The other day I was in class with the 9th grade, working on Trigonometry. A table group of boys that I would categorize as ‘experienced thinkers who tend to get a bit rowdy or distracted when it comes time to practice’ stopped me. In the process of doing a deep think on the derivation of the Pythagorean Theorem using Pythagoras’s Proof, one of them had all-of-a-sudden noticed that on his Trig Ratio sheet*, there was just a blank spot for the tangent of 90°.
Of course, we all know that 9th-graders aren’t supposed to know the answer to that question yet, so I told them all that I wanted them to get back to work and not ask questions that they aren’t supposed to know the answers to! Maybe one day they would find out, or maybe not.
I’m kidding. That would have been the worst possible thing a teacher could do in that scenario, right? Remember that The Medium is the Message, what students learn is not the content that we teach, what they learn is the methods through which we teach it. That approach, then, would not have taught students to think about the Pythagorean Theorem more deeply. It would have taught them to not ask questions sparked from real curiosity, or that only questions asked by a teacher or a textbook are real questions (see point 2 in this post), or that the only content that matters in mathematics is the content that I have to teach you in school (after all, they don’t know that they will learn the answer to this question in 11th grade). The last lesson is one that we are constantly struggling with, even today, as high schools still have standards that, while good, still miss out on a lot of important and useful maths; accordingly, it’s much easier for current teachers, who also grew up in an educational system founded on ‘the industrial model’ or ‘linear model,’ to unknowingly support their students adoption of the linear model as well instead of the ecological metaphor of education or ‘The Web.’
But back to the kids: this should be the dream moment for teachers everywhere. It’s one of those rare opportunities that we get to potentially plant a seed of thought that will grow and thrive. Moments like these are what makes learning come alive for us, right?!
So no, I didn’t tell them to not ask questions like that, or to get back to work, or that the answer was ‘above their level.’ Instead, I had fun asking questions with them and starting a mathematical dialogue.
“Hmm. That’s a good question! Why wouldn’t there be a value there? I am going to have to assume it’s because the author of the table just messed it up… or do you guys think there is an actual reason?”
“YES!” One of them said, “It’s just because 90° is the right angle! So why would you try to take sine or cosine or tangent of the right angle?”
“That makes sense – after all, the right angle wouldn’t have an ‘opposite’ and ‘hypotenuse’, right? Because they would be the same side? So then how would you be able to do opposite over adjacent?”
“Yeah!” They seemed satisfied with that answer. Oops – time for some contriving.
“Hmm. One other thing that I just thought about though…” I paused. And then, I tried to trick them with one of my typical tricks. I will often say incorrect things just to see if they catch me, like: so 8 x 7 = 64, right? This time I said “You guys know how Trigonometry is really about the study of circles, right?”
One of them nodded enthusiastically and said yes, but the other four all said “No they aren’t! It’s about triangles! You aren’t going to get that one by us!”
“Ok, you guys are getting better at that. But seriously, consider this: there’s a guy on a big, 50-foot radius ferris wheel at a carnival, and I know that the center point of the ferris wheel is 65 feet off the ground. But what if I wanted to know how far off the ground the person was at this point?” I said, pointing to the circle that I just drew on the board, right at about the traditionally-measured 40° mark. I added in a dotted, horizontal midline to the circle and said “Yeah, this is a good question. Figure this out for me and I’ll be back.” I went and did some conferring with other students and came back a little while later when I saw that they had filled in the circle with a triangle. They were excited to report that trigonometry actually could, kind-of, be about the study of circles, like I had ‘jokingly’ said earlier! But, they would have to know some more about the situation to figure out where the guy was. I pushed them on what more they would need to know, and they came up with: the radius of the ferris wheel, the angle of the person above the midline, and eventually the time it takes the ferris wheel to do one rotation (in service of finding the angle).
“Now, here’s an idea…” I said, “It seems like if you wanted to, you could create a graph that showed how high above the ground this person is for the entirety of the ferris wheel ride – however long that is…” And of course this is the moment that makes it all worth it. Their eyes actually got wide with realization. “Obviously we have work to do now, but that would be cool if one of you remembered that and proposed it as an alternative project when I give you the next project.”
And then, I walked away.
The point, obviously, is that The Web of Content metaphor has a very different end-game than the linear model. My hope is that these sorts of interactions plant seeds, not make kids into experts in trigonometric functions or even into experts on any of the other obscure mathematical topics that don’t get used anymore in a modern world dominated by Data Analysis, a world where being able to calculate a definite integral by hand is just not that impressive compared to being able to understand what integrals are and how they relate to derivatives and how to program a Differential Equation into Matlab for the purposes of modeling a complex economic situation. I want them to know that knowledge is not ‘pre-existing’ in the world – it comes from somewhere! Namely, from people who create it by observing the world and recognizing patterns that are that way because there is no other way they could possibly be! I want them to know that they are just as capable of ‘discovering’ concepts on their own as they are capable of learning those same concepts from a book. I want them to go through the process of creating their own language and symbols to keep track of how they are going to model their new discovery of ‘height above the ground while on the ferris wheel,’ and to then come to realize that all mathematical symbols, no matter how complicated-looking they are, came from people who had a similar desire to communicate their thoughts. And, I want them to then ‘discover’ that what they had ‘discovered’ has already been discovered and there is already a language and symbols through which people talk about it, and to go through the process of reconciling their symbols and language with that of the ‘standard’ way of talking about ‘height above the ground for guy on ferris wheel’ aka Trig Functions.
This is a very different end-game than ‘I want kids to have mastery over a*a+b*b=c*c’. It will take patience and lots of moment-to-moment distinction. But I think it’s sure worth it. How are you going to plant some seeds this week?
*Why do they use a trig ratio sheet? I’m glad I interpreted that you asked, because it’s important! I write more about it in Mathematics and Why, II.