Teaching teachers

How do you get students fired up about fractions? Reinvent math class.

PHOTO: Alex Zimmerman
Math Lab co-founder Peter Cipparone walks teachers through the morning's lesson before students arrive.

Earlier this week, Peter Cipparone was ending a meticulously planned lesson when the student he’d called to the board suddenly slipped. Kris, a rising fifth grader, was, Cipparone later said, a “plant” — selected because Cipparone felt confident he could clearly explain the fractions problem at hand to the rest of the class.

But when Kris got to the board, his understanding seemed to vanish. He was supposed to be placing the fraction five-thirds on a number line, but instead he marked off five and one-third, almost four points too far.

With just a few minutes left in class, held at the High School for Fashion Industries in Manhattan, Cipparone had to make a split-second decision: Should he offer the correct answer, or let students leave for the day without a clear resolution?

It was a tough call, but Cipparone knew he had backup. Unlike a traditional lesson, where educators teach in isolation, in this case, Cipparone was surrounded by 21 other teachers who would pore over every element of his teaching once the class ended.

The lesson is part of a New York City program, now in its second year, that is tasked with teaching two groups of people simultaneously: A group of rising fifth-graders like Kris, who spend two hours each day learning about fractions — and a group of educators from across the city who have signed up to improve their own teaching.

The program, known as the New York City Math Lab, gives educators an opportunity to perform the surprisingly rare work of watching each other teach. It also lets them hone an approach encouraged by the Common Core learning standards that many of the participating teachers said they have struggled to perfect: Getting students to learn math not just by practicing problems, but also by having conversations.

“With rules-based teaching, if you don’t understand why you’re doing what you’re doing, then that rule only applies to that very specific type of problem,” explains Kim Van Duzer, a teacher at Brooklyn’s P.S. 29 who co-founded the lab program along with Cipparone and Kate Abell, who have both taught in New York City schools. “If you have an underlying understanding of operations and the number system, you can build on that to answer other sorts of questions — that’s what reasoning is.”


For a week during the summer, Van Duzer, Cipparone and Abell host roughly 25 educators with a range of teaching experience who watch the trio perform a sequence of lessons on fractions, sitting on the sidelines of a large classroom as the class works. Then, when the students leave, returning to a summer program offered by the Hudson Guild, the teachers talk about what they saw.

The Math Lab’s emphasis on learning math by talking and thinking about it is clear almost as soon as the students enter the room. A list of the class’ “math community agreements,” posted on a board, reminds students to “add onto each other’s thinking” and “analyze and observe each other’s work.”

Teachers observed students taking in Tuesday's lesson.
PHOTO: Elizabeth Green
Teachers observe students taking in Tuesday’s lesson.

To help students internalize that philosophy, Van Duzer led an activity called “convincing a skeptic,” where students were asked to fold pieces of green paper into squares one quarter the size of the original and then convince their partner that the new shape was, in fact, one-fourth of the original.

Some students struggled to articulate why the squares they folded were one-fourth of the whole piece of paper. “Sometimes my partner asked questions I didn’t understand,” one student admitted. But encouraging students to challenge each other’s ideas paid off later that morning.

After introducing the idea of representing fractions on a number line, Cipparone asked students to begin thinking about whether eight-sixths is greater than one. Nine and 10-year-old skeptical voices quickly emerged.

One student declared that eight-sixths is less than one, only to be told by someone sitting nearby that he had the numerator and denominator confused. The ensuing debate ended when the first student admitted his mistake and leapt at the chance to offer a correct answer in his own words.

Many of the teacher observers said the lab was the first time they were able to watch another educator teach consecutive lessons. Watching others teach, they said, helped them become more confident in their own ability to guide students through conversations about fractions.

“You can read a [lesson] plan, but they do so much more that’s not in the plan — how to elicit students’ thinking, or how to get them to respond to each other,” said Julie Heller, a teacher at Brooklyn’s P.S. 29, where Van Duzer also teaches.

Laura Burns, a 20-year veteran teacher from P.S. 73 who also attended last year’s lab, said the experience gave her ideas about how to tweak her instruction, including posting a schedule outlining the elements of the day’s lesson so students have a sense of where they are headed.

“I can just watch and see everything – the language, the materials, the way they’ve laid stuff out. They’ve agonized over every decision,” she said.


As teachers reflected on Tuesday’s lesson, a debate of their own emerged. They began wondering about how Cipparone handled what the group would begin calling “Kris’ problem” — the moment that morning when Kris misplaced five-thirds on the number line and Cipparone had to make a split-second decision about whether to correct him before the students left for the day.

Deirdre Flood, a teacher at Brooklyn’s P.S. 11, said it could make sense to end the lesson ambiguously if “every single [student] made a decision before they left, so they were thinking about it on their way out.”

Another teacher wondered whether withholding the correct answer might leave Kris feeling tricked when he inevitably discovered his answer had been wrong all along. Meanwhile, Abell, one of the Math Lab’s facilitators, suggested a different approach altogether.

“It feels like a terrible thing to not let him finish his idea,” she said. “The question for the community is, ‘What is Kris thinking?’ Because he’s thinking something that makes sense.” Instead of debating whether to correct him or not, she argued, teachers should encourage other students to explain why he came to equate five-thirds for five and a third — because he mistakenly considered five as the whole, rather than the number of parts.

For his part, Cipparone never pointed out Kris’ error that morning. He threw the question back to the class for a quick debate. And even though some students argued their classmate was wrong, Cipparone stood back and facilitated.

“Sometimes you have questions and you don’t get the answers all at once,” he said, just moments before students filed out. “If you’re not convinced, that’s OK!”


Colorado schools are getting a major bump in the state’s 2018-19 budget

Students waiting to enter their sixth-grade classroom at Kearney Middle School in Commerce City. (Photo by Craig Walker, The Denver Post)

Colorado’s strong economy has opened the door for state lawmakers to send a major cash infusion to the state’s public schools.

As they finalized the recommended budget for 2018-19, the Joint Budget Committee set aside $150 million, an additional $50 million beyond what Democratic Gov. John Hickenlooper had asked for, to increase funding to schools.

“We believe this is the most significant reduction in what used to be called the negative factor since it was born,” said state Rep. Millie Hamner, the Dillon Democrat who chairs the Joint Budget Committee.

Colorado’s constitution calls for per pupil spending to increase at least by inflation every year, but the state hasn’t been able to meet that obligation since the Great Recession. The amount by which schools get shorted, officially called the budget stabilization factor, is $822 million in 2017-18. Under state law, this number isn’t supposed to get bigger from one year to the next, but in recent years, it hasn’t gotten much smaller either. 

But a booming economy coupled with more capacity in the state budget created by a historic compromise on hospital funding last year means Colorado has a lot more money to spend this year. In their March forecast, legislative economists told lawmakers they have an extra $1.3 billion to spend or save in 2018-19.

The recommended shortfall for next year is now just $672.4 million. That would bring average per-pupil spending above $8,100, compared to $7,662 this year.

Total program spending on K-12 education, after the budget stabilization factor is deducted, should be a little more than $7 billion, with the state picking up about $4.5 billion and the rest coming from local property taxes.

The budget debate this year has featured Republicans pressing for more ongoing money for transportation and Democrats resisting in the interest of spreading more money around to other needs. The positive March forecast reduced much of that tension, as a $500 million allocation for transportation allowed a compromise on roads funding in the Republican-controlled Senate. That compromise still needs the approval of the Democratic-controlled House, but suddenly a lot of things are seeming possible.

“We knew we were going to have more revenue than we’ve ever had to work with,” Hamner said of the status at the beginning of the session. But that presented its own challenges, as so many interest groups and constituencies sought to address long-standing needs.

“The fact that we’ve been able to reach such incredible compromises on transportation and K-12 funding, I think most members will be very pleased with this outcome,” Hamner said. “Where we ended up is a pretty good place.”

The big outstanding issue is proposed reforms to the Public Employees Retirement Association or PERA fund to address unfunded liabilities. A bill that is likely to see significant changes in the House is wending its way through the process. The Joint Budget Committee has set aside $225 million to deal with costs associated with that fix, which has major implications for teachers and school districts budgets.

The Joint Budget Committee has also set aside $30 million for rural schools, $10 million for programs to address teacher shortages, and $7 million for school safety grants.

The budget will be introduced in the House on Monday. Many of the school funding elements will appear in a separate school finance bill.

Going forward, there is a question about how sustainable these higher funding levels will be.

“It does put more pressure on the general fund,” Hamner said. “If we see a downturn in the economy, it’s going to be a challenge.”

outside the box

Program to bring back dropout students is one of 10 new ideas Jeffco is investing in

File photo of Wheat Ridge High School students. (Photo by Nic Garcia/Chalkbeat)

Jeffco students who drop out will have another option for completing high school starting this fall, thanks to a program that is being started with money from a district “innovation fund.”

The new program would allow students, particularly those who are older and significantly behind on credits, to get district help to prepare for taking a high school equivalency test, such as the GED, while also taking college courses paid for by the district.

The idea for the program was pitched by Dave Kollar, who has worked for Jeffco Public Schools for almost 20 years, most recently as the district’s director of student engagement.

In part, Kollar’s idea is meant to give students hope and to allow them to see college as a possibility, instead of having to slowly walk back as they recover credits missing in their transcripts.

“For some kids, they look at you, and rightfully so, like ‘I’m going to be filling in holes for a year or two? This doesn’t seem realistic,’” Kollar said. “They’re kind of defeated by that. As a student, I’m constantly looking backwards at my failures. This is about giving kids something like a light at the end of the tunnel.”

Jeffco’s dropout rate has decreased in the last few years, like it has across the state. At 1.7 percent, the rate isn’t high, but still represents 731 students who dropped out last year.

Kollar’s was one of ten winning ideas announced earlier this month in the district’s first run at giving out mini-grants to kick-start innovative ideas. Kollar’s idea received $160,000 to get the program started and to recruit students who have dropped out and are willing to come back to school.

The other ideas that the district gave money to range from school building improvements to comply with the Americans with Disabilities Act at Fletcher Miller Special School, from new school health centers to a new district position to help work on safety in schools. One school, Stott Elementary, will create a “tinker lab” where students will have space and supplies to work on projects as part of the school’s project-based learning model.

The Jeffco school board approved $1 million for the awards earlier this year. It was an idea proposed by Superintendent Jason Glass as a way of encouraging innovation in the district. This spring process is meant as a test run. The board will decide whether to continue investing in it once they see how the projects are going later this spring.

Officials say they learned a lot already. Tom McDermott, who oversaw the process, will present findings and recommendations to the board at a meeting next month.

If the board agrees to continue the innovation fund, McDermott wants to find different ways of supporting more of the ideas that educators present, even if there aren’t dollars for all of them.

That’s because in this first process — even though educators had short notice — teachers and other Jeffco staff still completed and submitted more than 100 proposals. Of those, 51 ideas scored high enough to move to the second round of the process in which the applicants were invited to pitch their ideas to a committee made up of Jeffco educators.

“We’re extremely proud of the 10,” McDermott said, but added, “we want to be more supportive of more of the ideas.”

McDermott said he thinks another positive change might be to create tiers so that smaller requests compete with each other in one category, and larger or broader asks compete with one another in a separate category.

This year, the applicants also had a chance to request money over time, but those parts of the awards hang on the board allocating more money.

Kollar’s idea for the GED preparation program for instance, includes a request for $348,800 next year. In total, among the 10 awards already granted, an extra $601,487 would be needed to fund the projects in full over the next two years.

Awards for innovation fund. Provided by Jeffco Public Schools.

The projects are not meant to be sustained by the award in the long-term, and some are one-time asks.

Kollar said that if that second phase of money doesn’t come through for his program, it should still be able to move forward. School districts are funded per student, so by bringing more students back to the district, the program would at least get the district’s student-based budget based on however many students are enrolled.

A similar program started in Greeley this fall is funded with those dollars the state allocates to districts for each student. So far, eight students there already completed a GED certificate, and there are now 102 other students enrolled, according to a spokeswoman for the Greeley-Evans school district.

But, having Jeffco’s innovation money could help Kollar’s program provide additional services to the students, such as a case manager that can help connect students to food or housing resources if needed.

And right now Kollar is working on setting up systems to track data around how many students end up completing the program, earning a high school equivalency certificate, enrolling in a college or trade-school, or getting jobs.

Helping more students on a path toward a career is the gold standard, he said, and what makes the program innovative.

“It’s not just about if the student completes high school,” Kollar said. “It’s are we making sure we are intentionally bridging them into whatever the next pathway is?”