I Will (Probably) Not Meet My Students Halfway

There's a saying that good teachers meet their students halfway.

I'm not sure that is true.

Don't get me wrong. I think it is critical for teachers to meet their students academically...somewhere. I just don't believe the meeting point needs to be defined with that much specificity.

If I have a student who is capable of going 80% of the way toward learning a concept, meeting that student halfway is a disservice to the student. I should only meet them at the 80-20 mark. If I have a student who is only capable of going 25% of the way toward learning that same concept, meeting that student halfway isn't sufficient. I should meet that student at the 25-75 mark. My responsibility to each student is different for the exact same concept. So why should it look the same for each student in my classroom?

It shouldn't.

Our classrooms are filled with students who come in with varying ability levels, various levels of self-worth, various levels of self-doubt, and various levels of self-respect. All of which will affect what our students are able to accomplish on their own day-in and day-out. This obvious but underutilized (or perhaps poorly utilized) truth illustrates the inherent problem with the "halfway" philosophy; it will rarely be accurate. And if it isn't accurate, it isn't helpful.

Here's an overly technical illustration of this problem from Calculus.

In Calculus, students study the concept of the average value of a function on an interval. For example, consider the function f(x) = 0.5 − 0.5cos(πx) on the interval [0, 2]. A graph of the function on this interval is shown below.

The graph of f(x) on the interval [0, 2].

Depending on how astute you are at spotting symmetry and understanding its consequences, it may surprise you to find that the average value of f(x) on [0, 2] is 0.5, even though f(x) = 0.5 only twice on this interval at x = 0.5 and x = 1.5.

The average value of a function is based on the idea of the definite integral from Calculus. The definite integral is typically introduced as being a measure of the signed area between a curve and the x-axis (or y-axis depending on the variable of integration). In this case, the area of the region bounded above by the red curve and bounded below by the x-axis is exactly 1 square unit. The area of the rectangle whose base is the length of the interval and whose height is the average value of the function should also be 1 square unit. And this is exactly how calculus is used to calculate the average value of the function. Find the area under the curve and divide it by the length of the interval.

The "halfway" philosophy is based on the idea that teaching should look like an average of the values. In other words, assuming that this image accurately depicts what is going on with f(x):

The area of the rectangle using the average value.

While the area of the shaded region is equivalent to the actual area under the red curve, it does not accurately depict f(x). The rectangle over estimates f(x) for some x-values and underestimates f(x) for others. Similarly, meeting students halfway will ultimately result in over-serving students who are capable of going beyond halfway on their own and under-serving students who are not getting to halfway on their own. 

A more accurate depiction of f(x) would be to simply show what we mean by the area under a curve.

The area bounded by the curve and the x-axis.

[End overly technical illustration of this problem from Calculus.]

So I offer an alternative.

Meet each of your students at that point where they no longer believe they can continue on their own. And from that point, help them get further than they first thought. When they start to believe that they can do more on their own, back off and let them continue. Each student will need you at different points along the journey, for different reasons, and for different amounts of time. Be o.k. with that.