, see§5.2 and [40].

Whether mathematics is “really” an abstraction of the physical world, as in(1) above, is a pointless philosophical debate. Cognitively this actually seems tobe backwards: It seems to be more effective to think of the physical world as,roughly speaking, a murky implementation of a fragment of mathematics. Con-temporary mathematical practice reflects this by splitting real-world applicationsinto two parts: modeling (essentially a translation into symbolic form suitable formathematical analysis); and mathematical analysis of the model.

Conventional wisdom in elementary education rejects the separation of modelingand analysis. As suggested in (4), word problems are thought of as a differentformat rather than a different activity. Neuroscience investigations that acceptthis, e.g. [17], [48], are ineffective. They misinterpret neural evidence that thesereally are different activities, and find behavioral equivalence because they followedthe educational practice of restricting to problems with trivial mathematical core.A critical comparison with more complex problems should give a very differentpicture; see §7.2 for a proposal.

[33] provides an extreme example. This study compares two strategies for youngchildren working extremely simple problems that require a size comparison. Oneuses a symbolic translation to connect to innate number sense, the other uses pic-tures to enable visual comparison. The strategies do not apply to other problemtypes, and the use of innate abilities is a dead end.

4.3.1. Summary. Educational theory has many hidden assumptions, and many arewrong. Accepting these uncritically can render cognitive neuroscience studies inef-fective or misleading. In fact the most immediate benefits from educational neuro-science may come from dispelling some of these errors.

A good rule of thumb is: anything philosophical is doubtful. Some educationaluses of philosophy verge on the hilarious: see [39] for use of Wittgenstein to defendeducation against a caricature of neuroscience.

5. Mathematics and learning

The author feels, particularly after developing the examples in later sections,that a cognitively-oriented understanding of learning difficulties of real students isthe primary qualification needed for a genuinely productive neuroscience collabora-tion. It would also be useful to understand modern cognitive strategies that enablehumans to do mathematics. These are explained in this section.

5.1. Teaching vs. diagnosis. The mainstream educational community, and teach-ers at all levels, are more focused on teaching than learning, or in other words moreon information delivery than diagnosis of problems with receiving the information.There is a good reason for this: teaching is essentially concerned with learning ingroups, and resource constraints forbid much individualized attention2.

By diagnosis we mean one-on-one sessions, usually initiated by the student,and intended to isolate and fix a specific difficulty. The goal is to provide a brief,targeted intervention that will enable the student to resume working on his own. To

2See [41](c) for further discussion.


accomplish this the helper should listen more than talk, and not jump to conclusionsabout the difficulty. Expert teachers find both of these difficult. Diagnostic workprovides a complex, fine-grained and individual view of learning. This view is ratherdifferent from the teacher’s perspective and from standard educational theory, andseems considerably more relevant to cognitive issues and neuroscience.

Diagnosis in this sense is (in a nutshell) the procedure used by the help staffin the Math Emporium at Virginia Tech, a facility providing computer-based andcomputer-tested mathematics courses to around 10,000 students per semester, andnow in its thirteenth year; see http://www.emporium.vt.edu. The author has spentover 1,000 hours in diagnostic work with students in the Math Emporium, and foundit far more revealing than thirty years of classroom teaching. This is the primaryexperience drawn upon in developing the proposals later in the article.

5.2. Modern mathematics. Professional practice changed profoundly in the earlytwentieth century. It became better adapted to mathematics and consequently morepowerful, but other aspects of the change are more important here.

Modern mathematical methods are more systematic, deliberate, and precise,and less dependent on intuition and heuristics. A curious consequence is thatstrategies for human use have developed: systematic methods admit strategies;intuition either works or it doesn’t.

It is significant that mathematical human-use strategies evolved without con-scious direction or understanding. Up through the nineteenth century mathematicswas quite influenced by philosophy, but the early twentieth-century transition in-cluded a break with philosophy. Because the strategies evolved without interferencethey could adapt, in ways we do not understand, to human abilities and limitationsthat we also do not understand. Cognitively-oriented study of these strategies cantherefore reveal quite a bit about human cognition [40].

At present these cognitive strategies are used by only a few tens of thousandsof professionals, but since they address general cognitive issues they should, inprinciple, be a rich resource for new educational practices. They certainly could be arich resource for educational neuroscience. The discussion in §3.1, 4.2 and exampleslater in this article should illustrate this point. However explicit understanding ofthese strategies is extremely rare, and in the end less vital than the diagnosticunderstanding of learning problems discussed above.

5.3. Educational hostility. Mathematics education remains modeled on obsoletepractices of the nineteenth century and before; see [40] for a detailed discussion.This is no doubt one reason it has had trouble improving on nineteenth-centuryoutcomes. It is also part of the reason so few students make the transition fromschool mathematics to the twentieth century.

Attempts to introduce a bit of modern mathematics into education, for instance‘new math’ in the US, have been failures. The big problems were not mathematical:the large-scale dissemination of ‘new math’ was so poorly designed and executed itmight have failed even if the goal had been to give away candy. However the educa-tion community saw the failure as proof that modern mathematics is suitable onlyfor freaks. Mainstream educators remain deeply hostile to modern methodologies.The hostility includes human-use strategies: one of the most powerfully effective isthe concise self-contained definition, but this is universally rejected by educators.

http://www.emporium.vt.edu

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A consequence of this hostility is that mathematicians who become involvedin pre-college education are required to “check their weapons at the door”: buyinto the idea that nineteenth-century methods are somehow kinder, gentler, andmore appropriate. This is easier than one might think, because the features thatmake modern mathematics powerful are internalized, not articulated. In any casethe result is that mathematicians involved in mainstream education have the samebiases and drawbacks as educators, regardless of their mathematical credentials.See [7] for an example.

5.4. Theoretical incoherence. Theories of mathematics and mathematical prac-tice are as incoherent and inconsistent as educational theories. One reason is thatthe philosophers and historians who might develop explicit theories are still con-cerned with pre-modern practice, and their incoherence reflects incoherence in ac-tual practice that made change necessary. Ironically, modern practice is inaccessibleto philosophical investigation because it is more effective: it enabled a rapid increasein technical difficulty that made it opaque to outsiders.

Mathematicians writing about mathematics are as incoherent as philosophers.They have internalized the methodology so well that it has become transparent,and it seems to be a general principle that people cannot figure out how they dothings that they do well. Somewhat like birds and flying.

The point for the present discussion is that current descriptions of mathematics,no matter what the source, are not good resources for neuroscience3.

5.5. Summary. It seems that the qualification most important for neurosciencecollaboration is a diagnostically-based and mathematically sophisticated under-standing of cognitive learning problems of real students. This is rare. An under-standing of human-use strategies of modern mathematics could also be valuable,but is even more rare.

The main point is that the relevant expertise will not be found in educators, inmathematicians involved in mainstream pre-college education, in the philosophy orhistory of mathematics, nor even in the writings of mathematicians themselves. Aconsequence seems to be that ideas must be evaluated directly on their own merits,rather than on the credentials of the proposers.

6. Sample collaborations

The goal is to give examples of experimental proposals whose outcomes couldhave significant educational consequences. These illustrate opening moves by poten-tial mathematical-educational collaborators, addressed to cognitive neuroscientists.This section describes design criteria, and provides background for the examples.

6.1. Structure. Proposals from potential collaborators should be sensible of theneeds and limitations of neuroscience, and informed by neuroscience studies, butneed not be specific about the neuroscience. It would not take much imaginationto add quite a bit of detail about brain regions of interest, imaging techniques, etc.The audience makes this unnecessary, and this is their job anyway.

Cognitive, mathematical, and educational issues should discussed in detail. Specif-ically:

• The experiment should address a specific and credible cognitive concern.

3The author hopes [40] will be an exception.


• The design should maximize the likelihood of clear and useful outcomes.• The relationship to mathematical structure and goals—particularly long-

term goals—must be made clear.

These are responsibilities of the mathematical-educational partner in a collabora-tion, so are features a neuroscientist should look for in a proposal.

6.2. Cognitive interference, outline. The two examples in §7 concern interfer-ence between subtasks of a task. In a nutshell, mixing or switching between sub-tasks can reduce success and limit scope of application. In some cases proceduresor algorithms could be reorganized to separate such tasks, but this is constrainedby mathematical structure and the need to avoid introducing new cognitive diffi-culties. The challenge is to identify interference severe enough to need change, incircumstances where structure permits it.

6.2.1. Polynomial multiplication. The example in §7.1 concerns conflict betweenthe organization and calculation aspects of multiplication. The experiment usesmultiplication of polynomials by high-school or beginning college students, com-paring performance with standard (mixed-task) methods and a task-separated al-gorithm. The immediate outcome should be better methodology for polynomials.A long-term goal is better methodology for multi-digit multiplication in elemen-tary mathematics, and this is taken up in §8.1. The polynomial version provides asimpler (in some cognitive senses) and slower (for imaging purposes) model.

The context for the study is task switching between two basic task types. Thereis an extensive literature on mechanisms and costs of switching in very simple tasks,see the review [36]. The ACT–R computer model [2], [4] has been extensively testedand refined and seems to model some elementary tasks reasonably well, c.f. [45].The work done so far is only marginally relevant, however. It corresponds, roughlyspeaking, to discrete behavior of matter at atomic scales, while we are concernedwith statistical behavior at significantly larger scales. There are also large-scalephenomena that either are not consequences of small-scale effects, or that revealqualitative differences between effects that are indistinguishable at small scales.

A useful general conclusion from small-task studies is that our thinking is es-sentially single-track. If we switch from task A to a different task B but knowwe will shortly be doing A again, we usually cannot economize by keeping task-Ainstructions loaded but off-line. Instead we have to flush task-A material, load B,and when B is complete, reverse the procedure. Further

• “Flushing” task A may involve inhibiting task-A instructions, not just emp-tying a buffer. Some errors (e.g. adding instead of multiplying) result fromincomplete inhibition. Further, residual effects of this inhibition can slowor complicate reloading for the next A task.• Repeated switching reduces the effectiveness of working memory [34]• In contrast, following an A task by another instance of A usually requires

less reorganization and has lower costs.

Algorithms with subtasks ABABAB . . . usually cannot be reorganized as AAABBBbecause each subtask requires the outcome of the previous subtask. When it is pos-sible, however, this should have cognitive benefits. The multiplication proposal isof this type.

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6.2.2. Word problems. The example in §7.2 concerns cognitive interference betweenthe modeling and analysis aspects of word problems. The experiment comparesperformance and neural activity in the standard (mixed-task) approach and a task-separated modeling approach. The short-term goal is to show that the educationalapproach is counterproductive. The longer-term goal is to explore ways to use wordproblems effectively in elementary education.

Educators see word problems as essentially mathematical; a different formatrather than a different activity (see §4.3 for discussion). As a result educatorsencourage a gestalt approach in which students “develop strategies” to work di-rectly with the formulation of the problem. Students find this hard, and accessibleproblems have either mathematical or modeling component (or both) trivial.

Mathematicians and professional users of mathematics split real-world applica-tions into two steps: ‘modeling’ translates physical data to a self-contained symbolicformulation called ‘the model’, and then the model is analyzed mathematically.These two steps use very different methods and, technically, the modeling stepis not mathematical. Diagnostic experience with students suggests that modelingand analysis are also quite different cognitively. Mixing seems to cause interferenceconsiderably stronger than that seen in multiplication, and success in science andengineering—and the mathematics courses that prepare for these—requires use oftask-separated modeling.

Higher-level experience thus suggests that school children find even trivial wordproblems hard because the approach used is inconsistent with human cognitivestructure. The experiment is designed to explore the nature and strength of theinterference. The greatest significance, however, would be a clear demonstrationthat there is a problem. Many elementary educators have powerful philosophicaland emotional commitments to the mixed-task approach.

6.3. Subliminal learning and reenforcement, outline. We are concerned withlearning that takes place during an educational activity such as a lesson or assign-ment, but that is invisible to the student, and frequently to educators as well. Thereare two variations: subliminal learning from the content; and learning that dependsin an unrecognized way on the structure (e.g. kinetic or verbal) of the activity.

When there is a single way to do things then they come as packages. It isneither necessary nor possible to disentangle exactly how learning occurs. Newmethods—especially technology—cause these packages to come apart and impor-tant subliminal learning may be lost. For instance “find 365× 86” requires a lot ofneural activity when done by hand, and rather less when a calculator is used. Isthis activity pointless, or is there subliminal learning that is lost with calculators?

Diagnostic work with students suggests that there is quite a lot of subliminallearning in by-hand elementary mathematics that is absent from current calculatorcurricula [41](d). The goal of the experiments is to fix this: understand severalinstances well enough to design programs that use technology and also providethis learning. Curiously, this should also enable improvement of traditional pro-grams. Subliminal learning is usually an accidental and inefficient byproduct ofby-hand work. Understanding should enable more efficient approaches, either withor without technology.

6.3.1. Subliminal algebra in integer arithmetic. This experiment concerns sublimi-nal internalization of algebraic structure from by-hand integer multiplication. The


point is that the symbols we write to represent numbers are symbols, not numbers,and by-hand arithmetic involves a lot of symbol manipulation. Students seem tointernalize some of the algebraic structure used in these manipulations.

The place-value notation presents integers as polynomials in powers of ten, withsingle-digit coefficients. For instance 438 = 4 · x2 + 3 · x1 + 8 · x0, with x =10. The standard algorithms for multi-digit multiplication essentially multiply thecorresponding polynomials and then evaluate at 10.

The experiment in 8.1 has two parts. The first compares neural activity in 3×3-digit multiplication by hand, and with a calculator. An objective is to see to whatextent the hand work recruits neural regions used in algebra, and more specificallyin polynomial multiplication.

The second part explores the use of a task-separated algorithm modeled onthe polynomial algorithm of §7.1. The first version is for hand use. It requiresmore writing than the traditional algorithm but should display the structure moreclearly and be easier to use accurately. It may also give a way to learn single-digit multiplication subliminally rather than by explicit memorization. The secondversion uses a calculator, but limited in a way that still requires expansion anddisplay of algebraic structure. The objectives are to assess potential cognitivebenefits by comparing neural activity with that associated to standard by-handmultiplication.

6.3.2. Kinetic reenforcement in graphing. This experiment concerns reenforcementof internalization of geometric structure of function graphs, by the kinetic aspectof by-hand graphing. In non-technology programs, both assignments and testingrequire drawing by hand. In programs using technology student work has visualoutcomes, and testing is also usually visual (choose the correct graph among anumber of alternatives).

Diagnostic experience is that many graphing-calculator trained students cannoteither verbally describe or qualitatively sketch standard curves. When they do tryto draw pictures they often reproduce a calculator display, to scale, with typicalpoor choices of range and microscopic features of interest. In other words theyhave not internalized the qualitative geometric structure. It seems that the kineticaspect of drawing powerfully (and subliminally) reenforces learning of qualitativestructure, and some students seem unable to learn without this reenforcement.

A general context is that serious learning benefits from, and often requires, activereenforcement. Recent studies ([43], [44]) report that young children do not learnfrom videos. To learn vocabulary, for instance, they must say the word, not justhear it. Verbal reenforcement seems to be more effective when ‘social cognition’facilities are engaged by the presence of an attentive human, and this may be theprimary mechanism in some cases. None of this should be a surprise. Children inrural areas learn their local dialects but usually not (in the US) standard English,even though they hear as much or more standard English on television. Similarly,what a child sees makes far less impression that what he draws or writes himself.

A closely related problem is that educators often confuse attention or ‘engage-ment’ with learning. The baby-video studies mentioned above report rapt attentionbut no learning. Similarly, most educational approaches feature activity that en-gages students. In the computer-based Math Emporium program mentioned in§5 we have found that in most cases movement on a computer screen interfereswith learning, even though it usually triggers alert attention. Indeed, movement

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may reduce learning because it draws attention, in which case some engagementstrategies may actually be counterproductive. These are issues that urgently needneuroscience investigation.

The experiment uses two versions of a brief lesson on qualitative graphs of sumsof functions. The first versions uses a typical visual computer-graphic approach, andthe other a hand-drawing approach. Students are quizzed using version-appropriatemethods: visual multiple-choice in the first case, drawing in the second. Finallythey are tested with the opposite methodology.

The questions concern similarity and differences in neural activity in the twomodes, and transfer of learning from one mode to the other. Diagnostic experiencesuggests that kinetically-reenforced learning should transfer, visual learning usuallywill not. This experiment is more complex than the others because the questionsconcern neural activity during learning, not just during use of a learned procedure.

7. Cognitive interference

Mixing different tasks often slows and degrades performance in both. It seemslikely that such interference has a neural basis, and understanding this should enabledesign of algorithms and procedures better adapted to humans use. The proposalsaddress two instances in which interference has been observed: multiplication andword problems. See §6.2 above for an outline.

7.1. Cognitive interference in multiplication. There are two important casesthat use essentially the same algorithm: multi-digit integer multiplication in ele-mentary school, and polynomial multiplication in high school and college. We beginwith polynomials because:

• the polynomial version is actually a bit simpler because there are no over-flow problems associated with converting polynomial-like outcomes intoplace-value integer notation;• the separated polynomial tasks take long enough to be imaged by fMRI,

and this is unlikely with integer multiplication;• more-extensive scratch work can be use to correlate cognitive and neural

activity;• high-school or college students are more consistent and cooperative exper-

imental subjects;• arithmetic skills of older students are already well-established and stable,

and should produce clearer and more consistent signals; and finally• this is the context of the author’s experience.

Another reason to begin with polynomials is that the problem seems to involvea genuine limit on human ability: experienced professors of mathematics seem tohave as much trouble as students with the mixed-task algorithm, and get as muchbenefit from the task-separated version. This should mean that the underlyingneural issues should be relatively uniform and clear. In contrast, the interferenceexperienced by children with multi-digit integer multiplication can be eventuallybe managed, so may have a developmental component (see, however, §??). In factthere are likely to be a number of different difficulties, and before we can assessany of them we must understand the adult endpoint. Further, the proper course ofaction may be unclear. If the problem is only developmental then finding ways tospeed development would probably be more useful than tinkering with algorithms.


The integer case is discussed further in §8.1.

7.1.1. Sample problems. These examples show escalating conflict between organi-zation of the polynomial structure, and coefficient arithmetic. Coefficients are con-trived so individual operations are easy; difficulties come from mixing rather thanfrom individual operations.

1) Write (3x2) ((2− a)x3) as a polynomial in x.Note that “simple arithmetic” in coefficients may include symbols, to emphasizethat we need transparent internalization of structure (associative, distributive etc.),not just number facts. This example has one coefficient operation and one polyno-mial operation: they are perforce separated and there is little conflict.

2) Write (3x2 − x + 5a) ((2− a)x3) as a polynomial in x.The result has three terms. The standard practice is to do coefficient arithmeticas each term is generated, so there are two arithmetic interruptions of the poly-nomial procedure. There is relatively little interference, partly because there arefew interruptions. Another reason is that the structure of first term provides atemplate for sequential organization of the task. Minor interference is suggestedby more-frequent sign mistakes with the −1 coefficient on x in the first term, ascompared to errors in isolated arithmetic tasks.

3) Write (3x2 − x + 5a)(x3 + (2− a)x2 − a) as a polynomial in x.

Simple expansion gives nine terms, with eight interruptions for coefficient arith-metic. Moreover the data is a 3 × 3 array so a strategy for organization as asequential task must be devised. Finally, terms with the same coefficient have tobe collected and combined. The success rate is low and errors in both organiza-tion (missed terms) and arithmetic are common. The difficulty comes from thealgorithm rather than the problem itself, however, as we see next.

7.1.2. Task-separated algorithm. The basic plan is to separate different tasks ascompletely as possible. In polynomial multiplication, organizational work relatedto the polynomial structure should be completely separated from coefficient arith-metic, and multiplication and addition separated in the arithmetic. This is illus-trated with problem (3) above.

Step 1: A preliminary scan shows that the output will be a polynomial ofdegree 5. Set up a template for this:

x5( ) + x4( ) + x3( ) + x2( ) + x1( ) + x0( )

Step 2: Scan through the possible ways to take one term from the first factorand one from the second. For instance fix a term in the first factor and stepthrough the terms in the second; move to the next term in the first factorand repeat, etc. (For an alternative scanning strategy see 7.1.3 below.) Foreach such pair, note the total exponent and record the coefficients in theappropriate place in the template. Do not do any arithmetic. The result is

x5((3)(1)) + x4((3)(2− a) + (−1)(1)) + x3((−1)(2− a) + (5a)(1))+

x2((3)(−a) + (5a)(1)) + x1((−1)(−a)) + x0((5)(−1))

The second term in the coefficient on x4 is (−1)(1), from −x in the firstpolynomial and x3 in the second. This is not evaluated to give −1: even

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completely trivial arithmetic requires a momentary change of gears, andwatching for an opportunity to do it is a distraction.

Step 3: Do multiplications:

x5((3)(1)︸︷︷︸3

) + x4((3)(2− a)︸︷︷︸6−3a

+ (−1)(1)︸︷︷︸−1

) + x3((−1)(2− a)︸︷︷︸−2+a

+ (5a)(1)︸︷︷︸5a

) + . . .

Step 4: Do additions.

x5((3)(1)︸︷︷︸3

) + x4 ((3)(2− a)︸︷︷︸6−3a

+ (−1)(1)︸︷︷︸−1

)

︸︷︷︸5−3a

+x3 ((−1)(2− a)︸︷︷︸−2+a

+ (5a)(1)︸︷︷︸5a

)

︸︷︷︸−2+6a

+ . . .

7.1.3. Other problem types. This algorithm considerably extends the range of prob-lems and examples accessible to students. Full calculation of larger products canbe done, but a better task is to adapt the algorithm to give partial results:

• Find the coefficient on x3 when (5x3 + 3x2 − x + 5a)(x3 + (2− a)x2 − a) iswritten as a polynomial in x.

The terms with total exponent 3 are obtained as follows: begin with x3 in thefirst factor, x0 in the second, and record the coefficients. Move to the next lowerpower in the first factor and the next higher in the second, and so on. The resultis (5)(−a) + (−1)(2− a) + (5a)(1). Doing the arithmetic gives a− 2.

7.1.4. Experiment. The subjects should be high-school students who have beensuccessful in a standard algebra curriculum, or students in first-year college calculus(i.e. not remedial). Proficiency with problems like (2) above might be used as acriterion. They should also be screened for dependence on calculators for basicarithmetic (see below). They are asked to work problems similar to the one above,using standard methods. They are then taught the task-separated version, andafter enough practice to become familiar with it, they are imaged working similarproblems with this methodology.

• To keep the picture clear the arithmetic should be kept minimal. Multipli-cation of multi-digit integers, for instance, would produce a small versionof the entire process.• Half the problems should have numerical coefficients, half have symbols in

the coefficients (as in the example).• Subjects should be told that accuracy is more important than speed (see

the comments on error-detection in §3.3).• Scratch work should be videotaped and time-stamped, for correlation with

imaging results.

fMRI should provide general information about the areas used and the degree ofusage, c.f. [29, 45]. It would be quite useful to know if there is a MEG or EEG sig-nature associated with major task switching in the task-separated versions, c.f. [47].Targeted DSI imaging may help identify what to look for because it reveals con-nectivity structure [25].


7.1.5. Calculator version. The core experiment concerns students with good man-ual arithmetic skills. If resources permit, it can be expanded to include studentswith similar proficiency but who use calculators for numerical work. Let them usecalculators as they like during the trials, and record this use. Expected differencesare described below.

7.1.6. Analysis. The basic plan is to look for neural and performance differencesbetween the standard and task-separated versions. The expectation (based on diag-nostic work with students) is that performance should be significantly better withthe task-separated version, and the hypothesized reason is that the task-separatedversion reduces interference caused by arithmetic interruptions of the polynomialorganizational task. There is not enough information to support speculation on thenature of the interference. Some possibilities:

• The uninterrupted polynomial task may involve fewer brain areas, reflectingits simpler mathematical structure, and this contributes to effectiveness.Or,• the task-separated algorithm may provide a more effective way to use paper

as working memory that cannot be overwritten by subtasks. [4] considersonly simple tasks with little or no competition, but suggests that workingmemory, or “buffers” may be a key concern.

A qualitative picture should emerge reasonably quickly. It might be possible todirectly explore interruptions and their short-term neural consequences by carefullycorrelating imaging with scratch work.

The above is the basic plan. We now discuss potential complications and refine-ments.

First, there may be a sub-population with substantially better performance withthe modified algorithm. The goals are algorithms that benefit everyone when usedas the standard approach, but it is unlikely that everyone will benefit when theyare used as a retrofit. The cognitive interference under investigation should be thedominant difficulty in the high-performance group.

Calculator arithmetic requires a significant attention shift and input/output pro-cessing, and there are a great many discrete arithmetic tasks in these problems. Itis hard to imagine that calculator use could become so transparent that this wouldnot be a source of interference. The prediction, therefore, is that students whoactually make substantial use of calculators during the trials will have low successwith any form of these problems.

Note that subjects cannot be screened in advance for this because the controlexperiment (using standard techniques) becomes impossible after the modified al-gorithm is taught. However it would certainly be useful to find (after the fact) apredictor of success.

Next, if at all possible, individual variation in the task-separated version shouldbe investigated. Currently the statistical techniques used to analyze data have abuilt-in assumption that everyone does these things in essentially the same way.Variation is treated as noise. The data showing that multiplication facts are storedin the angular gyrus using verbal memory, for instance, demonstrates that this isthe dominant mode. But is it really true that no-one uses visual memory for this?Understanding variation in successful learning is essential for understanding all the

26 FRANK QUINN

barriers to success, and the separated tasks may be long and uniform enough topermit this. Note that students who use calculators will have significantly differentcharacteristics.

The number and nature of mistakes made is probably a more significant indi-cator than time required to complete the tasks. Time measurements will be moresignificant for comparing different tasks done by one individual than for comparingdifferent individuals.

It will be very important to see effects of symbols in the coefficients. The hypoth-esis suggested by behavioral data is that students who have effectively internalizedthe symbolic structure of arithmetic should show little or no difference in eitherperformance or neural activity. There is some support for this in very simple tasks[4] [49]. Conversely, students who have not internalized this structure, or who thinkof symbols and numbers as essentially different, will find symbolic coefficients sig-nificantly more difficult and this should be reflected in neural activity. It seemslikely that most calculator users will be in this group.

A final caution: some scenarios require a large number of subjects:• Calculator-users are very likely to have different characteristics. This is

an important issue but to explore it would require doubling the number ofsubjects.• The key group will be those who quickly get benefit from the modified

algorithm, and numbers must be large enough to ensure this group willbe reasonably big. The proportion in this group probably depends on thesource of subjects (e.g. high school or college) and rigor of pre-screening.This (among other things) cannot be predicted and should be determinedwith a pilot run.

7.2. Cognitive interference in word problems. The modeling and analysiscomponents of word problems seem to interfere when mixed, and this interferenceis often very strong. This is explored through comparison of student work usingstandard (mixed-task) and modeling (task-separated) procedures. See §6.2.2 fordiscussion.

7.2.1. Sample problem. The following have the same mathematical core.Food version: A basket contains six loaves of bread. Half of these are put

in another basket that already contains nine loaves. Then one-third of thetotal contents of the second basket is put in the first. How much breadends up in the first basket?

Social version: Jen and Brad have six loaves of bread. Brad takes half withhim when he leaves to share everything with Angelia, who already has nineloaves. Jen’s lawsuit against Brad and Angelia is settled by giving herone-third of Brad and Angelia’s bread. How much bread does Jen end upwith?

Money version: A basket contains six dollars. Half of these are put in an-other basket that already contains nine dollars. Then one-third of the totalcontents of the second basket is put in the first. How much money ends upin the first basket?

These are easy to model and solve, but difficult with the gestalt approach becauseinterpretation and calculation are mixed.


7.2.2. Task-separated (modeling) version. Let A denote the bread in the first bas-ket, with subscripts 1, 2, 3 corresponding to the three times. Bi similarly denotesthe bread in the second basket. Translating the data for the beginning state gives:

A0 = 6, B0 = 9.

Changes that give the second state translate as:

A1 = A0 −12A0, B1 = B0 +

12A0.

Finally changes that give the third state give:

A2 = A1 +13B1, B2 = B1 −

13B1.

This is a symbolic form (model) suitable for mathematical analysis. After doing afew of these they become immediately recognizable as short recurrence relations.

Analysis proceeds in two stages; first substitute in two steps to reduce to anumerical problem:

A2 = A1 +13B1 = (A0 −

12A0) +

13

(B0 +12A0) = 6− 1

2(6) +

13

(9 +12

(6))

and finally do the the arithmetic. See §7.2.5 for discussion of cognitive and concep-tual features.

7.2.3. Experiment. The subjects are high-school students who have been successfulin a standard algebra curriculum. They are imaged working problems similar tothe ones above. They are then taught the task-separated version, and after enoughpractice to become familiar with it, they are imaged working similar problems withthis methodology. They should be asked to give the model as part of the solution(to ensure actual separation), and some problems should ask only for the model.

In both trials, problems to be worked should be interspersed with controls inwhich students are asked only to identify problem type (food, social, etc.).

Finally, subjects should be interviewed before and after the imaging trials. Pre-trial questions would concern attitudes toward word problems (enjoy, dread, etc.),neutrally probe reasons (actually interesting, easy grades because the math is triv-ial, believe teachers’ assertion that they are important, etc.), and ask the subject’simpression of his general competence and success rate. Post-trial questions wouldinclude feelings about task separation (helps, is a waste of time), and assess changesin interest and feelings of competence.

There are two points to the interviews. First, is there a correlation betweenreduced cognitive interference and increased interest or confidence? Second, themain justifications for word problems are motivation and relevance, because theanalytic tasks are trivial. It is therefore important to assess how a proceduralchange might effect these.

7.2.4. Analysis. Unseparated work should show extensive activity, probably includ-ing prefrontal recruitment to sort out confusion from interference. Active areaswill probably depend on the nature of the problem, and different types should beanalyzed separately to see this. The social version, for instance, should engageneural structures devoted to interaction with others of our species. Comparisonwith type-identification versions should reveal activity specific to the mathematicaltask. Questions:

28 FRANK QUINN

• Do some types interfere with mathematical activity more strongly thanothers (i.e. have lower success rates)?• Do different types lead to differences in the mathematical components, as

revealed by subtracting type-identification responses?• Is there systematic variation, for instance sex differences in responses to

social versions, or socioeconomic level effects in responses to food or moneyversions? If so, how do these correlate with success rates?

Subtasks in unseparated work will have irregular timing and sequencing, and willbe hard to image. This is not a big problem.

Task-separated versions should show clearly-defined shifts between modeling andanalysis. Questions are:

• How do the areas and degrees of activation compare to the non-separatedversions? For instance, are the same areas used, just in sequence ratherthan simultaneously?• Modeling has some symbolic activity, and this should be revealed by sub-

tracting type-identification responses. Where does this take place, and isit essentially the same for all problem types?• The symbolic aspect of modeling seems not to interfere with other parts of

the process, as long as no analysis is done. Is this true on the neural level,or does it reveal interference too mild to be obvious?

7.2.5. Further discussion. The immediate cognitive benefit of the task-separatedversion is that translation and analysis are both routine and reliable, and can beextended. Adding another layer, for instance if Brad goes back to Jen and there isanother redistribution of bread, could easily be done in the task-separated versionbut would be a serious challenge with the gestalt approach.

Modeling also has conceptual benefits. The model displays the mathematicalstructure as a recurrence relation rather than a sequence of arithmetic operations.Similar models describe superficially different problems, showing the underlyingunity and demonstrating the power of abstraction. It can be connected to othermethodologies, for instance vectors and matrices: set C = (A, B) and the modelbecomes

C0 = (6, 9)

C1 =(

1/2 01/2 1

)C0

C2 =(

1 1/30 2/3

)C1

Multiplying the coefficient matrices gives a direct description of output from inputand enables exploration of the relationship. Is there an initial distribution thatleads to exactly the same final distribution? In another direction, one can also seehow a large number of “players” could give a cellular automaton.

Finally, modeling can be a rich activity even when students cannot analyze themodel. For instance as soon as ‘rate of change’ is introduced they could model phys-ical systems as differential equations, and then see computer graphs of solutions.Among other things, this might motivate learning the analytic techniques.


8. Subliminal learning and reenforcement

Human brains are complex, and the relative lack of integration in children’sbrains means early learning has additional complexity. The fact is well-known butmany of the details are invisible to adults. The proposals concern subliminal learn-ing of algebraic structure in by-hand arithmetic, and reenforcement of qualitativegeometric structure in by-hand graphing of functions. Both of these are usuallylost in calculator-oriented programs. The goal is to understand these well enoughto design programs in which subliminal learning and technology can coexist.

8.1. Subliminal algebra in integer multiplication. The first part of the ex-periment compares multiplications done by hand and with a calculator. This isto establish bases for comparison in the second part, and to compare the by-handactivity with algebraic manipulation. The second part compares two versions ofa task-separated algorithm: one by hand, and one with primitive computationalsupport. See the discussion == for explanation.

8.1.1. Experiment, part one. Subjects should be high school or beginning collegestudents, with reasonable facility with both calculators and hand arithmetic.

The tasks are to find 3×3-digit products (e.g. 946×735) either by hand using themethod they were taught in school, or with a calculator, as directed. Answers shouldbe written in either case. They should be told that accuracy is more importantthan speed.

8.1.2. Discussion, part one. The number of digits is chosen so by-hand work willfully engage the algorithmic structure but not be overwhelmed by written interme-diates.

Neural activity in the calculator case should be input/output and translationof digits to key presses. Little or no numerical or symbolic activity is expected.By-hand multiplication should show input-output, number-fact recall, and organi-zational activity. The interesting questions concern the organizational activity anderrors; see the discussion for part two.

8.1.3. Experiment, part two. Subjects are taught to use a task-separated multipli-cation algorithm that uses multiplication of polynomials and a final assembly (seebelow for an example). The experiment has two versions:

• Use the algorithm to reduce 3×3-digit products to 1×1-digit products andadditions. Carry these out by hand.• Use the algorithm with 2-digit blocks (see 8.1.5) to reduce 6×6-digit prod-

ucts to 2×2-digit products and additions. Carry these out with a calculatorthat is restricted to 2× 2-digit products, and arbitrary additions.

8.1.4. Single-digit algorithm. The place-value notation describes integers as poly-nomials in powers of ten with single-digit coefficients. For example 946 = 9x2 +4x1 + 6x0, evaluated at x = 10. Multiply using the polynomial algorithm of 7.1.2,then evaluate at powers of ten. To avoid actually writing polynomials, make theexponent explicit by writing it over the digit. For instance to compute 946 × 735write

29

14

06 ×

27

13

05

30 FRANK QUINN

Collect coefficient products for each total coefficient 0–4:

104(9 · 7) + 103(9 · 3 + 4 · 7) + 102(9 · 5 + 4 · 3 + 6 · 7) + 101(4 · 5 + 6 · 3) + 100(6 · 5)

Do the coefficient arithmetic:

104(9 · 7︸︷︷︸63

)+103 (9 · 3︸︷︷︸27

+ 4 · 7︸︷︷︸28

)︸︷︷︸55

+102 (9 · 5︸︷︷︸45

+ 4 · 3︸︷︷︸12

+ 6 · 7︸︷︷︸42

)︸︷︷︸99

+101 (4 · 5︸︷︷︸20

+ 6 · 3︸︷︷︸18

)︸︷︷︸38

+100(6 · 5︸︷︷︸30

)

Assemble the pieces by writing them in offset rows and adding:

0 3 01 3 82 9 93 5 54 6 3

sum 6 9 5 3 1 0

The left column contains the exponent, which is also the offset.

8.1.5. Block algorithm. Multiplication using 2-digit blocks begins by expressingintegers as polynomials in 102 with 2-digit coefficients. For instance 638521 =63x2 +85x1 +21x0, with x = 100. Multiply as polynomials, do the coefficient arith-metic, then combine (evaluate at x = 100) by writing in offset rows and adding, asabove.

If integers are written in 2-digit blocks, then 6× 6-digit products become 3× 3-block products. The two versions of part two therefore have the same algebraicstructure.§3.1.1 of [41](a) suggests using this algorithm for school group projects. The

objective is to use ordinary calculators (i.e., that handle 9 or more digits) to multiply15-digit integers, by breaking them into 4-digit blocks.

8.1.6. Discussion. The objective in the single-digit version is explore cognitive suit-ability for multi-digit multiplication in early education. It should provide clearersubliminal exposure to the underlying algebraic structure, and in particular whenthe template for polynomial multiplication would already be in place when studentsget to algebra.

Comparison with the traditional algorithm suggests that the traditional versionhas been optimized for production use by minimizing the writing needed. Theexpanded algorithm here is less efficient, but this is not a problem if it has cognitivebenefits because production arithmetic is no longer done by hand. An interestingsecondary question is how people eventually learn to use the traditional algorithmfor integers with reasonable facility, but have so much trouble with polynomials.This may provide a probe of the limits of complexity of operations that can behandled internally.

The two-digit block version begins exploration of ways to deal with primitivecomponents in algorithms. If the two-block version evokes the same activity asthe single-digit version, except for recall of multiplication facts, then memorizationmay not be necessary for subliminal algebraic learning. The key here is distraction.Memorization gives access to simple arithmetic without serious breaks in attention.This is quite important: when first learning a mathematical procedure we usuallycontrive the arithmetic to be simple enough that it does not interrupt or confuse


the procedure. Once the procedure has been internalized then more-complex com-ponent calculations can be managed, eg. by accumulating them in a task-separateddesign, or by having enough detail in written work to enable resumption after in-terruptions.

Standard calculators are not a satisfactory alternative to memorizing single-digit multiplication because they require distracting attention shifts. This mighteventually be avoided with a carefully designed computational environment, see[41](b). The analog for the polynomial-product version would be computationalsupport capable of handling coefficient calculations but not the whole product.

Another possibility is that the algorithm might provide a context for subliminallearning of single-digit products. Students could be given a multiplication table ona card, and assigned multi-digit products. They could use the card for single-digitproducts that they do not remember. They get subliminal exposure each time theylook at the card, and faster work is a payoff for remembering rather than lookingit up. Entries on the card would be designed for recall: for instance ∗7, 5; 35 for7× 5 = 35. Cognitive considerations:

• Short (e.g. omit “equals”),• Begin with the operation (∗) to reduce confusion with +7, 5; 12• Denote multiplication with ∗ rather than × to avoid verbal awkwardness

(“times seven”) and confusion of the symbols ×, +.• Only have products a×b with a ≥ b, to reduce the total number and enforce

internalization of commutativity.Students should be told to say the entry out loud each time they use the card(e.g. “star seven five; thirty five”) for verbal reenforcement (see 6.3.2 and the nextsection). It seems likely that with careful design, children could internalize thesefairly quickly and painlessly.

8.2. Kinetic reenforcement of geometric structure. Qualitative geometricstructure is used to explore questions about functions, and to clarify the quantita-tive information needed for specific questions. For example the curves y = ax2n fora positive and n a positive integer, all have pretty much the same shape. We cansee, for instance, that a straight line will intersect any of them in either two points,one point (when they are tangent), or no points.

We want to compare a purely visual approach with one that includes reenforce-ment. The comparison is done by cross-testing so the precise questions addressedare: how well does kinetic learning transfer to visual testing, and how well doesvisual learning transfer to kinetic testing. In fact actual use of qualitative structurerequires hand drawing, so the crucial question concerns visual to kinetic transfer.

The role of neuroscience is to throw light on the mechanisms (or non-mechanisms)of transfer between domains. To what extent does learning in one mode recruitactivity in regions that are used in testing the other mode? Does recruitment,or lack thereof, explain success or failure of transfer? Answering these questionsrequires imaging the learning activity, not just the testing.

8.2.1. The experiment. Subjects should be non-remedial first-year college students,as above. The study design depends on the number of subjects that can be tested.

If the number is twenty or fewer then students should be pre-tested to assesscompetence in the two learning modes, and assigned to the variant correspondingtheir strength. In other words, students from largely-visual technology programs

32 FRANK QUINN

should be in the visual track, and students from traditional programs should be inthe kinetic track. There should be about the same number in each track.

If the number is significantly greater than twenty then students should still bepre-tested for reference purposes, but then assigned to tracks at random. This wouldallow assessment of cross-training. Do visually trained students adapt reasonablyquickly to kinetic training, for instance?

Training sessions should last between 30 and 60 minutes, with at least threeshort quizzes to reenforce learning and familiarize students with the quiz format. Itshould be possible to repeat at least the first subsection if the corresponding quizoutcome is unsatisfactory. Students should be imaged during the training sessions.Students in both tracks should be able to do scratch work, and this should berecorded. See below for sample materials.

Next, students should be imaged taking quizzes, in a one or two day window atleast three days after but within a week of the training session. The first quiz wouldbe in the mode in which they were trained, to assess retention by comparison withthe final quiz of the training session. The second quiz would be in the other mode,to assess transfer of learning.

Genuinely qualitative internalization should include some abstraction and pro-vide flexibility. The later quizzes should be slightly different from the lesson mate-rials to probe this.

8.2.2. Discussion. It seems likely that there will be significant differences in learn-ing and transfer between the two modes. Quantifying this would require much morecareful controls and larger numbers, but this experiment should suggest explana-tory neural mechanisms that could substantially sharpen design of followups. Forexample:

• When kinetic students take visual tests, to what extent is the transferinternal, or external? External transfer would use visual comparison witha scratch sketch, while internal would presumably require communicationbetween kinetic and visual regions, probably mediated by activity in theprefrontal cortex.• When visual students take kinetic tests (i.e. are asked to draw something),

does the learning transfer, or does the output look like a reproductionof a recalled visual image? (Sketches by students trained with graphingcalculators are frequently reproductions of a calculator display.) How doesneural activity reflect this?

If kinetic reenforcement is important for durable qualitative learning, then along-term goal is to find ways to incorporate kinetic reenforcement in technology-based programs. This experiment should help make a start on this.

8.2.3. Materials. The experiment requires learning something unfamiliar but rea-sonably accessible. The proposal is to explore how the shape of a monomial (y = xn)is modified by addition of a lower-degree polynomial. This subliminally includesthe qualitative similarity of the families y = xn for n even, and for n odd.

• The visual version is illustrated (as usual) with graphics generated by com-puter or calculator. Quizzes are visual multiple-choice.• The kinetic version is illustrated by videos of hand drawing. Quizzes require

drawing.The following illustrates visual lesson materials:


odd - degreemonomial

sum

line, negativecoefficient

Figure 1: sum of y = xn, n odd, and a line with negative coefficient.

Roughly, adding a line with negative coefficient tilts the graph a bit to the right.For very large values of x the two graphs are essentially the same.

The following illustrates a visual test item:

1

2

3

4

Figure 2: The solid line is the graph of a cubic monomial. Which of the functions1–4 is the sum of this and a quadratic with negative coefficient? Which is the sum

with a line with positive coefficient?

A corresponding kinetic test item would be: “sketch the graph of a cubic mono-mial with positive coefficient. On the same graph, sketch the sum of this and aquadratic monomial with negative coefficient.”

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