The Elephant in the Room

As I have been preparing to start teaching in OMG just a few days, assessment has been on my mind; I want/need some sort of formative assessment to get a feel for what understanding exists within this new (to me) community of learners.  What sort of sense-making has taken place, what models do they use, what is their common language?  I know the scores on a textbook test being left for me are going to tell me diddly.  I have developed strong beliefs about what school should look like (and why)— students actively making sense of ideas, learning from and with each other, valuing, making visible, and actively promoting their thinking— and have focused much of my formal and informal professional learning on how to create such a culture.  Yet when it comes to assessment that actually supports my pedagogical ideals, I feel a tad undereducated.  And I need answers STAT.

It does not make sense to invest time in building a safe and equitable thinking and learning culture, to empower students to deeply understand and connect mathematical ideas, to develop and apply calculation skills meaningfully…in short, to make thinking and learning the currency of school, and then not let them spend it at assessment time!  Whenever we evaluate student work with points, grades, or even levels of proficiency (yes, I said that), we send a completely different message about what and who is valued and the purpose of school:  grades and “right” answers and the students who know how to get them.  This is NOT what I want!

What does make sense to me is assessment that is rather indistinguishable from the regular activities of learning, something that involves students in a meaningful and reflective manner.  Something that they actually value because it is FOR them, is designed to both reveal and represent their current understanding to them, not just me.  This IS what I want!

Crazy?  I don’t think so, I just need help in making it a reality.  I feel like there is a ginormous gap in education conversations around assessment (and it onerous sibling, grading).  Not sure of the reason for this deficit.  Overlooked? Avoided? Too mandated? Ignored? It concerns me that much of what little I’ve read assumes/accepts testing and grading as natural and necessary parts of the Game of School.  Sure, there’s some clarity around terminology— formative vs summative, assessment vs testing— as well as some examples of “how” (such as proficiency rubrics, or not using zeros) but not so much when it come to the really, really important question:

WHY?  

Figuring out Why requires us to deeply examine and unflinchingly question still-prevailing status quo practices and compare them to our beliefs and values.  My gut tells me that assessment and grading are not in line or caught up with current practices that are shared in progressive face-to-face and on-line education communities, and therefore, send a conflicting message that undermines change.

Of course, I may just be completely ignorant and you will now kindly steer me to some excellent resources.  Until then, I’m going to do what I always do:  Make Shit Up figure out/find out what assessment that supports, promotes, and honors a thinking and learning community looks like, try it out on some real, live students, and learn.

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Sometimes you send up a flare and the Universe notices.

Last post, I said I wanted to belong.  Guess what happened?

I was recently offered and (duh) accepted a longish-term sub job at a school where I have previously taught.  While the search for a permanent teacher continues, I will be teaching four different ‘levels’ of middle school math for six to nine weeks.  I am SO EXCITED for this opportunity to put into action many of the ideas, values, and beliefs I have reading about and reflecting on over the last three years!  I really do feel wiser, and that feels good. 

This past week, I’ve been cramming.  Carefully read three more chapters of  Creating Cultures of Thinking.  Finally perused  Geoff Krall’s thoughtful blog series  from the summer, which I kept meaning to get to but never did.  Have finally had a couple brief yet valuable twitter experiences. 😆 Revisited several pertinent and inspirational blog posts:  Mark Chubb’s “Never Skip the Close”,  Sara van der Werf’s “Name Tents with Feedback”  and Fawn Nguyen’s “First two days of school” .  I’ve reviewed the routines in Illustrative Math , a stellar curriculum I upon which I am planning to lean heavily.  (Need something similar for Algebra, hint hint!)  Somehow I hope to get in some training in Notability and Google classrooms as well.  I don’t start until the 18th, but much of my time before then will be given to prior commitments.  Most of my anxiety circles around wanting to be fully prepared hahaha, wondering how to deal with the early and long hours and exhaustion, and related to that, what I need to do to keep family time sacred and create/sustain a reasonable sense of balance.  

I have a tendency to overthink (which may look at times like procrastination).  There are so many options and many decisions to make, and am working to gain focus and determine what my educational “big rock” priorities are (student relationships and student learning!), what would be nice, and what to let go, and what battles I can’t fight right now.  Decisions need to be made so I can move forward.  So far, this is where I’m at:

 

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Although students at this school know each other well— its a small-town K-8 AND its six weeks into the year— I don’t know them at all as people or as learners.  Nor do I know what sort of classroom culture I will be inheriting (although I have some sneaky suspicions).  I’m trying to figure out the right the balance between developing a thinking and learning culture and moving forward, deeply, with content.  Usually when you’re a sub, you have to work with the established culture.  However, since I am going to be there awhile, I definitely want to invest time in developing culture, even for this relatively short period, even though it may be dismantled when I leave.  

How could I not?  

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“The circle must be broken.”

 

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I have read and re-read and started processing Chapters 3 and 4 (Language and Time, respectively) in  Creating Cultures of Thinking by R. Ritchhart.  Mind blowing.  (Download it on iBooks to read and discuss with me!)  I am beginning to see some recurring themes: beliefs, values, choices, and messages.  Passive vs Active student roles (my personal soap box).  Its about CHANGING THE PARADIGM so that students and teachers experience different learning stories.  (If interested, my visual notes are below.)

Serendipitously, Dan Meyer  recently blogged about teacher’s beliefs and how our own experiences as students shape our pedagogical choices.  He asks,

 “What experiences can disrupt the harmful messages teachers have internalized about math instruction?

I appreciate this question for two reasons.*  a) the numerous responses were varied and offered sensible ideas I recognize as useful and timely as I find myself gradually drifting toward a mentoring role and b) the question moves us beyond information/opinions about beliefs and pedagogy toward at least a partial yet practical solution to a very real problem: lack of significant and sustainable change.  If a person in education (any position, right? This is not just a math thing!) has not yet had a disruptive experience that “breaks the circle”, reading or being told about a different paradigm is not too likely to yield a shift in values.  If the horse isn’t thirsty, it’s just not going to drink.  Or it may believe a constant state of thirst is the norm and blindly accepts it.  Or the horse feels threatened by a top-down decision that it must drink this water, now. Or it might be really OK with a drink if it just could make sense of what the hell that meant and looked like and had genuine and ongoing support in making it a reality.  

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Change is imperative, and paradigm-altering experiences are definitely one way to get there.  Creating Cultures resonates with me because my beliefs already align with the author’s because, fortunately, I DID have a disruption in my learning experiences that impacted my beliefs. Cycle broken, singing now possible. Also, I tend to be rather reflective and highly interested educational cultures.  But that’s just me, and obviously everyone is different.  What if this (or another) book was assigned and the plethora information within it overwhelms me so I just keep doing what I always have done?  The wonderful suggestions solicited by Dan (video recording with reflective analysis, joining a math teachers circle, imagine something different and plunging in, becoming a learner to gain perspective, shifting from talking to listening, etc.) all seem to share a common trait:  these teachers are the horses that already recognized they were thirsty and went looking for water on their own.  It seems first-hand experiences are the most powerful, but I wonder about colleagues who are not at the point of seeking water (yet).  Not that they are intentionally staying away, they really do care (because they’re people, not horses); honestly examining individual or group experiences, actions, messages, and practices is an emotionally challenging endeavor and knowing so may be enough to want to avoid it. 

What I wonder is, what are possible, effective ways to invite horses educators over for a drink, so to speak, so that many more disruptive experiences can take place?

I invite you to share your suggestions in the comment section.  This is a genuine question I have, so I am looking forward to your insights!    

PS.   I find it interesting that for some commenters to Dan’s post, drill-based math instruction became the question to discuss.  Hot Topic #358.  Interesting reading, to be sure.  Like all other highly-debated beliefs in education, there seems to be too much polarized “This OR That” going on, and not enough This AND That, carefully and thoughtfully balanced to promote learning.  We’re not just emoji yellow OR Tardis blue— it is much more likely that most (if not all) of us are some lovely shade of green.  Which means instead of an all-or-nothing tug of war more productive discussions (and PD**) could focus on examining beliefs, with honesty and without threat of judgement.  Our beliefs shape our actions, our actions send clear messages, both subtle and obvious.  What messages are being sent?  Are they supportive of or detrimental to learning experiences students have today?  

*Also note how Dan’s question was open-ended and accessible to everyone, and how he refrained from sharing his thoughts until he had respectfully listened to other voices.  I see what you did there, Dan.  Smooth move.  

**If you’re part of a group that examines culture in your classrooms, department, or school you could call yourselves….Culture Club!  (If you don’t get that lame joke, you’re too young.  Look it up.)

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Urgent Questions

For more time this afternoon than I’d care to admit, I’ve been attempting to organize my ongoing and somewhat rambling thoughts around teaching, learning, and mentoring.   Much of my reflecting goes on in my head and never gets formalized somehow via blogging or journaling; worse yet, I don’t have much opportunity to actually talk with other educators about these kinds of questions that, IMO, really, really matter.

This graphic is a product of my organizational efforts, and it became my first ever #MTBoS tweet. Amazingly, I immediately received a comment and a ♥️ . Well, that was cool, but I have more to say 😉

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It is my belief that these urgent questions (and their natural sub-questions, including WHY?) should be seriously, honestly, and throughly addressed, ideally as a school, or at least as individual classroom teachers. How we respond speaks volumes about our pedagogical beliefs, the culture of our schools or classrooms and, for better or worse, most definitely impacts students and their learning.

I invite you to consider any or all of these questions, and to share your thoughts in the comments. Or on twitter, if you can find my tweet! (Sorry, I don’t know how to direct you there…) Over time, I will do the same.  ♥️

A Graphic Conversation

Last October I was out of town during the NW Math Conference held in Portland, OR. I was pretty bummed, especially when I read that  Fawn Nguyen  was the breakfast keynote speaker. OK, “bummed” is not anywhere strong enough. I ended up getting up at 5 that morning to drive back to Portland (do you know how freakishly dark it is along I-84 at 5 in the morning?) just to go to that breakfast. The food was meh, but Fawn was lovely, wonderful, smart, amazing, funny….as expected; then I hightailed it back to my other commitment.

Months later, I am finally getting around to writing about something she included in her presentation that stood out for me. (There were many somethings. Plus, she made me cry at the end.)

From Jordan Ellenberg’s book,  How Not to be Wrong, she shared and spoke about this graphic:

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I have re-constructed it with blanks, because I think it is really cool, discussion-worthy, and relevant for ALL subjects:

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Right?

More recently, I came across this graphic on Steve Bohnam’s  blog:

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Which I also modified, because I am going with the premise that more discourse and sense-making can take place if there is a little less information.

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I think both of these these graphics could launch some amazing conversations around and examinations of practice.  Notice and wonder, kids!  As always, please share your thoughts; comments are open.

Half-baked is better than nothing.

Like you, I have a zillion half-baked thoughts and ideas going on in my head.  Stuff that I’m reflecting on, stuff that I find interesting, perplexing, important, and would love to talk about.

One of the things that keeps my thoughts from showing up here is a silly yet persistent notion that whatever I post needs to be Complete and Polished.  Insightful.  Worthy.  Intelligent.  Helpful.  Because that’s how Everybody Else’s blogs look to me.

But if I believe (as I do) that the primary purpose of this blog is for my learning and growth, then it stands to reason that it is completely acceptable for me to share thoughts that are still rough, still in need of additional reflection, and definitely in need of feedback (hint hint).  Writing helps me focus and gain some clarity, and lack of some imagined perfection or level of “doneness” should not prevent me from posting.  Right?

With that said, here’s a taste of what’s rattling around in my mind of late:

Calculating is not mathematics.
Spelling is not writing.
Decoding is not reading.
Memorizing is not learning.

So what is?

IDEAS.

Noticing, wondering, questioning, exploring, making sense of, using, testing, revising, expressing, connecting, analyzing, creating….

When a classroom or school or societal culture values performance and test scores, then teaching and learning evolve around that which that can be easily tested and graded.  Facts and rehearsed processes.  Right and Wrong answers.  Sort to accelerate and remediate.  Rewards and punishments, smart and…below grade level.

The development and questioning of ideas is messier, less quantifiable, harder to teach, harder to nail down.  It’s much more difficult to describe a students growth over time than it is to rank them.  More challenging (and rewarding!) to work with a student’s competencies and current understanding than to fault them for their deficits and errors.  A great shift in values needs to take place; teachers and students will spend their time and efforts differently.  What does this look like?  What’s my role?  How much time will this take? Yikes, what about the risks?!

Teaching is complex.  Learning is complex.  Learning about teaching is complexly complex.  Formal and informal professional development tends to focus on examining, questioning, and improving what teachers and students do and say in the classroom.  Planning and launching lessons, selecting worthwhile tasks and activities, anticipating student responses, questioning strategies, orchestrating discussions, making connections, closing the lesson….perplexity, curiousity, intellectual need, genuine engagement….active learning culture, growth mindsets, metacognition…. ALL REALLY REALLY GREAT and REALLY REALLY IMPORTANT and REALLY REALLY NECESSARY.

Yet the Student Learning Experience encompasses more than “The Lesson”.  What about homework?  What about assessment?  Grades?  What about __________? Without examining and questioning and improving ALL components, without implementing changes simultaneously, progressive efforts become at best undermined and at worst derailed and rejected. What’s the point (asks a student) to make sense of these ideas or persevere on this task, if they only thing I will be tested on for a grade (the only thing that matters) will be whether or not I can calculate the right answer?  Why should I be curious?  Why bother making connections?  Explain my reasoning? Transfer ideas?  Develop relational understanding?  Just tell me the trick/hack/rule.  That’s all I need to survive.

That’s what my dad did.  That’s what my grandma did.  That’s all math is.

Why Limit Opportunity?

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.

IMG_2416                             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.

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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.

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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.

 

 

 

If you 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.

IMG_2418As 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.

(Fin.)

 

* 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.

Write this down…

I’m wondering….again.

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.

A Picture is Worth…..Everything.

I spent about an hour yesterday playing with geometry tiles.  I was:

  1.  Fully engaged and enjoying myself.
  2. Experiencing OMG, WTF, and AHA moments.  (AKA actively learning.)

The inspiration for my mathematical play time was this photo from  Sara Vanderwerf’s recent post  on Stand and Talks, a bold routine she uses to increase the number of students talking about math.   Really great stuff and worth your time to read it through.

IMG_2094But back to the photo, which is her example of using a Stand and Talk to introduce a new idea, in this case, the Pythagorean Theroem.   My eyes perk up (is that a thing?) because I have been pondering about this very topic.  Specifically, wondering about what activities/tasks would place students in the active role of sense-making.

The first thing that popped into my head was….wait, trapezoids?  (Thank you, Sara!) Then I started some noticin’ and wonderin’ from my teacher-y perspective and also imagining/guessing what students might say.  The more I noticed and wondered, the more I realized how FULL OF MATH IDEAS this one photo is.

I find this exciting.  Really, really exciting.  Can you imagine starting here, with this one little image, asking students what they notice and wonder, and then letting them run with their curiosity?  Isn’t it joyful to know in your heart of hearts that this is so much BETTER than, “Hey kids, here’s a little something called the PT, copy it in your journal/onto your pre-made “doodle” notes, plus copy these three examples of how to get the right answer, and do this worksheet”? How superior this single photo and all its potential are to “I do, we do, you do” and /or anything else that keeps students in the passive role?

It didn’t take me long to drag out my tub of tiles and start messing around with them.  TOTALLY what I would want my students to ask beg to do.

The questions I decided to guide my play were:

First, can a proportional, scaled-up version of each polygon tile (that I have) be built with only that particular tile?
If so, why? Is there a pattern?
If not, why not?

Second, can any three similar polygons be used to make any type of triangular “hole”? Why or why not? Which ones make right triangles? Why? Is there a pattern??

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I started with squares.  Don’t know why, maybe because that’s what is already familiar to me.  No surprises, the areas are the square numbers, 1, 4, 9, 16…..

So I moved on to the equilateral triangles, wondering/kinda expecting that I will make triangular numbers.  Imagine my surprise IMG_2119(my OMG moment) to find SQUARE NUMBERS AGAIN!  Holy shit!  NOT what I was expecting!  Now I feel compelled to keep exploring.  COMPELLED.  (Imagine your students feeling compelled!!)

I quickly confirmed that the next in the series takes 16 triangles, and moved on to IMG_2125the rhombus.  YES, the areas are square numbers AGAIN!  Which gets me reconsidering my understanding of square numbers and forming conjectureish questions:  Can square numbers be made out of any polygon, not just squares? These are all regular; what if they are irregular?

I’ve been reading Tracy Zager’s fabulous book Becoming the Math Teacher You Wish You Had.   She starts Chapter 7, Mathematicians Ask Questions, with a quote from Peter Hilton: “Computation involves going from a question to an answer.  Mathematics involves going from an answer to a question.”  Brilliant distinction.  Looks like I’m living Chapter 7.  Know I want this for my students!

Remember, one photo.

I finally get to the trapezoids, which are more challenging to tile and require some analysis.   I managed to construct one, but the number of tiles it took (7) was a red flag for me, so I  examined it closely to verify if it was proportional or not.  It wasn’t, so I kept playing, keeping theseIMG_2126 two beliefs in my mind: a proportional trapezoid was possible, and it could be made out of 4 tiles.  And lo and behold….

TA DA!  “Square” Numbers!  (Whew!)

 

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By now, I have even more questions.  By now, I have reviewed and applied my prior understanding of proportionality and similar figures, without ever touching a worksheet.  By now, I’ve taken risks, persevered, reasoned, looked for patterns, used vocabulary, formed conjectures, and justified.  All from looking at one picture and allowing myself both time and pleasure to play and be curious….and I haven’t even begun exploring that thing with the triangular “hole”.   Yet.

Finally, the hexagons.  And….What?!?  You can’t even make a similar hex without using some non-hex friends!  WTF!

New question!  WHY NOT?

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One.

Little.

Picture.

Competencies

Its been so long since I’ve blogged that I had to look up my password.

One of the many reasons for my little hiatus is that there are plenty of UH-MAZING blogs out there– that people actually follow and read– making what I think and wonder and write about rather…superfluous.

For example, Ilana Horn’s insightful, intelligent, inspirational blog.  Informed and intriguing.   I recently started following, and back-read several posts she has written on status in the classroom, in part because I believe know status is deterimental to equitable learning yet is created and deeply ingrained and even actively perpetuated in the Old School system/institution of teaching and learning.  She explains it all  very clearly in a series of posts:

Status: The Social Organization of “Smartness”

Seeing Status in the Classroom

What Does it Mean to be Smart in Mathematics

Recognizing Smartness and Addressing Status in the Classroom

This morning (since I kind-of overdid it hauling bark dust yesterday), I decided to chill a bit and create a list of competencies I value in my classroom.  Not besides “fast calculations and right answers”, but instead of.  A definite and requisite shift in classroom currency if one is striving to achieve an active and equitable learning culture.

In no particular order….

Curiosity.
Perseverance.
Observation.
Students in the role of sense-makers.
Connections between mathematical ideas.
Connections between representations and models.
Clear communication of thinking (the WHY), even if incomplete or unsure.
Active and intentional listening to all peers.
Self-reflection.
Metacognition.
Multiple strategies and solution paths.
Gaining insights by making mistakes.
Willingness to revise thinking and understanding.
Great respect for the value of every person, their learning, and the strengths they already have.
Genuine Questions and Wonderings.
Collaboration in learning as a community.
Flexible thinking.
Creative thinking.
Visual/alternative representations of reasoning and ideas.
Connections between multiple representations.
Connections between different strategies.
Patience.
Perplexity.
AHA and WTF* moments.
Growth.
Active awareness and regulation of learning.
Attention to reasonableness of solutions (yours and others’).
Private time to think (and respecting it).
Critique of thinking, reasoning (not people).
Critical and deep thinking.
Understanding the thinking of others, even when it differs from your own.
Respectful disagreement.
Respect for (and celebration of) strengths and strategies that differ from one’s own.
Genuine/legitimate peer support in learning.
Growth mindset.
Engagement and involvement.
Willingness to start even if you are not sure.
Equitable collaboration.
Consideration of ideas other than your own.
Ability and willingness to adjust your reasoning/opinion and change your mind.
Learning from peers.

Wow, that list is a lot longer than I expected.  Which would you add, revise, or omit?  Why?

Here’s what I might do with such a list.  At the beginning of the school year, cut it up and have students in small groups sort them into 2-5 or so categories, their choice.  Sorting activities are a worthwhile way to get kids talking to each other, voicing opinions, making choices.  Listen in, because you’re finding out about them, too.  Notice common choices as well as different ones.  Ask groups to explain their categories to you.

Then, as a whole class, share and discuss.  Ask them to notice things.  I have NO IDEA what will happen here, but I’m wondering if anyone will notice that “right answers”, “smart”, “good grades”, “fast thinking” and those types of competencies typically over-valued (and detrimental to learning) are MISSING.  So are generic behavior-type rules, like arrive on time, do your homework, pay attention….Will they notice the focus on inclusiveness and learning instead of on first and fastest?  Will they identify with some of them?  I’m really curious about how kids will sort these and what they will say!  Finally (if there are enough common themes?), use their input to develop a SHORT list of classroom norms that recognize and support these valuable competencies.

 

 

*Probably should change this to WTH What the Heck, or HIW Hold it, What!? Or some such thing more socially appropriate, right?

WTF moments are not moments of frustration, though.  They are moments of realizing something is amiss, some reasoning, intuition, or process is not going the way you expected, or the solution make no sense.  Disequilibrium and perplexity reside here.  In a sense, these moments are insights, too, a realization that an adjustment is needed;  understanding WHY one path works and the other does not paves the way to the bigger insight (AHA!) and gains in learning and understanding.  When students people share their thinking, they tend to leave the WTF moments out and share only what worked, saving face and strengthening the currency of “right” answers.  However, in a healthy, inclusive culture of learning, WTF moments are valued as an important and natural part of the learning process, worthy of sharing, even celebrating!   “First, we thought….because….then we. saw…realized…tried….because….figured out….learned….”. Even “First we tried….because…not working……and now we wonder….not sure….have some questions…..”