Lately, I’ve been thinking about the way we as teacher talk about students and their understandings and skills and their struggles. We often describe students as “low” or “high” depending on their performance on formal assessments (but we often fail to acknowledge that those are snapshots in time, informed by all of the students’ experiences). I’m sick of hearing teachers talk about “filling in the gaps” or how their students are always making the same mistakes. Maybe we need to stop viewing student mistakes and misconceptions as barriers to learning and start seeing them as the necessary rungs of the ladder to understanding instead. Some students can scale the ladder quickly, getting boosted up through certain misconceptions by prior/alternate experiences with math, whereas some students need to build up more muscles before they can move past a certain misconception. And sometimes, students appear to have passed a particular rung because they’re able to do other math, but then it later becomes apparent that the muscles needed to avoid falling on that rung (making that mistake) were underdeveloped and need more strengthening. So when we see students making “the same mistake” over and over again, we need to ask ourselves what experiences do they need that will create the kinds of cognitive dissonance that will unearth the limitations of their current understanding and grow their brains?
Right now, I’m focused on the “conceptual errors” students make in the predictable most common mistakes. I’m not concerned with the copying errors or rounding errors or even the computational errors (unless they’re related to the concept at hand).
I have spent a lot of time analyzing student thinking closely by detailing what do students know and not know based on exit tickets (low-stakes problems solved at the end of class and collected. I give feedback, but no grades on these). I’m currently in my 8th year of teaching middle school math, and I always notice patterns in the kinds of mistakes students make. There are certain trends that I can tell students in class that I can predict the ways they might go wrong even before they start! I even draw their attention to the connection between these “most common mistakes” (MCMs) and the wrong multiple choice answers (the distractors) – those other options aren’t random! They’re usually carefully chosen to highlight a student that has chosen the incorrect strategy to solve a problem.
Lately, I’ve been rethinking the fact that we call these mistakes at all. If they’re so predictable, maybe they’re not the same as other kinds of mistakes. Maybe they’re actually a necessary phase of the learning process.
I first heard about the Landscape of Learning from Kara Imm, but my understanding is that Cathy Fosnot developed it.
When you look at it, it’s made up of big ideas, strategies, and models that build on each other. There’s no one linear path through the landscape, and sometimes students will have mastered some of the big ideas, strategies or models, but not others. While some of these build on others (i.e. you probably can’t use double and halve if you don’t know how to double), other strategies, big ideas, and models are independent of each other, and depending on a student’s prior experiences, they may each have different strengths than other students.
I’m now wondering if these predictable misconceptions also belong on the landscape for learning as stepping stones because they inform our recognition of student understanding (and misunderstanding). These predictable mistakes are often ways that students overgeneralize “rules” or mis-apply new ideas in the wrong contexts. For example, every teacher who’s taught fraction addition to students has probably seen someone take 1/4 + 1/3 and get 2/7. And we can see WHY students are getting that number! But by calling it a mistake, we’re not recognizing the way in which that misconception builds on addition of whole numbers; we’re just acknowledging how it falls short on recognition of fractions as numbers. I am proposing that we shift the conversation here from saying “What’s lacking in this student?” to “What can this student do?” and then build on where the student currently is. If we recognize these predictable mistakes as stepping stones that some students need to have access to the main roads through the landscape of learning, then we are better able to see the pathways that students need strengthened in order to master big ideas, strategies, and models.
We can better understand why students hold certain misconceptions (and thus what experiences they need to complete their understanding) by seeing which big ideas, strategies and model were only partially connected to this misconception.
Going back to my previous example of 1/4 + 1/3 = 2/7, we might recognize that students don’t seem to realize the big idea of needing a common whole to add or subtract fractions. We might see that the clock model might help students move through this misconception, because they know 1/4 of an hour is 15 minutes and 1/3 of an hour is 20 minutes, and they can see on the clock that the combination of that is 35 minutes (or 35/60) of 7/12 of an hour. (In this case, because of the numbers I chose, the clock model works better than the money model, but for other numbers such as 1/4 + 1/5, that model would work better). The connections between these ideas and the fraction bars model may be less obvious – but that’s another model that students might rely on when adding fractions with unlike denominators. Additionally, a student could use landmark fractions to help them estimate that 2/7 does not make sense, since the sum of 1/3 + 1/4 should be greater than 1/2. All of these different connections can provide possible pathways from the misconception (add numerators and add denominators separately) that might help guide students back onto the landscape of learning and away from this common mistake.
I also think that if we understand that students are having a difficult time with addition of fractions with unlike denominators, this might help us step back and question whether the student has mastered adding fractions with common denominators – whether they understand that they’re just counting up pieces or if they are still adding the denominators. For example, when they see 1/3 + 1/3, do they say 2/3 or do they say 2/6? That will inform where the disconnect in their landscape is and help a teacher figure out the necessary experience for the student to gain this understanding.
In thinking about the class discussions I facilitate, I often think posing the student mistakes, asking them to make sense of what someone might have been thinking if they made that, and then discussing where it falls apart in the reasoning or why we need to do something else – and then being able to convince our skeptics – results in a much richer classroom discussion where students are actually growing.
I think we need to reframe the way we discuss mistakes as wrong answers and see them through the lens of stepping stones on the learning journey. We need to honor their importance in growing our minds, and that without those mistakes, we never realize that there is more need to grow because we can’t see the limits of our own understanding. According to the definition of mistake, it is “an action or judgment that is misguided or wrong.” I think calling these ideas mistakes indicates that they’re wrong and misguided when in reality, they’re often necessary stepping stones that are pushing us on the trajectory of learning. I think it’s time we stop calling them mistakes, and start naming them as partial understandings and consider how they connect to the understandings we want to cultivate.