You know how sometimes TV shows begin at some point in the middle of a story arc, right at the point of high drama (a door opens to aliens, the heroine at the brink of death with no escape in sight….) and then on the screen you see “8 hours earlier…” or “one week ago” and you’re taken abruptly back to the calm beginning, still knowing where its all going to lead?
Ya know what I mean here?
I’m going to do that now.
Point of high drama in my day: WTF!
Far, far too many hours ago….*.
Recently, colleague “Jackie” and I looked at the opening lesson of her new curriculum, in which students review HOW to change a ratio/fraction into a decimal.** The “real world” application included is so faux its not even funny. Seriously, if you want to know which wrench (with fractional measurements) fits which bolt (with decimal measurements) on your bike, you are not going to grab pencil and paper to set up proportions! You’re not even going to use a calculator and divide. You are just going to try them until you find one that works. OK, maybe a little funny, in a sad sort of way.
Students are next prompted to change a repeating decimal into a ratio by GIVING THEM step by step instructions showing HOW, using algebra (Cue wah-wah sounds.) No connection AT ALL to the strategies just reviewed, no reason to do this other than to comply. What?!
After we finished gnashing our teeth and pulling out our hair, we started thinking about how to approach this content differently. That is, how to generate a need to convert repeating decimals and create a headache around the process that would have kids begging for some aspirin. All while resting the responsibility of sense-making on the shoulders of students. Where. It. Belongs.
We decided we’ll begin with a Which One Doesn’t Belong? to activate some prior knowledge and vocabulary.
The thought is, through some discussion and probing questions, students could arrive at these questions:
What are rational numbers?
Which of these are rational? Why?
Does every ratio/fraction have an equivalent decimal version? (Why/Why not?)
Does every decimal have an equivalent ratio/fraction version? (Why/Why not?) What about that repeating decimal…?
Next, we’ll give them some time to wrestle with converting repeating decimals, then when they ask for salvation, show them The Aforementioned Algebra Process in its entirety, without explanation, and have them work in small groups to 1) identify what is happening, 2) ask questions and 3) make sense of it.
Commercial break and time passes. Jackie and I part with our vague plan and our fingers crossed, and I sit down to think about it some more. Because vague does not sit well with me. Naturally, I end up overthinking it all evening and again the next day, which is my problem with trying to make a silk purse out of a sow’s ear. I also spent some time thinking about how I would incorporate Socrative, a tech teaching tool totally new to me filled with potential that I am dying to try. (More on that another time. Maybe.)
At some point I remember that I am only going to be in Jackie’s room for one short day and that this lesson is going to take several. I’m over-creating for my minor role. To focus, I decide to make the WODB above, and just to make sure I understand The Algebra Process and to anticipate student difficulties/misconceptions, I give it a go.
I’ve intentionally included alatta steps so students can (hopefully) dust off algebra skills and increase the chance of sense-making. I also used 1/3, since they may already be familiar with this equivalence.
give a mouse a cookie give an 8th grader a process to make sense of, they’re going to want to try it out on another repeating decimal. Well, at least that’s what I wanted to do. Maybe they will, too.
As I finished, I remembered that Jackie mentioned something about the process always involving 9’s. Now I see why.
Does this mean, I wonder, that EVERY repeating decimal’s fraction version has a 9 (or 99 or 999, depending) in the denominator? Let’s find out!
I also see an opportunity for students to notice the pattern and make a conjecture. An opportunity that would have passed me and students by had I never attempted to makeover this lesson because the student sensmaking in it is nonexistent.
Do you feel how close we are getting to that opening drama here? Truthfully, I was really enjoying where this was going; my childhood math experiences did not include this type of exploration, and it is FUN. Seeing an opportunity I did not know was there is exciting. I imagine students might think this pattern is just another neat-o/mysterious math trick and stop there. Unless you insist they test their conjecture….
And now, a word from our sponsor.
This entire explorative experience and the inevitable WTF moment will never happen if students are merely asked to imitate ad nauseam a process they don’t understand, followed by a test and a grade. If the culture of a classroom (and its supporting curriculum) revolves around “standard” algorithms and “right”answers instead of noticing and wondering, curiosity and perplexity, student-centered sense-making, and celebrations of WTF and AHA moments, then our students are being robbed of opportunities to see the beauty and humanness of math, are being denied a chance to know they are mathematically capable, and are less likely to grow into curious and creative people who can develop viable arguments and critique the reasoning of others. Life skills, for sure.
Where was I? Ah, yes, testing a conjecture.
Which I did. And ended up with 0.99999…. = 1. A fabulous WTF moment, I must say.
Notice the purpose of converting repeating decimals shifted from performing a rote process (booooring) to students uncovering something Big (exilerating). It does not matter that you will probably not be able to resolve their angst over this issue; in fact, it is OK to discuss a bit, argue a bit, consider a bit, and then leave this perplexing moment…a bit unresolved. (If you google it, you’ll find a lot of arguing. Infinity is difficult to nail down.) It is sufficient for students to learn that it is in excatly these kinds of moments where humans need to make sense of something that does not make sense that new ideas are born and learning happens. Zero. Place value. Fractions. Negative numbers. Irrational numbers. Imaginary numbers, for heaven’s sake! Infinity (and beyond).
You get the idea.
* Due to the fact that I am trying to help Jackie make over a less-than-satisfactory “new” curriculum; see my previous post about finally understanding why having an exemplary curriculum is a much better situation.
** The presentation of this topic is so very rote and unexciting it will do a great job of keeping kids hating math.