Tuesday, March 20, 2012

Another Addition to the Banned List

Awhile back I wrote a post detailing some of the phrases and teaching strategies I would like to see banned from our nation's math classrooms. There are always little things that make me wonder why a teacher chose a certain strategy ("When you multiply by 1 it stays the same because 1 looks like a mirror, so it reflects back the number." Do you think that kids couldn't reason out why multiplying by 1 doesn't change the value?). But there's been a big one coming up lately that's kind of been driving me out of my mind.

"Reducing" Fractions
I understand why we say reduce, especially when accompanied by the phrase "to its simplest form," but I what do you think of when you think of something being reduced? You think of something lessening, you think of it having not as high of a value, you think of it not amounting to as much as it originally did, etc. Kids in the US have a lot of trouble with fractions, particularly with the concept that fractions represent a specific type of comparison of a part to a whole. When one 'reduces' a fraction, an important conceptual aspect is that even though the numerator and denominator change in a way that lessens their respective values, the fraction as a whole does not change in value at all. To say that a "reduced" fraction is equivalent to its original form is kind of an oxymoron--I'd love to see a sale where the reduced prices are equivalent to the original prices. When we use the word "reduced," are we surprised that kids don't understand fraction equivalence?

So what's the better option? Simplify. I love this word for a few reasons. First, it's preferable to "reduced" because it makes sense that a simplified version of something still has the same meaning/value as the original--now it's just in an easier-to-understand form. Second, I want kids/people to realize that any value can be represented in infinite ways, but we prefer some ways because they are easier to work with, compare, interpret, manipulate, or use in a given situation. Would you want to go to a restaurant where a sandwich cost $1-3^(0!)+300/(4x10)? Mathematically, there's nothing wrong with that price; realistically and emotionally, it's just annoying. If we simplify the price (not reduce because the restaurant still wants the same amount of money), we get a value that's more useful for our brains because the simplified value is easier to compare to what we already know about sandwich prices. Is  $1-3^(0!)+300/(4x10) expensive? Cheap? I have no idea until I simplify it to a value that looks like the other values I could compare it to. It's the same reason why we change the form (not the value) of a fraction when we need to add it to another fraction, or why we sometimes factor a quadratic into a binomial and sometimes multiply a binomal into a quadratic, or why we convert linear equations into slope-intercept form. Math is only useful if we can make meaning from it, so we manipulate values, expressions, and data sets until we can mold them into something that makes the meaning easier to find. We're not reducing anything when we rewrite 32/40 as 4/5; we're simplifying it into an equivalent form that's more useful.

Wednesday, March 14, 2012

Happy Holiday

About a month ago, I got this in the mail:

Unfortunately, I did not heed the urgency of Atlas Pen & Pencil Corp's envelope and once again Pi Day has snuck up on me before I even realized it. But thanks to the nerdy blogs I read and even nerdier company I keep, next yet I will be prepared with an arsenal of Pi Day-related fun. Here, for your celebratory enjoyment:

This is so freaking cool. Is it worth the time to cut out all those apple and pie crust slices? Probably.

Thanks to this useful allocation of government time and money, I now know what "squaring the circle" means. Dear State of Indiana: do not try to legislate mathematics because it is above the law. Or, more accurately, mathematics is the law.

My students were super-excited that they are going to graduate in very important year, Pi Day-speaking. For the only time in any of their lives, it is their senior year of high school when we will get to have a completely legitimate Pi Day on 3.14.15. Some were more excited than maybe they should have been.

Finally, in case you forgot to celebrate today, fear not. There are many more mathematical holidays yet to be turned into Hallmark occasions and/or crappy math lessons. Some seem to be taken a little more seriously than others.

Happy 'Sorry Your Math Teacher Didn't Serve You Baked Goods (Mostly Because She Wants You To Learn The Correct Spelling Of The Greek Letter)' Day!