Friday, November 17, 2017

Understanding Standard Deviation

Long story short, I was presented an opportunity to teach standard deviation and I jumped at the chance. There are times as a coach that I wish I were back in the classroom. I do miss working with students daily. Anyway, here's what I did.

To start, I wanted students to think about the center of a data set. I asked them to individually consider each of the following and provide a typical number that best represents the situation.
I intentionally avoided using the words average, mean or center. Sometimes when I use a "mathy" word students hesitate. I'm not sure why, but I have my theories. I know I get better responses when I use the word typical so I went with that.

As we discussed each one, I wanted to draw out some important characteristics: Minimum value? Maximum value? and Are the possible values close in range or spread out?
I used the first situation as a contrast to the others to bring out variability. There are 24 hours in a day and this doesn't vary given you are talking about a day on Earth. The other situations have some variation in the data. That's what we are talking about today: variation in a set of data.

Going to the Movies
The Justice League movie is releasing next week and we are very excited about getting to see the movie. Imagine you are in the theater and settling in to watch the movie. There are people all around you and you begin to wonder...
  • How old are they?
  • Are they all the same age? If not, what is the typical age?
  • How close are the other ages to that typical age?
Here's what we got from that question.
The typical age is 26 years old with many people between the ages of 16 and 36. (Okay, I had to prompt them to think beyond the age of 30. At first it was 5 yrs old to 30 years old.)

Okay, so I've got them thinking about measure of center and variability. They've even put some values to that variability. Now, let's collect some real data.

Estimating Time
So here's why I wanted to teach this lesson. I really, really, really enjoy playing with this site and like how its simplicity captures students' attention.

Ever waited to go on vacation and it seems like time slows down but when you are on vacation the time goes by quickly? While the passing of time is constant, our estimation of how much time has passed varies. I wonder how close are we at estimating 10 seconds? Let's find out.

Each student was asked to estimate 10 seconds and record the amount of time to the nearest tenth on a sticky note. (Because this is a small school, I collected data earlier that morning from a variety of teachers and students. This allowed us to have enough data to analyze.) We built a histogram from the data. Great opportunity to assess student understanding of values within each interval.

The conversation started broad by asking what they noticed about the data and if it helped to answer our question about how close we are at estimating 10 seconds. I directed the conversation towards getting closer to standard deviation (without saying standard deviation, yet).
The students were developing a good sense of center and the values in relation to the center. Students talked about the estimates being 1.5 seconds less than 10 and 1.5 seconds more than 10.

This 1.5 you are using to describe how far away the values are from the center was a number mathematicians wanted to calculate. It's not the full range of the data but a value that can tell us how far some of the data is from the center. This number is called the "standard deviation".

Illustrative Math Task
Students were given the Illustrative Math Task Understanding the Standard Deviation. I used Part 3 as the formative assessment to gauge student understanding.

Up Next
The next lesson is to calculate the standard deviation and work on interpreting it. I like the EngageNY lesson Algebra 1 Module 2 Lesson 6 Interpreting the Standard Deviation.

In case you were wondering, because, let's face it, I was wondering, how close were the estimates for mean and standard deviation? The mean was 10.4 seconds and the standard deviation was 1.9 seconds.

Monday, November 6, 2017

Reflections on NCCTM17

The North Carolina Council of Teachers of Mathematics has their annual Leadership and Math Conferences this past week.It was a whirlwind three days with fabulous speakers including Jennifer Bay-Williams, Peg Smith, Juli Dixon, William McCallum, Joliegh Honey, Jennifer Wilson, ....  The list goes on. Anyway, I always enjoy the conference and wanted to share a few highlights.

First, I had the privilege to present with one of my teachers, Kim Clark. We presented Factoring Using the Area Model. It is very rewarding to work with a teacher who pushes you to grow because she wants to be better. I have learned a lot from her over the years and was thrilled to get to present this session with her.

This is a slide from our presentation. Did you know that the diagonals of the area model have the same product? Using that fact, we asked participants to find the possible area models. This question is the lynch pin of the investigation we did with students to help them develop a strategy for factoring when a is not 1. It's pretty cool and effective.

Second highlight was hearing Jennifer Bay-Williams talk about Becoming Fluent in Developing Procedural Fluency. This is an area of growth and learning for me. I am an avid supporter and believer that conceptual understanding is critical to students understanding and making sense of mathematics. What I am learning is that it takes specific and intentional instruction to help students develop that understanding into procedural fluency. I am also learning how to better define fluency as more than just quick.

The last highlight I'd like to share was the session from Dr. Valerie Faulkner on Opportunity, Equity & Agency: How do our grouping practices mediate student sense of mathematical identity? She makes a very convincing argument about how grouping in education (high and low students) just doesn't make a lot of sense. The statement I have been using to summarize her talk is one she made.

"We are surprisingly bad at evaluating what a person can and will do even if we have a vested interest in doing so." 

We should stop trying to separate students using some abstract high and low characteristic of student ability. We must begin to talk about students in regards to what they know and when they need to work on.

As I shared, the three days were a whirlwind. I am excited about new friendships developed and being able to connect with teachers from across the state. Maybe I'll even start blogging more often. Haaaa.....Haaaaa.....Haaaa. No promises but always a goal.