If you've been following this blog for a while, you'll know that I have strong opinions about the misconception that "math is only for the gifted." I've written about the importance of endurance and hard work several times:

A Math Blog? Say What?

A Math Blog? Say What?

Real Talk: Math is Hard, Not Impossible

Real Talk: Math is Hard, Not Impossible

Hey There Grad Student, You're in Good Company, part 1

Hey There Grad Student, You're in Good Company, part 1



Graduate School: Where Grades DOn't matter

Graduate School: Where Grades DOn't matter

A Ramble About Qualifying Exams

A Ramble About Qualifying Exams


Naturally, these convictions carried over into my own classroom this past semester as I taught a group of college algebra students.

Whether they raised their hand during a lecture and gave a "wrong" answer, received a less-than-perfect score on an exam or quiz, or felt completely confused during a lesson, I tried to emphasize that things aren't always as bad as they seem:

  • "Wrong" answers are good because 1) they mean you're actually thinking and 2) they often lead to interesting discussions which then lead to the "right" answer. 
  • Receiving a poor grade means you can identify exactly what you need to work on. This enables you to devise a precise plan-of-attack to improve any weak spots. Sticking to that plan means your chances of succeeding in the future are higher.
  • Being confused gives you an opportunity to dig deep into the material and hone in on the underlying ideas. The result of such diligence is a thorough understanding of the material that you may not have had otherwise.

Looking back, I'm pleasantly surprised by my students' progress. Quite a few started off rough but steadily improved during the course. Even better, I wasn't alone in this observation. My students saw it as well. Taking a cue from MAA president Francis Su, I had decided to use our last exam to reinforce this idea that there is value in the struggle. I gave my students the examples listed above and asked them to describe an instance where they thought a struggle turned out to be a bright spot.

Here are just a few of their responses:



My exam #1 grade was low, danger zone low. This was such a bright spot because it helped me understand that college is nothing like high school. You have to study to get a good grade, you have to do the homework and even work with a tutor and go to office hours. I adjusted to college, I started studying.... I made sure I was learning, and with this 'aha!' moment... I realized that this is college.

This student failed our first exam, improved steadily during the semester, began speaking up in class more, and earned an A+ on the last exam. I'm tickled pink!



On my first exam, I received a really poor grade. I was ashamed and worried…. This was a bright spot because the ‘scare’ I had of failing the second exam like I did the first one made me study harder, go to office hours, and ask questions. In a way, it was good that I failed because it opened my eyes and made me seek for help.

This student also failed our first exam but improved on the following one by 45 points. I'm impressed!



I struggled on a recent quiz... using trigonometric functions. When I saw my grade I felt extremely disappointed, but it helped me focus more on that section. Now I have a better understanding of that specific section. The point is, it's okay to not get the 'perfect score.' The low grade I received motivate[d] me more to do more practice.

Indeed, this student mastered all the trigonometry problems on the following two exams. Woo!



The bright spot about struggling means you have to use more brain power to think about what mistakes you are making....  When receiving wrong answers, it is better to discuss [them] because then you realized the parts you were confused about which leads to finding the correct solution.

Right on. Couldn't have said it better myself.



I have learned that simply doing the HW can save me a lot of grief and stress in the future.

I thought this was great, coming from a C student who was well aware that his time management skills weren't up to par. I think he figured something out, because his work on our final exam was stellar.






Now, truth be told, I'm not entirely sure that my students' results were a byproduct of our classroom environment. Perhaps I just happened to be blessed with a group of highly motivated kids? In either case, I thought this was worth blogging about.

I'm reminded of what Carol Dweck, Standford University professor and advocate of the "growth mindset," said in a recent interview as she reflected on her 40-year career: 


"...when students had more of a growth mindset, they held the view that talents and abilities could be developed and that challenges were the way to do it....

Setbacks and feedback weren't about your abilities, they were information you could use to help yourself learn."


She also noted that "a student's mindset was at the foundation of whether [he or she] loved challenges and persisted in the face of failure."

This goes hand-in-hand with a comment Andrew Wiles recently made on the "state of being stuck" that those learning new math often experience:


"It's like training in sport. If you want to run fast, you have to train. Anything where you're trying to do something new, you have to go through this difficult period. It's not something to be frightened of. Everybody goes through it.... 

I really believe that most people can really get to quite a good level in mathematics if they're prepared to deal with these more psychological issues of how to handle the situation of being stuck."


Even Cédric Villani noted that

"Mathematics, in some sense, will always involve a little pain."

For Villani, that pain led to a Fields Medal! For my students (and myself!), I hope it will at least lead to a stronger will to endure and a deeper appreciation for mathematics.