Arrrrghhh….don’t you hate it when students write down answers that are ridiculous and seem oblivious to the ridiculousness of it. I am at least happy when they write me a note that says “I know this can’t possibly be right but I can’t find my mistake…”. Then I know that they are at least reflecting on what they have done.
We have had quite a few conversations with teachers lately about students judging the reasonableness of their answers and how it is a source of frustration for many of us. So this is something I have been thinking about over the past couple of weeks.
Is it really their fault that they don’t do this enough? I have been reflecting on my own practice and specifically thought about:
- How often did I ask my students to consider and discuss what might be a reasonable answer before they began a problem?
- How often did I give them opportunities to share strategies for estimating what might be a reasonable answer?
- How often did I give them questions that ended up with unreasonable answers (because I just chose the numbers quickly without thinking) and when a student questioned the reasonableness I told them just to ignore it because I just made it up.
I also thought back to my own experiences with estimating in elementary school. I don’t ever think I was taught strategies for estimating. I actually recall, being asked to estimate when adding two 3-digit numbers. I think it was suggested that we round the numbers first and then add them to find an estimate. Unfortunately, the only strategy I really had for adding 3-digit numbers together was the traditional algorithm so it seemed ridiculous to me to round the numbers, line them up, add them together and then do it all over again with the actual numbers to get the exact answer. So I just would add the numbers together using the traditional algorithm and then round my answer and write that down for my estimate.
What started me thinking about this was the day we were in the grade 5/6 class from our last post. The teacher started the the lesson by asking the students the following question:
” A parent brought in 100 timbits for our class of 22 students to share. Would every student get at least 2? Would every student get 10?”
He had students think about the first option – Would everyone get at least 2? Students shared their strategies for checking this: some skip counted, student by student by 2’s, others were comfortable just doubling 22 and saying that there would definitely be enough. When asked if everyone would get 10, some students just multiplied 22 by 10 since it was a nice number to multiply by, while others counted by 10’s around the room and realized that they would hit 100 too soon.
This approach allowed students to say that the answer should be between 2 and 10. Several students talked about how they could narrow it down more and were comfortable saying that it would be between 4 and 10.
I started thinking that this is a nicer way to think about estimating. Why when I was estimating in elementary school did I always have to come up with a single number for my estimate. In fact, I think that is part of what bothered me – I knew that no matter what I wrote down it would be wrong. I think I would have been much more comforable coming up with a range of what a reasonable answer would be: “my answer should be between 100 and 150….”.
Dan Meyer’s Three Act Math videos encourage this idea by often asking students to make predictions and write down an answer that they know will be too high and too low.
We tried the You Pour, I choose problem in a class yesterday. However, this problem starts by asking students to guess which glass has the most pop (or soda in US english 🙂 ). I tried to get students to write down their guess on a whiteboard. It was really difficult to get them to commit to one. I still think that having to write something down, that might possibly be wrong, is what was holding them back. It is amazing how the fear of being incorrect in math, even when just estimating or guessing, stops students (and adults) in their tracks.
So, this is something I am going to be working on and thinking about more.
Intentionally talking with students about what a reasonable answer might be, before they begin the problem, and sharing strategies for coming up with their estimate. I also want to test my theory that, when possible, asking students to come up with a range for their estimate, instead of a single number, might make them more willing to commit to putting something down on paper before they begin.