Posts

Comments

Get Posts by E-mail

Better Online Math

tl;dr version

Currently, online math websites comprise video lectures and machine-scored exercises.

For several different reasons, online math websites should add an introductory challenge that activates a student's intuition and intellectual need. The video lecture should then be directed at satisfying that particular intellectual need.

Here's an example. Let's make this happen.

tl version

Online math sites are quickly defining math down to a) watching lecture videos and b) completing machine-scored exercises. I'm not going to re-litigate whether or not that definition of mathematics is as good as what we find in the best classrooms in the highest-performing countries. (It isn't.) Instead, I'm going to take this online model for granted and ask how we can make it better.

What should we improve? It isn't the lectures.

For some time there, I was meeting with founders who were pitching their startups as "Khan Academy plus [x]" where x was anything from better graphics, better lesson scripting, a face on the screen, or multiple choice questions embedded in the video. (Here's basically the entire set of [x] at once.) I don't believe there's much value to add there. The Mathalicious lecture videos are beautifully shot. TED-Ed pairs their lecturers with world-class animators. Woodie Flowers wants to see Katy Perry and Morgan Freeman narrate these videos (I think he's at least half serious) and my suspicion is that we have reached a point of diminishing returns on the efficacy of lecture videos. Once we passed a certain point of coherence and clarity, watching Drake rap over a combinatorics lecture animated by the Pixar team just isn't adding a helluva lot. If math were only about clear and coherent lectures, we could probably close up shop here in 2012. Thankfully, there's more interesting work to be done.

So what should we improve? It probably isn't the exercises either.

The machine learning crowd seems very impressed by the millions of rows in their databases which represent the clickstream of hundreds of thousands of users. That clickstream can tell a teacher how many hints the learner requested, how long she spent on a given problem, whether she's more apt to score well on machine-scored exercises in the morning or evening. But what the learner and her teacher would really like to know is what don't I understand here? And machine learning has added very little to our understanding of that question. So there's certainly value to be added there but I'm pessimistic that machines are in any position right now to evaluate a written mathematical assessment at anywhere near the skill of a trained human.

So what should we improve? We should improve what happens before the lecture.

Currently, the online math experience begins with a lecture. The implicit assumption is that students need to be talked at for awhile before they can do anything meaningful. Not only is that untrue but it results in bored learners and poor learning.

Dan Schwartz, a cognitive psychologist at Stanford University, prefaced student lectures with a particular challenge [pdf]. He asked students to do something (to select the best pitching machine from these four) not just to watch someone else do something. Those students then received a lecture explaining and formalizing what they had just done. Those students scored higher on a posttest than students who were pushed straight into the lecture without the introductory challenge.

I'll show you an example of how this could work online. Head to this website and play through.

Let me explain what I'm trying to do there. First, any student who knows or can intuit the definition of "midpoint" can attempt that opening activity. It's an extremely low bar to clear. The lesson will ultimately be about the midpoint formula but we haven't bothered the student with a coordinate plane, grid lines, coordinate pairs, or auxiliary lines yet. Save it. Keep this low-key for a moment.

Once the student guesses, she sees how her classmates guessed, which queues everyone to wonder, "Who guessed closest?"

We've provoked the student's intellectual need and set her up with the kind of introductory challenge that prepares her for a future lecture.

So we move into the lecture video, which has several goals:

  1. It references the introductory challenge explicitly. The point of the lecture is to bring some resolution to the conflict we posed in the introduction: "Who guessed closest?"
  2. It offers a conceptual explanation of the midpoint formula, not just a recitation of procedures.
  3. It explains very explicitly why we use abstractions like x-y pairs and a coordinate plane. This satisfies John Mason's recommendation that we become much more explicit about the process of abstraction.

After the lecture, the student sees the original problem, now with x-y pairs and a coordinate plane. No longer does she simply guess, aim, and click. She calculates. There's are blanks for the answer now. We have formalized the informal.

The student calculates the answer and finds out how close she was. We should also throw some love on the closest guesser who may be a student who doesn't usually get a lot of love in math class.

After that resolution, we ask students to practice their skills, but not just on automatically generated clones of the same problem template. We give them the midpoint and ask them to work backwards to one of the original points. That's essential if you want me to have confidence in your assessment of my student as a "master" of the midpoint formula.

That's it. An intuitive challenge that precedes a lecture video that explains how to resolve the challenge. That'll result in more engaged learners and better learning.

Other examples?

And on and on and on. There isn't a recipe for these challenge but I know two things about all of them:

  1. Math teachers have a stronger knack for creating these challenges than people who haven't spent years fielding the question, "Why am I learning this?" fourteen times a day.
  2. These challenges are more fun when they're social. It's one thing to see my own guess at the midpoint. It's another thing entirely to see all my classmate's guesses next to mine. We need the Internet to facilitate that quick, cool social interaction. It just isn't possible with bricks and mortar alone.

Current online math websites have managed to scale up the aspects of decades-old math learning that few of us remember fondly. We can tinker around the edges of those lectures and exercises, adding a constructed response item here or a Morgan Freeman narration track there. Or we can try something transformative, something that draws from the best of math education research, something that takes advantage of the Internet, and makes math social.

BTW:

  1. I made that site for my final project in Patrick Young's summer Front-end Programming course at Stanford, which, as I mentioned previously, was a pile of fun. If I can make that site in a couple weeks with a thimbleful of programming knowledge, I'm eager to find out what your team can do with its acres of talent and piles of VC funding or non-profit donations.
  2. This isn't real-world math. I thought initially to pull in some tiles from Google Maps and set up a scenario where the student had to place a helicopter pad exactly between two cities. I don't think it matters. Students ask "Why am I learning this?" because they feel stupid and small, not because they want you to force a context onto the mathematics. I'm trying to demonstrate that here. Everyone can click on a guess. No one feels stupid and small. Any context would be beside the point.
  3. Cost-benefit analysis. Too often we apply a benefit-benefit analysis to edtech. But there are clear costs to the model I'm suggesting here even apart from the cost of the technology itself. There were at least five different moments over that five-minute lecture where I wanted to stop, pose a question, or have students work for awhile. We lose that here. I acknowledge those costs. We may still come out ahead on benefits if we can scale this up cheaply. "Pretty good" times millions of students may outweigh "great" times thirty.

2012 Nov 7.

A couple of useful tweets.

I'm concerned the competitive vibe appeals only to males, but FWIW this is exactly the kind of reaction I'm trying to provoke.

It isn't! It's passively discouraged, which is a huge bummer. I can think of at least two ways a student might think about the midpoint and how to find it. You can take half a side of the right triangle and add it to the point with the smallest value or you can subtract that half from the point with the largest value. Those multiple methods and the discussion about their equivalence are to be prized and they're lost in the video lecture format. They're lost. That's absolutely a cost and not a small one.

2012 Nov 12. Mr. Samson reminds me of the Eyeballing Game, which has been nothing if not an enormous inspiration for the work I'm doing here.

2012 Dec 7. David Lippman does this discussion a favor and creates an environment where the video pauses for student input. Discuss.

Featured Comment

Michael Serra, author of Discovering Geometry:

Curiosity and engagement will always trump "real world" applications. Games, puzzles, being surprised or caught off guard with something new and trying to find out why, these are big tools in our teacher toolbox.

58 Responses to “Better Online Math”

  1. on 07 Nov 2012 at 10:38 amZachary Wissner-Gross

    Motivation up front is great! But why not then break up the subsequent lecture into steps that the students themselves can solve?

  2. on 07 Nov 2012 at 10:47 amKarl Fisch

    I need to think about this more, but two quick thoughts/questions:

    1. I’m not completely sold that any context would be beside the point. I think for some students it would be, others it wouldn’t be. Not sure there’s any way to solve that (other than making two videos).

    2. You said there were lots of places you wanted to “stop, pose a question, or have students work for a while.” While you obviously can’t make them do that, could you give them prompts to do that in the video? “Okay, now I want you to think about this – pause the video – a little wait time – then hit play again.” Any possibility there, or would that just interrupt the flow?

    As always, good stuff.

  3. on 07 Nov 2012 at 10:52 amZachary Wissner-Gross

    Asking students to pause sounds like a good first step. But ultimately, I think we’re going to need software beyond YouTube/Vimeo. Although I heard YouTube was working on a Q+A format for education-themed videos…

  4. on 07 Nov 2012 at 11:01 amKarl Fisch

    OK, think I identified the nagging question I had. What’s your end goal here? You clearly acknowledge that you don’t think this is as good as what good teachers can do in the classroom (basically all of what you’ve done here, without the drawbacks you cite), yet still seem to think that “we may still come out ahead on benefits if we can scale this up cheaply.” Can you expand on that a bit?

  5. on 07 Nov 2012 at 11:27 amToby Schachman

    Thanks for this, Dan, and great job with the interactive prototype!

    Something seems a bit off about the rhythm of the whole sequence. I think it’s because the video, to me, feels much longer than it should be. I see what you mean that there are points where you want to pause it and do an activity.

    For example, it would be great to pause it once the grid is down, and then from there pose the challenge, “How does this grid help us figure out where the midpoint is?” In other words, just putting the grid down produces a *visual insight*. We then want that visual insight to spread to the rest of our mind, until it hits the algorithmic/verbalizeable part. How can an online learning environment support this process?

    I can imagine that in a live classroom, you could have students work together in small groups to collectively suss this visual insight into a verbal strategy for calculation. They could then present their strategies to the class. This, I believe, is a huge weakness for online environments: they can’t support communication beyond the channels that are programmed in. Worse, you can’t take an online class and then go to lunch with your classmate and let the lesson bloom into the rest of your life.

    On the other hand, the greatest strength of the online medium, and one we don’t fully exploit, is that it potentially allows knowledge construction to be shared in entirely new ways. For example, what if instead of watching a video explaining how to find the midpoint, each student had to make their own video explaining how to find the midpoint. Afterwards, the student could watch her classmates’ videos and compare and discuss. Of course, the tools we have for frictionlessly manipulating video are too immature at this point, but I think there’s potential in having each student somehow express their understanding to each other, seeing and learning how other people think about problems.

    One question for you: you state that you become confident that a student understands the midpoint formula when she can, given one of the points and the midpoint, work backwards to find the missing point. I agree with you. But I’m wondering about the thought process that led you to this. Are there, in general, ways that you take an idea and ratchet it up one level of creative abstraction so as to ensure that a student has true understanding?

  6. on 07 Nov 2012 at 11:44 amKathy Sierra

    Just one of the many powers of this approach is that it becomes a highly engaging loop where the post-challenge lecture and exercises begin to drive the next challenge. “Hey, wait, well than what about…..?” and on it goes.

    After all, this is what so many great games do: the levels are not arbitrary, but rather the next iteration where now that you have achieved THIS new power/weapon, oh, look, now there is an even more clever enemy to defeat that requires you exercise your newly-acquired powers… And so on…

    Of course, your title on the chart could just be… Better Learning, period. I realize you’re making a specific comparison to current online models, but your heading somehow makes it seem like that Challenge-Lecture-Exercise model is somehow specific to online math, when it is the model for better learning.

  7. on 07 Nov 2012 at 11:52 amKathy Sierra

    Also, zeigarnik effect.

  8. on 07 Nov 2012 at 12:03 pmKarim

    I agree that the landscape is saturated with videos meant to teach students math, and that iterating on that — even with an Oscar cast — is only marginally useful.

    That said, a point of clarification: the Mathalicious video series wasn’t intended for students. Rather, we created them to offer a general audience a different way of looking at the world, much like Freakonomics and Mythbusters. Of course, there was an educational motivation — to highlight for teachers the types of conversations they might have with their students; and to provide them with [free] classroom lessons to help facilitate that — but as far as the students were concerned, the videos themselves didn’t even exist.

    Which brings up an interesting question: what is the role for video vis a vis students, if any? It’s scalable, sure, but is it any good? Indeed, I continue to wonder this about online learning itself. Even if we built the best online learning environment possible, will it get us any closer to where we need to go, or will it nibble around the edges and at the same time detract from the more important conversation that countries like Finland et. al. had years ago?

  9. on 07 Nov 2012 at 12:41 pmJason Buell

    Manu Kapur has been doing work on “productive failure.” (Ill structured problem solving in groups first, followed by sharing, then lecture). I feel like he’s largely replicating Schwartz’s work but he frames the results in terms of Cognitive Load Theory and directly at Sweller et al. IIRC, the essential argument is that the invention time allows us to create enough framework to reduce our cognitive load when we receive direct instruction and, like Sweller, ignores motivation entirely.

  10. on 07 Nov 2012 at 12:41 pmReese McLean

    I am currently in my first year of teaching math online. As I have been exploring the different curriculum that are available it is clear that there is a huge chasm between sound pedagogy and what is being used. They all follow the model of Lecture->Practice, the best at least having interactive practice.

    I love the prototype and know that this is where we need to head.

  11. on 07 Nov 2012 at 12:44 pmAudrey McLaren McGoldrick

    Explorelearning.com is an example of a math/science website that doesn’t fit your description of lecture + exercises. Instead of videos, there are interactive gizmos that allow students not only to answer questions, but to make up their own.

  12. on 07 Nov 2012 at 1:02 pmJason Dyer

    I’ve used (in class) an educational video with narration by Morgan Freeman.

    While you obviously can’t make them do that, could you give them prompts to do that in the video? “Okay, now I want you to think about this – pause the video – a little wait time – then hit play again.” Any possibility there, or would that just interrupt the flow?

    I’ve done this before (as have others) and it definitely feels like a hack. (Didn’t some of the ed videos of the 70s have a “pause here for class discussion” bit?)

  13. on 07 Nov 2012 at 1:10 pmDavid Wees

    I think you are missing a verb that will make your online model even better – discuss, perhaps before and after watch?

    I have more thoughts on this. I’ll try and post something in much more detail on my blog before the end of the week.

  14. on 07 Nov 2012 at 1:35 pmMichael P

    This is great. Brings together a lot of what you’ve been writing about lately, as far as ed-tech goes. But I want to hear more about your interpretation of Dan Schwartz’s paper.

    Guessing the midpoint is just waaaay easier a task then creating a metric for a pitching machine. The way that you’re talking about intellectual need, it sounds like your take on the pitching machine problem is that it created a narrative need. Guessing the midpoint certainly does that. But maybe the results have less to do with narrative need (it tells a coherent story) and more to do with the strength of that need.

    How committed am I to finding a procedure for computing the midpoint if I’ve just offered a guess? Answer: more than if I hadn’t, but less than if I was in the thrall of a difficult problem.

    Guessing gets buy-in, but minimal buy-in. That’s OK in the classroom, when I can take that minimal buy-in and buy myself a more difficult problem (“What information would you need?”) that further cascades until I have full intellectual commitment to the problem. (Ha! Like that happens every day.)

    So let’s guess before watching videos. But let’s not expect more buy-in than that challenge earns us. And let’s continue to stress about how to pose more challenging problems on the internet.

  15. on 07 Nov 2012 at 1:57 pmSean Wilkinson

    Just wanted to weigh in on the issue of pausing the video or not. When I watched this, I was very surprised to see you ask a question (“How many triangles are there?”) and then quickly provide the answer yourself. You definitely lose a chance to have the student figure that out for herself.
    But the alternative (the video auto-pausing and prompting for student input) has its own drawbacks. I was a student in Thrun & Norvig’s Artificial Intelligence MOOC, and they embedded all sorts of questions into their lecture videos. Often, these pauses seemed more jarring than helpful.

  16. on 07 Nov 2012 at 1:59 pmAllison Krasnow

    I LOVE this.
    Here’s why:

    I am experimenting with the notion of a flipped classroom, given the constraints that the vast majority of my struggling students don’t have computer access outside of my classroom. But I really like the idea behind a flipped classrooom and how it shifts the role of the teacher while IN the classroom with students.

    So, step one, was to obtain more technology. Check. Through 3 grants, I now have 11 netbooks in my classroom.

    I have a tablet PC so I am able to use screencasts to teach certain topics or to create a reteaching screencasts around topics where specific students struggled on a test. This has allowed me to differentiate a whole lot better after tests as I can target specific kids to listen to a specific screencast for some reteaching.

    What I really like about the midpoint lesson that you linked to this post is that I can visualize how some students could be learning about calculating midpoints via your type of model….posed challenge->lecture->practice question while other students are in a smaller group with me doing more open-ended problem solving.

    With your model, there’d be way more buy in and engagement, but the model still allows for students to learn while relatively independent from me, freeing me up to do deeper problem solving work with kids. Using your model of online math as a center that students rotate through on a weekly basis gives me a whole new way to conceptualize my week and my role. Sign me up.

  17. on 07 Nov 2012 at 3:15 pmAvery

    Schwartz’s pitching machine challenge is new to me. Awesome! Reminds me of the challenge to measure the squareness of different rectangles.

  18. on 07 Nov 2012 at 3:20 pmSean McClinton

    How will the effectiveness of this tool be measured?

  19. [...] Today, Dan Meyer, hit again on the video-lecture then practice problem routine. His proposal is to introduce an idea, a challenge, a problem that the student has the ability to get involved with prior to listening to the lecture. In my view, this is another way to build that conceptual foundation so that students do resort to “just split it into two equations and solve” algorithmic thinking. If we can build a way to include the challenge and then evaluate conceptual knowledge – well, just pass that idea along to me and I’ll get back to you from atop my mountain of cash. Share this:TwitterFacebookLike this:LikeBe the first to like this. [...]

  20. on 07 Nov 2012 at 4:05 pmTom

    Funny I was just finishing a blog post on Khan Academy and procedural vs. conceptual knowledge and specifically why the “video lecture, then exercises” model is insufficient. In my example, my students were able to spit out the procedure for solving an absolute value equation with ease. But when I asked them to describe what |x – 3| = 4 means, that was a different story. The problem with Khan, et al, is that question never even gets asked.

    http://mrwardteaches.wordpress.com/2012/11/07/what-khan-academy-cant-do/

    I really think the use of an introductory challenge is a great way to make sure you are not losing the foundational concepts – the “why” of the procedure you’ll eventually get to.

    Dan, my question would be where/if/do you see a way to assess conceptual knowledge in your online model?

  21. on 07 Nov 2012 at 7:44 pmDavid Lippman

    For a question like this, having them guess first is great! For other topics, like exponent rules, the approach of injecting questions into the video is a nice approach. It both allows them to do guessing of answers, but also allows them to attempt to execute a skill without having just seen a worked example to mimic. That’s the exploration / try it out step that is usually missing in video lecture.

    Udacity does this in their courses, and I found it an engaging approach as a student who gets horribly bored from lecture videos. And it certainly doesn’t have to just be multiple-choice questions like Coursera does; Udacity, Stanford’s Class2go, and my site MyOpenMath all allow a wider range of questions. (ex: http://www.screencast.com/t/S5auxo9AdH)

  22. on 07 Nov 2012 at 8:57 pmDan Meyer

    Zachary Wissner-Gross:

    Motivation up front is great! But why not then break up the subsequent lecture into steps that the students themselves can solve?

    The problem with the lecture isn’t that it’s too long or that it conjoins too many steps, it’s that all those steps are completely linear. The process is on rails. But real problem solving is rarely on rails.

    Karl Fisch:

    You clearly acknowledge that you don’t think this is as good as what good teachers can do in the classroom (basically all of what you’ve done here, without the drawbacks you cite), yet still seem to think that “we may still come out ahead on benefits if we can scale this up cheaply.” Can you expand on that a bit?

    There is enormous demand for learning that doesn’t require a classroom or a flesh-and-blood teacher. I can either sit back and continue saying, “A small classroom and a good flesh-and-blood teacher trump everything else.” But then that demand will only be met by people who don’t understand math education. So I’m shelving my reservations for a minute.

    Toby Schachman:

    Are there, in general, ways that you take an idea and ratchet it up one level of creative abstraction so as to ensure that a student has true understanding?

    Generally I find it useful to have students work a problem from both directions so I flip the known and unknowns around. Previously you were searching for the midpoint. Now you know the midpoint. There are other useful strategies I’m sure.

    Kathy Sierra:

    Also, zeigarnik effect.

    Always nice when Kathy Sierra stops by to add a scholarly citation to the mix.

    Michael P:

    Guessing the midpoint is just waaaay easier a task then creating a metric for a pitching machine.

    This was my adviser’s critique of the connection to Schwartz also. The wind-up needs to involve more. I wonder about posing a second blank where we ask students to speculate about a solution strategy. “What information would be useful? What would you do with it?” The danger here is in ratcheting up the cognitive load to the point that it overwhelms the learner, but it may be worth trying to locate that boundary.

    Sean McClinton:

    How will the effectiveness of this tool be measured?

    Pre-test / post-test for a start. Given the trouble your friends at KA have getting students to watch the videos, it would be interesting to see if more students watch the lecture video given the initial challenge than they would have otherwise.

    Tom:

    Dan, my question would be where/if/do you see a way to assess conceptual knowledge in your online model?

    That’s enormously challenging given the constraint of machine-graded exercises. The exercises might have to expand to problems and be graded by a human.

  23. on 08 Nov 2012 at 5:02 ammr bombastic

    A few critiques:

    I think it is much more intuitive to draw two congruent triangles as opposed to using similar triangles. One triangle with the hyp. going from the lower point to the estimated midpoint. Another going from the estimated midpoint to the top point. This is much more convincing visually.

    I am guessing that you lied in the video when you told us the reason for the placement of the grid. I think you specifically did not want one of the points at the origin (because you want to develop the distance formula) even though that would be a natural and simpler choice.

    What if a student’s method is to trace the line with their finger & count the number of vertical lines their finger goes through. If their finger goes through 18 vertical lines, then the midpoint is where their finger went through the 9th vertical line. This method is a natural approach for people and works if you have a grid. In other words, maybe the kid is thinking they have a simple approach that is better than your complicated approach and more or less ignores the video.

  24. on 08 Nov 2012 at 6:49 amPhilip Seris

    All of the commentary is great and I am glad to see that there is discussion in this area about online curriculum. I truly believe that the value in video is two-fold:

    1. There is a lot of help that could be offered to students who need to move through the lecture at their own pace and have issues with taking notes at the speed of the class.

    2. Showing is awesome. Math is a lot of visual that is sometimes lost.

    Both of those things being said, these problems have existed for years and have been solved in a variety of ways, not the least of which is student avoidance of math and math complexity all together. Harnessing digital curriculum and enticing the mind with something that can be seen and played with initially and then abstracted (as you often speak to) is a valuable goal. This goal could be an asset to advancing technology in mathematics education.
    Another benefit of curriculum like the one you showed is the ability to collect information and put it all together in one place. As students guess and play, we could collect information on common problems, assertions and misconceptions, leading to vast analysis of student tendencies. This could lead us to better pedagogy, digital or otherwise. One of the most powerful assets of this online learning process could be a “social mathe-media” of sorts, which could help us collect data about how effective different methods are in teaching a given topic.

  25. on 08 Nov 2012 at 9:26 amjoseph

    ” “Pretty good” times millions of students may outweigh “great” times thirty. ”

    You’d better trademark that before KA takes it as their new slogan.

    I wondered about students who just count boxes and divide by 2. The division process is the same, but getting the numbers from counting or looking at coordinates is very different. I have seen that when each grid mark equals 3 instead of 1. Students still treat it like 1 box instead of 3 units. That’s fine, until the graph’s x-axis counts by 3′s and the y-axis counts by 10′s. I guess the midpoint will still work out, but they will run into trouble when it comes to calculating slope.

    So anticipating the less-correct methods that students would use and helping them turn that into the better-for-later method is the only thing I would add to this otherwise great concept. I agree about excluding a word problem’s unnecessary motivation in this early stage of instruction. The social challenge is more motivating than where to put a pretend helipad.

  26. on 08 Nov 2012 at 11:02 amJames Key

    Dan, with you all the way on your principles, as usual.

    One small detail: there are many points “between A and B.” Clearer to say “exactly halfway between A and B.”

    On instruction: the student participates on that very bottom rung of the ladder: guess the midpoint and click to respond. But there are so many other opportunities for students to participate:
    • How long is the bottom of the triangle (run)? How do you know?
    • How long is the height of the triangle (rise)? How did you get your answer?
    • What is the midpoint of the base of the triangle? How’d you get it?
    • What is the midpoint of the vert leg of the tri? How’d you get it?
    • Guess the x and y-coor of the midpoint of the slanted segment, and explain how you got your answers.

    These are the questions I want my students to grapple with in my lesson on the midpoint formula. I expect a portion of the class to be able to answer all these independently – enough so that they can share with the whole group and we can have a good little discussion.

    In my view, answering the set of questions above is the lesson — and the students need to do it themselves, not hear a lecturer do it. Your sample lecture is an improvement on existing media in that 1) you have them participate at the outset and 2) you are explicit about things and motivate things. That being said, I think your model doesn’t go nearly far enough. To learn how to think mathematically, students need to struggle with the tasks higher up the ladder (i.e. more than just “click yer guess here.”)

    “It explains very explicitly why we use abstractions like x-y pairs and a coordinate plane. This satisfies John Mason’s recommendation that we become much more explicit about the process of abstraction.”

    Agreed, but I think you can reduce your word count by about 90%. If you just said, “I’m gonna lay down a coordinate grid, which makes it possible to be definite about the location of the midpoint, cuz now we can attach coordinates to our two points which were sort of floating in space before,” I think we’re good here. If students have no experience whatsoever with grids, they may need more. On the midpoint lesson, I think a minimally explicit formulation will do the job.

    On proof: you did a sidebar explaining how the midpoint of the slanted part relates to the midpoints of the legs of the triangle. In my view, you could do better by relegating this to the top of the ladder – hence leave it till the end. Make them take your specific numbers and answer my list of questions. Then make them generalize. Then make them practice. Then as a sequel – pose the task about proving that the midpoint of (x1,x2) and midpoint of (y1,y2) give the midpoint of the slanted part. I don’t even do this task in my honors classes anymore, because there are enough rungs below (and beside) that task to keep them plenty busy!

  27. on 08 Nov 2012 at 12:00 pmmalcolm

    I’ve watched videos/read books where I’m asked to pause or put the book down and do an activity. It always feels forced and out of context. Mostly I just ignore those remarks now and keep going. In my opinion, it would just interrupt the flow and reduce your traffic. Just sayin’

  28. on 08 Nov 2012 at 1:53 pmBethany

    I totally agree with James Key. I like that you don’t actually give the midpoint formula, but you could easily cut your lesson to about two minutes, and save the proof for an extension. Less talking-at and more action!

  29. on 08 Nov 2012 at 2:07 pmmalcolm roberts

    My experience (in higher education) is that many students, once the novelty of a different approach wears off and they also work out that the answer/procedure is coming (eventually) anyway, just sit back and wait for the teacher to explain. I suspect this is because these students have only the assessment task in mind and for that the teacher explanation will be enough to get them through.

    My attempts at providing motivation have not been nearly as well thought out and executed as yours but I think that assessment, as well as the tasks, needs to be considered.

  30. on 08 Nov 2012 at 4:08 pmMax

    Interesting that the commenters are homing in on the idea of leaving the proof out of the video. Was the proof the main point of the video — “hey, it’s not just about getting the midpoint. We could do that pretty well by guessing. It’s about convincing everyone you’ve got the one and only midpoint, no doubt about it!”?

    I wonder if some of the discomfort with the proving we’ve got the midpoint aspect, vs. the how to calculate the midpoint, is that the task you’ve perplexed students with is the task of how to find the midpoint, not how to convince others yours is the best. I’m not sure of a way, in the absence of some sort of moderation and semi-synchronous conversation with classmates, to have the “whose argument for their midpoint is best?” task be the one that perplexes students.

    Perhaps voting among a few canned, incomplete explanations (didn’t Veritasium or someone show that we learn more from video that contains our misconceptions and deals with them explicitly?) Or something “choose-your-own-adventure” style where you select from a few different abstractions (add a grid, add a triangle, add 2 triangles, add circles, add rulers) and then look at one or two expert arguments.

    It seems to me that guessing where the midpoint is motivates a lecture on how to calculate the midpoint, and choosing whose defense is better motivates a lecture on how to prove the midpoint formula.

    And I wanted to mention http://k12meets.com as a solution to someone’s idea of having kids make and compare their own videos. The idea is, record yourself answering a question, rate others’ answers, hear an expert critique the top X answers, and then watch an expert’s answer.

  31. on 08 Nov 2012 at 7:09 pmmr bombastic

    I tend to agree with those pointing out the easiness of the initial task – the guess is more or less reflexive – no thought & no tension involved.

    What if you started with several unevenly spaced points in a horizontal line and asked for a point in the middle. Put them on a number line & show that the average value seems to be in the middle. Make the middle point a fulcrum and the rest weights if you want to show you really have the middle in some sense.

    Next put several unevenly spaced points that are in a line with a positive slope to it. Again ask for the placement of a point that is in the middle of all of these. Then ask for some sort of method to finding the middle. Show that the (average x-value, average y-values) seems to do the trick.

    Now go back to the two point case & prove that the midpoint between two points is: (average x-coord, average y-coord).

  32. [...] dy/dan » Blog Archive » Better Online Math [...]

  33. on 09 Nov 2012 at 5:47 amRobert Berkman

    Oy, I’m getting a little tired of all this technical talk. If you want to know what’s really holding is back in mathematics education, you should read Darryl Yong’s excellent (and concise) memoir about his experience teaching high school mathematics at a public school in California. The last thing on his mind is the logistics of online mathematics. Read about it here: http://www.ams.org/notices/201210/rtx121001408p.pdf

    On another note, while you’re all tweeking the logistics of online learning, has anybody given consideration to the actual students you are attempting to reach? To those who have forgoten, I offer this commentary: http://bltm.com/blog/?p=197

    Finally! What are we going to do about Annie Murphy Paul, who insists that if only our students knew their computational facts, we would instantly catch up with the Chinese on international tests of mathematical achievement? Read about this ridiculous idea here:
    http://ideas.time.com/2012/11/08/why-kids-should-learn-cu-cursive/?iid=op-main-lead

  34. on 09 Nov 2012 at 6:58 amDan Meyer

    mr bombastic:

    What if a student’s method is to trace the line with their finger & count the number of vertical lines their finger goes through. If their finger goes through 18 vertical lines, then the midpoint is where their finger went through the 9th vertical line.

    This is intuitive, but needs to be proven. Why does the vertical midpoint necessarily correspond to the midpoint of the line segment? That requires proof.

    joseph:

    I wondered about students who just count boxes and divide by 2. The division process is the same, but getting the numbers from counting or looking at coordinates is very different. I have seen that when each grid mark equals 3 instead of 1. Students still treat it like 1 box instead of 3 units. That’s fine, until the graph’s x-axis counts by 3′s and the y-axis counts by 10′s.

    Or if the points are too far apart to easily count the units between them.

  35. on 09 Nov 2012 at 7:04 amjoseph

    “Or if the points are too far apart to easily count the units between them. ”

    So maybe to fight the line-counters, you progress to where the points are too far apart to count, or you remove the lines and keep the coordinate pairs as the only information available.

  36. on 09 Nov 2012 at 8:46 amChris Robinson

    joseph:

    So maybe to fight the line-counters, you progress to where the points are too far apart to count, or you remove the lines and keep the coordinate pairs as the only information available.

    I think the only good solution is to make the points too far away to count. If we have been explicitly talking about abstraction with our students, they should understand the power of laying down a coordinate grid to help climb the ladder of abstraction. Removing the lines and just leaving the coordinate pairs may result in a “plug-and-chug” with the midpoint formula. After they have developed a conceptual understanding of midpoint, then this may be fine, but until that point, let’s let them struggle with the problem and develop their understanding.

  37. on 09 Nov 2012 at 8:59 amR. Berkman

    This is reminiscent of a Dover book called “Taxicab Geometry,” which covers problems like this. Ideally, I would start by posing it as a problem about a city which has two firehouses and where would you put the boundary line on a map so that the closest company is dispatched to the fire. What would come out is that the it would yield a line that at each point is equidistant from both firehouses. This would lead into a discussion of what the different points “mean.” That is, some points are equally distant from the two firehouses, but they are both “equally far away,” which there are others which are equidistant, but are “equally close” to both firehouses.

    In my experience, starting with simple cases (like two points that have identical x and y coordinates) and then moving on to slightly more complex cases (what if they were separated by 5 units vertically and 5 units horizontally), and then move on to even more complex situations. It seems to me that they will look at different cases and understand that in some cases (like a shared x or y coordinate), all they need to do is calculate the difference between the two locations, and then cutting it in half. The goal is to distinguish between cases which have different properties, and then use the appropriate tool to figure it out. Too many students use a “one tool solves all” approach to mathematics.

  38. on 09 Nov 2012 at 9:58 ammr bombastic

    @dan, You seem to misunderstand what this student is doing. This student sees the line segment as a ruler, and has reduced the problem as you posed it to finding the midpoint of a RULER! They are not thinking about the horizontal or vertical midpoint when they do this. They are only looking at the segment itself. There is nothing to turn a student off quicker than ignoring their simple solution in favor of your complicated solution. And, the proof is easy.

    Simple solutions must be rewarded. The proof that the intersections of the vertical grid lines and the line segment are evenly spaced is an immediate application of the Pythagorean Theorem (I move right one square creating a leg of length 1, I move up whatever the slope is creating a leg whose length is the slope & the hyp is the distance between the intersections on the line segment).

    I realize you have already pointed out the limitations of taking a pre-determined approach. Just pointing out a plausable case with negative consequences for the student.

  39. on 09 Nov 2012 at 11:00 amKarlaG

    I just recently started teaching for a totally virtual public school, 8-10 math. I’ve been trying to find ways to make the units more inquiry-driven. Can you share what program you used to create the midpoint activity? Specifically, I’m interested in how you made the “Guess” component. Thanks!

  40. on 09 Nov 2012 at 1:20 pmKevin Hall

    The activity you’ve created is great, but it doesn’t seem to perform the function I need online math tools to perform. I guess it would be nice if online math included explorations, but honestly, I want the explorations done in class, because I think explorations are a social endeavor. Doing them online strips some of the opportunities for discourse (“John, does Meghan’s comment support your theory or not?”). What I really want out of online math is to outsource all the crappy drill practice that I have to get my kids through, so I can have more time for 3 Acts and sense-making in class. I never have students do online exercises before we’ve done an in-class exploration on the topic anyway, because online tools aren’t for instruction, they’re for practice.

    That’s why I think the online tool with the greatest potential is something that covers the drill practice, but comprehends and checks each step of a student’s work, not just the final answer. As far as I know, the only commercial software that does this is the Carnegie Learning stuff. It’s not as good as it needs to be, but it clearly could be fairly soon. If you don’t believe me, try the Triangle Properties sample exercises at this link:

    http://mathseries.carnegielearning.com/demos/

    Carnegie Learning software has a big weakness that teachers may be able to help with. It traces student work through the solution process and determines the likely rule a student is applying at each step. In problems with multiple solution paths, it can trace whichever path the student decides to take. It can also be programmed to recognize the likely incorrect rule a student is applying when a step is wrong.

    But–and here’s the bummer–the software doesn’t DO ANYTHING HELPFUL when it recognizes the specific rule the student misunderstands. It usually throws up a jargon-y hint like “It looks like you think the absolute value of a number is always its opposite. However, the absolute value of a number is its distance from 0. Distance is always positive. Therefore (blah blah blah).” This kind of feedback is completely useless to struggling students.

    Communities of teachers could create mini online lessons that remediate specific student misconceptions. The software can (already!) identify which misconception a student has. Perhaps Dan’s format of Exploration–Video–Practice would be useful in this remediation.

    My main point is: I would want a community of teachers to use Dan’s format to focus on mini-lessons I can assign to students when I see they’re making a common mistake, such as thinking (x + y)^2 = x^2 + y^2. True explorations can be tech-enabled, but I still want them done in class.

  41. on 09 Nov 2012 at 2:08 pmjoseph

    @ Chris Robinson…
    I think we’re going the same way on this. I just didn’t elaborate enough the first time around.
    Taking away the grids and only leaving the coordinate pairs will lead to “plug and chug” solution-finding, which isn’t the way to teach. But I think that’s where we expect them to finish at the end of the mastery of the Find The Midpoint skill. Or their ability to do that is at least one evidence of mastery. Doing something like the suggested, “Here’s one point, here’s the midpoint, now find the other endpoint,” is another way that they can demonstrate mastery.
    Removing grid lines isn’t something we should throw in along the way. If they want to count, let them. Then go up the ladder by keeping the grid, but spreading the points out to take the counting strategy away. Then they will realize the value of Dan’s method and how it is better than their previously-fine method. Once they understand how to use the coordinate pairs, via Dan’s lesson, they can demonstrate mastery by working with another level of information loss, when the grid is removed.

  42. on 10 Nov 2012 at 6:26 amMr. Samson

    After I read this post, I thought of a game Ze Frank (zefrank.com) posted a few years ago. After trying it I quickly became obsessed for a few months. It took me a long time to find the game but here is the url http://woodgears.ca/eyeball/. The game consists of moving a point to complete various tasks such as; given two lines create a parallelogram, bisect and angle, find the midpoint of a line, create a right angle, and other great geometry tasks. I think they could be a great step 1 to your activity or maybe a post activity that leads to other geometry treats.

    Thanks for inspiring me to become an educator.

  43. on 10 Nov 2012 at 7:18 amjosh g.

    Well, that was an unexpected.

  44. on 11 Nov 2012 at 6:58 amKarl Fisch

    I generally agree with the folks who say that asking students to pause feels artificial but, for what it’s worth, Popcorn Maker is a new tool that would make it easier to do that – http://blog.ted.com/2012/11/11/remix-web-video-with-popcorn-maker-launching-today/

  45. [...] Better Online Math For several different reasons, online math websites should add an introductory challenge that activates a student’s intuition and intellectual need. The video lecture should then be directed at satisfying that particular intellectual need. – Dan Meyer Addressing the video-lecture followed by procedural practice mantra of online math websites. Share this:TwitterFacebookLike this:LikeBe the first to like this. [...]

  46. on 11 Nov 2012 at 3:18 pmAndrew Stadel

    My favorite part of the video is the first task at 4:48 you assigned students by saying, “Write down a rule that will work for [finding the midpoint of] any two points.”
    This creates ownership of the exploration and suggests there is a rule, but not necessarily just one rule. I’m a huge fan of rule/pattern exploration. When exploring the midpoint this year, I had a group of students find the midpoint by applying the slope of the line and using half the slope. They also used the slope to find the other endpoint, given the midpoint and one endpoint. Great ingenuity, using pre-loaded skills.
    My constructive feedback would be the timespan between introducing the 2 points and assigning the task. It seems lengthy without any opportunity for students to give more input (or maybe you have this in mind). The AA Similarity property might not be discussed until later on in the curriculum and therefore could be a huge distraction for a struggling student. I love the brief AA recap, don’t get me wrong. I will definitely use this video with exploring AA Similarity in class later this year.
    However, I believe the video could be paused anywhere between 2:00 and 2:30 and you challenge students to input what they notice. You ask students two key questions, maybe rhetorical. One, “How many triangles do you see here?” and two, “What’s special about these two triangles?” I think allowing them two chances to provide answers would benefit their continued engagement.
    Lastly, I would pause a third time after 3:00 to 3:06 has you saying, “We know one relationship between two sides there.” and ask students, “What’s this relationship?”
    You’re seriously onto something here Dan. I appreciate you throwing this our way and are so open to feedback. It only helps the construction of our teaching tools and in the end, student understanding.

  47. on 12 Nov 2012 at 9:06 amChuck

    Yes well if you want this to happen then teachers are going to have to create it. How about organizing a group of tech-savvy teachers who want to create a curriculum for one course – say Algebra 1 – that is adapted to common core, interactive and also has exercises that are immediately ready to go (for teachers). If teachers could organize something like that it would be better than any other curriculum out there and it could begin to show the bigwigs that our talents are not only under-appreciated, but more importantly, they are grossly undervalued.

  48. on 12 Nov 2012 at 10:00 amDave Major

    Yes well if you want this to happen then teachers are going to have to create it. How about organizing a group of tech-savvy teachers who want to create a curriculum for one course – say Algebra 1 – that is adapted to common core, interactive and also has exercises that are immediately ready to go (for teachers).

    This. A hundred times this. I think the problem lies with the fact that all we can do at the moment is throw up something different and cause discussion – I don’t think anyone (including traditional publishers) have something solid and workable yet; they just got in first with a means to spread their wares widely.

    Right now, I can’t help but feel we need loads of teachers coming up with contrasting ideas, so we (as teachers and idealists) can find a bricolage that works.

  49. on 12 Nov 2012 at 12:26 pmAndrew Browning-Couch

    ‘Students ask “Why am I learning this?” because they feel stupid and small, not because they want you to force a context onto the mathematics.’

    This is such a profound statement. Sometimes this frustration turns itself into disruptions and behavior issues (the “If-I’m-in-the-principal’s-office-then-I-don’t have-to-answer-that-question” Syndrome), but for some students it really does surface in the “Why?” question. Thanks for helping me think deeper about my classroom.

  50. on 12 Nov 2012 at 1:05 pmMax

    Just wanted to fully disclose that when I recommended http://k12meets.com, there’s no financial relationship between them and the Math Forum (where I work). K12Meets was founded by Drexel Law School folks, and the Math Forum is part of Drexel’s School of Ed. But we aren’t financially or personell-wise connected. We do think their technology is cool and we may use it in the future for some online coursework and professional development, and would love to partner with them on future projects, but my recommendation was not intended to promote any interests of the Math Forum or Drexel University.

  51. on 13 Nov 2012 at 12:45 amRoberto

    @ddmeyer I mainly back your approach.
    Now a few notes (hope this helps improving the discussion here):

    - how many times a student is definitely NOT interested in the challenge of guessing the midpoint ? So he/she is way LESS interested in grids, coordinates, axis, equations etc etc. ! And way^2 less interested in knowing the solution but his/her marks in Math ONLY

    - how is it possible to *meaningfully* link your Math challenges to those challenges the students see as *meaningful* ? Not simply *real life* challenges but meaningful challenges !

    - my second question leads to what I think is the real bottleneck in today’s education: the constraints of overwhelming CURRICULUMS !

    - closing remark: is there any *feasible* way in today’s pressured classrooms to go *deep* instead of going *shallow* ?

    Thanks so much for your great work, Dan.

  52. [...] for the self-promotion, but here’s an example (and write-up) I prototyped today in which I’m trying to marry Dan Schwartz’s cog psych research with [...]

  53. on 26 Nov 2012 at 3:23 pmTwo Guys Walk Into a Bar… « wwndtd

    [...] says that for online math (a.k.a., the Kahn Academy approach) to be more effective, you first need an in-class hook. The lecture will be that much more palatable if you know you’re going to get something out [...]

  54. on 04 Dec 2012 at 8:52 amDavid Lippman

    I took a stab at adding interactivity to the video in the example, automatically pausing and popping up questions during playback. Would love feedback. http://drlippman.blogspot.com/2012/12/building-on-better-online-math.html

  55. [...] After authoring this post, I caught up with my feed reading and realized Dan Meyer had also emphasized putting something BEFORE the online lecture to activate students&#8217….  I’m happy to be on the same page as Dan Meyer and others (see comments), even if it means [...]

  56. on 07 Dec 2012 at 3:35 amDan Meyer

    @David, provocative. Thanks for putting it together. There were a couple of different options here:

    - video plays through, no questions.
    - video plays through, narrator asks questions, no input mechanism
    - video plays through, software asks questions, input mechanism.

    The second troubled me because there wasn’t any kind of accountability or press. Students would come to disregard teacher questions or take them unseriously.

    I was worried about the third (yours) because I figured students would become irretrievably stuck at different moments. But you may have fixed us all up here with the option that lets students continue past an incorrect submission onto the rest of the video. So long as the teacher sees those submissions, I think we’re all in good shape.

  57. on 20 Dec 2012 at 12:51 amMichael Serra

    THANK YOU DAN. FINALLY SOMEONE HAS THE COURAGE TO SAY WHAT NEEDED TO BE SAID:

    You wrote: Students ask “Why am I learning this?” because they feel stupid and small, not because they want you to force a context onto the mathematics.

    Yes, my experience as well. This constant call to give them content in a “real” context so that we think we satisfy their question of “Why do I need to learn this” is to be challenged. Middle school students working on percentages aren’t going to become more excited or engaged if we give then problems about mortgages on their homes.

    Curiosity and engagement will always trump “real world” applications. Games, puzzles, being surprised or caught off guard with something new and trying to find out why, these are big tools in our teacher toolbox.
    Good seeing you the other day.

  58. [...] Furthermore, this low-tech method will all have been done without going into faux “real world” midpoint applications. As Michael Serra commented at dy/dan: [...]