Sue Van Hattum will be leading a webinar tomorrow to counterbalance the one I facilitated a week ago. Sue is the lo-fi to my hi-fi. Sue’s thesis is that we can have the engagement and challenge of WCYDWT without the multimedia.
This is undoubtedly true. Consider Polya, who offered engaging, challenging problems without degrading himself by walking up a down escalator. “Into how many parts will five random planes divide space?” for instance, is challenging and engaging and offers points of entry to learners of all abilities.
So an open question: what’s the point of multimedia? If it’s just amusing – which is to say, engaging in the worst, most superficial way –Â I can do without it. My sense, though, is that the feature common to all of these problems …
- into how many parts will five random planes divide space?
- how long will it take Dan to walk up the down escalator?
- how many tickets are on the roll?
- how long will it take to fill up the water tank?
- how fast is the runner?
- what is the killer’s shoe size?
… is that they “reveal their constraints quickly and clearly.” They’re Twitter-sized queries that unpack into full-bodied mathematical investigations.
Multimedia lets us reveal constraints quicker and more clearly, though that isn’t a given. Multimedia can have low information resolution. (I’m talking about your stock photography, your dogs in bandanas, etc.) But the information resolution on this single image of a ticket roll blows me away. When you put the quarter next to the roll for scale, the problem literally reveals its own constraints. The learner can gather any information she wants – circumferences, radii, diameters, ticket dimensions – without the teacher having to write or say anything.
I suppose I’m trying to slip Sue a question in advance: how do I reveal the constraints of the same problem that quickly and clearly without the multimedia?
josh g.September 3, 2010 - 1:07 pm -
Put an actual roll of tickets in front of them.
Dan MeyerSeptember 3, 2010 - 1:25 pm -
Like, one on every desk? Or just one in front of the class? If just one, who gets to come up and take measurements? What are the other students doing at that time?
MichaelSeptember 3, 2010 - 1:31 pm -
What if the presentation was analog? Students were actually given the tickets or taken to an escalator or given a tank to literally fill with water and given a stop watch or . . . Even typical textbooks have “multimedia” elements in them.
How did the great Greek Mathematicians tackle some of the toughest math problems without multimedia?
Could true analog experience be more pure than any digital presentation/experience?
Colin GrahamSeptember 3, 2010 - 4:58 pm -
Surely a computer is only one medium, so multimedia implies using non-computer stuff too… ;-)
ruthSeptember 3, 2010 - 5:45 pm -
well, multimedia is an encompassing term that includes past and current media (being plural of medium) so it should include ‘traditional’ forms as well? besides, modern technology is a help rather than essential?
David CoxSeptember 3, 2010 - 5:54 pm -
I’m not sure I have a grasp on what exactly it means for a good problem to reveal it’s constraints quickly and clearly, but I do know that if I took my students to a theater with an escalator, calculating how long it tool to go up the down escalator wouldn’t be on their radar unless I explicitly asked the question. Multimedia allows us to bring a lot of noise, propose a question without having to ask it and makes the learner decide what is needed to solve it. Not sure how to do that on a static page. That’s why curricula like CPM were ahead of their time. There was no way to withhold information in a book.
Sue VanHattumSeptember 3, 2010 - 5:58 pm -
I hope you’ll bring your question up tomorrow, Dan, and then tell me whether the problem I present does that – reveal its constraints quickly and clearly.
Coming up with good problems is definitely a difficult art.
DebbieSeptember 4, 2010 - 4:28 am -
I feel it’s less to do with the strengths of Dan’s videos and images and more to do with the weaknesses of textbook problems. Any written question or diagram in textbooks I’m familiar with give the values you need and rarely one single piece of surplus information.
When using an artefact or video or photo I think we should seek to provide too much information so that students have to tease out the pertinent information themselves.
RichardSeptember 4, 2010 - 4:50 am -
Here is a relatively low-tech problem I’m considering for my geometry class this year:
See the cartoon linked below (the rainfall part) . What would you like to know?
ClimeGuySeptember 4, 2010 - 5:10 am -
Your question hints at another pro-con argument (discussion if you will) about tech vs. non-tech. That’s a smoke screen. The better question is what can I do to motivate the powerful idea (i.e. slope formula) that will engage kids and get them intrinsically focused on what I would like my kids to get out of my lesson. This can be done hi-tech, lo-tech, or no-tech. I love to use technology (which includes most multi-media) to motivate because it has this incredible power to engage and motivate quality learning. But like any other learning object it can fall flat on its face if not done well.
I’m sorry I will miss today’s Math 2.0 session. I’ll be on my way to do time on the beach at the Jersey shore this week, but I’ll pick up the archive as soon as find some Wifi.
Dan MeyerSeptember 4, 2010 - 5:31 am -
@Sue, see, I have no doubt that a problem can reveal its constraints quickly and clearly without the aid of photo or video. Polya illustrates that. My question is: how do I do that with problems like the ticket roll or the water tank or the runners or et cetera, without using multimedia?
Maria DroujkovaSeptember 4, 2010 - 6:22 am -
@Josh, @Dan (#1, 2) – My reply was identical to Josh’s – use the physical object itself. Of course physical media isn’t embeddable, linkable, remixable or reproducible in the ways digital media is. My answer to the concern of scarcity/reproducibility is, “Don’t aggregate learners in huge groups that can’t share an object sensibly.” Even “World of Warcraft” – a good model of scalability, and a digital world – caps most coherent content experiences at 5 to 25 people at a time.
There is one thing you can’t reveal constraints without, and media isn’t it. It’s geeky love. It leads to immense amount of attention to the features of the subject, and to noticing and categorizing properties by importance. This includes intrinsic (to the subject of love) ways to reveal constraints.
For example, paper folding provides great problem-posing opportunities to those who love it. Reference: http://www.youtube.com/watch?v=bzF_YldL-jU
sylvia martinezSeptember 4, 2010 - 8:43 am -
@dan I know you would agree that multimedia certainly can’t be the vehicle for every interesting problem, so maybe it’s just the right medium for the problems you find most interesting?
Mr. LovellSeptember 4, 2010 - 9:44 am -
Multimedia lends a (more tangible) richer experience of the problem than word-only questions.
More importantly, most students IMHO don’t care about planes in space (maybe rockets in space) – where’s the buy in?
These are not math undergrads who’ve declared their love for math is so strong they want to pursue it for 4 years. If you’re teaching a class of gifted / advanced / eager students who love to debate and dissect abstractions then fine, but for the regular high school kid – make it relevant.
Dan MeyerSeptember 4, 2010 - 10:54 am -
@sylvia, I hope I’m controlling my selection bias better than that. I think multimedia is the right medium for the entire genre of “real-world application problems.” It isn’t the best, but I don’t know another medium that splits the difference between cost and quality so well.
Right now, in Sue’s session, we’re solving the problem: “what is the fewest number of cuts you’d have to make to a circle to split it into seventeen pieces?” This is an interesting, challenging problem that requires no multimedia, which is fine as far as it goes, but how far does it go?
sylvia martinezSeptember 4, 2010 - 11:12 am -
I agree that multimedia lends itself to a certain class of problems. And that you’ve done some really good exposition of that, especially in contrast to the way textbooks spoonfeed process.
I guess my biggest qualm is that the “real world” is 3D, and multimedia is essentially a 2D representation of that world. I completely understand that it matches the math we teach in school, so that works. But there are some things (like the paper folding example above) can’t be done unless you get some paper out and fold it yourself. Now, whether that translates to the math we teach kids… that’s a whole ‘nother question.
That said, I’d be happy to see ANYTHING interesting done in most math classrooms. This conversation is quibbling about what the edge of the envelope looks like in changing the way kids learn math. It’s fun to talk about with the folks who are immersed in the attempt to change, but honestly, I am NOT complaining about anything you are trying to accomplish.
Maria DroujkovaSeptember 4, 2010 - 11:18 am -
@Sylvia – the chances of “anything interesting” happening are larger if technology is present in any form.
JuliaSeptember 4, 2010 - 3:28 pm -
Planes in space or ticket rolls – I don’t see the buy in. And I did spend 4+ years in university math, and I do enjoy the beauty of the completeness of real numbers, and many other abstract things. I just don’t think this type of specific problems are interesting.
Honestly, by fighting so hard to make math interesting (content wise) for the students, we’re only at most making it slightly more tolerable. I think it’s the process that’s the key, and the ticket roll image is a great layout of constraints and a base for questions – so it opens up for some real problem solving. A well-crafted mathematical problem (planes in space, for those who care about such things) does the same, without the multimedia. But maybe the multimedia puts a different twist on things and variety is nice?
I just don’t think we should overestimate the importance of students posing their own problems, buying in, as it were. If they do, great. If they don’t, OK. Mostly, in math, they do it to please us. I teach psychology as well as math and it’s worlds apart. In psych, they’re dying to know things. In math, at best they’re enjoying the challenge of problem solving. I don’t see that changing, multimedia or not.
Sue VanHattumSeptember 4, 2010 - 5:11 pm -
I am so glad to see this discussion. I’m just soaking it in.
I’ll keep trying to pose whatever intriguing problems I can, in my classes, and when I get the “smart” classroom, I’ll do a few of Dan’s snazzy ones, and see how they work for me. I’m looking forward to adding those into the mix, but I know I don’t love them like Dan does.
To me, it’s not which is better, but which you resonate with more. I think the teacher’s enthusiasm makes a huge difference in helping students get over the hurdles (of real thinking) that math poses.
Maria DroujkovaSeptember 4, 2010 - 5:29 pm -
@Julia – A tiny minority of people are hotly interested in any particular context at any given time, including math. So, “math as its own context” – aka intrinsically interesting problems – will probably interest 3-5% of people, just like any other context. Others won’t do it if they have a choice.
Good news is that we can, first, use metaphors – math in other contexts – and second, help people choose their own contexts, because math is everywhere.
I just blogged about it a couple of days ago, with examples and diagrams: http://www.naturalmath.com/blog/its-own-context/
sylvia martinezSeptember 4, 2010 - 8:20 pm -
@maria – in my imagination, any technology makes things more interesting. Unfortunately I’ve seen some really bad uses of technology that make me temper those sentiments.
I was at one meeting at a nationally known ed-tech group and they shared their math lesson plans that were supposedly tech-infused. They were so horrific it would have been funny if it weren’t sad.
One example had kids use Microsoft Word Art to make geometric shapes that they could then print out and write the names of the shapes on them. That one made me laugh out loud, unfortunately, I was the only one laughing.
JuliaSeptember 4, 2010 - 11:43 pm -
@Maria – Yes, I completely agree. Just like to add that far more than the 3-5% do enjoy problem solving – it’s simply intrinsically fun to see oneself grow, especially together with other people. That’s why I’m calling for more effective problem solving guidance, instead of searching for interesting content.
Maria DroujkovaSeptember 5, 2010 - 3:26 am -
@Sylvia – Lol, people find ways to make anything boring and meaningless. I feel, however, that some tools raise the number of interesting events happening, on the average. For example, Scratch from MIT, whenever people use it; it’s hard to spoil it completely, for several reasons.
@Julia – What percent of people choose non-applied problem-solving as their regular (at least weekly) activity, selected among all the other contexts freely? This is my definition of “like.” By this definition, my estimate is 3-5%
There is a tremendous amount of coercion around math, because it’s used as a gatekeeper left and right. Remove that, and the picture of who does what changes rather significantly.
Sue VanHattumSeptember 5, 2010 - 9:24 am -
>What percent of people choose non-applied problem-solving as their regular (at least weekly) activity, selected among all the other contexts freely? This is my definition of “like.” By this definition, my estimate is 3-5%.
Maria, don’t the people doing Sudoku and KenKen – and other puzzles – count? And people playing chess? I’d expect the total to be higher than 5%. (Although perhaps not more than 10%. And of course I’ve been known to be wildly, inaccurately optimistic.)
Maria DroujkovaSeptember 5, 2010 - 9:48 am -
@Sue I don’t consider puzzles like Sudoku or games like chess “non-applied problem-solving” for several reasons. I have an essay on problems vs. puzzles somewhere, if you are interested. If we include these classes, the number is higher, of course, and it’s a matter of definition whether to include them.
bbSeptember 6, 2010 - 6:46 am -
Maria, can I read your essay about problems vs. puzzles?
Sue VanHattumSeptember 6, 2010 - 7:50 am -
Me too. (I’d like to read your essay, Maria.)
Until about two years ago, I wasn’t doing much mathematical problem-solving, certainly not as a regular/weekly activity. If there was something interesting that came out of what I was teaching, I’d play with it. But that didn’t happen often.
It took finding a community of people online, who enjoy thinking together, to get me into doing math for fun on a regular basis. Before that happened, I played free cell and sudoku when I felt like exercising that part of my brain.
The biggest change for me was going to the Math Circle Summer Institute. For a week, I got to have this kind of community, and it was in person!
josh g.September 6, 2010 - 8:05 am -
Sorry I’m late to replying. My thought for this one was, have students in groups of 4-ish, and have one roll of tickets per group (or one for every two groups if they’re hard to come by). That should strike a good balance between accessibility for the students vs cost.
Don’t get me wrong, I agree that good use of media is a great tool to make data and reality accessible in the classroom. But the roll of tickets example is relatively easy to swap in direct access to reality. I tend to think that if you can bring in the real thing without too much fuss, that’s going to trump the indirect representation most of the time in terms of engagement and usefulness.
Obviously this doesn’t work with everything. I’m not bringing in a half-dozen hoses and giant water jugs. Plus, it may be harder to set those specific constraints that make the problem interesting (ie. pressing the pause button at the right time). And with an actual roll, you have to convince them that they’re not allowed to just, you know, unroll it or something.
Also, reality is easier for students to work with as they don’t have the intermediate step of dealing with scaled photographs. (Which is either a pro or a con, depending on what you want them to practice / demonstrate.)
RebeccaSeptember 8, 2010 - 9:03 am -
A few of you came close to actually stating the fallacy of composition I see with multimedia: It, too, is a representation of a reality. The multi-media representation of tickets or escalators (which I personally love) is still a step away from the physical reality. If our goal is to get as many students involved, multimedia is one way. Another way is having smaller teams/groups work on different physical objects, perhaps in a linear fashion (each group works on objects 1-5, in some pre-determined order), where they can compare their own approaches and outcomes.
And learning which consumes the product during the learning experience (cut up a paper circle…oops, wrong number of pieces…cut up another one…oops, wrong again…cut up a third one…) teaches some valuable lessons about why we need to be able to represent some objects in order to measure, learn, test them.
When possible, use the physical objects. When that becomes impossible (which is often more a limitation of our creativity than a genuine restraint) or harmful (we don’t destroy functioning buildings to measure them), use the best representation possible…assuming the technology is available to your students. And it still is not in many schools.
Dan MeyerSeptember 8, 2010 - 7:01 pm -
IhorSeptember 9, 2010 - 7:41 am -
Sylvia wrote: “I guess my biggest qualm is that the “real world” is 3D, and multimedia is essentially a 2D representation of that world.”
That reminds me of the difference between live theater and film. I almost always prefer film to theater which is maybe why I like multimedia so much.
@Sylvia – I dont know if Gary told you but I’m “interviewing” him on Math 2.0 Elluminate on 10/13. Maybe you want to join us for what promises to be a lively discussion of our vision of math learning in this Web 2.0 world?
ChristineSeptember 11, 2010 - 5:26 pm -
“If the only tool you have is a hammer, then everything looks like a nail”. Just as a carpenter knows when to use a hammer or a nail gun, as teachers we need to evolving our repertoire of tools so that we can bring out what tool is appropriate to maximise learning. I think it is our job as educators to look at the way our students learn (visual, auditory, kinasthestic..) and teach to that, with passion!
Maria DroujkovaOctober 1, 2010 - 11:16 am -
@bb @Sue I totally forgot to link the problem vs. puzzle write-up I put up that day three weeks ago. Well, here it is: http://www.naturalmath.com/blog/problems-and-their-camp-followers/