I am currently tutoring a student who is really lacking in math. At first I thought she was just resistant; perhaps she thought that her teacher had no choice but to pass her. What truly stunned me is that she is going to tenth-grade and has very little common sense with numbers.

I once asked her what is 3 minus 1/2, and she could not answer it. So, I drew three circles in front of her and shaded out one half. Now I asked how many circles are left, and she still could not answer it ....

An episode like this is not a rarity with her. Moreover, too often she would give responses that even a third-grader would know were nonsense. (What's half-way between 19.5 and 20? She said 9.) According to her mom, she is fine with her other classes. That's why I am wondering if a person can have issues with learning math specifically.

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    As Joel Reyes Noche mentioned, it really sounds like dyscalculia. If you feel that your student might have this kind of learning disability, you should let the parents know and suggest that they consult a psychoeducator (or anyone qualified to diagnose dyscalculia; this may vary depending on the country you live in). If she indeed has dyscalculia, then it might be possible for her to have access to better resources for her math class. For example, she could perhaps have the right to use a calculator during exams. – orion2112 Jul 26 at 5:28
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    Suggest you add the tag struggling students to your question so that more people who might be familiar with this kind of problem will find it. – Amy B Jul 26 at 5:35
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    @JoelReyesNoche Would you add dyscalculia as an answer to the question? – Tommi Brander Jul 30 at 10:43
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    @TommiBrander, I was hesitant to write an answer because I do not specialize in learning disabilities. But it seems that no one else has answered, so I'm writing my comment as an answer as you suggested so that the question is not left unanswered. – Joel Reyes Noche Jul 30 at 13:19
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    This doesn't directly address the question, so I am making it as a comment. How does your student deal with something very concrete like money: You have $3. You spend 50 cents on a popsicle. How much do you have left? Some students who seem to struggle mightily with arithmetic do better when it is contextualized in a familiar way. – paw88789 Aug 2 at 15:27
up vote 10 down vote accepted

The inabilities described in the question don't strike me as either particularly unusual nor as clear signs of a neurological learning disability. They strike me as more likely the passivity and mental shutdown that one frequently encounters in students who, whether consciously or unconsciously, are hiding a sense of incapacity, inability, confusion, etc.

Many students learn arithmetic operationally, as a series of rules for obtaining an answer, without internalizing the significance of the rules or their outcomes. Such a student neither has the habit of associating some mental model (e.g. the circles) with an arithmetic problem nor in many cases is able to relate such a model, posed by someone else, with the problem at hand.

Often students are not asked to consider whether their answers are reasonable (in some qualitative sense). Often they are rewarded (in grading) for performing certain operations correctly, whether or not the outcome is reasonable (with respect to extra-mathematical considerations). So a student who does not think much about what halfway means might easily obtain an answer that is less than either of the averaged numbers. That's not a sign of a learning disability, it's a sign of someone who hasn't learned how to connect arithmetic operations with common sense (in any case, common sense is mostly learned).

As an innumerate student progresses through the educational system, the psychological need to hide inadequacies becomes stronger and stronger, and the possibilities to correct basic misunderstandings become rarer and more remote. The algebra teacher often won't have time, energy, or inclination to teach arithmetic to a student who supposedly has passed several courses that require knowing arithmetic.

It's an extreme case, but once I taught a fourth year university student who was functionally illiterate (how could such a situation occur? This was in the US, and he was the starting tailback on the football team) and largely innumerate. He was adept at coaxing hints from the teacher, and at concealing his ignorance (of which he was fully and quite self-consciously aware). It took considerable care and effort to develop with him the confidence necessary to deal with the problem openly and honestly.

My own suspicion is that far more students have difficulties like this than most teachers care to admit. However, to detect such problems, one needs to interact with such a student personally, individually, and with the confidence sufficient to break down the barriers that prevent honest realization of the psychological obstacles (and the institutional context often makes that difficult or impossible). At the university one experiences that most students who struggle with calculus do so because they don't know basic algebra, even basic arithmetic. (A typical example is that students know that the logarithm and exponential satisfy functional identities, but can't remember if it is $f(xy) = f(x) + f(y)$ or $f(xy) = f(x)f(y)$ of $f(x + y) = f(x)f(y)$ and so guess.) Surely for students struggling in high school mathematics the common difficulty is this sort of inability to relate basic arithmetic to something other than a chain of operations aimed at getting points on an exam.

  • TY sir for your response. I'd like to ask her mom to get her a grade-school level exercise book. Do you think it's a good idea? That way, she will have an easier time in connecting math operations with common sense. – Andy Tam Aug 3 at 14:01

I do not specialize in learning disabilities, but it's possible that your student has dyscalculia. From Wikipedia:

Dyscalculia is difficulty in learning or comprehending arithmetic, such as difficulty in understanding numbers, learning how to manipulate numbers, and learning facts in mathematics. It is generally seen as the mathematical equivalent to dyslexia.

It can occur in people from across the whole IQ range – often higher than average – along with difficulties with time, measurement, and spatial reasoning.

The Wikipedia page gives some examples of signs and symptoms. For example,

Surprisingly, students with dyscalculia often do exceptionally in writing, reading, and speaking.

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