Category Archives: Learning

Enhancing Growth Mindset in Math – Learning together

We asked:

How might we, as a community of learners, grow in our knowledge and understanding to enhance the growth mindset of each of our young learners?

As a team, we have completed Jo Boaler’s How to Learn Math: For Students and have shared our thinking, understanding, and learning.

Blending online and face-to-face learning, we worked through the Stanford units outside of school so that we could explore and learn more when together.

This slideshow requires JavaScript.

Here are some of the reflections shared by our team.

As a teacher my goal is to help children approach math and all subject areas with a growth mindset. It is of utmost importance that my students truly know that I believe in them and their ability to succeed!

Everyone my age should know that you should never equate being good at math with speed. Just because someone is a slower problem solver does not mean that they are a weak math student. Rather, sometimes the slower math thinkers are the strongest math thinkers because they are thinking about the problem on a deeper level. Being good at math is about being able to think deeply about the problem and making connections with it.

When talking to yourself about your work and learning new things, reminding yourself that you can try harder and improve is critical to potential success.  People are more willing to persevere through difficult tasks (and moments in life) when they engage in positive self talk.  

Mistakes and struggling, in life and in math, are the keys to learning, brain growth, and success.

Thinking slowly and deeply about math and new ideas is good and advantageous to your learning and growth.

Taking the time to think deeply about math problems is much more important than solving problems quickly.  The best mathematicians are the ones who embrace challenges and maintain a determined attitude when they do not arrive at quick and easy solutions.  

Number flexibility is so powerful for [students]. I love discussing how different students can arrive at the same answer but with multiple strategies. 

Working with others, hearing different strategies, and working strategically through problems with a group helps to look at problems in many different ways.

“I am giving you this feedback because I believe in you.”  As teachers, we always try to convey implicitly that we believe in our students, and that they are valued and loved in our class.  However, that explicit message is extraordinary.  It changes the entire perception of corrections or modifications to an essay–from “This is wrong, you need to make it right” to “I want to help you make this the best it can be,” a message we always intended to convey, but may not have been perceived.  

Good math thinkers think deeply and ask questions rather than speeding through for an answer.

Math is a topic that is filled with connections between big ideas.  Numbers are meant to be manipulated, and answers can be obtained through numerous pathways.  People who practice reasoning, discuss ideas with others, have a growth-mindset, and use positive mathematical strategies (as opposed to memorization) are the most successful.

We learn and share.

#ILoveMySchool

VTR: Sentence-Phrase-Word to dig deeper into Standards for Mathematical Practice

From Making Thinking Visible: How to Promote Engagement, Understanding, and Independence for All Learners:

Sentence-Phrase-Word helps leaners to engage with and make meaning from text with a particular focus on capturing the essence of the test or “what speaks to you.” It fosters enhanced discussion while drawing attention to the power of language. (Ritchhart, Church, Morrison, 207 pag.)

Screen Shot 2014-10-19 at 7.26.45 PMWhat if we read and learn together, as a team? How might we develop deeper understanding?

Screen Shot 2014-10-19 at 7.29.46 PMAs a team of learners, we first read Make sense of problems and persevere in solving them independently and highlighted a sentence, phrase, and work that resonated with us.  In round robin fashion, we read aloud our selected sentence so that every member of the team heard what every other member of the team felt was important.  Just the act of hearing another voice read and callout an idea was impactful.

After completing the Sentence-Phrase-Word Visible Thinking Routine for Make sense of problems and persevere in solving them, we asked everyone to take another Standard for Mathematical Practice to read and markup, highlighting a sentence, a phrase, and a word.

Screen Shot 2014-10-19 at 7.36.57 PM Screen Shot 2014-10-19 at 7.37.09 PM

We divided into teams where each of the remaining Standards of Mathematical Practice were represented.  Each learner shared the SMP that they read highlighting a selected sentence, phrase, and word. My notes are shared below. I was amazed at the new ideas I heard from my colleagues when using this routine.

Screen Shot 2014-10-19 at 7.40.51 PM

Seek diversity of thought. Listen to others.  Hear differently. Promote engagement, understanding, and independence for all.

Learn.


Ritchhart, Ron, Mark Church, and Karin Morrison. “Sentence-Phrase-Word.”Making Thinking Visible: How to Promote Engagement, Understanding, and Independence for All Learners. San Francisco, CA: Jossey-Bass, 2011. 207-11. Print.

 

Visual: SMP-5 Use Appropriate Tools Strategically #LL2LU

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

I can use appropriate tools strategically.
(CCSS.MATH.PRACTICE.MP5)

Screen Shot 2014-09-12 at 6.42.42 PM

Level 4:
I can communicate details of how the chosen tools added to the solution pathway strategy using descriptive notes, words, pictures, screen shots, etc.

Level 3:
I can use appropriate tools strategically.

Level 2:
I can use tools to make my thinking visible, and I can experiment with enough tools to display  confidence when explaining how I am using the selected tools appropriately and effectively.

Level 1:
I can recognize when a tool such as a protractor, ruler, tiles, patty paper, spreadsheet, computer algebra system, dynamic geometry software, calculator, graph, table, external resources, etc., will be helpful in making sense of a problem.

Suppose you are solving an equation.

Are you practicing use appropriate tools strategically if you use the numerical solve command on your graphing calculator?

nsolve

Or what about using your calculator to substitute values of x until you find a value that makes a true statement?

Screen Shot 2014-09-14 at 4.07.28 PM Are you practicing use appropriate tools strategically if you use a computer algebra system to explain your steps? 

Screen Shot 2014-09-14 at 4.09.09 PM

Or what if you use the graphing capability of your handheld?

Screen Shot 2014-09-14 at 4.10.14 PM

Consider each of the following learning goals:

I can explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution, and I can construct a viable argument to justify a solution method.  CCSS-M A-REI.A.1.

I can solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. CCSS-M A-REI.B.3.

I can explain why the x-coordinates of the points where the graphs of the equations y=(x) and y=g(x) intersect are the solutions of the equation f(x)=g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. CCSS-M A-REI.D.11.

Does use appropriate tools strategically depend on the learner? Or the learning goal? Or the teacher? Or the availability of tools?

[Cross posted on Easing the Hurry Syndrome]

Visual: SMP-3 Construct Viable Arguments and Critique the Reasoning of Others #LL2LU

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

I can construct viable arguments and critique the reasoning of others. CCSS.MATH.PRACTICE.MP3

But…what if I can’t? What if I’m afraid that I will hurt someone’s feelings or ask a “stupid” question? How might we facilitate learning and grow our culture where critique is sought and embraced?

From Step 1: The Art of Questioning in The Falconer: What We Wish We Had Learned in School.

By learning to insert feedback loops into our thought, questioning, and decision-making process, we increase the chance of staying on our desired path. Or, if the path needs to be modified, our midcourse corrections become less dramatic and disruptive. (Lichtman, 49 pag.)

This paragraph connects to a Mr. Sun quote from Step 0: Preparation.

But there are many more subtle barriers to communication as well, and if we cannot, or do not choose to overcome these barriers, we will encounter life decisions and try to solve problems and do a lot of falconing all by ourselves with little, if any, success. Even in the briefest of communications, people develop and share common models that allow them to communicate effectively.  If you don’t share the model, you can’t communicate. If you can’t communicate, you can’t teach, learn, lead, or follow.  (Lichtman, 32 pag.)

How might we offer a pathway for success? What if we provide practice in the art of questioning and the action of seeking feedback? What if we facilitate safe harbors to share  thinking, reasoning, and perspective?

Screen Shot 2014-09-02 at 8.51.01 PM

Level 4:
I can build on the viable arguments of others and take their critique and feedback to improve my understanding of the solutions to a task.

Level 3:
I can construct viable arguments and critique the reasoning of others.

Level 2:
I can communicate my thinking for why a conjecture must be true to others, and I can listen to and read the work of others and offer actionable, growth-oriented feedback using I like…, I wonder…, and What if… to help clarify or improve the work.

Level 1:
I can recognize given information, definitions, and established results that will contribute to a sound argument for a conjecture.

How might we design opportunities for intentional, focused peer-to-peer discourse? What if we share a common model to improve communication, thinking, and reasoning?

[Cross-posted on Easing the Hurry Syndrome]

________________________

Lichtman, Grant, and Sunzi. The Falconer: What We Wish We Had Learned in School. New York: IUniverse, 2008. Print.

SMP3: Construct Viable Arguments and Critique the Reasoning of Others #LL2LU

Screen Shot 2014-09-01 at 5.14.27 PMWe want every learner in our care to be able to say

I can construct viable arguments and critique the reasoning of others. CCSS.MATH.PRACTICE.MP3

But…what if I can’t? What if I’m afraid that I will hurt someone’s feelings or ask a “stupid” question? How may we create a pathway for students to learn how to construct viable arguments and critique the reasoning of others?

Level 4:
I can build on the viable arguments of others and take their critique and feedback to improve my understanding of the solutions to a task.

Level 3:
I can construct viable arguments and critique the reasoning of others.

Level 2:
I can communicate my thinking for why a conjecture must be true to others, and I can listen to and read the work of others and offer actionable, growth-oriented feedback using I like…, I wonder…, and What if… to help clarify or improve the work.

Level 1:
I can recognize given information, definitions, and established results that will contribute to a sound argument for a conjecture.

Our student reflections on using the Math Practices while they are learning show that they recognize the importance of construct viable arguments and critique the reasoning of others.

Jordan says “If you can really understand something you can teach it. Every person relates to and thinks about problems in a different way, so understanding different ways to get to an answer can help to broaden your knowledge of the subject. Arguments are all about having good, logical facts. If you can be confident enough to argue for your reasoning you have learned the material well.jordan quote

And Franky says that construct viable arguments and critique the reasoning of others is “probably our most used mathematical practice. If someone has a question about a problem, Mrs. Wilson is always looking for a student that understands the problem to explain it. And once he or she is finished, Mrs. Wilson will ask if anyone got the correct answer, but worked it a different way. By seeing multiple ways to work the problem, it is easier for me to fully understand.”

franky quote

What if we intentionally teach feedback and critique through the power of positivity? Starting with I like indicates that there is value in what is observed. Using because adds detail to describe/indicate what is valuable.  I wonder can be used to indicate an area of growth demonstrated or an area of growth that is needed.  Both are positive; taking the time to write what you wonder indicates care, concern, and support.  Wrapping up with What if is invitational and builds relationships.

Move the fulcrum so that all the advantage goes to a negative mindset, and we never rise off the ground. Move the fulcrum to a positive mindset, and the lever’s power is magnified— ready to move everything up. (Achor, 65 pag.)

The Mathy Murk has recently written a blog post called “Where do I Put P?” An Introduction to Peer Feedback, sharing a template for offering students a structure for both providing and receiving feedback.

Could Jessica’s template, coupled with this learning progression, give our students a better idea of what we mean when we say construct viable arguments and critique the reasoning of others?

[Cross-posted at Easing the Hurry Syndrome]

_________________________

Achor, Shawn (2010-09-14). The Happiness Advantage: The Seven Principles of Positive Psychology That Fuel Success and Performance at Work (Kindle Locations 947-948). Crown Publishing Group. Kindle Edition.

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?

Observation of Practice – Learning Together

What if we lend another our perspective?

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?

Screen Shot 2014-08-24 at 6.19.40 PM

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?