Whether or not you were a “math kid” growing up, you likely got a wrong answer at one point. While some students are able to see that they simply made a mistake or didn’t understand one piece of the puzzle, other students see it as further reinforcement that they are just “bad at math”. And it’s not their fault, it’s how we’ve all been conditioned to treat incorrect answers in mathematics. For some reason, we can see how someone may have gotten an odd idea from a book in English, or mixed up an event in History, but not when it comes to math.

One of the reasons this is the case is because we don’t have sufficient language around incorrect answers in mathematics — typically we just see “wrong, incorrect, invalid, etc.”.

I’ve looked at millions of student answers to math questions. I’ve spent months trying to understand how students arrive at these answers, and what these answers can tell us about what a student understands. I’ve measured the probability of students getting the correct answer years down the line based on the answer they gave initially. I’ve broken answers down into detailed maps of mathematical understanding, to illustrate that if even one small piece of knowledge is taken away, it can break the entire chain and lead to an incorrect answer. But there is one powerful framework I’ve been experimenting with lately that has really helped me make sense of answers in a new and profound way.

The framework is thinking about answers as Conceptually Correct and/or Computationally Correct. Let’s first dive into the ideas behind computations.

## Computationally Correct and Incorrect

Let’s look at two problems. The first is a simple problem about finding the number of tentacles on seven octopuses. The second is about adjusting a recipe for higher volume.

We gave the octopus question to third graders and commonly see answers like 64, 49, 55, 57 etc. What is going on here? Taking a look at their work, we see students at that age are commonly drawing octopuses and then counting the number of tentacles. So, I tried to use that same method. I drew an octopus, counted the number of tentacles, 8, and then drew an octopus, and counted 8, and drew an octopus, and counted… and drew…

This is what I came up with.

If you take a look, you’ll see that I drew 8 octopuses… There are only 7 octopuses in the question! A little bit about me… I was a stellar math student, studied statistics in college, worked on a trading floor helping to build high frequency trading algorithms, and now I’m working in math education… and here I am getting 7×8 wrong using a common student method. And I would bet that I’m not an anomaly, that many of us would make this mistake.

Drawing/counting the wrong number of tentacles, or the wrong number of octopuses. The issue with this is not my understanding of how to solve this problem, rather it’s the method I used. In this scenario, students are simply using a strategy that is difficult to execute for the problem at hand. Counting is a useful tool, as is drawing, but drawing 7 octopuses with 8 tentacles each, leaves plenty of opportunities for a simple mistake (maybe it’s more difficult for me than it is for the artists in the room).

For that reason, I would say these students aren’t wrong, per se. They did arrive at an incorrect answer, but it’s because they either don’t know or don’t feel comfortable with multiplication, which is a much easier way to solve this problem. So they chose to count, I call this a “Suboptimal Strategy”.

Now let’s take a look at the second problem, about a lemonade recipe.

One the the most popular student answers we saw was 12.5 cups of water. For a while, I couldn’t quite figure out how students were arriving at this answer. Luckily for us, some students explained their thinking quite well!

“Double and then add a half”. They understood the concept here, to increase the ratio in multiples, but they add 0.5 instead of multiply. Perhaps they aren’t comfortable with decimals, or maybe they believe that decimals are a totally different concept. In this case, I would call this a “Procedural Error”. The student totally gets the ideas here, and has a process to solve it, but their process has a bug in it. So, it’s not that the student is “bad at math” but they are just using a flawed process. The way to fix this type of error is to fix the process with the student… NOT to have the student cement in this flawed process by practicing worksheets where they will do use this flawed process over and over again. (For more on process errors, I recommend: Error Patterns In Computation by Robert Ashlock)

Finally, let’s look at a type of error we all make, “Mistakes and Typos”. Here we may see a student do something like 340 – 60 = 180 on one question, but easily compute similar subtraction problems without making that same mistake.

Before we move on though, we need to address how to tell the differences between these concepts, because these are all somewhat related.

How can you tell the difference between a Process Error And A Mistake?

Great question! This is a pretty simple one. At the end of the day, you can make the determination with a couple more questions. In our example of a process error, ask the student to multiply a couple other decimals with whole numbers. What do they get? Was it a one-off problem? Or is it a systemic issue? Same goes for the “340 – 60 = 180”. What do they get for 330 – 70, 430 – 50, etc.

Isn’t miscounting the Octopuses just a mistake?

Yes, it is. However, the key thing in that scenario is that there is a different strategy that would lead to significantly less mistakes. So, when addressed, it provides the students with a significantly higher degree of consistency rather than calling it a one-off mistake.

That addresses computational errors! In my experience, computational errors are the much smaller problem. Most students make mistakes every once in a while, but most of the time, the issue lies with Conceptual Understanding.

## Conceptual Errors

Let’s take a look back at a new problem. This one is about saving up money for a new bike.

The most common conceptual error that we see comes from the answer “17”, which accounted for about 10% of students in this sample. You can get 17 as your answer by dividing \$340 by \$20 per day. If a student does this, it means they just looked at this problem as a proportional relationship, but ignored Raul’s starting balance of \$60.

Much less common, is the answer “20”. Students who submit this answer know they need to use the starting balance, but end up using it the wrong way, adding \$340 + \$60 to get \$400, then dividing by \$20/day.

In both of these cases, their problem isn’t computational accuracy, but rather understanding how to use the different information in the question. We categorize these different approaches to the problem as “Ways of Thinking”. Each way of thinking creates a different opportunity to engage each student.

In the first couple of answers shared here, you might re-engage students by having them think about how starting with \$60 changes things. “If you had \$60 now, and wanted to buy a \$50 video game, how many days would you need to work?” The goal would be to help them connect the dots and build a foundation for understanding linear relationships in real world problems. But these students only represent about ⅓ of students who got an incorrect answer on this problem.

Other students had some computational errors, but the majority fall into a “Way of Thinking” that we call, “Assorted Calculations”. In this way of thinking, students mostly seem to lack a good way of making sense of the problem. They do a variety of calculations to get an answer. Anything from adding all the numbers in the problem together, to multiplying them, etc. Sometimes, this is caused by using a process to identify key words in a problem that hint at a certain computation. Other times, students know they need an answer and they feel that guessing what to do is better than leaving it blank.

In these cases, one way to re-engage students is not to just talk to them about the starting balance, but rather to re-engage them on how to make sense of what the problem is asking. Or to talk to them about breaking apart the problem into their units and understanding how units work when adding, subtracting, multiplying, or dividing them.

That’s a brief talk about Conceptual Errors!

## Recap

Inside of all of these different Ways of Thinking, Conceptual Errors, and Computational errors, we can see how each student brings something unique to the table. If we want to re-engage students, it doesn’t make sense to treat them all the same. Every student understands the world a bit differently, thinks through problems a bit differently, and sometimes hears things a bit different than how they were taught. By labeling everything as either correct or incorrect, we disempower students by failing to help them see that they can do math. It also takes power away from our amazing teachers, who when equipped with the right information, can make an even bigger difference in their classrooms than they already do.

That is a brief look into how I think about errors in mathematics. From a high level, we have computational and conceptual errors. The perception in math education is generally that there are a lot of computational errors, but in my experience 90% of all errors are actually conceptual. If we can start looking at mathematical errors with more nuance, not only can we re-engage students intelligently, but we can also help students see errors as something to understand and build upon, instead of just further evidence of them not being “not a math person”.

If you haven’t heard enough yet, check out How I Think About Correct Answers In Math.

Quick Reference to Understanding Math Errors:

Conceptual Errors

Ways of Thinking

Break down answers into how students thought about the problem. What did they understand, how can we re-engage them to help them see the question/solution in a new light?

Assorted Calculations

Correct computations, but may indicate that the student had trouble making sense of the problem. How can we re-engage these students into a practice of sense making?

Computation Errors

Suboptimal Strategy

Are students using an outdated tool? Using counting/drawing for multiplication, subtracting for division, etc. Do they not know the more advanced tool? Or are they just nervous to use it?

Procedural Errors

Is a student getting the same type of problem incorrect in a similar way each time? How can we take some time with the student to allow them to demonstrate how they solve these problems, and how can we point out where the mistake is being made?

Mistakes/Typos

Does a student normally get problems like this correct? But now you see them arriving at a strange answer? Depending on the student, it may simply be time for some compassion, we all make mistakes! Or if these small mistakes are occurring often, perhaps suggesting a habit of checking their work at various steps of their problem solving process.