Usually once a month, someone posts a mathematics word problem online (FB, Twitter, email, text) and asks people how they solved it. Most often, I might scroll past it because I do a lot of math for a living and might wanna just look at a cat video for the moment. (Check out #BongoCat BTW). The other reason why I might scroll past is that I know me. Somtimes, I’ll take the risk, try the task, and post my answer, but then I would inevitably get something wrong: I’d read the problem wrong, I’d neglect a detail, I’d get lost in some minutiae that wasn’t really relevant to the actual purpose of the task. Last week was no different. I quickly fell into the trap that I knew I would when solving a problem… I overlooked a part of the problem, submitted my answer, and quickly retreated when I saw that I was wrong. Will someone come and take my Ph.D. from me? No. Did I stew about it and wonder how I got it wrong? You betcha. Did I decide to write about it? Yep. That’s my new thing.
But whether or not I got the right answer is not the point of this post. We don’t always answer math problems correctly the first time, every time, and with speed/efficiency. The point of this post is to look beyond the right answers and consider how would someone get another answer.
To me, that’s the challenge of learning to teach mathematics that I try to stress with my students. It’s great when students give you the right answer that matches perfectly with the problem you asked, but what happens when they don’t?
Do you dismiss their answer and quietly show them how to solve it so they can see where they went wrong? (Answer: Please don’t do that.) Do you carefully consider their solution and say “I am curious about your thinking… Tell me more.” And this “I am curious about your thinking” statement can be said even if the teacher sees a correct answer (my students HATE THAT). Being curious about students’ thinking doesn’t always have to highlight a wrong answer. The “I am curious about your thinking” statement can be a more helpful response to students when they share their thinking for a number of reasons.
First, no one wants to be wrong. And no one wants to be wrong in public. Unless there’s an assigned job as an antagonist in the classroom, students don’t want to be wrong (or be seen as wrong) because how they see themselves (and others see them) as being competent in mathematics is tied to how they perform in math. This isn’t breaking news though. The spaces that we create can send messages to students about their potentials and identities as those who do mathematics (Stoehr, 2016; Zavala, 2014). For example, if you got the right answers in school, you were likely to be rewarded for these answers by teachers and your fellow classmates (who likely looked to you for help and support). It’s human nature. Maybe they called you smart. Maybe they said you were just naturally good at it. So then you developed a positive identity about yourself and math.
And the opposite is also possible. I can’t tell you how many times I’ve heard from someone, “I did terrible in school, especially in math class. That’s why I am just not a math person. I am just bad at it.” If we think of ourselves as beings that can constantly evolve in our thinking, then neither of these scenarios should be messages that we communicate to students… not all students who seem smart get the right answer and some students offer an answer that will really surprise you as a teacher. But nonetheless these messages persist and they continue to creep into our everyday lives… even on social media.
For your consideration: The Case of Angela and her Sand dollars collection.
The following problem was posted online by my cousin from a college course she is taking and I thought it was fascinating:
Solve the problem: On her first day of vacation, Angela found 2 sand dollars on the beach. She put them in an old sock. The next day she found 4 sand dollars, and she put them in her sock. On each day of her vacation, Angela found 2 more sand dollars than she had found on the day before. On what day did she have 42 sand dollars in the sock?
Before you scramble to get the right answer, stop. Just stop. Read the question and see if you can imagine Angela doing this on a beach. What would be the process that she might take? What images come to your mind as she collecting these sand dollars? What catches your attention about the problem? What makes you curious about it?
Ok, now let’s approach the problem. If you get an answer, cool. If not, that’s ok too. And if you haven’t figured it out, getting👏 the👏“correct answer”👏ain’t👏the👏purpose👏of👏this👏post.
Consider the following three proposed solutions (student names are pseudonyms and randomly selected) and let’s unpack each.
Colleen’s Proposed Solution 1: Day 21 is when Angela gets 42 sand dollars. In this solution, Colleen says, “I started with 2 and then 4. I added 2 to each day because I read that ‘Angela found 2 MORE sand dollars than the day before.’ 6 is two more than 4. 8 is two more than 6. So then I just continued the pattern until I got up to 42.” The student offers the following graph and y=2n (where n is the number of days) to explain their thinking.
Questions about Colleen’s strategy: What makes you curious about this thinking? What about the solution can be identified in aspects of the problem? What questions would you ask next?
Niki’s Proposed Solution: Day 20 is when Angela collects 42 sand dollars. Niki says, “I see that she starts with 2, but then the second day is 6 because she adds TWO MORE from the previous day (she collects 2 + Day 1 onto the existing two she already has in her sock.) So then if I just continue this pattern of adding 2 more each day, then I get 20. And I can show that this is 6 + 2(n-2) if we don’t count day 1. But in looking at Colleen’s graph, mine looks a little off. Maybe I am wrong. There’s a bump in mine. Maybe that doesn’t matter?”
Questions about Niki’s strategy: What makes you curious about Niki’s thinking? What about Niki’s solution can be identified in aspects of the problem? How can you see that there might be connections between Colleen and Niki’s strategies? What questions would you ask next?
Dina’s Proposed Solution: Day 6 is when Angela collects 42 sand dollars. Dina says, “I started off with 2 and then whatever I added with each day is going to be 2 more. It’s not going to be linear because she is collecting the sand dollars from the day before and I am not adding on the same amount with each day. I can actually see a pattern within the pattern from day 2. But hold on… my picture looks REALLY off from Colleen and Niki. My graphs aren’t supposed to show curved lines (see, I can’t use my ruler to perfectly connect the dots) so I am pretty sure I am wrong. And can Angela reasonably collect that many sand dollars in 6 days? How big is this sock that she’s dragging around, Miss??”
Questions about Dina’s strategy: What makes you curious about Dina’s thinking? What about Dina’s solution can be identified in aspects of the problem as it was posed? Did it match how you imagined Angela would collect the sand dollars? How can you see that there might be connections with Colleen and Niki’s strategies and Dina’s? What are aspects of the problem that Dina seems to be grappling with? What questions would you ask next?
So now that we’ve considered these three solutions, I’ll tell you that Dina’s proposed solution is correct and… is NOT the one I gave online. I offered Niki’s solution and was thoroughly embarrassed that I had missed the “Angela found 2 more sand dollars than she had found on the day before.” And I missed how she collected the sand dollars from the day previous. But again… whether or not I got the right answer👏ain’t👏the👏purpose👏of👏this👏post.
The purpose of this post is to remind ourselves once again about how the notion of status (a perceived social ranking that tells some they are more valuable than others (Cohen & Lotan, 2014) creeps to (re)position us as having more mathematical talent than another. If someone has a PhD, then they must have the right answer (which isn’t always true). If someone quickly got the correct answer, then they must be really really smart (which could be true but we could also have more deliberate thinkers who take longer to arrive at their answer). But this experience reminded me once again to help my teachers learn how to pause and think:
What risks are my students taking as they share their solution? Do they feel safe to share?
What claims can I make about what they DO know even if it might not be the answer I was expecting and/or matches the actions in the story?
How reasonable is the context of this story—what kind of beach will have 42 sand dollars on it?
If teachers only look for answers like Dina’s, we might neglect the ideas of Colleen and Niki’s, even if they don’t match with the actions in the story. Think of the amazing conversations that can happen when a teacher might ask: How might I change Angela’s problem so that Colleen is right? So that Niki is right? How might I analyze this graph to make sense of the connection between the answers?” 🤔
[And shout out to Desmos for making a website where teachers can create graphs really quickly!]