I was working with a student this past week, introducing concepts related to algebra through this open-ended task from Illustrative Mathematics:

  1. Evaluate each of the following expressions:
    1. 2(4−3)2
    2. 2(6−3)2
    3. 2(8−3)2
  2. In the expressions above, what parts are always the same? What part changes?
  3. We can write an expression to represent all of the expressions above if we use a letter for the part that changes. Write an expression using the letter x that could represent all of the numeric expressions above.
  4. What value of x would make your expression evaluate to 98? 

The student began by calculating 32, then subtracting 9 from 4 and multiplying the difference by 2. Their answer? -10.

When we consider the order of operations, we can see that this is not a correct evaluation of the expression. Of course, I noticed this, but I wanted to take an inquiry-based approach to this students’ misconceptions.

“Interesting,” I said. “You know, when I evaluated the expression, I got 2. How might it be possible that I got 2 instead of negative 5?”

The student looked back at the problem, and they paused. My question was really making them think.

“Oh, you did 4-3 first!” they replied. “Wait a second… is this like PEMDAS?”

“Oh, you know about PEMDAS?” I replied.

“Yea,” they replied. “I made a video on it last year.” The student rattled off the order of operations automatically.

The minute I noticed the child’s misconception, I could have reminded them about the order of operations and then watched them evaluate it. But I didn’t do that–and it was for a very specific reason.


Independence matters most in the learning process–and not the brand of independence where students necessarily do everything by themselves. Instead, we center the type of independence where students are not over-reliant on others. Learning is a communal activity, and it should be. In communities, individuals contribute what they can and take only what they need from others. Seeking unnecessary help treads into the territory of over-reliance. While it’s natural for students to take the path of least resistance, we must lead them down the longer, more challenging path. We must train our students how to embrace the discomfort of cognitive dissonance, ask questions, and try to answer their questions on their own.

How do we achieve this? We achieve this by asking questions–and lots of them.

In the algebra example above, I could have simply recited the steps of PEMDAS to my student and reminded them that they must always follow them. Instead, by asking questions, the student to retrieved this information on their own, strengthening their independence as a learner and increasing the likelihood that they’ll remember it next time.


Nowadays, I’m doing a lot of one-on-one instruction, but when I am working with groups, I do my best to teach them how to ask each other questions, too. I first realized the power of this when observing my colleague, Meghan, teach math. She facilitated discussions in math beautifully, and she did it mostly through asking questions. But she went beyond asking the questions on her own: she encouraged her students to ask their peers questions, too.

Copyright Humanizing Distance Learning (France, 2020); adapted from Meghan Smith

In order to make this happen, Meghan had a list of questions on an anchor chart that students could use during math discussions. While they didn’t have to choose from the list, they could use the list as a scaffold. After seeing how effective and student-driven her math discussions were, I immediately adapted this for my classroom.

With some routine-building and teacher support, my math discussions transformed. Suddenly, students were asking one another to prove their methods, and they were providing constructive feedback to one another on how to improve their methods. I realized that these discussion prompts didn’t need to be confined to math, though. They could be used at any time in the school day.

They could be using when engaging in peer writing conferences. “Have you considered adding dialogue to this section to make the story come alive?”

They could be used during reading groups, too. “How has your thinking changed about the character since the beginning of the story?”

Truly, they can be used in any area of the school day, as these discussion prompts leverage questioning as a means for self-discovery, encouraging students to share in the responsibility of constructing knowledge.


Distance learning has challenged us all. It has helped us see just how accustomed we are to learning on behalf of our students. Personally, it helped me see just how much I was hovering over my students; it helped me see just how much I engendering dependence in them. But most of all, it helped me see that, in some ways, we should always be teaching from a distance–and no, not in the sense that virtual teaching should be a standard, 21st-century pedagogy, but instead in the sense that if we teach too closely, we build dependence in our students.

We must ask ourselves: when we shouldering most of the responsibility of constructing knowledge for our students, who are we serving? We’re certainly not serving our students, especially if our collective goal is lifelong learning. And we’re certainly not serving ourselves, as it creates an unsustainable workload. The reality is, by constructing knowledge on behalf of our students, we’re actually just serving the system, attempting to take the path of least resistance to learning so that our students’ test scores can raise.

We must resist this urge. We must change the way we teach. We must let learning be messy. We must be patient and wait while our students construct knowledge on their own–because when we do, we not only teach for independence, we teach for liberation.

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