Category Archives: Standards for Mathematical Practice

SMP5: 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)

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 Experiments in Learning by Doing]

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

Posted by on September 14, 2014 in Standards for Mathematical Practice

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?

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.

[Cross-posted on Experiments in Learning by Doing]

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

Posted by on September 8, 2014 in Standards for Mathematical Practice

SMP3: 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 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 relationship.

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 on Experiments in Learning by Doing]

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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.

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 x4–y4 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.   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. 3.OA Patterns in the Multiplication Table 4.OA Multiples of 3, 6, and 7 5.OA Comparing Products 6.G Same Base and Height, Variation 1 A-SSE Seeing Structure in Expressions Tasks

Animal Populations

Delivery Trucks

Seeing Dots

Equivalent Expressions

Leveled learning progression posters [Cross posted on Experiments in Learning by Doing]

SMP7: Look For and Make Use of Structure #LL2LU

We 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 Experiments in Learning by Doing]

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.

Visual: SMP-1 Make sense of problems and persevere #LL2LU

What if we display learning progressions in our learning space to show a pathway for learners? After Jill Gough (Experiments in Learning by Doing) and I published SMP-1: Make sense of problems and persevere #LL2LU, Jill wondered how we might display this learning progression in classrooms. Dabbling with doodling, she drafted this poster for classroom use. Many thanks to Sam Gough for immediate feedback and encouragement during the doodling process.

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 Experiments in Learning by Doing]

SMP1: Make Sense of Problems and Persevere #LL2LU

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

I can make sense of problems and persevere in solving them.  (CCSS.MATH.PRACTICE.MP1)

But…What if I think I can’t? What if I’m stuck? What if I feel lost, confused, or discouraged?

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 find a second or third solution and describe how the pathways to these solutions relate.

Level 3:

I can make sense of problems and persevere in solving them.

Level 2:

I can ask questions to clarify the problem, and I can keep working when things aren’t going well and try again.

Level 1:

I can show at least one attempt to investigate or solve the task.

In Struggle for Smarts? How Eastern and Western Cultures Tackle Learning, Dr. Jim Stigler, UCLA, talks about a study giving first grade American and Japanese students an impossible math problem to solve. The American students worked on average for less than 30 seconds; the Japanese students had to be stopped from working on the problem after an hour when the session was over.

How may we bridge the difference in our cultures to build persistence to solve problems in our students?

NCTM’s recent publication, Principles to Action, in the Mathematics Teaching Practices, calls us to support productive struggle in learning mathematics. How do we encourage our students to keep struggling when they encounter a challenging task? They are accustomed to giving up when they can’t solve a problem immediately and quickly. How do we change the practice of how our students learn mathematics?

[Cross posted on Experiments in Learning by Doing]

The First Day of School

What do you do on the first day of school?

Our teachers did a lot to promote growth mindset, learning math using the math practices, and norms for teamwork.

More of us asked students with which statement(s) they agree, from Carol Dweck’s Mindset. We were pleasantly surprised at how many of our students have a growth mindset towards mathematics (statements 3 and 4).

We will continue to work on promoting mindset and making our classrooms a place where errors are welcome – where making mistakes and correcting those mistakes is evidence that students are learning. Some teachers showed one of Jo’s videos about mindset and mistakes. We are also encouraging students to sign up for Jo Boaler’s course through Stanford, How to Learn Math: For Students.

Several of us asked students what it looks like when your team is working well together.

There is no yelling or fighting.

Everyone contributes; we help each other.

Create a variety of ideas and listen to all ideas. We build off of one another’s ideas.

Constructive Criticism− if someone gets the answer wrong, don’t lower their self-esteem by saying stuff like “”Ha. you got it wrong!””.

Agree to disagree respectfully.

Expand knowledge, learn life skills, and be open minded.

We have stimulating conversations.

We make progress.

Teams also work better wearing matching shirts.

Synergy. The whole is greater than the sum of its parts.

We debuted our Learning Mode poster so that we make students more aware of how they should currently be learning – alone, with a partner, with a team, or participating in a whole class discussion. The best moment in my second class was when I looked up and noticed that a student sitting near the front had changed the learning mode for us based on my verbal instructions. The class had already nominated her to be in charge of the poster!

One of our other teachers used video clips from the Big Bang Theory to help students have a better understanding of the Math Practices. You can find them under “Going Deeper with the Big Bang Theory”.

Many of us gave our students a copy of the Math Practices Poster that we have hanging in our rooms.

Another teacher assigned each team a Math Practice and asked them to make some kind of visual representation of the Math Practice. They hung the posters, and then all of the teams viewed the posters and matched the practice to the poster. Another class “judged” all of the posters for each practice at the end of the day and voted on which one will be hung in the teacher’s classroom. Can you tell which practice is represented by each poster?

All of us started with open-ended, low floor high ceiling tasks so that every student had access to starting the task. One student told her teacher that “geometry is going to be fun”. Another student told his teacher that he was so glad they did something during class besides going over policies.

This year’s journey is off to a good start, and I am thankful for the good company of my math department along the way …

Why I Believe in CCSS

If we take a segment 2 in. long and put it with a segment 3 in. long, we end up with a segment 5 in. long: the part plus the part equals the whole.

But as I reflect on what CCSS have done for my students, my coworkers, and me, I lean less towards Euclid and more towards Aristotle in my beliefs: the whole is greater than the sum of its parts. We are better together than we are alone. Mississippi educators are better with educators from all over the U.S. than we are alone. Mississippi students are better when their teachers are learning alongside other teachers all over the country than they are when we only share within our own school and state.

What has happened in my classroom over the past two years of implementing CCSS regarding student ownership of the mathematics we are learning is more visible than I would have predicted. The positive changes are not only undeniable; they are important, and they have been transformative for my students and for me.

The CCSS for mathematics are anchored by the Standards for Mathematical Practice, which is how we want students to learn math. The first math practice is to make sense of problems and persevere in solving them. How many times do you have to make sense of problems and persevere in solving them in the work that you do? How did you learn to make sense of problems and persevere in solving them?

Most students that I’ve encountered think that math is about a teacher demonstrating how to work a problem and a student mimicking the teacher. They are surprised on the first day of class to encounter problems that not only can be solved in more than one way but for which they are encouraged to solve in more than one way.

Another math practice is to construct a viable argument and critique the reasoning others. Most students that I’ve encountered think that math is about getting an answer. They are surprised when I encourage them to talk about how they are getting an answer – and even more surprised when I ask another student whether they agree or disagree with the process. We learn math by talking about how we are doing math. We even learn math when someone discusses a wrong method, and we make sense of the misconception. We celebrate learning from and correcting mistakes.

Another math practice is to look for regularity in repeated reasoning. If you took a high school geometry course, you learned about 45-45-90 and 30-60-90 special right triangles. In how many of your classes were you given the opportunity to think about the triangles formed by cutting a square in half on its diagonal or cutting an equilateral triangle in half on its height?

Were you given formulas for the relationships between the legs and hypotenuse of the right triangle (as you can see that I used to do from the picture of the transparency from which I used to teach)?

Or were you given an opportunity to make sense of the relationships – and maybe even figure them out yourselves by looking for patterns and talking about what you notice with other students?

Another math practice is to look for and make use of structure. What is the formula for the area of a triangle? Why? Can you calculate the area of a trapezoid? Looking for and making use of structure is about providing students opportunities to make sense of the formulas and equivalent expressions that we have traditionally given them and asked them to memorize without providing them the opportunity to make sense of why.

I won’t spend our time here talking about how the standards are not curriculum. If you don’t know what this means, maybe it would be helpful to hear that CCSS simply tells us that the Pythagorean theorem is a topic for introduction in grade 8 and then students learn more about it in a high school geometry course, but CCSS doesn’t tell us on what day of the year to teach the Pythagorean theorem or how to teach the Pythagorean theorem. As a teacher, I have full control over what standards I teach when and how I teach them. The standards are “the floor, not the ceiling”. They are the minimum that we expect our students to do in order to graduate as college and career ready. Students will still have the opportunity to take calculus and/or dual enrollment college math courses while they are in high school.

The Math Practices have been transformative for my students and for me. And what’s better is that CCSS says that all students will learn math using the math practices. Not just those who have a good teacher. Not just those who go to a good school. CCSS allows me to learn alongside educators from all over the United States who tweet and blog about their efforts to provide opportunities for their students to make sense of mathematics. I use free sites such as Illustrative Mathematics and the Mathematics Assessment Project that make it easy to search for tasks and lessons by standard.

And so the journey to help my students make sense of mathematics using the CCSS and the Standards for Mathematical Practice continues … And you are invited to come see it in action any day. Just be forewarned, my students will expect you to fully participate in the lesson – no one is just a spectator in our math class.

Use Appropriate Tools Strategically – The Student

One of the Standards for Mathematical Practice is use appropriate tools strategically, and one of my calculus students sent me the following reflection about this practice.

KH has reached a point in mathematics where the functions that we use have become part of what she thinks of as available tools to use when solving a problem. She was referring to problems where logarithmic differentiation was helpful. She was solving differential equations, and after integrating it was helpful to make both sides of the equation the power of e. Her comments struck me as something I want to remember as the journey continues …