Seeing Patterns

This week I covered some basic statistics with my Algebra I students. I love the brief discussion on week 6: day 4 of Dr. Zadina's workbook. She addresses the fact that the brain is designed to see patterns and to see differences in patterns. This is math in a nutshell- the study of hidden patterns all around us. (Cool article here). So at what point does math get so hard for some students? If the brain literally wants to see patterns and give us a red flag when there is a deviation from the pattern, why is math content sometimes so difficult grasp? All of this made me start thinking about ways that I can present the content to my students in a way that allows them to see patterns and develop the relevant processes from the patterns that they are seeing. Keeping in mind that I simply don't have time to allow my students to "reinvent" concepts on their own, I decided to try a few things that would allow them to come up with at least come up with a process on their own. I know that so often we just hand our students a process or a list of steps and show them how to work the problems, and let's face it- it's the easiest way for us to make sure that we taught the material. But this week I was determined to try.

We started the week by quickly reviewing mean, median, mode, and range. The majority of my students remembered these concepts with no problem. So then we moved on to discuss outliers- what are they and why are outliers relevant to a set of data? More importantly, how do they affect the data? I would show the students a set of data and tell them the outlier. It was up to them to tell me why the particular value was an outlier. I thought this was a great way for their brain to recognize a pattern- the outlier is much less or much greater than the rest of the data set. For example, if all of the numbers in the data set are around 30 and one value is 1,000 this is a deviation from the rest of the data. The students quickly picked up on this and they understood that this value was the outlier. They also realized that this value sometimes greatly affected the mean of the data. If all of the values are near 30, then adding that 1,000 to the data would make the "average" appear higher than it really is.

On the day that we covered standard deviation, I made an effort to give my students the chance to determine the process we needed to use to calculate the standard deviation. I gave the students a handout with a couple of completely worked out examples. I gave them some time to work in a group to try and figure out what was happening with the data set in order to determine the standard deviation. It would have been so easy for me to give them a list of steps and show them exactly what to do. I just felt like it would be much more meaningful for them to try to come up with that process on their own. So after 15 minutes or so of discussion, many of the students were coming up with various processes. I gave them a new example that wasn't worked out, it simply had a data set and the answer to the standard deviation. I asked them to try their steps out on a new example to see if they came up with the correct standard deviation. After a while, we came together as a class to fill out a six-step graphic organizer.

As we filled in the steps, I had the students explain things like "we figured out that you take the square root of 16 because 4 squared gives us the value in the previous step... so then we made sure it worked for both examples." Doing it this way probably took more time than if I had just handed them a set of steps at the beginning of class, but they were SO excited that they figured this out on their own. There was discussion, friendly competition, some disagreement, and it required the students to justify their answers and analyze patterns.