Failing isn’t the portal to mathematical bliss due to some skewed, personal philosophy. Failure is the portal to mathematical bliss because that is its ENTIRE HISTORY.
If we could catalogue every person’s experiences with mathematics from the beginning of time, the overall narrative would be ripe with failure.
Mistakes. Misconceptions. Locked doors. Blind alleys. False paths. Incorrect assumptions. Wrong turns. Calculation errors. Oversights. Dead ends.
No solution is and has been the most common meeting point of everyone who has ever done mathematics — including every mathematical giant in the subject. Success in terms of proofs and theorems are merely the punctuation marks. There is a whole lot of text — disjointed at that — which often precedes these discoveries and epiphanies. It’s great to recognize someone like Andrew Wiles, but a richer story would be to rightfully include the thousands who fell horribly short in proving Fermat’s Last Theorem.
The way that mathematics gets presented is much like the classic iceberg model.
Math anxiety stems from many places. One of them is the almost unconscious and false idea that mathematics is only worthwhile if the journey of understanding ends with some kind of resolution/solution/success — usually in a time slot of 45 minutes.
Below is a textbook I used around 20 years ago for a HS course here in Ontario. Read the passage about the problems in this section. That message was for students. That message is universal. It applies to every teacher from K to 12.
Days or longer…sigh.
Solving math problems for correctness with speed. This is the most damaging idea we(not we, but our curriculum)directly/indirectly promote in mathematics. How damaging? Not only do students have math anxiety, but so do teachers — most of them being elementary. They need support and reassurance that their journeys in math knowledge is not static and invaluable to the collective voice of math education.
It is wonderful that we are sharing and reflecting about math at conferences/PD sessions. But, bottom line, we — the teachers — have to be doing the mathematics. English teachers read new books all the time. Math teachers should be doing new problems all the time. If you have time to read books on pedagogy, then you have time to try and solve a math problem or two. Whether you get them correct is almost immaterial. By doing new problems and not old ones, teachers will have real empathy for the struggles and failures students will come across. But, more than that…aren’t we collectively curious/excited about math.
Or, are we only excited about teaching it and watching students do the math.
We have created a culture where these bumps along the road don’t have the time to marinate and become woven into the personal exploration of mathematics. Not only is it alright to fail, it is a prerequisite for doing mathematics.
If you are not failing at mathematics, then you are certainly not doing mathematics. And, that can only become a pervasive idea if teachers themselves feel jubilation through the failing process and not the societal-confirmed anxiety.
In the universal spectrum of mathematics, I suck at it.
I don’t want to talk about my domain of knowledge in K to 12(even that needs vast improvement). I want to joyfully speak about how many things I just don’t understand, get confused by, and repeatedly fail at grasping. I didn’t start to have real confidence teaching calculus until probably my 10th time teaching it. I still don’t fully comprehend mathematical induction. And boy oh boy, I don’t think I will ever, ever get my head around the calculation of something like Graham’s Number.
Hopefully, one day, Ben Orlin will draw something again about Graham’s Number. But, instead of warning us, he will wholeheartedly invite us to revel in our misunderstanding.
Mathematics is not a spectator sport — for anyone. Pull out a pencil and do a problem that seems out of reach. Keep all your scribbled attempts and doodles. Don’t crumple them up like I used to as a student. They represent critical parts of our journeys.
Kintsugi, is the “Art of Precious Scars”. Broken pieces of ceramic are welded with gold. Mathematical failure is our broken ceramic bowls. Celebrating them is the gold.
Fail well, my friends. Fail well…