No point(s) needed.

Once in college I was docked a point on a math test because I did not use the variable the problem told me to use. What a rebel.  My work was otherwise fine, mind you, just had the “wrong” variable.

I tell you this because early this morning, maybe a tad too early, I read Fawn Nguyen ’s recent post  “Scoring an Ordered List” ; at first I took what she was saying seriously because to me, everything she writes is solid gold. (You should probably read it to get where I’m going with this.) I gushed about her in my last post, for heaven’s sake.  Why is the great Fawn obsessing with points?  I started feeling some panic.  WHY WHY WHY is she obsessing with points?  I was flummoxed.

Then the caffeine kicked in and it dawned on me. She’s just messing with us. Having some fun with obsessing with points, and inviting us to join in, math-geeky.

Or maybe she’s making a point about points, in which case she has gone platinum.

Awarding and denying points is what many teachers do, across all content areas, even if other aspects of their pedagogy employ Best, or at least Better, Practices. Somewhere between arbitrary and intentional, they determine points per problem, points per quiz, per essay, per whatever. Preferably the points add up to something that can easily be turned into a percentage so letter grade convertions are a snap.

Why?  Because it’s what what they know, what they believe they are supposed to do, its what everyone expects. It’s another one of those pesky, unexamined “givens” in the dystopian Game of School. I don’t know if that’s the right use of dystopian here, but I like it so its staying.

Determine total points possible, deduct points for wrong-ness, calculate a score, assign a grade.  So math-y.

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You can find scores of graphics like these online, which supports my statement about the Game.   I find the emojis particularly unsettling horrifying.

This poster is hanging in a school I frequent: ‘Grades are not given, they are earned.’

Bull. Shit.

I digress. Returning to Fawn’s scenario, suppose a teacher decides that the problem was worth X points, and to receive said points, every number has to be in the correct location. Black and white, not even one shade of grey. That is certainly the way a computer program would determine whether so-called feedback should be “Awesome! You Rock!” or “Incorrect.” Of course this means that if Kat writes the numbers greatest to least, she would be told she is ONE-HUN-DRED-PER-CENT-WRR-ONG, striking yet another blow to her growth mindset.

An alternative (that I am sure Fawn and many other awesome teachers use) would be to skip the points, look at work holistically, and ask Kat a question or two. The goal is to understand her thinking, to gauge what she understands and where her misconceptions may lie. Kat either needs a reminder to read and follow directions with greater care or a conversation around the words least and greatest. Easy peasy.

Just for kicks and giggles, here’s another example: Suppose I’m checking to see if my students can correctly follow order of operations to simplify expressions. Dougie clearly demonstrates mastery of this skill— nothing out of order in his work— yet it contains a minor calculation faux pas or two. (Been there, done that. Lost points.) I am NOT going to deduct points or even determine he is not meeting expectations. I am going to PASS Dougie on that particular skill, period.

There’s a significant difference between 1) using student work to inform both the teacher and learner so meaningful feedback can happen and learning can continue, and 2) using student work in order to pass judgement. Which should be valued, empowering all students to think critically, creatively, to make sense of (insert content area here), or rewarding some students for “correct” imitation and memorization? The first requires, if I may, a balanced, healthy, and inclusive teacher-student learning relationship; the second also involves the student, but is unhealthy and detrimental to learning because it also involves power.

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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.

What’s an Ideal Student Learning Experience?

I am seriously grateful and excited to be a participant in Geoff Krall’s  online course, Mathematical Anthropology.  Nothing beats being able to listen to (and learn from) voices other than your own, even if virtually. So much to process! From all over the world! (And it’s free!)

For one assignment, we read  this article and chose a quote for reflection. While perusing comments, I was especially struck by this exchange:  5A751E09-7F08-4CC7-B131-E07EAD20DC81

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!

Mad to Glad

IMG_2520I’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.

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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 IMG_2521about 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!

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.