October 10th, 2008
Since I have been diving into the research about open ended problems, and have been experimenting with them in my classroom, I decided I would again try to start my math class with a warm up activity that involved an open ended question. Again, I thought I would spend maybe ten minutes on this and then carry on with the intended lesson. Again, I was in for a surprise. I read it somewhere in the mountain of literature I have been swimming in lately that using this type of approach and asking open ended questions may often lead to places you would never dream of, to areas of the curriculum that is at the moment, untouchable. That is exactly what happened today.
I began class with an opening question on multiplication. I posed the question: Tell me what you can about 13 X 52. I wanted to gauge students understanding about number and the concept of multiplication of two, two digit numbers. Asking students to tell me everything they know about this sentence, I am opening it up so that everyone can enter the problem and everyone can offer some input. Compare this with asking students to find the product of 13 x 52. This open-ended problem allows all students to engage, which will give me vital assessment data into what and how much they know about important concepts. This assessment data will then guide my instruction as it allows me to see where exactly students are with their understanding and will guide me to plan my instruction beginning with where they are and leading them to where they need to go. I will use this to ensure the lessons I teach and the problems I pose in the coming days are right for the student, open so everyone can enter the problem and experience success, but also open enough so that I can extend the problem, or change it in some way to challenge the students.
So, once I posed the problem I gave students a cue card to write their responses on. As always, I gave everyone time to at least get something down on paper and then, as a class, we began looking at some student responses. I invited students to come to the board and tell the rest of the class what they knew about this multiplication sentence. I called on the more reluctant students first to ensure they would have something to contribute. They stepped to the plate and offered some very insightful responses. Some said things like, the product is greater than 100, it is an even number, it is a three digit number, and so on. When one of the kids said it was an even number, well, of course, I couldn’t resist. I then asked the question of how they knew it was an even number. The response, and I remind you this came from a low average student, was something along the lines of they seen that the ones digit was a two and knew anything multiplied by two was even. Holy smokes! This led into a discussion about if this was always true, which lead us to writing a conjecture (which I explained to students). They wanted to prove that whenever you multiply an even number with an odd number, it will always be even. I said go for it. I decided to leave that conjecture to the students and gave them the opportunity to take it home and work on it to try and prove or disprove it.
The discussion then turned to the student response of the product being a three digit number. Again, I asked how they knew this and some were not sure. This spawned a great discussion about place value and conditions of what the numbers had to be for the product to be a three digit number. This lead to writing another conjecture. Students wanted to know if when you multiply two- two digit numbers will the product always be a three digit number.
From here, which was now about 35 minutes into the class, we discussed this conjecture and began talking about numbers and how to go about multiplying 13 X 52. We discussed things like breaking the numbers down and multiplying the number parts to get the product. Before long, and in the middle of this discussion, a hand goes up – this caught my attention as this particular student does not usually offer input or comments in the middle of class. When I addressed her, she simply stated that she had disproved the second conjecture. I asked her to come forward and show the class how she had done this. At this point I did not care that she was not engaged in the conversation about multiplying parts of numbers! She came to the board and wrote out a multiplication sentence that read – 99 X 99. She quietly said in a matter of fact way that the conjecture was not correct because if you take the two largest two digit numbers and multiplied them together, the product would be greater than a three digit number. Kind of taken aback, I quickly asked her why she chose the two largest two digit numbers. She, just as quickly said “well, miss, if this conjecture was true for all two digit numbers, the greatest two digit numbers when multiplied together would give you the highest product, and this product is more than three digits because 99 can be rounded to 100 so 100 X 100 is greater than 999, so that’s what I did and I know that this does not work.
So much for a ten minute warm up! The point I am trying to make is this – if I were to simply ask the students to calculate the product of 13 X 52, I would not have been given this wonderful opportunity to engage in this level of discussion with this class, nor would they have wanted to talk about, let alone, work through this level of content. This is just another example of what using open questioning in math can do. These students walked away with a stronger sense of number sense and hopefully in multiplication. I walked away with achieving so many other outcomes than intended, and came away with very strong evidence of where students thinking is and a clear sense of where my teaching needs to go.
No comments:
Post a Comment