Saturday, May 14, 2011

Which problem is "just right" for you?

Today on my Twitter-stream there was a great deal of discussion about how we use homework in math class. Being a university math educator, I have more freedom than most K12 teachers when it comes to assigning homework but that doesn't mean that I am any less concerned about this topic. Homework is a important memory for many math learners and we teachers need to consider carefully how we use it.
If our schools require us to assign homework from the text but we have some freedom in what it looks like, it is time for us to work our magic as problem solvers. John Golden and I team-taught a course for preservice teachers a few years ago and modeled some ideas of adding to pre-existing items from a text. We called them "just right" problems in reference to the NCTM article Vygotsky and the Three Bears. Here's what the original items looked like:
And here is how we tried to make them more thinker friendly:

Using these existing textbook items to demonstrate the processes associated with doing math. Pick one or two of the following to explore:
• Do either the odds or the evens – your choice. Why did you pick the evens (odds) to work on?
• Look over all the items. Which five do you consider the easiest? What makes them easy? Which five do you consider the hardest? What makes them hard?
• Pick an addition item that is just right (not too hard and not too soft). Solve the item using two of these three methods: using manipulatives, drawing a picture, or developing a real world context. Compare the two methods you selected.
• Pick a subtraction item that is just right (not too hard and not too soft). Solve the item using two of these three methods: using manipulatives, drawing a picture, or developing a real world context. Compare the two methods you selected.
• Pick one item to solve and write a metacognitive memoir that describes your cognitive efforts.
• The answer to one of the items is nineteen-twenty-fourths; which item is it? (Be sure to keep a record of your thinking)
• Which answer is closest to one? How can you be sure? (Be sure to keep a record of your thinking)
• Put the items in order based on their answers from least to greatest. (Be sure to keep a record of your thinking)
• Write your own ‘just right’ problem related to this review.

Which problem is "just right" for you?

Is there another problem we could add to this list?

1. What do you think of something like, "Calvin got an answer of 27 for question XX, but the answer key says that that's not right. Did Calvin make a mistake? If so, what was it? If not, why does the answer key disagree?"

2. shiftingphases,
Nice! I might tweak it a bit by asking the learner to respond to Calvin directly. Then, the learner has audience AND purpose in the activity.

Your's is definitely a problem I will add to this list. Thank you.

3. I absolutely love these questions and would like to try them in my class. Can you tell me if you have to model student responses to these type questions, though? I teach Algebra 1 and I've noticed the most of the students refuse to think for themselves when I ask questions that require them to think. I am afraid that if I gave an assignment like this, most of them wouldn't even attempt it. Any advice???

4. I like to challenge my students to find out WHY their result might be different from that of another student or group of students. If the question is open-ended, there is no one "correct" response; if question is not open-ended, then the students have explored reasons for inconsistency on their own......much can be learned by understanding why something DOESN'T work. Mistakes are good IF there is a WILL to explore what went wrong.

5. Becky,
You've got it right - approaches to answering these problems will take modeling. Take any one of the problems and chances are we modeled solution methods in prior lessons before putting them on an assignment.

For example, in the "answer is nineteen-twenty-fourths" problem most learners think this means trying to solve every item until you get that particular answer. We acknowledge this as a possible approach but wonder if there is something that requires less brute force. If learners don't come up with a more reasoned approach, we share our ideas using a think-aloud. After all, teachers are a part of the learning community.

Bottom-line, this is not the first time our learners have seen many of these problems. Therefore, they tend to choose those problems they can be successful on. I hope that helps.

6. I often take an approach similar to shiftingphases.com I tend to be more 'evil' by giving the students all of the answers (except maybe the T/F question in this case!) and telling them exactly one answer is incorrect. In the types of problem you set out here, the 'thinkers' can often decide which answer ought fit each question before they start working out each individual answer.
For example, if two of the possible solutions are above 37, then one of them must be for q10 and the other must be the single incorrect response...
Thinkers will then check that all of the other answers are correct. Guessers will probably not! I've found this can be a good way to encourage thinking about approximate size to see if an answer is reasonable, and it also gives the calculation part of the problem a 'purpose'.

7. Those are some great tasks! I especially like the "closest to one" and "nineteen twenty-fourths" ones. It's awesome that you shared these with your pre-service teachers, both as mathematical questions and as assignment types. Hopefully the choice aspect will trickle down another layer to their own future students. Having students pick from in-the-same-arena but hard-in-different-ways tasks is both fruitful and motivating. I'll be using some of the above and things inspired by them with my fifth graders this year. I'm intrigued by the pairing of "do the odds" and "find the one with answer x." Thanks for sharing!

8. Especially in later school years it could be both useful and interesting to inquire from students:

1) Which exercises did they think were there to teach some new small quirk and which were there just to reinforce some known method?
2) Which tasks do they think would be something that could come up in a test? What is crucial and what is not? It would be nice if more students knew better what the curricula are about...

9. David, this is a great approach to making homework or classwork from a text more of a thinking exercise.

I see that you, like me, teach preservice teachers, for whom these are great questions. Could younger students in elementary classes cope with your questions, do you think? I'd like to hope so, but have you tried it with youngsters?

1. I have used this approach with some fifth graders with success. They enjoy the novelty of doing something different with their homework and having some say in their assignments. The questions (and number of choices) would certainly differ given the learners' prior experiences.

Some of our preservice teachers have also tried this method with their learners. They have found it to be rewarding, as well. But this is often occurring in a non-traditional classroom setting.

With that said, no, I have not done any formal research with this approach.