Trying to get students to make the leap from concrete to abstract is often the most difficult transition in thinking. In 7th grade, we learn to write algebraic expressions and equations. Some students make the leap seamlessly, understanding that "some number" indicates to use a variable. Others struggle, wanting the actual number to write. They push buttons on calculators, bang on their heads, try to work out the problem, when in actuality, all that is expected is for them to write an expression or equation using variables to represent the unknown parts. No matter how many times I tell them, show them, give them examples, when turned loose on their own to try the process, they insist on sticking numbers in their work instead of variables.
I wonder if this is the critical point where we go wrong in math. Is this the actual turning point where we lose them?
Even working with small numbers of students in my math classes, I find it difficult to figure out ways to meet each child where they are and take them through the learning process. It is kind of like taking 5 students to the track to run a 4 minute mile. Some of them, no matter what we do, will never make the target. If I take them over to the high jump, no matter what I do, some of them will never be successful in clearing that bar.
Maybe if we waited another year, they might make the grade. Maybe not... The fish is NEVER going to climb the tree.
But the reality of my classroom is I am expected to make that fish climb that tree... make the young man who still thinks concretely, whose brain has not matured to comprehend the abstract, make him see that using a letter for an unknown number is the secret to success.