So far, we’ve seen what it takes to prepare an exploration, a graphing progression, and a discussion about what kind of questions can be centered around an exploration. To wrap up our month, I’m going to extrapolate from a specific example to a general framework for explorations.

A General Framework for Explorations

One of the most fundamental aspects of an exploration is the materials the students use. In the graphing exploration, the materials mainly stayed the same, index cards or paper pie slices and tape. Other explorations, however, can be greatly varied based on the material. For example, say I wanted to lead an exploration about how textures impact painting. If I wished to alter the materials, I would change what they used. Maybe one day we would paint on standard paper, then silk to see a smooth texture, and then painting on tree bark to see how that differed. Conversely, we could paint every day with a different substance mixed into the paint (rice, sand, and then flour) to see how the texture of the substance affected their art. Especially with young children, they are very sensitive to changes in the physical materials they use and benefit from these varied exposures. In my experience, these are the easiest pieces of an exploration to change.

A second aspect of an exploration that can be altered is the constraints placed on the students. Constraints, as generally understood, are restrictions on how students can use their materials. This is an overlooked aspect, as teachers generally only restrict the final product students can create or the general amount of time that can be spent on an activity. This is shortsighted, as there are incredibly nuanced and powerful changes that can result from properly applied constraints. In the graphing exploration, the class had constraints based on who they could vote for, how they voted, and the representation of their votes. To continue with the painting and texture example, students could paint with their eyes closed to see how the slick paint felt when spread over the rough paper. While the distinction between materials and constraints may be nebulous at times, it remains a valuable lens through which to view explorations.

The final aspect of an exploration is the focus of the students, which is directly impacted by you the educator. Through your questioning, you help students realize what they should be paying attention to or thinking about in an experience. To be clear, students can and will surprise you by noticing things you never expected, but it is also important to plan an exploration around key questions and vocabulary. For example, in the graphing exploration, I drew students attention to the relationships between the numbers of votes instead of just who had more. In the painting and texture example, the focus would include questions like “How does this feel different than that” and “How did this texture affect your painting”. Focus work would also include highlighting vocabulary that would be useful, like ‘rough’ or ‘smooth’ in the texture example. This questioning and vocabulary should expand as the exploration progresses, encouraging the students to think more deeply or analytically about the process.

Some Examples of the Framework in Action

In general, I have found it best to alter either the products OR the constraints day to day, not both. This allows the students to more easily reflect on a specific change from the prior activity. This is not a hard and fast rule, just a general guideline.

In order to provide a launching off point for future explorations and help explain the three aspects of an exploration, here are couple of examples of explorations and how their aspects can be modified.

Building

-Materials: unit blocks, legos, paper towel tubes, rocks

-Constraints: goal (height, representation of specific object, volume), time limits (15 seconds, 30 seconds, 1 minute), blindfolded, only using one hand

-Focus: “Is it easier to have a wider base or a narrower base?”, “Do you think you’ll be able to build as much in 30 seconds as you did in 1 minute?”, balance, symmetry

Color

-Materials: shading paint (a single color with black and white paint to alter shade), colored paper, stained glass (tissue paper on a light table), magnetiles and flashlights

-Constraints: painting in colored lenses or light, painting in dim light (colors appear washed out and gray), colored shadows

-Focus: “How did you make that color, since I didn’t put out any orange?”, “Why doesn’t this look as red as it did on the white paper?”, shade, blend

Hopefully, with this framework and these examples, you’ll be able to take a great idea and expand it into a full-fledged exploration. After all, there’s nothing wrong with doing something fun!

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.

The difference between an activity and an exploration

Exploration based learning may be all the rage, but not every Pinterest post is an exploration. Explorations are intensive, thoughtful investigations into a concept, while an activity is a solitary project, isolated from any surrounding work. In my Pre-K/K classroom, the emphasis is on explorations that take anywhere from a week to a month or more. In this series of posts, I will explore the groundwork I did to prepare for an exploration of graphing, the work we did with graphs for that month, the benefits of this exploration, and wrap up with some guidelines about how this mindset can be extrapolated to inform other explorations.

The seed of most of my math explorations comes from Little Kids – Powerful Problem Solvers by Andrews and Trafton. While I believe this book contains many great ideas, I’ve often felt it did not do enough to provide a full plan and instead simply explored a single activity. Therefore, when I read that our January exploration was going to be centered around graphing, I was certain that there were ways I’d need to expand it.

Laying the groundwork

I knew we’d be working on graphing in January, so I started several months earlier by seeing what the class already knew. They voted for whether the movie or book version of How to Train Your Dragon was better. After the vote, the class made a bar graph based on the votes cast. While they were working and during our discussion, I took careful notes about what prior knowledge the class already had.

Once the graph was completed, I questioned the class to see how well they could analyze the data. The whole class could tell who won and lost, which was reassuring. Most of the class was also able to tell how many people voted for book and how many voted for movie, another encouraging sign of comprehension. Further probing, however, revealed a number of weaknesses. No one was able to determine how many people had voted or how much the movie had won by. This, therefore, provided the baseline for the scope of my exploration.

While it is not necessary to have a full questionnaire before initiating an exploration, it is vital to check what prior knowledge the students have. You will almost certainly not have to cancel an exploration based on what the class already knows, but it is this tailoring that allows students to best grapple with the material later on.

Setting the scope

If you were going to attempt this exploration, it would be critical for you to determine what goals are appropriate in your setting and time frame. Students younger or you have a shorter time frame? Consider emphasizing the recognition of the larger picture of the graph (who won, what were the categories, etc). Students with a more developed mathematical ability? Push them to create graphs in groups or independently and spot errors in a graph based on the data set. Using this as a capstone to previous graphing explorations for advanced students? Have the class compare multiple representations of the same data to determine the ideal format or study misleading statistics.

With this range of goals in mind, I prepared for this exploration by determining what educational skills I wanted for the class to achieve. By the end of the month, I wanted my students to be have two related skills:

-Tell what information a graph is trying to depict.

-Be able to visualize how a set of numbers would look when graphed.

These were the two goals that all the work within this exploration would be working to address. As we move through this month, these goals will be the lens through which all of the activities will be evaluated.

This scalability is not unique to graphing. Many, if not most, topics in mathematics can be easily adjusted up or down. The final post this month will detail how to do this, but rest assured the teaching moves discussed here are generally applicable.

Creating a Sequence

This is one of the trickiest parts of the process: what is the actual work the students are going to do? I’m a worrier, so the trap I find myself falling into is making a timeline: Monday do this and ask this, Tuesday this and that, etc. However, since the goal isn’t me being able to do these activities, it is unwise to make too precise of a timeline. Instead, it is more efficient to make a sequence of topics or activities the class can cover. This way, I have a vision of where we can go and what to prepare, but I’m letting the class determine the pace. For this exploration, I decided to start with bar graphs and move into pictographs in the third week. In addition, it is useful to consider the breadth of questions that can be asked. This vision of exploration preparation will be explored more in the final blog post this month, so check back for a fuller explanation.

Check back next week as the rubber hits the road and we actually start graphing!