Category Archives: Ask Don’t Tell

#SlowMath: look for meaning before the procedure

In her #CMCS15 session, Jennifer Wilson (@jwilson828) asks:

How might we leverage technology to build procedural fluency from conceptual understanding?  What if we encourage sketching to show connections?

What if we explore right triangle trigonometry and  equations of circles through the lens of the Slow Math Movement?  Will we learn more deeply, identify patterns, and make connections?

How might we promote and facilitate deep practice?

This is not ordinary practice. This is something else: a highly targeted, error-focused process. Something is growing, being built. (Coyle, 4 pag.)

What if we S…L…O…W… down?

How might we leverage technology to take deliberate, individualized dynamic actions? What will we notice and observe? Can we Will we What happens when we will take time to note what we are noticing and track our thinking?


What is lost by the time we save being efficient, by telling? How might we ask rather than tell?

#SlowMath Movement = #DeepPractice + #AskDontTell

What if we offer more opportunities to deepen understanding by investigation, inquiry, and deep practice?

Coyle, Daniel (2009-04-16). The Talent Code: Greatness Isn’t Born. It’s Grown. Here’s How. Random House, Inc.. Kindle Edition.

HMW walk the walk: 1st draft doesn’t equal final draft

In her #CMCS15  session, Jessica Balli (@JessicaMurk13) challenges us to consider how we might redefine mathematical proficiency for teachers and students. Are our actions reflecting a current definition or are we holding on to the past?

How might we engage with the Standards for Mathematical Practice to help all redefine what it means to be ‘good at math’?

Do we value process and product? Are we offering opportunities to our learners that cause them to struggle, to grapple with big ideas, to make sense and persevere?


Do we value our learners’ previous knowledge or do we mistakenly assume that they are blank slates? What if we offer our learners opportunity to show what they know first?  How might we use examples and non-examples to notice and note and then revise?

What if we take up the challenge to walk the walk to prove to our learners (and ourselves) that a first draft is not the same as a final draft?

Number Talks: how AND why

Listening informs questioning. (Berger, 98 pag.)

How do we know learning has occurred? How do we know how learning has happened? What if we pause and listen to learn?

If both sense and meaning are present, the likelihood of the new information getting encoded into longterm memory is very high. (Sousa, 28 pag.)

How would you add 39 to 67? Would you use the traditional algorithm? Would you need paper? How might we teach flexibility, sense making, and numeracy to build fluency and confidence?

Number talks are about students making sense of their own mathematical ideas. (Humphrey & Parker, 13 pag.)

How might we seize the opportunity to confer with our learners to see if they are making sense of what is being taught?

This is the challenge – and joy – of teaching by listening to students. (Humphrey & Parker, 13 pag.)

If interested in additional examples of number talks, both the how and the why, listen to Jo Boaler and her students from the Stanford Online MOOC How to Learn Math: For Teachers and Parents.

Do we believe our learners – every one of them – are capable of developing proficiency in mathematics?

How might we show what we know more than one way?

How might we continue to send the message I believe in you and mean it?

What if we listen to learn?

I am grateful to Kristin Gray (@MathMinds) and Crystal Morey (@themathdancer) for their leadership and facilitation as a dozen #TrinityLearns faculty participate in an online book club (#mNTmTch) for Making Number Talks Matter: Developing Mathematical Practices and Deepening Understanding Grades 4-10 along with over 600 educators across the globe.

Berger, Warren (2014-03-04). A More Beautiful Question: The Power of Inquiry to Spark Breakthrough Ideas . BLOOMSBURY PUBLISHING. Kindle Edition.

Humphreys, Cathy, and Ruth E. Parker. Making Number Talks Matter: Developing Mathematical Practices and Deepening Understanding, Grades 4-10. Portland, ME: Steinhouse Publishers, 2015. Print.

Sousa, David A. Brain-Friendly Assessments: What They Are and How to Use Them. West Palm Beach, FL: Learning Sciences, 2014. Print.

In context: review, new ideas, norms, and inquiry

Learning in context.  Answering questions based on our collected data.

How might we review what we already know and build upon it at the same time?  And, how are we teaching our learners about the social norms and the sociomathematical norms in the context of our community?

I love it when co-learning happens.  Kristi Story (@kstorysquared) facilitated another great lesson in statistics with our 6th graders this morning.  Our learners collected data to investigate statistical questions and distribution of data in terms of shape, center, and spread.

Collecting data (love this organization):

  • I usually spend about _____ MINUTES taking a shower or bath.
  • There is a total of _____ LETTERS in my first, middle, and last names.
  • There are _____ PEOPLE living in my home.

Collaboratively analyzing the data:

  • Data sets were collected for each question.
  • Each group was given one set of the collected data to organize and analyze.

Establishing both social and sociomathematical norms in context.

  • What if we collect data to answer statistical questions?
  • What if we grow as a community to continue to embrace a norm of challenging and questioning each other?
  • How might we take messy data and organize it?
  • How will we summarize the data to communicate center, shape, and spread?
  • How might we show what we know in more than one way?
  • What if we organize collected data and discuss the distribution of data in terms of center, shape, and spread?

Learners were not told to answer the above questions.  The questions and the necessary answers came up organically as the learners grappled with the data.

My Learning

I joined the group working on minutes taking a shower.  Here’s what it looked like.

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Here’s my messy attempt to organize and analyze the collected data.

We could compute the landmark data points.  We could quickly represent the data as a dot plot.  What happens when or if we want to represent the data using a box plot? I really didn’t know how to draw a box plot of this data since the median=Q3.

What can we learn by using technology to aid in the visualization process?


What if we leverage technology to show us more than we might see when we graph by hand?

What if we are intentional in our commitment to #AskDontTell inquiry approach to learning? How might we continue to teach the norm of challenging and questioning? What if we learn about and practice both social norms and sociomathematical norms in context as we learn in grow together?

Norms and Mathematical Proficiency.” Teaching Children Mathematics. National Council of Teachers of Mathematics, Aug. 2013. Web. 31 Aug. 2015.

WODB: #MathFlexibility and Construct a Viable Argument #LL2LU

Have you checked out It’s a #mustdo for developing mathematical flexibility and deepening learning.

No one threw rocks at me last week when I launched WODB with our entire teaching faculty, not just the math faculty. Actually, I think it was quite fun.

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How might we lend each other our observations and thinking? What if we improve the way we notice and note our observations? Can we help the young learners in our care to construct viable arguments and critique the reasoning of others? 

What if we try? Can we slowly hack away at the “one right answer” culture in our classrooms?

Suppose you choose which one doesn’t belong and it is different from the one I selected. Is it possible that we can both be correct? Can we construct a viable argument to make our case? In other words, can we say why we see things the way we do? Can we critique the reasoning of someone who sees it differently? Are we able to teach listening to and seeing another’s point of view?

Will listening to another add to our understanding and the flexibility of our thinking?

MathFlexibility #LL2LU
Do we apply what we learn?  If teaching very young learners, run – don’t walk – to check out Christopher Danielson’s A Better Shapes Book.

As a faculty, we played with WODB on Friday. On Monday, it was put into practice with our youngest learners.

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What might we learn if we risk trying new things? How might we learn and grow?

Be brave.



Learn by doing.

Bourassa, MaryWhich One Doesn’t Belong? N.p., n.d. Web.

Danielson, Christopher. Which One Doesn’t Belong: A Shapes Book. A Talking Math with Your Kids Production, 07 Jan. 2015. Web. 30 Apr. 2015.

SMP-8: look for and express regularity in repeated reasoning #LL2LU

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

I can look for and express regularity in repeated reasoning. (CCSS.MATH.PRACTICE.MP8)

But what if I can’t look for and express regularity in repeated reasoning yet? What if I need help? How might we make a pathway for success?

Level 4
I can attend to precision as I construct a viable argument to express regularity in repeated reasoning.

Level 3
I can look for and express regularity in repeated reasoning.

Level 2
I can identify and describe patterns and regularities, and I can begin to develop generalizations.

Level 1
I can notice and note what changes and what stays the same when performing calculations or interacting with geometric figures.

What do you notice? What changes? What stays the same?

Can we use CAS (computer algebra system) to help our students practice look for and express regularity in repeated reasoning?

What do we need to factor for the result to be (x-4)(x+4)?
What do we need to factor for the result to be (x-9)(x+9)?
What will the result be if we factor x²-121?
What will the result be if we factor x²-a2?

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We can also explore over what set of numbers we are factoring using the syntax we have been using. And what happens if we factor x²+1. (And then connect the result to the graph of y=x²+1.)

What happens if we factor over the set of real numbers?

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Or over the set of complex numbers? 

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What about expanding the square of a binomial? 

What changes? What stays the same? What will the result be if we expand (x+5)²?  Or (x+a)²?  Or (x-a)²? 

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What about expanding the cube of a binomial?  Or expanding (x+1)^n, or (x+y)^n?

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What if we are looking at powers of i?

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We can look for and express regularity in repeated reasoning when factoring the sum or difference of cubes. Or simplifying radicals. Or solving equations.

Through reflection and conversation, students make connections and begin to generalize results. What opportunities are you giving your students to look for and express regularity in repeated reasoning? What content are you teaching this week that you can #AskDontTell?

[Cross-posted on Easing the Hurry Syndrome]



What is a Fraction? … be flexible, use appropriate tools strategically

What if we use technology to visualize new concepts and interact with math to investigate and learn? What if we pair a process learning progression with a content learning progression?

By the end of this lesson, we want every learner to be able to say:

I can explain and illustrate that a fraction a/b is the quantity formed by a parts of size 1/b, and I can represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0.


I can apply mathematical flexibility to show what I know using more than one method.

We have completed Jo Boaler’s two courses – How to Learn Math: For Students, and How to Learn Math: For Teachers and Parents.  As a team we are working on our math flexibility with math learners of all ages.  We challenge ourselves to offer more visuals and additional pathways for success. How might we leverage appropriate tools and use them strategically?

Enter: Building Concepts lessons from Texas Instruments.  Kristi Story (@kstorysquared) used What is a Fraction? to review and assess what is already known with our 6th graders.

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To offer a glimpse of the learning experience, a copy of my raw notes from this lesson are below.

Kristi starts with The Number of the Day to chalk talk a number talk.

It is obvious that our students have an understanding of fractions, decimals and percents.  Kristi encourages students to and modeled making connections between different representations of 2 1/5, the number of the day.  Many students answered aloud and enthusiastically moved to the board to draw or write a different representation.  By using the chalk talk method, this number talk encouraged number flexibility and creativity and the number talk offered all learners the opportunity to expand their understanding and fluency.


Kristi launches the TI-Nspire software and the lesson What is a Fraction? and encourages our students to explore and investigate what the software will do and interpret the results.  This led to a side conversation about 1.5/3 and complex fractions.

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Kristi introduces the vocabulary of unit fraction. Interesting discussion and another chance for mathematical flexibility happens when students are asked to describe/illustrate what happens when the value of the denominator increases.  How does the number of equal parts in the interval from 0 to 1 change? What happens to the length of those parts?

Students clearly possess background knowledge of fractions, and Kristi challenges them to become more flexible in representing fractions.  Note: Many students are drawing circles to represent fractions.  In addition, we want them to draw number lines  and rectangles.

The discussion transitions to compare 3/5 to 7/5. Student answers included

3/5 is 3 copies of 1/5.
3/5 is a little more than 1/2
3/5 is 60% of the way between 0 and 1
3/5 is 2/5 back from 1
7/5 is 2/5 more than 1
7/5 is 3/5 less than 2
Both are 2/5 away from 1 but in different directions.

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Kristi and students use Think-Pair-Share to describe how they decided to explain their answer to the question Is 11/8 closer to 1 or 2? Kristi asks everyone improve their answer based on partner feedback. Kristi asks for volunteers to read their partner’s idea.

From me to Kristi:

I thought today was great! I love how you facilitated a discussion encouraging all learners to talk about math. My notes are attached.  Thank you for your willingness to pilot this software with our students.  I was glad to hear that you have enjoyed this start with fractions.

From Kristi:

Thank you for all the feedback. As I said yesterday, it was exciting to present fractions in a way that I think will make a difference in their understanding of fractions. I’m looking forward to continuing this series.

What if we use technology to visualize new concepts and interact with math to investigate and learn?

#LL2LU for What is a Fraction?

Level 4:
I can decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation, and I can justify decompositions by using a visual fraction model.

Level 3:
I can explain and illustrate that a fraction a/b is the quantity formed by a parts of size 1/b, and I can represent a fraction a/b on a number line diagram by marking off a lengths 1/from 0.

Level 2:
I can represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts.

Level 1:
I can explain and illustrate that a fraction 1/b is the quantity formed by 1 part when a whole is partitioned into b equal parts.

I can compare fractions by reasoning about their size.

Level 3:

#LL2LU for Mathematical Flexibility

Level 4:
I can analyze different pathways to success, find connections between pathways and add new strategies to my thinking.

Level 3:
I can apply mathematical flexibility to show what I know using more than one method.

Level 2:
I can show my work to document one successful  method.

Level 1:
I can find and state a correct solution.

What if we pair a process learning progression with a content learning progression?