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 …

  • Start with informal investigation, then move to formal language.
  • Teacher facilitation is key.
  • Where to Learn more:

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 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
    5. Specific to discipline/grade level
  • 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:

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

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

NCSM logo 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.”
    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).
    Tasks First… Examine the tasks in your instructional materials (Higher/lower cognitive demand tasks)? Do ALL students have the opportunity to grapple with challenging tasks. Examine the tasks in your assessments (Higher/lower cognitive demand tasks)?
    … 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!

Titanic Two-Way Frequency Tables

This outstanding lesson on two-way tables was born out of three sources: a fantastic resource from Illustrative Math, the award-winning movie, Titanic, and an empowering presentation by Chase Orton (@mathgeek76 ). At CMC-North, 2015, Chase shared the “Lived or Died” two-way table, which does a fantastic job of hooking students and introducing them to the concept of determining dependence from categorical data; plus he offered some follow-up activities. While Chase’s lessons claim to be for 8th grade standards, Illustrative Math provides lesson plans and student prompts for the high standards regarding two-way tables. The movie clips help frame the historical  contexts and the mathematical questions to be answered with the tables.

Here is how I put all these elements together into one lesson.

Titanic Day 1

Dual Objective: Reason Quantitatively in using two-way tables to  determine probabilities of survival.

1) 320 is what percent of 710?
2) What do you Notice and Wonder about this following 2-way frequency table?

The obvious notice was the number of men that died. The common wonder was in regards to the event that caused so many deaths. The conjectures ranged from disease to war. After the enthusiastic discussion, I shared this video clip from the movie, Titanic.

So now we know… the disastrous event represented in the data table was the sinking of the Titanic. I disseminated the handout that offered a new two-way table relating to that fatal day.

The students were very curious about why the total number of survivors and deaths were so significantly different from the previous table. We all concluded that this table strictly showed passenger data, and the other one must have included data for the crew as well.

The prompts today focused on the basic skill of calculating probabilities from the table, specifically, the probabilities of event A, event B, events A and B, event A given B. Namely,

P(A), P(B), P(A and B) and P(A|B)

For example,

  • If one of the passengers is randomly selected, what is the probability that this passenger was in first class?
  • If one of the passengers is randomly selected, what is the probability that this passenger survived?
  •  If one of the passengers is randomly selected, what is the probability that this passenger was in first class and survived?
  • If one of the passengers is randomly selected from the first class passengers, what is the probability that this passenger survived? (That is, what is the probability that the passenger survived, given that this passenger was in first class?)

Today was a great prelude to the next lesson on determining dependence.

Titanic Day 2

Dual Objective: Reason Abstractly in using two-way tables to  determine dependence between passenger class and survival on the Titanic.

I noted to the students that our target was very similar to yesterday’s in that we were still dealing with the same content (2-way tables), with the small change being in the Mathematical Practice. Today we were changing the Reasoning Quantitatively to Reasoning Abstractly. This meant that yesterday we focused on thinking about numbers, and today we were going to be thinking about relationships.

Warm-up: (yesterday’s skill and notation)
1) What is the probability that a passenger was in second class? P(A)
2) What is the probability that a passenger survived? P(B)
3) What is the probability that a passenger was in second class and survived? P(A and B)
4) What is the probability that a second class passenger survived? P(A|B)

After reinforcing yesterday’s lesson, I showed this second video clip from the movie Titanic.

This scene is an artistic interpretation of the treatment of third class passengers, but is this attitude towards people in steerage historically accurate? Did the wealthy receive preferential treatment in evacuating the ship? Even though we were not there over 100 years ago, we can still determine the truth, because we have data!

So, I disseminated handout #2 that offered the same two-way table as yesterday. The major idea here is to determine if the chance of survival of all passengers was different from the chance of survival for a first class or a third class passenger.  In other words, was the probability of survival dependent or independent of passenger class. Namely, is P(A|B) = P(A).

Since the general chance of surviving was 38%, but the probability of survival for first class passengers improved to 62% , while that for third class diminished to 25%, the students concluded that survival was indeed dependent upon class.

Then it was time for the students apply what we just learned to another question of dependence aboard the Titanic. I showed this video clip.

Again, we needed to test the validity of this artistic interpretation of history with data … and our new skills. (handout #3) Was it truly “women and children first?”

A colleague of mine, Kristan Morales (@KristanMorales1), did this lesson and asked the students to offer questions that can be asked from the table, and collected the responses in a google doc. Here is a small sampling of the student generated questions:

Titanic Morales

Day 3 … Chores & Curfews

Dual Objective: Use Structure in creating a two-way table and use the table to Judge the Validity of an Argument regarding dependence.

On the third day, I had the students practice with some more relevant (and less traumatic) contexts that Chase provided in his session. I loved how Chores & Curfews tied in Venn Diagrams to the probability conversation while requiring students to complete their own 2-way table.

All the work involved here empowered the students with the tools and information needed to determine the validity of the claim that those with chores were more likely to have a curfew.

The following are the various materials available for this lesson, from…
Chase Orton: Titanic Plus
Illustrative Math: Titanic 1, Titanic 2, Titanic 3
Chris Shore (Me): Titanic AllChores & Curfews

Personal Note: The executive producer of TitanicJohn Landau, was my pledge father in my fraternity at USC.

Bumping Airlines

Out of 615 million airline passengers last year, half a million were bumped from flights. 9 out of 10 of those were voluntary. What percentage of all booked passengers were involuntarily bumped from a flight?

Yes, I had fun at the expense of United Airlines’ most recent viral embarrassment, but I had two serious questions that I needed answered:

  1. Should I change my air travel habits?
  2. How many of my Algebra 2 students could correctly answer this question?

I had my class answer both questions for me. I started class by handing them the prompt and this now famous video clip:

United Air Clip

I then shared what I learned about the law in regards to this incident. United Airlines and law enforcement officials were legally in the right to remove the passenger from the plane. When overbooked, an airline has the right to randomly bump passengers, but they must first offer an adequate incentive for volunteers, which United did. These regulations are in the contract rules that we all agree to, but never read, when buying an airline ticket. The law also states that any passenger must comply with directions given by airline personnel or law enforcement officers. Since the unfortunate gentleman on the plane resisted the directions of the authorities, the airline and the police had the legal right to forcibly remove him.

It was the third part of the law, however, that was the most disconcerting for me. If an airline involuntarily bumps you, they must guarantee your arrival at your intended destination within 24 hours. But that is not good enough for me. I often travel to places where I am expected to be working with teachers early the next morning. A 24-hour delay would be far too late. So my new burning question is: Should I leave greater leeway in time when I am traveling? That is what I needed to answer. The students helped me think through it.

Nine out of 10 voluntarily bumped means only 10% of the 500,000 bumped passengers, or 50,000 passengers were removed involuntarily. That 50,000 out of 615 million is a whopping 0.008% of all booked passengers last year. So what does that mean in terms of my flight habits? How many times would I have to fly in order to expect being bumped at least once? 0.008% of what number equals one (1 = 0.00008x)? It turns out that I would need to fly 12,500 times. In over 40 years of an active adult travel life, I would have to board a plane nearly every day of my life to expect this to happen. Of course, probably and possibility are utterly different, so I could be bumped on my next flight, but I am not ready to start adding an extra day to every travel trip for such a small chance.

So how did my students do with this calculation? My prediction of one student was an underestimation. Five actually calculated correctly, with 3 others getting close, showing appropriate work. Why was such a simple math topic (calculating percentage) such a challenge for a group of 15 & 16 year olds? In talking with the students, I came to realize that this was the classic case of “making sense of problems.” There were multiple layers in unpacking the prompt,  as well as the added layer of interpreting such a small fraction of a single percentage point and the need to make a decision based on that numerical interpretation.

It is noteworthy to reveal that I gave this problem to four adults. All four answered correctly (0.008%), and all four struggled to make sense of what was being asked.

So how do we get students more proficient at making sense of problems that require basic math? Easy. We pose those problems more often. Which I intend to do.

Confirming Answers with Graphing Software

I added two components to a lesson task on rational equations. The first was an idea called Hint Cards which I discussed in a previous post. The second one was having students use Desmos to confirm numerical answers that they found algebraically. This lesson is worth sharing, because of what I discovered that the students didn’t know how to do and what they learned from the activity.

The students had just completed the  Optimum Bait Company task. I gave them 5 minutes to check their answers with Desmos. Mathematically speaking, they needed to confirm that for the equation, , the following were true:

  • C(1000) = 4.45
  • C(4000) = 1.3, C(8000) = 0.775, C(12000) = 0.6 and C(42000) = 0.26
  • The horizontal asymptote is y = 0.25
  • C(5600) = 1

I was struck by how utterly stumped they were on using a graph to check these answers, even though they possessed all the prerequisite knowledge necessary. They understood the context of their answers (The average production cost of 1000 lures is $4.45). They knew how to find a y-value from a graph given the x-value (C(1000) = 4.45). They also were experienced at graphing equations with Desmos (y = (4200 + 0.25x)/x), and they knew how to establish a domain and range for axis in context. While my students had each of these four connected skills, they were missing the connection between them all, so I embraced the teachable moment.

I started with the basic idea that the equation we need to graph was the one for the average cost:  C(x) = (4200 + 0.25x)/x. A major problem faced them, though, when they first graphed this equation… they couldn’t see the rational function in the default window.

Bait Desmos Blank

So my next move with the class, was to use the numeric results to determine the domain and range. The range was simple since we all of the monetary answers were between 0 and $5.00. The domain needed a bit more discussion because the one value of 42000 lures compressed the graph so significantly that the class thought it better to leave it out of the visible domain, so we agreed upon 0 < x < 13000.

Bait Desmos Window

From here I could have just traced with finger along the screen to show where a point with a value of x would be located on the curve, but I wanted to tie in the writing of horizontal and vertical equations, and solutions of systems. Therefore, I had the students enter the additional equation of x = 1000, and click on the point of intersection.

Bait Desmos 1K

The students were getting happier and more confident so we kept rolling by entering a table with the additional increasing values of x, representing the number of lures.

Bait Desmos Table

The table confirmed the students answers and supported their common response that the values were approaching a limit of 25 cents. This appealed to their sense of context that the cost per lure could not drop below the original 25 cents per lure. That made for an easy connection to the finding the horizontal asymptote of the rational function for which the degree of the numerator and denominator are equal, which in this case was also 0.25. So we graphed this as an asymptote. It turns out that it was easier to see how closely the curve approached the asymptote if we temporarily increased our visible domain to the 42000 lures.

Bait Desmos Asymptote

Finally, we entered the equation y = 1 in order to see the number of lures required to drive the average cost down to $1 a lure.

Bait Desmos 1 Dollar

This serendipitous exercise was amazingly productive for reinforcing understanding of graphing in general, as well as the connection between numerical values, algebraic formulas and context.

Hint Cards

Hint CardI added two components to a lesson task on rational equations.The first was an idea called, Hint Cards, shared by Michael Pershan at Shadow Con at NCTM 2015.  The second, which I will discuss in another post, was having students use Desmos to confirm answers that they found algebraically. There were so many surprising positives from this lesson that I have to share.

As always, lesson awesomeness starts with a good task. The class had studied the simplifying, solving and graphing of rational functions. It was time to write and apply them. My school’s Algebra 2 team decided on a common task titled Optimum Bait Company. I’m not sure where the task came from, but it offered the following context followed by six prompts.

My brother Matt owns Optimum Bait Company. Optimum Bait Company manufactures fishing lures. The monthly cost to run the factory is $4200 and the cost of producing each lure is an additional $0.25 per lure.

  1. If he produces 1000 lures in one month, what is the average production cost per lure?
  2. Create a function, C(x), that models the average production cost per lure.
  3. Calculate the average production cost per lure if he produces 4000 lures in one month? 8000 lures? 12000 lures? 420000 lures?
  4. As he produces more lures what price does the average cost of production approach? Why?
  5. If he wants the average cost of production to be $1, how many lures would he have to produce in one month?
  6. If he wants to make a profit of at least $4000 per month, what is the minimum number of lures he would have to produce if he sells every lure he produces for $4?

I was thinking that the students would need a lot of help on this, so I created a set of six Hint Cards. Each card gave some assistance for one of the prompts.

Front of Card

Back of Card

#1: Average Cost of 100 lures

Average = Total Cost/Total Number

#2: Create C(x)

Let x = number of lures

#3: Average Cost per Lure

C(4000) = (4200 + 0.25(4000))/4000

#4: Limit of Average Cost

The Ratio of the Leading Coefficients

#5: Average Cost of $1

C(x) = 1, instead of   x = 1

#6: Profit of $4000

Profit = Income – Expenses

As an incentive, I announced the following scoring system.

  • Like all other tasks, this will be worth 5 points.
  • There are 6 prompts. Every wrong answer to a prompt costs a point.
  • There are also 6 hints. Every hint used costs a point.
  • Yes, that means you either have one free pass on a wrong answer, or a free hint.
  • The only thing that you may ask of the teacher is for a hint card to a specific prompt.
  • 30 minutes will be allotted to complete the task.

I was pleasantly surprised how little they needed or wanted help. Many groups didn’t even take advantage of their free hint. In fact, for all of eight groups, I gave out only seven Hint Cards… total. The most common hints asked for were for #1 and #5. The most commonly missed questions were the last two. I suspect that if I had given more time on the task, students may have ask for more hints and given more effort on what I consider to be the hardest questions for the task. That’s a lesson for next time; there will definitely be a next time because of the benefits that resulted from the Hint Cards technique:

  • The time crunch spurred a hyper-focus in the students.
  • The level and intensity of the student discourse was heightened tremendously.
  • A common dynamic was having one student raise a hand for a hint, while another group member protested, “we don’t need it, yet.”
  • The average score on the task was 4.2 out of 5.

Hint Cards delivered a terrific learning experience for my students. One that I will be sure to give them again.

The 6 C’s of Claims-Based Grading

6-cs-picFor the past three years, I have been using a claims-based grading system in my math classes. Rather than using the traditional categories of Tests, Quizzes and Homework, or the standards-based categories such as A.REI.1 or Solving Linear Equations, my grade book is now comprised of the following claims-based categories that I refer to as the 6 C’s:

  • Concepts & Procedures
  • Critical Thinking
  • Communicating Reasoning
  • Constructing Models
  • Creativity
  • Collaboration

I call these “claims” because the first four of the six draw directly from my state of California’s testing system, The Smart Balanced Assessment Consortium. The SBAC exams and reports are based on four Claims for Mathematics Summative Assessment:


I figured that since the signers of my paycheck now expect me to impart these four abilities to students, that maybe my grade book should reflect these capacities as well.

I also know that the famous 4 C’s of 21st Century Learning are important skills for students to possess when they graduate our schools, therefore I thought that should be reflected in my grade book as well.


Two of these 21st Century C’s overlap with the SBAC claims. By choosing the phrasing “critical thinking” over “problem solving” and tweaking the SBAC phrase of Modeling and Data Analysis just a bit, I had my own 6 C’s of Claims-Based Grading.


This new grading system has demonstrated terrific benefits in the classroom for both my students and myself…

Student Focus & Reflection

Having the picture shown above displayed as a poster at the front of the classroom serves as a constant reminder to students as to why they are in the course. There is much more to math the just busting out algorithms. If they never have to solve an equation in their adult life, hopefully, they will understand the mathematical principles that they hear about in the news, be able to think and communicate in a quantitatively manner, interpret data and represent the story that the numbers tell, solve problems creatively and work collaborative to meet a goal.  Claims-Based grading keeps these ultimate purposes front and center in the students’ minds.

My students also have a grade sheet that reflects the 6 C’s on which they record the scores they received on each assignment. Any given assignment may have more than one score on it, much like what is done with standards-based grading, with each score being based on a 5-Point Rubric (to be shared in a future post). In other words, after each assignment, students are required to look at how they performed in terms of, say, critical thinking or constructing models, rather than studying for a test.

The portfolios in the class are structured around the 6 C’s as well, with the first six of the eight sections being the 6 C’s themselves. After each assignment is recorded, it gets filed in their portfolio in one of the sections that it was graded on. For example, if an assignment was scored on Communicating Reasoning and Creativity, then the student gets to choose into which of those two sections the assignment will be placed.


While a Traditional grading system focuses student attention on study habits, and Standards-Based grading focuses them on specific skills, Claims-Based grading focuses them on broader capacities that will serve them well as adults.

Teacher Focus &  Reflection

The greatest benefit of the Claims-Based grading system is how much it reminds me to teach and assess the capacities that I often forget. I naturally teach to conceptual understanding, critical thinking, communicating reasoning, and collaboration, but I need to be frequently nudged to present students with tasks that require them to construct models and create unique examples or solutions. For example, a group quiz will pose several claims-based problems on the same mathematical topic with a few cumulative questions as well.


The Collaboration grade is always a self-assessed grade by the group, with me holding the power to veto. Quite often, though, they accurately score themselves. This is not surprising since we score it according to the school-wide norms on collaboration.

Reflecting upon the results of the Claims-Based grading has great value to me also. Take my end-of-semester results for one class, for example. (Note, there appears to be a large number of assignments, but remember that each assignment may have multiple scores, like the quiz example above.)


With the exception of the collaboration grade, the scores appear to be fairly consistent. This is interesting since individual students do not show this consistency. They usually have a claim or two that lags the others. The numbers that give me the most pause are the number of items. The few number of collaboration scores is not a concern, because most of the assessments are individual anyway. However, I am assessing procedures twice as much as critical thinking, three times as much as communicating reasoning and constructing models, and five times as much as creativity. I’m not convinced this is an issue, but I’m not convinced that it is not one either.

There is another interesting phenomenon that has me reflecting on my practices. The final exam scores are far less correlated under this system than with my traditional system. In previous years, I would have only a handful students whose final exam score was different from their classroom grade significantly enough to raise or lower their course grade, and most of those would be an improvement in the grade. Under my new system, there is about a 45% volatility. That means that nearly half of the students score differently enough to change their course grade, with the number being split between raising or lowering the grade. I think this is because the district finals are so heavy on the procedural side, with absolutely no questions addressing modeling or creativity. Students who are strong or weak in the Concepts & Procedures category will then see a gap between their course grade and the final exam grade. I am keeping a careful eye on this dynamic as I move forward with the new grading system.

Moving Forward

For all the reasons that I have shared, I will be keeping this Claims-Based grading practice for a while. l see myself adjusting the system less, and using it to improve my instruction more.

Future Posts on Claims-Based Grading
  • The Claims-Based Grade Book
  • The 5-point Rubric
  • Value-Based Grading


The 10% Challenge

leinwandI’ve heard Steve Leinwand say that it is unprofessional to ask teachers to change more than 10% a year. It is also unprofessional to ask them to change less than 10% a year.

I love this thought that we need to always be growing as professionals, but that our growth needs to be realistic and sustainable. However, I’m also challenged by what 10% change looks like, especially if I present this idea to my fellow teachers.

10-percentHow do you quantify professional growth?
How can you see this 10% change?

Then it struck me. 10% equals one-tenth, which is one out of every ten school days. That means Steve’s 10% is calling for us to try something new once every two weeks. That seems very doable for everyone. Imagine what a math department would like a year from now if every teacher tried something new and effective every two weeks. That would be a total of 18-20 days, or about a entire month of innovative instruction for each teacher. That sounds, realistic, sustainable and exciting.

Let’s all embrace Steve’s 10% Challenge.



Recap: Greater San Diego

logo-gsdmcThe Greater San Diego Math Council resurrected its annual conference. After a two year hiatus, Jason Slowbe, Sean Nank and their Council colleagues did miraculous work to bring GSDMC 2017 to life. This Glorious Day was worth all their efforts.

Opening Session (Four Bursts)
Rather than one keynote speaker, four presenters gave brief talks.

Observe Me
pic-kaplinskyRobert Kaplinsky (@RobertKaplinsky),
Downey USD, CA

Robert made two strong points:
1) The #ObserveMe practice, which calls for teachers to invite others to observe them. The key here is that very specific feedback is called for from colleagues.
2) The need for teachers to gain new perspective. Robert shared the story of Febreze. It is a very effective product that initially had a tough time selling, because people were nose blind; in other words, they did not realize how badly their houses smelled. Similarly, teachers will not buy into professional development until they recognize the need for change.  Therefore, we really need to do the work on changing teachers’ perspectives on the results of their practices.

Music Cues
pic-matt-vMatt Vaudrey (@MrVaudrey),
Bonita USD, The Classroom Chef

Matt is well-known for his use of Music Cues to save on transition time in the classroom. In fact, he showed how as much as 21 hours of instruction time a year (a whole month of school!) can be saved with the use of these cues. In my own class, I personally use four of the cues that Matt offers in his Google folder.

Social Justice in Math Education
pic-susie-hSusie Hakansson (@SusieKakansson),

Susie told the story of “Carol” and all the barriers to accessing rigorous math courses that she confronted as an Asian girl. Then she revealed that “Carol” was really herself and the experience she had growing up in the American school system. She called for more equity in access for all students, particularly in STEM courses.  “Don’t let test scores, skin color, or adults low expectations to prevent students from taking rigorous math courses.”

The Converging Future of Math and Computer Science
pic-pierre-bPierre Bierre (@pierrebierre),

Pierre  drew our attention to the growing number of computer Science courses being offered on high school campuses. Pierre went on to also share how programming can be a terrific problem solving tool in math class. This was a good primer for the number of sessions at the conferences that dealt with programming in math classes.

These quick presentations set a terrific tone for the conference experience.

Math Coaches Panel 
Brenda Heil (@BrendaHeil)
Bethany Schwappach (@MsSchwappach)
Chris Shore (Me) (@MathProjects)


Brenda, Bethany and I each offered up a 10 minute introduction of our roles as math coaches and a particular point of emphasis for math coaches to focus on. The rest of the session was open to questions fielded by our facilitator, Sean Nank.  The conversation was rich, and I learned a great deal from my panel colleagues.

brenda-slideBrenda is a TK, K-1 Coach in Escondido. Her biggest point was avoiding the badge of same… “If I work with a math coach, it means that I suck.” She instead insisted that math coaching training should be advertised as a resource for everyone.

Bethany is a Technology Coach for El Cajon.  She did something interesting by surveying math coaches, prior to the conference, with the question: “What are some of the greatest challenges Math Coaches are assigned to tackle?” The number one response was differentiating professional development for teachers. So Bethany offered two terrific ideas. The first was a Badge System for online “Anywhere, Anytime” PD, much like the structure of online mastery courses for students. The other was the promotion of omnipresent communication of the math coaching program to teachers.



I am a Secondary Math Coach in Temecula.  I declared that a math coach’s job is all about relationships as I shared out how many people I deal with to my south (teachers I serve), my east-west (coaching colleagues) and my north (administration). Because of this, the most important question to ask any of them is “How may I best serve you?”


I also offered three axioms that I believe all math coaches should base their work on. Each of them are quotes from famous researchers.

  • Axiom #1, Dr. William Schmidt, University of Michigan: The greatest determining factor in the quality of the education that students receive is the decisions that the teachers make on a daily basis.
  • Axiom #2, Dr. Kenneth Leithwood, University of Toronto: Indeed, there are virtually no documented instances of troubled schools being turned around without intervention by a powerful leader.
  • Axiom #3, Dr. Maggie McGatha, University of Louisville: The meta-research shows that math coaches are effective. We see small bumps in years 1 & 2, and large spikes in years 3 & 4.

The second one seem to resonate with this crowd.


From the questions and conversation I learned that …

  • …no two math coaching job descriptions are alike. Everyone’s daily routine was unique, but we all had a common goal… improve classroom math instruction.
  • … most math coaches are tossed into the position with very little support and training. Everyone, including administrators, deem this job important, but seem to be figuring it out as they go along. It was awesome to discover that San Diego County offers math coaching training. This is an idea that should spread to other counties as well.
  • … everyone is optimistic. Math coaches acknowledge that education has a long was to go in improving math instruction, but that we have all seen significant progress despite the challenges.

Clothesline: Algebra, Geometry & Statistics
Daniel Luevanos  (@DanLuevanos) &
Chris Shore (Me) (@MathProjects)

I loved presenting with Daniel. He is a Clothesline Math enthusiast who has developed some terrific ideas, particularly on systems of equations.

We demonstrated fractions, algebraic expressions, linear systems, solving multi-step equations, vertical angles, special right triangles and statistics (average, range, standard deviation).

pic-twitter-clotheslne-gsdmcMy favorite moment was during the Call to Action when Daniel challenged the teachers to use the Clothesline to enhance their own understanding of mathematics. So I surveyed the room by asking “How many of you today learned something about mathematics itself, not just the teaching of it?” Ninety percent of the room raised their hand!


21st Century Conference Ideas
I also want to give a quick shout-out to the GSDMC President, pic-slowbeJason Slowbe, and the rest of the Council for their willingness to experimentation with some new conference formats:

  1. Opening Session Burst: Instead of one keynote speaker, four presenters gave brief presentations within the same hour as the MC greeting.
  2. Genius Bars: Presenters were made available outside of their sessions for participants to meet and ask questions.
  3. Panel Sessions: 3-4 panelists share brief introductions and presentations (15 min), then the remaining hour was open to question by the audience.
  4. Working Lunches: People received their box lunch (part of the registration fee) and then were allowed to sit in the session rooms. Many of these rooms had exhibit presentations.
  5. pic-philippClosing Session Reflection and Evaluations: Closing speaker. Randy Phillip (@rphilipp), asked us all to reflect on one idea that we would take back to our classrooms. After giving us time to ponder, he asked for volunteers to share out publicly. It was an excellent way to have participants reflect on their conference experience and increase the chances of us committing to improve our instructional practices.

Hidden Figures’ Lessons for the Classroom

hf-all-threeEveryone in the theater applauded as the credits rolled at the end of the movie, Hidden Figures, and for good reason. It is an amazing, humorous, educational, inspiring and important movie. It is a film that every educator and math student should see. This true story of the contribution of three black women to NASA’s launching of John Glenn into orbit is full of so many positive messages about math, science, patriotism and social justice that I am compelled to share my perspective as a high school math teacher.  Here are my reflections from this incredible movie.

On Math
Math is More Than Computing.
Yes, a team of nearly twenty African-American women was known as “the computers,” because in the days before calculators, computation was done by hand. However, the movies’ three protagonists, Katherine Johnson, Dorothy Vaughn and Mary Jackson, were all given assignments for which the math went far beyond simple computation. They had to visualize geometrically, draw graphs, generate equations, assign units, assess another’s work, creatively program and, yes, solve lots of problems. Much of the work that was glorified in the movie was the application of mathematics, not calculation with mathematics. In fact, the director the Space Task Group, Al Harrison insisted:

“This isn’t about plugging in numbers, this is about inventing the math.”

Intellectual work is valuable.
Several times in the movie there was reference made to the “work” done by the various NASA personnel.

Dorothy reference the number of people needed to run the new IBM:  “We’re going to need a lot of manpower to program that beast. I can’t do it alone. My gals are ready. They can do the work.” *

Al Harrison in response to a politician’s inquiry: “That’s the math we don’t have yet, gentlemen. We’re working on it.”

Ruth (Katherine’s colleague) on Katherine’s last day on the Space Task Group: “You did good work around here.”

Al Harrison after John Glenn was returned safely to earth: “Nice work, Katherine.”

Dorothy instructing the other women in reference to Katherine re-computing John Glenn’s critical re-entry coordinates: “Alright, give her space. Let her work.”

hf-workThe work referenced here is the kind of work that educators are called to teach in the 21st Century classroom. We math teachers are currently implored to replace meaningless busy work with relevant intellectual work.

Math put humans into space.
Katherine upon being questioned about how she knows that one type of rocket is needed over another:

“What’s there tells the story if you read between the lines. The distance from launch to orbit is known. The Redstone mass is known. The hf-mathMercury Capsule weight is known. And the speeds are there in the data…. The numbers don’t lie.”

That says it all.

The movie got the math and the science right.
Hollywood has a track record for flunking math and science in movies, however, in this one, they earn a stellar score.

Katherine as an 8-year old child prodigy: “If the product of two terms is zero, then common sense says at least one of the two terms has to be zero to start with. So, if you move all the terms over to one side, you can put the quadratics into a form that can be factored, allowing that side of the equation to equal zero. Once you’ve done that, it’s pretty straight-forward from there…”

Stafford: “The Atlas Rocket can push us into orbit. It goes up. Delivers the capsule into an elliptical orbit. Earth’s gravity keeps pulling it, but it’s going so fast that it keeps missing the Earth – that’s how it stays in orbit.”

A+, Hollywood.

On Math Education
All need to be encouraged to check their work.
When Katherine Johnson was asked by her new boss to check the work of the lead engineer, Paul Stafford, Mr. Stafford balked. In response to his objection, Harrison gave a speech about the importance of the task, and that no one is above having his or her work checked.

“Do I need to remind everyone…that we are putting a human on top of a missile and shooting him into space? It’s never been done before. And because it’s never been done … everything we do between now and then is going to matter: it’s going to matter to their wives, their kids, I believe it’s going to matter to the whole damn country. So this Space Task Group will be as advertised. And America’s greatest engineering and scientific minds will not have a problem with having their work checked.”


Yes, those engineers had to check their work because the boss said so, but the boss also gave them the reason why… because getting the answer right is important. If NASA engineers needed to be reminded of this on occasion, then so do our math students.

Much of the math that launched John Glenn into space is taught in high school.
Yes, the characters in the movie mentioned things like the Frenet Frame and the Gram-Schmidt, but many of the terms used and the equations shown in various scenes would be recognized by students in high school math classes across America.

Paul Stafford: “We need to move from an elliptical orbit to a parabolic path.”hf-paul

Katherine: “On any given day, I analyze the manometer levels for air displacement, friction and velocity and compute over 10,000 calculations by cosine, square root and lately Analytic Geometry.”

Our math students should know that they are actually learning rocket science!

There is opportunity for everyone in STEM fields. (Science, Technology, Engineering and Mathematics)
The three women portrayed in Hidden Figures were specialists in three different STEM fields: Mathematics, Engineering, and Programming. These are all fields in which we have a shortage of American citizens earning a degree, to the point that much of the classified work in this country is being done by people who have citizenship from other countries. The STEM community sees this as an issue and would very much like to see more Americans pursuing careers in these fields. With the typical STEM job offering twice the annual earning of a non-STEM job, there is a huge opportunity for economic advancement for low-income students entering these vocations.  If they are not choosing this on their own, then we educators should be doing more to encourage and support their election of these endeavors.

On Equity
The oppressed are not victims.
Hidden Figures is very much a story about victory over oppression, not victims of oppression. The victory was achieved by the heroines changing themselves, changing others, and changing the system…

In order to adapt to economic change, we must improve ourselves.
When Dorothy Vaughn finds out that the new IBM computing machine means that the human computers will be obsolete, she not only makes moves to position herself well in the new age of computers, she encourages her friends to do the same.

“It’s not going to matter soon. This IBM’s going to put us all out of work… Only one thing to do: learn all we can. Make ourselves valuable. Somewhere down the line a human being’s going to have to hit the buttons… We have to know how to program it. Unless you’d rather be out of a job?”


Dorothy then goes to the public library and checks out a book on Fortran (one of the original programming languages). She even reads it aloud to her sons on the bus, as a mother’s lesson in overcoming adversity.

This message is extremely relevant in the political climate today. Many jobs are being lost to automation and changes in the global economy. The promise of today’s politicians to “bring those jobs back,” is equivalent to thinking that the emergence of the information age, represented in the film by the IBM machine, could have been prevented in order to maintain the human computing jobs. It is equally as silly to think that anyone today can stop the evolution of the job market. Instead, we educators should be teaching students, and ourselves, to do as Dorothy did, and adapt to the new 21st Century economic environment.

They were strong women not just smart women.
These three women were more than simply a mathematician, engineer and programmer; they were wives, mothers and daughters. Katherine Johnson was a single, widowed mother who had to raise and support three children. Mary Jackson was a married mother who held down a job and attended night school. It takes internal strength to balance that kind of life.


They were brave women not just strong, smart women.
The three heroines each had a moment in the story in which they spoke truth to power.  In the engineering lab, the courtroom, and the office.  In each case, their courage effected change.

Black men are not thugs.
The two primary black male roles in the movie, the husbands of Katherine and Mary, were not the gangsta and drug-addict that is too often the portrait of black men in movies. Jim Johnson and Levi Jackson were both strong, family-oriented men of solid character.


Prejudice and Discouragement are sometimes found in your backyard.
The women of Hidden Figures were not only dealing with racial bigotry, but they faced sexism as well, even from their own friends. At the first meeting with her future husband, Katherine’s suitor puts a huge foot in his own mouth:

Jim Johnson: “Aeronautics. Pretty heady stuff. They let women handle that kind of- … I was just surprised something so taxing…”


video clip

Levi Jackson (Mary’s Husband commenting on her desire to become a NASA engineer): “All I’m saying, don’t play a fool. I don’t want to see you get hurt. NASA’s never given you gals your due, having another degree won’t change that. Civil rights ain’t always civil.”

Both men came around to offer full support of their ladies’ dreams, after their wives stood strong to their convictions. Sometimes the battles for equity must be fought in our homes and communities, not only against “them.”

Prejudice is sometimes harder to see now.
We no longer have colored bathrooms, colored bus seats, colored drinking fountains or colored coffee pots, but we do have colored schools and even colored classrooms. We know that schools are just as segregated now, as before Brown vs. Board of Education. The inequity in funding and support for the black schools means segregation by opportunity, which is more criminal than segregation by race. The roster of my own class of “at-risk” students is 90% populated by students of color, while those same groups of students make up only 54% of the school population. When I brought this to the attention of the administration, they were genuinely unaware, but instantly concerned. Statistics like this, which exist on paper, are harder to see and less humiliating, but actually more dangerous than a “coloreds only” sign. Therefore, we educators need to be more vigilant in exposing these numbers and in changing the practices and policies that they represent.

The Powers That Be must be part of the change process.
As much as the three heroines are rightfully credited with impacting change in NASA’s practices regarding women and blacks, we need to also recognize those members of power structure that aided them in their cause: Space Task Force Director Al Harrison who gave Katherine access to classified data and high level meetings, Astronaut John Glenn who went out of way to greet the black female computers and insisted that Katherine do the calculations on his re-entry, Polish engineer Karl Zielinski who encouraged Mary to seek her engineering degree, IBM Technician Bill Calhoun who gave Dorothy the opportunity to program the new technology and her boss, Vivian Michael, who promoted her to Supervisor.

People of all ethnicity and gender can contribute.
This story was about more than whites and blacks sharing the same bathroom. It was about the talents and contributions of people of all backgrounds. Katherine makes this very point in the first meeting of her and her future husband.

Katherine: “So, yes…they let women do some things over at NASA, Mr. Johnson. But it’s not because we wear skirts…it’s because we wear glasses.”


Racial equality is pragmatic as well as moral.
This important story of Johnson, Jackson and Vaughn is about more than women or blacks receiving a fair shake. It is about how one of the crowning achievements of America may not have been accomplished without them.

Vivian Mitchell (Dorothy’s Supervisor):  “Seems like they’re gonna need a permanent team to feed that IBM.”

I don’t want to question Al Harrison’s sense of social justice, but his tearing down of the “Coloreds Only” bathroom sign was as much an action of practicality as it was a display of righteousness.

Harrison “Go wherever you damn well please. Preferably closer to your desk.”


His primary purpose was to get an American into orbit, not to get a black woman to urinate next a white one, but he saw that the path into space traveled through an integrated restroom. Katherine knew it also traveled through an integrated boardroom. She was being kept out of key meetings, because of her gender and race, when she knew that her work was being hindered by the locked door. She stated her case for inclusion to her boss, not on a basic of equality, but on a basis of practicality.

Katherine: “I cannot do my work effectively without having all of the data and all of the information as soon as it’s available. Indeed to be in that room, hearing what you hear.”

When Harrison has to stand for her presence in the meeting, he did not say, ‘We need more black women in these meetings.’ Instead, he claimed,

“This is Katherine Goble with our Trajectory and Launch Window Division. Her work is pertinent to today’s proceedings.”

hf-meetingOne of the strongest points of the movie is that prejudice not only harms the oppressed, but it hinders all of us. Equity does not only make America more fair, it makes America better.

On American Patriotism
Black history is American history.
The story of Hidden Figures is not the only story of African-American mathematicians and scientist who have made terrific contributions to our nation. In fact, those lists are long and distinguished:

America always struggles to live up its ideals, but those ideals are America.
The names of the two spacecraft highlighted in the film are named Freedom and Friendship. These names seem a bit ironic when contrasted to the social controversies of the time. Many blacks and women in the ‘60’s would not have considered America to be friendly or free, however, gender and racial equity made huge strides towards these American values during that time period.

Mary: “I’m a Negro woman. I’m not going to entertain the impossible.”

Zielinski (engineering colleague): “And I’m a Polish Jew whose parents died in a Nazi prison camp. Now I’m standing beneath a space ship that’s going to carry an astronaut to the stars. I think we can say, we’re living the impossible.”

Mary would be happy to know that over 50 years later, not only are women of all ethnicities being allowed to become engineers, they are being actively recruited for such a career, and we in the public school system are being charged with raising them up. Again, this is not a matter of equality; it is a matter of practicality. America needs more engineers, and any antiquated systems of injustice will keep us from achieving our technological potential, and from reaching our highest ideals of Friendship and Freedom.


* All Quotes from Hidden Figures Screenplay by Allison Schroeder and Theodore Melfi, May 12, 2015 (Based on the book Hidden Figures by Margot Shetterly).

Innovative math lessons you can use in your classroom today