I am at the NCTM Institute about Infusing Reasoning and Sense Making into the Classroom in Orlando this week. I have been looking forward to this workshop since I heard about it at the NCTM National Conference at the end of April, especially when I read that Dan Meyer was the opening keynote speaker. I have to say I was not disappointed and was actually very pleased. If you ever have the chance to see Dan present, go. Well worth the price of admission.
Rather than sum up everything he talked about, I will hit some of the high points. Some what he presented are parts of blog posts he has done this summer.
Dan opened with the story of his favorite horse, Clever Hans. If you are not familiar, Clever Hans is the horse around the turn of the century who could predict the answer of any mathematical question. When researchers tried to figure out what was so special about Clever Hans, they figured out that if the person who questioned him didn't know the answer or wasn't visible to Clever Hans, he would miss the question. Basically, what Dan is getting at here is something most of us do as teachers is like what Clever Hans' questioners did - we give away whether a student is correct by any one of a number of visual or verbal cues. Ultimately, we are creating a bunch of Clever Hans' - students who can figure out if they're right, instead of Smart Hans', who can reason and make sense of what they are doing mathematically and find the correct answer on their own.
When Dan talked about the Clever Hans story and got to his point, I felt very convicted. I do this too. I need to get away from that and work on being less helpful. In a later discussion at my strand session (I'm in Intermediate Algebra), I was sitting at a table with Hank Kepner (past NCTM president). He was talking about a similar thing in his classroom when he taught. I asked him, what do you do when the student presents you with a correct solution. His response was to ask them why it was correct. He wants his students to be able to defend their answer and know why the answer is good - the process is more important than the answer. Between these two things, I realize that I really need to learn about questioning and how to be more effective in the classroom. Less Clever Hans trainer and more a guide. Need to figure that out, and soon.
Enough chasing rabbits for now and back to Dan. He then went through a bit about curriculum and talked about application problems - problems that are practical (when am I ever going to use this?) and/or explanatory (even though we aren't using math, math is using us). For practical, he uses the example of helping his parents figure out whether it was cheaper to drive and park at the SF airport or drive part-way to (I forget where), park and take the shuttle. For explanatory, he uses the basketball problem.
Another point Dan made was our textbooks are set up to connect problems to previously worked examples – our textbooks even tell us where to find the examples. Too fill-in-the-blanky. Don’t reduce math to filling in blanks and working fixed examples. This is something else I am guilty of. Give the example and problems are similar to them. I had already read part of his blog post on this and had already felt convicted by it. This reminded me that I really need to do something about this. By the time Dan was done showing the Little Big League clip about the house painting problem (the one where the team is trying to help the young star with his math homework problem) and Dan closes by stating that this clip is emblematic of what concerns Dan about math teaching, it has really hit home to me that I have been that math teacher that concerns him about math teaching. I have done all these things, and even worse, I have tended to skip the application problems in favor of getting through the material. But I also know that this doesn't work and I need to change. That's why I am here in Orlando. My sincere hope is that I learn how to make the changes I need to and improve my math teaching.
The next section of Dan's talk is about how to set up the application problem. He takes a typical problem and breaks it down. In most textbooks, you have an image (maybe some clip art), the setup of the situation, the given information, some steps (textbooks tend to either do the process for the student in the steps or make the problem more difficult), and finally the task. Dan breaks this down into 4 steps:
Visualize the problem (here's where the multimedia we are familar with comes in)
Abstract the problem (what information do we need to solve the problem - textbooks give them this)
Decompose the problem (textbooks do this for students all the time - these are the individual parts of the problem)
Verbalize the problem (this is almost always the last thing in a textbook and it should be the first thing)
My notes as Dan talked through this process:
Verbalizing the problem is the very end of the problem in a textbook but in our head at the very beginning.
In textbooks, problems are decomposed to the point that are bite-sized and each point is trivial.
If I suggest the formula to the students in the problem, both teacher and student lose. Let students decompose the problem.
Separate the tasks, just like they are in life.
We have made math boring and challenging in all the wrong ways. Math is boring when we just give the students the formula, they know how to use it and they have the answer. It is challenging in the wrong ways when students have no idea where to start because they don't know the formula.
Dan then walks through taking a textbook problem and revamping it. He takes a fairly typical perimeter problem (it involves building a fence with boards that are spaced 1/4 inch apart and it has a very typical drawn picture - you all know what I am talking about) and then proceeds to revamp it. He starts by finding a better visual using Google Maps. Then he draws in the fence and asks for any questions. You all know what the question is - the picture draws it out. Then he asks what do you need to solve the problem? He fills in that information and then you have at it.
Dan has blogged before about the whole idea of three acts to a problem. I previously got the 1st act and mostly got the 3rd act, but really struggled with the 2nd act. I guess I didn't really understand what all we provide in that part. I knew that the 2nd act was where the problem was solved, but I guess I wasn't totally clear on how much (or how little) information we provide to students. Watching Dan today transform ye old boring textbook problem into something more interesting helped make the whole thing click for me. It also made me feel that this was something I could do. I have read bits and pieces of his blog before and to be honest, it is a bit intimidating to see all this great and wonderful math that he is pulling together with multimedia and real world context. I tried to play around with the whole WCYDWT idea (see Wii Bowling) and didn't get anywhere I thought I would with it. The whole experience left me somewhat frustrated and still intimidated. Then I did a foray into #anyqs with the comparison of different maps for our Houston trip (which if I were to do again in my class I would revamp more towards what Dan did at the beginning with his parents' trip) and felt like I had a better grasp on how to do the 1st act of the problem. When I read his post on the three acts of the problem after I had done the Houston #anyqs thing, I was back to being confused. At the moment, I am at a point where I feel I can at least attempt this whole idea of finding something real-world and setting up a problem based on something I have seen in a text book. I am still not sure I am ready to come up with something totally on my own.
I do have to say that since Dan posed the original #anyqs challenge, I am finding more and more math around me. It pokes in my head when I least expect it (and sometimes that's a real curse, gosh darn it!). However, I have not gotten to the point where I can find that stuff and it's something accessible to my students. Maybe I just don't give them enough credit. I've never taught like this. I have said many times, I am very much a traditional teacher. Putting something in front of my students and looking for questions and having them solve it (and be successful) is not something I've done before. This year will be the first time I have done that to completion ever (and I have just finished my 19th year of teaching). Probably the biggest thing that scares me is that I don't feel like I know how to use these types of problems. Do I put them in front of my students before they have had instruction on the mathematics they need to solve the problem and let them have at it? If I do that, will they say to me at some point, "Mrs. Henry, we don't know how to do this. Can you teach us?" Do I let them come up with their own methods to do that, Common Core Curriculum be damned? (I think that's what I'm "supposed" to do - not just teach them whatever the skill is.) Where does the instruction part come in? And so the questions go on in my head...
Dan also did go through the pyramid of pennies problem that @dandersod originally brought to the #anyqs thread on Twitter in May or June. It was also really cool to see this problem discussed in the same manner. It was also good to see how Dan would present the information and the answer.
Although Dan never used the whole idea of "three acts" by name, his presentation (to me) went through the three acts. I did find it easier to grasp. At the end, he talks about verifying the answer and whether students "buy" the answer. The analogy he uses is if he's selling you a knife and he tells you that it cuts through anything. You say, I have an old shoe, show me. He says, it says in the pamphlet it cuts through anything. But you want to see it. He keeps says that the pamphlet says it does, so that's good enough. We do the same thing when we give kids the answer (or have them look in the back of the book). It's better to show them the answer.
All in all, an excellent presentation by Dan. I left it feeling that this is something I could do, which I haven't felt before and that I have a minimum starting point (my textbook). I also know I have other places I can look (internet, news, and of course, the world around me) - I just have to get better at it. Most of all, I have to not give up on it. That is going to be the hardest part. Thanks Dan, for an excellent presentation. Well done.