While discussing statistics, particularly the three basic measures of central tendency with some teachers recently I got the ubiquitous quick definitions of “most” for mode, “middle” for median, and “average” for mean. Upon probing with questions further the teachers soon began to realize that in our want to make things “easier” for our students we often allow too much leniency with the language of mathematics. We do not always “attend to precision.” This was especially evident as I questioned further about the median and the mean.

The median is “the middle” but only in the sense that if I am the median of a set of data I happen to be the piece of data that is central, by location, among the other pieces of data numerically arranged. Having them create a human set of data points allowed them to understand this more fully. We explored how the quantities beyond the center data person would need to be changed in order to truly affect the median. Often, if students are able to explore concepts kinesthetically, visually, and concretely prior to the abstract and symbolic, it helps support the development of deeper understandings of “why” is this data point classified as the median and what does that really mean. Not just how I find the median – “by counting to the middle”, as one teacher said.

The truly interesting conversation unveiled as the definition of “mean” was discussed. When I probed further and asked the follow-up question of “What is an average?” to their answer that a mean was an average, they of course replied by telling me , “You add up all the numbers and divide by how many there are.” So I then asked them if this defined the mean or simply told me HOW to find the mean? They had given me a procedure. We then had more discussion until we were finally able to come to an understanding of what a “mean” truly is.

Later during a reflection and debriefing time, one teacher admitted that he was probably at fault of allowing his students to give him the “how” and not the “why.” He was an older gentleman, but new to education. He said his life had been focused around the “how” as he used mathematics daily, but not really focused around understanding the “why” things worked the way they did. What about in your class. Can your students tell you the “how” and do you accept that, or can they tell you the “why” as well to show that they truly understand the mathematics they are using? How do we move them beyond just knowing the how, the procedure, to the understanding of “why” the procedure works. And not all students will make that journey during the same time frame. Hence the importance of differentiation and scaffolding instruction. More to come on that later…