All posts by Chris Shore

Creator of Lessons Trainer of Teachers Teacher of Kids Lover of Math

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.”

al-h-2

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?”

hf-women-computers

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.

hf-true-women

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 a gangsta and a 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.

hf-mary-husband

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…”

hf-stroll

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.
hf-john-glenn
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.”

hf-kathernine-hand-up

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.”

al-sign

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.
hf-friendship
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.

hf-women-nasa


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

The Election Pollsters Still Got It Right

election-forecastThere has been a great deal of Monday morning quarterbacking about how the 2016 Presidential election polls “got it all wrong.” Radio pundits like KFI’s John and Ken have been claiming that pollsters obviously don’t know what they are doing. There are three points to consider here.

1) Did the polls get it wrong?
2) Did the pollsters do something wrong?
3) What good math activity can we generate from all this fuss?

Here are some direct answers with, hopefully, simple, clarifying mathematical (not political) explanations.

The Polls Got It Right
The poll results were within the expected margin of error. In fact, four days before the election, Harry Enten of FiveThirtyEight wrote “Clinton’s lead is small enough that it wouldn’t take more than a normal amount of polling error to wipe the lead out and leave Trump the winner of the national popular vote.” In the end, Clinton still won the popular vote, by approximately 1.5% compared to the 3.3% predicated the day before the election, well within the normal margin of error. Gallup shows that, historically, the polls have been within 2%, on average, of the actual results, and within 1% half of the time, with the victories of Reagan in 1980 and Truman in 1948 being the most notable anomalies.

In fact, Nate Silver of FiveThirtyEight noted the day after the election that a 1% swing in Clinton’s favor across all states would have flipped the Electoral College tally.

Further support that the polls got it right comes from the understanding of probability. Clinton was given a 71% chance of winning on the eve of the election. That means that Trump had a slighter better chance of winning the election than he had of flipping heads on two consecutive tosses of a coin. When heads occurs twice when tossing a coin, should we all protest that statistics and polling are unreliable? This is why Nate Silver claims that the polls missed, but he did not say that they failed.

The Pollsters Did It Right
People have been willing to give more grace to the mathematics than to the mathematicians. Pollsters (those creating the polls, not the folks on the phone) have taken a great deal of heat for poor sampling, but these pollsters have been vindicated voter turnout numbers, because the pollsters surveyed registered voters, not guaranteed voters.

PBS‘s Michael Reagan writes that the data on actual casted votes reveals that Clinton had 2 million fewer voters than Obama did in 2012, while Trump had a slight uptick over Mitt Romney. Had voter participation been similar to the 2012 election, America would have had a different 2016 result.

Liberal filmmaker Michael Moore was extremely concerned just before the election about the lack of enthusiasm for Clinton versus the overwhelming passionate support for Trump. His concern turned out to be warranted.

A Good Math Activity: Secretary Clinton Attempts A Field Goal Kick
Given the information below from FiveThirtyEight, at what distance (in yards) would a field goal kicker in 2014 have the same chance of success as Secretary Clinton in the election of 2016.

election-percentage

Election Kickers.png

Spoiler alert: Approximately 48 yards.

Fortunately, if an NFL kicker misses a field goal attempt from just inside the 50 yard line, I still have faith in statistics and statisticians… and America.

Is Talent a Wall or Launching Pad?

In the quest for delineating the difference between fixed and growth mindsets, educators have created a plethora of lists for contrasting habits, beliefs and statements.

 

With my students, I summarize all of these lists and graphics with two simple pictures that pose a simple question: “Do you see talent as a wall or as a launching pad?”

Talent Wall

Carole Dweck‘s mindset research shows that self-perception of talent as a limit or as a starting point has a tremendous influence on student learning. The mindset is really about how people perceive their natural abilities and view the potential of their efforts, not just their level of effort.

In fact, this double-pronged image of wall vs launching pad helped me make sense of two things that Dr. Dweck has said regrading effort and growth mindset. The first is that “the most common misconception is simply equating the growth mindset with effort.” (Education Weekly, September 2015).  I didn’t fully grasp the issue that troubled her until I heard her say that it isn’t just the struggling math students who have a fixed mindset, but that even some of the more “successful students” have one as well. An example of an advanced student who possesses a fixed mindset is the one who believes that they cannot learn math (remember mindset is about self-perception), so they must compensate by studying and memorizing in order to pass the tests and get good grades.

This thought leads me to another of my favorite image comparing fixed and growth mindset… the scans of the brain of someone with a fixed mindset versus one with a growth mindset.

Brain Mindsets

brain-coldWhen faced with a challenge, the fixed mindset brain “goes cold.” It literally shuts down.

brain-on-fireHowever, when faced with the same challenge, the growth mindset brain “fires up.” It knows that more is being asked of it, so it kicks into high gear to meet the challenge, rather than duck it.

Some frozen brains walk away from learning by checking out or acting out. Other frozen brains circumvent the learning by grinding through the course with a hyper-powered work ethic. If good grades is the ultimate goal, then one of these fixed mindset responses is valued more than the other. If learning is the true prize of an education, though, then neither response is sufficient; instead, Dweck claims that students need to implement a “repertoire of approaches—not just sheer effort—to learn and improve.” After all, it is how one reveals to setbacks that reveals their true mindset.

To see some of my novice attempts at teaching growth mindset in math class, see the following posts: Nicki the Neuron, Neuron Stickers, Brain Surgeons & Wrinkle Sprinkles, and Neuron Problems and Classroom Norms, or click Growth Mindset in the tag cloud.

Ready, Set, Launch!

 

Nicki the Neuron

NickyAfter I gave a presentation about my use of Neuron Stickers, Brain Surgeon & Wrinkle Sprinkles at Twitter Math Camp 15, Julie Wright (@julierwright) sent me a tweet that directed me to a stuffed Neuron with eyes. So Bitchen!

I took her suggestion and hung it at the front of the classroom as a class mascot, naming it Nicki the Neuron, since Nicki is a name that is gender and ethnic neutral.

Julie Wright Tweet

 

Nicki was  a bigger hit than I expected. One of my students insisted on holding our new mascot during class.

Nicky n Fan

This inspired me. I thought to give Nicki temporarily to the group to which the most recent Neuron Sticker was awarded. I was concerned the boys wouldn’t receive this too well, but Nicki quickly became of badge of honor for the groups.

Nicky n Girl

Nicki is now part of the responsibilities of the Brain Surgeon and is generating a great deal of focus on the Process Reward System that I am implementing. Thank you Julie!

Neuron Stickers, Brain Surgeons and Wrinkle Sprinkles

Brain-SurgeonI was inspired at a Growth Mindset workshop by Jo Boaler and Carol Dweck. I knew I was going to be teaching a class of at-risk students, qualified by being on the socio-economic disadvantaged list and having struggled in 8th grade math. Rather than repeating in high school the math course that they failed in middle school, these students would taking Algebra 1 with me. If there is a group of students that need help shifting from a fixed mindset to a growth mindset, it is a group of at-risk students who have struggled in math. However, I did not want to just put a bunch of Power Point slides saying how they should believe in themselves.

So, I came up with three vehicles to develop growth mindsets in my students:

Neuron Stickers

Dr. Boaler emphasizes the plasticity of the brain. This means that the brain actual rewires itself when it learns, by forming new or strengthening current connects between brain cells. We also know now that the outer later of the brain thickens as we learn, much like muscles get bigger from exercise. These facts create a contemporary view of the brain that is in direct opposition to the conventional view in education in which the brain is a passive vessel to be filled with knowledge. These two views are best contrasted by the following images of the brain.

Brain Pair

The image on the left implies that we are building a brain. I love that idea so much that I enlarged the graphic to poster size and put it up on the classroom wall. I tell my students that is exactly what they are here to do … build their brains. We then publicly discuss the actions that help us build our brains in class, like…

  • Sharing mistakes publicly
  • Offering unique solutions
  • Asking clarifying questions
  • Making connections
  • Having an “Aha!” moment
  • Helping others

To encourage these and other behaviors that contribute to learning, I created Neuron stickers. This was easy, I pulled a drawing of a neuron from the internet and created a sheet that I could print onto a sheet of mailing labels.

neuron color

Neuron Sheet

Each time a student demonstrated action that promoted learning, the student receives a neuron sticker which they get to place on the Brain Poster. Once the poster is filled, I put up a new one and we continue honoring growth mindset throughout the year.

Brain Color Brain Pic Final

Brain Surgeon

Each day we designate a “Brain Surgeon,” who serves as a class leader for the day. I purchased this model of the brain to be given to the day’s Brain Surgeon.

Brain Surgeon Model

The role of the Brain Surgeon comes at the beginning and end of each class.

Opening Class Duties

  • Supervise preparation for class (getting materials ready)
  • Lead Drum Roll (Class Opener)
  • Reading of Instructional Objective
  • Placing Nicki The Neuron with the group who had it last during prior lesson

Closing Class Duties

  • Supervise clean up and storing of materials
  • Return Nicki The Neuron and the Brain
  • Lead Wrinkle Sprinkle 

Wrinkle Sprinkle

Each class concludes with a debrief titled “Wrinkle Sprinkle,” implying that learning adds a new wrinkle to the brain. (Note: Anatomically we know this is not accurate, though we know that the neurons make new connections and the outer layer of the brain thickens.) The brain surgeon calls on students who raise a hand to offer something that they learned that day. These Wrinkle Sprinkles are recorded on the 180Blogs on this site. Wrinkle

 

Neuron Problems & Classroom Norms in Algebra 2

Day 3, Fri Aug 12, 2016

A vs Don-stepmom-shoulderTarget: Recognize that  Voice = Choice when it comes to having a growth mindset as we solve problems about our amazing brains.

Entrance Ticket
I greeted the students at the door, but today I was checking homework. They only had to do one problem of their choosing from the Neuron Facts last night. If they did not have it, they had to quickly do one outside. Message sent: You are doing your homework in this class.

Growth Mindset
On the growth mindset web site they make a point of the “voice = choice,” meaning that we have a choice whether or not to listen to the fixed mindset thoughts that we all have, They give a 4-step breakdown of how to shift from a fixed to a growth mindset. I had fun soliciting the help of a very ancient visual of a devil and an angel on your shoulder.

Voices choices

Neuron Fact Problems
So then came time to practice recognizing the fixed voice and talking to ourselves in the growth voice, while doing challenging math problems. They already sit in groups of four, so I had them spend the rest of the period working through the Neuron Fact Problems, which I created from the Facts on the front side of the paper.  They were to call out any fixed mindset words or actions demonstrated by their partners. They actually did. I worked the room with Neuron stickers and Nicki. I honored about half the groups. I was pleasantly surprised at how well my crew worked.

During the lesson, as I worked the groups, I asked  one student how she got her answer, and she told me that she had copied from her partner. I praised her for her honesty, then paused the class and brought their attention to our classroom norms.

Norms

These were originally shared with us by Dr. Juli Dixon (@thestrokeofluck) in a math training at our district. They became very popular among our teachers. Our new principal has implemented them schoolwide, providing posters for every classroom. I drew the students attention, that we “Share, Don’t Copy.” When we share, one person explains, the other listens, then question follow if we don’t understand or if we disagree. If these three norms are occurring then writing down someone else’s solution is not copying.

After a half hour of solid work, we debriefed where we saw evidence of a fixed mindset and where we saw evidence of a growth mindset. This whole activity was very well received by the students. I gave them advance notice that Monday we will be debriefing their actual solutions to the problems.

Wrinkle Sprinkle

  • Share, Don’t Copy.
  • The equals about 3 lbs.

Introductions & Neuron Facts in Algebra 2


neuron vertical
Day 2, Thurs Aug 11, 2016

The Brain Surgeon
Today, we began my regular routine of designating a daily Brain Surgeon. Since this was our first day of the Brain Surgeon, I introduced the routines of the Drum Roll, Reading of the Dual Target, Music Cues, and the Wrinkle Sprinkle. The students seem to embrace the spirit of of it all.

Student Introductions
As with every new school year, I had each student briefly state their name and something interesting about themselves. When they were all done,  I recited all their names. That always impresses a class. Then I told them things about myself. I state that yesterday we started with math, because that is what we are all about here. But since I teach math to them, they are also important and I need to know who they are.

Growth Mindset
Most of our Course Teams across the district agreed to do some kind of growth mindset activity. Here was mine.

I started by summarizing the plethora of lists of fixed vs growth mind set statements with two pictures. I told the students that research in student learning is showing that self-perception of talent as a limit or as a starting point has a tremendous influence on their learning.

Talent Wall

Then I shared that scans of the brain of someone with a fixed mindset versus a growth mindset, shows something very interesting. When faced with a challenge, the fixed mindset brain “goes cold.” It literally shuts down. However, when faced with the same challenge, the growth mindset brain “fires up.” It knows that more is being asked of it, so it kicks into high gear to meet the challenge, rather than duck it.
Brain MindsetsNow it was time to test out where we see ourselves demonstrating  a fixed or growth mindset.

Neuron Facts
I gave the students the worksheet with the Neuron Facts on the front side. I found these on the internet and thought they would make for a good lesson since they highlight the amazing function of our brains. I added the subheadings of Fast, Crowded ,etc. I started with a common practice of mine Notice & Wonder popularized by Annie Fetter (@MFAnnie) of Math Forum.  My Gradual Reel-In process looked something like this:

  1. You Do: Independent response.
  2. Ya’ll Do: Each member of the group shares both their notice and wonder.
  3. We Do: Each group decides on one Notice and one Wonder from those shared. These get shared out by each group as I write them on the board.
  4. I Do: I summarize the major point(s) that I want all students walking out with. Here it was the process of Noticing and Wondering and how we facilitate group discussion in class… And of course how amazing our brains are.

The groups were then tasked with doing one problem together. Homework was to do one more.

Wrinkle Sprinkle
Tying into the concept of the plasticity of the brain, I joke that when we learn we get a new wrinkle on the brain. Each class then concludes with what we learned that day. The brain surgeon leads and records the discussion. The students today stated that they learned…

  • Negative thoughts shut down your brain
  • Speed of the brain cell
  • The amount of oxygen the brain uses

First Day – Algebra 2

Day 1, Wed  Aug 10, 2016

{I am new to Chaparral High School, having transferred within my district as a Math Specialist.}

Opening Quiz Alg 2 on the 6Cs: After greeting each student at the door with a high-5, I started the year by answering the transformation question: “How will you (the students) be different in June than you are now, because of my class?” I am still answering that question with the same 6Cs that I launched 2 years ago. My Claims-Based grading system and the students portfolios are structured as such also.

6 Cs PicAs I do with all classes each year, I gave the students the blank copy of the quiz below, and told them this was not to be graded nor was it a test of their previous knowledge. It was like a movie trailer of things to come, but I still wanted them to give me their best shot. I then gave them my standard 3-response speech.

As a mathematician I cannot always give an accurate response; I cannot always give a complete response; but I can always, always, always give an intelligent response. Blank is not intelligent.

I pressed them to give me something… numbers, equations, drawings … anything intelligent.

Opening Quiz Alg 2 Pic

They worked on these independently, then in groups, then as a class, followed by my summary. I wanted to model this process of “gradual reel-in” (as opposed to gradual release) right away, because I use it often.

During the class discussion, one senior claimed out loud, “This is the 5th time that I have taken this class!” (She had failed two semesters as a junior, then 2 semesters in summer school.) I told her that this year she will pass because, “You are that smart, and I am that good.” I had the students repeat this:

Me: “You are that …”
Class: …smart!”
Me: “I am that …”
Class: “…good!”

This was a set-up for the Growth Mindset discussion that was coming over the next few days. In the meantime, I hope I sent the message that I believe in them, and that I believe in my ability to teach them (The 3 Growth Mindsets).

The students brought some terrific energy. I’m so looking forward to my first year as a Puma.

Recap: Twitter Math Camp ’16

TMC LogoThe annual Twitter Math Camp is always amazing. This summer’s conference in Minneapolis, at Augsburg College. was no different. My great disappointment was only being able to stay for one full day this year, but the one day did not disappoint. 

As always, portions the Math Twitter Blogosphere (#MTBoS) rallied from around the country in genuine excitement to see and learn from each other after another year of digital friendship and collaboration. Thanks go out to Lisa Henry (@lmhenry9) for being the lead on this terrifically special event.


A “JUST ENOUGH” APPROACH TO INTERVENTION (Session)
Michelle NMichelle Naidu (@park_star),
Saskatchewan Professional Development Unit

A packed room on the topic of intervention was surprising to both me and the presenter, because the MTBoS dialogue mostly revolves around first instruction. The large audience is a testament, though, to the need for reaching ALL kids in the era of 21st Century Standards. Michelle is leading a very successful intervention program in Canada which is focusing on some basic premises:

Differentiating for All Students is like Cowboys Herding Cats, but “it’s a good feeling having the herd [of students] arrive on time without losing a one.”

Early Intervention on the Pre-Requisite Skills (Readiness) that are required to be successful in the current curriculum is the first and most important intervention move. Pre-Assessments on prior content are then necessary to help improve students’ chances for success. Back at home we call this Boot Camp. Michelle affirmed that this work is good, and also inspired me to go back to my site and push to make it a priority.

Unpacking Standards Collaboratively serves two purposes. (1) It allows you to throw out material that is not in the standards, which buys you time for intervention/differentiation (Grade Level)  and (2) It helps you focus on the pre-requisite skills needed for students to learn the new material (Readiness).

Intervening on Readiness = Differentiated Content
Intervening on Grade Level = Differentiated Product

SnowballI also saw an interesting take on the Snowball Activity. Students write down one comment and one question about a topic (notice and wonder), then wad up their papers and throw them around the room. Each student picks up a “snowball” and adds another comment and question. This is done again, until there are three of each. After the fourth toss of the snowballs, the students do not write, but instead debrief publicly as the teacher summarizes the comments and questions on the board. This is a strong way to have ALL students reflect on learning.


KEYNOTE: RACE, MATH AND WHAT WE’RE NOT TALKING ABOUT
Jose Vilson (@TheJLV)JoseV
educolor.org

Jose’s most solid point was that public conversation on math education reform often does not include educators, especially those teaching the marginalized. He accurately stated that if the medical system in America were being discussed on cable news, there would be a doctor on the show, but you never see a teacher on TV talking about education.

In many ways, Jose was calling us out to be activist on our campuses for the changes that we in the Blogosphere write so much about, particularly for students of color. He made a claim that really stuck with me: “We say that we teach math to all kids, but students of color are taught a different type of math than white students.” I know this is true on my campus, While my school is relatively diverse, the lower-level math classes are disproportionately populated by students with Hispanic surnames.

I asked a question of Jose, preluding it with a statement that prejudice on my campus tends to run more along income lines than racial lines (although, racism exists everywhere). Students are accepted and succeed as long as they behave like ‘these kids.’ So I asked, “How do you get teachers and staff to be more accepting of ‘those kids,’ so that they can remain authentically themselves and still learn?” Jose’s response was, “Teach the adults to recognize ‘different types of genius.'” I love that phrase! He went on to explain that kids in poverty are often times going to bring the norms of their own sub-culture to class, which is many times in conflict with the rigid, quite, patient, controlled environment of traditional school. If we can respect that and honor ALL students’ intellects, while also teaching proper social behavior, schools will break down a lot of walls and reach more marginalized students.


AudreySTUDENT-CREATED GEOGEBRAS
Audrey McLaren (@a_mcsquared),

Audrey showed samples of student work from her classes, in which she has students BUILD activities and graphs in GeoGebra and Desmos. The best example was Sticky Points. I love how the challenging of students to create the special points for a function like the x-intercept(s), the y-intercept, and the vertex demands that the students do some algebraically manipulation. The graph offers an immediate feedback loop until students do it correctly. This builds their algebra skills and conceptual understanding simultaneously. I’m using this idea in my class this year for sure.

Desmos Sticky points


WHAT IS MATHEMATICAL MODELING?
Edmund Harris (@Gelada) and Myself (@MathProjects)

Edmund Model.png
I thought Dr. Harris asked for “mathematician modeling!”

I was honored when Dr. Harris, of the University of Arkansas, asked me to present with him on Mathematical Modeling. Edmund and I have been friends since our first Twitter Math Camp (TMC13), and I always look forward to our laughs and deep mathematical conversations. Edmund wanted to share the theoretical meaning of mathematical modeling, and he asked me to add my take on how the teaching of it manifests in the classroom.

Logo Pear DeckWe started by surveying the audience on Pear Deck, prompting for their definitions of mathematical modeling. The vast majority of the responses fell into two categories:

  • Representing a Real-Life Situation
  • Applying the representation to make Predictions.

It turns out that these are quite accurate if we include them BOTH, but the two are not necessarily a comprehensive list, as Edmund explained.

The professor started by claiming that shepherds in the field used to count sheep by using stones in their pocket by which a small stone represents one sheep and a larger stone represents 20 sheep. This, he asserted, is an example of abstract modeling. (Leave it to the Brit to bring sheep herding into a math discussion.) Then he drew this diagram on the board:

Edmunds Model Diagram

Edmund teachingHe explained that you start with “something to be modeled,” (noticed he did not say a real-life situation) and then you create an abstract representation of it. This is a back-and-forth process of verifying the accuracy of the model’s description of the something as well as the “thing” that we want to do with it. (Use rocks to keep track of the sheep). So the audience was responses were spot on… collectively. Yes, modeling is Representation AND Application, but not necessarily just Representation OR Application. Furthermore, Edmond wanted to make it clear that modeling does not have to apply to only “real-world” examples. He claims that when we discuss the transformations of a family of functions, we are also modeling… using an abstract representation to “do something” to the original parent function.

Modeling Tweet Me

In my investigating of what is expected of school teachers when it comes to modeling, I studied the common core documents and found very persistent, clearly defined attributes of Mathematical Modeling:

  1. Modeling is a process.
  2. Modeling is a verb.

In other words, using a model that is already provided is a good and healthy step in the learning process of modeling, but it is not modeling itself unless the students are generating the model themselves.

Modeling Tweet Heather

Modeling Tweet Jasmine

Thank you, Edmund. It was a pleasure working with you, my friend. You always make math appear so joyful.


THE SIDE TALKS
I had several conversations throughout the Camp, but two that stood out were with …

Math Modeling
Edmund Harris (@Gelada), Brian Miller (@TheMillerMath) & Alex Wilson (@fractallove314)

TMC Bar ModelingThe first night of TMC16 was a huge social event by Desmos. Edmund, Brian, Alex and I had a beer-laden discussion about modeling that proved quite passionate (read as: table pounding, finger-pointing, and all in good fun B.S. calling). It was such a blast to throw ideas around with people of high intelligence, strong convictions and the deep desire to get this thing that we call teaching right. Cheers to changing the world one math lesson at a time.

Intellectual Need for VocabularyPic Dan M
Dr. Dan Meyer
 (@ddmeyer):
Dr. Meyer completed his dissertation last year. Knowing how much those with a doctorate enjoy talking about their research, and being truly curious about it, I ask him to share his findings with me. He joyfully did, including some of the back story behind it. In essence, Dan studied the effectiveness of giving students the academic vocabulary after first posing a task that required its use, rather than front loading the terms. He called this method Functionary. His study showed that the both Functionary (using the vocabulary to communicate) and Traditional methods (making flash cards to memorize definitions) were equally effective in teaching students academic language found on traditional assessments. The Functionary method, however, showed superior results when students were asked to communicate their thinking using the vocabulary terms or to complete less traditional (more CCSS-like) tasks. You can listen to the Defense of his Dissertation on Dan’s Blog


As always, I highly recommend this event to any math teacher. I hope to see you all at Twitter Math Camp in  Atlanta, July 27-30 2017.

How Deep for Teachers?

 

26570883 - flood level depth marker post with rain falling into the surrounding waterRecently, I conducted a training with a school district in West Virginia.  It was for new teachers (1st-3rd Year) in all subjects K-12. There were approximately 75 participants and 20 mentors. One of our activities dealt with Depth of Knowledge. I showed the typical D.O.K., but I wanted them to have a more meaningful experience with D.O.K. levels in mathematics.

DOK Chart

The activity I created was inspired by the work of Robert Kaplinsky. I love his Tools to Distinguishing Between Depth of Knowledge Levels. I particularly like his example of sums of whole numbers:DOK RK sampleThis is a simple and clear example of the D.O.K. progression. However, it does not show D.O.K. Level 4, so I contacted Robert and he directed me to this Problem Post of his, How Many Soda Combos are There on a Coke Freestyle Machine? 

DOK Soda

Perfect! I compiled these four into one document, scrambling the order, and asked the teachers to discuss the problems in their table groups and to assign a D.O.K. level exclusively to each one. I was intrigued at how different their responses were compared to what Robert (and myself) considered the problems to be. I noted that the group was a broad range of grade levels and subject areas, so I thought I would conduct the same activity with a collection of high school math teachers that I was scheduled to train in California  the following week. I was very curious if math teachers would view the problems differently than non-math teachers. Indeed, they did. However, they also disagreed with Robert and me. Below, are the all responses from the groups at each training, as well as Robert’s determination. Notice the variety of responses that was generated within each training.

DOK RK Response

The choices that earned  the most votes looked like this.

DOK RK tops

Notice that there is not a single example in which all three parties agree. I have no profound analysis of these results; I am simply sharing this very curious experience. I am still pondering the outcomes and their meaning many times over. So much so, that whenever I hear the phrase “D.O.K.,” I smirk and scratch my head.