Parts:

In the first two parts of this series I hope to have shown that 1) real learning and understanding cannot be built on shaky foundations and that 2) math is a language that students can appear to speak perfectly even when they only have shaky foundations. Or: they can’t learn it unless they really understand it, and we can’t know if they really understand it.
Ef !
What Convention Has Already Addressed
Even the most traditional math classrooms have word problems. The word problem can be as thin as a veneer of English over an equation: “If I have two apples and you have three more than I do, how many do you have?” instead of “find x if .” It can be as thick as a paragraph of English over an equation, or even, once you get in to advanced classes, two equations.
I mock word problems, but actually, they’re great. They go a long way towards eliminating that student strategy of memorizing and applying rules without understanding semantics. Even if you can solve by only memorizing “ifyouseeanegativenumberonthe sidewiththethenaddthatnumbertobothsides,” you can’t solve “If I have two apples and you have three more than I do, how many do you have?” that way. At least, not until your teacher has given you ten or fifteen of the same sentence structure. Word problems require students to do more translations in their brains and so are more convincing evidence that a student understands. Word problems connect some English fluency (which we assume the students have) with mathematical fluency (which we assume the students haven’t).
What Are Way Better Than Word Problems And What Riley Feels Like He Invented Even Though Obviously He Didn’t
The problem with word problems is that the teacher must prepare all of the information for consumption by the student. While the student must connect some English fluency with mathematical fluency, he is still just using symbols at an abstract level. We can swap out “apples” for “grapefruit” (mmm, please do) and “three” with “seven” and the thought process is really identical. If the student had to prepare his own information, then we could be sure, proof positive, that he understands what he’s doing. In a word problem, doing random subtractions and additions on the numbers three and two, you can come up with like three different answers (1/3 chance of being right). If somehow you could present the problem without giving away these numbers, the likelihood of random success would drop by a lot.
A Slightly Better Word Problem (more information given)
Instead of asking, “If a rectangular field is 120 yards long and 50 yards wide, how long is the diagonal?” (1/3 chance of random success plugging numbers randomly into Pythagorean theorem) you could ask,
“Look at the diagram of the field below. It has a circle in the center of the field that has a radius of 5 yards, and there are stripes across the field every 10 yards. The lawn is cut with a mower 2 yards wide that takes 60 passes to cut the entire field. In total, the field is 120 yards long, and 50 yards wide.
How long is the diagonal line across the field?”
This is a good attempt at lowering the chance of random success (now something like 1/45) without raising the difficulty of the fundamental question. Throw in the surface temperature of the sun and the period of Haley’s comet if you like. These questions require more mathematical fluency – more basic understanding – to answer correctly.
My beef with this method is that your questions are really confusing and muddled and it feels like you’re deliberately trying to confuse the reader (you sort of are). And you’re still producing all of the information here – your students are still just consuming the numbers you come up with. The only difference is that now they have to have a little taste to distinguish between your good numbers and your fishy numbers.
A Way Better Word Problem (less information given)
“Here’s a satellite picture of our pond. What is the distance between the two red points?
To help you, I’ve put a traffic cone in each of the actual locations indicated by the three points on the picture. There are 150foot tape measures in the closet.”
To answer this question, your students would have to make the ultimate connection: mathematical fluency with physical fluency. They already understand distance, and measuring it. They have to get that and in Pythagoras’ Theorem are distances, and can be measured.
The benefits of this type of problem go beyond lowering the chance of random success (now immeasurable, but obviously smaller). Since this problem connects to physical fluency, students are empowered to know about the reasonableness of their own answers. If they measure the two sides of the triangle to be 200 feet and 250 feet and come out with an answer of 1000 feet, their physical fluency will tell them there is a problem where their underdeveloped mathematical fluency could not.
An Aside: Riley’s Daring Stance on Relevance
When I talk about precision and deep understanding and other seeminglypedantic characteristics, I often get comments like that in the first post of this series:
“If we want to encourage student precision, we need to have authentic situations where it matters. We can’t just demand it because some day it might make a difference. That’s just not relevant.”
I agree with the sentiment that we can’t just demand precision and correctness. Some people think that students need to be able to use math in their outofclass lives for them to care about correctness in class, but with this idea I strongly disagree. Math doesn’t need to be “relevant” to be interesting – students simply need some degree of fluency to be interested. If you give someone something that they don’t understand at all they’ll feel totally helpless and confused (which presents as total apathy in students). If you give someone something that they feel total mastery of (like measuring distance) and then a question that they can almost answer with that mastery, their natural response will be curiosity and motivation. The connections they make between their prior fluency and their new learning will be stronger than any “relevance” you might have contrived in their real lives – just ask my students of five years ago who I forced through loan interest compounding formulas because it would be relevant to them soon.
And I mean, where are you going to find a relevant application for the Pythagorean Theorem anyway, you know?
The Big Conclusion
I recommend that all teachers try to explicitly connect new material to skills of which students already have complete, 100% mastery. It can be a basic skill like measuring distance, or a skill they learned last year like calculating distance given two points, or a skill they learned yesterday in class, but they have to be PERFECT at it, and they have to be able to recall it so fast that we’d be willing to call them fluent in it. When you connect new forms of expression with old fluencies, you give students the tools to 1) find flaws in their own reasoning, 2) extend their expression to include new meanings, and 3) remember their new skills with more conceptual connections. You extend that base of fundamental understanding I talked about in part 1 of this series, and you avoid the problem I warned of in part 2. And, to boot, your class gets more fun because everyone feels like they know at least part of what’s going on!
You can achieve these fluency connections by creating environments for your students to explore, or by properly crafting a word problem, or by following the WCYDWT example, and I’m sure there are many more. Whatever you do, as long as you intentionally include ways for your students to connect with your lesson on terms in which they are completely fluent, you’ll see interest, motivation, and test scores increase.
“And I mean, where are you going to find a relevant application for the Pythagorean Theorem anyway, you know?”
Construction workers use it all of the time. Check out my post on a way to use it in class similar to your cone and tape measure idea.
http://concretekax.blogspot.com/2010/08/mathicebreaker.html
Augh: how many of your students are construction workers? This is the kind of definition of relevance I writhe under – I don’t think math is interesting to kids if it won’t be useful in the next thirty minutes (and forget the next several years!).
I like your icebreaker activity because it creates its own constraints and its own questions. I think going outside and holding things and moving things makes it relevant – NOT the fact that maybe in a few years they’ll need to know the theorem.
What do you think? Thanks for posting!
Well, I disagree that math has to be relevant immediately. I agree when you say that a perplexing problem can be as engaging as an “authentic, real world” application. I think both are good ways to teach. The key to me is the approach of the teacher or the need for storytelling.
I have expanded the icebreaker into a theme for a whole geometry unit https://docs.google.com/document/edit?id=1aQCFEKYoizUsX00wVX75_rAdqgoovdJwryX80HwNbJI&hl=en&authkey=CJG654gM
What makes this lesson relevant to me is not that I will say to students “Hey this is how construction workers use math. Maybe you will be one someday.”
Rather it lets me start with a story of what I did this summer. It allows me to share my life outside of school with students. It helps me build relationships with students. I can almost guarantee that half a dozen students or more will tell me, “My dad works in construction.”
Also the questioning part is key. My approach is to challenge students to think of how construction workers solve the problem of layout. This is what makes the example motivating and interesting to students is that it is a good question. Also as you say it is accessible. They have the tools to lay out a rectangle without any knowledge of Pythagorean just by trial and error.
The question of how we check to see who made the best rectangle begs the students to use/seek the mathematical properties of rectangles and their proofs.
The point in using authentic examples is to demonstrate that math is legitimate outside of school in varied examples and not just college and engineering. But the technique is really the same as presenting a challenge problem.
It is not enough to just use real world situations, but to use storytelling to present interesting questions.
I think overall we are in agreement in our philosophies. Maybe?
I think you’re right when you say we basically agree. I overreacted to your first comment – so many people say “well profession xyz uses math, so that makes it relevant to our kids” that I’m blinded to subtleties around the statement.
The idea of storytelling as a method of engagement is an interesting one to me. It creates a context in the students’ minds and, if they’re in to the story, they’ll be in to the questions it brings up. I’m afraid, though, of overdramatizing my lessons for sheer entertainment value. Don’t get me wrong: I don’t think stories are only base entertainment. But I worry that when *I* focus on a story, I will get carried away with it and lose focus on the math. I’ve seen other teachers do that and it comes off (to me) as kind of lame.
And, you know, whatever. There are worse things than accidentally spending 10 minutes of a math class on something fun and unrelated to math. It’s just that I struggle.
Using examples from outside of class (e.g. construction projects) could also benefit students by encouraging them to look for mathematical situations all the time (not just in math class). I never though explicitly about that connection before. Hmmm.
Interesting stuff here! Thanks for posting your longer response.
Ugh. I had a horrible class today where it felt I did everything right, and it went so wrong. I built relevance for inverse functions by introducing a “find the inverse” problem in the homework on composite functions from previoud class – some students solved it, explained it, and the class seemed to agree and understand. The rest should have been a breeze but I think I stumbled on the flunecy thing you’re talking about. They had a really hard time understanding the concept of inverse function, and I pushed them into exercises without heeding the warning signs. It felt like a mental car crash. Now I’m wondering how to recover. Any tips?