Great Classroom Action

Jennifer Abel creates a promising variety of card sort activity:

Basically, after dealing the cards, the basic idea is for kids to pass one card to the left while at the same time receiving one card from the player to their right. The object of the game is to collect all cards with the same suit/type/category. Here are two examples that I recently created for next year.

Julie Morgan offers three sharp lesson-closing activities. My favorite is “Guess My Number”:

I choose a number between 1 and 1000 and write it on a piece of paper. Each group takes it in turns to ask me a questions about my number. The questions can vary from “is it even?” to “what do the digits add up to?” to “is it a palindrome?” (my classes know I like palindromes!) When a group thinks they have figured it out they write it down and bring it up to me. Each group is only allowed three attempts so cannot keep guessing randomly. I like this for emphasising mathematical knowledge such as multiples, primes, squares, etc.

Pam Rawson contributes to #LessonClose with both a flowchart that illustrates her thought process at the end of classes and then some example exit polls for both “content” and “process” objectives:

As a member of the Better Math Teaching Network, I had to come up with a plan – something in my practice that I can tweak, test, and adjust with ease. So, I decided to focus on class closure. Since I don’t have an actual process for this, I had to think intentionally about what I might be able to do. I created this process map.

Robert Kaplinsky offers the #ObserveMe challenge:

We can make the idea of peer observations commonplace. It’s time to take the first step.

Teaching Math in a Disaster Area

A reader writes from Ascension Parish in Louisiana where 40,000 homes have been impacted by a catastrophic flood:

The Red Cross is asking for certified teachers to help out in the shelters. What would you do if you were asked to help? What kind of activities could be done in a shelter with children and a minimal amount of supplies?

What would you recommend?

Given those constraints, I’d probably use the print-based portions of Jo Boaler’s Week of Inspirational Math along with a bunch of the Tiny Math Games we collected several years ago.

I’d probably cry a bunch, too, I don’t know, Jesus man what a horrible situation.

How 37 States Are Handling Teacher Shortages

Schools are back in session and districts and states around the United States are experiencing teacher shortages. (A superintendent in Idaho calls it a “famine.”) I searched Google News to help me understand how different states are handling the issue. I found that states are:

  • Increasing compensation and benefits, though not without asking for extraordinary concessions from teachers in some cases. (Related: The teacher pay gap is wider than ever.)
  • Lowering standards, as in Utah, where you can now become a teacher without any additional training beyond content certification, or in Pennsylvania, where they’re relaxing the content certification requirements themselves.
  • Abusing the H1-B program by hiring teachers from the Philippines, Spain, and Mexico.
  • Increasing recruitment efforts by reaching out to college sophomores and juniors more aggressively, by reaching out to former district graduates who might want to teach at their alma mater.
  • Increasing new teacher retention through mentorship programs, especially residency programs, which often pair new and veteran teachers, offer stipends, and require multi-year commitments.
  • Sending administrators back into the classroom, in one case.
  • Doing not really much of anything that I can tell, in another case.

This is to say nothing of macroeconomic trends that may require federal action. For example, the 2008 recession resulted in the layoff of newer teachers (hi!) in favor of veterans who are now retiring and it turns out those newer teachers didn’t stick around.

I don’t claim any kind of policy insight. I just find all of this interesting and a little terrifying. If the states are laboratories of democracy, we’re about to watch a number of lab experiments performed on our nation’s kids.

Increasing Compensation and Benefits


There’s a bill to raise teacher pay by 3 percent, and another proposal for 4 percent. The leader of the Senate, President Pro Tempore Del Marsh, R-Anniston, is working on a bill that could offer some teachers a higher rate of pay if they elect not to enter the teacher tenure system.


This year, lawmakers made some progress addressing the problem. They sent the governor a bill that would provide $500,000 for incentives to drive teachers to rural Colorado. The legislation would: Create education training programs through coordinators at colleges in rural parts of the state. Provide stipends to offset tuition costs for student teachers who agree to teach in rural schools. Establish programs in rural areas to identify high school students interested in teaching. Provide money to teachers in rural districts who pursue national certifications.


In Pinellas County, teachers can get up to $25,000 in incentive pay to work at low performing schools, which are currently under investigation by the U.S. Department of Education. But to get the full amount, they must have a master’s degree, be at the top of the pay scale, work 90 extra minutes a day, plus some Saturdays — and attend summer sessions. “There’s an ambulance waiting for them at the end of the rainbow there, because it’s almost undoable,” Gandolfo said.


The Blue Ribbon Commission on the Recruitment and Retention of Excellent Educators – a panel of 49 state office holders and educators – last year issued a report with suggested solutions for state legislators to consider in the 2016 session, however, only one related bill passed. It sets up the New Generation Hoosier Educators Scholarship, which offers up to $7,500 per year for students who commit to teaching in Indiana for at least five years.


Iowa College Aid has announced that 126 teachers were offered awards as part of the Iowa Teacher Shortage Loan Forgiveness Program. The average repayment was $5,493, for a total of $692,171.


The Legislature has supported the 2013 recommendations from Gov. Butch Otter’s education task force, which were created to provide Idaho with an unprecedented five-year plan to improve K-12 education and teacher satisfaction. One of those recommendations — the estimated $125 million dollar career ladder — increases teacher pay. Another piece — the state’s leadership premium package — designates additional money for hard-to-fill positions.


Moore said that his district has offered candidates for positions such benefits as a four-day school week, health benefits and a competitive salary, and candidates are still not willing to accept positions in rural areas.


The school district is currently in negotiations with the tribe in Owyhee to see if they will be able to provide housing for teachers. However, Zander acknowledge that finding housing in areas like West Wendover where there isn’t any teacher specific housing available may still be a challenge.

New Jersey:

“When trying to find in-demand teachers, districts can negotiate certain options such as initial placement on the salary guide, where the Superintendent does have the ability to recommend that a teacher be place on a higher level,” said Belluscio. “And there’s also the concept of signing bonuses for teachers who would be in a difficult to recruit area.”

North Carolina:

Bracy said part of the state’s teacher shortage has to do with fewer college students enrolling in education majors. Another part has to do with low pay, he said. “It sometimes makes the teaching profession not as attractive as others,” Bracy said. Governor Pat McCrory recently signed the state budget, which would allow a 5 percent increase in pay for teachers. That means the average teacher could make close to $50,000.

North Dakota:

The state’s teacher shortage loan forgiveness program has so many applicants that it can no longer fund all who apply, North Dakota University System financial aid director Brenda Zastoupil told the Education Standards and Practices Board on Thursday. This year, 628 teachers applied to receive up to $1,000 in forgiveness from their college loans. Last year, the number was 525 and the year before, 471. The program allows teachers to apply multiple years for a maximum of $3,000.


Voters will have a chance to help teachers out come November due to a proposed state question that will raise sales taxes by 1 percent statewide, which is expected to generate $615 million annually, with about 70 percent designated for a $5,000 pay raise for Oklahoma teachers and other funding for K-12 schools. Republican Gov. Mary Fallin also has discussed a possible special session that could restore some education funding.

South Carolina:

Frances Welch, dean of the College of Charleston’s Teacher Education Department, said she has seen student enrollment in her department shrinking for years now. She said the state needs to expand student loan repayment programs, such as the Teaching Fellows scholarship, and minority recruitment programs, such as Call Me Mister.


The Office of the Superintendent of Public Instruction asked the Legislature this year to raise the salary for beginning teachers, and add signing bonuses and other incentives to make the profession more attractive.


Yet, the salary issue cannot be ignored. According to the Wisconsin Department of Public Instruction, the average public school teacher had 14 years of experience and made a salary of $49,908 in 2013-’14. By comparison, the U.S. Census Bureau calculated that the average salary for a bachelor’s degree holder nationwide in the same year was $62,048.

Lowering Standards


Along with the substitute regulation change, the state board has proposed to eliminate the state regulations requiring that teachers be “highly qualified.”


He’s one of the many Savannah-Chatham schools hired through the Alternative Pathways to Teaching program. It’s a way those with a Bachelor’s degree in any field can become a teacher in Georgia, through a work-as-you-go certification program that takes one to three years to complete.


The board discussed the possibility of busing students to Stanfield, or using curriculum the ODE has established for small schools where an adult supervises classroom learning but does not have to be a licensed math teacher.


The prospect is sufficiently worrisome to Pennsylvania Department of Education officials that they are eliminating obstacles to becoming a teacher – for instance, by easing math requirements on the basic skills assessments that teaching students take as sophomores and have struggled to pass.


Utah has a severe teacher shortage, so it decided to do something about it. Under a new rule, schools can now hire people to teach who have no training in the profession. None whatsoever.

Abusing the H1-B Program


At the start of the summer, Goodsell hired a few recent graduates from out-of-state education colleges. Next, he began Skyping with high-school math and science teachers in the Philippines who had applied for work visas to teach in the United States. He selected 11.


Lewis said her district hopes to fill math, science and special education jobs with teachers from the Philippines. Capitalizing on connections of previously recruited Filipino teachers, officials conduct interviews via Skype and, if successful, navigate the long process of securing an H-1B visa.


Superintendents in Jackson Public Schools, Noxubee County, Holmes County, Meridian and Gulfport have all hired foreign teachers on temporary visas called H-1Bs, along with others. According to MDE, there are 451 teachers in the state with degrees from outside the country.


Dallas employs some 350 teachers from Puerto Rico, including some who have become administrators. The district’s H-1B visa program recently brought in 30 Mexican teachers who will stay for three years. Dallas ISD also takes part in the Texas-Spain Visiting Teacher Program, a J-1 visa initiative that each year brings about 75 Spanish teachers to the district for three years.

Increasing Recruitment Efforts


Wilson says with money tight, they’ve been all about hiring quality teachers instead of just filling their quote. He says that all starts with getting good student teachers from [Southern Illinois University].


Krysl says because of the hiring freeze, they looked for talent from colleges, towns, and states they hadn’t actively recruited from. That’s where they found teachers like Amy Shamp who moved here from Owatanna, Minnesota to teach at Curtis Middle School.


Murphy said the school system has been working to stay in contact with graduates interested in becoming teachers to return to Charles County. “We’re hoping to create that pipeline and connect with those students throughout the years,” Murphy said.


Education Minnesota is proposing several solutions to attract and keep good teachers. The union wants to build a pipeline program to get students interested in teaching as early as high school. It also wants to change the licensure process and provide a stronger financial incentive. Once teachers get into their careers the union wants more collaboration between educators and administrators, stronger financial benefits and more professional development.

New York:

“Our students need, and our districts need, creative bright young people to enter the profession,” he said. “The way to encourage that is to maintain high standards but for our policymakers to show that they value teachers and they respect teachers and they listen to the voice of teachers when they make education policy.”

Rhode Island:

“The unfortunate reality is fewer people today are choosing to enter the education field and lead our classrooms,” said Rhode Island Teach for America Director Heather Tow-Yick. “There are a number of factors contributing to this: we’re seeing an increasingly polarized public conversation around education; the broader economy is improving and people who experienced the national recession during their college careers are showing more interest in what they see as financially sustainable professions; teacher satisfaction has dipped steeply in recent years; and education has been deprioritized by most Americans when they rank major issues facing our country.”

South Dakota:

“I think it’s all about getting people to go into the teaching profession and that means talking to maybe junior and seniors in high school or freshman and sophomores in college,” he said.

Increasing New Teacher Retention through Mentorship


Ritter in 2012 founded the Arkansas Teacher Corps, which recruits the highest achievers from all majors and fields to teach in districts that struggle to hire and retain quality teachers. Corps teachers receive a $15,000 stipend, paid over three years, in addition to their regular teaching salary for working in such a district.


“More teachers are leaving, and fewer teachers are going into the profession,” said Corey Rosenlee, president of the Hawaii State Teachers Association, which represents 13,500 teachers. “We cannot find even emergency hires for these positions.” DOE officials contend that the hiring process is selective and that more focus on a mentoring program for new teachers will continue to boost retention rates.


In Richmond, a more intensive teacher residency is helping retain both new and veteran teachers, said Jan Tusing, a teacher leader-in-residence at VCU’s Center for Teacher Leadership, which runs the residency program. The residency program is an intensive multiyear process where new teachers are paired with a co-teacher. The program’s original goal was to provide extra help for new teachers, but it ended up helping retain the experienced teachers who served as mentors also, Tusing said. That’s a win-win for districts.


Ritter in 2012 founded the Arkansas Teacher Corps, which recruits the highest achievers from all majors and fields to teach in districts that struggle to hire and retain quality teachers. Corps teachers receive a $15,000 stipend, paid over three years, in addition to their regular teaching salary for working in such a district.

Sending Administrators Back into the Classroom

New Mexico:

“You have so many more schools than you did 10 years ago,” Armenta said. “We’ll do what we did last year, and that is, in those classrooms where we haven’t found teachers yet, we will send in administrators from city center again to fill in.” That means administrators who have desk jobs now but are still licensed to teach will be in the classroom.

Doing Not Really Much of Anything That I Can Tell

West Virginia:

“We’re looking at issues in terms of effort, in terms of teaching our teaching workforce overall, making sure our students are able to navigate properly with the numeracy aspects in the classroom, and making sure they can really solve the problems necessary. So there’s not just one silver bullet, there’s a lot of things in motion here.”

Featured Comment:


My only complaint is some of your pejorative language (e.g., “abusing work visas”; “experimenting on students”). You may have good reason to disagree with some or all of the responses, but it seems that this requires some argument rather than name-calling.

That these states are performing experiments on kids seems just empirically true to me.

I am, indeed, editorializing about the H1-B. That program was meant to staff hard-to-find individuals in specialty fields. While I agree that teaching is a specialty field, teachers aren’t hard to find so much as hard to compensate and treat in a way that’s fair. Teachers are, in my view, underpaid, undervalued, overworked, and deprofessionalized. The solution is to fix those working conditions, not exploit an immigration loophole to find people who will tolerate them.

Featured Tweets


a/k/a Oh Come On, A Pokémon Go #3Act, Are You Kidding Me With This?

Karim Ani, the founder of Mathalicious, hassles me because I design problems about water tanks while Mathalicious tackles issues of greater sociological importance. Traditionalists like Barry Garelick see my 3-Act Math project as superficial multimedia whizbangery and wonder why we don’t just stick with thirty spiraled practice problems every night when that’s worked pretty well for the world so far. Basically everybody I follow on Twitter cast a disapproving eye at posts trying to turn Pokémon Go into the future of education, posts which no one will admit to having written in three months, once Pokémon Go has fallen farther out of the public eye than Angry Birds.

So this 3-Act math task is bound to disappoint everybody above. It’s a trivial question about a piece of pop culture ephemera wrapped up in multimedia whizbangery.

But I had to testify. That’s what this has always been – a testimonial – where by “this” I mean this blog, these tasks, and my career in math education to date.

I don’t care about Pokémon Go. I don’t care about multimedia. I don’t care about the sociological importance of a question.

I care about math’s power to puzzle a person and then help that person unpuzzle herself. I want my work always to testify to that power.

So when I read this article about how people were tricking their smartphones into thinking they were walking (for the sake of achievements in Pokémon Go), I was puzzled. I was curious about other objects that spin, and then about ceiling fans, and then I wondered how long a ceiling fan would have to spin before it had “walked” a necessary number of kilometers. I couldn’t resist the question.

That doesn’t mean you’ll find the question irresistible, or that I think you should. But I feel an enormous burden to testify to my curiosity. That isn’t simple.

“Math is fun,” argues mathematics professor Robert Craigen. “It takes effort to make it otherwise.” But nothing is actually like that – intrinsically interesting or uninteresting. Every last thing – pure math, applied math, your favorite movie, everything – requires humans like ourselves to testify on its behalf.

In one kind of testimonial, I’d stand in front of a class and read the article word-for-word. Then I’d work out all of this math in front of students on the board. I would circle the answer and step back.


But everything I’ve read and experienced has taught me that this would be a lousy testimonial. My curiosity wouldn’t become anybody else’s.

Meanwhile, multimedia allows me to develop a question with students as I experienced it, to postpone helpful tools, information, and resources until they’re necessary, and to show the resolution of that question as it exists in the world itself.

I don’t care about the multimedia. I care about the testimonial. Curiosity is my project. Multimedia lets me testify on its behalf.

So why are you here? What is your project? I care much less about the specifics of your project than I care how you testify on its behalf.

I care about Talking Points much less than Elizabeth Statmore. I care about math mistakes much less than Michael Pershan. I care about elementary math education much less than Tracy Zager and Joe Schwartz. I care about equity much less than Danny Brown and identity much less than Ilana Horn. I care about pure mathematics much less than Sam Shah and Gordi Hamilton. I care about sociological importance much less than Mathalicious. I care about applications of math to art and creativity much less than Anna Weltman.

But I love how each one of them testifies on behalf of their project. When any of them takes the stand to testify, I’m locked in. They make their project my own.


Why are you here? What is your project? How do you testify on its behalf?

Related: How Do You Turn Something Interesting Into Something Challenging?

[Download the goods.]

A Very Valuable Conjecture


Here is a very valuable conjecture:

The spelling of every whole number shares at least one letter with the spelling of the next whole number.

Which is to say that:

  • “one” and “two” both share an “o”
  • “two” and “three” both share a “t”
  • etc.

Could that possibly be true for every whole number?

If I were starting a course on geometry or a unit on proof or an activity on deductive logic, I would introduce this conjecture very early in the process. Let me explain what I find so very valuable about this conjecture.

Deduction is hard. It’s an abstract mental act that adults find difficult. (See: the van Hiele’s and their levels.) Too often we rush students to that abstract act, rushing them past the lower van Hiele levels, and we ask them to argue deductively about objects that, to them, are also abstract.

I suspect that, to many students, those proof prompts read something like this:

Given that the base bangles are twice the tonnage of the circumwhoozle and the diagonalized matrox is invertible, prove that all altimeters cross the equation at Quito, Ecuador.

The word “prove” is weird. And, unfortunately, so is every other word in the sentence.

So I cherish opportunities to help students argue deductively with concrete objects, which is what we’re working with here, with the spelling of whole numbers. This conjecture also gives students several different angles on the proof act.

You can ask students to find a counterexample, for example, a useful strategy when first interrogating a conjecture.

Once students have tried several different numbers they may satisfy themselves that the conjecture is true. This is one of the naive proof schemes Harel & Sowder observed in the students they studied. When this proof scheme surfaces in conjectures about geometric shapes, it’s challenging to summon up one new shape after another to challenge the student’s proof by example. It’s trivial, by comparison, to summon up one new number after another and ask the student to check her hypothesis again.

At a certain point in this process, likely after you give several numbers in the millions, your students may transform in two ways:

  1. They’ll get tired of trying example after example. “Proof by examples means you have to try all the examples,” you can say, giving you both a moment to reflect on the need for a more rigorous proof scheme, like deductive reasoning.
  2. They’ll notice that every number in the millions shares an “n” with every other number in the millions. And same for the billions. And same for the trillions. And … same for the hundreds. And so on.

And suddenly we’re on our way to a proof by exhaustion, which is much more rigorous than a proof by example. Nice.

This conjecture also leaves ample room for you and your students to pose follow-up conjectures. Like, “Does it work for all integers, or just whole numbers?”

I saw the conjecture and saw its value immediately. This is a very valuable kind of conjecture, I thought. But I don’t have many of them. Do you have another you can trade?

[via Futility Closet.]

BTW. You’re worse off in at least one way now than before you knew the conjecture was true. Now, when you ask your students, “Could that possibly be true?” you’re going to have to pretend.

Featured Comments:


What I like about game strategies is you go from “what seems to work,” to “will this always work,” to “here’s why this does/doesn’t work” pretty seamlessly.

Paul Hartzer:

Is it true for any other language with an alphabet? It fails for German (5/6 using umlauts, 7/8 otherwise), French (2/3), and Spanish (7/8).

Michael Serra:

What letter(s) of the alphabet do(es) not appear in the spelling of the first 999 whole numbers? Prove it.

Lynn CP:

List all the factors of every number from 1-100. What do you notice? Which numbers have an even number of factors? An odd number? What do you notice about numbers with an odd number of factors? Can you prove which numbers beyond 100 will have an odd number of factors?