Recently I read a New Yorker article about Brian Greene’s multimedia project, which paid tribute to some of Albert Einstein’s discoveries. It was launched on the centennial celebration of his general theory of relativity. Brian Greene is one of the more famous string theorists due to the popularity of his books which have included Fabric of the Cosmos and The Elegant Universe among others. He is a brilliant physicist and mathematician. In the article the author relates a story about Greene being set to the task of figuring out how many inches it is from here to the Andromeda galaxy by his father… when he was five. Not typical five-year-old work. Which then segues into a “consolation” for mathphobics that even Greene struggles to comprehend the homework assigned to his children in third and fifth grade.
“It’s all about strategies—‘Come up with a strategy’— and my kids are, like, ‘I don’t have a strategy, I just know it.’” Really? He has time for infinity, but can’t seem to piece together strategies for how students can learn to work with numbers?
We don’t “just know” things. If we know them, we know them for a reason – we understand something about how the numbers go together. It is simply a matter of slowing down and saying “How do I know that?” This is unsubtle Common Core and/or “new math” bashing, and it drives me crazy. For one thing, the article begins with a love letter to Albert Einstein who was able to put ideas together in new ways – to see hidden relationships – to manipulate ideas. This is how we (I’m a fifth grade teacher) teach kids to work with numbers! We use strategies, we take numbers apart to look at how they are constructed and put them back together in new ways. Consider the (5th grade) problem of 36 x 25. Where does sense-making occur?
|Standard Algorithm||Strategy: Doubling and Halving|
|36 x 25 = 18 x 50 = 9 x 100 = 900|
Mathematics is an art form. It is a dance that the human mind does with ideas within the science of pattern and order. The strategies we teach elementary students are well known to mathematicians.
To those who do not know mathematics, it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature…If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in.
And the brilliant and revered Feynman was even outdone (at least once!) in his “basic” understanding of arithmetic. An amusing story was related in James Gleick’s book Genius: The Life and Science of Richard Feynman, where the Nobel laureate Hans Bethe (one of Feynman’s mentors) out-computes Feynman with… you guessed it: a strategy.
When Bethe and Feynman went up against each other in games of calculating, they competed with special pleasure. Onlookers were often surprised, and not because the upstart Feynman bested his famous elder. On the contrary, more often the slow-speaking Bethe tended to out-compute Feynman. Early in the project they were working together on a formula that required the square of 48. Feynman reached across his desk for the Marchant mechanical calculator.
Bethe said, “It’s twenty-three hundred.”
Feynman started to punch the keys anyway. “You want to know exactly?” Bethe said. “It’s twenty-three hundred and four. Don’t you know how to take squares of numbers near fifty?” He explained the trick. Fifty squared is 2,500 (no thinking needed). For numbers a few more or less than 50, the approximate square is that many hundreds more or less than 2,500. Because 48 is 2 less than 50, 48 squared is 200 less than 2,500 — thus 2,300. To make a final tiny correction to the precise answer, just take that difference again — 2 — and square it. Thus 2,304.
(Feynman quote and anecdote found here)
Then to add insult to injury, Greene goes on to say that humans are not wired for math, but rather that we are wired to avoid lions or catch bison. I’m so confused. Is this the voice of someone who understands the language of mathematics to be both universal and unique to humans (in that we invented all of the math we have)? If the human mind is “wired,” it is certainly wired for math.
If ESSA is going to back off of on its support for the Common Core, that should not be an indictment of the principles and standards that have supported sense-making in mathematics. The math we teach students today helps them make sense of numbers the way that mathematicians make sense of numbers. The ideas are important, the strategies for working with numbers are important—the way students learn how to solve problems is important. We have no idea what the future holds for our ten-year-olds, but we know that they will need to be competent in understanding how to strategize, communicate, and reason. A Google search can tell you how far the Andromeda galaxy is from here. If that number is not in inches, a Google unit conversion will finish the job.
Let’s do some real math.