I teach a calculus class. It’s the first time that I’ve done that. I have a story about them.

Yesterday, I introduced the idea of limits, which, according to them, was the first time any of them had ever had limits discussed in any way in a math class. After a short lecture-style introduction about the basic idea of limits, the notation, and some anecdotal comments, I unleashed them on this handout. I was not going to collect handout. I wasn’t going to fasten any points to this handout. They knew both of those things. I gave them 12-15 minutes to try the handout. This is where life gets interesting.

About 6 minutes into the time, I began walking around to look at some work. There are 23 students in the class. I estimate that 17 of them had, for as many as 10 minutes had done zero math, but had done a fantastic job of transferring the table from the handout into their notes. I was flabbergasted. They had spent 10 minutes drawing a table. These calculus students (who, by most traditional metrics represent the most confident and talented math students in our student population) wouldn’t try the math.

Their years of math class “success” had thought them two things that created this scenario: A. Stay busy, or at least *look* busy. Teachers get mad at students who sit around. And B. The only answers worth writing are correct answers. If you don’t know the correct answer, don’t write anything.

These students didn’t want to write a wrong answer. They didn’t know the right answers, so they opted to create incredibly high quality tables in their notes. Their explanation for this usually included the statement, “Well, I didn’t know how to do it.”

I responded with, “Of course you don’t know how to do all of this, I introduced it 10 minutes ago. I’m just looking for you to give it your best try to see where your thoughts are taking you right now.”

After about a 5 minute pep talk, they tried, and much to the surprise of most of them, most of their attempts where actually correct answers.

And make no mistake, this isn’t just calculus. This isn’t just the traditionally-successful math students. Anyone who has taught a math class has seen students who know that they don’t know how to solve a problem and would much rather leave the problem blank than put something down that is wrong. The wrong answer seems worthless.

It seems that we have conditioned our students into thinking that an answer on a page is an opportunity for judgement. If they write something on the page, I’m going to have the final say whether it is right or wrong. Wrong answers are bad. Grades go down because of wrong answers. This mentality would prefer to leave an answer blank. At least if it is blank, you can pretend that you simply need more time.

Instead, we should be showing our students that an answer on a page is the beginning of a conversation that ends with them learning something new. It could be that the answer on the page confirms that they have already learned (which makes for a short conversation). It could also be that the answer on the page demonstrates that more learning is needed, and the answer on the page is the window into the confusion, clues to the misconceptions or the missing understanding. When learning is incomplete, perhaps the BEST thing a student can do is show us his or her very best wrong answer.

But first we have to teach our students that anytime they put what is in their heads on paper, there is value. There is value in a correct answer and there is value in an incorrect answer. It’s true that they are valuable for different reasons, but they push toward the same end. The authentic learning of mathematics.

Perhaps if we can reestablish the value of an incorrect answer, we can do something about this incredibly pervasive fear of trying.

A wonderful piece, Sir. It’s almost impossible to create something new if you’re not ready to fail.

Fawn Nyugen suggests an activity called “My Favourite No” to use partially as formative assessment, and partially to build up kids tolerance to writing down work which might be wrong.

She has kids write down the answers to their problems anonymously on index cards. She collects all the cards, and flips through them until she finds a response with some issues in it and then puts it under the document camera for everyone to look at and discuss. I think she has them find what is positive about the mistake, and what needs work, try and figure out what the person was thinking.

Mindset is a huge issue in math with lots of people to blame for the current state of affairs in math ed.

You’re right that there is plenty of blame to go around. That also makes for a problem with an incredibly tricky solution. My sister (who is a researcher at U of Mich. hospital) says the same about her 22-year-old interns. They don’t seem comfortable taking risks. If they don’t know exactly what to do and how, they’re paralyzed.

So, there is something the “academic” mindset that is producing this as a side effect… At least I hope it’s a side effect.

Beautifully said, however, as you are well aware, with the school environment being so focused on testing (testing where the correct answer is the ultimate end result) learning to fail, to accept that not finding the correct answer is ok, learning to work through processes with no potential perfect end–these are not in the ethos of schools (elementary on up).

It’s a bizarre ethos, too. I read one of Sir Ken Robinson’s books which lays out (quite convincingly) that the way real, applicable learning happens tends to be directly at odds with the practices of practically all public schools in America.

This is why we advocate posing problems with unclear approaches from an early age and focusing on their solutions (plural) more than the answer (singular). For instance, this primary school octagon problem…

http://fivetriangles.blogspot.com/2013/09/99-defaced-stop-sign.html

…has neither obvious solutions nor an obvious answer (at least to any of us, it didn’t); the student has to try something in order to proceed. A try may lead to a dead end, in which case the student has to backtrack and try something else.

I really like the idea of perplexity. An obvious answer or obvious procedure shuts down exploration. Getting the students used to exploring gets them used to learning as a procedure as opposed to a knowledge transfer.

So true. I am a middle school math teacher. “we have to teach our students that anytime they put what is in their heads on paper, there is value” is to me the most important statement in the blog. Value is built by work, and it is impossible to do any kind of work without getting some benefit. That is why I show them homework from a college class years ago. I haven’t thrown it away because I don’t create junk. It took work to be neat and complete and now it is a resource for review and example. Students need to grasp that work is not one way to become good at something, or to develop your talents or your mind–it is the only way.