Tom Fisher has been an educator for 21 years. He currently teaches middle school at a small, independent school in Portland, Maine (USA). His favorite moments as a teacher are when students start working on concepts that seem unapproachable or too difficult, get started, find some momentum, and come to conclusions and understandings they once thought were beyond their abilities.
One of the main goals of Algebra I is learning how to balance equations, taking an equation and rearranging it so that it is in its most useful form. One of the semi-magical things you can do by balancing equations is to illuminate and quantify hidden relationships. For example, there is clearly a mathematical relationship between how far an object falls and how long it is in the air. A baseball dropped off a two hundred foot cliff will be in the air longer than the same baseball dropped off of a ten foot cliff. When I first started teaching Algebra I wanted to create a fun, challenging assignment in which struggle was normal and success was exciting. I wanted students to not only get practice with balancing equations and with persevering, but to glimpse a bit of the magic of mathematics.
In this assignment I run parallel to the base of our main school building at a speed that is as constant as I can make it. My students are lying on the edge of the roof approximately thirty five feet above. They are armed with water balloons which they try to drop on my head, one at a time, as I run by.
Their challenge is to draw a line on the ground we call the drop line. They drop their balloons when I cross the drop line. They are not allowed to time falling objects from the roof. Figuring out how far the line should be from the point of impact requires a lot of creative thought and practice with balancing equations.
Over the years this event has become a school tradition. Kids from preschool to eighth grade come out to watch, chanting, “Get him wet! Get him wet!” as each algebra student takes their turn.
Dropping water balloons on your teacher, being on the roof, and having an audience are all obviously aspects of play that make the event fun and lead to motivated students who are excited for their chance. However, I think it’s easy to overlook the other very important aspect of play at work. The students know that this math is quite difficult. The difficulty, and the time given to dealing with that difficulty, allows the students let go of expectations and to seek a positive outcome rather than trying to avoid a negative one (being wrong). Play is about seeking the positive and, because of play, they can approach this difficult task with excitement rather than dread.
Subconsciously, when I first started teaching math, I think I saw playful learning in math as a good way of challenging my strongest math students. These students got great satisfaction from finding hidden relationships between numbers or from starting with something messy and turning it into something simple. There were of course students who struggled with, or even feared math, as well as students who were “good” at traditional math focused on computation, but who wanted nothing to do with the less black and white world of playful learning.
But it doesn’t take a lot of time as a math teacher to realize that your primary job is not to make sure that a student can divide fractions or to be able to write an equation in slope intercept form. I’m not downplaying the importance of these skills. As teachers we have a responsibility to the children in our classes to make sure they are ready to confidently learn and apply what they have learned after they leave our classrooms. But the importance of these skills is secondary to the importance of developing a tolerance for confusion and a belief in our own abilities to solve puzzles. Math happens to be a great place to work on developing comfort with not instantly seeing the answer, with trying approaches even if you don’t know they will work, and with developing a methodical way of working through complex ideas.
As a young teacher it became pretty obvious that the biggest difference between the students who felt confident in math versus the students who struggled or who craved predictability was that the students who felt confident explored math, while the students who struggled or limited themselves were simply trying to avoid being wrong. Which is the cause and which is the effect is hard to say, but the longer I taught the more obvious it became that a key to teaching math was to create an atmosphere in which finding the correct answer is not always the normal, expected outcome. Unintentionally I think we often create the idea that there is only correct, which is neutral to faintly good, and incorrect, which is clearly bad.
I started to see the importance of discovery and confusion, in which struggle is neutral and finding a solution is celebratory. It became clear that while the strongest students might do this more easily, the students who were afraid of being wrong needed this approach even more.
It’s important to say that one of the disservices we do each other as teachers is to describe scenes of glowing excitement and to act like this is how our classrooms always work. This is not only dishonest, but has the effect of leaving many teachers questioning what they are doing wrong and why their classrooms are not these idealized sanctuaries of enthusiasm. Discovery and confusion do not work without a strong understanding of basic concepts. And while I do tend to believe that it is almost always better to structure learning in a manner that allows children to discover concepts rather than learning from direct instruction, this is not always flashy and can be quite mundane. But these more basic lessons can lead to math play, where struggle is normal and success exhilarating. And this is how we develop mathematical thinkers.
Full disclosure from me - Jen, your friendly blog manager - Tom is a current teacher at my daughter's school. We were lucky to have had him teach my son for three years...and have experienced this water balloon exercise first hand. He's kind of a rock star in our family.
Mathematical Explanation in Plain English
One of the things we emphasize throughout middle school math is the idea of explaining a problem using plain English before trying to apply any computation. In regards to the challenge of figuring out where to place the drop line this seems like a straightforward problem. Distance equals rate times time. For example, if you imagine a car traveling at 40 MPH (rate), and it travels for 2 hours (time), it will travel 80 miles (distance). Forty times two equals eighty. So, if they know my rate and they know how long the balloon will take to drop from their hand to a spot 5’8” (my height) above the ground, they can just multiply those to find out how far away from the impact spot they should make the drop line.
However, two things complicate the problem. The first is reaction time. There is a lag between when our brain notices it’s time to drop the balloon and when our fingers let go. It’s about 0.19 seconds for most of us. We figure this out in class and average the class’s reaction time, but it is always close to 0.19 seconds. The other, which is much more difficult to figure out, is figuring out how long the balloon will be in the air. Students were allowed to measure the height of the building, but not allowed to time how long it takes an object to fall off of the building. This is particularly tricky to figure out because a falling object does not have a steady speed. It continuously accelerates until it hits.
The Mathematics of the Explanation
Because we are trying to figure out the time that the balloon will be in the air, we start with the formula that time = distance divided by rate.
We know the distance, so all we have to figure out is the rate. The problem is that there is no constant rate. The balloon will accelerate as it drops. To get around this we can think of rate as average rate. After all, a car that travels 20 MPH for 10 minutes and then travels at 40 MPH for ten minutes, goes the same distance in 20 minutes as a car that travels 30 MPH the whole way. So all we have to figure out is the average rate.
But how do we know the average rate? This is a little tricky. Because the balloon accelerates smoothly rather than in stops and starts, there is some moment during its drop when it is traveling at its average speed. When it is higher than this spot, it is going slower than the average speed. When it has fallen below this spot it is traveling faster than its average speed. The crude diagram below might help picture the balloon’s speed as it falls. Letters and numbers in black show speeds slower than the average. Red shows the average speed and blue shows speeds faster than the average.
Imagine that the average speed is 5 (the units don’t matter in this hypothetical example). Notice that the initial speed would be 0 and the top speed (maximum velocity or Vmax) would be 10. Therefore the average speed would be half of the top speed. This will always be true. So if you can find the top speed, you can find the average speed. It will be the top speed divided by 2.
So now we just need to find the top speed. Well that’s just a matter of multiplying G (this is acceleration due to gravity which is 9.8 meters/ second squared or 9.8m/ s2) by how long the balloon will be in the air (time or T). But we don’t know time! This is where you would be stuck without algebra. But let’s try writing it all out.
We know R (rate) is Vmax (Maximum speed) divided by 2. So we can write:
If you cast your mind back to middle school math you will remember that when you divide by a fraction you multiply by its reciprocal.
We know the Vmax is G times T
If you cast your mind back to pre-algebra you may remember that you can do anything you want to one side of an equation as long as you do it to both sides of the equation. In this case we will multiply both sides of the equation by T.
When you simplify this and cancel out you get
But we don’t want to know what T squared is, we want to know T. So once again, we perform an operation on both sides of the equation. In this case we take the square root of both sides. Unfortunately the computer does not like to put exponents or fractions under a square root sign. But after you take the square root of both sides you end up with:
We measured the distance from the top of the roof to my head (D) and we know that falling objects accelerate at 9.8 meters/ second2 (G). So we now have enough information to find T, the amount of time the balloon will be in the air. We then add 0.19 seconds for reaction time and we know how much time should elapse between when a student’s brain tells their hand to drop the balloon and when the balloon will be at the height of my head. Because they have timed my rate of speed, and I have calibrated my speed to the expected rate, they can figure out how far away to place the drop line.