By popular demand (and because I need somewhere to organize this for myself), I’m going to put up a series of posts about what I learned at the NCTM conference. If you’re a math teacher, I hope this is as valuable for you as it is for me!
(If you’re not a math teacher, I promise I’ll include “NCTM Takeaway” in all the titles so you’re warned in advance. These are posts you definitely don’t have to be reading. I also promise I’ll return to regular posts soon enough.)
Did you know the US Department of Education has a website dedicated to research-based best education practices? Me neither. It’s at dww.ed.gov and my initial reaction is to be impressed. (I’ll admit I haven’t dug through it too thoroughly yet.) The fact that someone has been researching best practices and compiling them somewhere shouldn’t be surprising, but I’ve never thought to look for this before. They have plenty of reports, and they also have professional development resources including teacher interviews, sample materials, and guides for administrators/coaches. Here’s a sample, on connecting concrete with abstract in math class. It looks worth checking out.
I’m sure you can tell that my focus at the NCTM conference was problem solving. Here are my notes from the Doing What Works problem solving session. Some of it will seem obvious, but I found a couple of details to be very helpful. I should preface by saying that right at the beginning, they made a little jab at the typical four or five step Problem Solving Process. That’s always been the Holy Grail of any PD I’ve received on this before, so I was automatically surprised and intrigued when they started talking.
Definition of Problem Solving – any problem that has more than one solution (meaning “way of solving” and not to be confused with “answer”) and requires thought.
Key Components of Teaching
1) Prepare problems and use them in whole-class instruction. Problems should sometimes be routine and sometimes non-routine for your kids, allowing them the chance to be successful before they are really pushed outside of their comfort zones. It’s okay to go out of your way to ensure that students understand the problem, meaning you can address issues with context, language, or cultural differences. You might need to rephrase or completely re-contextualize to make a problem work for your kids. Make sure you’ve considered existing mathematical knowledge when you plan a problem. Mathinaz note – does this mean it’s okay when my kids are struggling and instead of reading aloud, I tell them the problem like it’s a story and add in lots of detail until it comes alive for them? It’s always helpful but feels a little like a cop-out for their understanding of word problems. Are these people saying I can do that?
2) Assist students in monitoring and reflecting on their own Problem Solving Process. Model your own thinking process and self-monitoring for them. You might need to explicitly teach them to do things that seem natural to you, like how to get themselves out and start over if they find that their first strategy doesn’t work. It could help to provide a list of prompts to help them monitor and reflect.
(What’s the story about? What’s the problem asking? What do I know? How can it help? What is relevant? Is this similar to anything I’ve done before? What are some strategies I could try? Is my approach working? If I’m stuck, can I find another way? Does the solution make sense? How can I verify? Did my steps work, or not? What would I do differently if I tried this problem again?) Mathinaz note – when I was working on those problems I posted yesterday, I realized that I naturally evaluate my work as I go, and I found myself thinking things like, “Why did I waste so much time doing that? If I see a problem like this again, it would be way faster if I’d just done this instead.” It’s so important in my own process, but I never considered teaching my kids to do it too!
3) Teach kids how to use visual representations. Heavily model what you would do, and use lots of think-alouds and discussions to teach kids how to visually represent situations. Diagrams are great for everyone, but they are especially powerful for ELL students. Don’t forget how to teach them how to convert back from visual to mathematical notation as well.
4) Expose kids to multiple strategies. You do need to explicitly teach different strategies, like Draw A Picture or Make An Organized List, and give them relevant problems so they can practice those strategies. Students should eventually be able to generate and share multiple strategies on their own, but that won’t happen at the beginning. To start, you can have students look at eachother’s work after they finish and try to re-do the problem using someone else’s strategy. Teachers can also create worked-out examples for different strategies on the same problem, and have kids compare and contrast the strategies you used. Mathinaz note – I’ve never thought of that last sentence, and I think it’s brilliant. I have taught lessons on various strategies, but never thought about how I could teach a lesson on comparing and choosing between strategies. Letting them see the details of more than one solution and discuss the merits of each is something I’m definitely going to try.
5) Help kids recognize and articulate mathematical concepts and notation. Describe relevant concepts from the problem and how they relate to the activity. Ask kids to explain each step used in a worked-out example and why it was important. Help kids to make sense of algebraic notation. Mathinaz note – you can tell I was getting tired by the end of this session, because I can tell those sentences are three different thoughts but can’t really elaborate on any of them now….