# Students most effectively learn math working on problems that they enjoy, not drills or exercises

Students learn math best when they approach the subject as something they enjoy, according to a Stanford education expert. Speed pressure, timed testing and blind memorization pose high hurdles in the youthful pursuit of math.

"There is a common and damaging misconception in mathematics – the idea that strong math students are fast math students," said Jo Boaler, a Stanford professor of mathematics education and the lead author on a new working paper. Boaler's co-authors are Cathy Williams, cofounder of Stanford's YouCubed, and Amanda Confer, a Stanford graduate student in education.

**Curriculum timely**

Fortunately, said Boaler, the new national curriculum standards known as the Common Core Standards for K-12 schools de-emphasize the rote memorization of math facts. Maths facts are fundamental assumptions about math, such as the times tables (2 x 2 = 4), for example. Still, the expectation of rote memorization continues in classrooms and households across the United States.

While research shows that knowledge of math facts is important, Boaler said the best way for students to know math facts is by using them regularly and developing understanding of numerical relations. Memorization, speed and test pressure can be damaging, she added.

On the other hand, people with "number sense" are those who can use numbers flexibly, she said. For example, when asked to solve the problem of 7 x 8, someone with number sense may have memorized 56, but they would also be able to use a strategy such as working out 10 x 7 and subtracting two 7s (70-14).

"They would not have to rely on a distant memory," Boaler wrote.

In fact, in one research project the investigators found that the high-achieving students actually used number sense, rather than rote memory, and the low-achieving students did not.

The conclusion was that the low achievers are often low achievers not because they know less but because they don't use numbers flexibly.

"They have been set on the wrong path, often from an early age, of trying to memorize methods instead of interacting with numbers flexibly," she wrote. Number sense is the foundation for all higher-level mathematics, she noted.

**Role of the brain**

Boaler said that some students will be slower when memorizing, but still possess exceptional mathematics potential.

"Math facts are a very small part of mathematics, but unfortunately students who don't memorize math facts well often come to believe that they can never be successful with math and turn away from the subject," she said.

Prior research found that students who memorized more easily were not higher achieving – in fact, they did not have what the researchers described as more "math ability" or higher IQ scores. Using an MRI scanner, the only brain differences the researchers found were in a brain region called the hippocampus, which is the area in the brain responsible for memorizing facts – the working memory section.

But according to Boaler, when students are stressed – such as when they are solving math questions under time pressure – the working memory becomes blocked and the students cannot as easily recall the math facts they had previously studied. This particularly occurs among higher achieving students and female students, she said.

Some estimates suggest that at least a third of students experience extreme stress or "math anxiety" when they take a timed test, no matter their level of achievement. "When we put students through this anxiety-provoking experience, we lose students from mathematics," she said.

Boaler contrasts the common approach to teaching math with that of teaching English. In English, a student reads and understands novels or poetry, without needing to memorize the meanings of words through testing. They learn words by using them in many different situations – talking, reading and writing.

"No English student would say or think that learning about English is about the fast memorization and fast recall of words," she added.

**Strategies, activities **

In her paper, "Fluency without Fear," Boaler provides activities for teachers and parents that help students learn math facts at the same time as developing number sense. These include number talks, addition and multiplication activities, and math cards.

Importantly, she said, these activities include a focus on the visual representation of number facts. When students connect visual and symbolic representations of numbers, they are using different pathways in the brain, which deepens their learning, as shown by recent brain research.

"Math fluency" is often misinterpreted, with an over-emphasis on speed and memorization, she said. "I work with a lot of mathematicians, and one thing I notice about them is that they are not particularly fast with numbers; in fact some of them are rather slow. This is not a bad thing; they are slow because they think deeply and carefully about mathematics."

She refers to the famous French mathematician, Laurent Schwartz, who wrote in his autobiography that he often felt stupid in school, as he was one of the slowest math thinkers in class.

Math anxiety and fear play a big role in students dropping out of mathematics, said Boaler.

"When we emphasize memorization and testing in the name of fluency we are harming children, we are risking the future of our ever-quantitative society and we are threatening the discipline of mathematics. We have the research knowledge we need to change this and to enable all children to be powerful mathematics learners. Now is the time to use it," she said.

Explore further

**More information:**"Fluency Without Fear: Research Evidence on the Best Ways to Learn Math Facts": youcubed.stanford.edu/fluency-without-fear/

**Citation**: Students most effectively learn math working on problems that they enjoy, not drills or exercises (2015, January 30) retrieved 18 June 2019 from https://phys.org/news/2015-01-students-effectively-math-problems-drills.html

## User comments

jesse_elickerR_ Craigenluc_mckehansenantialias_physorgIn the stated example it's more like: understanding of the principle tops regurgitation of facts (not surprising, as 'understanding' is much more versatile)

Where does it say "don't teach the basics"? All she says is: teach it in a way that MEANS something rather than just having pupils regurgitate lists. it's not "don't teach" but "teach the right way".

And in that assessment she's 100% correct as I can attest from tutoring kids with problems in math for quite a few years. Breaking them of the 'quick fix' of just replying with memorized values is a challenge - but essential for some kids to ever get a handle on math.

LWW"While research shows that knowledge of math facts is important, Boaler said the best way for students to know math facts is by using them regularly and developing understanding of numerical relations. Memorization, speed and test pressure can be damaging,"

Yes, knowing the basic math facts is important; I don't think anyone would argue that you don't need to know basic math facts to be good at mathematics. Justifying with research seems unnecessary. However, what's damaging to students' confidence is getting bad grades; if a student survives the memorization, speed and test pressure and ends up with a good grade, they're not damaged; they're feeling good about themselves. I tell my students all the time that math is hard only if you don't know it. If you know the material and understand it, you might think it was a hard road getting there, but you won't think at that point that it's hard.

LWW"No English student would say or think that learning about English is about the fast memorization and fast recall of words,"

Try reading an English passage where you don't know the meaning of any of the words. Most students would hate it, think of it as just busywork looking up words, and want to quit. The fast recall of the meaning of words is essential to reading, processing, and understanding the meaning of a whole passage. Without it, it's just a jumble. By the same token, the fast recall of math facts is essential to reading, processing, and understanding a mathematics equation or concept. My high school math teacher recommended we memorize the squares of all the numbers from 1 to 30. I did, and it made a lot of things much easier in algebra II and higher math because I could immediately recognize perfect squares without thinking about it. Factoring trinomials, the quadratic equation, factoring expressions with square roots..

rnelson696"At all ages, there are several ways to improve the functional capacity of working memory. The most central of these is the achievement of automaticity, that is, the fast, implicit, and automatic retrieval of a fact or a procedure from long-term memory."

Note "fast" and "automatic." Each author in the above report has outstanding scientific credentials. Ms. Boaler says they are wrong, but she is not a scientist. Why does Phys.Org claim is true what scientific experts in cognition says is not true?

antialias_physorgHowever, math is not a subject where 'automatic' (or 'fast') are figures of merit when it comes to assessing a student's ability.

Fast and automatic is stuff you need in the military or at poem recitals - not in places where you actually need cognition.

rnelson696"Fast" and "automatic" are essential in solving math problems because human working memory has an essentially unlimited capacity to recall and manipulate what is well memorized, but can only hold onto a few elements not well memorized, and can do so only for a few seconds. Fluency means fast, automatic recall, which is how we speak. Fluency is essential in solving math problems. That's what every cognitive scientist is telling us. Boaler's "speed does not matter" is anti-science and its practice will harm kids.

collegeprofLosikalc269I'm sure we all recall too well the feeling of sitting down at a desk pre-test and wracking our brains for that last formula. If you couldn't remember it, there was nothing you could do, and you left feeling defeated. With this new enjoyment-focused way of teaching, students will be able to derive formulas using their deeper understanding of numbers.

I love that Common Core de-emphasizes math facts and puts more of a focus on understanding how numbers work. This feels in sync with what our schools should be teaching, confidence, capability, and love of learning. Hmm… maybe I should sit in on a high school math class and get over my fear.

LosikLosikerg83erg83zosmNewsflash: With a PhD in math, i still think "six times 8 = 2 x (3x8)=2 x24=48" Takes longer to write out the thought process, but not significantly longer to THINK. The negligence was the fact that nobody ever taught me how to think this way.

Please sign in to add a comment. Registration is free, and takes less than a minute. Read more