Tag Archives: ruminations

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.



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 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.

Stats Are Like A Fist Full Of Worms

wormsAlgebra is like an ox. It does a lot of work for us, obeys our commands and remains very predictable.

Geometry is like a puppy. It’s fun to play with, doesn’t ask for anything other than your attention, and doesn’t promise anything other than that in return.

Calculus is like a horse. It is gorgeous to watch when it runs, exhilarating to ride and takes us places we have never been before.

Statistics are like a fist full of worms. They wiggle around and are hard to get a hold of. They live underground, so you always have to dig deep to find them. Once you get one, you don’t know what to do with it other than stick it on a hook and cast it out into the ocean in hopes that it brings you back something useful.

Get to the Core of The Core

apple coreThe Common Core curriculum can basically be summed up in the following sentence:

Teach your students to THINK and COMMUNICATE their thinking.

Thinking and communicating are the 21st Century skills. Many people believe that the skills of the future involve the competent use of technology. While it is true that using digital tools in school and the work place is the new reality, it is actually the proliferation of technology that makes thinking and communicating imperative in the information age. When all the knowledge of humankind is available at anyone’s fingertips, memorizing information becomes far less important than being able to construct, evaluate and apply it. You can Google information; you cannot Google thinking.

So the core of the Core truly is Thinking & Communicating.

To make my case for this, I would like to pose that the following equation

6 + 4 + 4 + 8 = 22

be adjusted to

6 + 4 + 4 + 8 = 21

Before you start shouting that everything you have read on Facebook about the Common Core is true, let me declare that I am using this equation simply as a teaching device, not a true mathematical statement. You will understand what I mean after I present my evidence.

6 Shifts

Let me start my case that the core of the Core is Thinking & Communicating with the 6 Shifts, which are best represented by the following document found at Engage NY.

6 Shifts

In essence, these shifts are redefining rigor. Old school rigor was defined as sitting quietly taking notes, and completing long homework assignments in isolation. The new school definition of rigor envelops the last 4 shifts on the list: Fluency, Deep Understanding, Applications, and Dual Intensity. The rigor is now placed on the students’ minds instead of on their behinds.

The shifts are also calling for balance. Dual Intensity insists on both procedural fluency AND critical thinking by the students at a high level. It is not about dual mediocrity or about throwing the old out for the new, but a rich coupling of both mechanics and problem solving.

Therefore, I make the case that:
                     6 Shifts = 21st Century Skills,
which are to
                     Think & Communicate.

4 C’s

Another list that is framing much of the Common Core dialoge is the 4 C’s. Resources for this list can be found at Partnership for 21st Century Learning (p21.org).

4 C'sThese C’s redefine school…

The old school definition: A place where young people go to watch old people work.

The new school definition: A place where old people go to teach young people to think.

… and they redefine learning.

The difference of old school vs new school learning can best be contrasted by the following images of the brain.

Brain Chillin   Brain Build

The image on the left shows a passive brain that just hangs out as we stuff it with esoteric trivia. The image on the right shows a brain being built, symbolizing its plasticity. We now know that when the brain learns, its neurons make new connection with each other. In other words, learning literally builds the brain. The 4 C’s  claim that this building involves the capacity of the students’ brains to Critically Think, Communicate, Create and Collaborate.

Therefore, I make the case that:
                     4 C’s = 21st Century Skills
which are to
                     Think & Communicate.

4 Claims

Smarter Balance creates it’s assessments based on 4 Claims. (I teach in California. PARCC has 4 Claims that closely align to those of SBAC.)


4 Claims

Notice that Claims #2 & 3 are explicitly stated as Thinking & Communicating, which also overlaps with two of the 4 C’s. Mathematical modeling is #4, which will be discussed later. I want to point out here that Claim #1 reinforces our idea of Dual Intensity from the 6 shifts.

There are two important notes for teachers about this first claim. 1) It says Concepts and Procedures, not just procedures. The students need to know the why not just the how. 2) The Procedures alone account for about 30% of the new state tests, so if we continue to teach as has been traditionally done in America, we will fail to prepare our students for the other 70% of the exam which will assess their conceptual understanding as well as their abilities in problem solving, communicating and modeling.

Therefore, I make the case that:
                     4 Claims = 21st Century Skills
which are to
                     Think & Communicate.

8 Practices

If you open the Common Core Standards for Mathematics, the first two pages of the beastly document contain a detailed description of the Standards of Mathematical Practice. Then at the beginning of each of the grade level sections for the Standards of Content you will find 8 Practices summarized in the grey box shown below.
8 practicesWhat do you notice about the list? Indeed, these habits of mind all involve Thinking & Communicating. While the content standards change with each new grade level, the practice standards do not. With each year of school the students are expected to get better at these 8 Practices. Notice that the first half of the list has already been included in the ones discussed previously: Problem Solving, Communicating Reasoning, Constructing Viable Arguments and Modeling. A case is often made that the other four are embedded in these first four. However one might interpret the list, “Memorize and Regurgitate” is not on there.

Therefore, I make the case that:
                     8 Practices = 21st Century Skills
which are to
                     Think & Communicate.

The Sum of the Numbers

So, as you can now see, the 6 Shifts, the 4 C’s, the 4 Claims and the 8 Practices are all focused on the 21st Century Skills of Thinking & Communicating. Therefore, I can finally, explain my new equation …


    6 Shifts
    4 C’s
    4 Claims
+  8 Practices
= 21st Century Skills

then 6 + 4 + 4 + 8 = 21!

None of these numbers represents a list of content, because the content changes brought on by the Common Core, while significant, are actually no big deal in the long run. A few years from now we won’t remember all the fuss regarding Statistics and Transformations, but we will all spend the rest of our careers learning how to teach kids to Think & Communicate.

I rest my case.

Making a Lasting Impression… Without Even Knowing It

Footsteps in the Snad

GUEST POST: Today’s article is written by Greg Rhodes, the co-founder, creative director, and overall tech guru for MPJ. Usually, he stays behind the scenes, but a recent email made such an impact on him that he just had to share it with our readers.

I’ve been out of the classroom for a long time now, over fifteen years. But prior to my career transition, I was a math teacher like many of you. During those years, I did my best day in and day out to help students think logically and solve problems creatively… and maybe even have some fun in the process. But did I ever think that any of my hands-on lessons or outdoor projects made any lasting impression on my students? Not for a second.

But all of that changed one day when I received an email from Andy, a former student of mine. It left me absolutely speechless.

Hi Mr. Rhodes!  This email is likely to be out of left field since I haven't seen you in about fifteen years (assuming this is the correct "you"), so I apologize for potentially appearing to be an internet stalker.

Background:  I took geometry with you at Trabuco Hills High School in 1996/97 as a freshman and then struggled with algebra 2 honors the following year.  I am currently (after a rather circuitous journey) in a single subject credential program for chemistry at a local state college.

I have found myself bringing up your geometry class over and over again in class discussions of late, and reflecting somewhat extensively upon that time in my life.  Now, as I write the TPA 2 that is due this Monday, I just wanted to take a moment to tell you that it was a good class, and that it made a lasting impression.

So, thank you.

Wow! Fifteen years later and he’s still thinking about my little geometry class… and even discussing it with his classmates. In my wildest dreams, I never could have imagined that my teaching would leave such a lasting impression on any of my students.

So, here’s a word of encouragement as you prepare to go back into the classrooms (or as already there): Keep pursuing great teaching. Keep asking yourself how to make your lesson better, how to help your students understand deeper.

You are making a difference in the lives of your students. Never forget that. Some will thank you with a card or an coffee mug on Teacher Appreciation Day, and others may never say a thing. But don’t let that stop you from being the teacher they remember with fondness fifteen years from now.

Millionaires and Their Cars?

Car FerrariMy teenage son is preoccupied with three things these days: water polo, his girlfriend and expensive cars. He has fantastic talent in water polo, and has a wonderful girlfriend. He does not have an expensive car.

Currently, he is saving for his first car and knows that he will have to start with a used, low-end model, but he dreams big. He is always talking about Ferrari’s and Rolls-Royces. We like to talk about them together and point them out on the road whenever we are driving. He is convinced that he will own one someday. When I respond to his talk of grandeur, I want to sound like the Encouraging Dad (“Terrific, what kind of successful job do see you see yourself having, so you can afford that kind of car?”), but I worry that I sound like the Practical Dad (“That’s nice, it might be more realistic to set your sights on a cheaper car.”). The truth is that my words usually come out somewhere in the middle, which led to our very interesting math conversation the other day.

On a long drive back from a water polo game, we were talking about reasonable incomes (Practical Dad ruling the moment). He is a Junior in high school, so his interest is peaking about how much money is to be made as an adult. The conversation went like this:

Me: Guess what the average annual income is in America.

Preston: I know, because we talked about this in History class. $36,000 a year, but if I make $100,000 a year, a can save half of that and buy a rich car in five years.

Me: Do you know what percentage of Americans make over $100,000 a year?

Preston: 25%?

Me: It is actually about 4%. I know several people who make that kind of money. None of them drive a Ferrari, so you are going to have to make more than that. (Encouraging Dad trying to break through.)

Preston: If I made a million dollars a year, I could buy it in one year and still have enough to live on.

Me: With enough left over to care of me and your Mom. That would be awesome, but you are going to have to do something special, because less than one-half of one percent of Americans make a million dollars a year.

Preston: It has to be more than that. Look at how many rappers there are making bank.

Me: And think about how many are making just a normal living or how many are standing on a street corner singing while they hold their hat out for tips. Very few earn “Checks that look like phone numbers.”

Preston: Look at how many millionaires we know.

Me: I would say less than 5, off the top of my head.

Preston: Yeah, see?

Car LamborghiniInspired Math Question #1: If you know 5 millionaires, what percentage is that of all the people you know?

Inspired Math Question #2: If one-half of one percent of the people you know are millionaires, how many people would that be?

Preston: I bet there are at least a million millionaires in the country.

Inspired Math Question #3: Given that there are 300 million people in the U.S., and that 75% are adults, would one-half of one percent of American adults be more than a million people? (to be estimated while driving without a calculator)

Me: I am guessing that we are both correct on this one.

Preston: I still say it has to be more than that then. (Whether I am encouraging or practical, I am still Dad, so he must win!) Look at how many expensive cars we saw just today. There was a Ferrari, a Lamborghini and a Bentley.

Me: Yes, and think of how many other cars we saw today.

BentleyInspired Math Question #4: Approximately how many total cars might you see driving on a freeway for an hour on a Sunday afternoon? (must explain your reasoning on this one)

Inspired Math Question #5: If you see three expensive sport cars on that same trip, what percentage of all the cars would that be?

As we arrived home, Preston was still seeking victory. He is very good with mental math, so he knew where I was going with all the number crunching. In order to get the upper hand, he needed to bring in an expert, and what better expert in the world of teenagerdom to call upon than the internet? He Googled on his smart phone, “How many millionaires are in America?” and got an answer of over 3,000,000. He loudly reveled in glory. I countered with the age-old math argument of the importance of definitions. In this case, there was a difference between annual income and net worth. He was having no part of it. He was to busy flexing and bragging to Mom about how he just “owned” Dad in a math debate.

Theorems to Teach By

Teaching TheoremI found this in some old files. I compiled these thoughts 17 years ago more as inspirational thoughts than scientific edicts, but my long teaching career has proven them all to be true for me, so I thought I would share. They deal with classroom management, student rapport, and grading. It is written in the vernacular of a math teacher, because old habits die hard.

The Triple Bird Principle

    1. The Pigeon Theorem
      When feeding pigeons, if you thrust your hand out and chase after the pigeons, they will fly away. If you sit calmly and hold out your hand invitingly, they will eat out of your palm.
    2. The Mother-Chick Corollary (by Isaiah Thomas’ mother on her death bed)
      There is no such thing as teaching, only learning. Just as a mother bird can’t teach her chicks to fly, she can only love and nurture them, and allow them to do what they were born to do.
    3. The Eagle Paradox
      Eagle chicks learn to fly by being pushed out of the nest by their mother.

The Dewey Principle

“It is folly to believe that the only thing that your students are learning is what they are studying at the time.”

The Push-Pull Principle

Leadership is distinctly different from and just as important as management. You are FIRST among EQUALS.

The Contract Principle

You don’t need to be your students’ friend. You MUST be their ally.

The Mediocrity Principle

    1. The Equilibrium of Rigor
      Teachers do not allow too many students to succeed, nor too many to fail; both assessment and instruction are adjusted until the results are “just right.”
    2. X Equals Two Aspirin
      Only teachers guarantee their own professional mediocrity. Doctors do not insist that a certain portion of their patients die, allow only a few to be healed, nor do they impose minor complications upon the rest.

The Power-Influence Dichotomy

“To influence is to gain assent, not just obedience; to attract a following, not just an entourage; to have imitators, not just subordinates. Power gets its way. Influence makes its way.”
— Richard Lacayo. June 17, 1996. Time.

The Smart is Sexy Christmas Story

Ray Bradbury
Ray Bradbury
August 22, 1920 – June 5, 2012

I share this story as a gift to my classes every year just before Winter Break. It is an engaging tale that has proven to be as inspirational to others as the true events originally were to me. Central to the story is my unique interaction with Ray Bradbury, author of Fahrenheit 451. This past year, Mr. Bradbury died, so I felt it appropriate to commit my three-decade long oral tradition to writing. The theme of the story is about leaving a legacy. Here is my tribute to a great American legacy.

As we all know there are three phases in life. In high school, cool is sexy; in college, smart is sexy; after college, rich is sexy. Since this true story involves a girl and a holiday gift while I was in college, I have thus dubbed it The Smart is Sexy Christmas Story.

I was a freshman at USC attending a philosophy discussion class. There were seven of us sitting in an arc being led by a young grad student. He was asking us to share out the topic of our term papers. I went first and spoke about Aristotle’s Nicomachean Ethics. I didn’t pay much attention to the others after that, because I was too focused on the gal sitting at the other end of the row. She was cute, petite and I had a mullet that tapered to a thin braided pony tail. So hot! (This was the Pat Benatar era after all.)

When it was her turn to finally speak, she shared that she was perplexed by how Socrates handled his own death. Socrates is well-known for being executed for teaching the youth of the day about democracy. His government let him choose his form of execution. Socrates elected to drink hemlock. On the day of his execution, Socrates held up the cup of Hemlock and toasted, “By doing this, you will forever immortalize my teachings!” He then chugged the poison and died.

Socrates meant that if the government had simply let him do his thing and pass on quietly, maybe no one would notice, but since he was being silenced by the powers-that-be, generations of people were going to want to know what he was saying. And here we are talking about him 2,000 years later.

Although this idea eluded the object of my attraction, it was time to join our professor in Mudd Hall with a few hundred others for our philosophy lecture. Although the teaching assistant said we would finish the discussion next time, I saw an opportunity to break the ice.

I was recently reading a book that I thought might help, Fahrenheit 451. It’s author, Ray Bradbury, did a guest appearance at my high school the year before, so I was inspired to read one of his works. The day before, I read a passage that particularly struck me. I thought it might illuminate Socrates’ words for my classmate, so I copied it down on a sheet of scrap paper and handed it to her on the way to class. She was enormously grateful.

The next time I saw her, my new friend said that she was going to have a present for me at the end of the semester. Sure enough, on the day of the final she handed me a framed sheet of paper. I noticed that on the page was typed the passage that I had written down for her. Since my mind was focused on the impending test, I didn’t exam it very carefully, though I did thank her for the sweet gesture.

After I was done writing about the wisdom of men in togas, I picked up the gift to take a closer look. I now noticed that the passage was typed on Ray Bradbury’s personal stationary … and it was signed by the man himself! “Good Wishes, from Ray Bradbury, Dec. 1982”

How?! I approached my benefactor and inquired as to how this came about. She claimed that Bradbury Frame 3what I did was the nicest thing that anyone had ever done for her. My first thought was “Yes!” My second thought was, “This girl has had a rough life.” She told her father the story, and her Dad also thought it was the nicest thing that he had ever heard of anyone doing for anyone else. (Dad must have had a rough life, too.) By unbelievable coincidence, her Dad was friends with none other than Ray Bradbury himself (no kidding) who also thought it was the nicest thing that he had heard of anyone doing for anyone else, so he typed up the passage on a sheet of his personal stationary, signed it, and gave it to his friend, to give to his daughter, to give to me.

And that was the last I ever saw of her. I cannot even remember her name, but I have cherished the gift to this day. When I became a teacher, I hung the framed passage on the wall of my classroom and have told that story every year at this time. And that was the extent of my story, for twenty-four years … until I personally met Ray Bradbury.

In 2007, my town built a new library. Ray Bradbury was scheduled to make a book signing appearance to commemorate the moment. The book everyone was promoted to read and bring to the Grand Opening was none other than Fahrenheit 451. I was so pumped. This was my opportunity to finally thank Ray Bradbury in person, so I pulled some strings and got a ticket to the exclusive event.

At the night of the opening, the literary icon’s much-anticipated arrival was delayed by rainy traffic. While two-hundred fans anxiously stood with copies of Fahrenheit 451 in hand, my smart-is-sexy gift drew quite a bit of attention. I must have told the story a dozen times while we waited. Eventually, we were all escorted to the room where Bradbury was to speak to the crowd. His delay was getting longer, so one of the organizers asked me to entertain the crowd with my story. I stood on the platform and reiterated my tale to a room full of Bradbury junkies. They loved it.

Shortly after I finished, Ray Bradbury finally arrived. He was very old and sick, so he had assistants escort him out in a wheelchair. Despite his infirmity, he spoke with humor and passion. For the next two hours, I listened to the greatest storyteller I have ever heard. He told story after story about events in his life that led to the writing or publishing of his various works. Like being a kid working in a carnival and meeting a man with tattoos all over his body, which led to The Illustrated Man. And how a lunch meeting with an aspiring new magazine editor led to the publishing of Fahrenheit 451 as a three-part series in the first few issues of … Playboy. The young editor was Hugh Hefner. The tales went on. We were all mesmerized.

When Ray Bradbury wrapped up his talk, we were instructed to line up for the book signing. This was my big chance, after a quarter of a century, to finally say thank you. I was so excited, but I found myself about 150th in line. It was late; Bradbury was sick; there was no way he was going to be there long enough for me to get to him. Then people around me started to take notice. They had all heard my story so they started letting me take cuts. Over and over again, I was being allowed to stand in front of the next person, and the next, until in a matter of a few minutes I was 10th in line.

I soon found myself face to face with Ray Bradbury. I knew I didn’t have much time. As kind as everyone was, they all wanted their turn as well, so I handed him my treasure and spoke fast.

“Mr. Bradbury, I have been waiting 24 years to thank you for this,” I started. He held the frame in his shaking hands, and as I rambled on about the girl and the friend of his, he read the passage.

Looking up with a smile, he said, ” That’s a really good quote!”

“Yes, it is,” I responded, “You wrote it!” I continued my brief recap of how he typed it up to give to me through his friend, whom I did not know.

To which he said, “I am a really good guy, huh?”

“Yes you are, sir, so I wanted to thank you.” As I showered him with words of gratitude, an assistant helped him pull the paper out from underneath the framed glass. Unbelievably, he autographed it again. I left Mr. Bradbury to the rest of his fans as I walked away with another amazing, unexpected gift from him.

The gift continues to hang on my classroom wall, and I continue to tell my Smart is Sexy Christmas Story each year. It is my Christmas present to my students, because it speaks about living with purpose and leaving a mark on the world. It is in that spirit that I end my story with what is known as the “Gardner’s Passage.” Merry Christmas to all.

Everyone must leave something behind when he dies, my grandfather said. A child or a book or a painting or house built or a pairs shoes made. Or a garden planted. Something your hand touched some way so your soul has somewhere to go when you die, and when people look at that tree or that flower you planted, you’re there. It doesn’t matter what you do, he said, so long as you change something from the way it was before you touched it  into something that’s like you after you take your hands away. The difference between the man who just cuts lawns and a real gardener is in the touching, he said. The lawn cutter might as well not have been there at all; the gardener will be there a lifetime.

Fahrenheit 451

Good Wishes!
Ray Bradbury
Dec. 1982

Ray Bradbury
Feb. 07

The What, the How and the Who (by Brad Fulton)

When I first began teaching math, I focused on what math skills needed to be taught. It was a changing time in California mathematics with the old guard and their traditional ways pitted against us young, progressively-minded newcomers who wanted to have students explore numeracy and solve the problem-of-the-week. At the end of each day, I slumped at my desk and asked myself, “Now that I have taught that concept, what comes next?” The what of mathematics consumed my thinking. We were moving from a pencil-and-paper age to a calculator-and-computer age, so it was a time of redefining the content of mathematics. Whereas my elementary teachers simply turned to the next page of the textbook, we were asking, “Is it necessary to learn to master long division with decimals?
How important is factoring in this digital age?”

Over the years the pendulum of mathematics swung back and forth between focusing on conceptual understanding or on computation. During these swings, I began to explore the how of teaching. What is the best way to communicate mathematical thinking and processes to the student? Do we begin with a problem? with manipulatives? with a procedure? The emphasis of the how over the what redirected my teaching. The what was still the destination, but the how was the vehicle that would get us there.

This idea was confirmed when I recently read Chris Shore’s blog on the “4½ Principles of Quality Math Instruction”. In the article, Shore noted that the top performing mathematics nations do not demonstrate a similarity in their instructional techniques. However, they do have common underlying principles. Although the what is taught in various ways from one high performing nation to the next, those nations share certain principles. These were the same principles that govern successful math teachers everywhere: Commitment to a few high standards, initial instruction in concepts prior to procedures, good questioning strategies and high accountability.

The focus on how I taught mathematics occupied much of my planning time. However, as the years wore on, I found that I was tiring of the profession. I was floundering through my fourteenth year in education when I learned the lesson that changed my teaching forever. Misbehaving students were making every day a nightmare. I was ready to quit when one morning I decided I had had enough. No longer was someone else going to determine what kind of a day I was going to have. From now on I was going to have a good day regardless of what the students tried to do. I started smiling at their snarling faces. I talked kindly even when they were surly. I commented on their new shoes, asked about their interests, and commended their
accomplishments. The transformation produced immediate results as I connected with my students for the first time.

That’s when I learned that who I taught mattered much more than what I taught. In the past, I had always tried to learn more about math. Now I tried to learn more about my math students. I discovered that students don’t learn math from mathematicians; they learn from people who care about them, and for the first time, they had my heart as well as my head.

There is a wonderful children’s book called The Velveteen Rabbit in which a brand new stuffed toy is given to a boy. Through the years, the boy loves the rabbit and holds it so closely that eventually all its fur is rubbed off. When he grows, the old rabbit is tossed out in the yard. That night the rabbit awakens for the first time to find that he has become real because he was loved. I thought I was a teacher when I first held my brand new teaching credential and state-adopted textbook. Over the years, my students pulled me so close to their hearts that all my hair has since been rubbed off! They grew up, moved on, graduated, and left me behind, but in the process they loved me and I loved them, and somewhere along the way, I became real. We can’t teach the what until we have answered the how. And we can’t teach at all until we know the who.

Brad Fulton, teaches in Redding, CA. He runs Teacher to Teacher Press, with Bill Lombard. The materials and lessons they offer stress conceptual understanding and problem-solving in very creative ways. Brad is an experienced and engaging presenter. I had the chance to sit down to lunch with him at the CMC-South conference this month. We were two old dogs talking about how “we have been doing common core long before they called it common core.” The conversation of how math education has progressed over the last two decades and about the “half principle” led me to invite Brad to write this guest blog.

Math Saves ASB Election

Last spring, I was walking through the front office of school when I was pulled into the ASB office. The director and his secretary appeared fearfully perplexed. They had the results of the election for ASB President, but could not determine a winner, so they needed the help of a math teacher. The winner was scheduled to be announced in less than an hour. Here was there quandry:

They advertised that the election would be based 20% on the Interview, 20% on Teacher Evaluations and 60% on Student Voting. Those results stared at them from a white board on the wall.

   Interview              Evaluations           Voting
Candidate #1       Candidate #1       Candidate #3

However, they could not determine from this data who won the election. I asked if they they had all standings in all three categories.They affirmed that they did and added those results to the whiteboard.

   Interview             Evaluations           Voting
Candidate #1       Candidate #1       Candidate #3
Candidate #2       Candidate #2       Candidate #1
Candidate #3       Candidate #3       Candidate #2

With this information, I posed to weight the election categories much the same way that I do my gradebook. Each candidate’s score was the sum of the three products of the Candidate’s place in the category (3 being highest score, 1 being lowest) and the percentage given that category. Thus,

#1 = 3(0.2) + 3(0.2) + 2(0.6) = 2.4
#2 = 2(0.2) + 2(0.2) + 1(0.6) = 1.4
#3 = 1(0.2) + 1(0.2) + 3(0.6) = 2.2

Candidate #1 wins!

I know there are several other mathematical methods to determine the winner. I invite those solutions in the comments below. My purpose for sharing this antecdote is to pose the question. “Why did it take a math teacher to figure this out?” These were intelligent, educated people struggling with this problem, “Why did they need a mathematician to resolve it?” The math involved is not high level, nor is this type of problem in any math curriculum that I experienced, so “How was I more adept at solving it than they were?”

I am going to surmise (inviting rebuttal), that it is my experience in solving math problems, not my content knowledge of math, that led me to the solution. I definitely have taken more math courses than the Activities Director, yet I did not need Trigonometry or Calculus to crown a champion. I needed simple problem solving skils in a mathematical context. So The Election Problem spurs me to pose more “problems” and fewer “notes and exercises” to my students.