This is a series about “developing the question” in math class.
I’m proud of Graphing Stories. That was the first math lesson that drew in any serious way on my video editing hobby. That was the first math lesson that alerted me to the enormous value in sharing curriculum with teachers on the Internet.
I’m unhappy with the project now. I look at it and see the product of a math teacher who is eager to get to the answer of how to graph a real-world relationship and less interested in developing the question that leads to that answer.
If you watch Adam Poetzel’s graphing story, in which he slides down a playground slide, here’s what you’ll see:
- A title announcing the quantity we’ll be recording: “height of waist off ground.”
- A gridded graph that shows the scale you’ll use. It runs up to 10 feet.
- A video of Adam sliding.
None of this is Adam’s fault, of course. That’s my editing.
Here’s how I’ve been doing a better job developing the question lately in workshops.
- I play the video of Adam sliding.
- I ask participants to tell their neighbors everything they saw. “Don’t miss a detail,” I say, and I’m always surprised by the details participants recall.
- I play the video again and I ask the participants to tell their neighbors their answer to the question, “What quantities could we measure throughout the video?” People suggest all kinds of possibilities. Speed, distance from the left side of the screen, height, temperature.
- Then I tell them I’d like them to focus on Adam’s height. I ask them to tell their neighbors in words what happens to his height over time.
- We share some descriptions. People compliment and critique one another. Then I point out how difficult it is to describe his height over time in words alone.
- Only then do I pass out the graphs.
The difference is immense. It takes an extra five minutes but participants are much better prepared to make the graph because they’ve spent so much time thinking about the relationship in so many informal ways. So many more participants walk away from the experience feeling like valued contributors to our group because the questions we’ve asked require a wider breadth of skills than just “graph relationships precisely.”
That’s the benefit. Again the cost was only five minutes of class time.
The most productive assumption I can make about any question I pose to a student is that a) there are questions I could have asked earlier to develop that main question, b) there are interesting ways I can extend that main question. In other words, I try to assume the question I was going to ask is only a thin middle slice of the corpus of interesting questions I could have asked. Tell yourself that. Maybe it’s a fiction. Maybe you use the entire question buffalo every time. It’s a useful fiction in any case.
Next: Let’s make a resolution.
BTW: Kyle Pearce got here first.
ihor CharischakAugust 15, 2014 - 5:42 am -
Interesting. This fall Im going to be volunteering in an alternative high school (for students who didn’t function well in a typical school environment) and plan to do the activity with them. To be honest I’m a bit skeptical about it working with kids who don’t have the habits of mind that I think you are assuming here, but I’ll suspend judgement. Im looking forward to getting back in a classroom so it should be fun.
Bryan AndersonAugust 15, 2014 - 8:37 am -
I found this resource last year, and I loved using it with my students. I agree with you and Kyle and have already done the “editing” of the edited movies. One thing I would say is that keeping a few videos in your current state is good for scaffolding. The most powerful part of this lesson became the fact that my students wanted to then design and create their own videos- something that creates this thinking without even posing questions to students. They decided what activity they wanted to video, what the variables were to be graphed and displayed, and edited on their own. They also decided to leave out the intro parts, not due to limitations on our technologybut because they also determined it is more intriguing to watch a video and analyze it.
I truly hope you revisit this project and make tweaks to the videos, this is a great resource for students.
Bryan AndersonAugust 15, 2014 - 8:39 am -
I also teach students who have problems outside of school (at a juvenile center), I believe you will be surprised what kind of interest and enthusiasm this project will generate.
Ihor CharischakAugust 15, 2014 - 8:44 am -
I look forward to being surprised.
Cathy YencaAugust 15, 2014 - 4:47 pm -
It probably won’t surprise you that I think your new-and-improved Graphing Stories experience described here would make a fantastic Nearpod remix.
Harry O'MalleyAugust 15, 2014 - 11:22 pm -
Here is some very recent mathematical activity I have engaged in.
1. I learned that GeoGebra could make Voronoi diagrams, so I made one and played with it. Then I realized that they would be great tools for analyzing the free space around players in team sports. So I built one for soccer with 11 players to a side and used it to design some possible passing schemes. Then my brother and I went out with a ball and acted a few of them out, which gave us a few more ideas that we acted out in the GeoGebra sketch, and then back outside, etc. The GeoGebra sketch is below:
2. After seeing a post of yours that used an image from the website Daily Overview, I browsed that site and found the images beautiful. From there, I link surfed to the site of Alex MacLean, who is a renowned aerial photographer. I selected one of his photos and sketched it in GeoGebra. This sketching process proved very rewarding, as I learned how to make the path of a quadratic function (or any function for that matter) meander across the screen like a river instead of tracing its pure, smooth quadratic path, by using the sine function as a mediator. The link to the photo as well as the sketch are here:
3. I created a graph whose nodes were people: myself, my wife, my kids, my parents, my wife’s parents, our siblings. Then I created edges connecting those members who had a deep loving connection. The perspectives I gained from building and playing with this graph were astounding and deeply personal. I realized that, although there are five members in my immediate family, there were actually ten loving connections among us. I had never perceived that before. I realized that before my three kids were born, there were only 18 connections in the diagram, but because they are there, there are 38! Three new people, 20 new connections. Kids really do unite families. The graph is below:
I only take the time to summarize this activity so that there can be some examples to refer to when I say that mathematics is not about answering questions. It is about building and finding. It is about taking your experience and structuring it so that you can learn more about it and how it relates to you.
In none of the above examples was there a final question to answer like “What is the minimum distance the left midfielder would have to pass the ball in order to guarantee that one of his teammates receives it and not the opponent?” or “Which route is shorter, across the middle, or around the circle?”
It is true that specific questions can help to organize action in learners, but having students continually experience math as the act of answering specific questions gives them a highly misconceived image of what math is and can be for them.
Much of your work is good because it provides the learner with opportunities to structure their experience mathematically. But you are constraining yourself too much by trying to find ways that all of that structuring can be put to service answering some typical “math-style” question. Free yourself. Let the structures be the goal.
Harry O'MalleyAugust 16, 2014 - 10:25 am -
With regard to his own improvements to Graphing Stories @Dan writes: “participants are much better prepared to make the graph because they’ve spent so much time thinking about the relationship in so many informal ways”
Regarding these “informal ways” that learners can use to get to know a situation they are modeling, I find it useful to break them into 4 categories. When making sense of something, learners should…
3. hear about
…whatever it is they are making sense of. These are natural, powerful ways humans learn about everything in their world.
To see this intuitively, imagine the new iPhone came out and you’re really interested to know more about it. What would satisfy your curiosity? “Let me *see* it!”, your brain would scream as you scurry to Google to look for images. Then you’d want to read what others are saying. You’d ask around and read reviews and reactions online. In other words, you’d want to *hear about* it from other people. Next, you’re off to Best Buy. Why? To put it in your hands, turn it every which way, look at it up close for detail and far away for whole picture aesthetic at will, and feel the weight and texture. In other words, you’d be dying to *handle* it. And finally, you’d want to *use* it. Put your favorite playlist on and take it for a run; move between apps; take it in and out of your pocket…
When I’m designing tasks for learners or doing math for pleasure and purpose, I make sure to engage in each of these four learning postures.
Dan MeyerAugust 16, 2014 - 12:18 pm -
Thanks for your thoughts here, Harry. I do wonder if your mathematical activity was as aimless as you represent it. At each stage it seems you had a goal, whether that was just to “make a Voronoi diagram for myself” or “apply this somewhere.” It seems to me that while your activity had a certain structure, it was unstructured enough to allow different and new questions to take root. I’m arguing that your activity and mine differ by degree, not by kind. I’m also curious how you develop curriculum for this kind of unstructured questioning when curriculum by definition is structured questioning.
PS. You shouldn’t be wasting all of this thoughtful analysis in the comment section of some guy’s blog. Think about getting an online home of your own and posting it there.
Harry O'MalleyAugust 16, 2014 - 1:00 pm -
Using the learning posture checklist above, I noticed two weaknesses to the Graphing Stories activity, even with Dan’s new updates. Although the video allows learners to *see* the sliding activity, and Dan’s built in neighbor conversations allow learners to *hear about* it from others, the nature of the video medium means we can’t *handle* or *use* it. Here’s where structuring comes in.
Instead of the goal of the activity being an end product, the static height vs. time graph, let it be a spatially tangible model, a structure, created in GeoGebra. Here’s what I made in response to this newly defined task:
To start, I sketched the height vs. time graph using the freehand function tool and called it p(x). Then I also sketched the horizontal position from the starting point vs. time with the freehand function tool and called it q(x). Then I made a slider for the time variable called t, ranging from 1 to 15 seconds, and created a parametrically defined point (q(t), p(t)) and boom, now I’ve got a spatially tangible version of Adam’s slide experience that I can *handle* and *use*.
Tracing the point while animating the slider, t, allows us to perceive Adam’s relative velocity throughout the sliding experience in one image, here:
Then I started playing around with transforming q(t) and p(t) with the intention of finding ways to creatively reimagine the experience. The point (2*q(t), 2*p(t)) traces out a path that suggests an extreme version of the slide design (could be fun), while (0.3*q(t), 0.3*p(t)) suggests including a toddler slide in the park. The point (q(2*t), p(2*t)) traces out the same path, but twice as fast. Timed slide races, maybe?
This version of the activity blurs the lines between using context to understand mathematics and using mathematics to understand context. They become one and the same.
Harry O'MalleyAugust 16, 2014 - 1:25 pm -
Thanks for the response, Dan. I certainly don’t claim that my activity was aimless. It was driven, passionately driven in fact, toward goals, as you said. They are driven, I think, as much by my desire to be autonomous, competent, and feel related to as by the completion of a well defined task.
As for curriculum, what I am developing is not question based, it is design based. There are certainly well defined objectives, like “create a dynamic point in GeoGebra that moves through the path that Adam’s waist moves through in the video as a function of t, where t is the time from the beginning of Adam’s journey to the end of it.” But the purpose of the task is not simply to complete it (although that is certainly one the the activities that drives learning). It is also to better understand Adam’s sliding experience and be able to interact with it in creatively valuable ways that are unique to each learner.
As for my own home, I’m working on that now. Thank you for the encouragement. I hope you won’t mind if I continue to interact with you guys here until I get mine up and running.
Adam PoetzelAugust 16, 2014 - 5:45 pm -
Just a quick, fun story….
The day I shot this video of me sliding down the slide, I went to the park early in the morning when the park was free of people. As I took several “takes” of the video – trying to get the timing right, adjusting the camera, etc – a few moms and their young children started to arrive at the park. Needless to say, I received some strange and slightly unnerved looks by the protective mothers wondering why a grown man was recording himself go down the slide again and again. After I got the video I needed, I started to leave and walked past one of the mothers. Wanting to explain, I said “I was just making a video for a math lesson”. The look on her face told me she now thought I was even more strange than she had previously assumed :)
Dan MeyerAugust 16, 2014 - 7:00 pm -
My hero …
ChrisAugust 16, 2014 - 10:28 pm -
Nice change, I’ll do that this year, thanks. Like Bryan my students made their own videos which was great fun and led to many useful, enjoyable mathematical discussions, some really weird ideas and one very creative use of Minecraft.
Bob LochelAugust 21, 2014 - 5:28 am -
Dan, I used graphingstories last year with my 9th grade Algebra 1B class; students who were preparing for the PA Keystone exam. The first few go-arounds were rough, and I used a document camer to share student works and encourage discussion. We had sme rich math conversations, but like many classroom hooks the novelty wore off after a few daily stories as openers.
Here was my follow-up, which provided valuable information regarding function misconceptions. I would provide a graph, then challenge students to write a plausible story which could be modeled by the given graph. Usually, I was looking for a story which made use of the intercept and slope, but some students would look at starting and end points. Other students would simply place the given numbers in a story without concern, and we would critique all of these. These story-writing experiences were done through the year, and were adapted as we learned new functions, function notation, domain and range. Also, it was easy to bring in vocabulary like independent, dependent, domain, range.