Teaching elementary school math just got trickier, or at least deeper.

The new Common Core State Standards require students to demonstrate a deeper understanding of math concepts, which means teachers will have to change how they teach those concepts too.

The new standards, adopted in California and 44 other states, have ushered in a whole new set of academic standards for math, with significant changes in the early grades – kindergartners must now be able to count to 100 by the end of the year, for instance, rather than 30. Second graders will no longer learn multiplication tables; that’s now a third grade task. And geometry standards are now less about identifying and measuring shapes and more about building and deconstructing them.

Common Core standards for kindergarten through 3^{rd} grade math also require students to demonstrate a greater depth of understanding than needed under the previous California State Standards, established in 1997. While the old standards were often criticized for an excessive reliance on memorizing certain facts or procedures, the new standards routinely call for students to solve problems that require a strong grasp of mathematical concepts and to explain their reasoning.

“Most people think math is computation at the elementary level – drilling them in the skills,” said Jeanie Behrend, an education professor focused on math education at California State University, Fresno. “Math is really about application and problem solving.”

#### Major challenge

Making this shift in the classroom – from focusing on computation to focusing on problem solving strategies – stands to be one of the biggest challenges teachers face as they work to implement the new standards, Behrend said. Students will take practice tests aligned to the new standards this year in preparation for the first statewide assessment tied to Common Core in spring 2015. While some teachers are ready to tackle the new standards, others said they have received scant training in how to teach math as a hands-on, exploration based activity.

“It’s easy (for teachers) to focus on memorization of facts and memorization of procedures without really identifying the important mathematics” behind them and teaching those concepts to students, Behrend said.

Memorization has never been the best way to learn math, Behrend said, but it was often enough to meet many of the old standards.

The new standards, adopted in California and 44 other states, have ushered in a whole new set of academic standards for math, with significant changes in the early grades – kindergartners must now be able to count to 100 by the end of the year, for instance, rather than 30.

For example, California’s 1^{st} grade students have long been required to “commit to memory” all the ways of adding two numbers to come up with sums between two and 20. While the Common Core standards do require 1^{st} grade students be able to solve addition problems with sums between two and 20, the requirement that they commit the equations to memory has ~~ been moved to the 2nd grade. Students can now use multiple strategies – drawing a picture or using building blocks as counting aides, for example – to find their answers and demonstrate proficiency.~~

Strategies based on the theoretical underpinnings of mathematics are also emphasized in the new standards. First grade students may not have to memorize as many addition equations, but now they must be able to understand and demonstrate the use of the commutative property of addition, that is: *6 + 2 = 2 + 6.* Previously, this idea was not introduced until 2^{nd} grade.

*(See chart below comparing the old California State Standards to the new Common Core Standards.)*

By removing memorization standards and requiring teachers to cover fewer topics over the course of a year, the new standards are also meant to encourage teachers to spend more time on the underlying concepts behind mathematical concepts.

Changing how math is taught in California elementary schools could be critical to student success. California 4th graders now rank 46th in the nation in math, with only 33 percent considered proficient, according to scores on the 2013 National Assessment of Educational Progress.

#### Teaching the teachers

The shift in what students need to know is requiring teachers to learn new concepts, as well.

In teacher Jenny Aguirre’s 1^{st} grade classroom at Ayer Elementary in Fresno, students were learning about addition during a class at the start of the school year. Aguirre has taught addition many times in her 12 years as an elementary school teacher, but she was going at it a bit differently this year.

Each of Aguirre’s students had a yellow laminated mat and eight poker chips. The number eight was written at the top of the specially designed mat, followed by space for an equation and two equally sized boxes. The goal was to split the chips between the two boxes to find different ways of combining two numbers to make eight.

Seated toward the back of the room, Eliana Garcia, 6, put five chips in one box and placed the three that were left into the second box. Next, she wrote an equation to illustrate her finding: *5 + 3 = 8*

The goal of the activity was to allow the 6- and 7-year-old students to discover an important property of arithmetic on their own. They knew the answer was always going to be eight because the number was written on every mat and they had eight chips. But they were learning that there are multiple ways to add up the chips to equal eight. Most students started by splitting the chips evenly into two groups of four, but quickly progressed to trying new combinations and flipping the order of their equations.

Leon Thao, 6, even discovered that *0 + 8 = 8. *

Aguirre was so impressed by Leon’s work – a big milestone in the understanding of addition in 1^{st} grade – that she asked him to come up to the front of the room and show his classmates how he’d figured out what happens when you add eight to zero.

In years past, Aguirre said, she would have started with traditional addition facts and left the lesson with the mats, which was not technically required under the old 1^{st} grade standards, to later in the year. She also might not have asked students to work in pairs or spent time letting a student haltingly explain his mathematical reasoning. This year, she’s doing all of those things in an effort to comply with the new Common Core standards.

Aguirre said she’s excited about the new standards, but she’s not completely clear on how they will change what she’s expected to present in the classroom.

“There’s not (been) a lot of training on how to teach (Common Core) math or on what we should expect of the children,” she said in August. “This is just a total shift for us.”

Fresno Unified has since begun their effort to train teachers in Common Core by offering several district-wide trainings as well as more focused trainings for teachers in small groups.

#### Killing ‘drill and kill’

Third grade teacher Jaime Button enrolled in a three-year training program offered to teachers in her district by the Shasta County Office of Education, in the hopes of improving her math instruction.

“The saddest thing is that (my students) could parrot back answers, but they didn’t have a real understanding” of the math, said Jaime Button, a 3rd grade teacher at Alta Mesa Elementary in Redding.

Button, who teaches at Alta Mesa School in Redding, said she has felt pressured over the last decade to use a “drill and kill” method to prepare students for state standardized tests – drilling students repeatedly on specific skills without spending much time explaining the underlying concepts. This clashed with what she’d learned when she was training to be a teacher two decades ago, she said, but her school and district leaders wanted teachers to cover all of the 3^{rd} grade material included under the California State Standards and she didn’t feel she had time to explore other methods.

“The saddest thing is that (my students) could parrot back answers, but they didn’t have a real understanding” of the math, she said.

Button said she has been thrilled to find that the new training is encouraging her to return to a more hands-on way of teaching math, one in which students can puzzle out math problems using objects or drawings or by working in teams, rather than following along in a book or filling out a worksheet. She’s also pleased that she’ll be required to cover fewer topics, allowing her to spend more time on complex ideas.

In another Fresno elementary school, kindergarten teacher Joe Dawson worked with his students on counting in late August. He sat at a small table in his classroom at Robinson Elementary School watching his students try to count sets of 20 plastic bears he’d given each of them. Though some students counted the bears easily, others made all sorts of mistakes: counting some bears twice, skipping others, pointing to each bear but saying the numbers in the wrong order.

Juh’Ziyah Atchinson, 5, was one the students who couldn’t seem to get it right. Dawson, who has been trained in a method of teaching mathematics called cognitively guided instruction, was more interested in understanding why Juh’Ziyah kept counting bears twice than he was in correcting her. The idea behind this method is for teachers to build on a child’s existing knowledge about math to guide them to the correct answer, rather than quickly correcting them. Instead of simply telling Juh’Ziyah to only count each bear once, Dawson started a conversation with the little girl to determine if she even understood that basic counting rule.

#### Path of discovery

“I’m not an imparter of information,” Dawson said later. “I want my children to discover.”

Getting them to discover often means conversations where Dawson simply says out loud what he’s seen a child do – “I noticed you counted a few (bears) twice.” Observations like this are meant to make a child reflect on what she has just done and hopefully learn something new by being directed to focus on the actions called out by the teacher.

In Juh’Ziyah’s case, Dawson concludes that she does know objects can only be counted once but is having trouble keeping track of which bears she’s already counted. He shows her a tool to help with that: Organize the 20 bears into several short rows, rather than trying to count a messy jumble.

Megan Franke, an education professor and teacher trainer focused on math education at the University of California, Los Angeles, said research shows that teaching math like Dawson does is more successful than teaching methods focused on memorizing right answers. The Common Core elementary school math standards, Franke said, are based on that research.

“The math practices (in Common Core) are asking kids to explain their mathematical thinking, to represent their ideas and to make sense of other people’s ideas,” Franke said. She hopes these principles encourage a shift in how math is taught that will ultimately change what children know about math.

Dawson is confident that teaching math based on these principles will help Juh’Ziyah and his other students meet the Common Core Standards. In previous years his students have exceeded the old state standards, he said.

By the week before Thanksgiving, his confidence proved warranted: Juh’Ziyah was not only counting to 100 by ones (i.e. 1, 2, 3…) and 10s (i.e. 10, 20, 30…), she was solving three-part word problems using subtraction.

“If Juh’Ziyah had 15 cookies and wanted to give three to one friend, three to another friend and two to a third friend, would she have enough?” Dawson said he asked the girl in November. “And if she had enough, how many would she have left?”

Juh’Ziyah, who hadn’t been able to count to 20 three months ago, was undaunted. Using small objects to represent the cookies, she nailed it: Yes, she’d have enough cookies to share, she told Dawson, and she’d have seven left.