cognitive neuroscience and mathematics education cognitive neuroscience is concerned with the neural

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    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 education has 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 is a 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.


    Introduction 2 1. Background, and outline of part one 2 2. Technical difficulty, and consequences 3 2.1. Technical difficulties 3 2.2. Needs of neuroscience 5 2.3. Implications for application 6 3. Ineffective neuroscience 6 3.1. Multiplication 6 3.2. Solving equations 8 3.3. Errors 8 3.4. Summary 12 4. Pitfalls in education 12 4.1. Lack of scientific skills 12 4.2. Lack of content sophistication 13 4.3. Hidden cognitive errors 15 5. Mathematics and learning 16 5.1. Teaching vs. diagnosis 16 5.2. Modern mathematics 17 5.3. Educational hostility 17 5.4. Theoretical incoherence 18

    Date: Draft, November 2010.



    5.5. Summary 18 6. Sample collaborations 18 6.1. Structure 18 6.2. Cognitive interference, outline 19 6.3. Subliminal learning and reenforcement, outline 20 7. Cognitive interference 22 7.1. Cognitive interference in multiplication 22 7.2. Cognitive interference in word problems 26 8. Subliminal learning and reenforcement 29 8.1. Subliminal algebra in integer multiplication 29 8.2. Kinetic reenforcement of geometric structure 31 References 33


    The original goal of this article was to explore potential applications of neuro- science to mathematics education, and the kinds of collaborations that would make these possible. The conclusions, however, are discouraging. Good applications will be 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 educational communities. These conclusions are explained in §§1–5.

    The second part of the article (§§6–8) was developed as an antidote of sorts to the first part. As evidence that high-impact educational neuroscience is possible in spite of all the pitfalls, we give four detailed proposals for neuroscience experiments that 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 to development 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 than most cognitive activities, and in the late 1990s Stanislas Dehaene [12, 13] exploited this in a pioneering exploration of mechanisms behind number sense. Applications to 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 too far”. Instead, he recommended a two-stage approach with educational applica- tions undertaken by cognitive psychologists, and neuroscience providing input to psychology. 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 was that Bruer’s two-stage approach was not working: educators were using distorted popular accounts rather than solid science [23] [19], and psychologist intermediaries seemed to be ineffective. There is something to this: [51] for instance gives a


    generic, uncritical summary of neuroscience findings that could be used in all sorts of ways. On the other hand, by then many of the people doing education-related cognitive 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 than as psychologists or educators, and they had found that this brought more respect and influence.

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

    Section 2.1 explores what neuroscience needs from a partner. In a nutshell, the expense 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 it means. §3 illustrates how ‘edu-myths’ and lack of subject sophistication can render neuroscience studies ineffective. On the other side of the collaboration, complexity and ambiguity of outcomes means care and sophistication are needed to properly apply them. More than that, sophistication is needed to avoid misapplication that could discredit the approach. §4 explains why the education community cannot pro- vide such care and sophistication. Finally, §5 describes more precisely the expertise that—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 is conducted and on the feedback needed to make progress. It also makes good use of the outcomes a challenge.

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

    Next, brains are busy places, and signals relevant to the question at hand must be 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 about neurotransmitter activity, for instance. Serious imbalances in neurotransmitter activity are well known to effect cognition, and can cause differences in activity that show up in imaging. It is probably common for individual differences to effect


    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 for such effects. They must be treated as noise and this, no doubt, is one reason statistical 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-ray tomography, but the radiation is higher-energy and the target is an organ that 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 it does track, and whether it really correlates with glucose delivery, is a matter of debate. For practical purposes it may be more important to understand how 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 should be similar to PET data, and nicely complementary to EEG or MEG. Un- fortunately the high field strength makes it difficult or impossible to use these methods at t


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