How about a little math? A function is a relationship between two or more variables, where one input or one set of inputs to the function produces only one output from the function. For example, let’s consider a diagonal line as our function: f(x)=mx+b. If we choose a horizontal position for that line (the x-coordinate or x value), there is one and only one vertical address implied (the associated y-coordinate or the calculation of f(x)). If that line is perfectly vertical (m is a number divided by zero), it is no longer a function; if it’s perfectly horizontal (m=0), all inputs get the same output (b). In our youth we were introduced to functions when we first learned about x-y coordinate systems. And, we were tortured by functions if we ventured into higher levels of calculus. Excuse me while I take a moment.
A continuous function is one where a small change in the inputs gets a relatively small change in the output, and is drawn as a continuous smooth line. Just like our diagonal line from above, or that beautiful curve we see as normal and sounds like a bell. Move just a hair one way or the other in the horizontal direction and the vertical response moves just a hair as well. This kind of relationship is profound, especially when you think about how it might apply to our daily lives. Imagine if this was how our weight was controlled by calories. Just reduce your caloric input and you immediately weigh less. Yes! Of course, calories are related to weight, but not as a linear causal function (more on that later). But even so, these continuous functions suggest that even a small effort in moving the inputs does something to the output. Small, perhaps, but something.
Now let’s consider when a small change in the inputs yields a relatively large change, or jump, in the output. This is a non-continuous function, and would be drawn with a line that has a gap. One type of non-continuous function is the step-function. It is named that because it looks like a staircase when graphed. As you move horizontally nothing happens until you reach a certain point, then the graph jumps up or down one full stair. An example of a step function is the rounding of numbers to the nearest whole number. As you move from the input of 1 to 1.49, the output (the rounding of the input number) stays at one. Nothing has happened as far as the output is concerned. Then you get the input to 1.5 and the output jumps all the way to two! The small change from 1.49 to 1.5 has yielded an unusual jump in the output.
This type of relationship is more often how life works. We all have thresholds of tolerance and intolerance. When the threshold is reached, we respond, sometimes dramatically. This can work in reverse as well (stairs going down). We respond to some stimuli until we reach some sort of stability. Once our world is stable, even if it’s not perfect, we stop responding. My best example is my own home. When I bought this house, I knew there were only four things to upgrade: the upstairs, the downstairs, the front yard, and the back yard. The short version of the story is, once the kitchen became functional, the DIY efforts came to a grinding halt. My threshold was found, and I was sated.
Now think about your classroom, playground, hallways, lunchrooms, the school in general, or even your personal environments. Certain conditions get our attention and keep it until a stability is reached, and then we relax or move on to the next condition demanding our attention. Unfortunately, that stability is almost always imperfect.
I often refer to this phenomenon as minding the “phone ringing” metric. While the phone rings, I work to improve conditions, the moment it stops ringing, I can relax. School principals know this feeling all too well. If parents, teachers, superintendents, or school board members are complaining about something loud enough, they work to make it better and silence the phone. But once the complaints stop, all too often, so does the improvement effort. Not because the principal doesn’t want the best school possible, but because this phone has stopped ringing and they can now relax or redirect their attention to the one that still is.
Sometimes the phone never starts ringing, but stability has fooled people into their own step function. Did you ever wonder why schools that serve homogenous populations with a relatively high SES don’t get better test scores? Their kitchen is functional.
In the long run, this means that schools are hard pressed to pay attention to the continuous functions that describe more fertile grounds for improvement. The one that comes to mind is how the school and classroom learning environments work with learning outcomes. Just like calories and weight, they are related, but take a little faith that changing one will ultimately change the other. And since their relationships are continuous, even small efforts get gains. Statistically the gains might seem small, but to the one child that benefitted from the extra effort, it means the world.