The Slow Reveal

This year my colleague Regina Payne and I tried something new as we visited classrooms across our district – numberless graphs. Similar to a numberless word problem, you present a graph with no numbers and proceed to have a rich mathematical discussion as you slowly reveal more and more information on the graph. Early in the school year, I shared a Halloween-themed numberless graph, and I also wrote a blog post about it.

We briefly touched on this work in our session at the 2017 NCTM annual conference, and it’s been exciting to see my #MTBoS colleagues taking the idea and running with it in their schools! In case you don’t follow them – which will hopefully change after reading this post! – I want to share their work so you don’t miss out on all the great stuff they’re doing.

Kassia Wedekind

Kassia has written two wonderful blog posts about how she took our ideas and tinkered with them to create a data routine called Notice and Wonder Graphs. I like this name because it’s more inclusive than numberless graphs. When it comes to graphs, you might hide the numbers, but you could just as easily hide other parts of the graph first. It all depends on your goals and how you want the conversation to unfold. In Kassia’s first post, she shares this graph with students. Notice it has numbers, and little else.

Curious what it’s about? Then check out Kassia’s post. I’m betting you’ll be quite surprised when you reach the final reveal.

I love this routine for many of the reasons that I love Brian’s numberless word problems–it slows the thinking down and focuses on sense-making rather than answer-getting.

But I also love it because it brings out the storytelling aspect of data. So often in school (especially elementary school!) we analyze fake data. Or, perhaps worse, we create the same “What is your favorite ice cream flavor?” graph year after year after year for no apparent purpose.

I’ve decided to make it a goal to think more about data as storytelling, data as a way to investigate the world, and data as a tool for action. In my next two posts (YES, people! I’m firing the ole blog back up again!) I’m going to delve into the idea that we can use data to discuss social justice ideas and critical literacy at the elementary level. I’m just dipping my toe into this waters, but I’m really excited about it!

And Kassia did just that! So far she’s followed up with one post where her students noticed and wondered about a graph showing the percent of drivers pulled over by police in 2011, by race. I love how the graph sparked a curiosity that got her students to dive more deeply into the data. How often does a graph about favorite desserts or our birthday months spark much of any curiosity?

Jenna Laib

Jenna shared a numberless graph that immediately got me curious! This is one she created to use with 6th grade students.

I can’t help but notice a bunch of dots grouped up at the beginning with a just few outliers streeetttcchhiiing across almost to the very end.

Once she included some numbers, my first instinct was that this graph is about ages. Apparently I wasn’t alone in that assumption!

And then there’s the final reveal.

So why did Jenna create and share this graph? What was her mathematical goal?

I especially loved this observation about how her students treated the dot at 55 before they had the full context about what the graph is really about.

Chase Orton

Chase wrote a detailed post about how he worked with 2nd grade teachers to do a lesson study about interpreting graphs.

…there’s so many rich opportunities for meaningful student discourse about data.  That is, if it’s done right.  Most textbooks suck all the life out of the content.  Students need to understand that data tells a story; it has contextual meaning that is both cohesive and incomplete.  Students need to learn how to ask questions about data and to learn to identify information gaps.  In other words, students need to learn to be active mathematical agents rather than passive mathematical consumers.

Chase walks you through the lesson he and the teachers created and tried out in three different classrooms. I love how he details all of the steps and even shares the slides they used in case you want to use them in your own classroom.

He closes the post with a great list of noticings and wonderings about continuing this work going forward. Here are a couple of them about numberless graphs specifically:

• We need to give students more choice and voice about how they make meaning of problems and which problems they choose to solve.  Numberless Data problems like these can be be a tool for that.
• The missing information in the graph created more engagement.

A huge thank you to Kassia, Jenna, and Chase for trying out numberless graphs and sharing their experiences so we can all be inspired and learn from them. I can’t wait to see how this work continues to grow and develop next school year!

If you’re interested in reading more first-hand accounts of teachers using numberless word problems and graphs, be sure to check out the ever-growing blog post collection on my Numberless Word Problems page. I recently added a post by Kristen Acosta that I really like. I’m especially intrigued by a graphic organizer she created to help students record their thinking at various points during the numberless problem. Check it out!

Sink your teeth into data. Don’t just nibble.

Looking for math all around started as a challenge I made for myself and I’m realizing it’s becoming a full-fledged theme for my year. When I had to think of a topic to moderate this week’s #ElemMathChat, I started by asking myself, “What’s a topic we haven’t talked about since the chat started in August 2014?” After some brainstorming, I eventually came up with analyzing data. What a great topic for my theme! I don’t think I could throw a rock without hitting some data in the world around me.

In fact, as I was fleshing out the topic for the chat, I was regularly checking some real-world data online. After a long dry spell, we finally got some rain in Austin. And by “some rain” I mean a deluge. On a couple days last month it just kept pouring and pouring. Throughout each day it rained, I found myself checking our neighborhood weather station on Weather Underground to see how much rain had fallen. By the time October was over, we had received 10.3 inches of rain in my neighborhood! That simple piece of data became the catalyst for tonight’s #ElemMathChat.

I started digging into rainfall data for October, then rainfall data for other months, and finally I expanded my data dive into other cities in and out of Texas. When I was done, I had a spreadsheet full of various tables of data that I wanted to share in my chat. To make this chat work, I realized I needed to be intentional about how I shared the data in order to tell a coherent story. I also wanted to create a variety of data displays that would match the various data displays students encounter across grades K-5. As an aside, I think #ElemMathChat sometimes leans a bit heavy on content for grades 3-5, so I was trying to be mindful to show some graphs that could be analyzed in a Kinder or 1st grade classroom.

It took several nights to research, create graphs, and pull it all together to make a story, and in the end I’m proud enough of the final result that I wanted to capture it on my blog.

Before starting my data story, I shared the following guiding questions that tied into my primary goals for the chat.

My Data Story

Our story begins with the piece of data that started it all. I asked the participants to tell me what they noticed and wondered about this statement.

What do you notice and wonder?

Many people wondered how this amount of rain compared to other cities. Funny you should ask.

What do you notice and wonder as you look at this pictograph?

One thing I noticed is that I accidentally left the key off the graph. Oops! Each picture is meant to represent 1 inch of rain. Despite my mistake, several people liked that the missing key invited students to wonder about what the pictures represent. That sounds like such a wonderful conversation to me that I opted to leave the key off when sharing the picture in this blog post.

I had a little fun with this graph because I had to decide which cities to include. I decided to focus on other state capitals, but the question became, which ones? When I noticed how many start with A, I decided that was more interesting than picking random capitals. It just so happens that all the other capitals on this pictograph are all on the East coast, so I wonder if it would have been better to choose capitals with greater geographic diversity. In the end this is just a fun way to get our story started so I’m okay with what I chose.

Next we moved from cities outside of Texas to cities inside of Texas, specifically along the I-35 corridor from San Antonio to Waco.

What do you notice and wonder as you look at the October rainfall totals for these cities?

Now that I shared two different graphs, what questions could you ask students about these graphs? What math skills can students bring to bear to interpret and further understand the data in these two graphs?

One thing that we often do with graphs found in textbooks and tests is ask one question about them and then move on. How unfortunate! There’s so much rich information to dig into here. One of my key points for tonight’s chat was reiterating something I read by Steve Leinwand about mining data. Ask a variety of questions about data displays. Sink your teeth into them; don’t just take a small nibble.

The one thing that stood out to me and many others in the chat was how little rain San Antonio received. The difference between San Antonio and New Braunfels is quite striking considering how close they are to each other.

Other people felt that Austin’s rain wasn’t fitting with a general trend in the data. I didn’t want to get into it in the chat, but I’ve noticed the rainfall in my neighborhood tends to be less than other parts of the city. Our weather station recorded 10.3 inches for October but others in Austin clocked in at closer to 13 inches of rain. I thought about using the larger number, but because the catalyst for this whole story was my weather station’s data, I opted to stick with that. By the way, I don’t think it’s an issue with our weather station’s rain gauge. Over the years there have been many instances of rainfall in other parts of the city while my neighborhood in north Austin remains bone dry.

Now that we’ve looked at rainfall in and out of Texas, it’s time to drop a bit of a bombshell. With this new information, what story is the data telling so far?

Here’s what I see as the story so far: Austin received 10.3 inches of rain in October, which was a lot compared to areas outside of Texas, but fairly common for our area in Texas. Not only was this a lot of rain, but it also fell in a very short amount of time, 6 days.

Next, I asked for help. Now that you know it rained only 6 days in October, which data display would you choose to represent October rainfall?

Option 1

Option 2

Most people preferred option 2 because it shows the full picture of October. That was surprising to hear. In my mind, because we just saw the picture graph showing that it only rained 6 days in October, I didn’t feel option 2 was needed. I already know it didn’t rain on very many days, so why waste the space with all those days showing 0 inches of rain? Option 1 puts the focus squarely on analyzing the rainfall on the days where it actually rained. In the end there’s no “right” answer, it all comes down to how you justify showing what you choose to show.

We’re nearing the end of our story. There are two more graphs remaining. What does this next graph add to our story? What is one question your students could answer based on this data?

I love looking for relationships so here are the questions I came up with:

• Where do you see the relationship “three times as much” represented in this graph?
• Where do you see the relationship “half as much” represented in this graph?

I especially like wondering what students will come up with because both questions have more than one correct answer.

And now for the last graph. How does this close out our data story?

Here’s a follow up question for you. What could be the sequel to the story I just told? How could you and your students explore and tell the sequel? What other data stories could your students explore and tell?

I closed the chat, and I’ll close this post, with two key points I want everyone to take away from this conversation.