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*?

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.

]]>For one assignment, we read this article and chose a quote for reflection. While perusing comments, I was especially struck by this exchange:

My mind is blown by John’s suggestion that ALL decision-making from ALL stakeholders in education *including students* should be made with student learning experiences in mind. Not just students or their learning, but their **experiences!**

While attempting to wrap my head around this, I developed these notes:

What do you notice and wonder?

Currently, *I’m* wondering…

What is the learning experience for students in each of these scenarios? Which is “ideal”? Why? For whom?

In fact, what *is* an ideal learning experience, and won’t these vary depending on the person?

What are the *teaching* experiences?

For which of these scenarios are a variety of ideal student learning experiences **difficult** to achieve? Or not so difficult? Why?

What the **role** of each stakeholder/group in each scenario?

I’d like to know your thoughts!

]]>I’m pretty sure any 4 year old could put these pots in order. Dina, too.

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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!**

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.

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.

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.

(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.

]]>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!**

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**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

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.

]]>Eclipse-inspired art.

]]>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.

]]>- Fully engaged and enjoying myself.
- 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.

But 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??*

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 (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 the 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!

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 these 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!)

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?**

One.

Little.

Picture.

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