*love*the idea that you'd give the kids an open-ended situation and leave it up to them to show as much understanding as they can about the situation. Seems like something that works extremely well in physics -- a subject whose goal is to encourage kids to gain a multi-layered understanding of everyday situations using a combination of various "laws" and models. It's also really exciting to me because this process highlights (to the teacher as well as the kids) that conceptual understanding of the same situation will continue to build, as you accumulate deeper knowledge on how to drill down further into pertinent details. For example, take a situation where a kid is told that a hot air balloon flies through the air. At the beginning of the year, they might only be able to say that it's because the air inside the balloon is less dense than the surrounding air. A week later, they might be able to say that this is because the air is warm and warmer air is less dense. Another week later, they might be able to say that the warm air is less dense because the molecules have more kinetic energy and therefore create pressure and expand the volume of the balloon. Another week later they might be able to explain the heating mechanism and why hot air doesn't escape through the hole of the balloon. And even later, they might be able to predict what temperature the balloon would have to have in order to carry a certain amount of weight. (I'm not actually totally sure if they can do those calculations AND I'm pretty sure you'd teach those concepts at a faster pace. I'm just using those steps to roughly illustrate how an idea about the same situation progresses over time.)

It sounds like something I would like to try in my own classes, but two difficulties immediately come to mind:

1. Is it necessarily applicable to a math classroom? Although open-ended explorations in math are indeed possible and interesting, a vast majority of the math processes we practice in our classroom encourage kids to

*eliminate*cluttering information that they don't need. Whereas physics encourages the learner to take something very simple and expand it into something complex and multi-layered and to consider all factors involved, math (in my mind) is more like a funnel that gets rid of all of the cluttering details and focuses in on only the most important details. Even much of the WCYDWT stuff is inherently begging a certain question to be asked, and then more or less inherently requiring some specific math strategy (with some real-world messiness, of course).

Or am I missing something here and my view of the goals of math teaching is really much too narrow? (I'm thinking out loud here. Feel free to jump in. My mind really isn't made up about this one way or the other. After all, it's clear that the early mathematicians never limited themselves to thinking about the fastest way to get from point A to point B.)

Anyway, personally, I think there is always value in "playing around" aimlessly with a problem, even if it doesn't immediately lead you to something productive. Sometimes questions arise that way and other times solutions arise when you least expect them to, just because you've meandered your way through most of the issues. Past a certain age, all the problems that are worth solving are not solvable in your head anyway, so some playing around is usually necessary and giving kids these non-questions encourages that line of open-ended thinking.

2. Let's assume that my first point was moot and that this technique is entirely applicable to the math classroom. The beauty of this process, as I have described above, is that you can give the kid the same situation at the beginning of the year, the middle of the year, and the end of the year, and their understanding should continue to build upon itself and to encompass all existing knowledge, plus brand-new knowledge. What types of problems would I be able to give that tie together all of the things they would learn in the course of a year, so that they could demonstrate understanding at all different levels? (Here I feel that physics has another natural advantage -- a bunch of the ideas you learn in physics are all interrelated.)

I have no easy answers. I'm going to have to think about this one, so that I could possibly implement something like it next year.

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PS. Speaking of good test problems, I recently put a question on my regular Geometry exam that nicely incorporated some old and new concepts all in one shot. The question gave the kids 3 different y=mx+b equations to graph, and asked them to find the angles inside the triangle that is formed from these 3 lines. In order to do that, they needed to know: 1. how to graph lines (old knowledge), 2. how to use pythagorean thorem/distance formula to find edge lengths of the triangle (old knowledge), 3. how to use inverse trig to find the angles within the triangle (new knowledge). Synthesis!

initial thought, here. So, not sure how fully formed it will be.

ReplyDeleteI think that there is a way to do something like the goal-less problems in math, but it would look different than many of us are probably used to. I think the focus in a math class that uses goal-less problems would need to be on connecting ideas and using relationships rather than being extremely standards-driven. In other words, more focused on process standards than content standards.

The idea/example that came to my head was with modeling. If that is the big idea for the unit/month/semester/course/unit-of-time then you could look at a lot of different types of situations through a modeling lens. And then when students looked at a new situation they could start out by modeling it (graphically, in a table) and build off of what they already know, with tools they already have to learn about this new situation.

Oooo, I LOVE your test problem. Put a spin on it, have 3 or 4 different sets of 3 lines and have the kids pick out of a bag to see which one they get. Maybe not for a test problem, but for just a daily assessment.

ReplyDelete@betweenthenumbers A little more concretely, what about something like "Sofia left her house at 3:30pm to walk towards Jason's house at a costant 1 m/s, and Jason left his house at 3:35pm to walk towards Sofia's. He initially walks at 1 m/s, but every 5 seconds he speeds up by 1 m/s, until he is sprinting at 7m/s. Their houses are located on opposite sides of the park as shown in the scale drawing below." This problem involves linear patterns and velocity vs. time, distance vs. time, and d=rt, and basic ratios (in reading a scale drawing). Common algebra concepts... But, this is not really that interesting of a problem. Sort of just sounds like a word problem that got chopped off. :(

ReplyDelete@ER Thanks! I've only done the "hat" thing a couple of times, but kids did love the idea of picking things out of a hat. :) But, the problem for me is that as soon as they sit down to work on the problem, they usually realize that it's just another way to trick them into doing a regular algebra problem, so the "wow" factor wears off real fast with my kids. :P --More power to you if you have better luck than me keeping them charmed!!

@Mimi Avery over at Without Geometry Life is Pointless is trying "minimally defined problems." Here's an example. http://mathteacherorstudent.blogspot.com/2010/05/how-would-you-respond-to-minimally.html

ReplyDeleteI think in math, goal-less problems would be focused on defining relationships in a situation. It seems like if you gave a kid a situation (like the salt-shaker sliding along the table) and asked them how math related to it, the first impulse would be to just describe numbers or arithmetic. The first layer would be to describe relationships between concrete numbers, but the tricky part would be to push past that layer. So the next layer in the goal-less problem might be to define symbols for quantities about the situation (distance traveled, speed, change in time, friction force, angle of table, etc etc). And the next layer would be to come up with relationships between the symbols.

ReplyDeleteAnother key to goal-less problems is the use of multiple representations. So in math, you want to get the graphs, tables, equations, verbal descriptions, etc in there, too.

I'd be interested to hear if you end up trying something like this! When we finish spring break, I'll run this by my colleague who teaches a section of physics with me but also teaches two math classes and see what he thinks about goal-less problems in algebra and calculus.

@Jason Thanks for the link!

ReplyDelete@kellyoshea Maybe give the kids a word description of a pattern (in context), and kids can:

1. make a table (basic)

2. define all related variables and verbally state/summarize their dependencies (basic/intermediate)

3. graph relationships (intermediate? basic? maybe it depends on the topic)

4. write equations (intermediate / advanced, depending on the topic)

5. fully explain the meaning of the new numbers that emerged from Step #4 -- ie. the meanings of all calculated coefficients and constants. (intermediate / advanced)

6. state domain constraints of equations (intermediate / advanced)

7. further analyze inverse relationships, if one exists... or explain why the relationship has no inverse function. (advanced)

8. take two fully analyzed situations and draw comparison or combine them somehow to make further sense / draw further conclusion. Sort of like systems of equations type of stuff, where they can apply further math skills and make further interpretations. (advanced)

Thoughts?

Sounds like a good start. The hurdle is shifting the kids' focus from finding "the answer" to describing the pattern/relationships/situation. Once they understand the framework and practice it a bit, they will surprise you by including things you haven't thought of.

ReplyDeleteOne thing I have seen a bit of in my kids, but want to focus on more explicitly, is trying to use models or strategies from earlier material in a different context. Can you still analyze this problem in terms of momentum even though it isn't a collision or an "explosion"? I have sometimes stopped the kids from going down a path like that too quickly because it isn't "a momentum problem." But when I catch myself and stay out of the way, then they really start to play and figure things out. A lot of the time it is perfectly fine to use momentum (etc), and I want to focus on that sort of thing explicitly so that they can get a feel for when it is possible or not, easy or not, etc.

You can try doing some goal-less math problems on your own by just finding book problems that have enough information to play with if you get rid of the question. It's kind of fun to get into a "flow" type state of just seeing how much more you can show or find.