How can the argument be simplified? How can it be any clearer? Feel free to leave comments here or within the automated argument (click the deny buttons until you get to a text field to leave a comment).
A central tenet of standards-based grading is that specificity is a good quality of feedback. For example, “Timmy has an 50% in spelling and 100% in grammar” is better than ”Timmy has a 75% in English” because the latter is less specific. The relationship can’t hold forever, though. Do we want this as part of our grade report?
Hey, you know, maybe we do want this. I can come up with some rationalizations. But what about this?
I’m going to take a stand here: this is too much information. It’s overwhelming for the student – there’s so much to work on! If I did this level of granularity on my class I’d have 400 standards for the first semester alone. In a more subtle way, this is also a problem because we need our standards to be a little bit vague. We actually cannot distill every piece of content in our class down to a standard and test it. Maybe the one standard “spelling” is good enough.
Finally, a big part of SBG is letting go of the idea that you’re going to test everything anyway. Or that a student will be able to do every single thing you taught her to do. This has been a big part of me growing into a leadership role at school and at camp, too, actually. Communication isn’t perfect, and people aren’t perfect, and they don’t need to be. So, when you’re handing back Suzie’s essay, which had 45 spelling errors on it, maybe you mark eight of the spelling errors and don’t mention that she’s missing a comma on page five.
Which leaves us with a pretty big set of decisions: what will our standards be? I’ve written a guide about how I decided in my classes, but of course there will be many situations in which it won’t help.
An idea cooking in my brain: what role can students play in deciding what the standards will be? Maybe they can’t really know what’s important, and that’s our job as teachers to decide. Maybe we could give them the list of 400 things we wish we could give them, and they choose 30 for the semester. Maybe we just work up some directed inquiry activities and then talk with the kids about what the standards should be after the fact? Or maybe we have a base 15 required standards, and challenge kids to make their own after that. Hmmmm.
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.
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 “if-you-see-a-negative-number-on-the- side-with-the--then-add-that-number-to-both-sides,” 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 150-foot 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 seemingly-pedantic 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 out-of-class 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.
Math is like English and other such languages because it can express ideas. “The cat is over there” means that some theoretical cat is in some theoretical place that is not at the theoretical “here” (the sentence will probably be accompanied by a point or nod or glance of the eyes), and “2+5” evokes the idea of adding two with five (this sentence will probably be accompanied by some units or a diagram).
Being fluent in English means knowing a large list of words and knowing the rules for putting them together. You also must be able to recall the words and rules quickly enough to form ideas in your head when you take in the language and to express ideas from your head when you’re producing the language. Similarly, being fluent in Math means knowing a (much smaller) list of words and knowing the rules for putting them together. The languages are a lot alike.
Math has an incredible component that other languages do not have: a set of rules that operate above the sentence level. There is a part of math that operates not on the ideas that the sentences represent, but on the sentences themselves. The really incredible part: though these meta-rules do not consider the meanings that the sentences represent, the meta-rules do preserve those meanings.
This is a big deal.
For example, let’s say the number of apples I have is called . Now “I have 2 apples” is equivalent to “.” So far there is a direct translation, and English and Math are equally powerful, able to express pretty much exactly the same information. When we apply the meta-rule of math that says we may add any quantity to both sides of an equation, we can generate “,” which is roughly equivalent to “Three is one more than the number of apples I have.”
This is a big deal because the meta-rule did not know anything about the apples, but still managed to express a new, true idea about the apples! In contrast, there is absolutely no rule of English that says anything like “if you have a sentence that expresses a quantity, you may increase that quantity by one as long as somewhere else in the sentence you sort of balance it out by saying that that quantity has been changed by the amount of your increase.” English is mired down in the specific ideas that you’re talking about. English metaphors can be beautiful and very expressive, but always lose precision. Metaphors in Math are always perfect.
More powerful examples abound. If you can express a rate of change as a function, you can almost always apply the “integrate” metaphor to get a total amount of change. For that matter, if you can express any quantity with a function you can get specific information whenever you like. For example, if I have two apples but will get another apple every day, I can say that I have apples where is the number of apples I have and is the number of days from now and . “How many apples do you have 10 days from now?” becomes and Math has automatically given me the answer. If I only had English I’d have to count on my fingers and create another whole sentence practically from scratch, but Math generates the answer by operating strictly on the symbols involved. It doesn’t give a crap about the meaning.
“Riley, I think you’re missing the point,” one of my colleagues said in conversation. “The meaning is the important part. No one would care about math without that interesting semantic hook!” And indeed, many people see the meaning as so connected to the symbols that it seems like we are operating on the ideas themselves, not just the abstract numeral “2.”
BUT: it is possible to operate mathematical meta-rules without understanding the meaning behind any symbols.
AND: our students can learn how to operate the meta-rules without understanding why they work.
SO: we must be extra freaking careful about our definitions of “fluency” in our classes. We can test for English fluency by asking a student to write an essay, right? There’s no way to write an English essay without understanding what you’re saying, and in fact I would venture that English essays almost always convey an underestimate of students’ understanding. When someone tries to BS an English essay, you can tell: there’s no content. But it is freaking easy to write a Math essay (a proof, an application of a formula, whatever) without knowing what the hell you’re talking about. The writing of math is not enough to prove fluency. You can solve a right triangle perfectly, with 100% accuracy, without even considering the triangle. You’re just going to be screwed when you actually come to something meaningful.
As teachers, we must consciously decide what level of understanding our students will need in order to satisfy us. We’re teaching the laws of cosines and sines: do they need to know a proof of it? Do they need to be able to solve a triangle from a diagram? From a list of side lengths? From a picture of a flagpole with a shadow labeled on it? From a flagpole outside and all they get is a protractor and a measuring tape?
Reality check: we can’t test everything. We can’t know exactly what our students understand. We don’t have time to be intentional about every aspect of our teaching. And deciding what kids need to know is really important, and really hard, and even harder because of the decoupling between mathematical meaning and mathematical operations. Good luck!
Stay tuned for part three, in which I’ll explain some ideas I developed for linking mathematical fluency with fluency in other forms of expression.
Do you ever use words without quite knowing what they mean? And you sort of assume that others will natrilently understand what you’re talking about even though the word isn’t exactly what you want? If you just keep using it, it’ll start to take on the meaning you need it to, and you don’t know the precise word you need, so what the hell, you’ll just keep calling your favorite professors “erudite” until you eventually look the word up in a dictionary?
It sort of works. But until you look the word up, you feel awkward using it. You float it out in conversations and watch for reactions. Most of the time no one else can define it exactly either, and they get the gist of your message anyway, so conversation can proceed. The conversational ground is tenuous, but sometimes that doesn’t really matter – it’s easier to just continue.
I care about the phenomenon of building on ideas we’re not perfectly sure about. Can you guess where this happens four times a week in fifty-minute chunks? I’ll give you a hint: it starts with “R” and ends with “iley’s Algebra class.”
No one can build understanding on a foundation of vaguely expressed ideas. In our classes, we must not allow vaguely expressed ideas to be final products, because until those ideas are sufficiently specific our students will not be able to build on them. If a student offers an answer that is correct but too vague for our needs, we should avoid saying, “right, and…” and then supplying the rest of the answer ourselves or from another source. Doing that would give our student the impression that he was right when he wasn’t, and that vague answers are good enough. Instead, we should say “your answer is incomplete,” or, “we need something more specific,” or, “what do you mean?” In the end, before you think he knows what he’s talking about, he should give a complete answer.
Such a demand for specific clarity can seem overly picky, or even mean, but it’s vital because of a human ability to get the gist of things and to use that gist to reach conclusions that sound reasonable but are in fact bullshit.
Hilariously, I want to leave off here and try a different tack. Please keep the vagaries above in mind as you consider the following. Take two:
We are so smart that we can parse badly formed language (I can remv vwls frm mst of my wrds and you hv no prblm undrstndng me, I can skewer the definitions of words, and I don’t have to even use grammar that good). This is an incredible feat, but there are two major problems it causes:
- When we use imprecise expressions, we lose the benefits of generalization and specification, because both acts require very precise definitions, and
- When we use imprecise expressions, we lose the benefits of generalization and specification, but we don’t realize it, and we generalize and specify anyway. Incorrectly.
Now, it is hard to recognize a badly-formed sentence in a language when we are not fluent in that language. When we aren’t fluent, we don’t have that immediate reaction of something being out of place, like we do when I say, “what time are it right now?” And since our students do not have the fluency to automatically detect discrepancies in their math – to notice immediately that their units do not match or that their answer is an order of magnitude bigger than what they would expect – it is hard for them to recognize these badly-formed, imprecise statements in math. We cannot expect them to build any significant understanding until they are fluent enough to catch this sort of thing. They learn the formula for the area of a circle, and then apply it to an ellipse, because they haven’t internalized the precise definition of “circle.” My students say “x squared to the third” to mean and then think I’m just giving them a hard time when I insist they say “x cubed” or “x to the third power” - they really don’t see the big deal. So they fall especially in to problem 2 above – thinking they can build on concepts that they think they understand, and then being confused when their calculator spits out .
On the other hand, I got around alright in Italy having taken two years of high school Spanish. Sometimes all we need are nouns and hand gestures. In part two of this series, I’ll write about one key difference between math and spoken languages: that mathematical notation, not just the meaning behind the notation, can be manipulated to find more information. Stay tuned! But for now, let’s remember that we don’t need to start with number theory for everything; we can memorize some multiplication tables and an algorithm and then get a very thorough understanding of geometry and calculus without ever understanding (or even thinking about) the underlying mechanics of how that multiplication works. I conclude this post with a question: how do you decide where to draw the line? When do you say “this is fundamental and we need to understand it before we move on,” and when do you say, “you can sort of see how this works from this picture; now let’s move on?”
The Virtual Conference on Soft Skills has come to an end. Thanks for participating! The presentations will stay up indefinitely, so if you haven’t gotten a chance to read all of them yet, you don’t need to worry.
I want to give a special thank you to the presenters of the conference: thank you. Treating other people well is hard, and showing other people how to treat other people well is even harder. This is our job as teachers, and this is our jobs as adults; we have to create the world we want to live in, and we do it one classroom at a time, one student at a time, and, you know, one interaction-with-a-cashier at a time. We use these skills all the time, not just in our classrooms, and I’m grateful that there are so many people thinking about them.
To those of you who aren’t sharing yet, on a blog or in some other form, please start soon. We all need help, and we all have help to give. If you don’t feel like you have anything to share yet, well, start figuring something out. If you don’t think you’re qualified to share, well, you’re wrong.
There were 17 presentations in the conference, which is great. Pick your own topic and organize your own conference! This conference probably took me about 5 or 6 total hours to organize and administer. It was great publicity for me and everyone who presented… give it a whirl! It’s been fun. But now, I’m excited to get back to my usual blogging style and schedule. Next post: defining “fluency.” Ooooh!