Tag Archives: look for and make use of structure

Simple, Yet Not: Growing Patterns – Growing Mathematicians

What looks simple on the surface can be deceptively complex and elegant.

How might we teach our young learners to deepen their algebraic reasoning?

Let’s see what you think…

Unit 8: Cartesian Coordinate Plane, Two-Variable Equations, Graphing, and Regularity in Repeated Reasoning

  • graph on the Cartesian coordinate plane,
  • look for and make use of structure,
  • look for and express regularity in repeated reasoning,
  • use and connect mathematical representations?

Kristi Story, Trinity’s 6th Grade math teacher, set the above goals for student learning and selected what looks like a simple, yet is actually a deep task that aligns with these goals.  Providing opportunities for students to learn important mathematics content and to engage in essential mathematical practices are at the forefront of this planning.

Tasks that provide the richest basis for productive discussions have been referred to as doing-mathematics tasks. Such tasks are nonalgorithmic—no solution path is suggested or implied by the task and students cannot solve them by the simple application of a known rule. (Smith, 16 pag.)

Day 1’s Task is modified from Fawn Nguyen‘s Visual Patterns, pattern #10 puppies. Starting as a simple number talk, how many do you see and how do you see them?

Fawn Nguyen‘s Visual Patterns, pattern #10 puppies
Task modified from Fawn Nguyen‘s Visual Patterns, pattern #10 puppies

Obviously, 6th graders know that there are three puppies, but how do they see the three? Do they see two puppies in the top row and one puppy in the bottom row? Do they see two puppies in the first column and one puppy in the second column? Either way, they would write 2+1=3. To make their thinking visible, they circle the two and the one. Also, they might see a 2×2 square with one puppy missing and write 2×2-1=3.  It is a quick check about attending to precision, making use of structure (making visible what isn’t readily seen), and writing an equation.

Continuing the number talk, how many do you see and how do you see them?

Fawn Nguyen‘s Visual Patterns, pattern #10 puppies
Task modified from Fawn Nguyen‘s Visual Patterns, pattern #10 puppies

6th graders immediately know that there are five puppies, but how do they see the five? Do they see three puppies in the top row and two puppies in the bottom row and write 3+2=5? Do they see two puppies in the first two columns and one puppy in the third column and write 2+2+1=5 or 2×2+1=5? Do they see a 2×3 rectangle with one puppy missing and write 2×3-1=5?  Additional practice attending to precision, making use of structure (making visible what isn’t readily seen), and writing an equation.

Fawn Nguyen‘s Visual Patterns, pattern #10 puppies
Task modified from Fawn Nguyen‘s Visual Patterns, pattern #10 puppies

6th graders immediately know that there are seven puppies, but how do they see the seven? Do they see four puppies in the top row and three puppies in the bottom row and write 4+3=7? Do they see two puppies in the first three columns and one puppy in the fourth column and write 2+2+2+1=7 or 2×3+1=7? They might also see a 2×4 rectangle with one puppy missing and write 2×4-1=7.

The important reflection question is: Did I use the same structure for each of the figures, or did I make use of different structures with each figure?

Fawn Nguyen‘s Visual Patterns, pattern #10 puppies
Task modified from Fawn Nguyen‘s Visual Patterns, pattern #10 puppies

Using previously discovered structures, students predicted the number of puppies in Figure 4 and in Figure 10. Connecting to the algebra in their previous unit, they wrote a generalization for any figure number using their structure and reasoning. We found the following different expressions.

(n+1)+n. where n is the figure number
2(n+1)-1, where n is the figure number
1+2n, where n is the figure number

“These all represent the same pattern. Are they equivalent expressions?” asked Kristi. Using the distributive property, and combining like terms, they proved equivalence.

Committed to deep understanding for our young learners, Kristi asked students to graph (Figure Number, Number of Puppies) on the coordinate plane.

Trained to notice and note, our students were surprised to discover a linear pattern.

JH said, “Hey, to go from one point to the next, all you have to do is go up 2 and over 1.”
When asked, CJ interpreted the point (6, 13) saying “that means that there will be 13 puppies in Figure 6.”

My #ObserveMe notes illustrate more of the details and flexibility.

Our students graphed points and a line on the Cartesian coordinate plane, made use of structure, expressed regularity in repeated reasoning, used and connected mathematical representations, and deepened algebraic reasoning.

That’s a lot of Algebra I for a 6th grader, don’t you think?

Deep learning. Empowered learners.

Never underestimate the power of a motivated learner.


Smith, Margaret (Peg) S.. The Five Practices in Practice [Middle School] (Corwin Mathematics Series). SAGE Publications. Kindle Edition.

 

#SlowMath: looking for structure and noticing regularity in repeated reasoning #T3IC

At the 2018 International T³ Conference in San Antonio, Jennifer Wilson (@jwilson828) and I presented the following 90 minute
session.

#SlowMath: looking for structure
and noticing regularity in repeated reasoning

How do we provide opportunities for students to learn to use structure and repeated reasoning? What expressions, equations and diagrams require making what isn’t pictured visible? Let’s engage in tasks where making use of structure and repeated reasoning can provide an advantage and think about how to provide that same opportunity for students.

Here’s my sketch note of our plan:

Dave Johnston (@Johnston_MSMath) recorded his thinking and learning and shared it with us via Twitter.

Cross posted on The Slow Math Movement

Learner choice: using appropriate tools strategically takes time and tools

All students benefit from using tools and learning how to use them for a variety of purposes.  If we don’t make tools readily available and value their use, our students miss out on major learning opportunities. (Flynn, 106 pag.)

I’m taking the #MtHolyokeMath #MTBoS course, Effective Practices for Advancing the Teaching and Learning of Mathematics.  Zachary Champagne facilitated the second session and used The Cycling Shop task from Mike Flynn‘s TMC article.

screen-shot-2017-02-03-at-2-50-42-pm

You can see the notes I started on paper.

mtholyokemath-2-zakchamp

Jim, Casey and I used a pre-made Google slide deck provided to us to collaborate since we were located in GA, MA, and CA.  We challenged ourselves to consider wheels after working with 8 wheels.

Here’s what our first table looked like.

cyclingshop1

Now, I was having trouble keeping up with the number of wheels and the number of cycles.  So I did this:

screen-shot-2017-02-03-at-3-08-56-pm

This made it both better and worse for me (and for my group).

Here’s an interesting thing.  I’ve been studying, practicing, and teaching the Standards for Mathematical Practices. Jennifer Wilson and I have written a learning progression to help learners learn to say I can use appropriate tools strategically.

Mathematically proficient students consider the available tools when solving a mathematical problem. (Sage, 6 pag.)

Clearly, I was not even at Level 1 during class.  Not once – not once – during class did it occur to me how much a spreadsheet would help me, strategically.

8wheelsspreadsheet

The spreadsheet would calculate the number of wheels automatically for each row so that I could confirm correct combinations.  (You can view this spreadsheet and make a copy to play with if you are interested.)

When making mathematical models, [mathematically proficient students] know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. (Sage, 6 pag.)

With a quick copy and paste, I could tackle any number of wheels using my spreadsheet.  I can look for and make use of structure emerged quickly when using the spreadsheet strategically.  (I want to also highlight color as a strategic tool.) Play with it; you’ll see.

9_wheelsspreadsheet

[Mathematically proficient students] are able to use technological tools to explore and deepen their understanding of concepts. (Sage, 6 pag.)

screen-shot-2017-02-03-at-4-03-03-pm

There is no possible way I would have the stamina to seek all the combinations for 25 or 35 wheels by hand, right?

Students have access to a wide assortment of tools that they must learn to use for their mathematical work. The sheer volume of possibilities can seem overwhelming, but with time and experience, students can learn how to choose the right tool for the task at hand and how to use it strategically to reach their goal. (Flynn, 106 pag.)

Important to repeat, “with time and experience, students can learn how to choose the right tool for the task at hand and how to use it strategically to reach their goal.

For this to happen, we need to have a solid understanding of the kinds of tools available, the purpose of each tool, and how students can learn to use them flexibly and strategically in any given situation. This also means that we have to make these tools readily available to students, encourage their use, and provide them with options so they can decide which tool to use and how to use it. If we make all the decisions for them, we remove that critical component of MP5 where students make decisions based on their knowledge and understanding of the tools and the task at hand. (Flynn, 106 pag.)

To be clear, a spreadsheet was available to me during class, but I didn’t see it.  How might we make tools readily available and visible for learners to choose?

When we commit to empower students to deepen their understanding, we make tools available and encourage exploration and use, so that each learner makes decisions for themselves. In other words, how do we help learners to level up in both content and practice?

What if we make I can look for and make use of structure; I can use appropriate tools strategically; and I can make sense of tasks and persevere in solving them essential to learn for every learner?

How might we offer tools and time?

It’s about learning by doing, right?


Flynn, Michael. Beyond Answers: Exploring Mathematical Practices with Young Children. Portland, Maine.: Stenhouse, 2017. Print.

Flynn, Mike. “The Cycling Shop.” Nctm.org. Teaching Children Mathematics, Aug. 2016. Web. 03 Feb. 2017.

Common Core State Standards.” The SAGE Encyclopedia of Contemporary Early Childhood Education (n.d.): n. pag. Web.

A lesson in making use of structure from/with @jmccalla1

Jeff McCalla, Confessions of a Wannabe Super Teacher, published some really good thinking about collaboration vs. competition.  In his post, he describes challenging his learners to investigate the following:

Which of these product rules could be used to quickly expand (x+y+3)(x+y-3)? Now, try expanding the expression.

Product Rules

Jennifer Wilson, Easing the Hurry Syndrome, and I have been tinkering with and drafting #LL2LU learning progressions for the Standards of Mathematical Practice. I have really struggled to get my head wrapped around the meaning of I can look for and make use of structure, SMP-7.  The current draft, to date, looks like this:

What if I tried to apply my understanding of I can look for and make use of structure to Jeff’s challenge?

Scan 1

Note: There is a right parenthesis missing in the figure above.
It should have (x+y)² in the area that represents (x+y)(x+y).

What if we coach our learners to make their thinking visible? What if we use learning progressions for self-assessment, motivation, and connected thinking? I admit that I was quite happy with myself with all that pretty algebra, but then I read the SMP-7 learning progression. Could I integrate geometric and algebraic reasoning to confirm structure? How flexible am I as a mathematical thinker? I lack confidence with geometric representation using algebra tiles, so it is not my go to strategy. However, in the geometric representation, I found what Jeff was seeking for his learners.  I needed to see x+y as a single object.

How might we model making thinking visible in conversation and in writing? How might we encourage productive peer-to-peer discourse around mathematics? How might we facilitate opportunities for in-the-moment self- and peer-assessment that is formative, constructive, and growth-oriented?

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.