I have a friend with dyscalculia and was teaching her some some mathematics (namely, solving a linear equation, simplifying certain expressions, and what (affine linear) functions are).

She understood solving equations of the form $ax + b = 0$ by first adding $-b$ to both sides and then diving by $a$. Dealing with negative $a$ and with expression $b - b$ was something of a problem, but I hope she figured it out, also.

Adding slightly more complexity created more problems. For example: $2x + 3 = -7$. We subtract 3 from both sides, getting $2x = -7 - 3$. She has great trouble seeing that $-7-3 = -10$.

How to communicate and teach the concepts here? I tried using the thermometer analogy, explaining how $a - a = a + (-a) = 0$ and, somewhat poorly, that $-7 - 3 = - (7 + 3) = -10$. How to justify the last attempt in a useful way? What other models or intuitions are there for understanding the negative numbers and particularly summing them?

becauseas written it would be confusing. That and because it's always lots of fun to sneak in some general nonsense when teaching a general audience. (Agree that it's off-topic in general, though.) – Harrison Brown Dec 6 '09 at 0:35