Math Make-overs

Robert Kaplinsky wrote a post about open-middle vs open-end problems that got me thinking.
The punch-line is that there are simple make-over tricks that you can use to convert almost any problem into the type you need, whether closed- or open- (or half-open) middle, closed- or open-ended.

Note: some of this transformation thinking is clearly inspired by Dan Meyer's "remove the information" method.

I think this post is, mostly, intelligible without reading Robert's note, but everything will make more sense if you have particular problems (and problem-statements) in mind.

Closed vs open middle

Closed-middle problems tell the student how to answer the question, either directing them to a specific strategy or scaffolding a specific approach through explicit interim steps. Open middle problems leave it to the student to find (or select) their own strategy.

Perhaps there is a half-open-half-closed-middle where students are given a menu of strategies?

I conjecture that note that any problem can be made into either type. Specifically, the ones in Robert's post all can, including calculating 475/25. The main technique to change a closed-middle to an open middle is to remove any guidance about the strategy the students use to answer the question. Going in reverse is also possible (tell them what strategy to use, scaffold interim steps) but there are already too many closed-middle problem presentations, no?

Closed vs open-ended

Is there one right answer/end result regardless of which strategy is used? If so, then it is closed-end.

Again, almost any problem topic could be either closed or open-ended, depending on whether it has:
(a) how many interpretations are possible
(b) how much data is supplied and whether that data is consistent.

It is easy to be surprised by problems with multiple interpretations where we only expected one. Recently, Marilyn Burns wrote a post about fractions that seems to have one right answer, but actually depends on how we define our reference unit. Popular probability questions seem ripe for interpretation-based disagreements.

If the question has just the right amount of data or all the data is consistent, then it will have one common ending result and is closed-end. If there isn't enough data or "too much" (some data is inconsistent), then it suddenly becomes open-ended since students have to use other ways to fill in the gaps or make choices between inconsistencies.

In Robert's post, there is a question about hybrid cars that is open-ended because there are some assumptions the students need to make and there are elements in the data that aren't consistent, so choices need to be made. This could become closed-end by adding more data (removing the need for student assumptions) and doctoring the data to make it all consistent.

In contrast, his example of In-and-Out burgers could become open-ended by taking away data or making the data inconsistent. My favorite way to blow open the end would be to go way beyond the assumptions of simple extrapolation: let's order a burger with 1,000,000 patties! How would they price that?

By the way, you might think some questions simply can't become open-end. For example, calculate 475/25. Is it possible to transform this one?

Well...we didn't specify which base we are in. In base 10, of course, we get 19. In base 8, though, the answer is 17 with remainder 2! I'm sure cheeky students out there could find other innovative ways to interpret the question, if given a chance.

Which is better

Sure, we can change the problems from one form to another, but which is the better type of question? Open-middle, open-ended, of course!

Just kidding. All of them have their place and it depends on what you are doing and why. Though it conflicts with my personal preference, I even see a place for closed-closed questions where you want the kids to practice a particular strategy (closed-middle) and you need the consistency of a closed-end answer to quickly check that everyone has gotten to the same result.

Open-middle, closed-end can be good for generating discussions that compare and contrast strategies. That's logical, since the strategies are where you would (should) find differences from these problems. However, aren't some of the kids thinking: "we all got to the same place, I don't care that someone else took a different route." That was something I often thought as a student.

When you have enough time, I think open-open questions create really rich discussions that include strategy comparison. Our typical conversation is something like:

• Hmm, we got different answers!
• What did you do?
• I used method A and data Z
• Oh, I used method B and data Y
• If we used method A with data Y, what would happen? Is that even possible? Why or why not? Would we get the same as (A, Z)?
• If we used method B with data Z, would we get the same answer as (B, Y)? etc etc

You can see I have a bias for open-open questions. One last reason: kids get a lot of closed-end questions already, so I don't feel that I need to add more.