DesCartes Fly - Anchor math concepts with memorable funny story
Descartes' Fly Copyright 2010 Lee Tilson Goal: Make Math Concepts Easy to Understand and Memorable Example: A Simple Way to Remember (Anc ...more »
Descartes' Fly
Copyright 2010 Lee Tilson
Goal: Make Math Concepts Easy to Understand and Memorable
Example: A Simple Way to Remember (Anchor) the Concept of Cartesian Coordinates.
INTRODUCTION
Effective teaching anchors concepts. The more solid the anchor, the longer the concepts are retained. Without solid grounding, concepts seem to float away over time. The simpler the foundation, the easier it is to build upon.
Few of us recall details of classroom lectures from years ago. We may recall papers we authored, phrases that were repeated many times, or memorable classroom discussions. The only things I recall of several college chemistry courses are an experiment I designed to measure the amounts of caffeine in beverages and a nasty taste of some solution I that ended up in my mouth. Some stories seem permanently inscribed in my memory. The most memorable is a lively but humorous confrontation between a fellow student and a professor. Some lessons on which I have relied to solve problems through the years still available in my memory banks. The vast majority of classroom discussions seem to have wandered away.
What anchors have worked for us? Can we study the anchors that have kept old lessons available to us? By studying those anchors, can we devise new and better anchors to offer today's students?
What simple, intuitive explanations can we give to help students being introduced to Cartesian Coordinates? The exchange below offers a humorous confrontation as an anchor. It tries to make the basic idea of Cartesian Coordinates memorable. It is merely one way to introduce the idea of Cartesian Coordinates, the graph of the X Axis and Y Axis, that we use to visualize equations in modern algebra.
The exchange is very loosely based on the description of DesCartes' invention of Cartesian Coordinates handed down through the years. According to the versions of the legend I have heard, DesCartes watched a fly crawl across the ceiling while laying in bed ill. He realized that the position of a fly on the square ceiling tiles could be described with numbers. That insight led to "Cartesian Coordinates."
I have designed an imaginary confrontation between a teenage DesCartes and his frustrated math teacher. I want to build on this confrontation in future lessons. We can move the ceiling tiles to the wall or to the floor. We can follow this first exchange by allowing the young character of DesCartes to actually number the tiles and describe the position of a fly as it crawls around the tiles. We could invent lessons about measuring distances by placing the fly on a leash and taking it for a walk, with a tight leash being the line segment between our location and that of the fly. We could introduce the concept of a circle and its unique properties by letting the fly drive a car which is chained to a stake around the tiles. (The chain will keep the car at a fixed distance from the stake, and will be a circle.) The lights on the car can be used to generate lines that are tangent to the circle. My hope is that devising this anchor (explanation) will help them remember and build on those explanations.
The hope is that this funny and simple story will help students learn, and remember, the concepts we are trying to teach them. If you do not find funny stories like this helpful, let me know. What kinds of anchors worked for you? What lessons do you remember?
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Background
DesCartes has been at home ill for two weeks. The teacher is upset because DesCartes is behind in his homework and needs to catch up. Descartes is excited with his new idea of mapping equations and wants to explain it to the teacher. Focused on DesCartes' failure to do his homework, the teacher becomes condescending and exasperated. DesCartes is trying to share a brilliant insight.
DesCartes tries to tell his teacher about the idea of drawing math equations on a map to see what they would look like on a map.
Teacher:
"DesCartes, you have been out sick for two weeks. I sent some homework to your dad. Did you get it? Where is your homework? Did you do it?"
Young DesCartes:
"I was working on it when I had this great idea. I wanted to tell you about it. I thought of this new way to do the work you sent. Instead of solving the problems with equations and proofs, suppose we think of what these equations would look like on a map. "
Teacher: (interrupting)
"I just want your homework. Not your ideas, I told your dad to tell you that you had to do your homework. All the other kids in the class turned theirs in weeks ago. Where is yours?"
Young DesCartes:
" I'm working on it. I'll finish it tonight. I was working on it when I had this really great idea of a new way to solve these problems by drawing a map of the equation, a picture."
Teacher: (exasperated)
"Look kid. We don't draw pictures of equations here. Pictures are pictures. Math is math. Pictures are a part of art. Equations are part of math. Two different subjects. Got it? Pictures are not my department. Math is. If you want to draw pictures, go down the hall, third door from the end, ART ROOM. OKAY? If you want to draw pictures, well that is art. This is math. Got it?"
Young DesCartes:
"But I got this idea watching a fly on the ceiling."
Teacher:
"Well hold on. Let's consult an expert about this mapping a fly business. Now who would be an expert on pictures of math. What do you say kid? Euclid? Can we consult with Euclid?
Young DesCartes:
"Hasn't he been dead for a few thousand years?"
Teacher: (cocky)
"No problem. If flies can do equations, we can wake up the dead.
"OK. Hey, somebody go wake up Euclid from the dead.
( Teacher picks up his cell phone.)
"What do you know, my phone is ringing. Maybe it's my buddy Euclid calling.
(teacher opens up the phone and holds it to his ear, pretending to have a conversation with Euclid from the afterlife.)
"Hey Euclid. Yeah you Euclid. Yeah Yeah, I know. But listen up, I have this kid in class. He tells me he had this great idea. He wants to do math watching flies. He thinks flies understand math better than I do. , What's that?.......
(pause, listening on the phone)
"Yeah yeah, a fly.
(pause, listening on the phone)
"I am not kidding Euclid,
(pause, listening on the phone)
"Euclid, tell me the truth. Did you find flies to be good at math?
(pause, listening on the phone)
"Uh huh, okay
(pause, listening on the phone)
"Well, how many flies did you train to do math. Maybe it was you that trained that fly on DesCartes' ceiling.
(pause, listening on the phone)
"What's that? (pause)
You don't train flies? Never? Can I tell that to my boy genius who didn't do his homework? (pause)
"Thanks man. (pause, listening on the phone)
"What is that? you don't think flies could do math?
(pause, listening on the phone)
“Really, why is it that flies could not do math? (pause, listening on the phone)
“Really. No neurons? No brain? (pause)
Can I tell that to this boy genius here?
“Thanks Euclid. You too. What's that? Yankees in five games? OK thanks" (teacher hangs up cell phone)
Teacher (sneering at kid)
"Hey Genius. That was Euclid on the phone. Do you think he agreed with me or you? Do you want to guess what he said?
"Now where is the homework?"
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Homework assignment after this introduction. (UNGRADED)
Give the kids several pictures of ceiling tiles with a few flies on them. Ask them to try to find a way to describe the position of the fly in each picture with numbers.
Now the kids are engaged. They are not memorizing. Their effort in trying to devise a way to describe the position of the fly in each picture will, hopefully, ground their intuition. They will not be memorizing information described by others. There is an intuitive base. That is the anchor. That is a foundation on which Algebra can be built.
We can create a flat table like surface by "talking about" taking the ceiling down with tiles intact and putting in on a table. Different concepts can be generated by the activities of the flies as they live out their lives on the table.
Lines
Lines could be generated by having the fly roll a bowling ball that leaves a mark.
A concept of lines that helps develop the concept of the slope of a line.
Slope of a line
Alternately, lines could be generated by placing a ball on a tilted table, and use the tilt to describe the slope of the line.
Circle
Let the fly drive a car or bicycle that is tethered to a post.
Tangent
A headlight on the tethered vehicle could generate a tangent.
Y Intercept
If I can find some meaning for the Y Axis, and an event that occurs as it crosses the axis, that becomes a reason to look for the "y intercept"
Graphing a Parabola: y = x squared
If the table was tilted at a fixed angle, I think that rolling the ball in certain directions on a tilted table will generate a parabola.
Graphing an equation such as: y = x to the third power
I think I can generate this diagram on a graph if the table surface is rotating at a particular rate.
Comments:
Now, we have the attention of the kids in the class because they are watching this funny scene, the students are watching another student be a teacher. It is funny. This makes it easier to learn.
Next steps in this lesson:
Each child tries to explain the concept to the teacher. Students can write essays. Students can make analogies. Students are encouraged to think about what it means to "explain" something, and to try every available explanation to show the teacher.
We think of all the obnoxious responses the teacher can make to feign not understanding.
This time it is the students who are having to explain the idea.
Next Lesson:
As the graphs begin to deal with new concepts, we can move the ceiling tile. We can have square tiles put on the wall and floor. Descartes' fly might find a toy car to drive on the floor. Certain facts about the car might explain the way that it moves. For instance, if tethered, it might go in circles. Straight-ahead headlights on the car may be tangent to the circle.
The purpose is to ground the math lessons in intuitions they already have and or that they develop themselves so that they can more easily remember and use them in the future.
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