Those of us who care about math education and making it more engaging generally have strong opinions about “math drills” and the over-emphasis on standard algorithms (ie. doing pencil-and-paper arithmetic the way that one’s parents were taught to.) The progressive view is generally that while students *do* need to learn basic facts, they should also build understanding of what those facts mean so that they can tell where and when to apply them. They should be able to work with the ideas of arithmetic and algebra through multiple representations, and alternate algorithms are great.
(If you want to debate that view, that’s fine, but maybe not here right now. There’s a bigger point coming.)
So in that context, here’s the thought I had that struck an odd chord with me:
Drilling standard algorithms used to be the right answer.
I don’t know if the weight of that statement really hits by just reading it here, so let me unpack a bit.
When something new comes out, it changes how we perceive the previous tools. A classic example is how photography forever changed the role and perception of painting. Before photography, a painting or drawing was the only way to create an accurate, persistent image of someone or something. Post-photography, the role of the traditional visual arts in society had to be re-examined and a new form of art emerged.
But while most of us have at least a pop-culture awareness of “modern” art’s experimentalism, what we usually don’t think about is how our view of the pre-photography era has shifted as well. For centuries, painting was valued as a way to (re)produce images from life. If a painter had used tools such as lenses, measuring devices, and mirrors to produce the image realistically, this would have been seen as simply part of the trade and a valuable skill. Nowadays because the actual production of realistic images is trivial, we have instead romanticized the ability to produce beautiful images out of paint and hand-skill alone. Most people would assume that using light tables to trace, lenses to project or Photoshop to digitally paint over a photograph is somehow cheating. This can be evidenced even in the larger art world by the reaction of many art historians to David Hockney’s Secret Knowledge, a book detailing his theory that the shift to photo-like realism in the 1400-1500s and beyond was likely because of improvements in optics that artists could use to project images onto a canvas as a guide. Many of the historians view this as slanderous to the great artists of old, rather than a tribute to their ingenuity.
For 20th century math, the calculator and the computer were to pencil-and-paper mathematics what photography was to painting.
It is incredibly hard to look back at the past through the lens of the present without projecting the present backwards.
In my case, I “knew” that pencil-and-paper used to be the only option for doing math before the calculator. But this wasn’t enough for my imagination to really picture how different the workplace would’ve looked for those dealing in numbers.
The first time something went ‘click’ in my head to upgrade this was an old photo someone had shared online (which I wish I could find again!) of a business analyst’s office from I think the early 1970’s. The entire room was covered with hand-drawn charts. COVERED. Every wall. And we’re talking hand-drawn line graphs where sometimes they had to add an extra bit of paper at the bottom when things went out of the original scale.
Something clicked then – this was all they had. You had to learn how to draw graphs by hand because that was the only way you would have a graph. There was no Excel. The enormity of what this meant for the workplace, not just the math classroom, started to sink in. This wasn’t just an exercise, nor was it paperwork filed away. This was vital business analysis required to understand the direction of the company, and it was done in meticulous hand-drawn, possibly hand-calculated ink.
There were other moments like this – a scene from Apollo 13 where NASA engineers confirm a calculation by having three of them quickly scrawl on paper and then give successive thumbs up; teaching Accounting briefly and learning about the history of bookkeeping and how, yes, it was actual books and here’s how they were organized.
Workplaces used to need people who could not only do hand-calculations, but could do them in a recognizable, standardized way. And anyone working with numbers – anyone dealing with money, measurements, analysis – would be significantly slowed down in their day-to-day work if they couldn’t do those calculations quickly. Speed wasn’t the key to understanding, ever, but it may have been the key to getting the job done by a reasonable hour and going home.
In this light, of course students were being taught to do their math quickly! It was a job skill. Knowing your multiplication tables well may have been the difference between opportunities in ‘knowledge work’ industries such as accounting, engineering, and sciences or having those doors closed, not because math streaming closed those doors but because you wouldn’t get the job without it, even if you did understand the concepts.
I don’t really know this. I mean, by the time I was in the workforce I was already working with PCs and doing store inventory in a database. I don’t know first-hand what the everyday experience of pre-digital workplace mathematics was like. I turned forty this year. I’m not the oldest math teacher around, nor am I the youngest. But I feel like there’s a perspective on why things used to be done they way they were done that’s about to be lost as our most senior teachers retire right now. Even someone sixty-five today would’ve seen pocket calculators begin hitting the market at about the same time as they entered the workforce.
I feel like math education is barely settling into how to adjust to calculators, and still awfully wobbly on how to adjust to computers. I don’t mean that as a criticism, just an observation. This entire complex system of society, technology, curriculum, and classroom simply seems to have a slow and messy feedback cycle. But I wonder if it gets even harder once everyone who understood the context for the old decisions are around to help us evaluate whether they’re right anymore.