At the end of every marking period, my fifth-grade math students assemble a portfolio of their work. As part of that process, I ask them to reflect on what they think they have learned and what they still need to work on. By the third marking period each year, they get pretty good at mimicking the kinds of things they hear as feedback. They write they have “not yet mastered multiplying fractions.” They tell me that their “ability to explain their thinking clearly” has improved.
But by far the most frequent reflection is one that I know I never say—that their mistakes are the result of carelessness. One student has written that he needs to stop making “silly mistakes,” while another explains that she “rushes her work and makes careless errors.”
I know from my work as an instructional coach that teachers attribute poor performance to “careless mistakes” more often than any other cause. When we identify a math mistake as careless, what we usually mean is that we believe that a student understands a concept but fails to perform the computation accurately. But the more time I spend looking at the kinds of mistakes we might call accuracy errors, the clearer it becomes that there are a number of different causes for those mistakes.
It takes a lot of time and attention on a teacher's part to notice the real patterns in student work, and the more we stay neutral in how we talk about mistakes, the better classroom culture we create.
Mistakes, But Not From Carelessness
Often what looks like carelessness is in fact a lack of mastery. Sometimes a student knows enough to get started on a problem, but they haven’t sufficiently consolidated their understanding to the point where they can consistently apply what they know.
For example, a student might be able to solve a straightforward multiplication problem, but when they’re faced with solving one that requires multiple steps, they appear to have forgotten basic facts. A student who can correctly recognize that the reciprocal of ⅔ will complete the equation ⅔ x ____ = 1 may not see that the same answer will work for ⅔ x ____ = 22 - 3, even if they can simplify 22 - 3 to 1. This isn’t carelessness but a result of split attention. If the student hasn’t consolidated foundational skills, more complex problems become much more difficult because there are simply too many pieces for the brain to keep straight.
When we see these mistakes, we know that the student needs more work with the concept, but also more opportunities to see the concept in different contexts. We need to help them move from an isolated understanding to an integrated one. We also need to be clear that this isn’t an easy transition for students to make.
Often accuracy errors have nothing at all to do with mastery of content. Students with dyslexia, students who struggle with executive function, and students who have anxiety all tend to make frequent accuracy errors. My son is dyslexic and labels certain mistakes that cost him points off his high school tests as a “dyslexia tax.” He assumes that he’ll lose somewhere between 5 and 10 points per test for miscopying a number, losing a negative sign from one line to the next, misreading a symbol, forgetting a parenthesis, or failing to do all of a multi-step direction. Yes, a negative sign is important, but it’s disheartening to have understood every concept on an assessment and still score badly. What’s even more disheartening is being told it’s because you aren’t trying hard enough.
Saying that a mistake is a result of carelessness is an accusation that the student has not invested enough effort or care into their work, but we shouldn’t judge effort based on results. We can assess whether the student has achieved a stated goal, but we can’t assume that we can tell whether they really tried.
When we remove the judgement around why someone has made the mistake, we open up the conversation about what they did understand and what they were able to show. We also are likely to improve their sense of agency and advocacy. Improving accuracy is really difficult for many students, and they need to be encouraged, not derided for their effort.
Another Source of Error
There is one source of accuracy errors that I have an enormous amount of sympathy for. Students frequently make accuracy errors because, consciously or subconsciously, they know the answer doesn’t matter at all. A few years ago, I piloted a new draft of a textbook in a third-grade class, and the early version had an error that in retrospect was very telling. Students were told a farmer packed 5 bags, each with 12 apples. Then they were asked how many oranges the farmer had in total. Almost everyone answered 60.
They weren’t being careless; they simply knew the problem wasn’t really about apples and that whether it was apples or oranges wasn’t the point. Even if the question had asked for the number of apples, it would be hard to imagine that the students would honestly care whether their answer was accurate. We weren’t eating the apples. We weren’t curious about the situation.
There are many ways to make the accuracy of the answer matter more. Anytime we ask a question that a student truly wants to know the answer to, we’re likely to see a higher degree of accuracy. My favorite way is to create the need for an accurate answer. When we teach surface area and volume, we can ask students to calculate how much cardboard they need to make a box with a set volume. Then we give them that much cardboard and they make the box. If they don’t request the right amount of cardboard, they can’t make the box they want. Both understanding and accuracy are involved. Not everyone gets it right the first time, but everyone can see the point of the accurate computation.
Labeling work as careless isn’t good feedback—it obscures the source of the problem and prevents us from taking steps to help the student work on that source.