Next month I’m going to an interview event for an alternative teaching certification program. One of the things I’ll be doing is presenting a 20-minute “mini-lesson” to about ten other candidates and a couple of the interviewers.
I’m a math guy. The idea I’ve had kicking around my head for my lesson is a light introduction to complex numbers. The basic plan is to start off with a joke I got from right here at the SDMB: “What did the mathematician say after Thanksgiving?” The answer: “Oh, man, sqrt(-1/64)”. Then review (with audience participation) the rules of multiplying positive and negative numbers together, and squaring/taking the square root of a number, followed by presenting the problem of taking the square root of a negative number, leading to the introduction of the number i. Lastly, the answer to the joke is simplified to read “i over eight”.
Well, a few days ago I was informed that the lesson is to be targeted to a middle school audience. Crap. I took Intro to Algebra and Algebra I in the seventh and eighth grades, so a topic in algebra should be fine; but I’m pretty sure I was first introduced to complex numbers in Algebra II, in the tenth grade. I could be misremembering, though.
So, is my idea too advanced for a middle school audience?
I am one of those weirdos who thinks complex numbers should be taught a lot earlier than they usually are. But that’s because I think complex numbers are taught kinda stupidly.
To make the lesson accessible to middle schoolers, I like a different approach that avoids i until the end. Instead, I like to imagine them by analogy with fractions.
A fraction is just an ordered pair of numbers which you can write many ways, like “3/4” or “3÷4” or “three fourths,” etc. If you wanted, you could write a fraction like (3, 4)[sub]F[/sub] and work with it without even knowing what division really is. These pairs are just an extension of the whole numbers we already know, and just like the whole numbers they have their own rules of addition and multiplication. E.g. (a, b)[sub]F[/sub] times (c, d)[sub]F[/sub] equals (ac, bd)[sub]F[/sub]. And it turns out that any fraction where the second number is one is exactly equal to that whole number. And so fractions are just an extension of whole numbers that allow us to answer new kinds of questions, like “how do we divide three apples among six people?”
So complex numbers are just these things that are just like fractions: ordered pairs of numbers with their own rules for addition and multiplication, and you can do stuff with them without even knowing what this i thing is. For example, (a, b)[sub]C[/sub] times (c, d)[sub]C[/sub] equals (ac-bd, ad+bc)[sub]C[/sub]. And it turns out that any complex number where the second part is zero is equal to that real number. And so complex numbers are just an extension of real numbers that allow us to answer questions like, “how do we solve x[sup]2[/sup] + 1 = 0?”
Then you can introduce i and say how even though i[sup]2[/sup] = -1, really, i is just a bit of notation, analogous to a fraction bar or division sign, that tells us we’re looking at a complex number.
What I like about this approach is it allows you to show both things in parallel. These students already know about fractions and what kind of questions they can answer, so you can do a nice progression:
A new kind of question
So we extend the number system
We make the rules consistent with our existing number system
We use these new numbers to answer our new kind of question
Personally, I think it is a great idea for a presentation.
You sound like you know how to entertain, as well as teach - VERY important for a teacher of kids that age.*
Combine some good and bad English grammar phrases when doing math - for instance, say “I have two apples!” (Good English kids, simple math!)
Which might equal “I ain’t got no two apples! - Which is bad grammar, kids, but great math!” (explaining double negatives)
I am sure you can come up with a better example, but it makes kids laugh, teaches them both English and math and will impress a group of teachers.
*More power to you, and major kudos - but most teachers I know would rather lather themselves in honey and sleep in a bed of fire ants than teach middle school!
Yeah, it sounds like you can present complex numbers in a way that bright middle schoolers could understand. And you realise it is not a topic they would normally study, so you can mention that. Also, you might want to able to discuss *why *you would teach complex numbers at that stage. How would it fit in with the rest of the curriculum and/or help them understand other/future topics?
Anyway, I dislike teaching lessons to adults when they are designed for kids. Are the adults supposed to pretend to be kids? Are you supposed to teach exactly as you would teach the kids, or to some extent acknowledge you are teaching adults? You have my sympathies.
I have a graph theory lesson (Bridges of Konigsberg) that I have used a few times as a demonstration lesson for middle school students. If anything, the fact that it goes outside the curriculum was seen as an advantage, as long as the students were engaged and learning.
They won’t expect you to know what is appropriate for middle schoolers. They will want to see that you can be engaging, that you know your material, and that you can prepare.
That said, assume all your audience knows is basic math, because that’s probably true.
Also, the more you ask the “kids” to do things, the better. Current philosophy is that good teachers don’t perform as much as lead.
Parent of a normal, scores above average, not gifted middle schooler. Way over their heads. Parent of a gifted math team participating fifth grader who will be a middle schooler next year. Of the 150 kids in my daughters fifth grade class, you’d be at the level of five of them. Middle school is six, seventh and eighth graders.
They are introduced to complex things earlier now - i.e. easy single variable algebra in elementary school. Linear equations to find the intersection. But neither has seen square roots introduced at school. Negative numbers have been introduced, but grasped only by the math team kids, and i - not even close year.
I have no teaching experience. I’m quite nervous about this, actually.
When I think back to all the various teachers I had in middle and high school, it’s the ones who had a sense of humor that stick out in my mind as the good ones. I had the same teacher for Geometry (9th grade) and Trig (11th grade). Gawd, that guy was a bore. Think Ben Stein in The Wonder Years and Ferris Bueller’s Day Off. I dreaded going to his class. But my Algebra II teacher (10th grade) was awesome. He always made us laugh. That is how I want to be.
You misunderstand… the lesson has to be targeted to middle schoolers. I most definitely want to teach high school, TYVM.
I teach middle school math, and with the right amount of preparation my advanced kids could do this. It wouldn’t be my first lesson in August, but by the end of the year my accelerated kids are doing this. My regular pre-algebra kids wouldn’t have a clue about what you were talking about. However in interviews, unless you have a math teacher in the interview room, most administrators wouldn’t know what the math curriculum is. If you can reach them, it will be good.
One way to make this more “middle school” is to explain that your objective is to teach how to manipulate negative numbers and radicals, and that you think of the imaginary numbers part as kind of a teaser for high school math, a bit of enrichment for the brightest kids–that your goal is to have them understand the steps of the process.
Also, don’t say “here is this joke, now let me explain it”. Say “Here is something you can say after your next visit to Cici’s that will show how smart you are.”
I personally LOVED the rare lessons where advanced subjects were taught. It gave me a hint about why the boring crap we were learning actually mattered. i, and the complex number system is the perfect example.
Exponents and Square roots come pretty early in the math education, as I recall. But there was always this rule… you can’t take the square root of a negative number! No one ever explained why, or even hinted that the rule might be false, it was just a fact that you didn’t talk about. Same deal with dividing by zero… “why?” one might ask, and “because you CAN"T!” would be the standard reply.
So, give the kids a chance to understand a little bit more about math, and you will get a positive response. Teach them that you really CAN do something that you were told you couldn’t, and you might get their attention. Now, are you going to teach them about the Complex number system, and non-real roots of quadratic equations, and the relationships of pi, e and i? No, probebly not. But can you teach them about the simple procedural concept of taking the root of -1, yes! And they should be able to grasp the idea with a solid lesson plan. And the future nerds and dopers in the class will really enjoy it.
Free anectodote: When I was in Calculus, learning about volumes of solids of revolution, the recitation instructor showed us the infinite paint can problem… a paint can shaped like a particular line revolved around the x axis, from a point to infinity, could contain a finite volume of paint, but the surface area of that same can would be infinite, so you couldn’t paint the outside with any amount of paint. Next year, in Calc2, after learning about surface areas of solids of revolutions, I tackled the problem on my own. I never stopped thinking about the puzzle, because that instructor should us something “neat” with the boring material we were learning.
Why you can’t divide by zero (as I remember it being explained). Understand what division is: it’s creating smaller groupings, and if a group has 0 members, how can you make that smaller?
Is it all middle school (6-8 grade) and what’s the level of the school? Those are really important questions, as 6th graders are vastly different from 8th graders in terms of ability. I taught at a low level 6th grade once where the kids were a bit shaky on division. I hate to say it, because I was always frustrated by the molasses pacing of my classes in 6th grade, but now that I’m a teacher I understand that you have to teach to the average of the class. There’s really no way that complex numbers will be remotely grasped by an average 6th or 8th grader.
Most of your audience is going to tune you out. Sorry.
Yes, thank you for putting into words what I was thinking. This is what I meant by a “light” introduction - just explaining how to take the square root of a negative number, without going deeper into all that other stuff.
Something else I’ve thought of… I don’t think I’m going to use the word “complex” at all; I’ll just stick with “imaginary.” Nevermind the subject matter, I’m thinking kids would like “imaginary,” but be turned off by “complex.”
