We started by analyzing this problem.
As we discussed the problem, I heard "oh, that's not challenging" and "I know the answer" and some even calling out the answer...mostly saying 10 or 11. I moved forward, undeterred by my less than enthusiastic audience. I told them that I was not really interested in the answer but in the strategies and/or tools they would use to solve the problem. This is the list they came up with. I resisted any comments or judgement on their ideas busily recording their ideas on the board.
I asked each student to commit to a plan, then partnered everyone up as best I could. We grabbed the needed supplies and students got to work. I was disappointed at the lack of discussion between many of the partner groups, hoping for more 'scholarly' discussions and sharing of ideas. (Note to self...model and provide more opportunities for this.) Most groups finished quickly and were sure that their answer was correct... rejecting any suggestion to reread the problem and analyze further.
These are their solutions:
This group chose to create a chart. Their thinking is clear, showing the original 5 friends and each of their 5 friends. I love the correction!...adding Miss Spider and changing 30 to 31, revealing that these students checked their work and revised their solution. :)
These two solutions appear to have a similar misconception. After interviewing the kids, I think they counting the original 5 friends as groups of 5. What I love about this solution is that these students recognized the 'fiveness' of the problem and immediately went to showing groups of 5 and counting by 5's.
These solutions resulted in the same incorrect answer. Interestingly, these students were the ones who were calling out that 11 was the answer when the problem was first presented. I believe that once they thought they knew the answer, they went about showing a strategy that proved the answer that was in their head rather than analyzing the problem and designing a strategy for solving.
This solution exactly illustrates why I want to find 'meaty' problems for students to solve. I think these students are used to math being 'easy' and they often arrive at answers without actually having to think too hard. They need to be challenged more regularly so that they are in the practice of seeing analysis as a 'must' when solving problems (my job). I also think that 'reasonableness' needs to be a bigger part of our conversations as initial discussions around the problem resulted in most students agreeing that there would be "a lot of spiders" at the party. 11 is not really a lot.
This solution had potential in the early stage. A tree map is a very effective solution strategy for this particular problem. After interviewing these students, I think they simply didn't assign 5 to the original 5. Perhaps they also had the preconceived solution of 11 in their heads. Note to self: Teach visualization of the problem as an initial step in the process then check against reasonableness as a final step.
These two 10 frame solutions are the same. One group used 'real' manipulatives and the other chose a digital tool. They clearly saw the 'fiveness' of the problem. I must admit, using a 10 frame would never have occurred to me and my OCD tendencies absolutely LOVE the color coding of the solution!
As a class, we shared and reflected on solutions and strategies and tools. Each group shared their product and their thinking. During this time, I asked a few times, would anyone like to change their thinking...improve their work. Some were very quick to recognize their errors and articulated how they would change things but some held out to the bitter end. I didn't get the feeling that they were really attending to the other student's explanation of their process and matching it against their own, but rather, thinking that their answer was correct and just waiting to be declared 'the winner'.
We talked about really imagining the problem...slowly and carefully. The focus was on efficiently and effectively finding a reasonable and preferably a correct solution. The last step involved all of us sitting in a circle and modeling the problem with some plastic bugs. I led the modeling revealing my thinking step by step.
So, upon reflection, I am encouraged that all solutions, correct and incorrect, showed a correct representation of the number they came up with. Additionally, I saw students incorporating grouping strategies and counting by 5s as they worked. I am also pretty happy with the variety of tools used and that students were comfortable knowing what tools are available to them and where to find them in the room.
What I need to work on...Creating 'meaty' problems. If anyone has a resource for this, please share:) Also, I need to work on helping students visualize/analyze problems as a first step. This is something I struggle with because I don't want to go too far and make the problem easy but I do want to scaffold toward deep mathematical thinking and ultimate success. I also worry that sometimes initial conversations make misconceptions public, shutting down original thinking or depth of thinking. The early on calling out of 11 as the answer may have contributed to some students taking that number and running with it rather than independently analyzing the information.