Before our day had started I had 2 students anxiously waiting for me with all these papers in hand. They both started talking at once unable to restrain their excitement and questions, waving their investigations in the air.
Once they slowed down ,they both had decided that multiplying the whole number by the denominator then dividing by the numerator worked…with non unit fractions AND surprisingly with unit fractions as well. They had tried it with different problems and were able to use models to prove.
How exciting, students were actually hooked, went home and explored their thoughts…without the guidance of a teacher prompting them, and couldnt wait to get back together to talk about it. One student was very proud of herself, and her understanding and walked away from the conversation… happy and satisfied. However the other student remained.
He said, “This is so neat and I think I get it, but Mrs. Hogan I am not really sure it works with ALL numbers/problems.” Ok, so my heart starts racing, and I begin to panic. Where is he going with this, what if I can’t work him through it, what If I don’t understand what he is trying to explain to me. I take a deep breath, channel my inner “Caban” and say “Tell me more”.
Side note, channeling my inner Caban ” is a reference to our district math coach. I say it all the time for inspiration, and my students know what that means to me. She is an amazing, inspiring, math learner, math teacher and coach. She has supported me throughout my journey from lecturing, algorithming, worksheeting teacher, to where I am today. I am looking forward to continuing this journey, knowing I have her continued support for when it gets messy. Great lead in… messy…
My student says… I don’t think it works every time. He shows me his thinks from last night.
I ask him to…”tell me more” He said “I don’t know, I’m not sure how to explain, can I give you an example. 5 divided by three fifths”. He said, “that would equal 25 divided by 3, if my conjecture is true, and that just doesn’t work.” I asked “why”, and he just kinda looked at me. He replied “that would equal 8 and 1/3” . There was silence. I didn’t know how to respond. We sat there pondering in quietness. He said, “I can’t prove that with a model like I can with the other ones. Its messy”
I was so very proud of him, his perseverance, his willingness to struggle and be ok with not knowing for sure. I could relate, I knew the place he was in because I was there with him, unsure. I dont think I was as comfortable with the struggle as he was, and I really wanted to make it not messy for him.
I wish this story ended with a magic model that we figured out together, or that I was able to guide him to resolution. But that would not be real, or the truth. We stayed in the mess, together, knowing that he made conjecture that is true, but messy, and we cant prove it all the time right now. Here’s the magic, we both left our math talk feeling accomplished and proud of all the learning that had happened. Neither of us left defeated or felt like we failed te math or each other. We left excited to learn more about models, and really prove his conjecture.