Today I’ll talk about Sam. Sam was going into 7th grade with a C in math from 6th grade. The plan was for Sam to be tutored by me twice a week over the summer between 6th and 7th grades. In exchange, his parents would refinish the oak floors in my house. It was a win-win situation.

As is the case with most middle school students who are struggling in math, fractions are a real problem. Early in our first meeting together I asked Sam to solve this problem:

^{1}⁄_{2} + ^{1}⁄_{3} = ?

Sam answered ^{2}⁄_{5}

To Sam (as to many, many struggling students) the way to add fractions is to add numerators and then to add denominators.

They think that ^{a}⁄_{b} + ^{c}⁄_{d} = ^{a+c}⁄_{b+d}

To Sam, those fractions were just meaningless symbols. Otherwise, he could have seen that his answer couldn’t possibly be correct, as his answer of ^{2}⁄_{5} is less than ½. But to Sam, fractions (and most of math in general) was just meaningless numbers and symbols.

To help Sam understand fractions, I pulled out my handy-dandy fraction pieces. These are often plastic circles and pie pieces. Students like Sam need a concrete model, so that the symbols can make sense. Piaget was right, in my experience. The transition from concrete objects, where math is connected to the real world, to pictures, and then to abstract notation really works. Sam was able to figure out for himself that the only way to add fractions was using this formula, which made sense:

^{a}⁄_{b} + ^{c}⁄_{d} = ^{a+c}⁄_{b}

For the next few weeks, we worked on more complex problems, and Sam was cured!

Sam went on to get a B in 8th grade math the following school year!! A couple of years later I moved and my new house had oak floors that needed refinishing. I called Sam’s mom to see if he might want more tutoring, in exchange for refinishing my new floors. Sam’s mom reported that Sam was doing great in high school math, building on his success in 8th grade. So that one summer of tutoring got Sam on track for success! (I had to pay for the refinishing of the floors, but it felt great nonetheless.)