# What is a Fraction? – visual lesson

What if content isn’t the essential learning? What if content is just the vehicle to learn process?

Imagine these essential learnings:

I can describe and illustrate my thinking so that a reader understands without having to talking with me.

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

Our 5th graders just started a unit on fractions. What if we use fractions to teach our young learners to show their work and demonstrate flexibility of thought?

Lesson tic-toc

• Introduce the two essential learnings.
• 60 second quick write to recall what is a fraction?
• 60 seconds of pair-share to improve answers to what is a fraction?

• Regularly return to the essentials to learn and the associated learning progressions.  How are we doing? At what level are we right now? What is a next step?
• 60 second quick write to recall what are equivalent fractions?
• 60 seconds of pair-share to improve answers to what are equivalent fractions?
• How might we use technology for learning and investigation?

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How might we put process and product on even ground? What if we emphasize communication and flexibility and use content and skill as vehicles to show what you know and how you can communicate?

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.

Level 4
I can show what I know using words, numbers, and pictures.

Level 3
I can describe and illustrate how I arrived at a solution so that a reader understands without having to talking with me.

Level 2
I can describe or illustrate how I arrived at a solution so that a reader understands without having to talking with me.

Level 1
I can find a correct solution to a task.

# 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:

# Ask; Don’t Tell: Listen to Learn and Assess – #nspiredatT3

At T³, Sam and I also facilitated a 90-minute session titled Ask, Don’t Tell: Listen to Learn and Assess.  Here’s the program description and our simple agenda.

Ask, Don’t Tell: Listen to Learn and Assess Can we merge diagnostic and formative assessment to lead learning? How will TI-Nspire™ CAS Handheld action-consequence documents combined with the TI-Nspire™ Navigator™ System allow us to leverage technology to focus on learning? What if we used the ideas of simplicity and restraint when developing and leading lessons? What can be learned if we question our way through an entire lesson? is it possible to allow students to steer the lesson through their questions? Will listening to student questions help us diagnose, assess and chart a course in real-time? Can we lead learning by following their thinking? Will you come to this session and plan to serve as a student, an observer, and a questioner?

(15 min) Introductions and Ignite talk on Assessment (40 min) Sam facilitates Quadratic_Roots.tns, 3-12-3 protocol for questioning, and QuadInvestForm.tns formative assessment (30 min) Jill facilitates Leveled Assessment discussion

I used the same Ignite slide deck from yesterday’s session since our participants were not the same group of people.  Interesting for me…I did not give the same talk, but I used the same images.

Sam then introduced the Ask; Don’t Tell idea by modeling a lesson on the discriminant using the TI-Nspire Quadratic_Roots.tns file and the 3-12-3 protocol.

###### Want to explore the investigation? Here’s how:  Clicking on the screenshot should enable you to download the TI-Nspire document and open it if you have the TI-Nspire software on your computer.  Clicking on the Launch Player button should open a player file where you can interact with the document without having TI-Nspire software. (Be patient; it is a little slow to launch.)

Using the TI-Nspire document Quadratic_Roots.tns, facilitate a 3-12-3 protocol to generate student questions.

• 3 minutes: Independent investigation of the Quadratic_Roots.tns file.
• 12 minutes: Work with a partner to share questions, convert closed questions to open questions, and generate additional questions. Partners should identify their top 2-3 questions.
• 3 minutes: Use the TI-Nspire Navigator to collect each student’s top question.

Facilitate the class discussion of the lesson by responding to student questions from students as well as the teacher.

Following his “lesson,” Sam check for understanding using the leveled QuadInvestForm.tns formative assessment.

Again, great discussion from our participants.  Sam received good feedback about his assessment.  Participants shared strategies they have used to debrief student responses while using the Navigator.  I thought it was great that Sam opened the discussion up by asking for ideas from the participants.

After experiencing a leveled assessment, I facilitated a discussion about the philosophy and strategies involved in using this type of formative assessment.  The summary of this discussion was captured by Sarah Bauguss (@SBauguss).

I am grateful that Sarah took the time to tweet during the session.  Often I don’t really know what I conveyed. Having this series of tweets offers me another level of feedback.

For other examples of leveled assessments, see the following posts:

# Art of Questioning reflection – #NspiredatT3

My previous post shared the lesson plan for our 4 hour Art of Questioning session at T³. I want to share what actually happened, my reflections and what I learned.

We used Skype so that Grant Lichtman could join us and present with us.  I LOVED doing this.  Grant and I did this the day before in a shorter session for T³ instructors.

I did a better job (I hope) introducing Grant by reading from Step 1 – Art of Questioning of his book The Falconer: What We Wish We Had Learned in School.

Our educational systems have been constructed entirely around the goal of providing the correct answer to a question provided by an instructor or handed out on a standardized exam.  This system provides a form of valid comparison for the results of a group of students, and it provides a foundation of shared information amongst those who have followed a course of study.  Unfortunately, the real world, particularly the real world of the coming century, does not and will not work this way.  Our heroes are not defined by how well they answered canned questions or what they scored on their SATs precisely because these outcomes do not determine success in real-world situations.  The real revolution in education and training, if it comes, will be overtly switching our priority from the skills of giving answers to the skills of finding new questions.

Questions are waypoints on the path of wisdom.  Each question leads to one or more new questions or answers.  Sometimes answers are dead ends; they don’t lead anywhere.  Questions are never dead ends.  Every question has the inherent potential to lead to a new level of discovery, understanding, or creation, levels that can range from the trivial to the sublime. (Lichtman, 35 pag.)

Grant told two powerful stories of leveraging learner questions to facilitate learning. He made the great point that if you teach from student questions, you know someone in the room is interested in what is being discussed.

Then it got seriously interesting for us.  Grant facilitated an experience of questioning techniques while I drove a lesson (shown below) on the TI-Nspire.

##### Want to explore the investigation? Here’s how:  Clicking on the screenshot should enable you to download the TI-Nspire document and open it if you have the TI-Nspire software on your computer.  Clicking on the Launch Player button should open a player file where you can interact with the document without having TI-Nspire software. (Be patient; it is a little slow to launch.)

Grant reviewed the “big 6” types of questions and transitioned to another type of question – “What if”.  Here’s what the exercise looked like when I finished following his directions.  Remember, he could not see the linear investigation, and I could not see him.

Grant then signed off so that we could roll up our sleeve and get to work experiencing learning through the art of questioning.  I opened the next section of this lesson by reading from Step 0: Preparation of The Falconer: What We Wish We Had Learned in School.

The excitement of learning, the compelling personal drive to take one more step on the path towards wisdom, comes when we try to solve a problem we want to solve, when we want to solve, when we see a challenge and say yes, I can meet it.  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.)

Wow! Worth repeating:

Great teachers lead us just far enough down a path so we can challenge for ourselves.

So, how do we do this?  Sam accepted the challenge of modeling this type of facilitation of learning by leading a lesson.  We wanted the participants to experience the investigation, question generation, and learning. Sam chose to use the EllipseInvest.tns file show below.

Sam employed the 3-12-3 protocol:

• 3 minutes: Independent investigation of the EllipseInvest.TNS file.
• 12 minutes: Work with a partner to share questions, convert closed questions to open questions, and generate additional questions. Partners should identify their top 2-3 questions.
• 3 minutes: Use the TI-Nspire Navigator to collect each student’s top question.

Facilitate a class discussion by responding to student questions encouraging responses from students as well as the teacher.

It was awesome!  Sam knew that he was going to administer a formative assessment next.  As his peer observer, I could see his effort and questioning to guide the discussion through the participant questions to the essential outcomes of the lesson. Another point from The Falconer that is worth repeating:

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.

Sam then used EllipseForm.tns to model the leveled formative assessment idea.

Experientially, our participants could make their own determination of the value of this type of formative assessment.  Continuing his questioning technique, Sam prompted the participants to identify why the questions were at the given level.  Could they see a leveling up in the questions?  What did it take to move from one level to another?  The discussion was excellent, and Sam received strong feedback about his assessment design.  Yay!

We were at about the 2-hour mark in our 4 hour workshop.  I asked our participants if they could stand a 4-minute Ignite talk on assessment to set the stage for the next 2 hours of work and then we would take a break.

We resumed after the break by watching Dan Heath: How to Find Bright Spots.

Leveled assessments provide the opportunity to bright spot the work of every learner.  They come in the door saying I can do this, Ms. Gough; will you help me level up? Let’s take the challenge of highlighting what learners can do rather than what they cannot.

For the next hour, Sam and I watched, listened, and coached as participants worked to designed a leveled assessment on a topic of their choosing.  We displayed an example through the projector as a point of reference. Our participants asked for the template of the table of specifications.  All files linked on my previous post are .pdfs.  The table of specifications as a Word doc is shared below.

How great that our participants asked for a usable resource! Learn and share!

I was so pleased with the engagement and the collaboration of our participants.  There were so many good questions.  I challenged our participants to share their ideas – in any form by dropping their files in my Dropbox.  I’ve promised to zip all files shared in my Dropbox by Tuesday and share this on this post.

This morning I noticed this tweet…

I love having this type of feedback!

_______________

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

# The Art of Questioning: Leading Learners to Level Up – #NspiredatT3

Today, we are delivering a ticked, 4-hour session on the Art of Questioning at the Teachers Teaching with Technology International Conference in Philadelphia. What follows in the intended lesson plan for the session.  We are so excited that Grant will be joining us via Skype!

The Art of Questioning: Leading Learners to Level Up

“Questions are the waypoints on the path of wisdom.” ~ Grant Lichtman. This session will focus on the art of questioning and using the TI-Nspire™ Navigator™ System as a formative assessment tool. Work on becoming a falconer…leading your learners to level up through questions rather than lectures. Come prepared to develop formative assessment strategies and documents to share with students to help them calibrate their understanding and decode their struggles. Be prepared to share your assessments with others for feedback and suggestions

(10 min) Introductions – Jill reads from Step 1
(20 min) Grant discusses the Art of Questioning – The Falconer
(10 min) Grant and Jill use Linear TNS file to model What if…questioning
(40 min) Sam facilitates EllipseInvest.tns and 3-12-3 protocol for questioning
(30 min) Leveled Assessment using Navigator and discussion pedagogy
(10 min) Jill’s Ignite talk on Assessment
(15 min) Break
(15 min) Dan Heath’s Bright Spot video and quotes
(60 min) Work time – participants develop a leveled assessment
(20 min) Share and feedback session
(10 min) Conclusion and Challenge

As a system…

As a system…

Drop your files: http://www.dropitto.me/jplgough

On Tuesday, Jill will zip all files and link them to this blog post.

Our participants have been asked to bring their laptops so that we can build assessments using the TI-Nspire software.  We intend for participants to leave with an assessment ready for use next week.

# Design for Learning and Inquiry – PD Day #T3Learns #nspiredatT3

The tribe of T³ Instructors gathers the day before the 2013 T³™ International Conference to learn together. I am honored to be part of the team that has been asked to facilitate a PD experience for my colleagues and friends.  Our topic: TI-Nspire Advanced Authoring – Designing for Learning and Inquiry.

Below is the description as it was sent to the T³ Instructors and our tentative agenda. (Note: The links in the agenda below are password protected and available only for face-to-face professional development participants.)

TI-Nspire Advanced Authoring – Designing for Learning and Inquiry

Facilitators:
Ruth Casey, Jill Gough, Sam Gough, Jeff McCalla

From the PD signup document:
Design For Learning – 120 minutes
This workshop was successfully piloted last summer.  The focus was on the effective design of activities, using TI-Nspire to encourage student learning.  This session will give you an overview of the goals of the PD offering, and the instructional approach taken in the workshop.

Goals:

Materials Needed:

• Diagnostic Assessment sent via email to participants by Monday, March 4, 2013
• TI-Nspire Navigator for Networked Computers
• Chart paper and Markers for Storyboarding

Agenda:
(15 min) Introductions
Review Diagnostic Assessment Results
(30 min) Introduce the Anatomy of a Document
Anatomy of a Document and investigation
(60 min) Introduce Storyboarding
Instructor modeled storyboard and design of a lesson and Storyboard a concept using principles of  “Anatomy of a Document”
(15 min) Share the Design for Learning and Inquiry Google site

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. Can we begin to see ourselves as designers of learning experiences?

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.

These essential learnings are:

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.

# Art of Questioning – PD Day #T3Learns #nspiredatT3

The tribe of T³ Instructors gathers the day before the 2013 T³™ International Conference to learn together. I am honored to be part of the team that has been asked to facilitate a PD experience for my colleagues and friends.  Our topic: The Art of Questioning.

Below is the description as it was sent to the T³ Instructors and our tentative agenda.

The Art of Questioning – 120 minutes

This session will focus on the art of questioning.  Strategies will be discussed to assist learners to “level up” through questions rather than lectures.  Come prepared to develop documents to help students calibrate their understanding, and share your assessments with others for feedback and suggestions.

Goals:

• Introduce a protocol for facilitating learners to ask their own questions
• Introduce leveled assessment
• Participants develop a leveled assessment and share it with other attendees

Facilitators:

Jill Gough, Sam Gough, Grant Lichtman, Jeff McCalla

Materials Needed:

• TI-Nspire Navigator for Networked Computers

Agenda 2:

(10 min)            Introduce facilitators
(20 min)           Learn with Grant Lichtman via Skype
(30 min)           Art of Questioning using EllipseInvest.tns
(15 min)            Break
(20 min)           Introduce Leveled Assessment
(30 min)           Participants develop a product
(10 min)           Share session

Grant is going to share two powerful stories of teaching and learning through the art of questioning. He has also agree to take us through an exercise from The Falconer to illustrate the possibilities when asking quality questions.  <EXCITING!>

Jeff, Sam, and I will model methods of using TI-Nspire’s interactive capabilities to give rise to more student questions by putting our participants in the seat of the students.  We will share student-learner reactions, challenges, and successes. The overarching idea is posted at Practice seeking questions – #AskDon’tTell and a sample story is shared in the post Ellipse Investigation – #AskDon’t Tell.

We will recommend and read from several books (listed below) that have helped us frame our work. Our goal is to facilitate a workshop where the participants learn by doing.

From an earlier blog post:

1. Grant Lichtman‘s The Falconer: What We Wish We Had Learned in School,
2. John Barell’s Developing More Curious Minds, and
3. Dan Rothstein’s Make Just One Change: Teach Students to Ask Their Own Questions.
2. Practice, practice, practice…Remove the scaffolding:
1. Watch Dan Meyer: Math class needs a makeover and try it.
2. Stop creating slideuments.  If your TI-Nspire document, your PowerPoint presentation, or your worksheet has multiple pages, slides, or steps, eliminate lots! Create space for questions, investigation, and thinking.
3. Use Gamestorming games to develop techniques for learning to ask questions. I like Brainwriting3-12-3, and others.
3. Risk, reflect, revise:
1. Try it – more than once. One trial does not make an experiment.  Celebrate even small successes.
2. Have strong wait time, and have questions in your “back pocket” if prompting is needed.
3. Seek feedback from a trusted colleague. Engage in peer observations to help you see from another perspective.

# Ellipse Investigation – #AskDon’tTell

One more learning investigation from our experiment – see Circle Investigation – #AskDon’tTell and Parabola Investigation – #AskDon’t Tell.

Sam designed the following ellipse investigation (EllipseInvest.tns).

Look at the continued improvement in the learners’ questions:

Sam notes that his learners are beginning to ask more mathematically relevant questions. Isn’t this exciting? Can you imagine learning math this way?

I’m pretty sure there is a hyperbola lesson coming soon.

# Parabola Investigation – #AskDon’tTell

Continuing our experiment to lead learning by following and responding to learner questions, Sam designed the following parabola investigation (ParabolaInvest.tns) for his learners. (Note: Sam teaches in a 1:1 MacBook program, so these files are designed to be viewed and investigated in computer view rather than handheld view.)

Here are the questions generated by Sam’s student-learners.

Wow! What an improvement in the number and the quality of questions.

The importance of collecting many questions is critical.  If we answer questions in the order they are asked, we might not get to the most interesting or most critical questions.

Sam reported that he is very pleased with the learners’ questions and their engagement with the mathematics.  He says he is encouraged to continue this method of teaching because of the change in participation and interest from his learners. It is fun to teach and  facilitate learning this way! He says:

Kids should interact with these graphs instead of memorizing facts.  I want to continue offering investigations where they control points and identify patterns. It’s a great way to learn!

He is busy working on an Ellipse Investigation, and he is already planning a Hyperbola Investigation too.

# Circle Investigation – #AskDon’tTell

What if we facilitate learning episodes by following the learners questions? How might we set up opportunities for learners to explore and think PRIOR to a show-and-tell scaffolded lecture?

What if we gave learners the a TI-Nspire document and asked them to explore it for a few minutes? What if we asked them to jot down observations, patterns, and questions  that come to them as they play with the document?

So, here’s the hypothesis:  We can teach just as much (or more) by responding to the learners’ questions.  What if we tried an experiment with a one page Nspire document and a protocol for question generation?

Sam Gough, Algebra II teacher at The Westminster Schools, was brave enough to try this experiment with his learners.  Here’s the original plan:

Using the TI-Nspire document CircleInvest.tns, facilitate a 3-12-3 protocol to generate student questions.

• 3 minutes: Independent investigation of the CircleInvest.TNS file.
• 12 minutes: Work with a partner to share questions, convert closed questions to open questions, and generate additional questions. Partners should identify their top 2-3 questions.
• 3 minutes: Use the TI-Nspire Navigator to collect each student’s top question.

Using the TI-Nspire Navigator for Networked Computers, he sent his learners the circle investigation shown below and challenged them to interact with the document and record questions.

Using the 3-12-3 protocol combined with the ideas in Dan Rothstein’s Make Just One Change: Teach Students to Ask Their Own Questionshe coached his learners to investigate the TI-Nspire document and generate open-ended rather than closed questions. Below are the questions generated and submitted by his learners.

Pretty interesting, huh? Not bad for a first try.  Sam reported that he did teach what was in his original lesson. He was intrigued by the pattern question asked by his learners.  He said that he is concerned that when the lesson is more challenging than investigating a circle, he may not get the results he wants.  He also mentioned that the times might need to be 3-9-3 instead of 3-12-3.  I wondered if they might need more time as they improved as investigators and questioners.

Acknowledging that one trial does not make an experiment, we are going to try again tomorrow with a Parabola Investigation.