On Tuesday night, I blogged about a math activity that we did in class. Kelly McCrory, our fantastic math facilitator, responded to my blog post with this question:
Her comment got me thinking, and so on Wednesday morning, I created this new math problem that we did in class today:
This math problem addressed Kelly’s question, and I was very interested to see what the children did and if they could figure it out.
Students worked in partners with a bag of shapes (more than what they needed) and the math question. Just like with our previous problem, they could show their thinking on a white board, on the Livescribe Pens, or on various screencasting iPad apps.
When I started walking around talking to the groups of children, I noticed what I would have expected to see. Most of the students looked at the hexagon and the trapezoids, and figured out how to put two trapezoids together to make a hexagon. The majority of groups had more difficulty with the rhombuses. Some groups put the rhombuses down beside the hexagon and tried to move them around to create a hexagon. Other groups tried sticking the rhombuses underneath the hexagon until they made a hexagon shape of their own. Still other groups figured out to put the rhombuses on top of the hexagon.
There was one group that finished quicker than the other ones. I went to talk to this partner group, and I asked them, can you mix the rhombuses and the trapezoids to make a hexagon? How do you know? I walked away, and let them play. A few minutes later it was time to tidy up and discuss our findings, and that’s when this partner group came running up to me so excited: “Look, Miss Dunsiger! Look what we did!”
I was amazed! I created this problem, and I was sure that I knew the solutions, but this group did something that I didn’t expect. This group did something that I’m not even sure I could recreate. Wow! This is when “incredible” happens …
All of a sudden, our “reflect and connect” conversation changed. Now we looked at what this group created, and we discussed if this hexagon could replace the one in our picture and why. Below is a pencast of our conversation:
Now I know that I led the students to come to this conclusion. What could I have done differently, but still ended with the same result? The thinking that the children share is great, and I’m glad that they explain why they changed their mind, but I wish that I didn’t sound so “leading.” I’d love to hear some suggestions about changes that I could make!
While I know that I wasn’t perfect here, seeing a student push my thinking gave me an even better appreciation for the math problem solving process. I was genuinely thrilled when I saw this photograph, and I hope that I’ll continue to have more amazing problem solving moments like I had today. When have students surprised you by their answers? I’d love to hear your stories too!
Aviva
Hi Aviva,
You ask “when have your students surprised you by their answers” and I wanted to respond. It was when what was supposed to be a 5 min discussion in the library turned into a 40 minute math task! We participate in the Ontario Library Association Forest of Reading program and for the younger grades, there is the Blue Spruce Award. 10 picture books are nominated. The students get to vote for the winner. The program begins in January and the voting day is around the end of April. Every year, I find I’m scrambling to finish reading all ten books so they can make an informed choice when they vote. I had a grade 2/3 class come for their regularly scheduled library class and I asked for their opinion: how could I schedule in time to read all the books and do an activity for each book in the time I have? Suggestions were made and rejected and then the ideas and comments started spewing so quickly that we split up into impromptu math congress groups to try and figure it out. They also asked me plenty of important questions so they could gather more data to help them solve the problem – like “how long does book exchange take?” and “how long does it take you to read a book out loud?” Two groups came up with such clever plans that I am using one plan for the grade 1/2s, (I only see them for one library period a week) and one for the grade 2/3s (I see them twice a week – once for media and once for library). One of the boys came up with such a unorthodox but practical suggestion with his group that it blew my mind: why not read the book during library (followed by book exchange) and then use the media class to do the book-related activity (because it can fit with the types of things we do in media class)? Why didn’t I think of that?
Diana, thank you so much for sharing your story! Wow! This is fantastic. I love hearing stories like this, not only because of the solution but also because of the wonderful opportunity you gave these students. They used math with a purpose here. Way to go!
Aviva
In your email to me you said “I didn’t know I could enjoy math so much”. It made me laugh because I get that a lot. It really is about getting out of the textbooks and worksheets and not being afraid to explore. Don’t get me wrong, there are great problems, games, and activities to be found in commercial products, but choosing from different resources to fit the needs of the curriculum and the interests of the group makes lessons more engaging for students and teachers too.
When I started to teach this way it was so inspiring to see that I could make kids want to learn and to push themselves. It was rejuvenating. Kids wanted to be at school, they were engaged in learning. I see a lot of “learned helplessness” when I visit classrooms and I believe that is a result of us not being comfortable with a student not getting started right away on a problem so we guide (tell) them what to do. Kids know if they sit long enough they will be rescued. We need to tell less and question more. We don’t have to teach kids what we want them to know, we have to provide opportunities for them to explore and show us what they can do.
The reflect and connect (consolidation) piece of the lesson is where the teaching is and it is so powerful. Your expertise in guiding their thinking at the end of the lesson is beautiful to see in the clips you share with us.
I think I got off topic 🙂
Thanks for the comment, Kelly! I’m glad that you shared as much as you did. This has been the first year that I’ve really been consistent at using problem solving in the classroom, and I see a HUGE difference in my students. When I give them problems to solve now, they don’t just sit there and do nothing. They try things out, they talk among themselves, they make an attempt, they see if it works or if it doesn’t, and then they try again. They’re not happy until they figure out the problem, and they push themselves to think in new ways. All of the students love math, and more than ever before, I love math too!
I’m glad you’ve enjoyed the “conversations” that I shared here, and I can’t thank you enough for making me see and approach math in a new way. You’ve really guided me this year, and I know that my approach to teaching math (and just teaching in general) has changed for the better. Thank you!
Aviva
I just really love learning from you. I’m reading about math teaching strategies on Saturday morning in my pjs and I can’t tear myself away to go and do the dishes! Merci!
Thanks so much, Shannon! I’m glad you found this post helpful.
Have a great weekend!
Aviva
Aviva,
I love the use of real math language and way the livescibe was used by the students for their explanation. Hands on construction of the shapes make everything real world and relevant.
I so enjoy your blog and everything you do. You’ve created a wonderful environment for learning.
JoAnn
Thank you so much, JoAnn! It was the use of the real math language in the child’s explanation that I liked so much too.
Aviva
Aviva,
I’m a student in EDM-310 at the University of South Alabama. My blog is located here: Daniel Coker’s EDM-310 Blog and my class blog is located here: EDM-310 Class Blog. If it’s alright with you, I plan on summarizing my visits on the 12th of February on my blog.
I’m currently studying to become an educator, and it’s really a treat to watch how efficiently you use technology in your classroom, and how that in turn affects your students’ utilization of the information you present.
You said you sounded leading in regard to the discussion of the larger hexagon your students created, but I felt that your questioning wasn’t leading; it was focused enough to get your students to revisit their thought process, which allowed them to reason the problem out more fully.
You also wanted to know if there was anything you could do differently, while getting the same result, and that got me thinking – could an impromptu group session with those who think the larger shape would work in one group, and those who don’t in another, both presenting their ideas, lead to the same conclusion? I look forward to your thoughts!
Regards,
Daniel
Daniel, thanks for your comment! I really appreciate your feedback and your suggestion! I never thought of this small group approach, but it just might work. Then students could guide each other, and I could help meet the students “where they’re at.” I’m definitely going to try this!
Thanks again!
Aviva