When I began my observation days last year I was sitting in on a grade eight math classroom that was covering geometry. Students were working on calculating the area of a circle, and one task was the pizza problem. The pizza problem's reputation proceeded it, so I was surprised when it was simply a question out of the textbook that boiled down to whether you should buy two large pizzas or three medium pizzas. Students were provided with the diameters and prices for each size. I don't recall the specifics, but it worked out to there being about 180 cm² more pizza for the option that cost $5 more. More pizza, more money, and no surprises yet. But the question asked which you should buy---what is the better value?
Students were stumped by this, and some even made some very convincing arguments about how they wouldn't need that much pizza for their family, or how three medium pizzas would have more crust and fewer toppings. On it went, but much to my surprise no one figured out to use rates to solve the problem. Maybe it was because my mom took me grocery shopping a lot when I was little and I got to help her compare the unit prices of different sizes and brands, but it seemed so obvious to me. How much pizza do you get for a dollar?
180 cm² didn't seem like very much pizza for $5, so that couldn't have been the solution---it just wasn't worth it. And I agreed---it didn't sound like much! But no one ever tried to show me how much 180 cm² actually was. What does 180 cm² look like? I imagine it to be about one slice of a reasonably-sized pizza, and so I definitely wouldn't spend $5 on it. And this was where the activity delved into students' feelings about the solution.
The students intuitively knew that it wasn't a good deal to pay $5 for so little pizza, and it is this type of reasoning that is very important, yet underappreciated, in math. Math might deal almost exclusively in numbers, but those numbers have a context in the real world, and for students to say that they wouldn't pay that much for the extra pizza was a strong connection to self. But intuition in math goes well beyond pizza sizes because it is responsible for how we decide to approach problems with the information we have at hand. What do we do with our information? What results are we looking for?
Number talks are a great way to promote mathematical flexibility with mental math, but flexibility also applies to the procedures and formulas we use in all areas of mathematics. For example, we could use the sine law to solve for an angle in a triangle, but we might also use the knowledge that all three angles have a sum of 180°. And while this might seem overly simplistic, it is exactly these types of understandings that are lost by students when they focus on following formulas blindly. There are often better, faster ways of doing things, and when it comes to applied mathematics in areas such as chemistry or physics, there are many situations in which those beloved formulas don't hold. What then?
The Monty Hall Problem is one of those paradoxes that truly pries at human and mathematical intuition. Picking a door, we see ourselves as having a 1/3 probability of selecting correctly, and mathematically this is true, but the trick is that when a second door is opened to show a goat, the entire scenario is changed on us without our knowledge. People erroneously assume that there are two doors, one of which contains a goat, and therefore the probability of having a car is 1/2 for each. But what the latter example in the video shows is that the probability of all other doors condenses onto the remaining door after the goat is shown. There has been a lot of debate about this problem and it stems from the fact that mathematical and human intuition are divergent when that goat is revealed.
The one issue with the Monty Hall Problem as proposed in the video is that Monty Hall never allowed his guests to change their door. Contestants were allowed to take cash instead of sticking with their door, but were never allowed to actually switch. And while the chances of winning a car are higher if you could switch, what would be a good strategy if you weren't allowed to? Would it be worth taking the guaranteed cash rather than sticking with your 1/3 chance of winning a car? It's suddenly a much more human problem.
Intuition drives mathematics. It dictates how we understand and approach problems, and sometimes, as with the Pizza Problem, it assesses and breaks down the human components of math when put into context. Mathematicians may sometimes call it conjecture, but is it possible to just know that something is true without being able to properly articulate it? Why not? I am certain that Fermat knew his Last Theorem would hold despite not being able to offer any proof of it. And if others didn't feel the same way, then why would we have spent over 350 years trying to prove it?
It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers,
or in general, any power higher than the second, into two like powers.
I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.
---Pierre de Fermat, 1637
