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Visual: SMP-7 Look for and Make Use of Structure #LL2LU

How do our learners determine an equivalent expression to 4(x+3)-2(x+3)?  How would they determine the zeros of y=x²-4? How might we provide opportunities for them to successfully look for and make use of structure?

We want every learner in our care to be able to say

I can make look for and make use of structure.  (CCSS.MATH.PRACTICE.MP7)

But…What if I think I can’t? What if I have no idea what “structure” means in the context of what we are learning?

One of the CCSS domains in the Algebra category is Seeing Structure in Expressions. Content-wise, we want learners to

  • “use the structure of an expression to identify ways to rewrite it. For example, see x^4 – y^4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²)”
  • “factor a quadratic expression to reveal the zeros of the function it defines”
  • “complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines”
  • “use the properties of exponents to transform expressions for exponential functions”.

How might we offer a pathway for success? What if we provide cues to guide learners and inspire noticing?

Level 4
I can integrate geometric and algebraic representations to confirm structure and patterning.

Level 3
I can look for and make use of structure.

Level 2
I can rewrite an expression into an equivalent form, draw an auxiliary line, or identify a pattern to make what isn’t pictured visible.

Level 1
I can compose and decompose numbers, expressions, and figures to make sense of the parts and of the whole.

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.pdf of these visuals

Illustrative Mathematics has several tasks to allow students to look for and make use of structure. We look forward to trying these, along with a leveled learning progression, with our students.

A-SSE Seeing Structure in Expressions Tasks

[Cross posted on Easing the Hurry Syndrome]

SMP7: Look For and Make Use of Structure #LL2LU

Screen Shot 2014-08-24 at 4.43.58 PMWe want every learner in our care to be able to say

I can look for and make use of structure.
(CCSS.MATH.PRACTICE.MP7)

But…What if I think I can’t? What if I have no idea what “structure” means in the context of what we are learning?

How might we offer a pathway for success? What if we provide cues to guide learners and inspire interrogative self-talk?

Level 4
I can integrate geometric and algebraic representations to confirm structure and patterning.

Level 3
I can look for and make use of structure.

Level 2
I can rewrite an expression into an equivalent form, draw an auxiliary line to support an argument, or identify a pattern to make what isn’t pictured visible.

Level 1
I can compose and decompose numbers, expressions, and figures to make sense of the parts and of the whole.

Are observing, associating, questioning, and experimenting the foundations of this Standard for Mathematical Practice? It is about seeing things that aren’t readily visible.  It is about remix, composing and decomposing what is visible to understand in different ways.

How might we uncover and investigate patterns and symmetries? What if we teach the art of observation coupled with the art of inquiry?

In The Innovator’s DNA: Mastering the Five Skills of Disruptive Innovators, Dryer, Gregersen, and Christensen describe what stops us from asking questions.

So what stops you from asking questions? The two great inhibitors to questions are: (1) not wanting to look stupid, and (2) not willing to be viewed as uncooperative or disagreeable.  The first problem starts when we’re in elementary school; we don’t want to be seen as stupid by our friends or the teacher, and it is far safer to stay quiet.  So we learn not to ask disruptive questions. Unfortunately, for most of us, this pattern follows us into adulthood.

What if we facilitate art of questioning sessions where all questions are considered? In his post, Fear of Bad Ideas, Seth Godin writes:

But many people are petrified of bad ideas. Ideas that make us look stupid or waste time or money or create some sort of backlash. The problem is that you can’t have good ideas unless you’re willing to generate a lot of bad ones.  Painters, musicians, entrepreneurs, writers, chiropractors, accountants–we all fail far more than we succeed.

How might we create safe harbors for ideas, questions, and observations? What if we encourage generating “bad ideas” so that we might uncover good ones? How might we shift perspectives to observe patterns and structure? What if we embrace the tactics for asking disruptive questions found in The Innovator’s DNA?

Tactic #1: Ask “what is” questions
Tactic #2: Ask “what caused” questions
Tactic #3: Ask “why and why not” questions
Tactic #4: Ask “what if” questions

What are barriers to finding structure? How else will we help learners look for and make use of structure?

[Cross posted on Easing the Hurry Syndrome]


Dyer, Jeff, Hal B. Gregersen, and Clayton M. Christensen. The Innovator’s DNA: Mastering the Five Skills of Disruptive Innovators. Boston, MA: Harvard Business, 2011. Print.

 

The self-discipline to watch, wait, and coach

On our 4th day of cookie baking, AS taught me a couple of really great lessons about learning with my students.  Once again, by popular request, we were making Reese’s peanut butter cup cookies.  We make peanut butter cookie dough, roll it into balls, and cook them in mini muffin pans.  As they come out of the oven, we press mini Reese’s peanut butter cups into the center of the cookies.  Delicious.  My small extended family blazed through 8 dozen in two afternoons.

For the first 4 dozen, I made the batter and rolled the cookies.  Together we pressed the candy into the cookies as they came out of the oven.  No big deal. 

How often do our students watch us do the work to solve the problem or answer the question?   

Baking the second 4 dozen was a very different story.  My mother gave AS her very own measuring spoons, spatula, and mini muffin pan that bakes 1 dozen muffins.  Now she had her own pan; she was in charge.  It would have been so much faster for me to have rolled the cookies.  But, no…her pan; her cookies.  Her mantra: “I can do it myself!” 

 

So, I watched, waited, and coached.  Some of the balls were too small and would have been difficult to press candy into after baking in the oven.  Some were too big and would have blobbed out on the pan during baking.  She fixed most of these problems with a little explaining from me. 

Isn’t this happening sometimes in our classrooms?  It is so much faster and more efficient for the teacher to present the material.  We can get so much more done in the short amount of time we have.  But, how much do the children “get done” or learn?  When efficiency trumps learning, does anyone really have success?  How do we encourage “I can do it myself!”?  How do we find the self-discipline to watch, wait, and coach?

That was the story for the first 2 dozen cookies.  Can you believe that she would alter my recipe?  We cooked our second dozen cookies, and while I was busy pressing the peanut butter cups into my cookies, she decided that Hershey kisses would be just as good or better.  With no prompting (or permission) she created a new (to her) cookie.  Santa left kisses in her stocking and she wanted to use them. 

Does it really matter which method a child uses to solve a problem or answer a question?  Isn’t it okay if they use the lattice method to multiply?  Does it really matter which method is used to find the solution to a system of equations?  Shouldn’t they first find success?  Don’t we want our learners to understand more than one way?  Is our way always the best way? 

Was AS pleased with herself and her creativity?  You bet.  Were her cookies just as good as the original recipe?  Sure!  How can you go wrong combining chocolate and peanut butter?

Do we applaud the process that our learners use to solve a problem or respond to a question?  Do we praise them when they try something different?  Are we promoting and encouraging risk-taking, creativity, and problem-solving?

Can we find the self-discipline to be patient while learning is in progres, to watch, wait, and coach?  Can we promote and embrace the “I can do it myself!” attitude?