The simple answer to why math education has changed, “Common Core State Standards,” is only part of the story. Math teacher Christopher Danielson outlines the rest of the story in his book, “Common Core Math for Parents for Dummies,” and it goes something like this: Math education in America has evolved in response to concerns about our international competitiveness, first with Europe, and later, with Russia and its space program. Consequently, American math education prioritized the education of professional scientists and mathematicians who could get satellites in orbit and send men to the moon.
While we were busy chasing those lofty goals, we failed to educate most students in the basic foundations of math. To rectify this, the education pendulum swung back in the other direction, toward rote memorization. Cue the era of multiplication-table work sheets and timed math facts, tasks that still make up the bulk of elementary school math homework assignments.
Between 1989 and 2009, in large part because of the advent of No Child Left Behind, state standards and the testing necessary to measure states’ progress, math education became what Danielson refers to as a “mile-wide, inch-deep curriculum.” We teach many topics in each grade but at a superficial level. Math education became a series of skills served up in bits and pieces but never as part of a unified, mathematical whole.
Notably, we failed to give American children math sense, a natural and instinctive dexterity with numbers.
I was one of those children, despite having been educated in the top-ranked public school district in Massachusetts (Dover-Sherborn Regional High School). My mathematical education was characterized by drills memorization and instructions to accept abstract axioms and mathematical order of operations as “simply how it’s done,” concepts, my teachers promised, I would understand later. I dutifully followed their directions, memorized the steps and regurgitated on demand, but the understanding I had been promised never materialized. What I got instead was a raging case of math anxiety and the belief that I am not a math person.
With rare exception, most American children still receive a similarly counterproductive math education, one that produces adults who can recite multiplication tables but can’t make change when the cash register isn’t working, let alone view math as poetry.
“The highest achieving kids in the world are the ones who see math as a big web of interconnected ideas, and the lowest achieving students in the world are the kids who take a memorization approach to math. The United States, you won’t be surprised to hear, has more memorizers than any country in the world,” said Jo Boaler, professor of mathematics education at Stanford University, in a phone interview.
This chopping up of mathematical concepts, asserts Boaler, is where American math education fails children, and why Felix gets frustrated by her daughter’s math homework. Felix learned how to memorize, while her daughter is learning something much more valuable and useful: number sense, relevance and mental flexibility.
When the average teacher has about 200 separate math concepts or skills to teach in a given year, the connections between each piece disappear. “The kids don’t get to see them, and most teachers don’t know about them, either,” Boaler says. “When teachers are armed with the research about brain growth and [the reality that] everybody can learn math, it changes what they do. Teachers that are empowered with this research are doing amazing things. Really amazing things.”
Math coach Tracy Zager agrees. “It’s a phenomenal time to be a math teacher. We are in a time of great revolution and excitement, moving away from rote memorization and toward an understanding of process. It doesn’t mean that answers don’t matter, and it doesn’t mean that skills and memorization don’t matter, but when a student does something wrong, we want them to understand why,” she said in a phone interview.
In her book, “Becoming the Math Teacher You Wish You’d Had,” Zager writes, “Math is not about following directions, it’s about making new directions.” So I emailed her to ask about the direction she would take math education.
“We have to undertake the real work: high-quality, sustained, classroom-based professional development,” she said. “Doing it systematically would take money and time and belief in teachers as professionals.”
While teachers, administrators and education policymakers do battle over Zager’s question, Felix and her daughter need help today, with tonight’s homework assignment.
For that kind of practical advice, I returned to Danielson and his book, “Common Core for Math for Parents for Dummies.” Danielson suggests that parents stop giving kids easy answers and instead focus on asking these five essential questions:
- “Why?” and “How do you know?”
- “What if?”
- “Is it good enough?”
- “Does this make sense?”
- “What’s going on here?”
The questions, “Why?” and “How do you know?” require children to construct arguments, to justify their answers and to think about the reasons their answer may be correct. It’s not enough to know that 8 + 4 = 12, as Danielson writes, students must be able to articulate how they might figure out the sum of 8 and 4 if they do not automatically know the answer.
“What if” is a fantastic question to ask in any context, but in math, it’s particularly important. “What if” is at the root of play, experimentation, innovation and exploration. “What if” allows us to push students to contemplate questions beyond their immediate understanding and can fuel curiosity, deeper learning and intellectual breakthroughs.
The question “Is it good enough” gets at the concepts of estimation and precision. Is it good enough to say that .99 repeating is close enough to 1 to say that they are equal? Asking this question requires that students pay attention to units and attend to precision both numerically and linguistically.
“Does this make sense?” is a great question to ask at every step of the process, from choosing a path forward (“Does it make sense to add here?”) to the final answer (“Does that answer make sense?”) and gives kids the opportunity to pause, take stock and exercise judgment. Sometimes, of course, the answer to this question is going to be, “No,” and wrong answers can be just as useful as the right ones, Danielson argues, because, “A classroom climate that only values right answers is less likely to encourage students to persevere.”
Finally, the question, “What’s going on here?” helps kids look for the underlying structure of a problem. For example, “If you know that n is a whole number, then 2n is an even number and 2n + 1 is an odd number. What’s more, the expression 2n + 1 represents all odd numbers. This is the power of looking for and making use of structure — representing infinitely many things in a single short expression.”
I had given Felix a copy of Danielson’s book after we first met, so I called her to find out how she and her daughter are faring in math.
“Oh, honey, we are doing fantastic. That book was fantastic,” she said. “My daughter is doing great in math, and I can help her when she needs it. Plus, I get to feel smarter than a third-grader.”
This is where successful math education starts; with adults who know what questions to ask and who have the skills to help children discover their own solutions.