Math Concepts That Fry Young Minds

As I continue to tutor Math and write a book on the subject, I've gotten more and more familiar with the Mathematical concepts that seem to cause the most trouble at the various grade levels. There are quite a few of them (most of which I get into in the book) but two that have been coming up in my work a lot lately are negative numbers and linear equations.

Unlike fractions or word problems (two other members of that daunting list), though, I don't have any personal experience with ever struggling with Integers or line equations. So where it's easy for me to put myself in a student's shoes when he or she can't quite grasp how to divide one fraction by another or how to figure out which train arrives at the station first, I'm finding it more difficult to explain why 3 - (-7) is 10, rather than 4, -4 or -10 (as seem to be the most common results arrived at). I tend to refer the child to the number line in pursuit of the correct answer, but they don't seem to be getting taught that approach in school much anymore. I've tried variations like "subtracting -7 is like removing a debt of $7, meaning that you now have $7 more since you don't owe that $7 anymore" but that's admittedly a pretty circuitous route to take. In the end, I usually resort to flash cards and lots and lots of practice for the student.

With line equations, there appears to be a disconnect in the minds of some students as to what the equations even mean. In other words, they're taught that putting the equation in the form, y = mx + b, is important because they can then read the slope of the line (m, in the equation) and the y-intercept (b, in the equation) straight off the page. That much they tend to get. But they often struggle figuring out the x-intercept (since it's not staring them right in the face) as well as concepts like recognizing whether a point (x1, y1) is on the line or not, given the equation. The whole notion that a point is only on the line if you can plug its x-coordinate and y-coordinate into the equation (for the x and y, respectively) and have the equality maintained, just doesn't seem to be sinking in for some. Therefore, if you ask the question, "Is point (2,3) on that line?" they either look at you blankly or start to graph the line. All of which shows a fundamental lack of understanding about the relationship that exists between lines and equations, and that gap just compounds into larger problems once they're introduced to parabolas and other exponential functions. And yet, when I was taking Grade 9 Math way back when, it just seemed immediately obvious to me how equations with x and y in them worked, since we'd been working with the Cartesian coordinate system for awhile by then. So how do you get this point across if that particular penny hasn't dropped and equations still look like Greek to the student?

In my book I'll be confronting some of these questions and offering up what I can by way of solutions. At least I'm getting lots of real world experience dealing with what I imagine many parents are running into right now.