Teaching through problem-solving
While the bulk of my math workshop site seems to focus on workjobs, I do most of my teaching through problem-solving. As far as I'm concerned, that's the whole point of math. Kids naturally love games and puzzles, and so it seems to me that they should naturally love math. The trick is to present math as a puzzle to solve, rather than as a series of steps to follow.
Several years ago, I taught math using mainly problem-solving. This was the general architecture of my daily math workshop:
15 minutes whole group mini-lesson - setting the problem
25 minutes paired, small group or individual work - solving the problem using any strategy or manipulative, showing thinking through words, numbers and pictures
10 minutes whole group session - sharing solutions to the problem (two children/groups/pairs shared)
While I found that this was very successful (my success criteria: children enjoyed it, I enjoyed it, easy to plan and assess, children grew as mathematicians), I also found that sometimes extensions for early finishers weren't quite as powerful as I'd have liked. I also read a fantastic book by Marilyn Burns (About Teaching Mathematics) that described a math menu: a selection of math tasks that children self-select and complete in any order. I decided that that would fit perfectly into my teaching style and philosophy. So now, while I still focus on problem-solving, I've added workjobs to the mix and extended the independent math time a bit to make more room for them in my schedule.
Math Workshop mini-lessons
My mini-lessons now consist of:
- setting problems that children then puzzle through (this is the bulk of my mini-lessons)
- doing whole-group graphing activities (with a focus on problem-solving)
- teaching and playing games (again, I try to choose games that include strategy or logic, so there's a problem-solving slant)
- modeling and guided practice with new manipulatives (tied into - you guessed it! - how these can help us solve problems)
- working with the hundreds chart with a focus on patterns and numbers
When posing problems, I try to come up with problems that are interesting to the students, either because they're intriguing or because they relate to their lives. Throughout the year, I'll try to post problems that I pose to my first graders. To give you an idea of the problems that I've used in the past, here are 3 different problems that I used when teaching second grade:
number sense: We're going to the museum on Monday. Our class has 25 students and Mrs. L's class has 26 students. Each class is also bringing three parent volunteers. We've only ordered one bus. If there's enough room for 60 people on a bus, will we all fit?
fractions: My niece was supposed to come over to my house with three friends. I had four cupcakes ready for snack. The problem is that my niece actually came with four friends! I cut up the cupcakes and shared them, but I'm not sure if what I did was fair. If you were me, how would you have shared them?
measurement: I want to build a small fenced dog yard attached to my house, so that my dog can use the dog door and go outside during the day while I'm at work. I have enough wood to build a fence measuring 24 metres in all. What shape should I build to maximize the space for my dog?
Recording Thinking
We focus on recording our thinking using words, numbers and pictures. One of the big stumbling blocks is moving past "pretty drawing" and into representing thinking with quick sketches (ie. the difference between drawing a dog - complete with collar, leash and brown spot over one eye - and drawing four lines to represent four leg). I model this a lot at the beginning of the year. When circulating, I'll ask guiding questions (Is that helping you find a solution to the problem? Why did you choose to do it this way? Etc.) I also choose children who have made representational sketches to share during our closing circle.
I have a few different ways of structuring the independent work time:
paired problem-solving: Often children work with a partner and record their thinking with markers on large paper. Each child can only choose one marker. This keeps them both accountable, as I can see at a glance that both colours are on the page. It also helps move them away from "pretty drawing"; if kids have fewer colours, they're less likely to decorate their work with bubble letters and fancy pictures.
graffiti problem-solving: Sometimes I'll put chart paper with problems around the room. Children (either alone or in pairs) will move from chart paper to chart paper, adding their solutions to the ones already recorded. When doing this, I encourage them to read the other solutions first. Sometimes I challenge them to solve problems using a strategy or manipulative that is not yet represented on the chart paper.
problem-solving relay: My students LOVE doing this, although I don't do it all that often. Children work in pairs to solve a problem. When they've solved it, they come show me. If I'm satisfied with their work, I give them the next problem. To do this, I have to stay seated in the centre of the room and have children come to me. I can't circulate, so I would only do this at the end of a unit.
individual problem-solving: While my students can often choose between working in pairs/groups or alone, sometimes I require that everyone work alone to solve a problem in their math notebooks. Depending on my purpose, I can then use these solutions as formative or summative assessments.
placemat: Children work in groups of four. Each group has a chart paper divided in five, with one section in the middle and four sections around it. Each child has a section to record their thinking. We begin by working silently and alone. After a set amount of time, I stop the individual work and children have a certain amount of time to discuss and come to a concensus about a group answer to record in the middle of the chart paper.
Sharing Circle
Much of the student learning happens during the sharing circle at the end. Children learn so much from listening to one another's strategies and solutions. In order to focus students' attention, I choose one or two children to share based on the work that I observed. I've found that letting too many students share leads to them thinking more about what they're going to say than about what their classmates are saying. I spend a lot of time modeling, role-playing and practicing respectful and meaningful comments to children's presentations. We work on agreeing and disagreeing respectfully. We also work on commenting on something specific (I think it's interesting that you chose to...) rather than on something general (I like your answer.).
