I’m pretty sure any 4 year old could put these pots in order. Dina, too.
I’m pretty sure any 4 year old could put these pots in order. Dina, too.
I’m attempting to help a teacher friend of mine (Jackie) make over some lessons in her district’s new online curriculum. It’s a frustrating challenge; every lesson simply tells information and shows processes without any exploration or sense-making opportunities for the students. Very Old School, very passive (aka confused) students, very much perpetuating the myth that math is a jumbled bunch of random rules to memorize.
The topic in an upcoming lesson is irrational square roots. For Cathy Yenca’s very helpful online class Seeking Students Who Hide , I made a Socrative “quiz” to generate discussion about the relationship between the area and side lengths of squares, the rational roots of perfect squares, and some perplexity about the root of a “not-perfect” square. If you already have a Socrative account, the import number to share my quiz with you is SOC-30095225*. If you don’t have an account, get one now, I’ll wait. It’s FREE and fairly self-explanatory. Even I am figuring it out, and that’s something.
What’s important to note is that Cathy is showing us how this tech tool can be used for giving every student a voice, even an anonymous one. Anonymous is safe. No one gets to hide or opt out or dominate a discussion. So it’s not a quiz, its an equity tool, a real-time formative assessment tool. I chose to have my “quiz” (what should this be called instead?) be teacher-controlled and anonymous so questioning, discussion, exploration, and justification can happen in between each prompt, depending on what students say, ask, and need. I’m picturing having them draw perfect squares on graph paper (low floor), introducing them to the square root symbol, using area models to make sense of the length/area relationship, and challenging them to make whole number “not-perfect” squares (high ceiling).
The final multiple choice question about an area of 20 square units is meant to be the zinger. Four of the five choices can be justified, IMO, so I purposefully marked every answer “correct” when creating it so that the data we’d see as a class would be about the percent of students who chose each answer, NOT NOT NOT about which answer (or who) is “right”. I actually hope for quite a mixed bag, which is the perfect place to start an exploration into irrational roots of “not-perfect” squares.
*I’d love feedback from anyone who even just looks at this quiz. This is new territory for me and I am not sure I’m going to get to implement it. If you use it, even modified, let me know what happened! Here’s the bare-bones version; keep in mind something should be happening in between each question.
1. What are square numbers?
2. How do you find the area of a square?
3. Describe the relationship between the area of a square and the length of its sides.
4. T/F. √ 49 = 7 5. T/F √18 = 9
6-8 Solve each of these: a + √36 = –5, √121 – x = 7 , –14 = n – √64
9. If y2 = 25, what’s y? Explain.
10. If a square has an area of 20 square units, how long is each side?
UPDATE: Jackie and I have decided to go for the gusto and implement this Socrative lesson tomorrow! What excites me the most? Finding out what students say!
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.
I finally get it.
I have never understood the assumed necessity of having and using some big publisher’s packaged curriculum. You know the one, 15 pound student textbooks with a teacher’s guide painstakingly chosen by a committee every X years (or not, depending on the budget cuz they’re damn expensive). It seemed so Old-School-ish to me: curriculum = ultra traditional textbooks = lesson plans, Day 1, Lesson 1, Page 1. Here’s how you do it kids, practice it 5000 times, the test is next Thursday. Repeat ad nauseam. Yuck (with smug eye-roll). As a teacher, I rarely used the school’s textbooks except as an occasional resource. I already know what I have to teach, isn’t that the point of having standards? How I teach is up to me (thank goodness), and I chose to work towards constructing understanding via student-centered learning via worthwhile tasks. Prioritized student reasoning and valid arguments, focused students more on Why than How.
At least, that’s what I always aimed for. What a rebel. Consequently, I spent hours and hours (because I did not know about MTBoS yet!) developing my own curriculum, planning my own lessons, agonizing over details. H.O.U.R.S. Constantly creating, reflecting, refining. Continuously making shit up, learning from my mistakes (plenty of those!), trying again. The long hours and effort probably contributed to my early burnout, do ya think?
What I realized today is that I did not see any need to rely on a packaged curriculum because I had never met one I liked, one that I believed in, one that rang true for me, impressed and inspired me, one that I could follow without hesitation or makeovers or by sacrificing my convictions.
I seriously cannot say enough good things about Illustrative Math and Open Up Resources. OMG, this brand-spankin’-new 6-8 curriculum really delivers. It aligns so beautifully with my beliefs about teaching a learning mathematics that I could actually do Day 1, Lesson 1 without any qualms whatsoever. None, I’m not kidding. I get chills (they’re multiplyin’) just thinking about the implications.
NOW I get what you mean, Sadie Estrella (Math Nerd) when you insightfully advise us to rely on our good curriculums so we can put our precious time and energy into knowing our students and meeting their needs. I GET IT! Only now do I see and believe that it’s possible, because finally finally and oh joy of joys, here is a worthy curriculum that is going to hold me up when I lean hard on it. More than good. Glorious.
Plus, FREE is a very good price. Check it out NOW!
What is the purpose of note-taking?? (Specifically, in math class; still more specifically, in MS/HS math classes, where note-taking is more common, suddenly the more appropriate pedagogy.) Why are students expected to take notes? How essential are they, really, to learning? How much of it is about compliance (or The Way it Has Always Been) and how much about active, student-generated sense-making? Are notes primarily for recall of facts and rules, to practice processes and prepare for tests, or can notes aid in the construction of conceptual/relational understanding? What does it mean to “take notes”, and are some options better than others? Which ones? Why? Are there times when note-taking is appropriate and beneficial for the learner, and other times it is not? If so, when and why? Is there a difference between taking notes and making notes?
But before that, I noticed…..
I’ve heard teachers (including myself) repeatedly remind their students to take notes (Get out your journals! Write this down!) only to lament their lack of use. Like you, I’ve seen student notes range from dutifully copied examples to partial and chaotic scratches on a random page to nothing at all. I realize students do not, after all, magically know how to take or use notes just because they reach middle school. I’m aware that copying examples/filling in blanks ≠ understanding what the heck is going on, let alone why. I know some students are able to mimic processes well enough to maintain their status as the “smart” kids…and you know what happens to everyone else. I notice a lot of popularity with ISN’s and more recently, sketch or doodle-notes, both claiming to be improvements on “traditional” note-taking routines. There is a lot that intrigues me here, but I am cautious. Maybe even confused.
Here’s why. Much (all?) of the ISN and doodle/sketch-note materials I’ve seen via my brief ‘research’ online are teacher-generated, limiting student interaction to following directions/filling in blanks, keeping students passive and unburdened by any need to make sense of ideas. Happy and busy ≠ engaged in content and thinking critically; what risks are being taken, what fabulous mistakes are being made? How will there be any WTF or AHA moments?
My gut-feeling is that note-taking actually plays a more significant role in the culture of our classrooms that we realize, impacting everything from mindset to equity to assessment. The parameters and expectations we set communicate what and who we value in our classrooms and defines who takes the active or passive role, teacher or learner.
Mind you, I’m no expert on this subject; my teaching skewed toward less notes, more tasks, and my knowledge is limited. Note-taking does not seem to be a blog-worthy topic; I’m suggesting it should be. Surely I am not the only person asking these questions! There are plenty of blogs, mostly positive, about ISN’s, which seem to at least have the potential to include more than rules, examples, and definitions. I’m imagining how powerful post-exploration student-generated sketch notes would be…
I just want to push hard on some assumptions about notes and note-taking, start a conversation, ask more questions, and gain some insights. My gut is tellling me this is another component in education in need of a transformation. And to go eat a peanut butter cookie, but I am trying to ignore that part.
I still don’t understand twitter. That is, I get how it works, how one can write and read tweets, like/dislike tweets, comment on them, retweet, etc., (whether or not they contain truth or have value). All actions that seem available in all forms of social media, including blogs. I get that it is about sharing.
I am not inclined to have my personal or professional life revolve around social media. Maybe its generational, maybe it just me being selective or making priorities. Maybe my skepticism and reluctance are based on misconceptions and ignorance. Or all of the above. I know I am not interested in the self-centered, everybody-look-at-me aspect, although I don’t think that’s what people have in mind when they encourage me to get on twitter for professional purposes.
I guess what I don’t understand is, what DO they have in mind? What is it they are asking of me? Why do they feel this is important? What are the advantages and disadvantages? Is it possible to use twitter to have a worthwhile conversation? If so, how? I am not even sure what questions to ask about twitter that will convince me that it is worth my time and effort.
What am I missing, here?
PS. In the spirit of making it all about me (and to include a visual in all this boring text)… lookie what I did!
UPDATE (10 minutes later)
It occurs to me that maybe I should stop overthinking and just get a twitter account and start looking at what y’all are doing and saying.
Can you teach an intuition?
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The Writings Of Alfie Kohn
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Not everything that can be counted counts, and not everything that counts can be counted. -Einstein
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Not everything that can be counted counts, and not everything that counts can be counted. -Einstein
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