So as many of you know I am cutting out things I don’t need in a lesson or question (at least right away) in order to get students thinking and engaged in the question being posed to them. I ran across a great book, even before Dan Meyer mentioned it in his post. It is an old book called Problems Without Figures, written by S. Y. Gillan and published in 1909. The great thing about the book that I have seen myself is the stripping away of answer getting and the focus on problem solving. I gave my students the first problem of the book as bellwork yesterday.
If you know three times plus five times plus seven times a number, how do you find the number?
The first thing I will note is that it drove students crazy not knowing either the number or the total. I had many students argue with me that the problem was unsolvable, on the grounds that they needed to have one of those quantities. After that mutiny was quelled, my students divided into 3 camps.
Camp 1: My brain hurts
These students sat quietly and waited for others around them to find an answer or hoped I would discuss the problem as a class so they could quickly write something down and hand it in. They firmly believed there was no way they could solve the problem, and it took private conferencing of 1-2 minutes to get them to even start processing the problem in their mind. The odd fact I found is that in every class, these students were the ones to catch onto the idea and run so fast I could barely rein them in during discussion. I would say roughly 10-15% of kids from every class fell into this camp.
Camp 2: Looking at Multiplication
I had about 80% of my class in this camp. They at first were threw off by the language, but then would write out 3 3 3, 5 5 5 5 5, and 7 7 7 7 7 7 7. They would then add the numbers together to get their total. When asked how they came up with this method, they stated that three times meant three threes, five times meant five fives, and seven times meant seven sevens. They also thought of multiplication as repeated addition so they added up these numbers to find an answer. They were the ones who could call me over to check their “answer.” I would ask them what they thought, how confident in their work or answer were they? Many were unsure, thinking they did some good work but that they didn’t quite get the question. I agreed with their analysis, confirming that they had some good ideas going, but that they may need to reread the question and examine their starting point.
Camp 3: AlgeHeads
The remainder of my class (a very small amount) would somehow think of algebra representation. I would see a lot of 3x + 5x + 7x =? Some would go even father to say that it was 15x. That was the extent of the problem solving however, because they still felt they were looking for a specific answer instead of an explanation of how to calculate it. They were able to help guide the class conversation in working through the thinking and mathematics. They are the group that I feel has the best handle on the answer, divide by 15. They accept that you can generalize about a rule without having specific details.
The biggest challenge in this type of problem is that is focuses on student thinking, not answer finding. My students do not have academic stamina, and this type of challenge quickly frustrates them- causing many to give up sooner than if they are asked to find an answer. I do like the format of this question, it has cut out what I don’t need yet provided a challenge that created great student thinking. One great thing that came out of this is student understanding of multiplication, or lack thereof. Students only think of multiplication as repeated addition, so that limits their applications for it. I now understand why area and perimeter as such a problem for these students. If they associate multiplication with addition, then area and perimeter are adding tasks. I need to take time for students to explore multiplication and its different applications. Tomorrow’s problem has fractions, I can’t wait to explore student understanding in parts of a whole.