Good Data Day

Nothing revolutionary here. Just what turned out to be a nice sequence of experiences with a group of preservice elementary teachers. No handouts, just a story and pictures of what happened.

Previously... we measured some distances, from a whiteboard marker to an outdoor commons. We discussed technique and accuracy, and I shared a bit on with what elementary students struggle. They went off and measured, and then we shared data in a Google doc. (New to some, somehow.) Choice 1 we agreed on as a class. Choice 2 was up to their group. Distance to the bathroom, trash bins, and one group was obsessed with how do they measure a bus. Then we talked about how do we go from all those different numbers to a specific measurement. How tall is a desk? We want a single number! Of course, they first go to the average.



I always push on that. One, because you need to justify, man! Always. The other because the mean is so inappropriate for low sample size because of the power of outliers.

Then last class was a day to explore  measures of central tendency. I love human statistical displays and it was a gorgeous day. (October in Michigan.)

First sort: height.

They naturally lined up this way. We found the median height, with some good discussion about 19 students, which was the middle and what if it was between two people. When the middle height person stepped forward, we talked about if she was a good representation of the typical height of student in our class. People were a little uncomfortable about that. Especially at the tall end of the line.

Second sort: age.
First take, they lined up like they did for height. We found the median and discussed how well she represented typical. People were curious about the twins, who were gracious to answer age questions. I asked if it was true that boys tended to be older than girls, which led to a comparison of the significance of their position in the height chart vs their position in the age chart.

Then I asked them to line up spaced for their age. Start at 19, each sidewalk section was 4 months. "What if we're the same?" "Just line up." "Oh, it's a line plot!"   Students sorted themselves within a month by age. So when we determined the median, we talked about whether to count from the front or back of that month's line.  They thought that this showed the data much better because of the gaps, and were surprised how old it made that ancient 22 year old look.

Third sort: name length.

Finally split up the twins! For this one, we added unifix cubes. They made a stack of cubes the height of their name, first and last as it is on their birth certificate, and used that for comparison. The new question - could we use the cubes to figure out the mean? Would the mean be higher or lower than the median? "We could lay them all down in a big line and divide it by 19!" (Literal adaption of the arithmetic rule is pretty typical.) "What does that do?" "Could we all just share to have the same?" "We have these left over... what do we do with them?" Then we discussed if that meant the mean was closer to 13 or 14. Why was it higher than the median?

Fourth sort: shoe size.

I gave them the option of what to sort, but they were too shy to say anything. When I say shoe size, they jumped at it, and several said they were thinking about that. I like the ambiguity between men and women's, and the apparent diversity from shoe size not varying perfectly with height. I wanted to use the cubes again, but how do we do that with half sizes? Two cubes for each half size... that works! This time they sorted themselves into a line plot without being asked, working out their own spacing.

Again the mean was higher than the median. Will that always happen? How would it be different if we added 4 cubes to the men's shoes?

Starbursts
As they came back inside, I distributed Starbursts, very unfairly. Enter your data on the board before you sit down. DON'T EAT THEM YET. (Only one person did. Not bad.)

We discussed the challenges of using candy in class. Sugar, red dye, communicated values. (Why are prunes the only individually wrapped healthy snack?)

I asked them to make a line plot and pie chart of this data. The group that shared had a very nice way to make their pie chart:take the 80 starbursts, and think about 20 per quarter. Then half that is 10, half that is 5. "But there's 81?" someone asked. The class agreed that this wouldn't change the pie chart much.

Next was the hard discussion. If we want to be fair, how many should each person get? 2, since that's what most people got? But if people contributed, then there would be enough for 3 each. Wait a second, there's enough for four for each of us. With some left over! I tried to make the connection between this idea of equal distribution and the mean. Here's where we want to take outliers into account.

(Unfortunately, I showed them a graph of US income distribution at this point. Probably should have skipped that. It is one of the reasons I think statistical education is important - we can't understand our world and society without it. What do you think?)

Finally we looked at one more way to make a pie chart.

You can see two wrappers there. We compared this to the pie chart on the board, and they said that the lines really helped and they would add them. But they agreed that the drawn chart was very close.

After that we discussed how to share them fairly. I shared that this was real problem solving. The textbook problem is 81 starbursts, 19 students, how many do each get. The real life problem includes messy things like people wanting particular colors.   We did a quick check of color preferences by going to the corners of the room. One corner each for red (30%), pink (25%), orange (15%) and yellow (5%).  Center of the room for no preference (25%). A few methods were suggested, then someone said we should just do rounds of taking one. People agreed and we started. They wanted to make the pie chart for each one and who am I to argue.

So it was a good data day. Statistics and sunshine, who could ask for more?

Next class: starting the glyphs project!

Glyphs to Data to Display

Me.

Story of a 2 day lesson.

In the preassessment for my 6 week math for elementary class, it turns out that the students were pretty strong on the basics of statistics measurements and displays.  So instead of spending a lot of time telling them what they know, we went right to data collection.  One of my favorite ways to collect data is a glyph.  The first teacher I saw use one was Char Beckmann, but I think there's an old Teaching Children Mathematics (er, Arithmetic Teacher) article about them.  [Found it:  Cartland, Patricia E., What's in a Glyph?, Feb 1996, 324-28] Definitely one of my main faults as a teacher is trying to get too much out of a lesson, so beware that here.

Objectives:  TLW

• collect statistical information
• formulate questions
• organize and analyze data
• display data to address a question
• consider what features of displays are effective
• consider types of questions teachers ask
• review how the next day's test will be evaluated

Day 1: (40 min)
Schema Activation:  what would you be interested in knowing about your classmates?
They brought up music, sports, food, family background, and the like.  (I think of these as cultural identifiers.)

Focus:  introduce the idea of glyphs.  The handout has this information on it:

Glyphs

The circle on the other side of this page will be your face – but not the face you see in the mirror!  A face that tells much more about you…

Hair:  Put a hair on your head for each person living in your residence.  Curly if they are 18 or younger, straight if they are 19 or older.

Eyes: I purposely leave left and right ambiguous (like a mirror or a mask) because I want there to be issues in data collection, but

• left eye - favorite pet :   circle-dog, oval-cat, triangle-bird, spiral-fish, X-exotic or other, crescent    don’t like pets       
• right eye - favorite TV show:  rectangle-reality/game show, square-police/mystery, trapezoid-doctor/medical, rhombus-historical/documentary, kite-sports, Y-other (should be a chevron), crescent-don’t watch TV

Nose:  Make a shape with as many sides as books you read for fun last month.

Mouth  :If from outside the state, add a tooth for each year you’ve lived in Michigan.

• Smooth line: from Grand Rapids area
• Crooked line: from Michigan but not GR
• Rectangle: from another state in US
• Circle: from another country

1)    What other characteristics of a face could we use to ‘store’ information?

2)    What are some other categories of information that would be good or interesting to represent on a glyph?

Activity:

As a whole class they added categories for music (country, hip hop, chill, rock, classical, show tunes) as the left ear, food (italian/pizza, asian (incl. sushi), mexican, chocolate) as the right ear, and hobbies (sports, outdoors, shopping, games, etc.) as the eyebrows and came up with symbols -usually very pictographically, for each.  (It's okay for me that my starting questions are a bit boring,  as they ask about what they care about.  I don't have to do it for them.)  They completed their glyphs.

Reflection:  look over the glyphs of the whole class and share their reactions.
They remarked on how much they enjoyed making them, their interest in each others, and how cool it was to see them together.

Day 2: (2 hours)
Schema Activation: polish up your glyph (or make one if you were absent), move it to the back table, see what you notice about them as a group.  (Most people from Michigan, lots of pizza, music types remarked on, questions about living situation...)

Focus:
First we reviewed the types of elementary displays and covered any questions.  They asked about pictographs and boxplots.

Teaching note:  I like talking about questioning with statistics anyway, but they were really interested in what Jo Boaler brought up about teachers' questions at the last book club, so the timing was perfect.

Activity:
Pick a column.  Pick questions in that column so their numbers add up to four or more.  Answer the questions. (Display) means that a display is required.  Make a poster of your answer that includes your justification.

They collected data and dealt with interpretation issues, what to do with people who gave more than one answer, left/right, figured out pretty efficient ways to record their data, and started discussing display type.

After the posters were mostly complete they were passed around, and each poster was evaluated by each other group using our communication rubric: 0, 1/2, or 1 each for clear, coherent, complete, consolidated and content.  (Created with Coffey so the consonants all alliterate appropriately.)  This is how their exams will be graded the next day, also.

We came together as a group and discussed what makes for effective displays.  This is what they thought.

We also discussed the question types and made connections with reading.

• Literal.    Literal questions have answers that are found directly in the text or are answered by factual recall.  Examples:  How many people come from Michigan?  What was the main character's brother's name?
• Application. Application questions require computation or processing to determine the answer from the information at hand.  This usually is considered more mechanical than involving conceptual reasoning.  Example:  What's the average number of books read this month?  How long was Bilbo's journey?  
• Inferential.    Inferential questions require the answerer to create something unique, something that is implied by the information at hand.  Sometimes this is by combining prior knowledge and experience with literal information.  Inference may require students to imply, guess with support, or deduce.  Can be forward-looking.  Example:  How far would the average drive be from where people are from to Grand Rapids?  Why did the character do that?
• Analysis.    Analytical questions may require synthesis of literal information with information from other sources.  They typically require justification.  Sometimes analysis is referred to as synthesis.  It revolves around examining the information in and from the problem and solution.  Reflective in nature.  Example: Is there a correlation between favortie food and favorite music?  Why did the character do that?

Reflection:  Pick two of the following to address as a group.  Turn in your group response.

• What issues came up in data collection?
• How did you go from glyph to data?
• What was a useful form for recording data?
• How did you decide what graph to use?
• Any decisions you would make differently if doing it again?
• What do you see that was effective in other groups work?

Typically people thought about the teaching implications of what we did, but also thought a bit about the effective display idea.

Bonus:  better classroom decorations.