# 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.

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.

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 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.>

“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.
• 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.

“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.

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.

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.

# thinkering and applying – #MakerEd #LearnAndShare

On February 26, I participated in a workshop with Lindsey OwnVinnie VrotnyJaymes Dec, and Andrew Carle on Maker Education.  It was AWESOME! (You can read a summary of the details of the workshop on Lindsey’s blog post, #MakerEd at #NAISac14!) I applaud their plan, pedagogy, and execution. It was a real workshop with learner choice and learning by doing. Here’s a glimpse of the action:

My favorite of the experiences was the sewing station.  Using a strip of felt, snaps, an led, a battery, and some conductive thread, I created a wearable circuit. Now, I have to confess that I have, in my past, co-taught calculus-based physics to seniors.  While I was the calculus person on the team, I did quite well with circuits. I could read most problems, draw the circuit (in parallel or in series) and answer the question posed by the book.  Sewing my bracelet at NAIS was the first time I ever created, touched, designed a circuit. Amazing and sad at the same time.  How much more would I have understood about physics if I’d had the sewing experience first?

I wanted to have two leds on my bracelet.  In conversation with my 9-year old, she asked if her bracelet could have her name as well light up.  Trying to apply her ideas into my learning, here’s the next iteration in my learning:

I used 18 ct Aida cross stitch fabric and DMC thread to produce my bracelet.  I tried to capture the process in pictures.

I am grateful to  Lindsey OwnVinnie VrotnyJaymes Dec, and Andrew Carle for the experience at NAIS.

How might we connect ideas with our learners? How might we ramp up design and hands-on experiences to make additional opportunities for curiosity, creativity, critical reasoning, communication, collaboration, and control?

# #LL2LU Formative Assessment that Builds Confidence and Skill – #NspiredatT3

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.)

Our final session at T³ International Conference was, of course, my favorite of the sessions we offered.

Here’s the original plan:

.

I started with a personal story about actionable feedback and then gave the quick 4-minute Ignite talk on the foundational ideas supporting the Leading Learners to Level Up  philosophy.

We then went right to work.  Here’s what it looked like:

Responding to questions from participants, I shared the following additional resources:

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.)

________________________

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.

# Design for Learning and Inquiry – #NspiredatT3

Can we – do we – see ourselves as designers of learning experiences?

Today’s session offers  T³ International Conference participants an opportunity to go deep.  Friday’s sessions ran for either 60 or 90 minutes.  While there are 60 and 90 minute sessions at the conference today, participants may also elect to spend four or six hours in learn-by-doing sessions.

Here’s what we submitted for the program:

Design for Learning and Inquiry
Interested in more inquiry from student-learners? In this hands-on session we will focus on designing one-page TI-Nspire documents that promote investigation, learning, and inquiry.  Our goal is to learn by doing.  We want participants to be able to say at the end of this session:  1) I can exercise the ideas of simplicity and restraint when designing TI-Nspire learning investigations; 2) I can storyboard a learning investigation prior to beginning to design to streamline the concept and balance the information to be learned; and 3) I can create TI-Nspire documents to promote learning and inquiry. Bring your laptop with TI-Nspire Teacher Edition and sample learning targets or assessments that you will tinker with.

Here’s how it was printed in the program.

We want our participants to learn to design a one-page TI-Nspire document that promotes student investigation, learning, and inquiry.  Our goal is to discuss – experientially – the essential learnings for the summer workshop. We know we can’t do justice to a 2-day workshop in 2 hours.  We planned to go deep into one activity rather than cover the entire agenda at a rapid pace.

We encourage the idea of Storyboarding prior to launching in to designing with TI-Nspire. We are inspired by Garr Reynolds and Presentation Zen.  In particular we are going to try to avoid creating Nspire documents that are slideuments. For more information, please read “Slideuments” and the catch-22 for conference speakers.

Our hope:  At the end of this workshop, participants should be able to say:

• I can exercise the ideas of restraint and simplicity when designing learning investigations.
• I can identify what is important and remove what is not important.
• I can design where less is more visually – I can include only what is necessary to promote inquiry and investigation.
• I can design documents that are engaging and prompt questions and inquiry from the learner.
• I can storyboard a learning investigation prior to beginning to design to streamline the concept and balance the information to be learned.
• I can explain the goal of the activity and outline the expected learning outcomes.
• I can design a variety of dynamic constructions that are controlled by different inputs including points, sliders, and stored variables.
• I can design documents with a variety of outputs, which use color and strings to support opportunities for  visual connections.
• I can create TI-Nspire documents to promote student investigation and inquiry.
• I can enhance documents with conditional statements to make information appear and disappear as needed to enhance a lesson.
• I can apply TI-Nspire construction tools: geometry tools, scatterplots, data capture, etc. to create the investigation.
• I can use free points, restricted points, sliders, stored variables, etc. to control the actions in the document.
• I can use color, text boxes, strings, etc. as inputs and outputs to connect ideas and promote questions.

We have a plan which is shared below, but we are going to lead our learners by following their questions.

Essential learning: I can explain the effects of a, h, and k in the vertex form of a parabola.

Level 1: I can graph a parabola and use the interactive tools of TI-Nspire to shift and stretch the parent function to investigate graphs of parabolas.

Feedback:

• I like that learners can shift and stretch the graph to see the graph and function change.
• I wonder if the decimals are helpful or distracting.
• What if we created a document where the values of h and k are Integers?

Level 2: I can design a document to stretch and shift a parabola where the values of h and k are Integers.

Feedback:

• I like that the values of h and k are restricted to Integer values. I like that I can control the step of these values by changing the scale of the graph.
• I wonder if learners will connect the values of (h, k) shown in the ordered pair to the equation.
• What if the equation showed the numerical values of h and k rather than the symbols?

Stage 3: I can design a document to stretch and shift a parabola where the values of h and k are Integers and the function dynamically shows the numerical values of h and k as the function changes.

Here’s what we planned to do:

Here’s what our participants prompted us to actually do because of the question How might we use color?:

Stage 4: I can design a document to stretch and shift a parabola where the function dynamically shows the numerical values of a, h, and k as the function changes.

##### Final Nspire document shown above

We’ve left the last hour for application and coaching.  Participants are invited to create their own document. We are available for trouble-shooting and brainstorming.

Here’s the next challenge for curious learners: