This year, more than ever, my students are saying things like, “I’m not quite ready for the conference yet. I want to watch this video again”, “Are there any more practice problems?”, “Can you check this for me before I do more, to make sure I’m on the right track?”, “Can I explain this to you, to see if I’m understanding?” Every time I hear a question like this, I am thrilled that my students are taking charge of their own learning.

Nick’s Challenge

In one particular case, Nick was struggling to understand the limit definition of derivative. His first exposure to it was an applied activity where the concept of derivative was introduced through the lens of physics and instantaneous velocity. He seemed to understand this physical interpretation so I thought he was ready to move on to the mathematical definition. 

When I presented it to him, he was baffled. I worked out an example with him and it still didn’t make sense. I showed him graphs on Desmos. I derived the definition alongside a picture. I sent him an applet to play with. None of it was clicking.

A Different Approach

On Friday, I said, “Nick, I am so glad that you are working so hard to understand this. At this point, it seems that my usual tricks are not helping you make progress. I’m going to spend the weekend trying to figure out a different way that I might be able to present the concept to you.  At the same time, I want you to think about it too. What is your roadblock? What part of this doesn’t make sense? What parts can you make sense of?”

We both spent the weekend reflecting. On Monday, I had something prepared to show him, but I didn’t pull it out right away.

“OK, what did you come up with?”

Nick responded, “I think it’s the notation that is messing me up. I don’t get how what f(x+h) means. How is that different from f(x) + h?”

Wow! His insight gave me something to work off of. I tried multiple strategies for explaining this idea but he still seemed to struggle. Next, I tried something new. I gave him a textbook (which I rarely use) and had him read over the worked examples in the book, looking for patterns. Then, we talked again to see what he had observed.

Can I show you?

“OK, I think I’ve got an idea. Can I show you to see if it makes sense?” He asked.

I watched as he developed his own, new notation, (essentially changing (x+h) to a new variable, z and then putting the (x+h) back in later). It worked! He was finally solving the problems.

“Yes! This is finally clicking!” It had been over a week of “banging his head against the wall” and Nick was elated.

“Looks like you’ve got it!”

“I think so, but can you give me a few more practice problems so that I can make sure I can do it?”

Growth Centered Questions

Throughout this process Nick constantly checked in with his understanding and sought feedback by asking, “Can I show you?”, “Can I explain what I’m thinking?”. Questions like these indicate that our students are metacognitively aware of what they need in order to progress in their learning. They have noticed a deficit in their understanding and are asking for ways to fill that gap.

Questions like this also indicate that the students have internalized a growth mindset. They are no longer afraid of not knowing. They understand that they just haven’t mastered it yet. They realize that it is possible to be successful in their learning if they just have a little more time, a little more practice, and a little more feedback.

How should you respond to “growth” questions?

It’s easy to hear a question like, “Can we have another day to review?” and want to dismiss it. Maybe we’re falling behind our pacing guides and don’t want to spend another class period on this topic. Perhaps we suspect our students are just stalling because they don’t want to take the summative. 

Whatever the reason, avoid the temptation to move on. Our best response is to feed the students’ hunger for growth. Provide them with additional opportunities to watch, learn, read, and practice. Teach them to find more resources on their own. Prepare a review game. Build in more peer editing. Give them detailed and specific feedback on their work.

We want to encourage a mindset by which all learning is formative and continuous growth is possible. We should feel grateful when students show the initiative. When they do, we can ask them reflective questions that will help them further internalize their budding growth mindset. These might include: 

After you’ve given the students the support they asked for, urge them to reflect again upon whether or not they understand better now. Encourage them to think about what helped them get there. Was it just “brain muscle memory” from practice? Was it a visual representation? Was it the way a specific piece of feedback was worded that made things click? 

These questions will push our students to use metacognition even more deeply so that they will continue to self-evaluate and ask for more support when they need it. They also help students figure out what type of additional resources they need so that eventually they can seek those out on their own, gaining true independence and ownership over their learning.

If your students rarely ask questions about their personal growth, pause to notice that. Think about why this might be the case and how you can encourage them to do so.

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