Routines and Practices to Elicit Misconceptions in Math
In my last post, I shared 8 reasons why mistakes are valuable in math class and in learning mathematics. I also wanted to share routines and practices that we can use to elicit misconceptions AND to open our math classrooms up for more opportunities to use students' misconceptions and thinking about math to our (and really OUR STUDENTS') advantage.
After reading "Mathematicians Make Mistakes," Chapter 4 of Becoming the Math Teacher You Wish You'd Had by Tracy Zager, I generated a quick list of routines and structures that either I have used in the past or I know of as math routines that if I chose to capitalize on them more, would lend themselves to creating a classroom environment where mistakes are an accepted part of the learning process, mistakes are something that we can share, and something that we can utilize to learn from one another and deepen OUR OWN understanding of a concept.
I've actually walked away from reading Chapter 4 feeling that it is the MOST IMPORTANT chapter in the book--to me, learning to capitalize on students math “mistakes” and misconceptions changes everything AND it links to so many of the other chapters--encouraging risk taking, teaching/doing math with precision in mind, etc.
Now, as I read the rest of Becoming Math, I want to try to read it through the lens of how we can change our approach to students' mistakes to improve every aspect of our math teaching. It feels SOOO exhilarating and I'm not sure I can even capture the way this chapter has revolutionized my thinking in just a blog post!
Before I share a few ideas/strategies/routines, I want to verbalize the goal of these routines.
I want to create classroom environments where students take mistakes in stride and understand that mistakes are learning opportunities. I want students to feel comfortable with where they are on a continuum of learning--that whether "struggling" or "brilliant," they have something to offer the WHOLE class with their ideas AND they have SOMETHING TO LEARN from others. Being 100% right all of the time or ashamed of ourselves when we are wrong is NOT the goal of learning math. I want to develop the skill as a teacher to HANDLE STUDENTS' MISTAKES CONSTRUCTIVELY.
Wow! What a list goals!
4 Routines and Practices We Can Use to Elicit Students’ Misunderstandings and Capitalize on Mistakes
Writing about Math and Turn and Talk: when we have students explain to another person how to go through a math problem or solve it, we provide an opportunity to elicit misconceptions, gauge the level of the understanding and mastery, and see what language students have internalized about the topic. We can also use these written and spoken opportunities to help students master mathematical vocabulary--as vocabulary is NOT truly mastered until one can recall it and use it correctly in their spoken and written language.
Note: To really capitalize on using students' misconceptions and thoughts about math (when you use writing about math and turn and talk strategies), you HAVE to be on the lookout for INTERESTING COMMENTS and MATH LANGUAGE. You should note student ideas that you would like to use during your whole group time, ask the student if they will volunteer to share their work, or if you can share it and lead a discussion. I also highly recommend walking around with your cell phone and taking pictures of interesting student responses so that you don't forget what you saw. If you don't have time to discuss them right away, they can be turned into discussion prompts or used to construct a whole investigation into the "truth" of the idea. (PS—interesting DOES NOT MEAN “correct.” You are looking for misconceptions or inaccuracies in the ways students have used math language so that you can talk about the true mathematical meaning of what they have said and lead a discussion on what is truly meant by the statements or discoveries!)
Inquiry based/open ended questions: These types of questions lend themselves to more opportunities to elicit students' thinking and often lend themselves to a variety of approaches that are interesting for us to discuss in whole group to show students that there are different ways of thinking about a problem that can still be "on the right track" or get you to the correct answer.
Consider having students use a pen in math. WHAT?!? If students use a pen, rather than a pencil, they cannot erase their mistakes. You (and they) will be able to follow their train of thought more as they work through challenging problems and later attempt to explain the process they went through.
Allow students the freedom to hold onto their misconceptions until they are convinced otherwise (through whole-group discussion, working with a partner and sharing their thought process, working with manipulatives so that they can see the concept, etc). We don't need to flat out tell students they are WRONG when the issue is a conceptual misunderstanding. We need to DIG IN or help them DIG IN to figuring out why they think what they think.
--> In chapter 4 of Becoming Math, Tracy includes two classroom snapshots where children with incorrect mathematical answers were given the opportunity to WORK THROUGH their ideas until they got to the truth. One student benefited from listening to her classmates AND grappling with why she thought what she thought, while another student was given space during independent work time to play with manipulatives and continue working through his "proof" to see if he was correct.
I cannot recommend grabbing a copy of Becoming the Math Teacher You Wish You'd Had: Ideas and Strategies from Vibrant Classrooms (Tracy Zager) enough. The classroom snapshots that are included are so valuable. (Note: That link is an amazon affiliate link. This means that I am a participant in the Amazon Services LLC Associates Program, an affiliate advertising program designed to provide a means for sites to earn advertising fees by advertising and linking to amazon.com. Read my full disclosure here.)
So often, we teach the way we were taught, but we don't know a better way. Reading the language and the "teaching moves" other teachers used to bring out these positive mathematical learning environments really helps me envision how I can do this in my own classroom.
Tracy also provides us with recommendations on how we can create classroom environments where students are “free” (read feel comfortable and safe) to "speak their minds" about math.
How do we create environments where everyone can “speak their minds” about math?
By opening your dialogue with students using "thought-oriented" questions rather than ANSWER-focused questions:
You do this by saying things like "I want to hear some thoughts," and "What are you reasoning through?" vs "What answer did you get?" or asking yes/no questions like "Are these numbers equivalent?"
Frame Thinking as an Ongoing Process
Gather reasoning and thoughts from more than one student
Gather different approaches to the problem
Be genuinely curious about students' thinking and about math to set the tone
Allow students time to engage in productive struggle
Are you interested in stimulating your brain some more on thinking about the power of mistakes in math class? Be sure not to miss my first post on The Power of Mistakes in Learning Math where I share 8 reasons mistakes are a valuable teaching and learning tool.
If you want to listen to Brittany and my thoughts on the topic, you can join our Upper Elementary Math Teachers group on Facebook and watch our chat about Chapter 4: Mathematicians Make Mistakes.
"Mistakes are golden opportunities for students to examine and refine their mathematical thinking. Teach students how to react to mistakes. It is rare for students to know how to turn a mistake into productive learning and growth." pg 57 (Zager, Becoming the Math Teacher You Wish You’d Had)