Years ago, when beginning some work on percentage with some HSE students (they were called GED students at the time), I posed the following problem:

*Veronica’s math class has 25 students. If 7 of them identify as men, what percent of the class does not identify as men? *

I used it as a quick assessment to see what kind of understanding I could build off and what kind of misconceptions I could draw out. I gave it out and collected it at the end of the class before, so I had a week between collecting the students’ responses and our next class.

There was little innovation from the half of the class who answered the question correctly. Of the students who got it correct, they did so using a handful of procedures they’d memorized. That was mostly on me as a teacher, since the problem I used was closed and procedural. But again, my goal was just to get a quick sense of where the class was.

Far more interesting (and helpful to me as a teacher) was the half of the class who got it wrong. Those students who didn’t have any procedures memorized came up with some interesting strategies that proved very useful when it came time to explore percent concepts and work on the more open-ended problems I gave the class.

Here are a few examples:

Student A

18 (and 18%) was probably the most common incorrect answer. Most students calculated it by writing 25-7 = 18 (and a few instances of 7+18 = 25). The first thing I really appreciate is that this student wasn’t sure how to proceed, but she kept working and drew a picture to try to make sense of the situation. What does this student understand that can help her? True she didn’t get the right answer, but her method was a great model later when we explored the idea that percent means “out of 100” and looked at the problem in terms of proportional reasoning and groups of 25.

Student B

This student got the wrong answer, but she probably demonstrated the deepest understanding of percentage of anyone in the class. Which isn’t to say she *had* the deepest understanding, but that she *showed* the deepest understanding. The students who followed the rote procedure did not demonstrate anything about their understanding of percentage, only that they had memorized one appropriate method. Can you figure out what this student was thinking when she set up her visualization of this situation? Where did that 28% come from? And what are all those 20s?

I start with these brief examples of student mistakes from my class to convey my fascination with student thinking and my genuine curiosity in how students make sense of things and what misconceptions they have. As teachers, we often tell our students that we learn more from them than they do from us. I sincerely hope that is not exactly the case, since my students have much higher stakes in our developing a successful learning relationship. But I do appreciate the sentiment and agree that every class is an opportunity to learn how adult students make sense of and confuse the mathematics concepts and problem-solving we want them to learn.

I should clarify, when I say mistakes here, I’m talking about core or conceptual mistakes, not just errors in calculation (though those are worth drawing out and uncovering too since many of them are covering up a consistent misconception.)

In my experience, I have found most student mistakes and misconceptions are wrong, but they are logical. That is to say, they are coming from some place – perhaps a concept learned correctly being applied to a new situation incorrectly or inappropriately. There is a study that illustrates this point well. In general our experiential feeling is that the Earth is flat, because it seems flat as we walk. Having that preconceived notion, kindergarten students in the study were “taught” that the Earth was round. So what was their take away… that the Earth is shaped like a pancake. Because that is what happens when we treat students like they are empty vessels to be filled with knowledge. Our students know a lot and some of it is correct and some of it is not, so we have to draw out their preconceived notions and address them in our teaching. Otherwise we might be building on a faulty foundation.

It is understandable why some teachers try to avoid student mistakes in math class and move past them as quickly as possible. No one likes to make mistakes in an environment where we don’t feel comfortable doing so.

The problem is that making mistakes is one of the most useful things a person can do when learning something. According to Stanford Professor Jo Boaler, every time we make a mistake when learning, we grow a new synapse. Our brain actually grows. It grows when we make the mistake in the first place and it grows when we reflect on that mistake.

Often when a teacher asks a question and gets an incorrect answer, we keep calling on students until someone says the correct answer. Then we go over the method of the student with the correct answer. But when we do that, what are we conveying to the students who got it wrong about the value of their thinking? And if we have to ask several students before one can answer correctly, does that one correct answer balance out all the incorrect ones?

What if we reversed the process? What if the incorrect methods/answers are expected (drawn out), respected (celebrated as a discovery) and inspected (as a way to get at the math)? What if that mistake was treated as the discovery that it is and the class spent time understanding it? What might that look like? <<If you are interested in classroom activities that create a classroom culture that celebrates student mistakes, check out my review of “My Favorite No”.>>

Remember, any mistake one student makes is a mistake that any of our students can make next time. It is also may be a mistake a quieter student has made. It is also likely to be repeated by the student until they understand what they are doing wrong – we can’t just drop correct procedures on top of deeper conceptual misconceptions and expect them to take root – we’ll end up with a pancake shaped Earth.

Of course, addressing mistakes requires preparation on our part. As teachers, we need to be able to quickly look at student thinking and mathematical work, understand what is going on and figure out how to respond. That can be hard, but it is also something we can get better at if we practice, especially if we practice it outside of the moment in class when we have to think on our feet. I have found that while there are always unique mistakes that dazzle and amaze me, there are patterns in misconceptions that begin to emerge over time, which can help us predict student struggle and plan our lessons.

To help us with all of this, I’d like to offer Math Mistakes – a wonderful website edited by Michael Pershan, a middle school and high school math teacher.

The premise is simple – teachers send in photos of student mistakes they find fascinating, confounding and/or mysterious. Michael posts it on the site and then teachers can discuss in the comment section. Teacher comments tend to focus on analyzing the thinking, identifying the assumptions behind the work, strategies for responding, what the next steps could be, etc. Generally Michael offers a prompt when he posts the submitted student work.

Here are just a few examples of math mistakes from the site that I enjoyed thinking about:

- Here Michael Pershan looks at posts over the last few years and attempts to classify the student mathematical mistakes into categories.
- Students in this class were prompted to graph a system of equations that had more than one solution. Here’s what one group came up with.
- Combining like terms. The mistake is clear. Why is it so tempting? And how would you help?
- Algebra in geometry. Vertical angles and ninja equal signs.

I can see adult education teachers using the site in several ways:

- When you are going to be teaching a particular content in math, go to Math Mistakes and search for mistakes in that content area. Looking at the examples (and discussion) of student work will help you prepare for your own students.
- We are students… of student thinking. Spend some time every week looking at a student mistake or two and trying to make sense of them. Write a comment. Sometimes I read the most recent post, other times I search for a specific content area I plan on teaching, sometimes I just browse around a grade level until I find something that interests me. There is also a button for just reading a random post.
- Do the above, but with a colleague.
- Post samples of your students’ work and have online discussions with other teachers.
- Share some of the math mistakes with our own students and asking them to understand/explain/correct the mistake.

Currently the site is almost entirely samples of K-12 student mistakes. Those are relevant to our work with adult students, and to our practice of making sense of student mistakes. But I would love folks to submit samples of mistakes from adult education classrooms across the state and country. The site is a place to house interesting mistakes, and I’d love to add adult student thinking to the mix. The more mistakes we post, the more interesting and useful the resource will be.

To submit a picture of mathematical work, email **michael@mathmistakes.org**.

In the comment box below, please let me know what you think about any of this. How do you use student mistakes? How does the fear of making mistakes stymy our students learning? How can we help our students feel comfortable making mistakes in math class?

Thanks, Mark, for writing about something important that needs reinforcement: mistakes are GOOD for learning! I enjoyed your post alot.

I figured this out when I noticed that I learned so much more about how to help students when I examined their mistakes; if I learned valuable information, wouldn’t they? Mistakes are royalty in my class, and we welcome them with honor. I tell them, “When you make a mistake, and share it, we all learn from that mistake, freeing us to make other mistakes!”

The bonus: students get out from under that rock of shame about being “wrong”.

Added bonus: I get to own the mistakes I make anyway, and embody the idea of teacher as facilitator, not authority.

Thanks, Lisa. I appreciate your comment!

I remember when I started out teaching, I had a conversation with a group of more experienced teachers. We were talking about why students make mistakes in math. We had been talking for a while and had a long list of things like, “They don’t work on math outside of class” or “They don’t know their multiplication tables”. But then one of the teachers said something so simple that we had all been ignoring and it hit the conversation like a bolt of lightning. What she said was, “Students make mistakes in math because they are learning and everyone makes mistakes when they are learning something new”. It really shifted my perspective in a way I have never forgotten. I’m not sure how or when mistakes in math got such a bad rap, but too many students (and maybe teachers too) think of mistakes as a bad thing. As you say, they are good for learning.

Your comment raises another really important point – what we do with mistakes and how we treat them in class will go a long way towards determining how students feel about making them. There is real power in students (and teachers) thinking of mistakes as opportunities and valued contributions. I love the way you put it – “Mistakes are royalty in my class”. We only learn from the mistakes we make

What do other people think? How do you use mistakes in your math classes? What are some strategies for developing a classroom culture where mistakes are valued?

I like this idea of putting a student “mistake” – wrong answer to a problem- on the board – and as a group discussing it. What’s right and what’s wrong. Looking at the thinking in an active way. My advanced HSE math section is working on basic algebra and its relation to geometry. In fact we just worked on a problem involving vertical angles; one is 2x + 5; the opposite is 3x – 10. Set them equal and solve for x. However we found ourselves talking about what is a vertical angle; then what is an “angle”. Why are vertical angles equal. My assumptions about a “closed problem” led to all kinds of open-endedness. It was interesting to see where the discussion led, and what they didn’t know, and what assumptions they brought to a strange problem. By sharing it we felt we all (both advanced and less advanced) had a stake in the game. And just what is “x” anyhow, and its relationship to an “angle”, which we thought was a measure in “degrees” (someone mentioned a protractor)?

“In my experience, I have found most student mistakes and misconceptions are wrong, but they are logical. That is to say, they are coming from some place – perhaps a concept learned correctly being applied to a new situation incorrectly or inappropriately.”

This review has reinforced the need for a technique that I use in my classrooms which is a “Things I do Wrong List”. I encourage students to keep such a list so that they can begin to understand why they do the things that they do “wrong”. It also helps to dim the frustration of making mistakes by framing mistakes as something that is “normal” and essential to the learning process. Over time, students begin to anticipate their mistakes because they begin to understand more and more that there is a logic to it all. I will definitely be using the Math Mistakes website as a way to further build my understanding of student learning.

Hey Wynnie, you have a way with words – “dim the frustration” I can just see it! Thanks, Richard

And this comment is why, Richard, it is so wonderful to do what we do.

I believe this quote by Albert Einstein says it all, but I will have more to share soon.

“A person who never made mistakes, never tried anything new.”

One of the problems my students have encountered is reading a word problem incorrectly. A word problem might require for a 2 or 3 step solution. Students will respond with a partial answer. Another mistake is when they do not fully understand the problem or use the wrong mathematical operation. To remedy the situation, as I re-read the word problem with them, they highlight the important parts of the problem. Students must proofread the problem to see if all the questions have been answered.

Michael Pershan gave a great talk at the 2015 NCTM conference in Boston.

Here’s the description:

Imagine staring at a math problem that you just don’t get. You want help, but you want the right kind of help – something that gives you a chance to be smart and leaves you with a tool to apply to other problems. In this talk I’ll explain why most hints let our students down and how we can do better by our kids.

http://www.shadowmathcon.com/michael-pershan/