Pictographs and Pie Graphs

I’m going to start this week with a frank statement: I messed up this part of the exploration. When I was planning the scope of the exploration, I intended for the class to spend two weeks on bar graphs and two weeks on pictographs. I knew that pictographs are a struggle for students and I frankly didn’t have any idea how to implement pie graphs in a way that made sense. However, halfway through week 3, I realized how it could be done and altered the scope accordingly. I’m going to present the exploration as it actually went, but keep in mind if you’re teaching this that I would actually do pie graphs before pictographs, just because pictographs can be so challenging.

Pictographic nightmares

For as long as I’ve been teaching math, pictographs have been problematic. A pictograph appears simple enough, as it’s just a graph that uses pictures to represent the data (like a smiley face to represent a vote). However, students often struggle to interpret them correctly. This is particularly common because pictographs will often alter the key and make each image worth two or more votes. When this is the case, students will often count the pictures, not the votes. I knew, if I was going to teach this skill to my students, I would need a way to represent this to them and let them find ways to think through it.

I saw how the students appreciated being able to have their own index card and decorate their ballot in the first section. Therefore, I felt that tweaking the ballots but keeping that basic format would be appropriate. On the first day of pictographs, each ballot now came with half of a circle on both sides. The students then decorated each side to show which option they were voting for. When they attempted to cast the ballots by taping them up, however, I explained that each ballot needed to be connected to one of their classmates’ votes to make a whole circle. We also discussed how the last person might not be able to make a circle and what a ‘half’ of something means.

When it came time to talk, the class did surprisingly well. There was definitely confusion about the difference between how many CIRCLES were in a category and how many VOTES were in a category, but we talked through it. We also discussed why it was important to put the votes together into whole circles instead of leaving them floating around.

As the week went on, I provided less support when they were casting their ballots, which allowed them to talk together about how to combine their votes to create a whole picture. The class also talked about how important it was to color on both sides, since you didn’t know which side you would have to use on the board.

By Thursday I felt the class was ready for a greater challenge, so we transitioned from each picture taking two votes to each picture taking four votes. Once again, I simplified the questioning and provided more support, but the students adjusted smoothly and were able to vote with little support on Friday. Throughout this entire week, I kept many of the questions from

the previous two weeks, like which team won and how many people voted, while adding questions about how many pictures were created and how many votes it would take to finish a picture.

Pie Chart Parade

As I said at the beginning of this post, I really struggled with how to allow the students to create their own pie graphs, at least how to do them without the use of technology. After all, if one of the students was absent and didn’t vote, the pie chart would either need to have bigger slices or be left with a whole in it. Thankfully, I realized that I could create an ‘absent’ vote, which I would place when necessary. That hurdle overcome, I realized I was ready to start our final week of graphing.

To make the pie graphs, I cut paper-plate sized circles out of multiple colors of construction paper and cut it into 12 equal pieces. I also traced a paper plate on the board to provide a template. The students would then pick a pie slice that matched the color of their vote (blue for dogs, red for cats, as an example). That pie slice would then be decorated and arranged on the template into ‘teams’. Our discussion afterwards centered on seeing which team had more votes, how many votes each team got, and how many people voted in all.

As the week went on, I transitioned into providing less guiding on placement, which led to a great discussion about why the teams needed to be together, and offering more options/colors. This week was actually far less challenging than the pictographs, which is why I would recommend switching them if you implement this exploration.

Taking Stock

As we wrapped up our week of graphing, I felt the class had done a great job being able to compare numbers and read the data from graphs. I actually saw them going back and looking at the graphs we had already done and using their new understanding to think about them more deeply.

Next week, I’m going to wrap up the graphing exploration as well as provide some guidelines on how to structure an exploration. I hope you’ll join me then as we get ready to take what we’ve learned and start applying it to the classroom!

Welcome back to week 2 of making graphs with kids! Today we explore rolling out an exploration and pushing students to think more deeply about a concept.

Let’s Make Bar Graphs!

Since I knew the class had only limited exposure to bar graphs going into this exploration, we spent the first two weeks working on bar graphs. This blog will explain how I helped the class move from simpler to more complex graphing problems.

We started this exploration back in January, so I knew what my first question should be: did you like your presents or spending time with family? I knew this was a question the class would easily understand, have firm feelings about, and be excited to talk about. This conservatism is a key aspect my exploration philosophy: introduce novelty in small doses. If I am expecting them to do a completely new thing, then the rest of their thinking should be simple enough they can focus on the new piece. If I had asked them to vote on a conceptual topic, like their favorite character across multiple books, they would have been less able to focus on the graph.

The Rubber Meets the Road

When it came to the actual voting, I had several goals for the process. I wanted to make sure the students couldn’t accidentally ‘vote’ multiple times, that they were able to express themselves through their votes, and to create a physical graph that could be seen and examined over time. Therefore, instead of asking the class to raise their hands, each student wrote or drew their vote on an index card. That card was then taped into a blank bar graph grid I created out of masking tape on our board. I also wrote out sentence stems for the class; while few of the students were able to read the stems, some of them decided to copy them into their exploration journals.

The timing of the voting and discussion was also rather important. The question was on the board from the time the students entered the classroom, which lead to big discussions about how people were voting and why. While this may have led to some politicking or vote changing, the point of the exploration was not to find out how they actually felt. Instead, they were learning to predict how people would vote, how that would change the outcome, and even making sure that everyone understood the question at hand. After all, when I hear one student say to another “Student Z hasn’t voted yet but will probably vote here, so we’re going to win”, that is a great sign that the student understands how to read and interpret a graph.

As the week went on, we transitioned from simple questions about their preferences to more complex questions about the novel we were reading, Charlotte’s Web. I also started encouraging the class to have more challenging discussions about the graph, like seeing how many people voted IN ALL, how much did this option win by, and what would have happened if two people had voted differently. These questions pushed the class to see the graph as a living representation, not a static object to be observed.

Ramping Up the Difficulty

In the second week I transitioned to more complex graphs. Instead of choosing between two options, the class had to choose between three or four. I also took away the horizontal guides so they could see how having the votes misaligned made the graph harder to read. Finally, the class switched from vertical bar graphs to horizontal bar graphs. In keeping with that conservatism I mentioned, the questions I posed to the class reverted to easier questions about who won and how many votes each option got until they became comfortable with these variables.

Throughout this entire process, I continually asked myself how these changes would either

help the class read graphs better or give the class a more intuitive understanding of the relationships between numbers. This helped me ensure the questions and graphs were challenging the class and working towards their mastery goals.

This was just the first half of our exploration, check back next week to see how we took it to the next level.

As an adult, remember to use voting as a means of decision-making only if you are going to respect the outcome of the vote.  If you present two options (three at the most for preschool aged children and younger) be sure that they are options that you are OK with.

When asked to vote, many young children will put up their hand for every option.  This is OK and to be expected.  For some it is hard to choose, and for others, their participation in the activity is the act of putting up their hand rather than making a choice.

No matter how they vote, be sure to use voting as an opportunity for children to count out loud.

Do you use voting as a decision-making process with your children?

Who remembers this campaign?

According to the Online Etymology Dictionary, to Give a Hoot is

to call or shout in disapproval or scorn,” c.1600, probably related to or a variant of Middle English houten, huten “to shout, call out” (c.1200), probably ultimately imitative. First used of bird cries, especially that of the owl, mid-15c. R

So, if you care about or support something, you show this by hooting.  If you don’t care for it, you don’t give hoots.  Easy enough.

Instead of voting, why not encourage children to give hoots for their choices; i.e. “Give a hoot if you want to go to the playground,” or “Give a hoot if you want to sing the —- song.”  Rather than raising hands, children can hoot.  Encourage them to give 2 or 3 hoots if it is something they really want, and then explain that if they don’t give a hoot, (or 2 hoots as the saying goes) they should keep quiet. 

In this first example, children voted for their favorite foods by drawing a picture of the food and writing its name.  The graph is simple and easy to understand.  It meets my criteria for good graphing practice because it:

1.  Creates a data set that can be evaluated and revisited later.

2.  It provides easy visual clues that quantify the numbers of choices that can easily be interpreted by children.

3.  The children can see who chose what food so those choices can be explored further.

My only suggestion for improvement would be to create a grid on the board so that each space is evenly distributed.  Otherwise, you run the risk of children gluing their choices down with variable spaces between.  This can cause confusion for the children who are still puzzled by appearances. You want your graph to be clear.

This next one is not so good, for a variety of reasons.  At first glance, this might look like the others I have presented but once I describe the process you should see where the problems are.

The children were asked to vote for their favorite color – simple enough, right?  The teacher labeled the graph well with the words supported with an example of each color.  That’s where the good stuff ends.

The children were then asked to come up to the board and vote for their favorite color using a small manipulative to represent their choice.  The basket was filled with dinosaurs and cars and animals.  The children didn’t really struggle with making their choices, you can see that from the chart but….

1.  The graph cannot be revisited.  When moved, all of the small plastic objects fell off.

2.  There is no way to know who voted for each color.

3.  The confusion is multiplied because the small objects were also colorful.  Many of the children wanted to choose an object that was their favorite color, but it didn’t make sense to them that their object and their choice didn’t match.  This didn’t make sense to any of the children in the group.

4.  The objects are variable in size and were placed in uneven spaces on the board.  Again, I would encourage you to always make a grid with even spacing for the children.

And now let’s look at the not-good-at-all.

This activity was great.  The children measured one another using the nonstandard unit of blocks.  I loved this part.  

Here, the teacher has neglected to create a clear graph for the children to “read”.  It is language-based, doesn’t provide visual clues to understanding the data, and is altogether useless.  The children can’t really revisit it or make meaning out of the data.

This could be used to create another, more appropriate graph, but that gets away from the whole primary purpose of the activity.

Graphs should be made with and for the children.  Using the ideas from above, when graphing with your children be sure to consider how they can make meaning from the data collected and then provide many useful and developmentally appropriate clues to support their understandings.

Above, you can see that this group of children chose their favorite book between “Brown Bear, Brown Bear” and “Panda Bear, Panda Bear.”  Using name cards with the children’s names written carefully across the top, and then a small picture of each child in the corner, children voted by placing their name card under their book choice.

What is good about this?

1.  Children’s names are reinforced with their photographs.  Remember, many children can recognize their own names using a variety of clues, but they may not recognize any of their classmates names.  Using the above technique, all of the children can “read” the data using the photographs as additional support.

2.  The slots for names are evenly spaced.  There is a clear one-to-one correspondence between the cards and the slots.  One card per one slot.  This helps support the children when they count the results. This also means that the children won’t be “fooled” by the votes.  They can easily see which book received more votes.

3.  There are only 2 choices.  Often, teachers are tempted to think that “more is more.”  For children under 3 I believe that choosing between 2 options is entirely appropriate.  You will also find less hemming and hawing when the children make their choices.

4.  The “graph” remains in the classroom.  Children can go and revisit their data set after the activity is over.  Teachers can ask the next day, or the next week, “Who can tell me which book had the most votes?” and children can go over to their data set and revisit the graph and figure it out for themselves.

5.  The books are familiar and recognizable by sight.   The book covers are copied and reduced in size and are completely identifiable to even very young children.

6.  If done well, children can count how many votes each book received.  It is also possible that some children can figure out how many more Brown Bear received than Panda Bear by showing them they can count on from the bottom of the Panda Bear list.  This is very difficult to do, but you may have some children who are ready for this.

Next week, we will look at more graphing examples and get lots of ideas for activities you can do with your own children.

You might need to get creative with this idea, because it will be most meaningful if it actually happens tomorrow, when the grown-ups in the children’s’ lives are also voting.  Think of something in your classroom that needs to be decided on and create ballots for the choices.  Create a voting booth where children can take their turns filling out their ballots and putting them into the ballot box ( a simple shoe box with a slot in the top will do just fine).

Make it simple:  They can vote between a walk around the block or a walking trip to a local playground.  They can vote about a name for something in the classroom – a class pet, or a nickname for the teacher.  They can vote about their favorite flavor of ice cream.  Be sure to draw pictures on the ballots so they know what they are choosing between.

Once everyone has voted, take the ballot box out and together with the children count the votes.  They can see how democracy works when you explain that the most votes wins.