Cookies and math tend to go together in an elementary school classroom. And not always as reward for a correct answer. Teachers use them as conceptual props to explain an operation like division. It works intuitively enough with whole numbers. If you have 12 cookies and four friends, how can you give an equal number of cookies to each friend?
The trouble with this standby analogy comes when you divide by a fraction. You can have the same 12 cookies, but if you divide by ¼, the old approach won’t work. We don’t have friends who come in quarters.
“You have to think that you’re not dividing up the cookies, you are seeing how many times ¼ can fit into 12,” explained Juli Dixon, professor of mathematics education at the University of Central Florida and author of several books on teaching math. “It’s a change in thinking. We used to teach procedural math, but now students have to understand the ‘why’ as much as the ‘how’.”
Depth of understanding was hailed by its architects as a cornerstone of the Common Core, a set of educational guidelines for what students need to know in each grade in English and math that have been adopted in 43 states and the District of Columbia. The problem is that most elementary school teachers did not learn math that way, and many now struggle to teach to the new standards.
An April 2016 study of a large urban school in Georgia reported the frustration of many elementary level teachers. Only two out of ten teachers there said they were very familiar with the standards and one out of four reported no training on how to teach to them. If the Common Core is to improve the math education of U.S. students as intended, experts agree that teachers who are meant to get students excited about math and become proficient in its basic concepts need more help and support. Yet the exact nature of that support and how to provide it are debated.
“Teachers are working harder than ever!” Dixon repeated a few times, and experts are adamant to not blame the teachers. But possession of content knowledge comes up as a key reason for the frustration. Schools of education typically require students earning a degree in elementary education to take regular arts and science classes – such English literature, science, social studies, math, fine arts, and even physical education – as well as a “methods” class that teaches them how to present the material and to anticipate the common errors students make. The required math classes, according to Dixon can include college algebra and statistics.People jump to the conclusion that teachers are not smart and don’t know math. This is not true. There’s no relationship between how many math courses K-8 teachers take and the achievement of their students.
“Elementary school teachers are generalists,” said John Ewing, president of Math for America, a non-profit that offers fellowships to teachers to improve math pedagogy. “Their content knowledge is less than what a specialist would have so they don’t understand math in a broad way. Preparatory programs have to be more attentive and have a way to develop teacher expertise.”
Some schools of education have sought to address the problem by revising their curriculum and adding more math coursework. After Illinois adopted the Common Core in 2010, Northwestern University School of Education added another math requirement and revised its methods course. The University of Central Florida added a second class that shows future teachers how to teach Common Core content in addition to its methods class. The University of Wisconsin-Madison School of Education already had a requirement of four math classes and administrators were happy that others were catching up.
But Deborah Ball, dean of University of Michigan’s education school, disagrees that the tension teachers experience has to do with how much math they studied. “People jump to the conclusion that teachers are not smart and don’t know math. This is not true. There’s no relationship between how many math courses K-8 teachers take and the achievement of their students,” she said.
In her view, the preparation that is needed to teach to the new standards goes beyond learning more algebra and even beyond a regular methods class. What’s needed is a class geared specifically to guiding teachers through problem solving from various angles and making connections between number operations, just like students are expected to do.
That can come in the form of a class that focuses on content for teaching class or in the form of professional development, instruction that teachers receive on the job. Susan Lee Swars, co-author of the April 2016 study of a Georgia urban school and a professor of math education at Georgia State University, said she was called in to provide professional development for the school and ran the study to see what kind of help the teachers would need. It was the school’s second year of Common Core instruction, and only 7 percent of the teachers surveyed strongly agreed that they were prepared to teach the standards and many voiced the sentiment that they needed to “unlearn” math and relearn it again. Other teachers spoke of encountering a lot of resistance in the classroom when they tried to modify math class to be more about the process than the solution.You can have the same 12 cookies, but if you divide by ¼, the old approach won’t work. We don’t have friends who come in quarters.
When the standards first rolled out at Los Molinos Elementary School in Hacienda Heights, California, where he taught fourth and fifth grade for 10 years, Gabriel Ward said his students were so used to plugging in formulas it was a struggle. “Both my high and low achievers wanted to know, ‘did I get the right answer?’ But I kept pushing them to validate their answers with different approaches.”
Between fall 2012 when the standards were first enacted and spring 2015 when he left the classroom for a coaching position, Ward remembers teachers being pulled out a few days for professional development.
But Judy Fancher, the assistant superintendent of his school district — the Hacienda La Puenta Unified School District — felt the teachers needed more. Last summer, the district offered an intensive math clinic to its more than 1,000 teachers run by an educational group that Florida’s Dixon started. Teachers studied topics such as calculating the area of geometric shapes, how to solve word problems in the way the standards demand, and making connections between different arithmetic operations in order to understand how fractions, decimals and whole numbers all fit in the number sequence.
Funded by the school district, more than 200 teachers signed up for the three-day class. This summer, she already has 300 sign ups for the mid July clinic.
This program is in addition to a two-week summer math intensive for teachers of grades three to five that is funded by a grant from the California Department of Education and other daylong conferences. If a third-grade teacher were to take advantage of all the professional development offered, Fancher said she or he could have over 100 hours of help.
MAF’s Ewing suggested another on-the-job approach such as the one Finland uses. One teacher is trained as a math specialist and they tutor the other teachers to help them construct lesson plans. Ewing said that a school is a community with instructors of different skill levels. There’ll be a teacher less comfortable with a topic and you’ll have someone who can train them.
“I used to be against specialists because in time of budget cuts, they’re the first to go,” Ewing said. “But if you want to teach Common Core properly, we’ll probably move to a country of specialists. I’ve come around on this.”
[Cross-posted at The Hechinger Report]