On Explanation

There has been a fascinating conversation on Twitter and over at Dan’s blog in response to the article in The Atlantic, “Explaining Your Math: Unnecessary at Best, Encumbering at Worst” by Katharine Beals and Barry Garelick. I’m going to try and find some middle ground.

There are a ton of teachers going to bat defending explanation, but I think folks are conflating several different reasons to ask students to explain their work, and muddling the argument in the process. I’m going to offer three different reasons to ask a student for an explanation, and attempt to describe what they are useful for.

Explanation as formative assessment. From Lisa Soltani in the comments: “I do ask students to explain their mathematical reasoning because it helps me assess what they actually know, and because clearly explaining their reasoning in writing is a skill worthy of development, and one that students should be developing across disciplines.”

Here is a great example from Math Mistakes:
6thGrade-copier@saintannsny.org_20150331_124451-1-page-001

I am in a much better place to help this student because of their explanation — I know that they are leaning heavily on one representation of fractions, and would benefit from expanding that representation to calculate more accurately.

I’d like to offer criteria for what makes an explanation useful as formative assessment.

  • The explanation must reflect the reasoning of the student; if the student is trying to mimic the reasoning of the teacher, it is unlikely to be useful.
  • The problem should be deliberately chosen as one the teacher can make an instructional change in response to. Asking students to justify every answer is a recipe for poor explanations, in particular form the students whose thinking we are most interested in.
  • The problem should be one where a “false positive” is likely — the student could solve correctly with a strategy that will not transfer to a different, future context.

Explanation qua explanation. The Common Core asks students to “Construct viable arguments and critique the reasoning of others.” Seems like a skill worth developing, in and of itself. But let’s take a more nuanced view. In the comments, Ze’ev Wurman argues against the overuse of this strategy:

“The *abuse* of explanations that the Atlantic piece tackled, and that I commented on, occurs when explanations are required for essentially every — even the most trivial — homework and test items, or when “verbalism” is *demanded* even when proper clear symbolism has already been used by the student. This is what seems to occur in Common Core testing and in many Common Core classrooms and what I think Beals and Garelick — and I — find objectionable and counterproductive.”

Another favorite of mine is John Holt cautioning against what he calls “word shoving” in How Children Fail:

“We say and believe that at this school we teach children to understand the meaning of what they do in math. How? By giving them (and requiring them to give back to us) ‘explanations’ of what they do. But let’s take a child’s-eye view. Might not a child feel, as Walter obviously did, that in this school you not only have to get the right answer, but you also have to have the right explanation to go with it; the right answer, and the right chatter. Yet we see here that a “successful” student can give the answer and the chatter without understanding at all what he is doing or saying.”

And finally, Dan tries to find some middle ground:

“Understanding is the goal. The answer, and even the algebraic work, onlyapproximate that goal. (Does the student know what ’80’ means in the problem, for example? I have no idea.) Let’s be inflexible in the goal but flexible about the many developmentally appropriate ways students can meet it.”

This summarizes the purpose of explanation. In a follow-up post, folks in the comments are exploring the idea of problems that cannot be adequately answered solely through symbolic notation. But the goal is not for students to practice writing, it is for them to practice mathematical justification. We are robbing students of the chance to think mathematically when the teacher decides an explanation must happen in a certain way.

My criteria here are simple:

  • The problem must be a problem, not one in a set of exercises practicing a well-defined skill.
  • A written explanation is necessary if and only if the mathematical notation, or students’ knowledge of mathematical notation, is unable to express the ideas in the problem.

Explanation for learning. From David Wees in the comments: “Another reason that we might want to listen or read a student explanation of how they solved a problem is just so, in the process of articulating their solution, students may run into their own inconsistencies in their work. I have noticed, quite often, that students will give an answer that I don’t understand, and then when I ask them to explain what they did, in the middle of their explanation they say something like, ‘Oh, oops! Yeah that isn’t right. I mean this instead’ and revise their thinking.”

The argument here is that student explanations help them learn. I find it interesting that David uses an example of a student verbalizing their explanation, which I find much more likely to result in students finding an error in their thinking. But there’s a psychological basis for this. In Make It Stick, the authors describe the process of elaboration.

“Elaboration is the process of giving new material meaning by expressing it in your own words and connecting it with what you already know. The more you can explain about the way your new learning relates to your prior knowledge, the stronger your grasp of the new learning will be, and the more connections you create that will help you remember it later.”

Here we have an argument for the use of talking through content for students’ learning, but it gives us another useful condition — this is part of the learning process, and its purpose is connection to prior knowledge. It seems exceptionally difficult to make these complex but important connections solely through mathematical notation. And if our goal in math class is for students to gain knowledge they can transfer to new situations, this is exactly the type of thinking they need to be doing.

Here’s an attempt at criteria for this type of explanation:

  • Explanation for learning must happen during the learning process, when it is most useful to connect new learning to prior knowledge, not through practice or exercises after students have gained competence with a concept.
  • Explanation for learning is useful when students are likely to make an error, and an explanation can help them think through the reasonableness of their answer.
  • Explanation for learning is insufficient if students are simply restating what they did mathematically; it must connect to prior knowledge or an objective measure of whether their work makes sense.
  • Explanation for learning is particularly useful if it prepares students to share their thinking with a peer to move their collective knowledge forward.

Closing

That was a lot. I presented a bunch of conditions for explanation, which I think do a great deal to support Beals and Garelick’s argument — there are plenty of situations where a required explanation is likely to be superfluous, and incessantly requiring explanations for every problem students solve seems counterproductive. But I hope I’ve also outlined some conditions that underscore the value of explanation when we use it effectively. I will certainly continue to attempt to do so, hopefully with a bit more skill after reading the thoughts of a great deal of smart people and organizing them with my own.

2 thoughts on “On Explanation

  1. howardat58

    It is often forgotten that before “algebra”, and there was a “before algebra”, every mathematical statement was expressed, in speaking and in writing, in words, and that the algebraic way of writing is a symbolisation of those words. If we spent more time on the connection between the words and the symbolic representation the kids would be able to “read” the meaning of an algebraic statement, and “explanations” would be second nature. Simple example:
    The slope/intercept form of the equation of a straight line. What is the equation saying? In words – “A quantity depends on another quantity in such a way that when the second quantity has the value zero the first quantity has the value b, and for each unit increase in the second quantity the value of the first quantity increases by an amount whose value is m.”. Yes, this can be made more colloquial, but if the students cannot see it this way they are going to come unstuck very soon.

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  2. Hannah A.

    @howardat58 I really like what you have to say about relating it to the “before algebra.” I am in my senior year as a math major and secondary ed minor and am currently taking a “History of Calculus” course. Here we discuss the fact that math was not a written down language for quite a while. It wasn’t until more recently that symbols were associated with a meaning and the language of math was recorded. In addition to relating it to the foundation of math, I like the idea of students understanding and explaining what they did mathematically and being able to read the math. Now, in my History of Calc class, I realize that back in high school when I was constantly taking integrals and differentiating functions, I had no idea why I was doing what I was doing. But when doing this, I was still receiving full credit because I remembered the process and how to do it. Now comes the question, did I actually learn the material in high school Calculus? Part of the requirement for the need for explanations I think comes from what you consider necessary knowledge for the students. Or, in some cases, maybe what knowledge would help them better understand a topic? I definitely think explanations are necessary in the classroom and help students further their learning. Personally, I like incorporating them for an assessment piece to see where my students are at in their understanding and where their logic is.

    Reply

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