A Framework For Using Digital Media In Math Instruction

When we teach math we are helping our students establish a framework for interpreting the world. One of the worst ways I know to help them establish that framework is to print an illustration of a real-world scene in a textbook, write in only the relevant measurements, and tell the students in the text of the problem which formula or strategy to apply. This leaves a student helpless and unprepared (in the mathematical, analytical sense) should she ever encounter the world that exists outside the pages of her textbook.

So we instead bring digital media from the world into the classroom, simulations of the world as students experience it, artifacts which students can discuss and to which they can apply frameworks of their choice. In order to leave students capable and prepared for their encounters with the world, this media must be captured and presented very intentionally.

Capture

We must capture this media â€” an audio clip, a photo, or a video, for example â€” so that it most closely approximates the student perspective, so that the media appears as nearly as possible to be the world as the students experience it, rather than as some audio, photo, or video that has already been interpreted for them by a photographer or a cinematographer or an editor. It must be captured to allow for the most possible classroom uses, the most possible interpretations, and the most possible framework applications â€” allowing even for the application of useless frameworks.

• Frame wide on the scene rather than narrow. Capture the entire scene inside a static frame and add 5% to the margins. This forces a useful question on the student, “Where do I focus my attention?”
• Compose the shot parallel to the plane of action. Extravagant camera framing makes mathematical photogrammetry difficult but also makes the camera operator an interloper on the scene. An extreme low angle, for example, begs the question, “Who shot this?” when you’d rather the student forget about everything but the scene itself.
• Record ambient audio only, exactly as the student would hear at that scene.
• Use location lighting only, exactly as the student would experience at that scene.
• Maintain a fixed camera position. Use a tripod. A handheld camera allows the camera operator to impose her point-of-view on a student’s interpretation of the scene. We want the student to decide for herself what parts of the scene are relevant, important, or useful, not the camera operator.
• Record several alternate takes, changing variables, illustrating different iterations of the scene.
• Take a photo of a ruler or meter stick within the plane of action for reference.
• Do not edit the video. Edits invalidate timecodes, and, again, they impose an outside party’s interpretation of a scene on the student when we’d rather the student interpret the scene for herself. There are, of course, two unavoidable edits â€” where you start the clip and where you end it.
• Allow at least five seconds on either end of the scene, though more time is preferred. This will force students to decide when something relevant has begun and ended.
• No narration. This allows the teacher and students to determine the dialogue.
• Shoot the highest-definition video possible.
• Shoot at the fastest possible frame rate given the location lighting.

Presentation

The goal with classroom presentation is to eliminate the presence of interlopers, to eliminate everyone from the scene but the student, including the teacher, to whatever extent possible. The teacher exists here to scaffold and curate the artifacts, not to suggest, explain, or gesture.

• The artifact must beg an obvious, compelling, seemingly unsolvable question, a question which begs for more questions and for more information. If the teacher has to suggest, explain, or gesture in order to persuade the student of an artifact’s interest, then it isn’t compelling or obvious enough for classroom use. (This is the most challenging criterion in this framework. This is the criterion that begs most loudly for open sourcing. The technical aspects of this framework are fairly intuitive but the creative aspects of this framework are extremely challenging and demand collaboration.)
• Declare nothing and ask only two questions: “What is the next question?” and “What measurements do you need to answer that question?” Again, our ideal digital media shouldn’t require any introduction or explanation.
• Impose a mathematical framework on the scene only as students request it. This is where textbooks fail, imposing a grid or labeling points or establishing measurements before the student has even begun to process the scene. Once the overarching question has been introduced (eg. “will the water balloon land on the target?”) the students will see the need for measurements. (“We don’t know how high it was dropped.”) The teacher then plays the exact same clip with a measurement grid superimposed on the footage. The ideal digital media artifact consists, then, of multiple remixes of the same video or picture or audio files.
• It is essential to add a timecode, if for no other reason but for student reference (“Can you scan back to five seconds in? I thought I saw something.”) but also because the timecode will allow for time-based calculation. The timecode needs to take the format Minutes:Seconds:HundredthsOfASecond not Minutes:Seconds:Frames.
• Invite the students to estimate the answer to the chosen question. Take five student estimates and post them on the wall. This offers the students a low-stress opportunity to consider a correct range for the answer. It also invests them in the problem.
• Offer the students something tangible to manipulate or measure. A hard copy of a video still, for example, or a digital copy of a video still they can import into a dynamic Geometry system like Geogebra.
• The final element in one of these digital media artifacts must provide the payoff for the mathematical work. It needs to confirm, for example, that the water balloon did or did not land on its target, or it needs to flashback to the moment the water balloon was dropped, revealing the height of the drop. The classroom discussion can then turn to possible sources of error or calculations of percent error.

This isn’t gospel. Please edit the wiki if you feel so led. This simply makes the most sense to me of my last three years teaching.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.

1. Sarah

February 18, 2009 - 6:57 pm -

Understood this isn’t gospel, but still very intrigued. Because it is a different model. While it’s something you’ve been building to since I started reading, the past week has been showing a different tilt to the style than before.

My mental set-up of your class is changing and feeling more incomplete than before. (I remember the, “You know what good teaching looks like. Go do it.” line form last year.) Hope you get the release forms figured out soon, because I’ll watch that video as quickly as the internet connection lets me.

2. sam shah

February 18, 2009 - 7:03 pm -

I bet if you made a contest or have people sign up to create one of these, you would get some volunteers. I mean, I would, if I had someone encouraging/goading me and giving me some realistic deadline. And the knowledge that there are others out there doing the same… that’s really the motivation I need. It’d get me looking around me for inspiration for this stuff.

Oh! They’re playing one of my favorite songs of last year in the background of the the new 90210.

Back to reality… I’d probably need a good amount of time, to think and get the materials and learn how to use video/audio/photo software. Anyway, just throwing it out there.

To conclude, here’s the song: http://www.youtube.com/watch?v=hbIvC66lPBI&feature=related

Sam.

3. Chuck

February 18, 2009 - 7:14 pm -

Wow, this is some good stuff here Dan…Seems like you’ve been blazing quite a trail the past few days or weeks.

Anyways, novice question for you or whomever…How do you insert a grid on a still image quickly and easily? I’ve searched around and haven’t found much, other than using the grid features for slide design in MS PP, or how to insert gridlines into charts. But say I want to impose gridlines on an image on my PP presentation either via some photo editing software or on powerpoint itself, is there a good way?

4. It is great to have this all written up in one post. Thank you.

The idea is a bit intimidating, that math teachers need to be photographers/videographers/photo editors/video editors, but also exciting for people who like a challenge.

My other reaction is that the spirit of your Presentation narrative doesn’t strictly require a digital artifact to work. Or maybe it does in this time and place in history (Kids Today, blahblahblah), but, a class can be set upon this kind of path to learning by a crude drawing or sufficiently ill-defined question. People have been teaching this way since well before digital projectors!

5. Dan,
I need more time to think about the framework and think it is excellent to see all your recent work summarized. I teach physics and use a much less formal framework with videos in the classroom. I analyze them with Logger Pro from Vernier. Not free but really cheap and super powerful. I also recommend the physic blog Dot Physics where he does video analysis. Here is a good example from a Super Bowl Ad. http://blog.dotphys.net/2009/02/analysis-of-super-bowl-commercial/

6. Kevin

February 19, 2009 - 4:33 am -

Again, this is a great idea. There is so much media out there and so much you could create to enliven the classroom. I think there are some benefits of digital versus hand drawn in that it is an ‘exact’ copy of reality. A drawing may very well leave out ferns, extraneous buildings, shadows etc. An image or video has a lot of ‘noise’ in it that the students need to filter out. Along these lines though we could ask what physical objects we could do math on, like Kate’s blog on the bouncing ball. Same dialog “How, What, Why, Where, etc” between teacher and class.

Keep up the great work!!!

7. Dan, I think this post is marvelous. And because it’s marvelous, I wonder if you’d be willing to cut and paste your standardized exam score results again from last year right into this thread. (I’m going to do this on my own blog shortly.)
I desire to explore concretely the assertions in a recent TED talk by Mr. Gates.

8. Kevin of course there are benefits! I simply wanted to point out that he’s tapped into some larger truths of good pedagogy here, which have been around since Plato, at least.

9. Jeanette

February 19, 2009 - 7:44 pm -

I loved your ideas but if a lesson can’t be done within a realistic amount of prep time it’s not worth it. Within 1/2 hour I had a plan that may work for implementation. I used the BBC site http://news.bbc.co.uk/1/hi/sci/tech/7793211.stm
for the video.
I paused the video at a desired spot. I have a promethean board so I used the activstudio software to take a picture of the paused screen. I sent it to a flipchart page where I can use the tools to measure what I want to on the picture.

10. Dan Meyer

February 20, 2009 - 8:43 am -

@Chuck, yours is a pretty perplexing question. You’ve got guys like Sam, people champing at the bit, no shortage of mathematical knowledge or creativity, who will have a hard time producing these digital media packages because they don’t know how to create a grid in Photoshop and import it into a video editing program. Or they don’t own the (expensive) software in the first place. This thing needs crowdsourcing but the tools are prohibitively costly and difficult. It shouldn’t be this hard.

Kate notes that we’re merely tapping into some deeper truths about teaching here and Andy notes (in the wiki) that this sort of technique applies pretty generally to subjects besides math.

They’re both right, in doses. If we’re trying to simulate the world then photographic quality is, of course, preferred, but even more than that, I like what Kevin points out, that drawings â€” particularly the kind your textbook stocks in its “Applications & Extensions” section â€” tend to leave out the noisy details â€” shadows, ferns, etc. â€” and include only what’s relevant or useful. But our students will have no one but themselves to tell them what’s relevant or useful. (This is why, on the EXIF photo, I made sure to include not only shutter speed but aperture, lens size, and a bunch of other totally irrelevant details.) We need the noise, which photographs supply in far greater quantity than drawings.

Shorter summary: Kate is right. I think it’s important for anyone who would commit to this sort of project to recognize a) how this is just good inquiry-based instruction, but b) modernized and simplified in ways that were impossible even ten years ago.

Ken, shoot man, I’ve got, like, four wikis parked in my garage.

Total non sequitur as long as I’m turning blog comments into posts, by far the most satisfying part of this framework is digging back into some of my early work, even impulsive stuff like Graphing Stories, which I shot and edited on a weekend, and watching how closely it tracks to the framework I’d lay down here years later. For good or bad (and no one so far has suggested that any one of these guidelines are misguided) I have internalized this aesthetic, which gives me a lot of hope that I can shepherd some sort of curriculum to completion.

11. Just to clarify…I didn’t mean to sound dismissive or imply there was anything “merely” about it. It’s huge. Math teaching is largely not being done this way, and needs to be.