This was great! I think perhaps the most common way teachers (not merely in math) use this is, unfortunately, too limited and late. That is when they write distractors for multiple choice questions. Because we know the kind of mistakes kids are likely to make, we make sure our distractors seem plausible to students, so as to identify which students know their stuff most thoroughly.

But Dan's post today makes me realize what an opportunity I've been missing. I've long taken not a small amount of pride in how well I've taught my students to deal with mistakes, even getting excited over a particularly common error and having kids think through what "Matthew" was thinking when he got that wrong. But this is really great, asking them to actually SEEK possible misavenues of thought. I definitely need to try to work this into my teaching.

One challenge (and source of frustration!) for me has been trying to get students to make ANY guess when approaching a new problem. Afraid of being wrong, of course, but also I think a product of mathematical training where there's always a procedure and you must take no step until the whole procedure is laid out before you (I teach college, so my students have had years of such training before they see me.)

Asking for wrong answers seems like a really clever way to at least get some answers. Like with the Gummi bear question, asking for estimates, an upper bound and a lower bound, might get few responses but "What are some wrong answers" is really doing the same thing and might get more responses. And then asking "Which answers are MORE wrong?" could start you narrowing your range between upper bound ("too big") and lower bound ("too small") til you actually arrive at a useful estimate.

I ask, "Wrong answers only!" when students are nervous, and that does help. I've also asked students to make as many mistakes as they possibly can in one exercise, or make as bad a mistake as they can. These tasks require some meta-cognition: What makes a mistake bad? Where exactly can our math go horribly wrong, and how?

More broadly, it's important to ask students to generate "many" (three or more) examples of their own whenever they meet a new math object. They also need to make many NON-examples! (And counter-examples, if we are working with logic and proofs.) Making non-examples is similar to requesting mistakes, as pedagogical principles go. Both invite meta-cognitive reflection and increase mathematical confidence.

Someone inevitably does, and I find that helpful. My quick response is, "Haha, good one" - they KNOW it's ridiculous, and it's not trivial to know that, mathematically speaking. Plus, tension relief!

Love this framing, thanks Dan! Encouraging mistakes has many positive benefits:

1. It helps students focus on the reason why something is right instead of focusing on the answer itself. This helps them actually learn and be able to apply it elsewhere.

2. Encourages a bias toward action. If you are actually stretching yourself, you’re gonna fail. If you’re not making mistakes, you’re playing it too safe.

3. Teaches you to not place your identity in being “right.” Focusing too much on the product isn’t good for us. Much better to spend our attention building a rock-solid process.

Use the system of linear equations if helpful to move students to vorrect thinking based on theirs. I taught algebra one for 15 years. Knowing what a “solution” was can be a hurdle.

Right - this is pretty crucial IMO. "What is this question even asking for?" is a question lurking beneath lots of student confusion. And it's a great opportunity to ask for a wrong answer. The wrong answer helps students understand the SHAPE of a right one.

I’m thinking that there is a lot of merit in the approach. After students have identified erroneous answers, what are the steps suggested to move them to correct thinking?

Consider doing whatever you would have done without asking first for wrong answers, only now equipped with more resources, for example a student's stronger sense of the solution space.

Maybe worth checking out RightOn! an app where teachers have the class play games/competitions to spot the wrong answer and identify the underlying misconception. Sinclair Wu is founder...

This was great! I think perhaps the most common way teachers (not merely in math) use this is, unfortunately, too limited and late. That is when they write distractors for multiple choice questions. Because we know the kind of mistakes kids are likely to make, we make sure our distractors seem plausible to students, so as to identify which students know their stuff most thoroughly.

But Dan's post today makes me realize what an opportunity I've been missing. I've long taken not a small amount of pride in how well I've taught my students to deal with mistakes, even getting excited over a particularly common error and having kids think through what "Matthew" was thinking when he got that wrong. But this is really great, asking them to actually SEEK possible misavenues of thought. I definitely need to try to work this into my teaching.

Right there with you!

edited Nov 30, 2023One challenge (and source of frustration!) for me has been trying to get students to make ANY guess when approaching a new problem. Afraid of being wrong, of course, but also I think a product of mathematical training where there's always a procedure and you must take no step until the whole procedure is laid out before you (I teach college, so my students have had years of such training before they see me.)

Asking for wrong answers seems like a really clever way to at least get some answers. Like with the Gummi bear question, asking for estimates, an upper bound and a lower bound, might get few responses but "What are some wrong answers" is really doing the same thing and might get more responses. And then asking "Which answers are MORE wrong?" could start you narrowing your range between upper bound ("too big") and lower bound ("too small") til you actually arrive at a useful estimate.

"Give me your BRAVEST wrong answer" is language that I have found useful in the past.

I ask, "Wrong answers only!" when students are nervous, and that does help. I've also asked students to make as many mistakes as they possibly can in one exercise, or make as bad a mistake as they can. These tasks require some meta-cognition: What makes a mistake bad? Where exactly can our math go horribly wrong, and how?

More broadly, it's important to ask students to generate "many" (three or more) examples of their own whenever they meet a new math object. They also need to make many NON-examples! (And counter-examples, if we are working with logic and proofs.) Making non-examples is similar to requesting mistakes, as pedagogical principles go. Both invite meta-cognitive reflection and increase mathematical confidence.

Maria, when you ask for wrong answers only, do you ever have an issue with kids giving ridiculous (non-plausible) answers?

Someone inevitably does, and I find that helpful. My quick response is, "Haha, good one" - they KNOW it's ridiculous, and it's not trivial to know that, mathematically speaking. Plus, tension relief!

Love this framing, thanks Dan! Encouraging mistakes has many positive benefits:

1. It helps students focus on the reason why something is right instead of focusing on the answer itself. This helps them actually learn and be able to apply it elsewhere.

2. Encourages a bias toward action. If you are actually stretching yourself, you’re gonna fail. If you’re not making mistakes, you’re playing it too safe.

3. Teaches you to not place your identity in being “right.” Focusing too much on the product isn’t good for us. Much better to spend our attention building a rock-solid process.

Use the system of linear equations if helpful to move students to vorrect thinking based on theirs. I taught algebra one for 15 years. Knowing what a “solution” was can be a hurdle.

Right - this is pretty crucial IMO. "What is this question even asking for?" is a question lurking beneath lots of student confusion. And it's a great opportunity to ask for a wrong answer. The wrong answer helps students understand the SHAPE of a right one.

I’m thinking that there is a lot of merit in the approach. After students have identified erroneous answers, what are the steps suggested to move them to correct thinking?

Consider doing whatever you would have done without asking first for wrong answers, only now equipped with more resources, for example a student's stronger sense of the solution space.

Maybe worth checking out RightOn! an app where teachers have the class play games/competitions to spot the wrong answer and identify the underlying misconception. Sinclair Wu is founder...

Got a reference for the wong article?