Learning the Mean

One of my other favorite activities was the initial lesson. I think introducing them to the multiple methods of finding the mean is really beneficial. I loved doing with our group of students because we had three students and three methods and each student picked a different method as their favorite and most easily understood at the end.

The Lesson:

Authentic tasks related to understanding the mean. Your goal is to assess whether the students can think about the mean as fair sharing and also as balance point.

1. Use the sticky notes tasks from Kader or the GAISE report or Russle & Mokres (1996) as a starting point.

How many pets do you have?

1. Ask them for their information to start the data, then we will build the remaining data points around their data, assuming our predicted numbers will include their data.
   Mean:3    Sum: 27     9 data points

     x                 x
     x                 x     x

     x    x           x     x    

0    1    2     3    4    5    6    7    8

2. Use sticky notes on the board or table to record data for all the students to view (We used sticky notes to make our number line as well, and that was confusing, so make sure to have a separate number line so then can separate the two concepts in their head, or draw a number on the board)
3. (Maybe discuss the "average" before hand... what is an average? The students had trouble answering the question without really knowing what an average is) Ask each student to explain how they think we could find the mean/average of the data points. (Maybe indicate that you want them to manipulate the data to find the average- ours answered only verbally)
4. Observe how they get to the mean.
5. Use their ideas and manipulate them to fit balance point and fair share models.
6. Start with balance point by having them visualize the mean as the balance point (like on a scale) by moving the data toward the center so that it is “stacked” on the balance point. **Demonstrate moving it back out to start to gain the concept of the "balancing" on the scale. Tell them that at each point when you're moving the data, those are completely possible data sets. Show them how moving two- one spot on one side is the same as moving one-two spots on the other side... because it is still "balanced". This might be a good place to insert MAD too.
7. With fair share, we will change the representations from the number line to a 3D version of the data (we used Base10 blocks).  Each data point will have a “stack” with that many pieces and they will manipulate the stacks so that the blocks are “fairly” “shared” (or all towers are level) between the number of data points we have (maybe have them use a sticky note as a place holder to remember that that space needs a stack).

b. Questons to Ask:

i. Can you show/tell me how you are doing that?

ii. What questions do you have?

iii. How do this work?

iv. How has your thinking changed?

c. Several multiple-choice (Plickers-style) assessment tasks at the end that check for understanding.

2. What is a mean?

a) The middle of the data points

b) The average of the data points

c) The maximum minus the minimum

d) The most used number in the data set

    ii.   What is the fair share strategy of solving for the mean?

a) Deciding which number gets the most data

b) Sharing the mean with each data point to be fair

c) Rearranging the data so that each point is equal

d) Deciding if the mean is a fair number or not

 

  iii. What is the balance point strategy for solving the mean?

a) balancing all the data points on the mean

b) moving the data toward the center to find the mean

c) balancing the amount the data is moved toward the center to equal 0

d) all of the above

Anticipated Student Responses:

Correct:

-Add the numbers and then divide by the amount of numbers

Incorrect:

-Might pick the number that is used the most

-With fair share, they might forget to keep 10 stacks and make either less than or more than 10 stacks

-Might multiply the numbers by each other instead of add them

-Might try locate the median thinking it is the mean.


I definitely think this activity will go along with the GAISE report because it came from the GAISE report! I would like to find a way to, going along with the report, get students to pose their own questions at this level. I think I would be able to probe their thinking to get them to think out loud or ask each other questions. At the Math in Action conference during the Towers Task, we watched a video about it where they had each student show how they find the solution. The other students were asking questions about the students method and the student had to defend their method. While this tactic could easily go bad, I think it would be worth it to get the students asking each other questions and being able to explain what they are doing to show evidence of thinking.