Guest Blogger: Reviewing Math Problems for Understanding

The best professional development I’ve had has come from having the privilege of working with outstanding colleagues. Who needs to go to trainings when you have an amazing teacher right down the hall? One of these great teachers is Dianne Allan Garcia who is not only a genius (MIT and Harvard), an award-winning and results getting teacher (crazy percentage of AP Calculus passing rates – think Jamie Escalante), but also one of the most kind and caring people I know. I am so honored to introduce her as a guest blogger on The Sacred Profession . . . Thanks Dianne!

“In my early years of teaching I [thought]….becoming a good teacher meant mastering a set of delivery techniques and knowing all the answers to my students’ questions. In those years it had not yet occurred to me that good teaching hinged upon what I knew and understood about the learners themselves and about how learning happens.”

– Mark Church, teacher, as quoted in Making Thinking Visible: How to Promote Engagement, Understanding, and Independence for All Learners, p. 9

When I reflect on my first five years of teaching, I realize that my biggest missed opportunity in helping students develop mathematical understanding and proficiency was the time spent in class going over the answers to problems they had just worked on. I would usually talk through the problem step by step while writing the work out and then ask if anyone had any questions. Meanwhile, students who had gotten the problem right tried not to nod off while they had to listen to it again and students who had made an error had no idea what they did wrong and just copied in the right answer.

A few years of observing math classrooms made me realize that I had to completely reframe my approach to problem review. Instead of getting to the right answer, I now see reviewing problems as a way to uncover student misconceptions, help students understand what they did wrong, and give them an opportunity to practice explaining their thinking. Now that I’m back in the classroom part-time, I’m finding ways to make this happen and watching the magic as students engage in rich discussions about things they don’t understand. Here are some strategies that I’ve been practicing to review problems more effectively:

  • Show the answers right away. Remember the goal is not to explain the steps that you did and go through each problem. It’s to figure out as quickly as possible who made errors where and to start talking about them.
  • Write and discuss the most common mistake that you saw. Especially as you begin this process, students might not feel comfortable admitting that they made a mistake so I will often walk around as they’re checking their work and then I will say, “Here’s what I saw a few people write for problem x…what’s going on there? What are they thinking? Is this a valid approach? Why? Or why not?”
  • Ask questions to search for misunderstandings and encourage the process. I often use questions like: Which ones do you want to review? Who got a different answer? Who solved it differently? I also encourage and praise students who contribute a mistake that they made. I’ll tell them that’s a really common error that I’ve seen or that I made that mistake too when I first tried the problem (sometimes, I’ll admit, I’ve said this even when it’s not true!). Teach your students that the discussion is more important than having the right answer.
  • Make mistakes and put up wrong answers on purpose. Unpredictability promotes engagement and critical thinking so every once in a while I will purposely put up a wrong answer and then go through the standard questions (Who got something different? What did you do differently?…and, eventually, what mistake did I make? How do you know?).
  • Ask multiple students to re-explain the same idea in their own words. Once you’ve started uncovering misunderstandings and discussing where students went wrong, students should start making connections and developing new understandings. When this happens, I will often call on several students to re-explain the idea in their own words or will ask an analogous question of a few different students to make sure everyone’s gotten it.
  • Have students write in words their new understanding, preferably in a brightly-colored marker. After the discussion, I ask them to write what they just learned on their sheet or on a separate learning log. I remind them that writing it out in words is going to help them remember for next time, especially when you write it in marker. (This is not research-based, just my opinion really…and writing things in marker is fun!)

What other strategies have you used to review problems and promote understanding in your math classroom?


2 thoughts on “Guest Blogger: Reviewing Math Problems for Understanding

  1. esuid says:

    Wow. Isolating these strategies simply from reviewing your teaching is pretty impressive. What you are describing is the 2nd half of Modeling. (The 1st part is having students collect data. The 2nd part helps them do something with it.) I learned these tricks through a multi-day Physics Modeling Workshop that showed me how to teach, based on the Arizona State University modeling framework. You can find out more about that here:
    The most helpful thing I learned there was to play dumb and stop filling in the gaps that students left when they explain their work. Ask questions to point out the gaps and make them reconsider what they took for granted the first time around. And a big key to the modeling approach mirrors your “multiple students re-explain the same idea in their own words,” by having two groups complete work on whiteboards, then present them and compare similarities and differences in great detail.
    And to provide some evidence for the benefits of your strategies and modeling’s methods, this is a comparison between “traditional” physics methods and modeling methods: . I don’t know if “Modeling” is much of a buzz word in math education, but it’s definitely the biggest thing that I’ve seen for physics. I would highly recommend attending any math modeling workshops, if they exist, to see it in action.

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