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But Artichokes Aren’t Pinecones: What Do You Do With Wrong Answers?

I have very small children which means my life is measured by little games and distractions stretched across the day. “What’s that called?” is one of those games. Point at a thing and ask for its name. Do that for another thing. Hey — it’s almost nap time!

So recently we pointed at an artichoke. “What’s that called?”

“Pinecone,” one of the kids says.

a drawing of a pinecone and an artichoke

That’s a factually incorrect answer, which is the same as lots of student answers in math class. But when my kid calls a pinecone an artichoke, I have a very different emotional, physical, and pedagogical response than when a student says something factually incorrect in math class.

With my kid, I am fine with the error. Delighted, even. I am quick to point out all the ways that answer is correct. “Oh! I see why you’d say that. They both have the kind of leafy-looking things. They both have the same-ish shape.”

I find it easy to build connections from their answer to the correct answer. “But an artichoke is greener, larger, and softer. People often eat it and people don’t often eat pinecones.”

However, if I’m teaching a math lesson and a student answers a question about math incorrectly, my reflex is to become …

… evaluative … “What did I just hear? Is it right or wrong?”

… anxious … “Oh no it’s wrong. What do I do now?”

… corrective … “How do I fix this answer and this student?”

I find it much harder to celebrate and build from a student’s incorrect answer in math class than I do an incorrect answer from my kids about artichokes. The net result is that my kids feel valued in ways that the students don’t and my kids have a more productive learning experience than the students.

I can give lots of reasons for my different responses but I’m not sure any of them are any good.

  • This is my kid so I feel warmer towards his early ideas than I do towards ideas from kids I see for only a small part of the day.
  • This kid looks like me so I’m more inclined to think of him as smart and brilliant and wonderful than I am a student with a different race, ethnicity, or gender.
  • The stakes are smaller. What’s the worst consequence of my kid referring to an artichoke as a pinecone? That he doesn’t get invited back to the Governor’s Ball? Who cares. This will work out. I’m not preparing him for an end-of-course exam in thistle-looking stuff.
  • I know the content better. I can build conceptually from a pinecone to an artichoke much more easily than I can build from early math ideas to mature math ideas.

But I find that every aspect of my professional and personal life improves when I try to neutralize those excuses.

  • I am a member of faith and educator communities that help me dissolve my conviction that my kid is more valuable or special than your kid, communities that help me dissolve my sense of separateness from you. We are not separate.
  • I am working with a team to develop experiences in math class that lead to student answers that are really hard to call right or wrong, or ones that at least lead to lots of interesting ways to be right or wrong. I am learning that it’s more helpful to ask a question like, “How are you thinking about this question right now?” than “What is your answer to this question?” because the first question has no wrong answer.
  • I am trying to develop pedagogical tools that make use of differences between student answers to replace ones that try to reconcile or flatten them. Tools like “How are these answers the same and different?” or “For what question would this answer be correct?”
  • I am trying to learn more math more deeply so I can make connections between a student’s early ideas and the later ones they might develop.

I am thinking about this idea from Rochelle Gutierrez more often:

All teaching is identity work, regardless of whether we think about it in that way. We are constantly contributing to the identities that students construct for themselves …

Whether my kid calls an artichoke a pinecone or a student offers an early idea about multiplication, they’re offering something of themselves just as much as they’re offering a fact or a claim. My goal is to celebrate those early ideas and build from them so that students will learn better math, but also so they’ll learn better about themselves.

Featured Comments

Several people mention that we have more time to enjoy our kids and their thinking than we do students in math class.

2020 Jun 13. Other examples of early ideas about language from around my home.

  • “Getting tangled out” a/k/a “getting untangled.”
  • “Yesterday” as a placeholder word for any time in the past.
  • “Foots” and “Gooses” as the plural for “Feet” and “Geese”.
  • Them: What do cows eat? Me: Hay, I think. Them: No, horses eat hay.
  • 6 looks a lot like a lowercase “g”.
  • “After” is any time in the future. Me [beleaguered]: “We’ll do that later, kids.” Kids [combative]: “AFTER!”
  • “More taller” is coming up a lot.
  • These kids think that as they get older, they’ll get bigger and I’ll get smaller and turn into a baby.

2020 Resolutions

Meanwhile, Nepantla Teachers, a group of math educators focused on social justice in their work, asked several educators to contribute a resolution for the new year. Here’s mine:

I'm resolving to spend as much time next year thinking about student lives outside of school as I do their lives inside of school. Teaching and curriculum have enormous influence on student learning but the influence of those in-school factors is dwarfed by out-of-school factors like housing and food security. So I'm resolving to practice humanizing pedagogies and to protest school closures in my city, to create interesting mathematical activities and to urge my representatives to protect and expand social programs. I'm resolving to ignore the distinction between educator and citizen. 

Click through to read resolutions from thoughtful people like Carl Oliver, Hema Khodai, Idil Abdulkadir, Marian Dingle, Makeda Brome, and Tyrone Martinez-Black.

Estimation Isn’t Just Calculating Badly On Purpose

Here is a tweet I haven’t stopped thinking about for a couple of months.

I think it’s possible we should cut the student some slack here.

If the student has all the tools, information, and resources necessary to calculate an answer, we should be excited to see the student calculate it. Asking students to do anything less than calculate in that situation is to ask them to switch off parts of their brain, to use less than their full capacity as a thinker.

If we treated skills in other disciplines the way we often treat estimation in math …

… we’d ask students to spell words incorrectly before spelling them correctly.

… we’d ask students to recall historical facts incorrectly before recalling them correctly.

Estimation shouldn’t ask students to switch off parts of their brains or use less than their full capacity as thinkers. It should ask them to switch on new parts of their brains and expand their capacities as thinkers. Estimation tasks should broaden a student’s sense of what counts as math and who counts as a mathematician.

Estimation and calculation should also be mutually supportive in the same way that …

… knowing roughly the balance of yeast and sugar in bread supports you when you pour those ingredients exactly.

… knowing the general direction of your destination supports you when you drive with turn-by-turn directions.

… knowing the general order of your weekend schedule supports you when you carry out your precise itinerary.

Engaging in one aspect of mathematics makes the other easier and more interesting. That’s what Kasmer & Kim (2012) found was true about estimation. When students had a chance to first predict the relationship between two quantities it made their later precise operation on that relationship easier.

If we want students to develop their ability to estimate, we need to design experiences that don’t just ask them to calculate badly on purpose.

Create tasks where estimation is the most efficient possible method.

Take that worksheet above. Give students the same sums but ask them to order the sums from least to greatest.

Students may still calculate precisely but there is now a reward for students who estimate using place value as a guide.

Create tasks where estimation is the only possible method.

This is the foundation of my 3-Act Task design, where students experience the world in concrete form, without the information that word problems typically provide, without sufficient resources to calculate.

“Estimate the number of coins.” Estimation feels natural here because there isn’t enough information for calculation. Indeed, estimation is the only tool a student can use in this presentation of the context.

Meanwhile, in this presentation of the same task, there is enough information to calculate, which makes estimation feel like calculating badly on purpose.

Estimation isn’t a second-class intellectual citizen. It doesn’t need charity from calculation. It needs teachers who appreciate its value, who can create tasks that help students experience its benefits.

BTW

Featured Comment

William Carey:

One thing I love about calculus is is proceeds from estimation to exact calculation, and there’s no way to justify the exact calculations without working through the estimation first. We often think of mathematics as a discipline that proceeds deductively from perfect truth to perfect truth, but there are whole swaths of mathematics where the best way forward is to work from an answer whose incorrectness we understand towards an answer whose correctness we don’t yet understand.

Mark Betnel:

I agree with you, but I think it’s interesting to turn your non-math examples into better activities that reflect what we’re trying to do with “good” math estimation tasks.

Mr. K references Fermi problems, which fall really nicely in the category of “tasks where estimation is the only possible method.”

Theresa Clifford:

At the beginning of the year, I fill four jars around the room. One with M&M’s, one with eraser caps, one with cotton balls, and one with paper clips. They are all allowed a guess for how many in each jar. They enter their answer and their name on a slip of paper and place it in a collection jar. Whenever we come to a question where I want them to estimate first, I remind them of what they did when they first looked at the jar. I don’t tell them how many in each until the winter break – the suspense is awesome. Then in January I start with four new jars.

Joel offers an example of this kind of estimation exercise.

Fave Five

Five of my favorite articles from the last month.

Humanizing Math Class Means Teaching Math Like The Humanities

Here are a couple of terrifying tweets from my summer.

I saw those tweets and had to sit back and collect myself.

That’s because I know how well I’m served by my knowledge of mathematics, how that knowledge helps me find value in early student thinking, how that knowledge helps me connect and build on thoughts from different students that, without that knowledge, might seem totally unrelated.

This isn’t a critique of those two newly drafted math teachers at all. Most of my horror here results from the thought of being drafted to teach history after a career teaching math. So what can they do?

You’ll find lots of people in those threads recommending resources and curricula. But resources and curricula are only as good as the teacher using them. A developing teacher can make a good resource bad and an expert teacher can make a bad resource good. (This is why John Mason prefers to talk about “rich teaching” instead of “rich tasks.”)

So my own advice is for these teachers trained in the humanities to focus on their teaching, not the resources or curricula.

Specifically, I hope they’ll resist the idea that math should be taught any differently than the humanities. I hope they’ll resist the idea that only the humanities deal in subjectivity, argumentation, and personal interpretation, while math represents objective, inarguable, abstract truth.

Math is only objective, inarguable, and abstract for questions defined so narrowly they’re almost useless to students, teachers, and the world itself.

find the volume of an abstract compound shape where all side lengths are known

In social studies, an analogous question might ask students to recall the date of the Louisiana Purchase or the name of the king who signed the Magna Carta — questions that are so abstracted from their context, so narrowly defined, and so objective that they make no contribution to a student’s ability to think historically.

The National Council for the Social Studies describes what’s necessary for students of social studies:

Students learn to assess the merits of competing arguments, and make reasoned decisions that include consideration of the values within alternative policy recommendations. [..] Through discussions, debates, the use of authentic documents, simulations, research, and other occasions for critical thinking and decision making, students learn to apply value-based reasoning when addressing problems and issues.

All of which rhymes perfectly with recommendations from the National Council of Teachers of Mathematics:

Teaching mathematics with high expectations for all students in mathematical reasoning, sense making, and problem solving invites students to learn to identify assumptions, develop arguments, and make connections within mathematical topics and to other contexts and disciplines.

Teaching math like the humanities asks us to:

  • Broaden the scope of the problems we assign. We can always narrow the scope in collaboration with students but the opposite isn’t true. Students don’t have the opportunity to “identify assumptions,” for example, if we pre-assume every detail in the problem.
  • Focus on mathematical ideas that are big enough to be understood in different ways. Ask students to make claims that demand to be argued and interpreted rather than evaluated by an authority for correctness.
  • Celebrate novel student contributions to mathematics. History is made every day and so is mathematics. If our students leave our classes this year without understanding that they have had made unique and original contributions to how humans think mathematically, we have defined “mathematics” too narrowly. (For example, someone just decided to call this shape a “golygon.” If that person has the right to notice and name things, then so do your students.)

Instead of the worksheet above, show your students this video of a pallet of bricks and then immediately hide it.

bricks stacked in an interesting way on a pallet

“Does anybody have a guess about how many bricks we saw up there?”

“Did anybody notice any features about the bricks that might help us figure out exactly how many bricks we saw there?”

“Let’s look at the video again. Okay, what’s the most efficient way you can think to figure out the number of bricks.”

“How were you thinking about the number of bricks you figured out? What assumptions did you make?”

“Someone else got a different answer from you. How do you think they thinking about the number of bricks?”

“Here’s the number of bricks. What’s another question we could ask now?”

These questions rhyme with the kinds of questions you’d hear in a productive, engaging humanities classroom, questions which are no less possible in mathematics!

Humanizing math class means teaching like the humanities. And if you’re joining us from the humanities, please be generous with your pedagogy. We need all of it.

BTW: This is my contribution to the Virtual Conference on Humanizing Mathematics, a fantastic learning opportunity hosted by Hema Khodai and Sam Shah through the month of August 2019.