For the 2016 Blogging initiative, Week 3 had the title of Questioning. In the article, it then proceeded to give suggestions for topics to write about. In previous weeks, the prompts were few in number, but I found one that seemed to be a good fit for me. This week, even though there were more prompts listed, I knew I needed to buck the system and write on a related topic that didn’t apply to any of the prompts.

As mentioned before, I teach in an online school. But I haven’t always. I started in a classroom and most of my experience is there as well. One thing I noticed while in the classroom is that students are observant.

If a student provided an answer to me, often they could tell by my body language or my facial expression if they were correct or not.

Slowly over the years, I was able to better mask my excitement or disappointment. If the answer they gave was to the question, “What is the next step?” often I would say, let’s try that and see what happens. Then we work through the problem as if they were correct.

If they were indeed correct, we have success. If, however, their thinking was flawed, I think it was helpful for them to see why it was flawed, or what would happen if we tried their method. Not only helpful for them, but helpful for the handful of other students who agreed with their initial response.

Other times, they wouldn’t be “wrong” per se. But their suggestion wasn’t helpful. For example, if we were looking at the problem *x* + 5 = 13, they might say to “add 5 to both sides” for the next step. Mathematically, this isn’t wrong. The Addition Property of Equality says you can add the same value to both sides of an equation and not change the answer to that equation (both sides are still equal).

When we use their suggestion, our next step would look like *x* + 10 = 18. Again, thier step wasn’t wrong; it just wasn’t helpful. I think students benefited for seeing this as well, especially because they have the right idea, just the wrong execution.

So what does that have to do with questioning?

More recently (and part of this is helpful in an online environment where students cannot always see my facial expressions as we work together), I hve started to question everything.

If they got the answer wrong, I would ask how they got their answer.

If they got the answer correct, I would ask how they got their answer.

They couldn’t tell anymore if they were right or wrong when I asked that question. Many students who were wrong were able to find their error and correct themselves as they explained why they did what they did. Even if the student could not find their own error, by asking the question, I (as the teacher) could better identify their thought patterns and discern what the true problem really was.

Although I can’t find the exact resource that lead me to start doing this, I believe it was in a blog post by Marilyn Burns, probably talking about the Math Reasoning Inventory (MRI). Alternately, it may have been from her article “Looking at How Students Reason” (bottom right paragraph on page 29).

Completely unrelated, but funny story. When I was in school training to become a teacher, I had the opportunity to pick up some free resources. Two of the items I chose were the books The I Hate Mathematics! Book and Math For Smarty Pants. I mostly picked them for their titles and fun graphic inside, along with the well written content. Just the other day, I was looking though them again, and I noticed they were written by Marilyn Burns! Who would have thought that she would have had an influence on me at the start of my career, and even now well over a decade later?!

I do the same exact thing! Always asking why and how. The students get frustrated in the beginning, but slowly understand the reasoning behind it. Not only does it get them to rehash their reasoning, but opens up their thinking to us.

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When students are helping each other (before/after/during class), do you find them also asking those types of questions to their peers?

Another example where I have seen “explain your thoughts” work well is with things like number talks. I have tried a number talk with my teaching team, but never yet with students. Is taht something you have head about or tried?

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It’s about 50/50 when students ask questions to each other- either “what’s #7?”, or “how did you get #7?”

I have done Number Talks before, but more as a strategy than something formal in class. It’s great for students to see there is more than one route to solve a problem. I’ll always try to show different representations when explaining. For some of the lower level students, it gets confusing because they are so used to seeing math in only one way and try to combine both ways in solving. However, for a majority of students, it’s definitely beneficial.

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I love the “not wrong, but not helpful” distinction. It’s something I come across with my algebra 2 kids a lot. And I do the same thing. As soon as a student suggests an idea, we often work through it, even if I know it’s not going to be helpful or efficient. Sometimes they take a messy approach, but it’s one that works for them and so why not let them run with it? This happened with a precalc student in parametric equations. It’s almost always easiest to solve for t in terms of x, then substitute into the y equation… But I had a student for whom it made more sense to isolate t in both equations and then make them equal. I got no problems with that, even though to ME it seems harder.

And the expression/body language thing is a real struggle. I work with kids who are emotionally needy a lot of the time, so I give them a lot of reassurance, even during tests. It isn’t usually a problem, because the kids are so good about making their best effort and not trying to take advantage of any emotional support I’m providing them… But once in a while it comes back to bite me when a kid says something like, “I can tell by your reaction that I don’t have the right answer” and it’s something that’s easily fixable, like for the quotient rule in calculus they can’t remember if it’s supposed to be g'(x)h(x) – g(x)h'(x) or the other way around. Rare, but it does happen.

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I love that you practice this in Algebra 2, because I think it will be of great benefit to them when they are starting to try to solve trig identities. Sometimes we just have to try something, see how far it takes us, and back track if we need to. Math isn’t always “get it right on the first try.” Especially for “professional” mathematicians.

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I, too, fell into the ask-more-questions-if-they’re-wrong trap early in my teaching career, and have carefully trained myself to ask the same questions of students regardless of correctness. I like that you have students follow through with their thinking to have them see their own errors. Do you also have them identify errors in incorrectly done work a la Algebra by Example or something similar? I’ve found that as students get more comfortable finding errors in other peoples’ work it helps them work through their own errors as well.

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We do. The curriculum is provided for our teachers, so they don’t have to create it, but I can think of several examples where we will present work from two different students and ask them to identify which one is incorrect, and then explain why it is incorrect (or how to correct it).

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I was working with a stuent last night on matrices, and was reminded of this post. She came in with a question on 12-4, how to solve a problem using matricies. To do that, though, she needed the inverse matrix, which also required the determinant. I asked her if she knew how to find the determinant. “What’s that?” she replied.

Well, the topic of 12-3 was “Finding the Determinant and Inverse Matrices,” so I could tell that was where we needed to start, not 12-4.

Once we found the determinant for a problem was 2, we were trying to find the inverse matrix. She saw A^-1 as the notation for inverse matrix, so immediately put down -2. Wow. This is a PreCalculus class, so I would hope that she knew inverse functions was not the same as opposite sign (additive inverse). Once we straightened out that she wanted the multiplicative inverse (reciprocal), then we needed to do scalar multiplication. “What’s that?” Back to 12-2 which was Multiplication with Matrices — both scalar and matrix multiplication.

Finally we are ready to do 12-4. Or so I think. Once we found the inverse matrix of our given problem, then we needed to left multiply the inverse matrix with the matrix on the right-hand side of the equation. She tried to apply the same idea as scalar multiplication and didn’t get the correct answer. Back to 12-2 to learn how to do matrix multiplication.

What reminded me of this post is that, all along the way, I was asking questions. Unfortunately, I think I asked so many questions, she was frustrated by my questions and the process. Since we are an online school working virtually, I have had students get frustrated before and (I believe) log out or “lose connection.” Fortunately for me, she did stick around and we were successful to the point that, at the end, she rated herself a 10/10 for her ability to solve these types of problems in the future.

But I wanted to point out that while we do want to challenge our students, we don’t want to do so to the point of frustration. It is much easier to see that working with a student face to face than online, but one feature that I know I will continue to work on.

Say, this sounds like the beginning of a new post altogether!

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