Category Archives: Blog

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:

screen-shot-2017-03-04-at-11-51-40-am

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.

4-cs

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.

6-cs-pic

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.

specs-pic

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.

quiz-rats

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.)

results-chart

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.

calendar

 

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),
TODOS

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),
AlgoGeom

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)

panel-pic

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.

bethany-slide-badges

bethany-slide-comm

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?”

relationship-panel

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.

berray-tweet

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
pic-luevenos
Daniel Luevanos  (@DanLuevanos) &
Chris Shore (Me) (@MathProjects)
clotheslinemath.com

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

pic-daniel-system-1
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.”

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 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.

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

 

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.