Paul Lockhart wrote a brilliant essay called “A Mathematician’s Lament,” in which he competed the teaching of music to the teaching of mathematics. What if children spent years learning how to identify and describe colors and art forms but were not allowed to draw or paint until they got to college?
I was reminded of a critical description of the teaching of science in the late nineteenth century. Children learned to memorize the parts of a flower but never to study the natural world around them. All they knew of nature was what they memorized in dry textbooks.
Thanks to Steve Nelson for recommending Lockhart’s essay.
Diane Ravitch's blog 
Lockhart’s essay needs to be required reading for anyone who has anything to do with mathematics education. It is SEMINAL.
And a reading of that essay should be followed up with this:
I’ve long wondered why people don’t ask themselves this question:
Every American adult undergoes at least 12 years of math instruction. But if you go out and talk to adults, you find that enormous numbers of them are a) innumerate and b) HATE math. Why?
Well, here’s what I think is going on. What they’ve experienced in math class is not math but memorization of vast numbers of rules for manipulation of symbols (as Lockhart so brilliant illustrates). And since the Common [sic] Core [sic], this excruciatingly dull activity has been compounded by asking students to think in highly ABSTRACT ways at very early ages. Kids in grade three, the “standards” tell us, are supposed to grok the “concept of the variable.”
Kids are very concrete thinkers, though, in a Clever Hans Phenomenon sort of way, adults often attribute to them far more abstract thinking than they are actually doing. Asking young kids to think abstractly is like asking fish to climb a tree. FMRI studies have shown that the parts of the prefrontal cortex that are dedicated to abstract reasoning don’t even start toward their full development until around the age of 14 and aren’t fully finished until around the age of 26. There are exceptions, of course. We sometimes find little Gausses and Fermats in our classes, and these should be pulled out and given special instruction. They are rarer than diamonds and much more valuable to society at large.
But more most kids, I think, we would do well to eliminate mathematics instruction beyond simple addition and subtraction, multiplication and division, in elementary school, and replace it with lots of activity involving manipulation of shapes and patterns that will help to build that abstract thinking ability–those internal structure. THEN, when they are cognitively developed enough to handle it, start the formal Math instruction.
I think that if we did this, we would accomplish more in in High School alone than we did in that 12 years. Far more. Kids would be developmentally ready.
There’s always a hidden curriculum. The hidden curriculum of our Math instruction as currently done is, for vast numbers of kids, a) this is boring, b) it’s difficult, c) you don’t like it and aren’t good at it.
Ask American adults. Most will tell you that. I don’t like it and am not good at it. That’s what they have learned. An NEA study found that 60 PERCENT of American adults could not calculate a ten percent tip, even though all they had to do was move the decimal point. They might have done OK in school, but as adults, they fell back upon what they MOST learned: math is hard and boring and I’m no good at it.
The confusion about how bad little kids are at ABSTRACT reasoning of the kind required for math comes, I think, from the fact that they seem to be REALLY good at some thing that APPEAR to be abstract reasoning. But those are the things for which they are hardwired by their evolution to be able to do very well.
“we would do well to eliminate mathematics instruction beyond simple addition and subtraction, multiplication and division, in elementary school, and replace it with lots of activity involving manipulation of shapes and patterns that will help to build that abstract thinking ability–those internal structure.” I so agree with this statement. Basic math is important, and can be engaging and fun for elementary students to learn. Observing children for many years at the elementary level, I try to incorporate pattern blocks, geoboards, legos, unifix cubes etc, in my center rotations as much as possible during math. And I leave those manipulatives out during play time. It’s engaging and energizing for students to build, design and make patterns, fix puzzles. Learning through play is so natural, healthy and should be the norm for much of the school day at the pre- K-2 level, and possibly beyond those years – but I don’t have as much experience past grade 2.
Thanks, beachteach. And thanks for ignoring the obvious typos in my post. I have been saying this for years, but when I do, people usually look at me as though I am totally insane. LOL.
Clever Coleman
Clever Coleman counts to ten
On Common hoof, a clever man!
Clever Zimba, counts to twenty
Nays and brays and winnies plenty
You are channeling Mother Goose for sure, here, SomeDAM. Awesome.
Puzzles of all types are a good way to teach and learn mat at all ages..
Not just visual puzzles but logic puzzles as well.
Mother Goose
And Dr Seuss
Rhyme abuse
Is SomeDAM muse
No better teachers!!!
Bob, this is a blog not a book. Don’t worry about your typos. Correct only those that may affect understanding.
I would not blame boring and terrifying math curriculum and instruction only on Common Core. Math miseducation has a long history here and all over the world.
In theory, Common Core wanted to cure the data, formula and memorization driven math education which has had at least a century old tradition, making generations hate math.
CC simply stole the language of the Right Side, but ended up pushing math curriculum and instruction even further to the Dark Side.
No doubt, it’s time for Enlightment in Math Education. Design something which considers only what’s best for all children, and forget about the interests of politicians, universities, businesses.
Superb points, Mate!!! Spot on!!!
“Design something which considers only what’s best for all children, and forget about the interests of politicians, universities, businesses.”
Hell will freeze over before that is allowed to happen in the US.
The DNC is so concerned that Bernie will get the nomination that they are now floating the John Kerry balloon.
People like Tom Perez are just pathetic. If the DNC can’t find someone with a brain to lead them, at least they should find someone who can grow more than peach fuzz on his face.
Let’s not forget how many major candidates entered the race for president in 2016 for the Republican Party.
“Sanctioned by the Republican Party, these elections selected the 2,472 delegates that were sent to the Republican National Convention. Businessman and reality television star Donald Trump won the Republican nomination for president of the United States. A total of 17 major candidates entered the race.”
How The Republican Field Dwindled From 17 To Donald Trump
https://fivethirtyeight.com/features/how-the-republican-field-dwindled-from-17-to-donald-trump/
Pray that the Democrats are more sensible and will select one candidate that is not the same as Donald Trump.
“Overall, there were 29 major Democratic presidential candidates in the 2020 election. As of January 31, 2020, 18 of these candidates have dropped out of the race, and 11 major candidates are still seeking the Democratic presidential nomination in 2020.”
Therefore, in conclusion the Democrats now have 11 major candidates left in 2020 vs 17 for the Republicans in 2015.
When you are confronted by an Always Trumper crowing that the Democrats (sorry the Alt-Right would never use the proper term. They say leftists or libtards) cannot make up their minds, remind them about 2016, please.
Good point, Lloyd.
Love the article. Yes, mathematics is Art.
Thanks for the video, Bob.
The Deformers are working on that, too: “were not allowed to draw or paint until they got to college”
If the greedy, power-hungry, corrupt billionaire deformers get what they want, even college will be stripped of every class but the basic job training courses. College will be turned into a toxic pipeline that leads directly into corporate jobs without any elective classes.
Any class that teaches children and adults how to think and problem solve on their own will also be gone. No more creativity.
The autocrats want the rest of us to be mindless drones that spend all our free time texting, playing video games and eating the same way Donald Trump does … but without health care or a retirement plan.
Oh, and once we-the-people are no longer useful, we will be allowed/forced to die ASAP to get us out of the way.
BTW, here’s a great big, fat book that does the best job I’ve ever seen of illustrating sheer fun with the art of mathematics:
Gulberg, Jan. Mathematics: from the birth of Numbers. Norton, 1997.
Such delights! It’s not cheap (almost $50), but it’s worth every cent. It’s expensive because it’s huge (1,100 pages) and lavishly illustrated. But it can be consumed in bite-sized pieces. Here you will find ample, really ample, illustration of mathematics as joyful play and as art.
Also highly recommended: Kline, Morris. Mathematics for the Nonmathematician. Dover, 1985.
This same understanding can refer to the teaching and learning of reading. If we focus all of our instruction on phonics-the sounds individual letters make, we are depriving our students, especially those that struggle, the joy that reading brings! That’s why it’s important to make sure students get to READ engaging books that they can successfully manage with very little help. Matching readers to texts they can read with 95% profiency, through careful and effective assessments is the most important thing I’ve found as a reading teacher to guarantee growth. Phonics, subtext, vocabulary, and comprehension can all be taught using the content from rich texts at all levels. It’s the same principle of learning to ride a bike. You don’t spend excess time memorizing the parts of a bike-you get on a bike and start pedaling! And contrary to what Mr. Coleman believes, children love to talk about a book’s content and give their personal impressions and connections which in turn helps them “break the code” and gain more knowledge.
Most kids need some phonics instruction, but ALL OF THEM DEFINITELY need joyous experiences with books. Thank you!
A person arrives at the ability to do close reading because he or she has fallen in love enough with reading to read a lot and with particular authors and books to read them carefully. It’s all about intrinsic motivation. And, the sort of writing that is done in exercises based on the execrable, puerile, backward Gates/Coleman Common Core are completely unnatural–they don’t reflect the actual reasons why people read–to have vicarious experiences that are significant to them, to engage with the content, with what writers have to say. Having kids write five-paragraph themes about how the author’s use of figurative language affects to the tone and mood of _____ is a certain way to ensure that they will never give a hoot about reading. It’s idiotic, bizarre, backward. Tiem to through out the CC$$ (and all those state standards that are simply the CC$$ with state-specific names).
Reading is about a lot more than ‘exploding the code.’ The privatizers are bashing balanced literacy programs as inadequate. Balanced literacy, if implemented correctly, teach students to read, write and think. These so-called studies promoting phonics only are non-peer reviewed. Everyone agrees that understanding how to use the sound system is an important element of reading, but the privatizers want to force students to get a steady diet of phonics via computer instruction. Don’t fall for this obvious attempt to undermine meaningful reading instruction.
Agreed, emphatically.
Math is like a puzzle
Math is like a puzzle
Every piece must fit
Brain is like a muscle
Built to flex a bit
Outcome is a pattern
Math the means to end
Like the rings of Saturn
Mind is meant to bend
The Common Color
Common Color
Paint by number
Coleman standard for the schools
Fill in spaces
Absent traces
Outside boxes. Them’s the rules
To Mock a Billing bird
How to kill
The love of learning:
Follow Bill
And billions burning
So many delights from your pen, SomeDAM!!!
To Bill, the schlock-king bird,
What matters isn’t learning;
Math is data and so are words,
Treasures for his earning.
This is a wonderful essay Diane, thanks. I copied it to my mathematician/ musician hubby, who has occasionally tried to share this sort of vision.
I was a truly terrible math student (except good at geometry). Still, a decade in a biz job in early adulthood– where I used numbers all the time– showed me I had strength at mental arithmetic, and a good sense for order-of-magnitude (which is great for estimating, & also for spotting mental math errors ;-). It just sort of kicked in when I started playing w/all those numbers every day. You find yourself looking for short-cuts & efficiency, & learning that one can miss the boat entirely, crunching data w/a calculator, without an overall sense of the problem you’re attempting to solve, & brainstorming a few alternate approaches.
If anyone had suggested to me that that field of work required math ability, I would have shunned it. I came to it thro a back door common for women in the ’70’s – a secretarial job. I found what my boss did interesting, & volunteered to do some of his work instead of just typing it up; one thing led to another.
The “problem” was always, “Who has the best proposal for building this thing?” But every jobsite had different variables, & different mfrs had different approaches to meeting the design criteria. Number-crunching was required, but hardly the main focus, & you could even lie w/the number-crunching to favor a certain conclusion. The way we teach math puts the number-crunching at the center, with so little focus on the big picture [what are we trying to accomplish– what might be some viable approaches] that it gets lost in the shuffle.
It often happens, I think, bethree, that people discover in adult life that they have ability and interest in mathematics that they didn’t have when they were children. Children aren’t typically great at very abstract thinking, and they aren’t much interested in manipulating abstract symbols. When people are a bit older (young adults), they have become quite a bit better at abstract thinking, and if they have reason to return to mathematics, they find it a lot easier and more interesting.
I remember reading a report resulting from a US-China math-teacher exchange in the ’80’s. The Chinese critiqued our math pedagogy as being the rote learning of formulas which are then endlessly applied to varying sets of data. They contrasted their own methods. All that has stuck with me is a description of a typical math class for middle-school-aged kids. They were divided into teams, each with a basket of manipulables, and given a problem to solve. The solution required devising a formula. It went w/o saying that multiple approaches would work; there was no one right answer. After a couple of teamwork sessions, each team presented their formula to the class. Then the class discussed the pros & cons of each, & decided which was best and why.
I’ve been really impressed with the elementary school mathematics books I’ve seen from Japan. They teach a LOT less, and the books are very clear and sequential, with one idea building on the next. from a review of the wonderful Tokyo Shoseki textbook program Mathematics for Elementary School, which is available in English translation:
“Perhaps the first reaction one gets when looking at Japanese math textbooks is that they are thin, lightweight paperbacks with colorful cartoon illustrations. The total number of pages ranges from 120 to 210 per grade level. This is vastly different from the 500 to 600+ pages found in typical, heavy U.S. textbooks. (I couldn’t help thinking that if for some reason the number of hours devoted to math education were reduced in the U.S., we wouldn’t bother reducing the content.) This embodies the Japanese philosophy of teaching a few important math topics per year in depth instead of what is typically done in the U.S. — teaching many topics per year superficially. Japanese textbooks are also inexpensive and they are given to the children to keep.”
The organization and content of these came out of teacher-based Lesson Study–a practice that we really need to adopt here in the U.S.
In Lesson Study, teachers are given time from their schedules, each week, to meet with other teachers to share ideas and materials, go over their lesson plans, discuss what’s working and what’s not, etc. These are like quality circles in business–bottom-up continuous improvement rather than top-down.
Let’s note that Lockhart’s essay was written 17 years ago, and hence it’s not the criticism of anything brand new in math education. The extremely harsh, criticism of the various math topics at the end of the article are as applicable today as they were when I came to this country 35 years ago, and they are extendable to college math education and curriculum as well. We simply do not teach what needs to be taught and we certainly do not present math in the classroom as it needs to be presented.
Nothing essential has changed, except math textbooks have become prohibitively expensive and even heavier than they were in 1985.
As the article suggests, the culture of math education needs to be changed. Yeah, make it hands-on and playful, but not because these sell but because that’s the true nature of math, yeah make it less rigorous, because rigor almost always hides the essential ideas even from mathematicians. High rigor in math reasearch papers has been adopted in the last 100 or so years due to some shocking experiences in the foundation of math 150 years ago. But we know (and have known for at least 50 years), these foundational issues have no effect on the body of math we need to teach in schools, and hence we can throw away our paranoia, relax and present math so that it makes sense to kids of every age group.
Yeah, instead of the words “rigor” and “proof” and “precision” and “speed”, we are better guided by the expression “make sense”: present math to kids, to all kids, so that it makes sense to them.
After rereading the article, I think it’s important to emphasize that students do need to learn some abstract notions in math. After all, math is a language to describe some abstract objects, hence it has a vocabulary and a grammar. To get a sense of the language, think about what 2020 represent. It takes a darn long time to explain its meaning in English, doesn’t it? While its rigorous meaning doesn’t have to be understood by kids (while one of the crimes of CC is to demand this understanding and the ability to explain it), kids do have to have a sense of the meaning of 2020.