Promoting Fluency in Math: How to Convince Your Students it’s not Just About the Answer

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As I learn more and more about effective math instruction, one thing is clear:  Good math instruction has less to with your students getting the right answer, and a whole lot more to do with flexibility in approach, and strategy.

Try explaining that to your students though.

My students tend to focus on whether or not their answer is right, so when I ask students to explain to me what they did to get an answer students will often ask, “So, is it wrong then?”, or, “Just tell me what I was supposed to do.”

I remind them that them that I am far more interested in how they got the answer than I am in the answer itself, and that it’s their thinking that is important to me.  But their eyes glaze over, and I know I’ve lost them.  They just want to get the answer and move on.  Does this sound familiar?

Taking the Focus Away from Getting the Right Answer

This year, I started off with a math talk that Jo Boaler did with a class of 6th graders using a dot card.  I chose to start off the year with this one in particular because this one does a really good job of highlighting to my students that it’s their thinking that is important.

It manages to do this because the math itself is very easy, and the actual answer to the question is so mundane and uninteresting that their thinking gets to take centre stage.

When you see the question, you will see there there really is nothing to compete with the many ways students solve the problem.  They realize there are a variety of ways to get the answer, and that is what is interesting and important.  Students are engaged, and even fascinated by it.

I did this lesson as my first number talk of the year with my grade 5/6/7 split class.  For many of the students it was their first experience with a number talk.

The Lesson

Credit for this math talk goes to Jo Boaler.  A video of her teaching this lesson can be found here.

I ran my lesson very similarly to the way Boaler did.  I told my students that I was going to show them a collection of dots for just a couple of seconds and I wanted them to tell me how many dots they saw.  I explained that it was going to be shown for only a couple of seconds because I wanted them to do it without counting.

Instead of displaying a graphic with a pattern of dotcardpicdots on an overhead, I presented the dots on a piece of paper drawn by hand.  This ended up working well because some students struggled to explain how they got their answer and needed to point to the array to help them.  Walking over to each student with the dot card proved easier and less disruptive than having students get up and down out of their seats.

Best Practices

After showing the dot card for about 2 seconds I asked students to give me a thumbs up if they had an idea about how many dots they saw.  Once everyone had given a thumbs up, I asked a student to share how many dots they saw (they saw 7), and I checked to see if anyone got a different answer (no one did).

I then asked the student to tell me how they knew there were 7 without counting.  They initially struggled with how to explain, so I re-phrased the question:  I asked her what she saw to know there were 7.  She was able to tell me she saw a line with 2, then a line of 3, and another line of 2, which was 7 altogether.

I represented her thinking on the board, asked her if it accurately reflected how she got her answer, and put her name under it.  I asked if anyone else saw it the same way, and added their names.  Finally, I asked if anyone else saw it differently.  I repeated this until I had exhausted all of their ideas.  This is what the share board looked like by the end of the lesson:

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Thoughts, Reflections, and Next Steps

The conversation was rich, and students were engaged for a long period of time.  It laid the foundation for placing an importance on student thinking rather than the answer.

I think the engagement was high enough to refer to this lesson when students just want to get the answer and move on.  I may even create a visual and put the dot card with the question “How many dots do you see?” on a sticky note, put the answer “7” on another sticky note, and all of their answers on a large piece of chart paper and put it in the hallway to share with the rest of the school.

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