The Thrill Remains

Our experiences create our beliefs, from which spring our values, upon which we hang our decisions and actions and words.  

I wasn’t going to read it but I did.  I wasn’t going to write a response, but I am.  I wasn’t going to make it another opinion piece, but it is, with additional resources below.  

My life experiences as student of both music and math and as an educator have shaped my opinions about learning and inform my pedagogical philosophies.  I know first hand one can appear to be “successful” at math or piano by simply being compliant and willing to please, which can easily be mistaken for dedication.  When told exactly what to do, I could.  A’s in math, scholarships in music.  I also know first hand the thrill of realizing there was so much more, that instead of passively following directions, I could actively listen and notice and explore and question and make sense of and mess up and try something else and reflect and maybe most important of all, feel competent and joyful. 

Whenever anyone over-simplifies anything as intensely complex as teaching, learning, math, or music in order to justify a particular personal stance,  its a whopping red flag for me.  I cringe at math-music (or math-sports, etc.) metaphors that appear to convince but under scrutiny, fall apart.  I am skeptical whenever words like “drill”, “understand”, “practice”, “learn” and (good grief) “ingrained” are used liberally without any clear, consistent definitions by the writer, especially when said writer makes claims about what’s wrong with teachers (or kids) these days and here’s exactly how to fix it.  I am perplexed by beliefs that Only This is Right and Poo-poo to That (Its Not How I Had to Do It).  I know how easily the general public latches onto back-to-basics type opinions and then banters them about as utter truths, forgetting about bias, prejudice, and self-interest and heaping more myths and misconceptions onto the pile about teachers and teaching, learners and learning.  I cringe whenever motivations for transforming education are driven by competitiveness and better test scores.  And I feel quite, quite sad that someone who confesses to hating math as a child believes its OK for learning to lack pleasure— to the point of being painful— and that we should make our daughters (OK, sons, too) experience the pain we did, and expect them to be grateful for it. 

While some people have a NY Times op-ed piece or masses of twitter followers, I just have my nearly invisible little blog.  This does not mean my reflections (which give me clarity) and opinions are any less important. I’m just less influential.

I leave you with links to Mark Chubb’s response to the same op-ed piece that’s particularly thoughtful and constructive (and influential), and to Dan Meyer’s follow-up question that re-focuses an out-of-control discussion.  In my opinion, of course.  As always, any additional resources you find valuable and/or insights you include in the comments are greatly appreciated.

Mark Chubb: The Role of Practice in Mathematics Class

Dan Meyer: What Does Fluency Without Understanding Look Like?

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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!

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??

IMG_2118

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!)

 

IMG_2133.JPG

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?

IMG_2147

One.

Little.

Picture.

Open That Can, Already!

I had coffee today with a person I have never met before.  (Why is uninteresting.)  When she found out I had been a teacher, she said she felt school should not be about telling students what to think, but rather about teaching them how to think.

Why is this type of comment so refreshing to hear?  Is this view that uncommon?  Who, outside of education, shares it?  Who, inside, does not?

If I were in charge of PD at a school AND wanted to develop an empowering and effective learning culture for both students and staff, I would lay it on the line and ask:

What is the purpose of school?

Then invest time answering it together, even if it takes all year.  I bet that can of worms would reveal quite a bit about one’s staff and what they need to move their pedagogy forward.