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

January 30, 2015 by Clifton B. Parker
Stanford Professor Jo Boaler finds that children who excel in math learn to develop "number sense," which is much different from the memorization that is often stressed in school.

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 – the idea that strong are fast ," 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 "" 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 . 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: Math anxiety causes trouble for students as early as first grade

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

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18 comments

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jesse_elicker
1 / 5 (2) Feb 02, 2015
The real tragedy is that some young elementary school teacher is going to read this nonsense and his/her students are going to be set up for years of failure because they didn't bother to ensure that the kids had a chance to learn their multiplication tables. A teacher who has his/her students futures in mind will give them plenty of chances to use, learn, memorize multiplication tables...to the point where "six times eight" is an automatic "forty-eight". Anything less is gross negligence.
R_ Craigen
1 / 5 (2) Feb 02, 2015
It must take a really advanced degree in Education to be able to go from "The basics aren't everything in learning" to "Don't teach the basics, they'll do harm!" The logic here eludes us ordinary mortals.
luc_mck
5 / 5 (2) Feb 03, 2015
I think the previous comments are missing the point. The article is not saying that you don't teach the basics, nor does it devalue knowing your times tables. Instead, it is encouraging flexible and conceptual understandings that can be utilized across mathematics, not just singing a rote learnt multiplication table. There are very few situations where taking 2 seconds to multiply 7 by 10 and subtracting 14 to solve 7x8 would be to a student's detriment. Conversely, it is to a students gross detriment if they can instantly parrot that 7x8 equals 56 without also having the understanding that they can use that knowledge to solve 7x80, or indeed 7x78.
ehansen
5 / 5 (1) Feb 03, 2015
There are too many students who have decided they are "bad" at math because they were slow to learn math facts. I agree, the author is not saying to dump the basics but to teach it in a way that honors a range of ways to think about math. I hope many teachers of young students will read this and realize that classroom time can be spent on activities that increase number sense and algebra skills while learning math facts. It does not take an advanced degree to understand this. Just spend some time with middle school students who have not grasped any conceptual understanding and whose brains are now swimming with facts and rules that don't make any sense to them!
antialias_physorg
5 / 5 (1) Feb 03, 2015
The conclusion was that the low achievers are often low achievers not because they know less but because they don't use numbers flexibly.

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

It must take a really advanced degree in Education to be able to go from "The basics aren't everything in learning" to "Don't teach the basics, they'll do harm!"

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
5 / 5 (1) Feb 03, 2015
Dr. Boaler might be making this harder than it is; a little too much analyzing.

"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
5 / 5 (1) Feb 03, 2015
Can't say I agree with the English analogy either.
"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
5 / 5 (1) Feb 04, 2015
In 2008, in the NMAP Report, on page 4-5, five of the nations's leading cognitive scientists advised,

"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_physorg
not rated yet Feb 05, 2015
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.

However, 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
5 / 5 (2) Feb 05, 2015
The section of the Report of the National Mathematics Advisory Panel (NMAP),on how the brain works was written by David Geary, Valerie Renya, Robert Siegler, Susan Embertson, and Wade Boykin. Google those names. Check their credentials.

"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.

collegeprof
not rated yet Feb 08, 2015
I think the author (or the researcher) is confused about working memory. "Working memory" does not equal memorization. As others have said "working memory" is finite (and individual to each person). If the students have to do a problem which requires basic math, understanding the text, and the more advanced processes, then it is to their ADVANTAGE, to have memorized the multiplication tables (provided they cannot use calculators). There is only but so much information and processes that can be held in working memory. The more information is in LONG TERM MEMORY, and once something is memorized it is in LONG TERM MEMORY, the better it is for any type of problem solving. The basics need to be learned but students also need to understand the concepts. FWIW I learned my multiplication tables because Sister So and So made us write them 100x for punishments! I hated it at the time, and this is not the best method by any means, but I am grateful that I learned them.
Losik
Feb 08, 2015
This comment has been removed by a moderator.
alc269
not rated yet Feb 08, 2015
Confession: math scares me. I used to anxiously memorize what I needed to pass the test, and then promptly let the formulas float out of my mind. Instead of teaching our students that success is being able to quickly memorize formulas, we should be teaching the joy of learning and applying knowledge, exactly what math Common Core emphasizes.

I'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.
Losik
Feb 09, 2015
This comment has been removed by a moderator.
Losik
Feb 09, 2015
This comment has been removed by a moderator.
erg83
not rated yet Feb 09, 2015
I agree with the author's premise that schools place too much emphasis on rote memorization. However, it would be an oversimplification of this article to conclude that the author wants us to throw out math facts from our memory and stop teaching them all together. What this article is saying is that having these math facts committed to memory does not equal an understanding of math. If I recite the 8 times table perfectly, it could be that I have a good understanding of multiplication, can apply it to multiplying 8, and now can quickly recall the basic times tables without going through all the work. Or it could just be that I committed the strings of words, "Eights times one equals eight," "Eight times two equals sixteen," and so on to memory without any idea of what it all means.
erg83
not rated yet Feb 09, 2015
Math fluency is an important skill to provide to students in "our ever-quantitative" society as described by Boaler, with math being a critical skill in many jobs and throughout everyday life. Knowing math facts is important to that fluency. Otherwise, our productivity would be heavily slowed in problem solving from having to count on our fingers or plug simple calculations like 2 x 2 into calculators all the time. However, the overemphasis on memorization seems to mistake one part for the whole. Yes, we should teach students basic math facts, but those mean nothing if we don't also teach how to really work with numbers and solve problems. It'd be like drilling students on vocabulary without giving them a strong foundation in sentence construction so that they can actually use the words. It's time for schools to stop narrowing their focus to memorization and see the greater context in which math facts are just one of the building blocks in quantitative ability.
zosm
not rated yet Mar 10, 2015
... A teacher who has his/her students futures in mind will give them plenty of chances to use, learn, memorize multiplication tables...to the point where "six times eight" is an automatic "forty-eight". Anything less is gross negligence.


Newsflash: 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.

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