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I am a bit afraid to ask this, but the question has bothered me for some time now. I have a student in my analysis class having a medical certificate of dyscalculia. This entitles her to write tests and exams in a special environment and using extra time. This is ok and I do not have questions regarding this case. My question is more general.

What studies are there to show connections between advanced mathematical ability (needed for example to be a maths major) and dyscalculia?

Would it be wise to advise students with this illness not to choose mathematics as a major?

I know this is a sensitive subject and I do not want to open a discussion about it. Please, provide me links to medical/pedagogical literature supporting the practice (or opposing) we have at our university.

In many subjects if you have a medical certificate you cannot study that particular subject. Most notable in our university is sports. Clearly, mathematics and dyscalculia are not comparable to this, but what is the evidence supporting this feeling?

ADDED: Question clarified according to comments. Thank you for making this a better question.

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    $\begingroup$ Some studies on this would be interesting indeed. I'm curious if dyscalculia is a problem with numbers, or if such a student would also have troubles with, say, abstract algebra or (algorithmic) discrete mathematics. $\endgroup$ – Roland Apr 14 '14 at 14:00
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    $\begingroup$ @BenjaminDickman: It's not very surprising that dyscalculia and mathematics appear in the same papers - dyscalculia is usually noticed when doing mathematics (often in school). However, OPs question is far more specific and finding the right paper sounds like a needle in a haystack, considering the number of google hits. $\endgroup$ – Roland Apr 14 '14 at 14:15
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    $\begingroup$ This does likely not really answer the question but Blažková 'Teacher Training for Teaching Learning Disabled Individuals - Dyscalculia' in section 6 says "Many notable personalities had problems with mathematics in their childhood, nevertheless they achieved remarkable results later, some of them even in mathematics or physics." And then mentions Luzin and Hilbert among others. $\endgroup$ – quid Apr 14 '14 at 16:07
  • $\begingroup$ @BenjaminDickman: as Roland writes, I would like to have some expert guidance in the literature. What do I need to know as a university lecturer when one of my mathematics major students has dyscalculia? $\endgroup$ – András Bátkai Apr 14 '14 at 19:05
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    $\begingroup$ I just sent a cold email to this professor -- cehd.umn.edu/icd/people/faculty/cpsy/mazzocco.html Who seems to be an expert in dyscalculia (among other math acquisition related issues). Mostly focuses on early childhood stuff, but she should have a much better idea of what the existing literature says than I would by Googling. I gave her a link to the question so she might answer herself, but I'll keep you posted on her response if I get one. $\endgroup$ – LinearZoetrope Apr 15 '14 at 4:53
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I am not an expert in dyscalculia, and, in fact, my area of study in mathematics education has not predisposed me to focus on the idea of mathematical disorders. Because of this, I think I have a point of view that may at least offer a useful way to view these questions. [Note: there are a lot of conjectures here, but I think that is not unusual for questions of this type. I will try to be explicit about it.]

Item 1 Assumption: everyone can learn mathematics. As far as I know, there is no dispute about this. This does not contradict the possibility that some people may have deficits related to certain parts of mathematics.

Item 2: dyscalculia is specifically a deficit in learning arithmetic (Butterworth, Varma, & Laurillard, 2011).

Item 3: there is mathematical reasoning apart from arithmetic reasoning, such as quantitative reasoning described by Smith and Thompson (2008). Such reasoning can be used in the development of algebraic reasoning, apart from arithmetic.

Answer pt. 1: I do not know of specific studies you ask for, but with the items above, we can conjecture that it is possible for someone with deficits in arithmetical reasoning to develop mathematical thinking that relies heavily on quantity rather than arithmetic.

Answer pt. 2: I would suggest it is not wise to dissuade students against a mathematics if such deficits are real, causing students to rely on other skills. The reason being: a motivated student may well find new ways to think about mathematics, relying on different skillsets. That person may well discover ways of thinking about mathematics that contribute not only to mathematics, but to mathematics teaching and learning. We can't know for certain, but diversity in thinking has a practical value, apart from the idea of social justice. At least, there is an argument there.

Works Cited:

Butterworth, B., Varma, S., & Laurillard, D. (2011). Dyscalculia: from brain to education. Science, 332(6033), 1049–1053.

Smith III, J. P., & Thompson, P. W. (2008). Quantitative reasoning and the development of algebraic reasoning. In J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 95–131). New York, NY: Taylor & Francis Group.

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    $\begingroup$ Thank you. I think there dare not many students studying higher mathematics with dyscalculia, so it may be that your reasoning is the most I can hope for. $\endgroup$ – András Bátkai Apr 16 '14 at 7:08
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    $\begingroup$ Assumption 1 does not appear obviously true. E.g.: Are all people with severe intellectual disability, or hydranencephaly, able to learn mathematics? Related: matheducators.stackexchange.com/questions/11396/… $\endgroup$ – Daniel R. Collins May 27 '17 at 13:55
  • $\begingroup$ From the very little reading I've done, I'm not convinced by assumptions 2/3. The description of dyscalculia I read included things I see as quantitative reasoning, like not having any sense of orders of magnitude, (eg 'is this a plausible answer to get?'). $\endgroup$ – Jessica B Jun 4 '17 at 7:01
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I teach middle and high school math students. It is my understanding that dyscalculia is the math equivalent to dyslexia with reading. While I was in school for my teaching certificate, I was directed to this website. http://www.ncld.org/types-learning-disabilities/dyscalculia/what-is-dyscalculia?utm_source=dyscalculia-interest-email&utm_medium=email&utm_content=dearreaderimage&utm_campaign=dyscalculia-interest-email

The website gave a very good overview of what students diagnosed with dyscalculia may be experiencing. I have also found that students diagnosed with is may have the ability to overcome it just as students diagnosed with dyslexia.

Another website that I found that goes into more detail regarding college age students is: http://www.dyscalculia.org/college-dyscalculia

Through my education courses, it was unadvised to advise students on whether or not they should pursue certain goals in life, such as dropping out of school, taking certain classes, etc. I would assume that the same would apply to college level students as well. I was told that advise for students about their choices should be left to the adviser, teaching the students should be left to the teacher. If a teacher starts giving advise to the students, it may contradict something the adviser has already talked to the student about, or worse, it could interfere with a personal goal/dream that the student has, or a family based decision. You, as the teacher, also are unaware of whether or not the student is trying to achieve something, and your advise is what pushes them to do something bad, like give up on their goals, drop out of school, or even try to harm themselves or others.

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  • $\begingroup$ Thank you. You raise important points. $\endgroup$ – András Bátkai Apr 14 '14 at 21:17
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    $\begingroup$ I am not sure what (who?) do you mean by "adviser", but I find the sentence about "adviser vs. teacher" at least very controversial (not to use a much stronger word). It's totally against my conviction of who the teacher is: a more experienced human being, who should be a master (as in "a master and a student"). It's obvious that after all, it is the student who makes the important life decisions, and there may be many people to ask for advice (though I wouldn't give much advice not asked). Also, the one who ask should be aware that s/he may get many contradicting answers. $\endgroup$ – mbork Apr 14 '14 at 21:57
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    $\begingroup$ @mbork, just to clarify: in some situations, such as in the U.S. at both undergrad and graduate levels, a student has an official "advisor" who, in principle, is aware of the larger picture of the student's academic life, goals, subsequent career plans, and so on. "Teacher" would mean someone who has perhaps-intense, but isolated (to one year, one topic, once course) contact with the student. Unsurprisingly, yes, in all too many scenarios the official advisor sees the student all too rarely, and a teacher would have many more opportunites woth to observe and to advise... $\endgroup$ – paul garrett Apr 15 '14 at 0:38
  • $\begingroup$ @mbork, Mr. Garrett has clarified my point for me very well. However, I would like to elaborate on my meaning. In the U.S., teachers are strongly encouraged against giving advice to students. This is because, teachers are not privy to all the factors that influence a student's decisions or behavior. It is rather taboo for a teacher to suggest career paths or majors for students. We take more of an "encouragement of their chosen endeavors, whatever they may be" approach. It is also considered overstepping the boundaries of the adviser for a teacher to advise students on these matters. $\endgroup$ – user839 Apr 15 '14 at 18:05
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    $\begingroup$ I see. Thanks for the explanation, I didn't know that. All this, however, add to my (already strong) feeling that the U.S. is a very, very strange place... Just to clarify: as I said, as a teacher, I would very seldom suggest career paths for students; but I wouldn't hesitate if asked. (BTW: a few weeks ago I was in a very similar situation; a freshman student approached me and asked precisely for such an advice. I was honored and terrified, since I did not feel very much competent to give such advice. I was fine, however, with explaining him a few things he didn't know, so that he... $\endgroup$ – mbork Apr 15 '14 at 19:02
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Possibly relevant: In the first chapter of Where Mathematics Comes From: How The Embodied Mind Brings Mathematics Into Being, Lakoff & Núñez discuss a few examples of individuals who have suffered some kind of neurological damage that partially or completely disables their ability to perform numerical calculations. With respect to this question, the most interesting example is probably this one:

Not only is rote calculation localized separately from basic arithmetic abilities but algebraic abilities are localized separately from the capacity for basic arithmetic. Dehaene (1997) cites a patient with a Ph.D. in chemistry who has acalculia, the inability to do basic arithmetic. For example, he cannot solve $2\cdot 3$, $7-3$, $9\div 3$, or $5\cdot 4$. Yet he can do abstract algebraic calculations. He can simplify $(a \cdot b) / (b \cdot a)$ into $1$ and $a \cdot a \cdot a$ into $a^3$, and could recognize that $(d/c) + a$ is not generally equal to $(d + a) / (c + a)$. Dehaene concludes that algebraic calculation and arithmetic calculation are processed in different brain regions.

The citation to Dehaene (1997) refers to the book The Number Sense. The relevant passage from that book seems to be p. 199:

Up to now, this book has been concerned only with elementary arithmetic. But what about more advanced mathematical abilities, such as algebra? Should we postulate yet other neuronal networks dedicated to them? Recent discoveries by the Austrian neuropsychologist Margarete Hittmair-Delazer seem to suggest so. She has found that acalculic patients do not necessarily lose their knowledge of algebra. One of her patients, like Mrs. B, lost his memory of addition and multiplication tables following a left subcortical lesion. Yet he could still recalculate arithmetic facts by using sophisticated mathematical recipes that indicated an excellent conceptual mastery of arithmetic. For instance, he could still solve $7 \times 8$ as $7 \times 10 - 7 \times 2$. Another patient, who had a Ph.D. in chemistry, had become acalculic to the point of failing to solve $2\times 3$, $7-3$, $9\div 3$, or $5\times 4$. He could nevertheless still execute abstract formal calculations. Judiciously making use of the commutativity, associativity,and distributivity of arithmetic operations, he was able to simplify $\frac{a \times b}{b \times a}$ into $1$ or $a\times a\times a$ into $a^3$, and he recognized that the equation $\frac{d}{c} + a = \frac{d+a}{c+a}$ is generally false. Although this issue has been the matter of very little research to date, these two cases suggest, against all intuition, that the neuronal circuits that hold algebraic knowledge must be largely independent of the networks involved in mental calculation.

Regarding these passages, a couple of comments are in order:

  • I am not sure if acalculia and dyscalculia are the same phenomenon, or what exactly the relationship is between them. Nor is it clear from this brief excerpt whether the patient (the one with the Ph.D in Chemistry) developed acalculia in adulthood, for example as the result of some injury, or whether it was a condition he had all along. (Hittmair-Delazer's first patient suffered from a "left subcortical lesion", but the Ph.D in Chemistry is "another patient", of whom no details are provided.) The distinction seems significant, as there is (probably?) a big difference between someone learning algebraic reasoning despite suffering from acalculia, on the one hand, and someone retaining algebraic reasoning that they learned prior to the onset of acalculia.
  • Algebraic reasoning and manipulation is one thing, but proof-based advanced-level mathematics at the level of an undergraduate analysis course is (maybe?) something else.

The primary source for this is probably one of the following two papers:

HITTMAIR-DELAZER, M., SAILER, U., and BENKE, T. Impaired arithmetic facts but intact conceptual knowledge – a single case study of dyscalculia. Cortex, 31: 139-147, 1995.

HITTMAIR-DELAZER, M., SEMENZA, C., and DENES, G. Concepts and facts in calculation. Brain, 117: 715-728, 1994.

Both of these papers are cited in the references of yet another paper:

Dehaene, S., & Cohen, L. (1997). Cerebral pathways for calculation: Double dissociation between rote verbal and quantitative knowledge of arithmetic. Cortex, 33(2), 219-250.

I suspect any of these three papers might have additional references that could be helpful in understanding these phenomena.

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  • $\begingroup$ In this paper (PDF) by Stanislas Dehaene himself, he refers to his own earlier work as TNS and cites it as follows: Dehaene, S. (1997). The number sense. New York: Oxford University Press. A 2011 edition of TNS can be found online (PDF). Both works contain multiple references to "acalculia", including a description of possible causes. The index for Dyscalculia says "See Acalculia". $\endgroup$ – shoover Jun 2 '17 at 17:43
  • $\begingroup$ @shoover Thank you -- I have updated my answer to include additional information gleaned from your links. $\endgroup$ – mweiss Jun 4 '17 at 3:19
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I was once a member of an email discussion list (about some long forgotten statistics software). Once somebody answered a question, some other pointed out he did some silly arithmetic error in that response.

He responded that, yes, he had discalculia, and , for that reason, he had very early decided that he had to study mathematics, because, then, he only needed to calculate with letters, not numbers, thus avoiding his discalculia problems!

The point being, of course, that discalculia in itself do not necessarily say anything about mathematical talent or potential. It might be just an isolated problem with numerals.

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