A Partisan Nonpartisan Blog: Cutting Through Confusion Since 2012
“Why, indeed, do we have public schools at all? There are advantages to having an educated public, and there are at least arguments to the effect that the private sector will undersupply education. But that’s an argument for government subsidies or vouchers; it’s not an argument for the government to actually run the schools. The reason the government wants to run schools is so that it can control what is taught. I hope that makes people uncomfortable.”
– Steven E. Landsburg, Fair Play, p. 31
New Orleans’ Recovery School District is doing a lot of things right.
Ever since writing “There’s One Key Difference Between Kids Who Excel at Math and Those Who Don’t” with Noah Smith and “How to Turn Every Child into a ‘Math Person’ as a follow-up, I am on the lookout for ideas helpful for math education. Jessica Lahey’s article linked at the top gives a nice description of the Discovering the Art of Mathematics: Mathematical Inquiry in the Liberal Arts (DAoM) curriculum.
Among math professors, Stephen Strogatz has one of the strongest presences on Twitter. In the classroom, he uses the Discovering the Art of Mathematics curriculum. Jessica’s article describes an exercise about folding paper so that a scalene (irregular) triangle can be cut out with one cut that is well worth reading about. Here are some other excerpts:
Strogatz has discovered a certain thrill in rectifying the crimes and misdemeanors of math education. Strogatz asks his students, more than half of them seniors, to provide a “mathematical biography.” Their stories reveal unpleasant experiences with math along the way. Rather than question the quality of the teaching they received, they blamed math itself—or worse, their own intelligence or lack of innate talent. Strogatz loves the challenge, “There’s something remarkable about working with a group of students who think they hate math or find it boring, and then turning them around, even just a little bit.” …
Twelve years of compulsory education in mathematics leaves us with a populace that is proud to announce they cannot balance their checkbook, when they would never share that they were illiterate. What we are doing—and the way we are doing it—results in an enormous sector of the population that hates mathematics. …
If we only teach conceptual approaches to math without developing skill at actually solving math problems, students will feel weak. … You need to have technique before you can create a composition of your own. But if all we do is teach technique, no one will want to play music at all.
This is a long, but thoughtful and valuable read. Here is an interesting passage to whet your appetite:
… parents without a high-school diploma spent more than twice as much time each day with their children in the 2000s than they did in the mid-1970s, according to data from the American Heritage Time Use Study, marshaled by Harvard’s Robert Putnam. But parents with at least a bachelor’s degree increased their investment of time more than fourfold over the same period, opening up a gap in time spent with kids, especially in the preschool years.
The quality of time matters as much as the quantity of time, of course. In a famous study from the mid-1990s, Betty Hart and Todd R. Risley from the University of Kansas found large gaps in the amount of conversation by social and economic background. Children in families on welfare heard about 600 words per hour, working-class children heard 1,200 words, while children from professional families heard 2,100 words. By the age of three, Hart and Risley estimated, a poor child would have heard 30 million fewer words at home than one from a professional family.
I heard about this post from Matt O'Brien's Wonkblog article “Poor kids who do everything right don’t do better than rich kids who do everything wrong.”
Until I read this piece by Clay Shirky, I leaned toward a laissez faire attitude about students using their laptops during class. But Clay changed my mind. See what you think:
Given my efforts in “How to Turn Every Child into a ‘Math Person’" to figure out something helpful to say about math education, I was delighted not too long after to run across Barbara Oakley's Wall Street Journal op-ed
(As with all Wall Street Journal articles, if you hit the paywall, just google the title. The Wall Street Journal lets you jump the paywall if you come from Google.) Here is the key passage:
True experts have a profound conceptual understanding of their field. But the expertise built the profound conceptual understanding, not the other way around. There’s a big difference between the "ah-ha” light bulb, as understanding begins to glimmer, and real mastery.
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.
I also especially like her conclusion:
Understanding is key. But not superficial, light-bulb moment of understanding. In STEM, true and deep understanding comes with the mastery gained through practice.
For anyone learning math, the key to learning is patience–patience with both with the mental training needed to become good at working through details and with moments of confusion that come along the way. Believe me, those who do math for a living (as I do in important measure) face many moments of confusion in our work. What makes a mathematician is patience and persistence through those moments–and often hours or days, and sometimes months–of confusion, as well as the hours of honing skills for getting mathematical details straight.
Here is the full text of my 52d Quartz column, ”How to turn every child into a ‘math person,'” now brought home to supplysideliberal.com. It was first published on August 11, 2014. Links to all my other columns can be found here.
This is now my 5th most popular column, edging out “After Crunching Reinhart and Rogoff’s Data, We Found No Evidence High Debt Slows Growth.” See the whole list of most popular columns and posts here.
If you want to mirror the content of this post on another site, that is possible for a limited time if you read the legal notice at this link and include both a link to the original Quartz column and the following copyright notice:
© August 11, 2014: Miles Kimball, as first published on Quartz. Used by permission according to a temporary nonexclusive license expiring June 30, 2017. All rights reserved.
Last month, the US Math Team took second place in the International Math Olympiad—for high school students—held in Cape Town, South Africa. Since 1989, China has won 20 out of 27 times (including this year), and in the entire history of the Olympiad, the US Math Team has won only 4 out of 55 times, so second place is a good showing. According to the American Mathematical Association website: “team leader Loh noted that the US squad matched China in the individual medal count and missed first place by only eight points.”
Reading about the US Math Team’s performance in the Olympiad this year takes me back to my senior year of high school in 1977 when, having taken 9th place in the US Math Olympiad, I was invited to travel to the International Math Olympiad in Belgrade as an alternate to the 8-member US Math Team. I chose not to go to Belgrade because the Olympiad conflicted with the National Speech Tournament, where my team couldn’t have tied on points for first place without me—while the US Math Team won without needing my help. This profoundly shaped my perception of myself as a “math person.”
Left: an article from 1976 when Miles placed 23rd in the US Math Olympiad; top: in 1977 Miles placed 9th in the competition; bottom: questions from the 1977 USA Math Olympiad.
More than 36 years later, I have come to the view that almost everyone should think of herself or himself as a “math person.” In our column “There’s one key difference between kids who excel at math and those who don’t,” Noah Smith and I wrote this about the often-heard statement: “I’m just not a math person.”
We hear it all the time. But the truth is, you probably are a math person, and by thinking otherwise, you are possibly hamstringing your own career. Worse, you may be helping to perpetuate a pernicious myth—that of inborn genetic math ability.
Not everyone agrees with us. Noah and I got some pushback for our rejection of the idea that inborn math ability is the dominant factor in determining math skill. So I did some more reading in the psychology literature on nature vs. nurture for IQ and for math in particular. The truth is even more interesting than the simple story that Noah and I told.
Three facts run contrary to the idea that inborn mathematical ability is a dominant factor in determining whether or not someone is good at math compared to others of the same age.
First, it is a reasonable reading of the very inconsistent evidence from twin studies to think that genes account for only about half of the variation in mathematical skill among kids. For example, this 2007 National Institutes of Health Public Access twin study, using relatively transparent methods, estimates that genes account for somewhere in the range from 32% to 45% of mathematical skill at age 10. That leaves 55% to 68% of mathematical skill to be accounted for by other things—including differences in individual effort. (Other estimates of the percentage of variation of mathematical skill in kids due to genes range all the way from 19% to 90%. )
Second, a remarkable fact about IQ tests, including the mathematical components of IQ tests, is that every generation looks a lot smarter than the previous generation. This steady increase in performance on IQ tests is known as “the Flynn effect” after the political philosopherJames Flynn, who discovered this remarkable fact. The American Psychological Association’s official report “Intelligence: Knowns and Unknowns” says:
… performance has been going up ever since testing began. The “Flynn effect” is now very well documented, not only in the United States but in many other technologically advanced countries. The average gain is about 3 IQ points per decade.
At that rate, an IQ test from 100 years ago would put an average American today at an IQ of 130—in the top 2.5% of everyone back then. The American Psychological Association’s report goes on to say:
The consistent IQ gains documented by Flynn seem much too large to result from simple increases in test sophistication. Their cause is presently unknown, but three interpretations deserve our consideration. Perhaps the most plausible of these is based on the striking cultural differences between successive generations. Daily life and occupational experience both seem more “complex” (Kohn & Schooler, 1973) today than in the time of our parents and grandparents. The population is increasingly urbanized; television exposes us to more information and more perspectives on more topics than ever before; children stay in school longer; and almost everyone seems to be encountering new forms of experience. These changes in the complexity of life may have produced corresponding changes in complexity of mind.
In other words, although people a century ago were good at many things, many of them would have struggled with the kinds of abstract problems IQ tests focus on.
(As a simple example of how math standards have risen, my father tells me that when he was in high school, people thought calculus was too advanced for high school students. Nowadays, about one of every six high school students takes calculus in the US.)
Third (and I wish the research were clearer about this for math specifically), the fraction of differences in IQ that seem genetically linked increases dramatically with age. For children, about 45% of differences in IQ appear to be genetic, while for adults, about 75% of differences in IQ appear to be genetic. Think about that. How could it be that genes matter more and more as people get older—even though the older you get, the more environmental things have happened to you? What I think is the most plausible answer, is that the genes are influencing what people do and what they do in turn affects their IQ.
No one yet knows exactly how genes, environment, and effort interact to determine mathematical skill. In light of the evidence above, let me propose what I call the “love it and learn it” hypothesis. This hypothesis has three elements:
For anyone, the more time spent thinking about and working on math, the higher the level of mathematical skill achieved.
Those who love math spend more time thinking about and working on math.
There is a genetic component to how much someone loves math.
Despite emphasizing time spent on math as the driver of math skill, this can explain why identical twins look more alike on math skills than fraternal twins. Since time spent dealing with math matters, it allows plenty of room for the average person to be better at math now than a hundred years ago. And the effect of loving math on math experience and therefore math skill is likely to only grow with time.
If the “love it and learn it” hypothesis is true, it gives a simple recommendation for someone who wants to get better at math: spend more time thinking about and working on math. Best of all: spend time doing math in the kinds of ways people who love math spend time doing math. Think of math like reading. Not everyone loves reading. But all kids are encouraged to spend time reading, not just for school assignments, but on their own. Just so, not everyone loves math, but everyone should be encouraged to spend time doing math on their own, not just for school assignments. If a kid has a bad experience with trying to learn to read in school, or is bored with the particular books the teacher assigned, few parents would say “Well, maybe you just aren’t a reader.” Instead, they would try hard to find some other way to help their kid with reading and to find books that would be exciting for their particular kid. Similarly, if a kid has a bad experience trying to learn math in school, or is bored with some bits of math, the answer isn’t to say “Well maybe you just aren’t a math person.” Instead, it is to find some other way to help that kid with math and to find other bits of math that would be exciting for their particular kid to help build her or his interest and confidence.
The way a teacher presents a mathematical principle or method in class may not work for you—or, as Elizabeth Green suggested in the New York Times, the whole American pattern of K-12 math instruction may be fatally flawed. If you loved math, you would think about that principle or method from many different angles and look up and search out different mathematical resources, until you found the angle that made most sense to you. Even if you don’t love math, that would be a good way to approach things.
Many people think that because they can’t understand what their math teacher is telling them, it means they can’t understand math. What about the possibility that your teacher doesn’t understand math? Some people are inspired to a life-long love of math by a great math teacher; others are inspired to a life-long hatred of math by an awful math teacher. If you are unlucky enough to have an awful math teacher, don’t blame math for your teacher’s failings.
Cathy O’Neil—who blogs at mathbabe.org—describes well what I like to call “slow-cooked math”:
There’s always someone faster than you. And it feels bad, especially when you feel slow, and especially when that person cares about being fast, because all of a sudden, in your confusion about all sort of things, speed seems important. But it’s not a race. Mathematics is patient and doesn’t mind.
Being good at math is really about how much you want to spend your time doing math. And I guess it’s true that if you’re slower you have to want to spend more time doing math, but if you love doing math then that’s totally fine.
I was lucky to have a dad and older brother who showed me a bit of math early on, in a way that was unconnected to school. Then in school, I spent at least as much time on math when I wasn’t supposed to be doing math as when I was. It was a lot more fun doing math when I wasn’t supposed to be doing math than when I was.
For one thing, when I did it on my own, I could do it my own way. But also, there were no time limits. It didn’t matter if it took me a long time. And nothing seemed like a failure.
I spent a lot of time doing math. And very little of that math was done under the gun of a deadline. I spent some time on literal tangents in geometry and trigonometry. But I spent a lot more time on figurative tangents, running into mathematical dead ends. When Euclid told King Ptolemy “there is no Royal Road to geometry,” it had at least two meanings:
Everyone—even a king or queen—has to work hard if he or she wants to learn geometry or any other bit of higher math.
The path to learning geometry, or math in general, is not always a straight line. You may have to circle around a problem for a long time before you finally figure out the answer.
I feel acutely my own lack of expertise in math education for students younger than the college students I teach. Fortunately, there are a wealth of practical suggestions for teaching and learning math by others who know more than I do, or have a different perspective from their own experience.
Noah and I received many comments in response to our post but the comments I learned the most from were from these people, who let me turn their comments into guest posts on my blog:
Matt Waite: How I Faced My Fears and Learned to Be Good at Math
Marjorie Drysdale: Even When You Can Do Math, You May Not Love It
Jing Liu: Show Kids that Solving Math Problems is Like Being a Detective
Matt Rognlie on Misdiagnosis of Difficulties and the Fear of Looking Foolish as Barriers to Learning
In Green’s article “Why Americans Stink at Math,” she talks about how differently math is taught in Japanese classrooms, and how we should hope that we might someday get that kind of math instruction in the US. The key difference is that in Japan, the students are led by very carefully designed lessons to figure out the key math principles themselves. That kind of teaching can’t easily be done without the right kind of teacher training—teacher training that is not easy to come by in the United States.
But some teachers at least encourage their students to follow a “slow-cooked math” approach where they can dig in and wrap their heads around what is going on in the math, without feeling judged for not understanding instantly. Elizabeth Cleland gives a good description here of how she does it.
Even when a student is lucky enough to have good teachers at school, a little extra math on the side can help a lot. Kids who arrive at school knowing even a tiny bit of math will have more confidence in their math ability and will probably start out liking math more. Even quite young kids will be interested in a Mobius strip made out of paper where a special twist makes what looks like two sides into just one side. And putting blocks of different lengths next to each other as in a Montessori addition strip board is exactly how I have always pictured addition in my head.
A Montessori addition strip board. Image via jsmontessori.com
Extra math doesn’t all have to come from parents. In some towns, enough Little League soccer coaches are found for almost every kid to be on a soccer team. And even I was once drafted as a Cub Scout Den Leader. If people realized the need, many more adult leaders for math clubs for elementary and middle school kids could be found. In addition to showing kids some things themselves, math club leaders can do a lot of good just by checking out and sorting through the growing number of great math videos and articles online, as well as old-style paper-and-ink books.
I use Wikipedia regularly as a math reference. (There is no reason to think Wikipedia is any less reliable than the typical math textbook; textbooks are not 100% error-free either.) I have a post on logarithms and percent changes that is one of the most popular posts on my blog. (Maybe it is the evocation of piano keyboards and slide rules, or the before and after pictures of Ronald Reagan.) And Susan Athey, the first woman to win the John Bates Clark Medal for best American economist under forty, highly recommends Glenn Ellison’s Hard Math for Elementary School as a resource for math clubs. All of that just scratches the surface of the resources that are out there.
The obvious issue raised by the “love it and learn it” hypothesis is that some people may not start out loving math, and some may never love math. Acting as if you love math when you don’t may work, but it can be painful. So it is important to figure out what can be done to instill a love of math. Even if they only know a little math themselves, people who can get kids who don’t start out loving math to come to love it are a national treasure. As the brilliant business guru Clay Christensen (among others) has pointed out, in an age when lectures from the best lecturers in the world can be posted online, the kind of help students need on the spot is the help of a coach.
For too long, we have depended too heavily on overburdened math teachers who have remarkably little time in school to actually teach math, and whom the system has deprived of the kind of training they need to teach math as well as it can be taught. It is time for all of us to take the responsibility for learning math and doing what we can to help others learn math–just as we all take responsibility for learning to read and doing what we can to help others learn to read.
Most of us who participated as kids in a sport or other competitive pursuit remember a coach who got us to put in a lot more effort than we ever thought we would. Math holds out the hope of victory not just in a human competition, but in understanding both the visible universe and the invisible Platonic universe. There is no impossibility theorem saying there can’t be math coaches in every neighborhood who make the average kid want to gain that victory.
That is the end of the column proper, but I have also collected here as a postscript a few memories, ideas and suggestions that had to be cut out of the Quartz column to make the column flow well. I added some headings to make it clear where each bit fits in.
I spent at least as much time on math when I wasn’t supposed to be doing math as when I was: The teacher might have been talking about social studies, but I was finding the prime factorizations of all the numbers from 1 to 400 by writing “2 ×” for every other number “3 ×” for every third number, “5 ×” for every fifth number, etc.—and then repeating that process for every other even number, every third multiple of 3, every fifth multiple of 5, and so on). The prime factorizations I learned from that satisfyingly tedious task I distracted myself with in elementary school came in handy when I took my SAT’s. And to this day, the way I get a hotel room number firmly into my memory is by doing its prime factorization.
Nothing seemed like a failure: At one point I knew just enough algebra to know that doing the same thing to both side of an equation left it a true equation. So for a long time, I transformed equations endlessly with no idea at all of where I was trying to go with those equations. Later on, when I actually had a purpose in mind for what I wanted to accomplish with a bit of algebra, I was able to draw on all of that experience just wandering around in algebra-land. And because I knew what it was like to do math without having any particular objective, I was able to appreciate how important it was keep the objective clearly in mind when there was an objective.
Proofs on other topics to get kids ready for proofs in Geometry class: Many kids who do well with arithmetic and algebra have trouble with geometry class in middle school or high school. It is often very hard to understand the idea of a proof when can’t see any reason to doubt the proposition to be proved in the first place. It is much better to get kids used to the idea of a proof earlier on in a context where the proof tells them something that doesn’t seem obvious. My favorite is the proof that there are an infinite number of primes. (There is a whole page of Youtube videos to choose from on this.) And a lot of kids wonder if imaginary numbers are numbers at all. The proof that complex numbers with an imaginary components obey all the rules of arithmetic and algebra and therefore can be treated as legitimate numbers not only answers a question kids really have, but uses concepts from “The New Math” that confused many kids in the 1960’s in a way that is obviously useful.
Math resources I found useful:
The “Hungarian Problem Books” of challenging math problems
Resources to check out that might be good but that I don’t have any experience with:
Khan Academy videos are at least free! Here is a set of videos about the Khan academy, including a TED talk by Salman Kahn and a “60 Minutes” segment and a talk by Bill Gates about the Khan Academy.
Hands-on Equations for algebra
PhotoMath: “Point your smartphone’s camera towards a mathematical expression and PhotoMath instantly displays a correct result, also showing the step by step process it used to arrive at the answer.”Suggested by Niels Andeweg
Note: if you want to advertise your tool or method for math instruction here, I encourage you to advertise it in a comment that you post in the comment box below. When I moderate the comments, I will approve comments that advertise tools or methods for math instruction like that unless I have reason to believe there is something wrong with that tool or method.
Maryam MIrzakhani, First Women to Win the Fields Medal
Going beyond the usual news articles such as these two,
Quanta magazine gives a more in-depth treatment of the the work of the first woman to win a Fields Medal, which is aptly described as the Nobel Prize of mathematics:
I know about this article thanks to Mary O'Keeffe’s Facebook post on my wall. Mary points out that Maryam describes herself as pursuing what I have called “slow-cooked math” (most recently in my Quartz column “How to turn every child into a ‘math person’”):
Mirzakhani likes to describe herself as slow. Unlike some mathematicians who solve problems with quicksilver brilliance, she gravitates toward deep problems that she can chew on for years. 'Months or years later, you see very different aspects’ of a problem, she said. There are problems she has been thinking about for more than a decade. 'And still there’s not much I can do about them,’ she said.
Mirzakhani doesn’t feel intimidated by mathematicians who knock down one problem after another. 'I don’t get easily disappointed,’ she said. 'I’m quite confident, in some sense.
Her slow and steady approach also applies to other areas of her life.
Here is a link to my 52d column on Quartz, “How to turn every child into a ‘math person.’”
In the companion post below, I have collected a few memories, ideas and suggestions that had to be cut out of the Quartz column to make the column flow well. I added some headings to make it clear where each bit fits in:
I spent at least as much time on math when I wasn’t supposed to be doing math as when I was: The teacher might have been talking about social studies, but I was finding the prime factorizations of all the numbers from 1 to 400 by writing “2 ×” for every other number “3 ×” for every third number, “5 ×” for every fifth number, etc.–and then repeating that process for every other even number, every third multiple of 3, every fifth multiple of 5, and so on). The prime factorizations I learned from that satisfyingly tedious task I distracted myself with in elementary school came in handy when I took my SAT’s. And to this day, the way I get a hotel room number firmly into my memory is by doing its prime factorization.
Nothing seemed like a failure: At one point I knew just enough algebra to know that doing the same thing to both side of an equation left it a true equation. So for a long time, I transformed equations endlessly with no idea at all of where I was trying to go with those equations. Later on, when I actually had a purpose in mind for what I wanted to accomplish with a bit of algebra, I was able to draw on all of that experience just wandering around in algebra-land. And because I knew what it was like to do math without having any particular objective, I was able to appreciate how important it was keep the objective clearly in mind when there was an objective.
Proofs on other topics to get kids ready for proofs in Geometry class: Many kids who do well with arithmetic and algebra have trouble with geometry class in middle school or high school. It is often very hard to understand the idea of a proof when can’t see any reason to doubt the proposition to be proved in the first place. It is much better to get kids used to the idea of a proof earlier on in a context where the proof tells them something that doesn’t seem obvious. My favorite is the proof that there are an infinite number of primes. (There is a whole page of Youtube videos to choose from on this.) And a lot of kids wonder if imaginary numbers are numbers at all. The proof that complex numbers with an imaginary components obey all the rules of arithmetic and algebra and therefore can be treated as legitimate numbers not only answers a question kids really have, but uses concepts from “The New Math” that confused many kids in the 1960’s in a way that is obviously useful.
Math resources I found useful:
Resources to check out that might be good but that I don’t have any experience with:
Note: if you want to advertise your tool or method for math instruction here, I encourage you to advertise it in a comment that you post in the comment box below. When I moderate the comments, I will approve comments that advertise tools or methods for math instruction like that unless I have reason to believe there is something wrong with that tool or method.
Link to Kevin Remisoski’s Twitter homepage and his LinkedIn page and an article in the local news (picture above)
In response to our column “There’s One Key Difference Between Kids Who Excel at Math and Those Who Don’t,” Noah Smith and I received many comments. Some of them gave good advice about teaching math to kids. Among those was this email from Kevin Remisoski, who graciously gave me permission to share this with you:
I liked most of the points you addressed in this article, however, I feel you missed one very important and crucial point. There are a good number of people who also have dyscalculia. This disorder is often times undiagnosed, and while some estimate that 5% of our population has this disorder, I would venture a guess that this in some form could likely affect as many as 20% of our population. Many of the symptoms seem fairly common to people I have helped with math over the years and my wife didn’t even know she had it until I gave her an assessment after she was complaining about how she always switched numbers around. This assessment went over 30 questions that are common symptoms of dyscalculia, and she answered 27 of them with yes.
I myself always excelled in math. This wasn’t due to parental drilling, flash cards, or anything of the sort. This was purely genetic, and this also seems to be the case with my stepson, though I’ve been teaching him more advanced techniques for his age as well as basic physics (he is 8).
I think somewhere along the way, teachers forgot how to teach math and left it to the text books and the curriculum to teach for them while they assist the book in the learning process. I’m sure this is most certainly due to laziness, but most human beings are lazy and just want to get home at the end of their work day. I know I struggled with some of my teachers in my youth, because countless times I’d approached them with easier ways of solving problems from basic math in elementary school all the way through college. I just don’t understand why some teachers are so focused on only teaching one solution to a problem.
You see the problem with dyscalculia, is that it is not impossible to teach math to people who suffer from this disorder. You just have to be creative, even if that means providing creative and abstract solutions at times. I have taught my wife quite a bit simply by using unorthodox approaches to math.
With all of that being said, in a base 10 number system, I feel that it is important that children understand a few basic concepts:
If A + B = C, then C – A = B, and C – B = A. I am not suggesting teaching 5 year olds the concept of substituting numbers with variables, but rather that they understand this concept as much as they will later be expected to memorize their multiplication tables. One exercise, I’ve worked on with my stepson is repetition. We don’t play with flash cards or have any visual representation, because I feel that defeats the purpose of memorization.
So, instead I would go with the range of numbers 0-9, and have him add and subtract different numbers within that range to see the relationships between those addends, sums, subtrahends, and difference. One way I went about this is as follows:
1+1, 1+2, 1+3……2+1, 2+2, 2+3…..3+1,3+2,3+3, and so on until each starting addend was added to zero through nine (despite starting with one in this example).
After I was sure he was comfortable with this, I would then have him add 1+2 for example and then 3-2, and 3-1. I would continue to have him solve addends, and then solve the difference from the sum when one of those addends was converted to a subtrahend, and then solve for the second one.
You see, I don’t believe in waiting for children to memorize addition and subtraction on their own. They should be able to look at any two numbers and solve either the sum or difference just as easily as they breathe.
The same can be said for multiplication when they are ready. They should not only write their multiplication tables out ten to twenty times a day until they have them committed to memory, but much like in my above example, they should understand the following:
If A * B = C, then C / A = B, and C / B = A.
This is the way to teach children math. I also firmly believe that until a child has memorized their multiplication tables at least through ten as well as committed addition and subtraction to memory, that they should not be allowed to even learn how to use a calculator. Genetic dispositions, as well as disorders, can be toppled by the human brain’s efficiency at memorization through repetition. If we are to believe that the day may come when no one will utter the words “I’m just not good at math.”, we also need to believe that there is a better way of instilling confidence in those young minds. Without the fundamentals of understanding the basic building blocks as I’ve described here, it’s really no wonder why so many children bomb in basic math much less algebra, geometry, trigonometry, and calculus. If Einstein could find a way to overcome dyscalculia, anyone can.
Marjorie Balgooyen Drysdale is a classical soprano, music teacher, conductor, and the author of the book Tagalong Kid.
Not all of the emails Noah Smith and I received in response to our column “There’s One Key Difference Between Kids Who Excel at Math and Those Who Don’t” agreed with us. Some people said they had tried as hard as they could and still couldn’t do math. In a few cases, genuine dyscalculia might be at issue. But more often, I suspect the problem is with the quality of the math teaching. Elizabeth Green had a fascinating New York Times article “Why Do Americans Stink at Math?” a few days ago that pointed the finger squarely at the lack of adequate instruction for math teachers in how to teach math.
Marjorie Drysdale, who received a Master’s degree in Music from the University of Michigan, graciously agreed to share this email she made in response to Noah’s and my column. In addition to the issue of how math is taught, and where one is at when the math is taught, she points out that just because you can do something doesn’t mean you will love it. I agree. There are always tradeoffs in life, and time spent doing math is time away from doing something else that you may love more–maybe a lot more. But at least if you know how to do math, you can make the choice. And if math is taught well, what you learn will have some value for your life.
Dear Miles and Noah,
Regarding your essay, “There’s one key difference between kids who excel at math and those who don’t,” I have to disagree with your assertion that “For high school math, inborn talent is just much less important than hard work, preparation, and self-confidence.”
I was an excellent high school student. I worked hard, prepared, and was self-confident. I always made the high honor roll. I graduated 2nd in my class.
I went on to a competitive college and graduated with highest honors, phi beta kappa. I had two majors. I was always on the Dean’s List.
I went on to get a master’s degree and graduated with honors there, too. I became a professional musician. Supposedly, math and music go together. Not with me.
After geometry, I simply “didn’t get it." I took one more year of math—a course which, in the 60’s, was called "fusion." It was a combination of trigonometry and advanced geometry. That did me in. Until then, I had always earned A’s in math. I barely passed "fusion.”
It might have made a difference that I had skipped a grade and then was put into the “honors group”—an accelerated class. (In those days, classes were “tracked.”) In that class, we were taking courses a year ahead of our peers. Therefore, I was taking courses two years ahead of my peers. Perhaps I simply had a “readiness” problem.
I “hit a wall” and never went back to math. It had nothing to do with lack of effort, believe me.
Oddly, when I took my GRE exams after college and two years of work, my verbal and math scores were both in the 700’s. This truly surprised me. I hadn’t taken a math course in seven years.
I think that “readiness” in my case was more of a determining factor than “hard work.”
As I grew older, I understood it better, but I still didn’t like it. There are people who adore math. They light up about it. They enjoy it from the get-go. Therefore, I do think there is an innate element to these differences. Some people love math; others don’t.
I adore classical music. Most people couldn’t care less about it. The first time I heard it, I was hooked. That had nothing to do with hard work, either. Succeeding in it required hard work, of course, but the love came first.
I have been thinking more about the issues Noah Smith and I raised in our column “There’s One Key Difference Between Kids Who Excel at Math and Those Who Don’t.” So I asked permission to publish a few more of the comments Noah and I received by email. Here is a note I liked from Kate Owino:
I’d like to thank Quartz and Profs. Kimball and Smith for the wonderful article on math capabilities in kids released on October 27th, 2013. Reading it reminded me of my personal relationship with mathematics, as a subject and as a life/job-related skill.
I was born and raised in Kenya, and here the attitude toward math takes on a sexist connotation in favor of male students. It’s rare to hear of a female student saying that she excelled in math not only for the sake of passing the exams and getting into a good school or university, but also because she LOVES the subject.
From my personal experience, I was one of a handful of students in secondary school who fell in the latter category. This has proved (to date) to be a slight challenge whenever the topic of attitude towards math arises in a discussion with my female friends - they talk about how poorly they performed especially in secondary school, to the point where it comes across like they’re actually proud of the grades they got (Cs and below). I cannot contribute to the self-mockery because I got As all the way to my final exam…and the same applies to Chemistry.
Reading about the criticism-to-work-harder approach employed by students in China reminded me of my mother’s toughness towards my performance in math. From the age of 8 she would literally slap my wrists if I worked sloppily at a sum, and it was worsened by my teachers’ constant comments in my report book about my propensity to make careless mistakes.
As I look back now I cannot help but be proud of my love for math (and the sciences in general), even though I ended up pursuing a different academic path - studied literature in my undergraduate, and currently work in web content management. I hope to find a way to help do away with that sexist attitude in schools in my country especially since, as it was indicated in the article, poor attitude toward math makes many people lose out on critical life skills and lucrative career paths.
Thanks once again for the wonderful article. Have a wonderful week!
Image created by Miles Spencer Kimball. I hereby give permission to use this image for anything whatsoever, as long as that use includes a link to this post. For example, t-shirts with this picture (among other things) and http://blog.supplysideliberal.com/post/92400376217/math-camp-in-a-barn on them would be great! :)
I like Naomi Schefer Riley’s account in the Wall Street Journal of Ben Chavis’s math camp in North Carolina’s poorest county: “Math Camp in a Barn: Intensive Instruction, No-Nonsense Discipline” (googling the title of a Wall Street Journal article jumps over the paywall, so my link is to the search page). Naomi’s article illustrates two related principles I have written about. First, almost anyone can learn math with enough hard work and a can-do attitude, as Noah Smith and I write in “There’s One Key Difference Between Kids Who Excel at Math and Those Who Don’t.” Second, a key element of learning is simply time spent learning, as I write about in “Magic Ingredient 1: More K-12 School.” Lengthening the school year is one of the most straightforward ways to increase learning, especially in hard subjects. Naomi points out the arithmetic of math instruction:
From 8:30 a.m. to 4 p.m. Monday through Friday the children learn math, interspersed with some reading, physical education and lunch. Each gets 120 hours of instruction during the three weeks, equivalent to what they would get in a year at a typical public school.
Among many other serious problems with education in the United States, our attachment to the idea of summer vacation is an important one.
The “You, too, can learn math” column Noah Smith and I wrote (currently, my most popular column, by far) is flagged in one of the 7 points about raising kids in the link above. :)
Four of the main justifications for public education are
Public education is a particularly attractive form of redistribution, since unlike direct transfers to poor families, (a) education goes directly to the kids and (b) education tends to encourage, rather than discourage hard work.
But public education is seldom optimized as a means of redistribution. Richer school districts often have better facilities and supplies, and typically offer higher salaries to teachers. But even when nominal teacher salaries are equalized across districts by state law, the greater difficulty of teaching the kids who need teaching the most to catch up often means that disadvantaged kids get worse teachers. (And the lower desirability for a teacher of either living near those schools or commuting further also tends to lead to worse teachers for disadvantaged kids.) To even equalize teacher quality would require paying enough of a salary premium for teaching in difficult schools that the typical teacher would be indifferent between taking on the tougher job with that salary premium or taking on an easier job with a lower salary. And an argument can be made that the very best teachers (in terms of being able to motivate kids and teach the most basic and important concepts well) should be teaching the kids who are the furthest behind.
The chickens have now come home to roost. The failure of California to address the problem of teacher sorting led to a remarkable decision yesterday by a Los Angeles Superior Court. As reported in the Wall Street Journal article linked above:
In a closely watched court case that challenged California’s strong teacher employment protections, a group of nine students have prevailed against the state and its two largest teachers unions.
A Superior Court here on Tuesday found that all the state laws challenged in the case were unconstitutional. The verdict could fuel similar lawsuits in other states where legislative efforts have failed to ease rules for the dismissal of teachers considered ineffective.
The student plaintiffs in Vergara v. California argued that the statutes protecting teachers’ jobs serve more often to keep poor instructors in the schools—hurting students’ chances to succeed.
Citing the Supreme Court’s landmark 1954 Brown v. Board of Education “separate but equal” ruling, Superior Court Judge Rolf M. Treu wrote in his decision that the laws in the case “impose a real and appreciable impact on the students’ fundamental right to equality of education.” The decision also agreed with the plaintiffs’ arguments that the poorest teachers tend to end up in economically underprivileged schools and “impose a disproportionate burden on poor and minority students.”
It is easy to get drawn into the debate about the merits of teacher tenure. But I hope people don’t miss the problem of teacher sorting. If many of the worst teachers in a state ended up in the richest school districts, I think it would bring home to wealthy and influential voters the importance of school reform–in many dimensions. Then maybe poor kids would have a chance, as they not only got average rather than below-average teachers, but the average teacher and school performance improved.
Update: I received some great comments on the Facebook version of this post (which was really only a link to the post here):
Robert FloodPublic schools are not allowed to pay a premium in $ or in class size or load to induce teachers to go where their product is highest. So, to keep the best teachers, they bid with school assignments. My wife, in the last 10 or 15 years of her teaching career taught math to the best kids in Montgomery Co MD - maybe the richest country in the US. The kids were great and the parent support was superb - except for the nutsy parental “grade stalkers.” Her biggest problem with teaching was “up county” administration. The admin grew much faster than the # of kids or # of teachers.
My advice - let schools bid for teachers. I also think that when we see diseconomies of scale in administration, i.e., admin growing faster than the administered, , the teachers need to be consulted and help to design administration.
Chris KimballI recently read (in Malcolm Gladwell’s “What the Dog Saw”, although there’s surely a source behind that) that the range for least effective to most effective teachers runs from ½ (year’s worth of material in one year of class) to 1-½ (year’s worth of material in one year of class). Notwithstanding a raft of questions about the data (is it valid, what does the distribution look like, can the extremes be replicated, etc.) the extent of that range was a wake-up for me–making teacher sorting and selection a much more important issue than I had been thinking.
