Mathematics is Messy. But Are We Teaching That?
The entire history of mathematics has been about slow, repeated failure — leaving a glorious residue of misconceptions, incorrect ideas, and broken proofs. That is the only story of mathematics, and it perfectly encapsulates the collective humanness which has stained thousands of years of exploration.
Our own stories are just as messy. We have all had — hopefully — some sobering failure with understanding mathematics. Some of us probably have deeper scars of trauma — that were installed by education, not by mathematics. Schools have created a false and unsustainable narrative that mathematics is a sterile journey based on repeated correctness under the notion of speed.
This is formalized as tests. What could go wrong?
Learning mathematics under these conditions has been unhealthy. Learning mathematics under these conditions has been a synthetic facsimile of the organic nature of mathematics. Mathematics is dirt. If you are not playing in the dirt, you are not playing mathematics.
The best example of how beautiful the mess can be with mathematics is something as innocent as fact fluency. Seems pretty sterile, right?
Earlier this year, when I was teaching a 6th grade class at Dexter Learning, I gave the students these 4 cards from Albert’s Insomnia.
The rules were straightforward. Use to one to four cards(no duplication) and the operations of adding, subtracting, multiplying, dividing, factorial, roots, and exponents to create PEDMAS questions from 1 to…
Our 6th grade Math Recess class went to 200. There is actually a 4th and 5th grader in my class.
While we obtained over 90% of the numbers from 1 to 200, the story of our class lies in the interstitial space between these discrete objectives. We literally went through the Looking Glass, and seeing numbers through a different lens. Here are some of the highlights I would like to share.
My board work on Google Jamboard was intentionally messy, which makes mathematics more disarming, with these spontaneous bursts of scribble/rough work which give natural texture to our mathematical stories.
As we progressed through the numbers, certain numbers became “anchors” and were used repeatedly to generate numbers in close proximity to them. Numbers were considered anchors if we could only use 1 or 2 cards to get them. In the beginning, 24 — derived by 4! — was used quite often. Later on the number 84(7 x 12) became a fixture for many answers. And, once we got passed 110, the magic of 5!(120) allowed us to generate all the answers from 111 to 130 in a symmetrical fashion. The 3 and the 4 could be used to generate all the numbers for one to nine. As such, on one side of 120, we subtracted them. On the other side, we added them. We blew threw these twenty numbers, as a class, in about 2 minutes.
As well, on every board I circled the primes. So, not only were kids learning about primes, but they were beginning to realize that they might always be the tougher ones to get — because we would never be able to multiply two numbers to get them. We would have to land close and get lucky with the remaining numbers and required operations.
This is inherent. Almost primal. Students needed to play with numbers, mix them with all the operations available, and get creative to obtain some answers. 12^(root 4) to generate 144 became mundane to all very soon. It just became a portal to having students stretch their creativity and imagination. As we got past 150, students started searching for anchors in this space. One must realize that most had gone outside their comfort zone of the cliched 12 x 12 times table. Sure enough, 24 x 7 equals 168 was manifested. The 24 coming from 4!.
Something very cool happened in the 160’s. Seraphina, a 5th grader, went way out into orbit to see 720 to build the anchor 180 — she used double factorials. Two weeks ago, she didn’t even know that operation. Now she was wielding that operation like a Jedi Knight with a light sabre. But really, so many of them do now. Such great contributions from students like Givi, Bowser, wilmatuker, JMassey, and others.
Another inherent, built-in trait of this game. The more stubborn an answer appeared, the higher the resolve from the class to find it. Some of these numbers may turn out to be impossible, but that didn’t deter the class. This critical mathematical trait was developed organically and intrinsically. Worse part of each class was when it had to end…
This was probably the most satisfying thing to come out of the last two weeks. The encouragement from the students to each other for finding certain answers or getting close/making a mistake with one of the rules. There was no competition whatsoever. Students cheered each other on.
To witness this collectivism/energy, I invited Berkeley Everett and Nikki Rohlfing(United Kingdom) to join a stream. They both witnessed the buoyant energy of the class — which had only gotten higher in the second week. Some students contributed more than others, but all students were appreciative of the surprises, which often resulted in obtaining some of the higher numbers.
Quite often that student was me!
Today is a wrap up day of our two weeks. We will take one last look at the numbers that eluded us. We will share our whys as to why we became enthralled with numbers and our messy playing with them.
Before we close the book on this one, I will open up a rabbit hole for them. I will mention the number 27. That it could have been an anchor number with just a 3 and 2. I will wait for perplexity to come to a rolling boil.
Then I will drop the hyper-operation, tetration on them:)
I went into far more detail about this in my upcoming book. Math’s magic is right in that messy middle!