Thursday, April 16, 2015

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
      1. 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)
      2. (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)
      3. Observe how they get to the mean.
      4. Use their ideas and manipulate them to fit balance point and fair share models.
      5. 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.
      6. 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.
    1. 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:
-Add the numbers and then divide by the amount of numbers
-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.

Samples Jump Around

On the last day of working with our students, we did an activity to represent that samples jump around.  For our activity, we had 100 random numbers for 3 different cities: Zeeland, Grand Rapids, and Holland.  Each number was representative of the number of pets in a household of that city.  We had the students choose 10 numbers out of their data set.  We had them calculate the mean, the TAD and the MAD.  We didn't have a lot of time to explain the goals of the activity, as it got crunched into the last few minutes of our lesson.  I would really like to run the activity again for a longer session and touch on things like how samples jump around, the variability of each data set, the "one MAD rule of thumb", and other statistical concepts.

The lesson would be:

  1. Distribute random number samples of a couple different cities.
  2. Have the students randomly choose 10 numbers.
  3. Have students calculate the mean.
  4. Have students calculate the TAD.
  5. Have students calculate the MAD.
  6. Sort data into 3 columns on the board for each city.  Have students record their data for their city.
As a class, discuss the data.

Questions to Ask:
  • "What are similarities/differences overall?"
  • "How can we decide if there is a difference between the number of pets per household in each city?"
  • "Could we get the same mean or MAD in all three cities? What does that tell us about the population?
Introduce or Review "One MAD Rule of Thumb"
Discuss how samples jump around... "what does this mean for things like polls, like in magazines or online?"
Discuss factors of variability

Find newspaper, magazine or online polls and ask them to evaluate them for their classroom to see how likely the results are to happen again.

In my opinion, this activity would go along well with the GAISE report.  I think that this activity has a lot of real world relevance which is kind of the point of studying statistics, according to the report.  We want students to be able to thrive and I believe that understanding this information about samples and collecting data and studying data is really important.


On April 8th, I participated in a "twitter chat". This wasn't a traditional twitter chat, but it was pretty interesting none the less.  Since it was Spring Break, they posted a playlist of videos of michEDchat-ters talking about experiences in their classroom.

There were a lot of really interesting topics covered over the videos.  One of the points that I always find interesting is integrating technology into the classroom.  Mike Kaechele talked about using social media to collect data.  Erin Mastin discussed how she is incorporating new technology like “Genius Hour, Student Choice, 20% Time”.  Heidi Gascon works in a 1 to 1 environment and said it takes a lot of patience to make progress.  Another important point was showing students how to use the internet responsibly and safely.  One of the ways I use technology in my classroom is similar to how they use it in Traci Jackson's classroom.  One of the challenges she has found is working with kids from high to low at the same time.  They have been able to utilize technology to let kids who are higher work ahead on different applications or websites, and also kids who are lower can use them for more in depth practice or understanding.  One of the teachers I had in high school was actually on a video, Doug Ragan, and his story was actually really touching.  He had a student in his classroom who was out on extended absences because she had Leukemia.  Having a flipped classroom where they integrating technology for so much of the learning allowed to stay on track with what was happening in the class and when she came back she was excelling.

Another big point in the videos was removing the rigorous structure of the classroom.  Mike Kachele said that he feels education is too structured, too scripted, has too many standards and is too much of adults telling kids what to do.  We need to remove some of the set expectations to be able to let students develop their own focus and goals and learn to self motivate and lead.  We need to set our students up for success, as Dave Goodrich noted, which is part of the investment of being a teacher.  In order to do this, we can't be constantly telling them what to do or how to do something.  Learning is about being investigative and thinking critically.  We need to give the kids opportunities to connect with each other more and have meaningful discourse.  Jeff Bush mentioned this and I've seen it in my own classroom, by simply believing that our students can do something already starts them on their way to achieving it.  If they know one person believes in them, that could be all the motivation they need.  Jeremie Coplin talked about understanding their atmosphere and to allow students to excel on their own time.  He was referencing students with ADHD or ADD struggling in the large group classroom but being able to do really well in the correct atmosphere.

Another thing that is extremely important in teaching is instilling values that will help build the future of America, which Keith Tramper said.  Tara Maynard expressed that one of her biggest challenges in the classroom is combining students with all different backgrounds and getting them to be collective learners and support each other in spite of their inequalities.  We should not only be teaching our children to "tolerate" differences or accept them but teach them to embrace them in the classroom.  This can be experienced in all things from the color of their hair, skin, eyes to their home life to the way they think at problems to their strengths and weaknesses.  The same way we need to be infiltrating real world scenarios and lessons and concepts into the classroom daily.  The greatest benefit for students will be learning how to apply their knowledge to the real world.  How often do we hear people say "I'll never need this!"  Is it because we haven't given them the ability to recognize it in the real world?

I will take all of these ideas into my classroom in the future.  I loved getting to participate this way and thought it was really interesting.  When I've participated before in the more regular setting, I thought it was really cool to hear many different perspectives of teaching.  I will definitely utilize twitter for more opportunities for professional development and assistance as a teacher.