Wait. What? No one likes to fail. But being good at failing is a top skill, especially in mathematics. This is particularly true for advanced learners or students in gifted programs. In this article, we will take a look at why failing is a good thing, and provide some tips on how to help your students get great at it!
Science Got it Right
Your child is doing a science experiment. They make their hypotheses, set up their materials, and dive in. They wait patiently to see the expected reaction but nothing happens. Turns out the hypothesis is wrong, they did something in their experiment that made the reaction null, what do we do? Panic? Think that our entire life is over because we are going to fail this project? No. Not at all. We type up our report that says we were wrong and here is why. We say the experiment didn't work and consider deeply the processes involved to work as a detective and figure out where we went wrong.
That's right, in science - failing is cool. It isn't just cool, it is desired! Proving your hypothesis wrong, can often give you more insight than getting it right. A 'failed' experiment isn't real failure, it allows students to dig deeper into the concepts and learn more than they probably would have if everything went off without a hitch. So why isn't that the case in math?
While failing in math certainly doesn't seem "cool", it should be. Math and science are closely related and the same failed hypotheses in science make for excellent opportunities for growth and discussion in math! The push for perfection - especially with advanced learners, likely stems from the idea that in math there is always one right answer. And while that may be true, there are infinite ways to get there! Just like science, not getting to the expected result shouldn't spell out disaster - it should present an opportunity to dive deeper, learn more, and - if your program is any good - have so much fun along the way!!
Fermat's Last Theorem
Don't believe us? Let's take a quick look at one of the most impressive feats of our lifetime. The Pythagorean Theorem (if not familiar or interested in a beautiful visual approach check out the animation from MathBait) is the well-known formula a²+b²=c² which describes the relationship between the side lengths in a Euclidean right-triangle. There are many side lengths that satisfy this equation. We call these Pythagorean Triples and they include 3-4-5, 5-12-13, 8-15-17, and 7-24-25 triangles to name a few.
Things work out nicely when we are looking at squares. But what about cubes? Does a³+b³=c³? Fermat claimed that there are no (like none, zero, zilch) integers that fit this equation. Not only that, there are no integer solutions for any exponent greater than 2.
That isn't even the good part. When Fermat claimed this discovery, he added in the tiny margins of his book that he had "assuredly found an admirable proof of this, but the margin is too narrow to contain it". WHAT?!?! Get another piece of paper Fermat! (Marco explores this idea and in fact, his moniker, Marco the Great, is inspired by Fermat. Check it out in Marco the Great and the History of Numberville). It turns out Fermat had a bit of a history with making unsubstantiated "theorems" that later were credited to him after other mathematicians did all the heavy labor.
Anyhow, people tried for hundreds of years to find Fermat's proof. Then they moved on to try to find any proof of his claim. From 1637 when Fermat first scribbled his findings until 1993, not a single mathematician was successful. Enter Andrew Wiles. Using mathematics Fermat could not have possibly had any clue of in his time, Wiles readied himself to present at a conference. He couldn't believe he had done it, but there it was! As he presented the room was a buzz. This is the most exciting thing to happen to a room full of mathematicians in ages! What a triumph! Wiles had proven Fermat's Last Theorem!
But wait, I thought this story was about failing? As soon as Wiles presented his proof, others began tearing it apart. Was every statement truly the logical conclusion of the previous? It turns out, they were not. There was an error in Wiles' proof and for the next year he went out of the public eye to tackle his mistake. Wiles was ultimately successful, but in an interview with NOVA he discusses how he first started working on Fermat's Last Theorem as a teenager! Can you imagine how many times he tried and failed?
The Key Points
Real mathematicians fail everyday, every minute. We try things, they don't work. We try again. That is the beauty of mathematics!
In other subjects, we view 'failure' very differently. In English we turn in rough drafts, earn feedback and improve our work. In art we sketch and erase, until the proportions are just right. In science, we often discuss how a failed experiment is wonderful. In fact Edison is famous for saying,
"I didn't fail. I just found 2,000 ways not to make a lightbulb. I only needed to find one way to make it work."
The same is true in mathematics. Often not getting something right, gives us much more information - particularly in the early grades. Unfortunately, most students don't see it this way.
Teach Your Kids to Fail
What can we do as parents and educators? Teach your kids to fail! Celebrate failures as much as successes. A 'failure' is just another word for a 'learning opportunity'. We can change the narrative to help students to see that it is okay to not get the right answer. Not only does this open the door for students to become detectives, to investigate their steps and choices and deepen their understanding in the process, it also helps them to develop real-world skills and coping mechanisms.
Even the most advanced students will get something wrong in their lifetime. They are human. Help them to develop the skills that guide them on what to do next. Retrace your steps, challenge your assumptions, try again. This can take a lot of pressure off their shoulders, reduce any math anxiety, and make learning a much more enjoyable experience.
You might be wondering - what does this look like in math? As a society, most Americans learned memorization was the key to math (it's not). We are so used to right/wrong we might have trouble even imagining what this looks like!
We wish to answer the following question:
Here is our student's work:
What beauty! Look at their clear and concise steps. Everything appears to make sense, so what went wrong?
Students love to cancel, despite the majority not actually understanding what 'cancelling' is. (If this is your student, make sure to grab your copy of Marco the Great and the History of Numberville and they will be wielding new and magical powers before spring break ends!) They see the x-1 in both the numerator and denominator and decide to throw those out of their equation.
In their first step (the second line of the work shown above) the student multiplied everything by x-1. That's fair game. But what they did next, was not. They cancelled. They banished the x-1 from their equation. We can't do that!
You see, cancelling works because when you divide a number by itself, you generally get 1. 5/5=1, 8/8=1, π/π=1, and so on and so forth. Students most often learn the rule and not the reason, meaning when it comes to slashing things away - they do it without care. Importantly, 0/0≠1. When we have (x-1)/(x-1) we can only 'cancel' this if it is not 0. Turns out it can be zero, and it is zero exactly when x=1.
That normally means all that beautiful, hard work, is down the drain. The answer isn't right and that is all that matters in math. We need to change that! Is the answer wrong? Yes. Is this a beautiful display of work that demonstrates understanding of a ton of key topics? Also yes! Should this student feel like they 'failed'? Absolutely not!
Take a page from science. Here is a student's science report where everything failed.
Wow! What a difference. The student doesn't feel like they failed, they simply found 'how not to make a lightbulb'. They studied what went wrong to determine how movement impacted the results allowing the student to learn more about the process of photosynthesis.
At MathBait, we always want students to feel successful and we especially want them to learn from their mistakes, dig in to see what went wrong, and come out the other side with a deeper understanding. We utilize game-based learning to help encourage this new view of mathematics. Students feel much more comfortable losing a game. In fact, good games are never easy! They develop a grit and a desire to 'beat the game' and each time they play they learn a little more.
Helping Your Kids to Fail
What can parents do? Start by giving students permission to fail. Often grades, performance, college, and all the other stressors make students feel like failing isn't an option. And while we aren't recommending purposely failing an entire course, instead let your kids know it is okay to get something wrong.
1. Let your kids know it is okay to get a wrong answer
Don't stop there! Encourage students to understand and explain why they were incorrect. Push them to put on their detective hats, go back through the problem, and identify where they went wrong.
2. Encourage them to play detective and find the mistake
Finally, help them to understand why their step didn't work. This helps bolster their memory and understanding. In the example above, we'd explain that cancelling doesn't work when we have 0/0. Any time we cancel with an unknown value, we have to make sure we don't get a zero in the denominator.
3. Help them to understand why it didn't work
While we are not sure about totally redesigning math education to include science-like reports, we do know that allowing your kids to make mistakes and learn from them will reduce math related stress and anxiety, allow them to develop a deeper understanding, and help them to develop essential lifelong skills that transfer to other areas.
So next time your student brings home a test or assignment with questions wrong, don't get mad! Congratulate them on their failures, they are one step closer to being a mathematician!
Solving Fermat: Andrew Wiles. Interview with NOVA. www.pbs.org/wgbh/nova/proof/wiles.html