## Saturday, August 27, 2011

### Have you ever ... ?

The #mathchat topic for August 18th was "How do I get students to take the first steps in problem solving?" During the course of the chat, I asked the following:
Here are some of replies:

With the exception of @kassiaowedekind, very few details were provided. This is understandable given the limitations of Twitter, but I thought it might be worthwhile to continue the conversation using a venue that allows for more explanation. I will get us started with a couple of ways I have approached this question and then others can add their ideas in the comments.

The first approach I use is trying to avoid asking questions to which I already know the answer. I learned to do this while working with the first-graders in my wife's classroom. When I asked, "What is four plus three?" the first-graders typically responded with the expected answer and then waited for me to validate it. This was unrewarding to everyone involved. They were becoming reliant on me as the authority, and I really wasn't getting any insight into their thinking. Maybe they just knew the fact. Or they could have counted all, counted on, used doubles plus or minus one, or some other strategy. Or maybe they answered seven to every addition fact.

Instead, I decided to ask the first-graders, "How would you figure out four plus three?" Their answers to this were much more interesting to me. Although I had some ideas of how they might respond, I could not be sure and there were times I was surprised. Also, the first-graders always enjoyed sharing their thinking with me - much more so than just giving an answer.

The second approach is to share a problem I have not solved. I have some ideas of how to solve it but I have resisted the temptation to actually work on it. That way, when my learners ask if their solution is correct, I can honestly say, "I don't know." Then, they have to share their thinking, not just their answer, in order to convince me.

One of my favorite problems to ask the preservice teachers in my Teaching and Learning Middle Grades Mathematics course is, "When does 3/19 repeat?" We have tried graphing calculators and spreadsheet to no avail. As a result, we end up solving simpler problems looking for patterns and making conjectures. (I am told Wolfram Alpha can compute it but I refuse to spoil the fun - for now.)

Now it is your turn. Have you ever given a problem you didn't know the answer to? And if so, what was it and how did it go?

1. Yes, and I blogged about it:
http://oldmathdognewtricks.blogspot.com/2011/04/my-first-foray-into-wcydwt.html
http://oldmathdognewtricks.blogspot.com/2011/04/wcydwt-wii-bowling-day-one.html
http://oldmathdognewtricks.blogspot.com/2011/05/wii-bowling-day-one-revisited.html
http://oldmathdognewtricks.blogspot.com/2011/05/wii-bowling-day-two-frustration-on.html

2. "A lie detector test is known to be 80% reliable when the person is guilty and 95% reliable when the person is innocent. Assume we have a suspect from a group of 100 people, only one of whom has ever committed a crime. If the test indicates the person is guilty, what is the probability of
innocence?"

I found the formulation of this problem online, as do many teachers searching for problems. Ran out of time to solve the problem before class.

The kids were very unhappy I didn't "have the right answer".

I think this exercise would have been so much more powerful if we'd had a way to independently verify the correct answer. Not an answer key but some kind of real confirmation.

3. I think it's important to give students problems for which their is no answer, or at least not one answer.

Students seem to think that mathematics is about finding the correct answer, and not enough about communicating what we know and presenting possible answers, depending on what assumptions we make.

4. I wrote a simple Excel spreadsheet to convert fractions to decimals. http://dl.dropbox.com/u/39561574/Fraction_To_Decimal.xls

5. @mgolding: This would be a great problem to simulate, if some of your students know how to program. Here is a simulation that I did in Python: http://codepad.org/AvVW8Rtu . The results are close to the theoretical probability of 99/115.

6. I've asked several of my classes things like:
how much lego would you need to build a house?
How long will it take you to count to a billion?
Which is taller - a basketball net or £1million?
How many golf balls could you fit in this classroom?
How many balloons would the house in UP need to lift it?
How heavy is the school?
What's the longest word you can write that has one line of symmetry?

After a while, they've stopped stressing about the "right" answer and started looking for ways to justify their answer.

7. I think this is a very important practice and can teach students very important lessons. First of all, they need to learn that not every problem has one answer and one way to find that answer. Second this practice awards them the opportunity to learn that the teacher does not have all the answers and instead we can teach each other things.

That being said, I personally find it difficult to give students a problem I do not know how to answer just because I am such a planner. However, I have sometimes learned to ignore my discomfort and try this approach with students. It usually works out well and many times I like to try and solve the problem and work with them. I have mostly done this with critical thinking warm-up problems such as a Sudoku puzzle or some type of brain teaser.

8. There was a project in the 90's called Mega Mathematics which took a third approach: Offering students problems that haven't been solved by anyone yet.

As far as I can tell now, all the contributors have moved on to different projects, but their work is still available at:

http://www.c3.lanl.gov/mega-math/welcome.html