When Is Video Valuable?

The question that bugs me at all hours is “When is video / photo / print valuable?” This video is one minute long and gets me closer to an answer.

The intermediate value theorem says that because you picked purple when the purple slice was big and blue when the purple slice was small and because slices run continuously from small to big, there is a particular slice that makes you go, “Meh,” that’s exactly in between “I choose purple” and “I choose blue.”

I love that students have an intuition about that slice, an informal understanding of probability that we can develop into something formal. We can access that intuition with video by showing that small slice growing continuously into the big. How do you replicate that experience in print, a medium which does a bang-up job with static quantities but has something of a panic attack when those quantities change?

Featured Comment

Avery Pickford:

Know what I’d really love. For every student to be able to click their mouse (or some equivalent) when they would make the switch and to have this data show up on my screen right after the video was done.

2011 Nov 29. Evan Weinberg hacked together something that does what Avery described. The results surprised me.

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. Nice. Know what I’d *really* love. For every student to be able to click their mouse (or some equivalent) when they would make the switch and to have this data show up on my screen right after the video was done.

    Have fun with that tech challenge, Jedi Master.

  2. Maybe you get a class on todaysmeet.com (or something like it.) Get the whole group to type ‘NOW’ but tell them not to hit ‘enter’ yet – they will know when to do so. Count ’em up, or write a program to scrape the page for that info.

    The thing I see about print is that the good questions that really make you (and potential students) think really don’t change with time. The right question or photo doesn’t require multimedia to be interesting. I was intrigued by the still frame of the two colors in the video before I hit play. I could imagine conversations between students about what mathematical ideas might be in store just from looking at that image. Then you hit play and watch those preconceptions either melt away or be confirmed.

    Imagine seeing that picture in a book at first, a static image, seeing the video that gave it additional meaning, and then taking a moment to go back to the static picture. You never go back to looking at the picture in the same way, even though that picture itself hasn’t changed.

    I think there’s something there that is still very unique to print. I think it’s the same sort of reverse technological fascination that drives my students (who are remarkably techno-savvy) to fight over playing checkers or fuss with the colonial style metal puzzles on my bookshelves. Maybe in contrast to the over-stimulating pages of most textbooks currently available, there could be a move toward a “less-is-more” approach. Only the best (and the most necessary) gets printed – the rest is available online.

  3. I can only hope that some experience teaching will gradually get me to where I am able to come up with ideas like this because right now it would never occur to me to pose the intermediate value theorem this way. Of course, if you keep spitting them out, Dan, maybe I don’t need to, haha.

  4. Avery: Know what I’d really love. For every student to be able to click their mouse (or some equivalent) when they would make the switch and to have this data show up on my screen right after the video was done.

    Me too. I wish I were as comfortable and expressive with code as I am with video.

    Christopher Danielson: If you are teaching the class, do you (@ddmeyer) show the video? ‘Cause I don’t/won’t. I’ll use the idea but not the video. For me, video is valuable when it brings the world into the room in a mathematically productive way.

    Probably goes without saying I’m not going to use a video I narrated while I’m standing right there. This is just a representation of practice.

    In class, I’d cook up static slides in Keynote for each of the scaffolds (even odds, no payout; even odds, 100/100; even odds, 100/200; horrible odds, 100/200), flick back and forth between the last two, and then have an embedded video that turns horrible odds into even odds, just like you see / hear above.

    I can think of alternatives, but they don’t work as well as video or animation. Geogebra would be fine. If I only had static images to work with, though, I’d use my marker, move it along the circumference of the circle, and ask them to say when the odds were even. That’d be workable, but inferior. I think you’ve put video in too small a box.

  5. Hi Dan, after the teacher gets the data, there could be a student-led simulation.

    For example, here’s a Roulette simulation with 10000 attempts (video – no audio):

    In some simulator, you could also graph it, like this:

    You can play with it here:
    and select “La roulette”

    Google translate will help you figure them:

    It was built for this project with open ended questions:


  6. Dan:

    I think you’ve put video in too small a box.

    Or maybe I have too narrowly construed the term video when you mean to point to dynamic representation.

    GeoGebra on a Smart Board is actually a really nice alternative here-you and the students are looking at the same image-the same one you’re pointing at. You drag that point around the circle, stop when students are loudly cheering for their favorite, etc.

    To follow up on your “goes without saying”…do you show the video if I narrate it? I’m still guessing no.

    I just don’t feel that it’s video you’re pointing to with this example. No, this example (for me) is quite different from the escalator problem, or Coke v. Sprite, or etc.

  7. I love this concept… and I’d love it even more if at the end there was a wondering asking the student to choose a new value instead of the £200 (let’s say £1,000) and ask them to draw the slice. [great assessment/box ticking for the teacher]

    Sure there’s a coder out there who could program the slice growing/shrinking depending on amount, and then allowing the teacher to input multiple variables (£100, £200 and £300, etc). That would be cool.

    This is real similar to your 25% amounts of a square, then seeing one of the diagonals slide down the vertical. Both could be presented so utterly boring in a text book, but in these visual ways are so very captivating!

  8. Great stuff, Dan! My colleague Richard said “This about sums up why animation is such a fantastic tool for a math ‘textbook.’ ” We like your stuff – keep it up!

    Oh – I did have one particular thought about this video – I found myself thinking the animation would’ve flowed better with the initial posing of the problem, had it run in the opposite direction, from half to full… Perhaps it’s just how my brain prefers it, but I do find that when teaching with video (we use it a lot), little things matter – though this one could purely be my own pref. (I’d like to know if others found the same thing).

    Anyway, thanks again for the challenging, sharpening thoughts, as always!

  9. I love this video, and I think starting with a simple spinner with 2 equally-likely outcomes is a great way to start. I’d probably start with the $100s split evenly and ask students if they’d pay $50 to spin the spinner and receive the prize landed upon. Yes. Then $75. Yes. Then $100. (They may say “Why bother?”) Then $125. No. Discuss why. We expect the outcome to be $100.

    Then do the 50-50 $100/$200 split. Would you pay $100? Yes. The least you get is $100, anything else is gravy. Would you pay $200? No. The most you could get is $200, anything else is a loss. What would you pay? Here’s where I’d talk about how the probability ties in to “expectation”. If we expect $100 to hit one out of two times, and we expect $200 to hit one out of two times, what do we expect to happen if we spin twice? $100 + 200 = $300. If we expect $300 for 2 spins, what do we expect for 4 spins? $600. 8 spins? $1200. For each scenario, how much for 1 spin? $150.

    Continue with varying areas of the circle.

  10. I’m a little bothered by Brian’s explanation of expected value. I’m afraid many students will expect to have two spins come out to exactly $300, rather than $200 1/4 of the time, $300 1/4 of the time, and $400 1/4 of the time. Going up to 8 spins gets a binomial distribution that will be very difficult for most students to handle. To use Brian’s approach, you would have to have already established the additivity of expected value, but I thought that his was intended as a first lesson on expected value, not a review at the end of a unit.

  11. Hi Dan, this is purely a video technique question. How did you create the animated circle and spinner for this post? I love your videos and would like to do some similar things for my science class. Also, how did you overlay the the graphs for the Graphing Stories piece? They are fantastic! I use them in my physical scence class when we talk about motion.
    Thanks, John