“Let’s talk about the common misconceptions kids have with fractions.”

I cringe.

I, like many others, have been thinking a lot about the language we use to describe children in the mathematics classroom. Words matter. Words convey and confirm how we treat each other. When it comes to mathematics, seemingly innocent words have a lot of power.

The word that made me cringe was misconceptions. Why this word? When you say misconception, you are suggesting that someone is thinking about something wrong. If you are trained as a teacher to notice students’ misconceptions, you will find yourself listening for what they don’t understand.

Knowing what children do not yet understand is useful to a point, because it can give you ideas of what to teach. However, the balance of research in learning says that children are not blank slates. That means that they already know something, and odds are even if you have not yet taught a particular concept to your kids, like say subtraction above 10 with your first graders, that they know something about the operation of subtracting, situations when you separate and compare things, and something about counting and addition that will help them to make sense of what you want to teach. In short, as the authors of the Cognitively Guided Instruction series have said for years now, “Children are sense-makers.” They are always making sense of new information in relation to what they know.

So rather than looking for the holes in kids knowledge, teachers should be listening and looking for what sense-making kids are engaged in, and what makes sense to the student.

Let’s consider Julieta –  a 9 year old emergent bilingual student from Ecuador who has been in the United States for a couple of years (not a real person, an amalgam of kids I’ve met over the years). Her teacher approaches her during a lesson on division. Julieta is to show how to “share 12 cookies equally between 4 people.” She counts 12 counters, then proceeds to push them around on the desk until she has 4 piles, of 3, 3, 2, and 4.

Julieta’s first solution to 12 cookies shared among 4 people

Uh oh, her teacher is thinking, Julieta doesn’t understand that she has to make each pile have the same amount of cookies. Julieta has a misconception about how division works. The teacher then asks Julieta if she’s solved the problem. Julieta nods. The teacher now asks, “Is each pile the same?” And Julieta says “no”. The teacher, says “make each pile the same”, satisfied she addressed the misconception. Julieta takes a moment, then reshuffles her counters, dealing two out to four piles, and puts the rest off to the side.

Julieta responds to “make each pile the same”

Glancing at her work, the teacher says, “we’ll look at how other people solved it later and it might give you ideas” and moves on, making a mental note that Julieta doesn’t understand division.

What is the consequence of the teacher’s thoughts over time? Julieta’s teacher is going to start to form ideas of the kinds of mathematical people her students are, using whatever patterns of thinking she has been accustomed to. If her major sorting of students is who “gets it,” and who has misconceptions, it’s not out of reach to start calling kids “good” or “bad” at math. It’s a slippery slope towards unwittingly condemning students as deficient, or lacking skills.

Julieta’s teacher could approach her in another way. If she pays attention first to what kids are doing, and the sense they are making, Julieta’s teacher might first notice that:

1. Julieta understands that this situation calls for counting out 12 cookies, which she does, next
2. Julieta understands that 4 people are going to all get some cookies, and she does that well too, as everyone of four piles got some cookies, and finally
3. Initially, Julieta gave out all the cookies, and only when the teacher asked her a question that she may or may not have understood did she put some cookies aside.

So what can we conclude? Julieta understands a lot about partitive division: start with a total, give out the total to the number of sharers, use all the total. This is evidence of her current conception of division. Now, we can be curious about her thinking. Imagine if the teacher had approached her and said, “Tell me more about your 12 cookies?” Julieta might have explained that she gave them all out. Continuing to presume that Julieta is making some sense out of the situation, the teacher could have continued, “Tell me more about your piles.” Julieta might then have said, “I get three, and my sister gets 3, but my brother is small so he gets 2, and so my mom gets 4.” Ah-ha, the teacher might now say to herself, I see how she’s making sense of this situation. Now the teacher can make a choice on how to use Julieta’s excellent sense making to help her further her understanding of division, and also to talk with the class about how dividing up in mathematics isn’t always the same as how we divide up at home. And the mathematics goes on…..

By training ourselves to notice what students do understand, and what actions they take based on their understanding, we can challenge deficit notions of students. By making it our job to elicit and understand student thinking, we can help ourselves move away from the damaging language that labels some kids as “good at math” and others as “bad at math.”

Here’s my first challenge for us all as we prepare to get back into school mode: Let’s take a stand. Let’s all agree that there are no misconceptions in mathematics. There are current conceptions. And we can all revise our thinking with new evidence, leading to new conceptional understanding.

Here are some ways you can help yourself move from deficit to asset-based language in the mathematics classroom:

• When planning your math lessons, Instead of anticipating “popular misconceptions,” prepare to focus on what kids actually do, and how they describe why they did what they did. If you need to talk about what students understand about a concept, you can use the sentence starter The current conception seems to be…
• When circulating while students work, ask yourself how is the way this student is thinking useful in another setting? For example, Julieta was thinking reasonably about sharing cookies based on her experience that her little brother shouldn’t get as much as she, her sister, or her mom. This is perfectly logical outside of the arbitrary rigidity of school mathematics. This provides insight into current conceptions.
• Put the emphasis on difference, not deficit. Help students make sense of how different situations call for different kinds of reasoning. This will help us stop sorting kids into piles of getting it or not getting it, and help us think about what kids actually understand and how to support them to make new connections or go deeper into their own thinking.

I’ll also plug the book Schooltalk: Rethinking What we Say About -and to- Students Everyday, by Mica Pollock. I’m reading it right now with some people and so far it’s great. This quote from the introduction has a lot to do with what I’m saying about doing away with deficit talk such as “misconceptions”, and Mica Pollock ties attending to how we talk as a key part of equity, in particular in how we understand who our students really are:

“In our schooltalk, all of us sometimes repeat common, habitual comments about young people or their communities that are fundamentally inaccurate and under-informed. Such ‘scripts’ are the simplifying, familiar claim we reach for when we talk about young people and about education. Much of the time, these ‘scripts’ have a shallow grasp on facts about young people’s lives and instead adhere to premade ideas about young people. Dangerously, scripts misrepresenting young people in general can keep adults from fully supporting actual young people – and keep young people believing falsehoods about themselves and their peers.”

This school year, let it be the school year that we end deficit thinking and talk in all its forms, and that we all grow more aware of the impact of what we say and don’t say about and to students.

Sources:

Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B.(2012). Children’s mathematics: Cognitively Guided Instruction, 2nd Ed. Ports-mouth, NH: Heinemann

Pollock, M. (2017). Schooltalk: Rethinking What we Say About -and to- Students Everyday. New York, NY: The New Press.