# Recap: NCTM 2017

San Antonio, CA , April 2017
I have summarized each session with some simple (•) bulleted notes, red underline to encapsulate my major take-aways, and occasionally a brief italicized commentary.

#### Math Task Makeover with Desmos Activity Builder — Michael Fenton (Desmos), Jed Butler (Heritage HS), Bob Lochel (Hatboro-Horsham High School)

• The Big Take-Away = Use Desmos activities to generate intellectual need to learn the lesson objective.”
• Generate need for Graph of a Linear Inequalities ….

• Generate need for Definition of Ellipses …

• Teacher facilitation is key.

I’ve got to starting using the overlay function!

#### Numberless Word Problems in the Elementary Grades — Brian Bushart & Regina Payne (Round Rock ISD)

• The Big Take-Away = Have students make sense of word problems prior to computation by removing the numbers.”
• The origin: Press kids to stop just circling numbers in word problems and applying random operation.
• Not all day every day. It is a tool for sense making.
• Focus on the relationship and the operation, formal language, and what the question would be, not the answer.
• #numberlesswp

This makes sense for secondary grades as well.

#### Rich Tasks as Landmarks for Students to Use in Navigating Their Mathematical Learning Journey — Peg Cagle (LAUSD)

• The Big Take-Away = Students’ work on Landmark Tasks throughout the year that should be visible in the classroom so that students can map their learning.”
• We don’t take advantage enough of narrative in math class.
• “Imagine shrinking down an entire map to the size of an index card. All the details get lost and the map becomes unreadable. What are the landmarks that will help students navigate the mathematical landscape”.
• Peg presented the criteria for a Landmark Task …

• … and presented us with a LandmarkTask …

Tied Up in Knots: In your groups, measure the length, in centimeters, of the piece of rope that you have. Then tie a single overhand knot and remeasure the length. Repeat the process several times. Create a data table, graph and equation relating the number of knots to the the length of the rope.

• … then she analyzed the task according to the criteria …

• … and showed how this landmark was made visible in her classroom.

• The Speech Bubbles were created by the students to make comments on other groups’ work.
1. This is the second year in a row at this conference that I have seen Peg give a year-long, big picture vision of using tasks in the classroom.
2. This is also the third presenter who has mentioned some variation of the Speech Bubbles. Time to use them in my classroom.
3. Peg made a statement that has me thinking deeply and that I have quoted several times already: “Students have ample amounts of robust evidence that they are not good in math.” We need to help them overcome that.

#### Changing Teacher Practices: Transforming Teaching 101 to PD 101 — Audrey Mendivil (San Diego County)

• The Big Take-Away = Shift from Professional Development to Professional Learning.”
• 5 Principles of Effective PD
1. On-Going
2. Support during implementation
3. Model new practices
4. Variety of approaches and active engagement
• Shift from Professional Development to Professional Learning

• How to Change:
1. Small Steps. Stick to only 2-3 short term goals.
2. Rethink Our Norms:

• Why PD often FailsHow can we set-up for success?
1) Top-Down Decisions: How can you include teachers in the decision making process?
2) Little or no support in transferring ideas to the classroom: What support is available?
3) Idea that teachers need to be fixed: How are you communicating your why?
4) Lack of variety in delivery modes: How can you differentiate for teachers?
• Essential Elements. Audrey took us through a terrific activity for those who create Professional Learning experiences. She gave a sets of cards that were color coded, and asked us to work together to sort them into 4-6 groups, and then name the groups.
She then shared how she grouped them (which is what the color scheme was for). The idea was to take ALL the things that we want teachers to know and do and rather than create a checklist for them, cluster these concepts into Themes or Essential Elements and have teachers learn that.

1. This was yet another session at NCTM that focused on Vision and the need to put the WHY in front of teachers.
2. The re-structuring of the norms resonated with me. I’m still thinking deeply on this one. The norms drive the culture of the meetings, so they offer great leverage.
3. In her call to keep the list of goals short, Audrey discussed the need to set short-term, intermediate and long-term goals. This falls in line with the concept of “leading and lagging indicators.” Student data may take awhile to improve (lagging) so what are the improvements in teacher moves that we can credit to our PD (leading)?
4. The objective of the card sort activity gets at the heart of what I see killing most PD in districts … too many short-lived initiatives. Keep the broader concepts in mind. Bigger, slower moving targets are easier to hit.

#### The Struggle is Real: Tasks, Academic Status, and Productive Problem Solving — Geoff Krall (New Tech Network)

• The Big Take-Away = Developing a culture of productive struggle requires holistic vigilance on the relationships between Quality Tasks, Effective Facilitation & Academic Safety.

• Protocols for Problem Solving
1) Make it visual
2) Estimate Before Solving
• Record what students know…
vs what they are assessed on.
• Promoting Access:
Example: Make the smallest (or largest) difference by filling in numbers 1-9 no more than one time each.

I am challenged by Geoff’s two graphs of the linear regression of student growth. My Claims-Based Grading needs a little more work in the area of reflecting cumulative knowledge rather than recent learning.

#### Logarithmic Earthquake Project: An Algebra 2 Project with Real Applications — Tanisha Fitzgerald-Williams & Beverly Heigre (Notre Dame High School)

• The Big Take-Away = Have students view videos of earthquake damage and do their own research on Richter Scale, before formal presentation of calculating Magnitude difference with Logarithms.”
• Step 1: Research

• Step 2: Calculations

• Step 3: Student Groups make Presentations
• Note: Tanisha & Beverly also have students offer possible humane response to victims of earthquake presented.
• There is a google drive folder available that contains materials for this projects: goo.gl/Y197YR

#### Clothesline: The Master Number Sense Maker — Chris Shore (Me)

• The Big Take-Away = Number sense and conceptual understanding of current content can be taught simultaneously with Clothesline Math.”
• I presented the power of the Clothesline to teach Algebra, Geometry and Statistics.
• clotheslinemath.com
• #clotheslinemath

There were at least 5 sessions at NCTM Annual in which the Clothesline was a part or the focus of the presentation.

Fun Sidenote: The ceiling rafters and the carpet print of the convention center had the same Geometric Pattern. I am sure there is lesson to be created out of this.

There are videos of keynotes, ShadowCon and Ignite
at NCTM’s Conference 2017 web page.

The city of San Antonio enhanced an already fantastic trip!

# Recap: NCSM 2017

San Antonio, CA , April 2017

I have summarized each session with some simple (•) bulleted notes, red underline quotes to encapsulate my major take-aways, and occasionally a brief italicized commentary.

#### Knocking Down Barriers with Technology — Eli Lubroff (Desmos)

• The Big Take Away =  Differentiation should not mean different tasks for different students, but instead should offer different depths with same task.
• Technology can be used effectively to address Inequality, Disabilities and Differentiation.
• Marbleslides is an example of a high cognitive demand task that naturally differentiates.

• Our (math ed community) work of offering High Quality, Meaningful, and Relevant mathematics for ALL has never been more important. Non-routine Cognitive jobs are becoming a growing percentage of employment opportunities.

• Inequality:
1) 51% of American students are on free and reduced lunch, which equates to 25 million kids!
2) The two great equalizers: Mass Adoption of Technology and Public Education.
• Disabilities: For students with disabilities, technology opens doors that have always been closed. Desmos is currently working with a blind person to develop an audio feature that converts graphs to music and sound.
1. OK, OK already … I’ll finally start using Marbleslides!
2. I will also place more “extensions/challenge” problems on tasks to offer deeper differentiation.

#### Gut Instincts: Developing ALL Students’ Mathematical Intuition — Tracy Zager (Stenhouse Publishers)

• The Big Take Away: In mathematics, intuition is as important as logic, and like logic, needs be explicitly developed.
• “Intuitive experiences must be acquired by the student through his/her own activities – they cannot be learned through verbal instruction.” —  Erich Wittmann
• Following mathematical intuition needs to precede the logic. Logic then further encourages the intuition:

• Tracy had us practice this explicit development of intuition with a simple challenge and manipulative. “Can you try and discover the size of some of these other angles?” We were given the fact that the squares had 4 right angles, but no protractor. Go!”

• Intuition can also be developed through estimates before the algorithmic practice occurs.

#### Feeding the Brains’s (Affective and Cognitive) Subcommittees for Mathematics Learning   — David Dockterman (Harvard University)

The Big Take Away = The Brain Has 3 Learning Networks: The WHAT, the HOW and the WHY.”

1. The WHAT Network  = Different parts of the brain …
• Approximate Number System
All animals can picture quantity especially when comparing a lot vs a few. It gets harder as the ratio approaches 1:1
• Language Retrieval
Association between number and name (assigning “eight” to a collection of eight items). Math facts are retrieved in the language that they were learned.
• Symbol Procedure
Association between number and symbol (assigning “8” to a collection of eight items).
• Feeding the WHAT Networka) Connect the Visual, Symbolic & Linguistic parts of the brain through multiple representations
b) Make Sense through Coherence: What is learned today should be related to what was learned yesterday.

2. The HOW network = Executive Function

• Cognitive Flexibility
• Working Memory
• Concentration
• Emotional Control
• Feeding the HOW Networka) Incentives for inputs not output
b) Develop the Math Practices, focus on process, not results.
c) Notice: Define the desired learning behaviors and call them out during lesson. This video of the “Monkey Business Illusion” demonstrate how the human mind focuses intensely on what you want it to notice. What behaviors do you want students to focus on?
d) Nudge: Use Norms to encourage target behaviors. Compare past, less productive behaviors to current successful behaviors.

3. The WHY Network = The Affective State

• The 3 Mindsets of Why
-Purpose & Relevance
– Growth
– Belonging
• Feeding the WHY Network
a) Purpose & Relevance: It is more powerful connecting a cause (Self-Transcendance) than to a career, knowing math will help you make a difference in the world.
b) Growth: Give feedback like you are giving advice (vs judgement). Carole Dweck, “The brain is like a muscle. The harder it works, the stronger it gets.” This should be the norm. We have to believe in our students abilities before they will believe in themselves.
c) Belonging: Establish a culture where it is safe to learn (vs perform). We cheer each other’s growth.

Feeding the Entire Network Intentionally

• Manage teacher cognitive load – don’t look for everything all the time.
• Match the tasks to illuminating the beliefs, behaviors or knowledge and skills that you want to notice.
• Behaviors aren’t one-and-done; they need constant nurturing of the mindsets that drive them.
• Have the right food ready — anticipate, notice and respond.
• YOU and the adults have to believe.

In regards to praising inputs, not outputs. Dr. Dockterman shared a study on financial incentives. Monetary rewards for better grades showed no improvement; monetary rewards for the behaviors that leads to better grades (notes, homework, questions) showed significant improvement.

#### What Every Math Leader Needs to Know and Be Able to Model to Support the Classroom Development of Numerical Fluency —  Patsy Kanter & Steve Leinwand (AIR)

• The Big Take Away = Developing numerical fluency requires planning and active engagement. “

5 Steps to Implementing Numerical Fluency

1. Commitment: Fluency is more than speed. It takes ongoing, protected time and assessments to develop.
2. 10 Pivotal Understandings of Numerical Fluency:
1. All quantities are comprised of parts and wholes that can be put together and taken apart.
2. Numbers can be decomposed.
3. Storytelling is key, because vocabulary of the four operations is critcial.
4. Properties of Operations reduces memory load (like 29 x 25).
5. Requires discussion of alternate strategies.
6. 5 & 10 are cornerstones.
7. Understanding that 9 and (10-1) are the same quantity.
8. Groups or 2, 3, 5, & 10 are cornerstones of Multiplying.
9. Ideas of equality and equivalency are key.
10. Place value dominates fluency of large numbers.
3. Cement Number Fluency:
a) Concrete Representations
b) Verbal Representations
c) Pictorial Representations
d) Discussion & Justification
4. Calendarize: 10-15 minutes daily
For example …
Monday: Make it Number Sense Activity
Tuesday: Operational Practice
Wednesday: Word problems
Thursday: Making math connections
Friday: Counting patterns, Games, Puzzles (week 1) and Assessment (week 2)
5. Assess: Include Numerical Fluency (not speed) on assessments. For example, “Write everything you know about two.”
1+1=2
2 is even
2 is the only even prime number
2 is the square root of 4 ! 5-3=2
Multiplying by two doubles any number

Though the examples here were mostly elementary the 5 steps still apply to secondary, particularly to the planning of daily activities.

#### Knowing Your HEARTPRINT: Transforming the Way We Teach and Lead.  — Tim Kanold (HEARTPRINT)

• The Big Take Away: I define your heartprint as the distinctive impression and marked impact your heart leaves on others.”
• Happiness boosts our productivity and heightens our influence over peers.
• Engagement. From Gallup Poll:
31% are Engaged Teachers (seek ways to be better)
57% are Not-Engaged. (unlikely to devote discretionary effort)
12% are Actively Disengaged (intentionally sabotage)
• Alliances: Collaborate with other teachers.*
• Risk. Take chances towards the Clear Vision and using Clear Results (data) to guide journey for OUR students (vs my students).
• Thought. Are you a person of deep knowledge capacity and wisdom?

This was an inside peek at Tim’s recently published and highly acclaimed book, Heartprint. It was one of the most emotional, inspirational presentation I have ever witnessed. Tim really called us to focus on the core purpose of why we are teachers.
* I refer to Dr. Kanold as the ‘Prince of the PLC movement.” The ‘King’ was Rick Dufour, his friend and mentor. Tim dedicated this book to Rick who passed away this year.

#### 10 Instructional Tweaks That Every Math Leader Needs to Advocate For and Be Able to Model — Steve Leinwand (AIR)

• The Big Take Away = Strengthen daily classroom instruction with collaborative structures and coaching, monitored with high-quality, well-analyzed common assessments.”
• Tweak 1: Cumulative Review
Most effective strategies for fostering mastery and retention of critical skills is daily, cumulative review at the beginning of every lesson.
• Tweak 2: Fewer Mindless Worksheets
Never more than 4 problems on new skill,
Annual reread of Jo Boaler‘s, Fluency without Fear
• Tweak 3: Change Homework Structure
2-4-2 Homework
2 Problems = New Skills
4 Problems = Cumulative Skill
2 Problems = High Order/Justification
• Tweak 4: Daily Exit Slips
with 5 minutes to go, every lesson:

– “Turn and tell you partner what you learned today”;
– “Individually, on a sticky note, complete this task”;
– Launch next lesson with “On the basis of yesterday’s exit slip”

• Tweak 5: Higher Order Questioning
Why? How do you know? Can you draw it? What do you notice? How are the same? etc
• Tweak 6: More Substantive Student Discourse
You = Struggle, Explore & Share
We = Justify Compare & Debrief
I = Consolidate
• Tweak 7: More Productive Struggle
We need more DOK 2 & 3 tasks, and weekly opportunity for rich and robust tasks
• Tweak 8: Greater Use of Technology
Docu-cams, Class twitter accounts, Desmos etc.
• Tweak 9: Effective Intervention
Most are ineffective, because they do not change the approach.
• Tweak 10: Effective Collaboration
PLC’s don’t magically make a difference. It is their content and follow up that do.

I have been sporadic with my use of reflections and exit tickets at the conclusion of the lessons. Steve & Patsy have challenged me to be more intentional and diligent in these practices.

#### Building Leadership Structure — Denise Porter & Kathryn Flores (University Chicago STEM Education)

• The Big Take Away = A Math Instruction Improvement Program needs a long term plan.
• The case study that was shared is a project in which the non-profit group provided funding and support of the University of Chicago STEM .
• Video recordings of exemplary teachers teaching a lesson that was requested by teachers through PD survey.

This speaks to the power of observations and particularly the use of video to model best practices. The question is how to find the time to record and edit lessons, without outside finding.

#### Conceptual Understanding and Exploration in Mathematics Via Desmos.  — Eric Milou (Rowan University)

• The Big Take Away = Use of technology tools can support both learning of math procedural skill as well as advanced math proficiencies, such as problem solving and justifying.”
• Desmos allows you to:
– Engage with dynamic motion
– Create with animation and art
– Monitor with instantaneous feedback

#### Ross Taylor Past Presidents Session.

• Hank Hepner: Have a shared vision and question the assumptions that lead to unproductive beliefs.
• Steve Leinwand: How can we help our colleagues envision more effective pedagogy? Get teachers to observe each other and debrief every other week:
– What was impressive, what would you different, a call action?
– Video recording of exemplar.
– Focus on only two things, you pick one and I pick one.

More talk of vision and modeling of exemplars.

#### Developing High School Mathematics Teahcer Leaders  — Mike Steele (University of Wisconsin Milwaukee)

• The Big Take Away = Prepare teacher leaders with hypothetical scenarios.
• Directions: Brainstorm how a leader would address the challenges connected to the sample scenario. There are a group of parents in your school who are very unhappy about the math curriculum. They think the program holds back their students. They want their sons and daughters to be challenged, to be successful (generally based on grades or tests scores), and to be able to move ahead in grade level or math courses. The parents indicate that mathematics is a discipline that is learned in steps. The steps need to be clearly defined and when the students succeed in accomplishing the steps, they are moved to the next level or the next course.

#### High Leverage Leadership Actions That Improve Teaching and Learning.  — Diane Briars (NCTM, Past President)

• The Big Take Away = There are no quick fixes, and me must act on two fronts: Teachers & Administrators.”
• High Leverage Practice #1: Shift emphasis from Answer Getting to Mathematical Understanding
Once kids know how to do something they don’t want to understand why… You can’t take it back. Understanding facilitates initial learning and retention.
• High Leverage Practice #2: Collective Professional Growth.
Support Collaborative Team Culture. All kids are our kids. The unit of change is the teacher team.
• High Leverage Practice #3: Implementing the 8 Effective Teaching Practices (Principles to Action).
… Then focus on Discourse. What do you do after the students do the task? DISCOURSE to advance the math learning of the whole class. It’s not about doing the task; it’s doing the task for a reason, so planning is key, and planning with someone else is even better. Facilitate the anticipation phase of the planning.
• High Leverage Practice #4: Awareness of the Micromessages about math and students.
Comments like, “It is immediately obvious that…,” Effort-based vs intelligent-based praise. Observe number of interactions and nature of questions teacher offer based on gender or race.

#### Shrinking the Equity Gap in Secondary Mathematics Coursework: Implications for College Coursework and Completion.  — Dr. Amy Wiseman  & Chritine Bailie (E3 Alliance)

• The Big Take Away = Acceleration Works!”

• 8th Grade Algebra is Key.
• Auto Enroll and allow parent “opt-outs” rather than “opt-in.”
• PD for teachers on supporting “Bubble Students” critical.

Thank you San Antonio for an Epic Trip!

# Re-Cap: NCSM 2016

Oakland, CA , April 2016

I have summarized each session with some simple (•) bulleted notes, red underline quotes to encapsulate my major take-aways, and occasionally a brief italicized commentary.

#### Game-Based Learning: The Hype is Starting to Give Way to Some Surprising Substance  — Keith Devlin (Stanford)

• Big Take-Away = Start with the thinking (which is the more important), then follow with the notation.
• The “Symbolic Barrier”: Symbols are a terrific way to use mathematics, but a horrible way to learn them.
• The vast majority of our population is taught symbolic notation, yet most need mathematical thinking.
• Students using Dragon Box Algebra learn the Algebraic thinking needed for solving equations in 90 minutes. However, this ability did not transfer to paper/symbolic test, therefore, both are needed.
• We teach students to play music, before we teach them to read it. The same should be true of mathematics.

Personal note: I’ve had Dr. Devlin’s book, Goodbye Descartes, for almost 20 years; after his talk he signed it for me.

#### Developing Deeper Student Thinking  and Reflection — Patricia Rogers (Gilroy USD)

• Big Take-Away = Use “structured” student collaboration to enhance student reflection, and thus student thinking.
• Good collaboration needs to be: Regular, Brief, Prepared, Open-Minded.
• 3 Teacher Moves (Phil Daro)
• Student thinking made visible (to other students, not just the teacher)
• “Everyone Ready” (ALL students individually prepare themselves to share thinking.)
• “Make an Expert” (of a students who has viable strategy) then have the rest of the class “Turn and Talk” when productive struggle weakens in order to focus on targeted math topic.
• Classroom Discussions (Chaplin, O-Connor, Anderson)
• Wait Time
• Revoice (The teacher rephrases what the student just said.)
• Restate (Student(s) rephrase what a student just said.)
• Add-on (Student(s) extend or challenge another student’s conjecture.)
• Apply (Students apply their own reasoning to someone else’s reasoning …” just try it on.”)

I’ve seen the two techniques of revoicing & restating demonstrated a great deal lately and have now been challenged to bring these into my class more often.

#### The San Francisco USD Mathematics Teaching Toolkit: Changing the Practice Along with the Content — Glenn Kenyon & Kathy Bradley (SFUSD)

• Big Take-Away = Established Vision, Beliefs and Goals before building district curriculum

Vision
“All students will make sense of rigorous mathematics in ways that are creative, interactive, and relevant in heterogeneous classrooms.”

Beliefs
1. All students can and should develop a belief that mathematics is sensible, worthwhile, and doable.
2. All students are capable of making sense of mathematics in ways that are creative, interactive, and relevant.
3. All students can and should engage in rigorous mathematics through rich, challenging tasks.
4. Students’ academic success in mathematics must not be predictable on the basis of race, ethnicity, gender, socioeconomic status, language, religion, sexual orientation, cultural affiliation, or special needs.”

3 Goals
1. Help students express, expand and clarify their own thinking. 2. Help students to listen carefully to one another and negotiate meaning.
3. Help students deepen their reasoning.

“The teaching strategies in the SFUSD Math Teaching Toolkit are designed to support an inquiry-based approached to learning mathematics, with an emphasis on classroom discourse. This approach reflects the shifts of pedagogy required to promote the Common Core Standards for Mathematical Practice.”

• Unit Design Structure to incorporate tasks

1) Math Talks
(SMP#3. “Math Talks”, instead of Number Talks, so discussion can broaden {e.g. strategies for computing area})

(Model for close reading of complex math text)
This only runs 10-12 minutes. Take away the question to create a rich task.

3) Participation (Group) Quiz A technique to give public feedback on group work. Lists ways a student can contribute (“You can help your group if you can…. create a table, draw a diagram, listen to people’s ideas and ask questions, etc) Also publicly list teacher expectations (e.g. How groups … us shared space? ask question? explain thinking? etc)

• Video Exemplars & PD modules are available on district web site.
1. SFUSD has a PHENOMENAL math web site chalked full of resources for supporting teachers implement the vision and the curriculum. Check it out!
2. The description of their Group Quiz speaks to the need to explicitly teach students how to productively collaborate.
3. This was the first of three sessions that spoke about the importance of vision. It will be the predominant point that I take home with me from this conference.

#### Beyond Relevance and Real World: Talking with Teachers About Engagement in Mathematics? — Dan Meyer

• Big Take-Away = ‘Real World’ does not have to be real, just accessible and engaging.
• 62% of teachers surveyed : Greatest challenge is “unmotivated” students. Interesting that they didn’t say motivating students was the challenge.
• Question: Why don’t teachers spend more time developing good questions?
Teacher Response: “Because we don’t have the time.” (True that.)
Real Issue: “Lack of creativity. Giving the answers does not require creativity.” (True that, also, but ouch!)
• A stronger option than the typical “engaging images or context” in a textbook: Redefine Real World. A situation is in the process of becoming real to you if you are able to …

#### High School Coaching Model: Building Bridges Between Coaching and PLC Culture — Kris Cunningham & Jeanette Scott (Phoenix UHSD)

• Big Take-Away = Roll out PD through PLC teams.
• New initiatives first unveiled during PLC team meetings.
• Most powerful change agent was a lesson study. (1st day by 1 teacher, next day by all teachers)
• Most teachers took 3-4 years to show change; 4 of 5 teachers showed significant change within 5 years.
• There exists a Common Lesson Plan format for lessons studies and co-planning.
• Professional Development certificates tied to evaluations. (i.e. Professional Growth affects evaluation outcome.)

The fact that teachers took 3-4 years to show change aligns with Maggie McGatha’s research shared at last year’s NCSM conference

#### Practicing the Five Practices: Coaching Teachers to Use Student Work in Planning  — Max Ray-Riek (Math Forum)

• Big Take-Away = Walk teachers through the 5 Practices of Discourse with student work samples.
• Max shared with us the Teddy Bear’s Banquet pattern problem. He had us determine the Math Goal for the lesson, and then Anticipate the student responses.
• Max then offered 16 samples of true student responses (Monitor) and then had us Select and Sequence some of the responses for classroom discourse and share why. We were then asked to Connect the responses to the Math Goal.

This is a great training tool that can be brought into any PLC structure.

I also witnessed Max slyly counting on his fingers. This was his way of giving is all wait time on his prompts.

#### Smarter Balance – Making Connections: Eliciting to Acting on Evidence —  Judy Hickman (Director of Mathematics, SBAC)

• Big Take-Away = When the scoring focus is on Reasoning, students can still score full credit with a minor calculation error, if they show understanding.
• Do NOT put too much emphasis on Interim Assessments. As “snapshots” they will give you good information, but it will be an incomplete assessment.
• The authors of the exams were shocked that students answered so few questions correctly.

#### Four Keys to Effective Mathematics Leadership — Mona Toncheff & Bill Barnes  (Activating the Vision)

• Big Take-Away = Vision needs to be created by ALL stakeholders
• The Four Keys:

1. Establish a Clear Vision for Mathematics Teaching & Learning
2. Support Visionary Professional Learning for Teachers and Teacher Leaders
3. Develop Systems for Activating the Vision
4. Empower the Vision of Family and Community Engagement

This was the second of three sessions that spoke about the importance of vision. This one stressed the need to have all stakeholders (admin, teachers, classified staff, parents and the business community) in on the creation of the vision. Mona & Bill then asked, “If you were ask 10 people on your campus, ‘What is our vision,’ how many answers would you get?”

#### The Secret to Leading Sustainable Change: Vision, Focus, Feedback, and Action! — Dr. Tim Kanold (Turning Vision into Action )

• Big Take-Away = Set the Vision, Help people advance the Vision,  Celebrate Evidence that the people are advancing the Vision, and take Action on the feedback towards the Vision.
• Sustainable change requires evidence that the change is bigger than their opinions.
• Is the work you are doing formative? Meaningful feedback must be followed with results in action by the teacher or teacher team.
• Meaningful Feedback = F.A.S.T. Action: Fair, Accurate, Specific, Timely. Action from your feedback is required.
• Mary Beth call. Dr. Kanold told a story of when he was Superintendent of Stevenson HSD. He called a secretary at one of the schools, restated that ‘engagement’ was part their district vision, and asked “What does engagement look like in your job.” That’s keeping the vision in front of the people!
• The Popeye Moment: Change happens when the moment of moral courage vocalizes what Popeye often said, “That’s all I can stands, cuz I can’t stands n’more!

This was the third of three sessions that spoke about the importance of vision. The story of calling the secretary is tattooed on my brain. Dr. Kanold stressed that the vision should be posted visibly during every PLC meeting, and that any unproductive dialogue can be redirected with the simple statement, “How does this conversation advance this vision?”

#### A Math Coaching Package — Donna Lione, Rebecca Williams & Chris Shore (Me) (Temecula Valley USD )

My colleagues and I presented the framework for developing a comprehensive math program. The details of each of the 8 components will be posted as separate posts.

• Vision
• Relationships
• Humility
• Influence
• Passion
• Faith
• Focus
• A Plan

# Recap: NCTM 2016

San Francisco, CA , April 2016
I have summarized each session with some simple (•) bulleted notes, red underline to encapsulate my major take-aways, and occasionally a brief italicized commentary.

#### The Status Quo Is Unacceptable: A Common Vision for Improving Collegiate Mathematics — Diane Briars, & Linda Braddy, Christine D. Thomas & Dr. Uri Treisman

• Big Take-Away #1 = College failure rates are 55% higher than for more active forms of instruction.
• Big Take-Away #2 = The math ed reform movement is now reaching the post-secondary level.
• Big Take-Away #3 = The change must be institutional.

• The challenge facing the Math Ed Community (the dismal stats)
1) Only 50% of students earn A, B or C in college algebra.
2) Women are twice as likely as men to not continue past Calc 1.
3) While 20% of all Bachelors Degrees are awarded to Blacks & Hispanics, only 12% of Math Degrees are.
4) Math is the most significant barrier to degree completion in ALL fields.
• Innovation does not affect normative practice. Out of 81 different projects (2-3 yrs) connected to a grant or leader, NONE replaced normative practice, because they were based on faculty development, not institutional change.  Dr Treisman, “Institutional change is a bitch.”
• Historically, school system does change when necessary.

#### The Learning Mindset Movement and Its Implications for Addressing Opportunity Gaps — Dr. Uri Treisman (The Dana Center)

• Big Take-Away  = Besides Growth Mindset, there is Belonging Mindset and Purpose Mindset.
• “I find Algebra beautiful, but will it knock the socks off of a 13 year old. Algebra well taught should leave them barefoot in the park.”
• “Why do kids give up? Most of the work I do is confusing, cause no one gives me problems in the back of the book.”
• Growth Mindset = “Can I do this?”
Belonging Mindset = “Is this where I belong?”
Purpose Mindset = “Does this connect to who I want to be?
• Dr. Catherine Good:  Building Bridges to Belonging: Mindsets that Increase Participation, Achievement and Learning
• Build Belonging through effort & engagement, not talent.
• Positive Belonging Mindset = Assume they belong.
Negative Belonging Mindset = Need to be invited in.

#### Paper Cup + Gust of Wind = Yearlong Rich Task — Peg Cagle

• Big Take-Away = Revisiting the same task through-out the year emphasizes math as reasoning not simply answer-getting.
• Peg had us roll a paper cup on its side. She then left us to our own devices to answer several questions, each of which addressed a different mathematical topic throughout the school year.
• Day 35 Question: How can you convince a skeptic of the shape that the cup traces out as it rolls?
• Day 70 Question: How can you locate the center of the shape that the cup traces out as it rolls?
• Day 105 Question: How can you use a cup’s dimensions to determine the area of the shape it traces out as it rolls?
• “Efficiency is overrated: That is a concern after you learn something.”

#### Coding to Enrich ALL Math Classes — Jason Slowbe

• Big Take-Away = Coding in Math class helps teach the Math, not just the coding.
• Coding can be done on the TI-Calculator
• Can help students understand the meaning and power of mathematics. For example, Archimedes’ method for approximating the area of a circle.

#### Rich Problem Solving to Support Today’s Standards — Chris Shore (Teacher Created Materials)

I conducted a product promotion for Teacher Created Materials. The session was on Problem Solving and Linda Gojak’s What’s Your Math Problem Anyway? My presentation focused on the following questions about the teaching of problem solving, each of which I will answer in its own post:

• What is problem solving?
• Why teach problem solving?
• Who should learn problem solving?
• When should we teach problem solving?
• How should we teach problem solving?
• Where do we find resources for teaching problem solving?

# Interview: Dan Meyer on Using a Ladder to Clear a High Bar

Dan Meyer is as close to a celebrity that a math educator can get. We all owe him a debt of gratitude for making math nerds look cool. He deserves his rock star status because he is an amazing presenter, a thought-provoking writer, and an ingenious creator of math tasks.

Behind all the hype, though, is some serious substance. Dan’s ideas are akin to the spirit of MPJ’s lessons in that they seek to engage students in meaningful mathematics, and aspire to teach them high-level cognitive skills. Dan’s methods, though, have a unique twist that challenges teachers’ thinking. I know he has given me a great deal to think on. I hope he does here for you as well.

MPJ
From what we read on your blog, you are about two things: 3-Act Lessons and the Ladder of Abstraction. Let’s start with 3-Act Lessons. Give us the gist of what they are and why they are an effective tool in teaching math.

Dan Meyer
We make huge promises to kids about the power of math in their world. But then we give them these problems that don’t do justice to that power or to the world they live in. Three-act math uses photos and videos to show students a more faithful reproduction of their world and a more faithful reproduction of the practices of applied mathematicians.

MPJ
MPJ has preached for over a decade the need to have students create their own mathematical models (abstract generalizations). Describe your Ladder of Abstraction and how it applies to teaching mathematics.

Dan Meyer
The process of abstraction is extremely powerful and also not something I understood intuitively until I was a long way out of my secondary math education. Basically, whenever we turn the world into a table or an equation or a graph, we LOSE something. People don’t run at a constant rate. The Earth isn’t a perfect sphere. But we abstract a runner into a linear equation and the Earth into a sphere because those abstractions are perfect ENOUGH to help us answer questions. That’s an important part of modeling. Asking, “Is this model perfect enough?”

MPJ
With so many teachers reliant on teaching from the textbook, do you have any ideas on how to get these practices used more regularly in classrooms?

Dan Meyer
I tell teachers what I tell myself: whatever you’re going to teach today, whether it’s pure math or applied math, make sure students have some NEED for it. A better need than “I don’t want to fail this class.” And I offer them techniques for provoking that need. I also offer teachers a homework assignment, an exercise like push ups, to get better at provoking that kind of need: take a photo or a short video and ask people what questions they have about it, if any. If they don’t have any questions, retake the photo or video in a way that provokes more questions. That homework assignment has been incredibly helpful in my own growth.

MPJ
How well do your theories mesh with what is coming down the pike as the Common Core?

Dan Meyer
The modeling practice of the CCSS gets focused treatment in high school. I encourage all of your readers to study high school modeling (it’s only two pages) and ask themselves, “Are the ‘real-world’ problems I assign preparing students to clear this high bar?” Then Google “three-act math” and see if my work can help.

MPJ
What do you intend to prove with your PhD research?

Dan Meyer
I’d like to understand how any or all of this translates to online education.

# Dr. Jon Star Speaks HOT Heresy

Dr. Jon Star, of Harvard University, gave a mathematically blasphemous speech at the 21st Century Mathematics Conference in Stockholm, Sweden last year. The presentation was titled, Neuroscience and Cognitive Psychology of Mathematics. His heretical statement was that mathematics does not teach higher order thinking skills; only the teaching of problem solving actually teaches problem solving. The Math Projects Journal has always preached the teaching of mathematical substance, or what is now commonly known as higher order thinking skills (HOTS), so we reached out to Dr. Star regarding his research.

The belief that just by learning math one gets critical thinking skills is also not well-supported by evidence.

MPJ
You must know that your claim stating that math does not inherently teach critical thinking is very unnerving to the math education community.

Dr. Star
Just to be clear about my goals in the Stockholm talk, I was trying to argue the following:

First, the belief that math plays some sort of special and relatively unique role (as a discipline) in promoting what you refer to as HOTS (Higher Order Thinking Skills) is not well-supported by evidence.

Second and related, the belief that just by learning/understanding math, one gets critical thinking skills as well (e.g., two for the price of one, without explicit or even implicit attention to developing HOTS), is also not well-supported by evidence. Certainly in some instances this does happen, but it does not appear to happen in any widespread way for ‘typical’ students.

And third, given that we do want students to develop HOTS, rather than expecting/hoping that these just emerge as a natural by-product of learning/understanding math, it is essential that we think about how to explicitly promote critical thinking and problem-solving in what we teach and how we teach math. With respect to this last point, arguably generations of math curriculum and pedagogy reformers have sought this same goal – teaching math such that higher order thinking skills develop. But evidence and intuition suggests that this is very hard to do. But certainly we should continue trying…

MPJ
Is it math, per se, that does not impart the HOTS, or is it the way we teach math that is inept in imparting these skills?

Dr. Star
I would say that both content and pedagogy are important, but it seems that pedagogy plays an especially important role. If we want students to be able to transfer knowledge to domains outside of math class – apply reasoning skills that worked in math class to other kinds of problems – it seems necessary to teach with such transfer goals in mind. There are many different (at times competing) pedagogical visions for how to teach math such that this kind of transfer is possible. Some feel that the best approach is to engage students in certain kinds of reasoning and communication that are believed to facilitate application of knowledge to novel situations, and others feel that a certain amount of practice in applying concepts and skills is necessary for future transfer. I can see potential merit in both of these approaches, although empirically there isn’t a lot of good evidence to point us in the right direction.

I would say that both content and pedagogy are important, but it seems that pedagogy plays an especially important role.

MPJ
The 8 Common Core Standards of Practice imply that habits of mind can be taught. In your view, do these practices have value?

Dr. Star
I think that the Common Core practice standards are admirable goals. However, as noted above, I think we are still struggling to determine the best ways to achieve these goals pedagogically.

MPJ
Anecdotally, educated people think, communicate and behave differently than uneducated people. I believe research bears this out as well. Is this then simply a non-associated correlation (people who already have educated traits get an education), or does a quality education truly transform an individual?

Dr. Star
Certainly some people do develop problem solving skills merely by learning math. Some of these people developed (or would have developed) both math understanding and HOTS even if they didn’t have a classroom or a teacher – they could have done so by themselves on a desert island, so to speak. Most people, though, definitely need math training to learn math content, and they need explicit instruction in critical thinking to develop higher order skills as well.

Most people, though, definitely need math training to learn math content, and they need explicit instruction in critical thinking to develop higher order skills as well.

MPJ
What advice do you have then for classroom teachers in the quest for teaching higher order thinking skills?

Dr. Star
Try to identify the places in your lessons where you hope students are developing higher order thinking skills, and consider ways that you can be more deliberate and explicit in your pursuit of and assessment of these goals. For example, ask your students about any broader connections they are making from the mathematical content of the lesson. Give students opportunities to apply what they have learned in a lesson to other mathematical and non-mathematical topics. Let students know what you mean by phrases such as “critical thinking”, “problem-solving”, and “logical thinking”; give students examples of what these practices look like as well as tasks that allow them to develop and experience these important competencies.

Let students know what you mean by phrases such as “critical thinking”, “problem-solving”, and “logical thinking”; give students examples of what these practices look like as well as tasks that allow them to develop and experience these important competencies.

**** Dr. Star may be reached at jon_star@harvard.edu
****For more of Jon Star’s thoughts on Math Education, see this Scholastic video on YouTube.

# Enduring Cosmic Power

Today, I received one of the greatest compliments from a former student in the following post:

This one-time middle schooler, enrolled in my high school Geometry class seven years ago, is referring to my consistent overt effort to have students understand and appreciate the true potential of math. Ultimate Cosmic Power in an Itty Bitty Thinking Space goes beyond the cognitive easter egg hunt that the “answer getting” routine too often reduces math to. I don’t know how to respond to knowing that most students think it is stupid or crazy at the time, but Jorge’s words of enduring impact have me smiling today.

Hey Georgie,
I remember you as a bright, happy young person. Though I am pleasantly surprised that my teaching has lasted with you, I am not surprised that you have chosen a mathematical career path. Build a new world, young engineer!

# Graph of the Week (New Site)

I am getting the word out on this awesome site: Turner’s Graph of the Week. My friend Kelly Turner did a presentation at the Great San Diego Math Conference last spring and I loved her idea of having students analyze graphs from magazines and newspapers. These are mostly one-quadrant graphs with a natural context. This ties in directly to the Common Core’s call for applications and for reading non-fictional text. I was so impressed that I encouraged her to go public with the idea. I am serving as her megaphone.

Kelly does this activity once a week with her students, thus the name. The site offers several features:

• Graphs. You don’t have to find your own. Kelly has already posted 12, and will post more on the GOWS page of the site as the school year progresses.
• Submissions. If you like the activity and have graphs of your own that would serve others, email them to turner_k@auhsd.us. Kelly will screen the submissions and build the online collection.
• Templates.  There is a generic worksheet template with writing prompts to guide students in reading, interpreting and analyzing the graphs.
• Samples. On the home page Kelly will offer the graph that she is currently using for the week. Directly below that will be a student sample of the previous week’s graph.

Try it. If you like it, share with your colleagues. The same graph can be used at multiple course levels, with the level of questioning being adjusted to the level of students.

Kelly thanks for all the work on this. You have made teachers’ work easier and students’ education better.

# TMC13 Session Recaps

In my last post, I summarized the overall experience of Twitter Math Camp 2013 at Drexel University. Following is my recap of the sessions that I attended. This conference was unique in that I learned something significant in each session.

Geometry Break-Out #1, Megan Hayes-Golding @mgolding, GA & Tina Cardone @crstn85, MA

After the opening greeting, the first morning session was a choice of break-outs according to course (Algebra 1, Geometry, Stats etc). These were intended to be open-ended discussion/work sessions. In the Geometry session, there was an overwhelming need by the group to wrap their heads around the Common Core Geometry Standards. Megan & Tina wisely went with the flow, and had us jigsaw the standards in pairs and share out. It was enormously helpful for everyone. I was already very familiar with the standards, but I still learned something about the CC standards on constructions. Specifically, the standards not only call for the four basic constructions plus those involving parallel and perpendicular lines, but the students are expected to construct a square, equilateral triangle, and hexagon as well. This was time well spent, with the bonus of getting to know Edmund Harriss @Gelada, Jessica @algebrainiac1 and StephReilly @reilly1041.

Through out the weekend, I had extended conversations with Edmund from which I learned a great deal. Mostly because Edmund is a math professor and as he spoke of his work with the mathematics of tiling patterns, I felt my IQ rise just by listening to him. Much of our discussions centered around the American education, though. Edmund had an interesting perspective, because while he teaches at the University of Arkansas and also leads special math programs for gifted children, Edmund is British. From that experience, he had a great deal to share about “how to run standards based education correctly.” I hope he blogs about that soon.

“I Notice & I Wonder,” Max Ray @maxmathforum, PA

Max Ray is the “Professional Collaboration Facilitator” at the Math Forum at Drexel. In essence, he teaches teachers how to teach problem-solving. I had heard before of starting lessons with “What do you notice? What do you wonder?” This phrase, which was originated by Annie Fetter @MFAnnie, is intended to initiate student thinking on a rich and robust task. That seemed pretty simple, so I wasn’t anticipating much new learning here … Boy, was I wrong! Max started with a picture of 3 glasses and the phrase “What do you notice? What do you wonder?”

We were asked to ponder for a moment, then share our thoughts with our neighbors. (Think-Pair-Share).  “I notice they have different shapes. I wonder if they have the same volume. What kind of drinks go in each one?” Then he posted the picture of 4 graphs, and again posed the same questions: “What do you notice? What do you wonder?” The ensuing discussion resulted in everything from “I notice the graphs are different colors” to “I wonder if the graphs correlate with the filling of the glasses.” The thing that I noticed about this whole activity is that Max let us mull this over without offering a single number or formula. Nor did he offer a single answer to any of our wonderings. Two pictures and two questions occupied us for 15 minutes. In the era of rushing through content it was wonderful to be reminded that mathematics starts with an observation and a question. Speaking of questions, my group wondered what glass shape would correlate to the fourth graph… while Max stood at the front of room silently smiling.

“Practicing the 5 Practices,” Christopher Danielson @Trianglemancsd, MN

Christopher Danielson is a professor of mathematics at Normandale Community College and also teaches methods courses for elementary school teachers. He shared the research published in Five Practices for Orchestrating Productive Mathematics Discussions. In summary, the 5 Practices are:

Anticipating, during planning, student responses to the lesson prompt
Monitoring students repsonses during the lesson activity
Selecting which student responses are to be discussed publicly
Sequencing those student responses chosen
Connecting the responses to each other and to the mathematical ideas

Chris emphasized that the first and last of these are the two most troublesome for teachers. Chris modeled all these principles by conducting a math lesson on fractions. He knew what the issues would be with the context. He called us specifically by name to present our responses in an order that allowed the discussion to develop from simple ideas to more complex. I was particularly impressed on how he asked us to compare and contrast the various strategies. This is where I personally saw that I needed to bolster my own efforts on connecting ideas in own my class discussions. I walked away with the understanding that while any class discussion is better than none, there truly is an art form to doing class discussion right.

“5 Ways to Boost Engagement,” John Berray, @johnberray, CA

I have to say that the number one way to boost engagement is to teach like John Berray. The joy that he has for the material and for his students was just bursting out of him. With that said, John had 5 other ideas on increasing engagement:

1) Turn the Mundane on its ear
2) Jump on the timely
3) Bring in the outside world
4) Unlikely objects arouse wonder
5) Spill some paint

Translation: 1) Make it fun, 2)Tie math to current events, 3) Use the internet, particularly video, 4) Be goofy, 5) Connect the material to kid’s lives.

Geometry Break-Out #2

Our group reconvened with a few new people joining in. It was especially nice to See Peg Cagle @pegcagle after so many years. While the first day was a working session, this day was all about discussion. The group really wanted to talk about how to teach all the standards we listed in the previous sessions, while instilling the CCSS Practices. Teachers shared their various ideas, experiences and techniques. There was also a question on grading practices that revealed the dark side of the MathTwitterBlogosphere … We can be a very opinionated bunch. The hot topic for us was standards based grading. This turned out to be a benefit to the new teachers in the room or to old teachers with open minds, because quite a variety of ideas and positions were shared. It was an engrossing conversation, because no matter the positions taken, they were all shared with a passion for teaching students rich mathematics. The end of session came way to soon.

“Still Keeping it Real,” Karim Kai Ani & Team Mathalicious, @Mathalicious, VA

Mathalicious offers engaging, innovative math lessons with a focus on “real-world” applications. Karim @karimkai led us through two Mathalicious lessons that were solidly based in mathematics and loads of fun. The first, Datelines, tied the age of potential dates to systems of inequalities. The age gap on a date becomes less of an issue as people get older. For example, a 24-year old dating a 20-year old is less awkward than the 20-year old dating a 16-year old. This is an engaging topic for teenagers that Mathalicious sets to a graph and poses critical questions according to a given rule on dating ages. Like I said … solid. The second lesson, Prisn, used Venn diagrams to analyze the probability of being wrongfully flagged by the governments PRISM program for mining data. This lesson was about as relevant as any can get. It allowed for rich non-partisan conversation on how much error the public will accept. As I told Karim, these lessons are sexy, but have a lot of substance. At the conclusion, he generously gave the TMC participants a free trial subscription to Mathalicious. I intend on checking out more of their work.

“Getting Students to Think Mathematically in Cooperative Groups,” Lani horn, @tchmathculture, TN

This one was very special for me, because Dr. Ilana Horn was such an influence on the teacher collaboration model that we have implemented at my high school for the last 9 years. Back in 2004, I was about to be the Math Department Chair for a new high school and was speaking with Jo Boaler about collab models for teachers. She told me that the person to contact was Lani Horn at the University of Washington (She is now at Vanderbilt in Tennessee). A week later, I happened to be vacationing in Seattle, and Lani was kind enough to give up time to a stranger and talk about her doctoral research. She was gracious as well as knowledgeable.

So I was excited to see her again and share how her information helped lead my crew back home to be one of the highest performing schools in the county. She was pleased to hear the news. Her session this time was on student rather than teacher collaboration. The specific model she shared is known as Complex Instruction (CI), in which students are grouped heterogeneously, with intentional methods to have all students participate. The focus of Lani’s session was on how academic status affects student engagement during group work. She was very intentional in telling us that participation is hindered by this perceived status about smartness, which is too often defined in math class as “quick and accurate.” To help make it safe for everyone to participate, the teacher needs to redefine smartness by acknowledging and rewarding “good questions, making connections, representing ideas clearly, explaining logically, or extending an idea.” Lani shared a video of a group of students working on a math problem, and asked us our thoughts regarding each students level of participation. She also asked us to analyze the teachers interaction and prompted us for alternative responses. This analysis of the work done by each student debunked the conventional wisdom that non-participatory children are lazy, stupid or shy. I had learned as much from Lani Horn on this day as I did in our first encounter.

Due to another engagement, I had to fly home early from the conference so I did not get a chance to attend the last session on Friday or any on Saturday. I heard I missed some great stuff,  which I don’t doubt.

# Rich and Robust

I recently had the pleasure of learning from Tim Kanold of Stevenson High School fame. I heard him speak on several occasions last fall, and he kept saying that we need to involve students in “rich and robust tasks.” He was addressing the Common Core‘s call to the Standards of Practice. These practices can be summarized by saying that the Common Core is demanding students to think and to communicate their thinking. This can’t get done by taking notes from an overhead and doing the odd problems in the textbook. It gets done by purposefully deciding that students are going to solve rich and robust problems rather than simply watch their teacher complete examples of algorithms.

There is still a time and place for direct instruction and guided practice; but that should not be the complete experience for students, which is what we unfortunately find in the vast majority of American classroom instruction. For quite sometime, MPJ has been producing what we hope to be rich and robust tasks. Due to the growth of the internet, the availability of such rich and robust tasks has expanded tremendously. There are many exotic islands of innovation among the seas of tradition, but the blogosphere has made these islands less remote. Below I have listed a few, alongside my paraphrasing of the some of the Common Core Practices. This is not a comprehensive list by any means, however, I encourage you to take a few minutes and peruse these lessons in order to get a quick taste of what I think Kanold means by Rich and Robust.

Listed here are some additional sites that offer rich and robust tasks. {Note: I will be happy to update this list with any reader-submitted links, subject to review.}

The activities listed above obviously are not your typical math lessons. For good or for bad, the mathematical frontier created by the experiences highlighted here would make for a far different academic education than the gauntlet of lectures that most of us remember from school.

Now, I am going to assume that while the thought of introducing these large-scale examples into one’s repertoire is exhilarating for many, it may be terrifying for some. Let me ease those hearts by saying that rich and robust can be done on a much smaller scale. For example, we could simply ask the students: “Is x times x equal to two x or x squared. In other words, which of the following statements do you think is always true, if either: x·x = 2x  or x·x = x2?”

The CC Practices call for students to construct viable arguments and critique the reasoning of others. If your students stare back at you in silence with this question, then you will know why the Common Core Practices are so needed. If you answer the question for them, then they will watch you participate in a rich and robust activity, while they again participate in mundane note taking. For those that believe that this prompt is too elementary for any course above Algebra, let me assure you that it is not. I posed this very question to my International Baccalaureate students. A handful chose incorrectly, while several “could not remember.” When I asked the rest of the class, which was comprised of some of the brightest seniors on campus, no one could justify their correct answer. The best I got was that they “remember someone teaching us that once.” A simple question turned out to be far more rich and robust than it should have been, but it was a worthwhile day. {Try this one and get back to me.}

I must say here that I am grateful for my math education; it was far better than not having one at all. However, admittedly it was not rich and robust. The question is: Will we make it so for our students? It will take a conscious decision on our part to give our students a different educational experience than most of us had. So ask yourselves: When was the last time that you immersed your students in a rich and robust task? When is the next one planned? Has the time between those two dates been far too long? Are we up to the rich and robust task of offering rich and robust tasks?