Years ago, I saw a stand-up show by the comedian Greg Behrendt. There's this one joke that I think about a lot.
He talks about the frustration of playing Pictionary with his wife. He gets a card, and it says that he has to draw the word "dry." He has no idea what to do, so he draws a bunch of waves and raindrops with a big X through them.
His wife says something like "No water!" and that's close, so he nods, but then she just keeps saying "No water! No water!" and he's stuck.
So then she starts yelling at him "Draw something different! Draw something different!"
So what does he do? He draws the exact same thing, HARDER.
This round of Pictionary does not end well.
I think about this joke a lot because I have a bunch of students in some form of math remediation. They tested poorly on fractions, so they are in a class to help them strengthen their skills in fractions.
The problem is, much of this remediation is just a slowed-down version of the exact same teaching methods that were ineffective for these students in the first place.
Their fifth grade teacher told them "To add fractions with different denominators, first you must find a common denominator and then convert each fraction to an equivalent fraction with that common denominator. Then you may add the numerators."
And now their eighth grade math remediation teacher (usually me) is telling them "To add fractions... with different denominators...first you find a common denominator... repeat that back to me, common denominator...then convert each fraction to an equivalent fraction..."
In other words, we are drawing the exact same thing, HARDER.
I think this type of remediation is ineffective because it assumes that the students just need to refresh their procedures and practice, practice, practice. But I don't think that these students are bad at adding fractions because they have been given insufficient opportunities to practice. In fact, I would say that procedural practice is something they've probably gotten too much of.
These students are the ones who can't get by on memorizing rules. They have to understand the concept of what a fraction is before they can manipulate it arithmetically. And one thing that is clear to me is that most students, even the ones who can add and multiply fractions, do not fully understand what a fraction is. I have many students in my class who can consistently add, subtract, multiply, and divide fractions, but cannot place the fraction 2/5 on a number line, or tell me whether 3/5 or 4/9 is greater.
I think it's time we draw something different.