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

**I can reason abstractly and quantitatively.**

**(**CCSS.MATH.PRACTICE.MP2**)**

But…What if I think I can’t? What if I have no idea how to contextualize and decontextualize a situation? How might we offer a pathway for success?

We have studied this practice for a while, making sense of what it means for students to contextualize and decontextualize when solving a problem.

Students reason abstractly and quantitatively when solving problems with area and volume. Calculus students reason abstractly and quantitatively when solving related rates problems. In what other types of problem do the units help you not only reason about the given quantities but make sense of the computations involved?

What about these problems from The Official SAT Study Guide, The College Board and Educational Testing Service, 2009. How would your students solve them? How would you help students who are struggling with the problems solve them?

*There are g gallons of paint available to paint a house. After n gallons have been used, then, in terms of g and n, what percent of the pain has **not** been used?
*

*A salesperson’s commission is k percent of the selling price of a car. Which of the following represents the commission, in dollar, on 2 cars that sold for $14,000 each?
*

In our previous post, SMP-2 Reason Abstractly and Quantitatively #LL2LU (Take 1), we offered a pathway to* I can reason abstractly and quantitatively. *What if we offer a second pathway for reasoning abstractly and quantitatively?

Level 4:

I can create multiple coherent representations of a task by detailing solution pathways, and I can show connections between representations.

**Level 3:
**

*Beginning***:
**

*Middle***:
**

*End***:
**

Level 2:

I can periodically stop and check to see if numbers, variables, and units make sense while I am working mathematically to solve a task.

Level 1:

I can decontextualize a task to represent it symbolically as an expression, equation, table, or graph, and I can make sense of quantities and their relationships in problem situations.

What evidence of contextualizing and decontextualizing do you see in the work below?

[Cross-posted on Easing the Hurry Syndrome]

Filed under: #LL2LU, Algebra, Assessment, Connecting Ideas, Learning Progressions, Questions Tagged: Illustrative Mathematics, SMP-2, Standards for Mathematical Practice ]]>

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

**I can reason abstractly and quantitatively.**

**(**CCSS.MATH.PRACTICE.MP2**)**

I wonder what happens along the learning journey and in schooling. Very young learners of mathematics can answer verbal story problems with ease and struggle to translate these stories into symbols. They use images and pictures to demonstrate understanding, and they answer the questions in complete sentences.

*If I have 4 toy cars and you have 5 toy cars, how many cars do we have together?*

*If I have 17 quarters and give you 10 of them, how many quarters will I have left?*

Somewhere, word problems become difficult, stressful, and challenging, but should they? Are we so concerned with the mechanics and the symbols that we’ve lost meaning and purpose? What if every unit/week/day started with a problem or story – math in context? If learners need a mini-lesson on a skill, could we offer it when they have a need-to-know?

Suppose we work on a couple of Standards of Mathematical Practice at the same time. What if we offer our learners a task, Running Laps (4.NF) or Laptop Battery Charge 2 (S-ID, F-IF) from Illustrative Math, before teaching fractions or linear functions, respectively? What if we make two learning progressions visible? What if we work on making sense of problems and persevering in solving them as we work on reasoning abstractly and quantitatively. (Hat tip to Kato Nims (@katonims129) for this idea and its implementation for Running Laps.)

Level 4:

I can connect abstract and quantitative reasoning using graphs, tables, and equations, and I can explain their connectedness within the context of the task.

**Level 3:
**

Level 2:

I can represent the problem situation mathematically, and I can attend to the meaning, including units, of the quantities, in addition to how to compute them.

Level 1:

I can define variables and constants in a problem situation and connect the appropriate units to each.

You could see how we might need to focus on making sense of the problem and persevering in solving it. Do we have faith in our learners to persevere? We know they are learning to reason abstractly and quantitatively. Are we willing to use learning progressions as formative assessment early and see if, when, where, and why our learners struggle?

Daily we are awed by the questions our learners pose when they have a learning progression to offer guidance through a learning pathway. How might we *level up* ourselves? What if we ask first?

Send the message: you can do it; we can help.

[Cross-posted on Easing the Hurry Syndrome]

Filed under: #LL2LU, Algebra, Assessment, Connecting Ideas, Learning Progressions, Questions Tagged: Illustrative Mathematics, SMP-2, Standards for Mathematical Practice ]]>

- Read Reflecting on My Learning 1.0.
- Watch The Future of Publishing (shown below). How might we reframe or reverse the way we are seeing, learning, thinking, and acting?

- Reflect, write, and post. Read and comment on posts from at least two others inourMyLearning 1.5 cadre. You might consider using the following protocol for your comments:
- I like…
- I wish…
- I wonder…
- I want to know more about…

**BONUS:** If you have written and published for other websites or magazines, cross post your work on your blog as artifacts of your writing and contributions to the learning of others. (Examples: *Falconry: I believe in you* is posted on Experiments in Learning by Doing and on Flourish.

- MyLearningEdu 1.5 (week 3) – Learning Together
- MyLearningEdu 1.5 (week 1) – Learning Together
- MyLearningEdu 1.5 (week 2) – Learning Together

Filed under: Professional Development Plans, Questions, Social Media Tagged: #MyLearning, Dorling Kindersley Books, reflection ]]>

(sketch by @katonims129)

How might we experiment and learn together about creativity, communication, critical reasoning, and collaboration? What if we risk, practice, and share to make our thinking visible? How will we grow and learn if we practice and accept feedback?

As you can see from the email above, Kato Nims and I have been experimenting with sketch noting or doodling to take visual notes since the beginning of the school year.

Twenty-four of our colleagues responded that they would like to participate on Thursday with several more asking for another session next week because of carpool duty. The little experiment turned into a bigger experiment.

Eighteen of us gathered in the Art room at 7:20 this morning and another six met this afternoon. We watched Kiran bir Sethi teaches kids to take charge and sketched.

We shared our sketches and ideas in small groups and debriefed the experience. We will try again next week. I wonder who might take action on this experiment in other venues to learn with others.

Click to view slideshow.Resources shared in our session:

- What we learn from doodles – CNN
- The Doodle Revolution – Sunni Brown
- Mapping Inner Space – Nancy Margulies and Nusa Maal
- Visual Thinking – Nancy Margulies and Christine Valenza
- The Sketchnote Handbook: the illustrated guide to visual note taking – Matt Rohde
- Visual Meetings – David Sibbet

Filed under: 21st Century Learning, Connecting Ideas, Creativity, Professional Development Plans, Questions, TED talk Tagged: doodle revolution, Kiran Bir Sethi, sketchnoting, Sunni Brown ]]>

- Read
*Categories vs Tags*and Silvia Tolisano’s (@langwitches)*Anatomy, Grammar, Syntax & Taxonomy of a Hyperlink*. Return to your previous post(s) and add both categories and tags and improve any hyperlinks if you have not already done so. - Watch Matt Cutts: Try something new for 30 days (shown below). What habits should we practice? What habits are we modeling and teaching? What habits do our student-learners want to acquire? How can we make reflection part of the habit of schooling?

- Reflect, write, and post. Read and comment on posts from at least two others in ourMyLearning 1.5 cadre. You might consider using the following protocol for your comments:
- I like…
- I wish…
- I wonder…
- I want to know more about…

This course is designed to build teacher experience, confidence, and understanding of reflection, digital portfolios, and feedback. Strategies employed in this course will be hands-on and digital development practices for reflection, self-assessment, learning, feedback, and growth.

At the end of this course, participants should be able to say:

- I can use reflection as a formative assessment and self-assessment tool.

- I can develop and utilize journaling and e-portfolios.
- I can use authentic peer-to-peer and self-assessment practices to inform professional growth and learning.
- I can design processes that can be used in a classroom to promote and celebrate self-reflection for learning.

- I can integrate technologies that enhance self-reflection and asynchronous communication.
- I can facilitate authentic peer-to-peer and self-assessment practices to motivate growth and learning.
This class will meet asynchronously throughout the semester from August 1 through January 1. Participants will document their learning on their professional blog. Participants will collaborate, learn, and share by commenting on the blogs of others participating in this course.To earn 2 PLU credits (Georgia Department of Education), participants will

- establish a professional portfolio to document the journey of becoming a more reflective teacher.
- demonstrate fulfillment of required activities by posting completed work and reflections to individual blogs.
- model connectedness by reading and commenting on the reflections of others in this course.
- practice offering warm and cool feedback in constructive, kind, and purposeful ways using suggested protocols.

Filed under: Professional Development Plans, Questions, Social Media Tagged: #MyLearning, matt cutts, reflection ]]>

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

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?

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

Are you practicing *use appropriate tools strategically* if you use a computer algebra system to explain your steps?** **

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

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*=*f *(*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]

Filed under: #LL2LU, Algebra, Connecting Ideas, Learning, Learning Progressions, Questions Tagged: SMP-5, use appropriate tools strategically ]]>

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

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

But…What if I think I can’t? What if I have no idea what are appropriate tools in the context of what we are learning, much less how to use them strategically? How might we offer a pathway for success?

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.

We still might need some conversation about what it means to use **appropriate** tools **strategically**. Is it not enough to use **appropriate** tools? Would it help to find a common definition of **strategically** to use as we learn? And, is *use appropriate tools strategically* a personal choice or a predefined one?

How might we expand our toolkit and experiment with enough tools to display confidence when explaining why the selected tools are appropriate and effective for the solution pathway used? What if we practice with enough tools that we make strategic – highly important and essential to the solution pathway – choices?

What if apply we 5 Practices for Orchestrating Productive Mathematics Discussions to learn with and from the learners in our community?

**Anticipate**what learners will do and why strategies chosen will be useful in solving a task**Monitor**work and discuss a variety of approaches to the task**Select**students to highlight effective strategies and describe a why behind the choice**Sequence**presentations to maximize potential to increase learning**Connect**strategies and ideas in a way that helps improve understanding

What if we extend the idea of interacting with numbers flexibly to interacting with appropriate tools flexibly?* *How many ways and with how many tools can we learn and visualize the following essential learning?

I can understand solving equations as a process of reasoning and explain the reasoning. CCSS.MATH.CONTENT.HSA.REI.A.1

What tools might be used to learn and master the above standard?

- How might learners use
**algebra tiles**strategically? - When might
**paper and pencil**be a good or best choice? - What if a learner used
**graphing**as the tool? - What might we learn from using a
**table**? - When is a
**computer algebra system**(CAS) the go-to strategic choice?

Then, what are the conditions which make the use of each one of these tools appropriate and strategic?

[Cross posted on Easing the Hurry Syndrome]

________________________

“The American Heritage Dictionary Entry: Strategically.” *American Heritage Dictionary Entry: Strategically*. N.p., n.d. Web. 08 Sept. 2014.

Filed under: #LL2LU, Algebra, Assessment, Connecting Ideas, Learning Progressions, Questions Tagged: 5 Practices for Orchestrating Productive Mathematics Discussions, American Heritage Dictionary, Mary Kay Stein, Peg Smith, SMP-5, Standards for Mathematical Practice, use appropriate tools strategically ]]>

- Read What Do You Know About Creative Commons by Bill Ferriter and/or How to Attribute Creative Commons images in your blog properly?.
- Peak Learning Exercise – “Think about your own life and the times when you were really learning, so much and so deeply, that you would call these the “peak learning experiences” of your life. Write a rough draft: Tell a story (you may include pictures, symbols, or other icons, too) about this peak learning experience, and respond to the question, “What were the conditions that made your high-level experience so powerful and engaging?”(adapted from 21st Century Skills: Learning for Life in Our Times, Trilling and Fadel, 2009; used as choice in Writing-Is-Thinking CFT Pre-Institute Assignment with Bo Adams in 2012.)
- Post one of your Peak Learning Experiences on your blog.

At the end of this course, participants should be able to say:

- I can use reflection as a formative assessment and self-assessment tool.

- I can develop and utilize journaling and e-portfolios.
- I can use authentic peer-to-peer and self-assessment practices to inform professional growth and learning.
- I can design processes that can be used in a classroom to promote and celebrate self-reflection for learning.

- I can integrate technologies that enhance self-reflection and asynchronous communication.
- I can facilitate authentic peer-to-peer and self-assessment practices to motivate growth and learning.

Filed under: Professional Development Plans, Questions, Social Media Tagged: #MyLearning, Derek Sivers, reflection ]]>

**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?

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.

Filed under: #LL2LU, Assessment, Connecting Ideas, Learning, Learning Progressions, Questions Tagged: #LL2LU, formative assessment, Grant, learning, learning progressions, make viable arguments and critique reasoning of others, questions, reflection, SMP-3, Standards for Mathematical Practice, The Falconer ]]>

**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.*”

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*.”

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.

Filed under: #LL2LU, Assessment, Connecting Ideas, Learning, Learning Progressions, Questions Tagged: #LL2LU, construct viable arguments and critique reasoning of others, Happiness Advantage, learning, learning progressions, questions, reflection, Shawn, SMP-3, Standards for Mathematical Practice ]]>