When I began planning my last two units for my general and honors classes, I noticed a topic that I never recalled from my middle school mathematics education: Mean Absolute Deviation (MAD). Part of my scant recollection was due to the pacing I encountered in my early adolescent years (I can barely recall learning any probability until 8th grade), but the topic was also something new with the implementation of the CCSSM.
The topic is straightforward enough. MAD is the average distance of data points from the mean of the data set. Akin to IQR for the median of a data set, MAD provides a measure of the spread/clustering of data. A small MAD indicates less variability (clustering of a lot of data around the mean). A large MAD indicates greater variability (a wide spread of most of the data from the mean).
Cool, but how can students develop a solid conceptual understanding of MAD instead of blindly memorizing the computational steps and viewing MAD as just another random number they got with calculator kung-fu?
My attempt to tackle this challenge is multifaceted and today was the first day of the topic for my general classes, so I’ll probably update this post to reflect future success or attempts at success.
I started today with some simple dot plots I had students in my study hall generate:
First, I asked students a series of questions to get them thinking about the distribution of the data. How would you describe this data? Is the data spread out or clustered? What’s the range of the data? How many people plotted their answer? What do you expect the mean to be? What about the median? Do you think there are any outliers? Why or why not?
With answers to all of these questions firmly indicating the students were reading and interpreting the graph, I projected the same dot plots with additional information added:
I proceeded to use a mixture of explanation and questions to get students to see MAD as a way to see spread and identify potential outliers. First, I said the MAD for the shoe size data was close to 1 and I marked 1 below and above the mean. I asked students if most of the data was between the green marks, That was a big affirmative. Then, I cut to the other dot plot and asked students what was different about that MAD. Students noticed it was larger, then a student offered that most of the data was also inside the green marks on this plot. YES! I got them noticing, so I reeled them in with the secret: MAD is related to the spread of data, but most of the time a lot of data should be inside the bounds we make by adding and subtracting it from the mean. I asked a couple follow up questions to solidify the understanding, then I heard the question I was waiting for, “Mr. Hall, you gave us the MAD. How do we find it?”
I primed the engine, but now the students wanted to pull the starter cord. I love when these moments happen.
The rest of the period was spent working the calculation skill, but a lot of students seemed confident with the concept by the time we got to a gallery walk of practice problems. I even heard a student telling another, “That answer can’t be right, Most of the numbers are close to the average.” A lot of students in my class still need another day to work out the minutiae of the process, but today was promising to see the growth of ideas over blind calculation.
I hope to pick up the conversation and put a different spin on the topic tomorrow when I use Nathan Kraft’s equilateral triangle activity. It will be a nice reinforcement activity for students with the small amount of data points and I’m hoping it helps students further develop an understanding of MAD as a measure of accuracy and consistency. I tried something similar during my student teaching with high school students last year, but I never thought about MAD when I asked them to pick the best equilateral triangle!
My first year is almost over, but I’m glad I was able to get my students thinking a lot today!
Update (5/14/15): The equilateral triangle activity worked really well! The small amount of data helped students who were struggling gain skill with the process and everyone got to see a different side of MAD. After sharing that MAD was giving us a numerical ranking for the triangles students were drawing, I related the idea back to the spread of data. For some students, I think the idea that lower is better kind of threw them off a little, but most students were able to interpret MAD correctly during a later practice activity.