Practice, Practice, Practice, Practice: Repetitive Work In Math Is Good
Engineering professor Dr. Barbara Oakley, who was recently on my radio show discussing her terrific new book, "A Mind For Numbers," has an op-ed in the WSJ on how we should be teaching math:
I'm now a professor of engineering, but in my mid-20s I was an artsy language lover who had flunked her way through elementary-, middle- and high-school math and science. What I discovered when I started over at age 26--first tackling remedial middle-school math and then working my way toward a Ph.D. in systems engineering--is that a conceptual understanding only gets you so far.Conceptual understanding has become the mother lode of today's approach to education in science, technology, engineering and mathematics--known as the STEM disciplines. However, an "understanding-centric approach" by educators can create problems.
Today's Common Core approach to teaching STEM is at least superficially appealing. The goal of placing equal emphasis on conceptual understanding, procedural skills and fluency, and application is laudable. But as with any new approach to teaching, the Common Core builds on the culture that's already there. And the culture that has long reigned in STEM education is that conceptual understanding trumps everything. So bewildered math teachers who are now struggling to teach the Common Core are leaning on the old thinking, which has it that if a student doesn't understand--in the "ah-ha," light-bulb sense of understanding--there's no way she or he can truly become expert in the material.
...As research by Alessandro Guida, Fernand Gobet, K. Anders Ericsson and others has also shown, the development of true expertise involves extensive practice so that the fundamental neural architectures that underpin true expertise have time to grow and deepen. This involves plenty of repetition in a flexible variety of circumstances. In the hands of poor teachers, this repetition becomes rote--droning reiteration of easy material. With gifted teachers, however, this subtly shifting and expanding repetition mixed with new material becomes a form of deliberate practice and mastery learning.
True mastery doesn't mean you use crutches like laying out 25 beans in 5-by-5 rows to demonstrate that 5 × 5 = 25. It means that when you see 5 × 5, in a flash, you know it's 25--it's a single neural chunk that's as easy to pull up as a ribbon. Having students stop to continually check and prove their understanding can actually impede their understanding, in the same way that continually focusing on every aspect of a golf swing can impede the development of the swing.
"The important thing is to understand what you're doing, rather than to get the right answer."
Rex Little at September 22, 2014 11:18 PM
Rex, hope your comment is at least a little sarcastic. At some point the right answer needs to be important too. I really, really want the engineer designing the bridge I will be driving on to correctly calculate the size and strength of the support members. The I-35W bridge in Minneapolis is a good example of what can happen when a wrong answer survives into the final product.
bkmale at September 23, 2014 7:58 AM
That "ah-ha moment" comes through repetitive practice.
Conan the Grammarian at September 23, 2014 8:11 AM
I think Rex posted that because it's so idiotic -- as a "rather than."
I want the person who understands AND can get the right answer to design the bridge.
The more practiced you are at something, the more it becomes second nature. Oakley writes about chunking in her book, how your brain, when you really learn certain bits of of related information, collects them together into a chunk, where you don't have to think about each little piece anymore. This allows you to graduate to more complex and newer information. And to use the information you've learned without exerting a lot of energy.
I really do recommend her book. It's not just about learning math and science; it's about how to be more productive at anything requiring creativity and learning.
Amy Alkon at September 23, 2014 8:49 AM
When I was in elementary school, it was clear that there were some students who understood math while others just couldn't "get" it, no matter which method the teacher used to explain it. The fact that Dr. Oakley "started over" at age 26 would indicate that it can take some people at least 20 years to grasp how math is supposed to work.
Fayd at September 23, 2014 9:06 AM
It didn't take her "20 years to grasp math"; it took effective ways to learn. She figured them out back then and found that they are supported by neuroscience and other research.
Amy Alkon at September 23, 2014 9:20 AM
Rex is quoting the redoubtable Tom Lehrer -- you need to listen to the song at the link, it's a classic.
As someone with dyslexia who struggled with arithmetic but went on to major in mathematics, I can't agree with Dr. Oakley enough. It's much easier to move onto the higher level concepts when once the arithmetic is embedded in your bones. My poor nephew is being subjected to this common core crap, and went from loving math and being very proficient, to hating the way math has now turned basically into a writing exercise. He can't just give the answer, he has to write word sentences. This is supposed to make math easier for those with better verbal skills to embrace math, but can't you make it an either or, rather than forcing kids to get math easily to waste time on BS?
Longtime Lurker at September 23, 2014 9:35 AM
The saxon math books are some of the best math text books I have ever used. There is no color. There is nothing fancy. Each new chapter includes old questions (~25%) so you keep practicing and remembering how to do those problems. I wasn't homeschooled. But my parents forced me to do saxon math at home. While I hated it at the time I fully intend to inflict it on my kids. I wouldn't be the engineer I am today without it.
Ben at September 23, 2014 11:05 AM
Fayd,
It doesn't take 20 years to learn math, the reason children struggle is because there is not enough practice. When my daughter was in the 4th grade we realized that while she got the concepts presented there was not enough practice to go along with it. She might do 20 problems a week and they all looked alike. It hit me when I asked her a rounding question while we were in the grocery store and the zero threw her for a loop. We went back to basics and started with daily math sheets....even in summer... and now she is at the top of her class in Pre-AP Geometry. At our house we like to say, "That a day without math is like a day without sunshine." We live in a high tech society and if you can't compete you will be left behind. As parents, it is our job to make sure she is not one of the kids sitting in the dust.
Sheep Mom at September 23, 2014 11:25 AM
For anyone interested, Khan Academy is the best free source of math tutoring. Their videos are great and they keep tweaking the site to improve it. For instance, your child can do practice problems, but has to wait 15 hours before they can take the mastery test. The kids hate it, but it is a way of making sure they have actually retained the information. There are also some really great sites that will generate worksheets for you. One that we use costs a 100 bucks, but is so worth it. It allows you to create a worksheet that targets practice in areas of weakness. These tools and the hours spent with them have really made the difference for my daughter.
Sheep Mom at September 23, 2014 11:48 AM
"True mastery doesn't mean you use crutches like laying out 25 beans..."
It does if you want to become a true bean counter.
The Former Banker at September 23, 2014 6:38 PM
That is true, but trivially.
When my son and daughter were kids (they are now in college) my wife and I subjected them to flash-card math-fact torture. They would know the answer to 7x3 as they know to breathe, and damn the concepts.
As it turned out, my daughter is immune to mathematical understanding; my son intuitively gets it. She will never be an expert mathematician. But the relentless grounding in math facts allowed her to be good enough to be near the top of her class, even if she did so by rote. Without the basics, she would have floundered.
Grammar rant: when speaking collectively, use the plural, because it avoids that hackneyed "she or he" crep. "... if students don't understand ... there is no way they can become truly expert in the material."
It is astonishing how many professional writers are immune to the obvious.
Jeff Guinn at September 23, 2014 10:21 PM
Amy I have been looking for a book like this forever.Thank you. I just started reading it and will write back when I'm finished. Thanks again for all you do, you truly are an Advice Goddess!
George McEwen at March 21, 2015 5:16 PM
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