Introduction to Mathematical Literacy

I was graciously invited on Monday to give a ten-minute lesson on literacy practices for part of our Professional Development session yesterday. Here’s my best attempt to capture this lesson in writing! I hope you find it informative.

Always in big woods when you leave familiar ground and step off alone into a new place there will be, along with the feelings of curiosity and excitement, a little nagging of dread. It is the ancient fear of the unknown, and it is your first bond with the wilderness you are going into. You are undertaking the first experience, not of the place, but of yourself in that place. It is an experience of our essential loneliness, for nobody can discover the world for anybody else. It is only after we have discovered it for ourselves that it becomes a common ground and a common bond, and we cease to be alone.”
-Wendell Berry,
The Unforeseen Wilderness: Kentucky’s Red River Gorge

Today we are faced with the task of answering the question “How does one effectively teach literacy?” in the next ten minutes. So let’s hop-to. 

To discuss literacy, let’s start with the foundations: it’s questions and it’s metaphors. First, the question: What is literacy? Well, in one sense it is being able to read and write. But that’s obviously not what we’re talking about here. We’re talking about the other definition: competence in a specified field. Media literacy. Financial Literacy. Mathematics literacy. In order to understand a specified field, we are faced with a conundrum as learners… but a catastrophe as educators. The catastrophe is this: in order to understand a specified field, we must be able to ‘speak the language’ of that field, if you will. But, of course, to speak the language of a particular field is to share a common bond with another… and this requires that you have discovered the world for yourself, and then expressed that through language. To demonstrate, let’s start where one would naturally when exploring literacy: mathematics. 

I’d like to teach you how to multiply and divide. Sorry, I messed up with my language: I’d like to teach you how to play with numbers. It’s fun! 

Post-It Notes come in packs of three stacks each. I want 24 of them. How many packs should I buy? HOLD ON. No answers here yet. You are in second grade and will be doing some discovering. [Pass out the manipulatives to count it out]. I’m now making you all go through the act of counting these out to discover the world for yourself: that 24 divided into groups of 3 makes 8 packs. And then I’m making you do similar types of problems for the next couple weeks, at least. 

AND THEN, and only then, I stop our whole little crew of math learners, and I say… hey, guys, we’ve been doing this thing where we want/have a lot of something and want to get enough of it / spread it evenly among a few people for a while now, and it takes a long time to say that. Should we just decide to call it something? Like … how about “division”? 

And this is the conundrum, right? In order to teach literacy, we must in some way experience doing the thing, and then get to the point of understanding the thing well enough to want to save time and name the thing, aka ‘to abstract it.’ And THEN, we have to understand that the people who ‘invented’ the field abstracted it by naming it or symbolizing it differently than we might have, so we have to learn their language! And just naming something is a first-order abstraction, right? Calling it “x” might be considered a second-order abstraction, but the point remains that mathematics is nonetheless just a language on it’s own: a way of expressing (aka communicating) patterns! 

Like this simple pattern: 

My friend Eric says not to drink soda, so instead I’ve been following my friend Chad’s advice and drinking beer! My favorite beer comes in 16 fluid-ounce tall-boy cans – packs of six – whereas Chad’s favorite beer comes in 12 fluid-ounce cans. How many fluid ounces do I get when buying a six pack of Chad’s vs. my beers? 

Well, if I assume that we are now adults again and we have learned the language of mathematics through “multiplication,” but perhaps still don’t really like to do multiplication, we’re not happy about these big numbers. That’s probably because there is an underlying problem with two things: our underlying metaphor of what multiplication is (memorized rules vs. sets of items), and the amount we’ve discovered aka “played with numbers” over time. But 16 times 6 is like 10 times 6 – 60 – plus 6 times 6 – 36, which is like 60 plus 30 – 90 – plus 6 – 96. 12 times 6 is like 10 times 6, 60, plus 2 times 6, 12, so 72. 96 fluid ounces is 72 plus ten (82) plus ten (92) plus 4, so 24 more fluid ounces in my 6-pack than Chad’s, and then from there we begin to compare the costs vs. ounces. 

What about this, no magic black boxes that produce the right answer of any sort – only logic please: 

Screenshot 2020-01-23 at 8.08.11 PM

What I find often is that most people know what this means. Ask them to discuss and they start conversing about the ‘right answer’. The conversation tends to be the interesting part. “OK take this number, and blah blah blah” or they use the number itself to talk about it. Well, what happens when we want to create a mental schema for this situation, aka “a rule”? Well, then it might be useful for us to ‘name’ parts of this statement – the numerator and the denominator; the dividend and the divisor… why do we have so many names for the same concepts/processes? Because just as we named ‘division’ before out of a need to do one thing, we then had a different need later and this is the dumb system we ended up with. Again – nobody can discover the world for anybody else – we must first discover it for ourselves, and then we may understand the common bond of developed conventions, and why they are necessary even if they are confusing. Anyways, back to this process of playing with numbers – if we were to simply think of ‘moving’ the decimal to the right in both the numerator and denominator of our first mental math problem, we can see that it’s just 56 divided by 8, which is 7. If we have our mental schema of division down (from earlier), then I can also use my understanding that division is asking how many times can 0.8 go into 5.6, we can use fingers to confirm that our answer of 7 is actually correct: 1 is 0.8, plus a 0.8 is 1.6, then 2.4, 3.2, and we can tell that indeed 7 is a reasonable answer. We can posit a rule now: if I ‘move’ the decimal the same number of spaces right in the numerator and denominator, my answer does not need a decimal-place adjustment. Following the same process for the next two mental math equations, we can discover that 70 and 0.7 are the answers and posit the corresponding rules about which way to move the decimal place on the resulting answer. In this way, we rely on a confidence that we can logic our way to the right processes in math, avoiding the constant fear that we have simply forgotten the rule, or memorized it backwards, and thus are doomed to be failures.

All this to say that trying to teach multiplication by first calling it multiplication is something we should be discussing… despite the fact that it takes far longer to develop this sort of knowledge, I personally think it is far more effective, and I would go so far as to argue that it’s the niche of Expeditionary Learning… one of the principles we are founded on… OH, and also what “literacy” means. Thus, my metaphor for literacy is ‘discovering the world for oneself first (in an abbreviated manner – that’s the job of the educator), and then learning the language through which to discuss it. 

Homework: Chapter 8 of Postman’s Teaching as a Conserving Activity

After reading, answer the reflection questions: What can you do during lessons in the next week that will help students to better understand and have mastery over “the language of” a particular subject? How does your ‘to-do’ item help to support the ‘discovering the world for ourselves first’ metaphor?


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