Flunecy (Part 2 of 3)

  1. Building on imperfect or incomplete understanding – where is the line?
  2. Math as a language whose symbols take on their own meaningful attributes.
  3. Connecting math with forms of expression with which students are already fluent.

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 a.  Now “I have 2 apples” is equivalent to “a=2.”  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 “a + 1 = 3,” 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 a(d) apples where a is the number of apples I have and d is the number of days from now and a(d)=d+2.  “How many apples do you have 10 days from now?” becomes a(10)=12 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.

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