Category Archives: Ask Don’t Tell

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?

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

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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 wodb.ca? 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.

Risk.

Experiment.

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.

AND

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.

2+1:5

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?

 

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 Jennifer Wilson (Easing the Hurry Syndrome) and I published SMP-1: Make sense of problems and persevere #LL2LU, I wondered how we might display this learning progression in classrooms. Dabbling with doodling, I drafted this visual for classroom use. Many thanks to Sam Gough for immediate feedback and encouragement during the doodling process.

Screen Shot 2014-08-16 at 1.21.17 PMI 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]

#LL2LU Fractions – we are smarter than me & modeling C’s – #MPVschool & #TrinityLearns

A new definition of strength: Can we learn together? What if we collaborate, ask for feedback, and lean in to leverage expertise and perspective of others?

If we truly believe in communication, collaboration, and the other C’s, how are we – as lead learners – modeling and taking action?

<Note the timestamps in the following communication, collaboration, critical thinking, and problem-solving.>

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“Hear” snippets of Nicole’s thoughts as she is developing the assessment shown above:

    •  I’m  writing a mathematics unit for a grade level that I have never taught to learn, to  help my team, to help our young learners.
    • This is hard.
    • I’m trying to model backwards design unit planning (Grant Wiggins hung the moon, most recently evidenced by his math blog post today). Stage 2 (How will I know when they have learned it?) must come before Stage 3 (the learning plan). Teachers should have access to the assessments (formative and summative) at the beginning of the unit.
    • Our learning outcomes are all I have to work with.  Reading these standards in depth helps me some, but I need feedback.
    • I heart Google.
    • The “I can…” statements need to be student-friendly. They will be directly related to the standards-based rubric we will need to create.
    • I’ve worked through several leveled assessments as collaborations with classroom teachers, but I have yet to write one independently.
    • Wait, why am I writing this independently? It’s nearly midnight. I’m sending this to Jill.

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“Hear” snippets of Jill’s thoughts as she gave feedback and edited the assessment shown above:

      • Wow…Such good work.
      • Level 1 “I can decompose a figure into equal parts. I can name each part.”  
        • I wonder if decompose is a 3rd grade word. (I do not know.)  I also wonder about “partition” as a 3rd grade word.
        • I wonder if you are having a resolution problem with the shapes in Level 1. The image shown is a rectangle, not a square.
        • I wonder how successful a child can be partitioning the circle without having the center marked and using a compass.
      • Level 2 “I can represent a fraction on the number line when some fractions are given to me.“  
        • Can we eliminate the word “some” and/or simplify?
        • What if we say I can represent fractions on a number line?
        • What if we add number lines to identify fractions before asking students to take action on number lines? Just this month, Jennifer Wilson and I presented on conceptual understanding of fractions and the new way to convey a consistent story using number lines. 
        • My TI-Nspire software and the fraction lessons will give me number lines. I’m not sure about mixed numbers and partitions past 1, but Nicole will know.  At least adding a visual might help.

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Nicole thinking:

How on earth did Jill create this fancy number line in a Google doc? I like her train of thought here but think the visual at it stands now will be too hard for grade 3 students.

Jill’s thinking:

Right. Number lines too hard. Would it be easier if we think together now that we are both awake?

Below is a copy of the next iteration of this assessment after a Google hangout discussion and co-learning conversation.

How might we collaborate, ask for feedback, and lean in to leverage expertise and perspective of others?

A new definition of strength: We are stronger than me. Learn and share!


[Cross posted on Curriculum Reflections] 

#NCSM14 Art of Questioning: Leading Learners to Level Up #LL2LU

What if we empower and embolden our learners to ask the questions they need to ask by improving the way we communicate and assess?

Great teachers lead us just far enough down a path so we can challenge for ourselves. They provide us just enough insight so we can work toward a solution that makes us, makes me want to jump up and shout out the solution to the world, makes me want to step to the next higher level.  Great teachers somehow make us want to ask the questions that they want us to answer, overcome the challenge that they, because they are our teacher, believe we need to overcome. (Lichtman, 20 pag.)

On Monday, April 7, 2014, Jennifer Wilson (@jwilson828) and Jill Gough (@jgough) presented at the National Council of Supervisors of Mathematics Conference in New Orleans.

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Jill started with a personal story (you’re letting her shoot…) about actionable feedback and then gave the quick 4-minute Ignite talk on the foundational ideas supporting the Leading Learners to Level Up  philosophy.

Our hope was that many of our 130 participants would help us ideate to craft leveled learning progressions for implementing the Common Core State Standards Mathematical Practices.  Jennifer prompted participants to consider how we might building understanding and confidence with I can make sense of problems and persevere in solving them. After giving time for each participant to think, she prompted them to collaborate to describe how to coach learners to reach this target.  Jennifer shared our idea of how we might help learners grow in this practice.

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.

 Participants then went right to work writing an essential learning – Level 3 – I can… statement and the learning progression around this essential learning. Artifacts of this work are captured on the #LL2LU Flickr page.

Here are the additional resources we shared:

How might we coach our learners into asking more questions? Not just any question – targeted questions.  What if we coach and develop the skill of questioning self-talk?

Interrogative self-talk, the researchers say, “may inspire thoughts about autonomous or intrinsically motivated reasons to purse a goal.”  As ample research has demonstrated, people are more likely to act, and to perform well, when the motivations come from intrinsic choices rather than from extrinsic pressures.  Declarative self-talk risks bypassing one’s motivations.  Questioning self-talk elicits the reasons for doing something and reminds people that many of those reasons come from within. (Pink, 103 pag.)

[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.

Pink, Daniel H. To Sell Is Human: The Surprising Truth about Moving Others. New York: Riverhead, 2012. Print.