I've been busy writing what amounts to a fifty-page research paper on the teaching of mathematics. It has been eating up much of my time and distracting me from my blog, and I do it gladly. On top of that, Google has decided to delete old videos next month, forcing me to transfer my blog videos to You-Tube and download my private ones. It's a good excuse to attach labels and revamp the blog. The timing stinks!
Wednesday, I tutored an intelligent young woman taking college algebra. She was struggling with factoring polynomials, so we spent over an hour working through problems. My primary goal in tutoring math students is to put them on a search for meaning. If they understand why they are doing what they are doing, they will be more likely to remember it. Even if they forget, meaning helps them reason their way through a problem. If I sniff any hint of wavering, I will ask them why something is true.
My friend had to solve 9x⁵ - 9x³ = 0. She had no problems with step one, dividing both sides of the equation by 9 to get x⁵ - x³ = 0 and knew to pull out x³. When asked what x⁵ divided by x³ was, she made the fatal mistake. She raised her eyebrows and questioned, "X squared?"
Me: "Are you sure about that?"
Her: "Our teacher said you subtract."
Me: "Did she explain why?"
Her: "No. She's old school. She just tells us how to do it. She doesn't have time to explain why."
Sigh. She is smart. She is perfectly capable of understanding why. People who disrespect motivated students enough not to explain why bug me. So, we headed down the path of meaning, peeling back her uncertainty until we reached something solid.
Me: "What does x⁵ mean?"
Her: "You times x by itself 5 times. You know, x times x times x times x times x."
Me: "Good. Any time I'm unsure about a procedure, I start thinking about meaning. If you freeze on a test and forget whether to add, subtract, multiply or divide, you can always fall back on meaning. Write it out the long way."
So, she wrote x⁵ ÷ x³ = (x ∙ x ∙ x ∙ x ∙ x) ÷ (x ∙ x ∙ x). Then I showed her how that is just like saying x ∙ x ∙ (x ÷ x) ∙ (x ÷ x) ∙ (x ÷ x). Then her face lit up, "Oh! Then you can cancel and get 0."
Believe it or not, that is a misunderstanding because we throw around words without meaning and precision and end up confusing students. I responded, "No. Lots of students do that. Let's go back go meaning. What does x divided by x mean?"
Her face went blank. Yes, I know I'm a pain, but this is important! Math makes sense when taught properly. So, I peeled the onion back further. I said, "Sometimes, it is easier to think about numbers. What does 5 divided by 5 mean?"
Another blank stare. The way we teach math focuses on doing, not thinking deeply. I explained, "Dividing means putting objects into equal groups. Suppose you had to share 5 cookies with 5 people. How many cookies would each person get?"
Her: "Oh, 1!"
Me: "What if you shared 10 cookies equally with 10 people?"
Her: "They'd each get 1."
Me: "What if you shared a million cookies equally with a million people?
Her: "They'd get 1!"
Me: "Now, let's get back to x divided by x. What does x mean?"
Her: "I don't know."
My friend answered correctly without realizing it. I explained to her that we use x to represent a number we don't know. It's a placeholder that means a number that we don't know. Having placeholders allows us to set up relationships between known and unknown numbers and manipulate them to figure out the unknowns or refine those relationships. I added, "We have a number of objects and the same number of people. We'll call that number x. If we have x pencils to give to x people, how many pencils would each person get?"
Her: "It's 1."
Me: "What is x divided by x?"
Her: "It has to be 1."
Then, everything fell into place, and she understood:
x⁵ ÷ x³ = (x ∙ x ∙ x ∙ x ∙ x) ÷ (x ∙ x ∙ x)
= x ∙ x ∙ (x ÷ x) ∙ (x ÷ x) ∙ (x ÷ x)
= x² ∙ 1 ∙ 1 ∙ 1
We had to peel the onion for only a few more glitches. My friend said she had a much better understanding. It disappoints me to know how much rote, meaningless instruction is happening in the math world.