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This content was originally published by Education Week as a two-part Q&A.

The debate over what early math should look like and what should be included in the Common Core State Standards for early mathematics has been contentious, to say the least.

We asked Bethany Rittle-Johnson, a professor of psychology and human development at Vanderbilt University in Tennessee, to explain the most important elements of early math based on her research into how children in preschool through first grade learn about the world of numbers.

Rittle-Johnson said there are specific things educators and parents can do in the early years to lay the foundation of children’s mathematical reasoning. We’ve highlighted seven of them below.

1. Children need to understand the concepts undergirding mathematics and memorize basic math facts.

Rittle-Johnson called the debate over whether kids should be taught concepts or drilled on math facts a “silly argument.”

“The evidence is pretty clear that children really need to do both things,” Rittle-Johnson said. For example, she added, “if you have to spend all your time figuring out what two plus three is, then you can’t notice relationships between number pairs.”

What’s important is to help kids learn facts and strategies in ways that enable them learn math rules, like order doesn’t matter in addition. While 2+3=5, it’s also true that 3+2=5. “It’s not very effective to just tell kids that, though,” Rittle-Johnson said. “They have to have experiences with it.”

Once they do, she said, they’ll begin to look for patterns and structures on their own.

2. How practice problems are structured makes a difference.

We can’t just hope children pick up on math rules without any hints, Rittle-Johnson said, so how teachers lay out their practice problems makes a difference. For example, a kid is far more likely to pick up on the rule that two numbers, added in any order, will always have the same sum if two problems are placed next to each other on a worksheet. An example: putting 2+3=? next to 3+2=?

“If the problems are randomly distributed, that’s not going to help as much as if they’re together,” Rittle-Johnson said.

3. Understanding the equal sign early is important.

“Kids actually make this very smart inference that just happens not to work very well in the long-term,” Rittle-Johnson said. They assume “that the equal sign means: ‘Add up the numbers.’”

Kids make that mistake because “the vast majority of problems they see through second grade are addition problems,” Rittle-Johnson explained. As they add subtraction and other operations to their repertoire, they may expand the definition of the sign to “Get the answer.” Again, smart, but incorrect, Rittle-Johnson said. And that mistake could set kids back quite a bit when they start middle school math.

“We have really clear evidence that it messes them up when they get to algebra, where you need to understand the equal sign means each side is the same as the other side,” Rittle-Johnson said.

Related: Why is this Common Core math problem so hard?

One possible solution again makes use of how teachers present practice problems. Instead of presenting a set of problems that involve adding 2 — 4+2=? and 5+2=? and 6+2=? — teachers can present a set of problems that add up to 6 — 4+2=? and 5+1=? and 3+3=? Research shows that second graders taught this way “learn the arithmetic facts just as well, but they understand the equal sign a lot better,” Rittle-Johnson said.

4. Using hands-on materials to learn basic math is fine, but don’t overdo it.

Using “manipulatives,” or hands-on materials, to teach math in the early grades can be effective, but Rittle-Johnson cautioned that these tools can trip kids up as well.

“I did this work with patterns with cute plastic bugs and sometimes the kids just wanted to play with the bugs,” Rittle-Johnson said. “Manipulatives can distract kids or get them to pay attention to things like color when it’s irrelevant.”

Research shows that kids need to move on from manipulatives at a fairly young age, Rittle-Johnson said. By age 5, children should begin learning about symbols in math — both the actual numerals and the symbols for addition, subtraction and equality.

“Kids can actually learn a decent amount from abstract symbols,” Rittle-Johnson said.

5. Preschool math should be taught purposefully.

Teaching math in preschool does not mean sitting kids at desks and drilling them on math facts, but singing a few counting songs about monkeys on beds isn’t enough either. It’s important to teach math concepts directly and purposefully, Rittle-Johnson said.

Teachers should take advantage of opportunities to point out real-life math situations in the classroom whenever they can, according to Rittle-Johnson. For example, a teacher might say, “I see that there are three dolls and four children. Can everyone have a doll? How can we share?” Or she might encourage numeracy practice during regular classroom activities by saying things like, “Let’s count the blocks as we pick them up from the floor. How many blocks were we playing with?” Or she might push kids to explain their reasoning by saying something like, “You’re right, it’s a triangle. How do you know it’s a triangle?”

In addition, teachers should have a strong math curriculum in place, Rittle-Johnson said. “You might hear the word ‘curriculum’ and think, ‘Oh, my God,’ but, you know, they’re preschool curriculums,” she said. “They’re not second grade curriculums that were just dumbed down.”

6. Numbers are very important in preschool — and that doesn’t just mean kids need to know how to count.

“When children count objects they don’t [necessarily] know that the last number they say indicates how many there are,” Rittle-Johnson explained. Memorizing the sequence 1, 2, 3, 4, 5 and pointing to objects as they recite that sequence is an important task for young children, she said, but it’s not enough.

If a child has just counted four objects and you ask him how many he’s counted, he’ll often just count again, rather than repeating the last number, she said. Or, if you ask that same child to give you four of something, he might just give you a random handful. Both of these actions are signs that the child has memorized how to count to four, but doesn’t fully understand what four is.

“Just because your child can count objects and recite the count-word sequence, as we call it, doesn’t mean those words have any meaning to them,” Rittle-Johnson, said. Children “need to understand four objects, the verbal number four, then, eventually, the written symbol four.”

7. Patterns should probably be part of the Common Core State Standards for kindergarten math.

Rittle-Johnson’s most recent research has focused on patterns and whether or not learning about simple, repeating patterns (such as A, A, B, B, A, A, B, B) in preschool has an effect on later learning in math. Both her research and that of others now show that children who understand how to decipher rules governing repeating patterns do better on unrelated math tasks in later years.

When the Common Core State Standards were written, this evidence linking pattern knowledge to later math outcomes didn’t exist, Rittle-Johnson said. Now that it does, she thinks understanding repeating patterns should be added to the Common Core State Standards. Such an outcome would likely please many early-grade teachers, who have traditionally spent a lot of time with patterns, she said.

There’s one big caveat though: Rittle-Johnson isn’t satisfied with basic patterning exercises. Having a child copy a red-blue-blue-red-blue-blue pattern, for example, doesn’t teach him much about math, she said. It’s better for children to work on “pattern abstraction.” That’s when teachers “make a pattern with one set of materials and ask kids to make the same kind of pattern, but using new materials so that they have to really be thinking about that rule,” she said.

So, if a child is shown the red-blue-blue-red-blue-blue pattern, she could be given a set of circles and squares with which to re-create it: circle-square-square-circle-square-square.

Rittle-Johnson says pattern abstraction exercises are also an excellent way for parents to help their young children learn math.

“It’s very accessible to 4-year-olds,” she said.

BONUS: For parents who are anxious about math (and even for those who aren’t), Rittle-Johnson recommends Bedtime Math, an app that delivers one age-appropriate math problem a night to do just before or after story time. Research has shown that using the app lowers the math anxiety of both children and parents and contributes to kids’ better understanding of early mathematics.

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