Which of these product rules could be used to quickly expand (x+y+3)(x+y-3)? Now, try expanding the expression.
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?
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?
What if we focus on what is happening in classrooms in purposeful and focused ways? What if we model and embrace formative assessment of our practice?
What if we add additional feedback loops in our culture? How and when do adults in our schools receive formative feedback? If I have a question about my practice, how do I and from whom do I seek feedback? If, as a school, we are studying formative assessment, self-assessment, and peer feedback, how are we practicing? Do I blog, journal, or keep a portfolio of my learning? What might I want to learn? Are my students learning?
Reading, Research, and Questions
How might we learn more about our practice? What if we team to discuss questions, concerns, and strengths of our learning environment, classroom culture, and planned learning episodes? What might we learn if we observe each other and discuss what we see, experience, and design?
What if we reflect, self-assess, and coach peers using the following protocol:
As a result of this observation of practice and feedback loop, which aspects of my teaching do I feel are bright spots?
As a result of this observation of practice and feedback loop, what questions do I have about my own teaching?
As a result of this observation of practice and feedback loop, what new ideas do I have?
In other words, will I see myself in my colleagues? Will I recognize effective strategies that we both use? Will I observe strategies that I might like to try? Will I want to know more about the instructional design? Will we ask each other questions where we need support?
As we piloted this 1-PLU course last spring, I enjoyed the observations and writing the reflections. I liked the emphasis on bright spots and questions about my own practice. However, the most powerful part of this learning experience was the debrief after each lesson. I was wowed by the questions, the vulnerability, and the humanity of discussions.
What if we shift the focus of peer observations from observing our peers to observing the products of their work – the actions of students?
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
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.
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]
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.
To pursue bright spots is to ask the question “What’s working, and how can we do more of it?” Sounds simple, doesn’t it? Yet, in the real world, this obvious question is almost never asked. (p. 45, Heath and Heath)
…“buoyancy”— a quality that combines grittiness of spirit and sunniness of outlook. (Pink, 4 pag.)
What if we broadcast bright spots of learning? What if we intentionally observe our community and culture through a lens that some might call rose-colored? How might we collaboratively and creatively tell the story of what is most important? What if we document and share small moments?
As we have seen, even the smallest moments of positivity in the workplace can enhance efficiency, motivation, creativity, and productivity. (Achor, 58 pag.)
At the end of this 1-PLU course, each learner should be able to say:
How might we learn more about our community and each other? What if we continue to develop a culture and a habit of positivity, bright spots, and buoyancy?
Achor, Shawn (2010-09-14). The Happiness Advantage: The Seven Principles of Positive Psychology That Fuel Success and Performance at Work Crown Publishing Group. Kindle Edition.
Heath, Chip, and Dan Heath. Switch: How to Change Things When Change Is Hard. Waterville, Me.: Thorndike, 2011. Print.
Pink, Daniel H. (2012-12-31). To Sell Is Human: The Surprising Truth About Moving Others (p. 4). Penguin Group US. Kindle Edition.
People see mathematics in very different ways. And they can be very creative in solving problems. It is important to keep math creativity alive.
and
When you learn math in school, if a teacher shows you a method, think to yourself, what are the other ways of solving this? There are always others. Discuss them with your teacher or friends or parents. This will help you learn deeply.
I keep thinking about mathematical flexibility. If serious about flexibility, how do we communicate to learners actions that they can take to practice?
How might we narrow what separates high achievers from low achievers? If number flexibility is a gateway to success, what actions are we willing to take to encourage, build confidence, and illuminate multiple pathways to success?
As a team of interested math learners, we will spend 10 hours (1 PLU of credit) learning together using the following outline as our course of study.
In order to share our reflections, we will use a copy of the Enhancing Growth Mindset in Math Google doc to record, expand on, and share the reflections from the Stanford MOOC and our thoughts and connections to the CCSS.
It is my hope that each teacher-learner will share their reflections with everyone in the group or at least one other member.
How vulnerable will we be? What if we share what we know and don’t know and learn together?
…we know that what separates high achievers from low achievers is not that high achievers know more math, it is that they interact with numbers flexibly and low achievers don’t.
I wonder how many times I’ve taught “the one way” to solve a problem without considering other pathways for success. Yikes!
After completing Lesson 4 from How to Learn Math: for Students, A-Sunshine, my 4th grader, asked me to solve another multiplication problem. I wondered how many ways I could show my work and demonstrate flexibility in numeracy. The urge to solve this multiplication problem in the traditional way was strong, but how many ways could I show how to multiply 44 x 18? How flexible am I when it comes to numeracy? Is the traditional method the most efficient? Are there other ways to show 44 x 18 that might demonstrate understanding?
How might offer opportunities to express flexibility? Will learners share thinking and strategies? How will we facilitate discussions where multiple ways to “be right” are discussed? What if we embrace Smith and Stein’s 5 Practices for Orchestrating Productive Mathematics Discussion to anticipate, monitor, select, sequence, and make connections between student responses?
If my solutions represent the work and thinking of five different students, in what order would we sequence student sharing, and are we prepared to help make connections between different student responses?
How is flexibility encouraged and practiced? Is it expected? Is it anticipated?
And…does it stop with numbers? I don’t think so. We want our learners of algebra to be flexible with
The list could go on and on.
How might we narrow what separates high achievers from low achievers? If number flexibility is a gateway to success, what actions are we willing to take to encourage, build confidence, and illuminate multiple pathways to success?
I wonder how each of my teammates will use this with student-learners. I am curious to know student-learner reaction, feedback, and comments. If you have feedback, I would appreciate having it too.
What if we are deliberate in our coaching to encourage learners to self-assess, question, and stretch?
[Cross posted on Easing the Hurry Syndrome]
As a Leadership Team, we designed an agenda for celebration, learning, and teaming.
We use Google docs and spreadsheets to communicate, collaborate, and choose time slots for learning. The tweets shown are linked back to the source if more detail is wanted. There is a quote from each of the nine summer reading books to offer a snippet from the books not chosen by any member of our community.
How do we celebrate our culture? How might we leverage digital tools to communicate, collaborate and offer choice? What if we up the ante on our infusion of the 4 Cs?