Math: Applying What We've Learned

On the first day of our new math groups, I handed out a sheet of math problems to the group. It was a hodge-podge of items that included questions like this:

Divide 30 by ½ and then add 10. What is the answer?

At first, almost everyone came up with 25, which is, of course, incorrect. (If you're confused, you'll find an explanation at the end of this post.)

At one point, a student called out in mock anguish, "JASON, these are RIDDLES, not MATH!"

While there was definitely a riddle-like aspect to some of the problems, they all required students to read carefully, think both critically and creatively, and really consider exactly what it was that was being asked of them. These are vital skills when it comes to solving problems, whether they're based in mathematics or not.

This is one of the main ideas I'm trying to convey in math group: in the real world, no one will ever ask you to solve a traditional math problem presented as they are in textbooks. For example, outside of school, I've never, ever had someone say to me, "Hey, Jason! Given 5x²+12x-98, plot a parabola. Your job depends on it!" In order for math to be truly useful, you must learn how, when, and why to apply it to real world situations.

For this reason, I've been stressing the importance of developing strong problem-solving strategies in our math group. Too often, students get bogged down with how quickly they're progressing through math. For example, one day I challenged the group to calculate the volume of our very irregularly-shaped classroom as part of a larger problem. While most of the class was energetically taking measurements and arguing about how to deal the odd angles of our room, one student was hanging back. When I asked why, he replied, "They're doing 7th grade math. I'm not in that book yet." When we sat down and broke the larger problem into smaller chunks, he quickly realized that he did have the skills, but because I hadn't posed the question in the same way that his textbook presented things, he assumed that it was beyond them. I want our students to start thinking beyond their textbooks. Textbooks can be a vital part of math education, but it's merely one component of an education in mathematics.

To address this, I frequently greet the math group with a warm-up challenge. Some are more challenging than others. One day, I gave them this classic scenario:

Danielle, her alligator, her monkey, and her parrot have to cross a rickety bridge. She can only take one animal at a time, or she risks falling on deadly cheese (she's lactose intolerant, a fact that was suggested by a student). The alligator can't be trusted with the monkey or the parrot, and the parrot can't fly because of a broken wing. How can she get all three animals across the bridge safely?

On this occasion, most students figured out a reasonable sequence of events relatively quickly, and we moved on to other things. Last week, however, solving our warm-up ended up taking up the entire hour of math, and we ended up calling in Karl's math group to help us out:

Al says: Bob is lying.
Bob says: Carl is lying.
Carl says: Al and Bob are both lying.
Who is lying? Who is telling the truth?
There is only one possible answer.

This logic puzzle generated a ton of discussion and numerous suggestions on how to solve it. In the photo above, there are two different white boards and numerous pieces of scratch paper being used to puzzle it all out. We eventually got the correct answer. See if you can as well!

Karl's class returned the favor by inviting us to participate in an electoral math exercise. They created a map of the United States using Karl's amazing coffee mug collection (he has one from every state in the union). First, all of the states most likely to vote for Romney were put in one group, then those most likely to vote for Obama were put into another. The battleground states were left on the floor, representing a number of different possibilities for electoral victory for each candidate. Not only did it shed light on our political process, but it also made real world connections about where each candidate has been campaigning lately, and why. 

Engaging our math groups with this problem-based approach encourages meaningful connections with the material being covered, which brings us closer to our goal of true mastery of mathematics.


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Back to our original problem:

* The question is worded specifically: Divide 30 by 1/2 and then add 10. Most people read it quickly and divided 30 in half, and got 15. Of course, 30 divided by 1/2 (or, stated another way, 30 divided by .5) is 60. So the answer to the question "Divide 30 by 1/2 and then add 10" is 70.

If you need help solving the other problems, ask your child about it!