What would you do, if you were designing high school math from scratch?
Well… probably not what we do now.
What in Noether’s name is going on here?
Why do we teach so many obscure technicalities, and so few practical facts?
Who the heck designed this monstrosity?
Nobody Designed Mathematics Education
I’ve come to believe that even this simple question—“who designed this?”—rests on a flawed assumption. The broad thing we call “the math curriculum” isn’t really “designed.” Rather, like all educational institutions and systems, it is shaped by a hailstorm of competing forces:
Over time, each of these parties tugs and prods at the curriculum, reshaping it to suit their needs. No single author writes the curriculum. Nor, even, do multiple authors reach a clear and coherent compromise. Instead, the curriculum is perpetually being nudged and tweaked, eroded and built up, by various actors who share no unified vision.
Math education isn’t like a skyscraper, designed by a single architecture firm.
It’s more like a mountain, shaped by many competing geological forces.
That’s why it’s so hard to find a clear sense of purpose in math education as a whole. Is it about pulling kids into STEM? Imparting broadly useful job skills? Cultivating critical thinking? Ensuring compliance and rule-following? Providing a neutral arena for students to compete for college admissions?
To varying degrees, the answer to all of these is “yes.” Math education doesn’t have one purpose. It has many—sometimes aligning, sometimes conflicting. Together, they form a knotted ball of twine that even the nimblest fingers struggle to tease apart.
Pleasing All the People, All the Time
Sounds like a teeming pile of contradictions, right?
Oh, I’m just getting started.
Take one actor from my list above. Put them under a microscope. You will not find a single goal-oriented organism, but instead a messy conglomerate, pursuing several contradictory goals of its own.
Start with universities. They seem to want:
Depending where you go on campus, you’ll find voices singing very different tunes about the nature of math education. There’s no way to make them harmonize. Instead, the math curriculum simply does its best to incorporate the whole scattered, dissonant bunch.
And that’s just universities! Think about a far more diverse and decentralized group, employers. Depending what business you’re in, you may want math education to cultivate in your employees any mix-and-match combination of the following traits:
When critics demand that math education provide more “real-world,” “professional,” or “practical” skills, they’re missing a crucial fact: the “real world” doesn’t agree what skills are valuable!
I haven’t even touched on the people actually delivering this education: teachers. They exert their own pressures, preferring to teach things that are:
The paradox? These are frequently opposites! “What I learned in school” is typically rote and spirit-crushing. “What I find fun and rewarding” is often far more creative and exploratory. Many teachers find themselves torn between what they know, and what they value.
Shelter from the Elements
So what should we do about this lumpy mass we call American mathematics education?
Well, let’s be frank. It’s hard to remake a mountain. Sure, you can rearrange the rocks, but the forces of geology will soon return it to its former state. You can’t fight the wind, the rain, and the grinding power of the tectonic plates.
Instead, the best you can do is to build a cozy little shelter.
When I read a piece like the recent hit The Wrong Way to Teach Math, I nod along to the criticisms. Andrew Hacker has a coherent and lovely vision for how to teach mathematics. But to treat his work as a blueprint for all of mathematics education is to make a category error.
A great class can no more become a national curriculum than a beautiful home can become a mountain.
Good education needs unity and clarity of purpose. My transcendent experiences in the classroom all came when an expert teacher built a gorgeous class, where every component—instruction, assessment, discussion—fit together in service of well-defined goals. A good class is a masterpiece of architectural design.
But “design” is precisely what the policy world—with its thousand competing forces and colliding actors—can never supply.
Teachers like Hacker provide their students with something beautiful and precious. They offer shelter from the punishing forces that shape educational institutions—the lack of resources, the need for sorting, the pressures of political orthodoxies. They give students a rare chance to engage in that most wonderful activity: learning.
But, to borrow a Silicon Valley verb, such visions cannot “scale.” It’s a mistake to think that educational progress means uniting behind a singular, coherent purpose, because our purposes are not singular, and will never cohere. One architect cannot remake the mountain.
Instead, what education needs are many architects, scattered across the mountainside, building shelters for the students under their care.
We cannot build a new mountain. But we can build our homes upon it.
If kids can balance a checkbook when they get out of school, they will be better educated than the US Congress.
It occurs to me that nothing in nature can be described by our box-like 3-D world. There are many dimensions, of sound, texture, color, temperature, etc., as well as time. The mountain will last far longer than the man-made structures built upon it.
Also, the problem you describe seems to cross all disciplines. To approach math as problem-solving seems reasonable. What measurements are necessary to bake a cake, knit a sweater, build a box out of wood, get value for money at the grocery store? These skills show the relevance of math in daily life. They educate the hands along with the minds and can be fun, too.
But it’s 2016, What is a checkbook? I haven’t seen one for a decade. Adding numbers by hand – yes until addition is understood and to check that the answer your computer gives you is the right order of magnitude and then never again. Seriously, why does anyone over 10 need to add by hand?
They should be building a spreadsheet, learning about why compound interest is great if you save and awful if you borrow. They should be learning about how to gather information on what they spent, analysis it and plan a budget. And making charts, charts are great, but making 100 from a spreadsheet and deciding which is most useful rather than drawing them by hand (urgh) that teaches them that the angles never add up to 360º when you do it by hand (that could just be me).
If you’ve ever knitted a sweater, you will understand why it’s important to know the basics. I still remember the multiplication tables and use algebra and geometry on a regular basis to solve problems when I’m designing and building things. I’ve never had a use for trigonometry and calculus in adult life, because of the career path I’ve chosen.
Those without the basics are easily swindled. What happens if the electronics break, as they so often do? What happens to the people who can’t afford this electronic gadgetry? Electronic money transactions are traps for the unwary. I still have a checkbook and balance it regularly, just to make sure the bank hasn’t pulled some new stunt without my awareness. It just started charging $6 per month for checking accounts under $1500, with the notice in fine print. Too bad if you didn’t know. The overdraft fee is $39.
It sure isn’t just math. See Pete Seeger:
What did you learn in school today,
Dear little boy of mine?
What did you learn in school today,
Dear little boy of mine?
I learned that Washington never told a lie.
I learned that soldiers seldom die.
I learned that everybody’s free,
And that’s what the teacher said to me.
Chorus
That’s what I learned in school today,
That’s what I learned in school.
What did you learn in school today,
Dear little boy of mine?
What did you learn in school today,
Dear little boy of mine?
I learned that policemen are my friends.
I learned that justice never ends.
I learned that murderers die for their crimes
Even if we make a mistake sometimes.
Chorus
What did you learn in school today,
Dear little boy of mine?
What did you learn in school today,
Dear little boy of mine?
I learned our Government must be strong;
It’s always right and never wrong;
Our leaders are the finest men
And we elect them again and again.
Chorus
What did you learn in school today,
Dear little boy of mine?
What did you learn in school today,
Dear little boy of mine?
I learned that war is not so bad;
I learned about the great ones we have had;
We fought in Germany and in France
And someday I might get my chance.
Chorus
This. Because no one pays attention to school committee elections.
Students generally ask, “Who invented this?” I work with a math teacher who alway answers, “Math wasn’t invented. It was discovered.”
But aside from that fun comment, in today’s world, math education is more important than ever. We live in a world with tiny computers connected to the internet living in our pockets. Kids expect instant gratification for everything. Higher level math forces them to persevere through multi-step problems… Well, forces those who want passing grades.
Whether “this” in math class was invented or discovered depends on what “this” is. The Pythagorean Theorem was discovered. Using the letters a, b, and c as the variables for the standard expression of the theore was invented. Too often math class is focused on learning the code of the literate mathematician (invention) and not on revealing the patterns found in the bedlam (discovery).
Good point.
Fabulous.
I often say that people who offer pat solutions saying things like, “People should just . . .” I stop listening the minute I hear “just.” Obviously, they’ve not thought through the problem if they think the solution is “just” anything. Thanks for this clear explanation of why there are no easy answers and why the discussion is so important.
Love it. Love your blog. Thanks!
Wonder, curiosity, application, invention or…..our current system?
Have you read Scott Alexander’s “Meditations on Moloch”? Your argument is very similar – that the problems in our world arise mostly because our systems are not designed by humans.
But I think math education might actually be possible to scale. At least, I’m going to be interning at Google this summer and working on that problem.
I have a serious quibble with the attached article. it provides and example that in Alaska 98.9% of people have phones and in Arkansas 94.6 have phones, and what is the better way to represent this. The author suggests that scale that goes to 0 and makes 98.9 and 94,6 look almost equal is the better picture. But, I see the numbers and say, “A person is Arkansas is 5 times more likely to not have a telephone. That is a huge difference!”
It makes me think that this author doesn’t grasp the very practical statistics he thinks we need to teach.
Hacker is a polisci professor who has decided he knows how to teach math, and has started doing so. In the intro to The Math Myth, he gives several CCSS Math standards, verbatim, as problems a student might be asked to answer. He has not thus far impressed me.
Those numbers do not say what you claim.
The phrasing is a bit sloppy perhaps (and the state is wrong), but the numbers can be accurately phrased as, “On a per capita basis, five times as many people in Arkansas lack telephones as in Connecticut.”
Yeah, I think that’s a good point. Flip the graph to “percentage of people without telephones,” follow the convention of starting your y-axis at zero, and suddenly you get a dramatic difference, instead of an almost imperceptible one.
Same data, same customs of graph-making, yet two very different graphs!
You say a “bit sloppy”. I say a “huge difference” 🙂
https://simpleslothslife.wordpress.com
This is one of my favorite posts of yours so far. I’m afraid to comment any more because it could easily turn into a rant… but I liked your point about scaling especially. Cheers. 🙂
No man. Stop enabling people who don’t understand maths, people like Andrew Hacker.
While I agree to some extent with your post, where is the pursuit of maths for discovery?! Where is exploring one tiny stream in Topology, or Lie Algebras, or some other highly specialised area of the humongous river that is mathematics?
Complex numbers weren’t discovered because of that we-need-to-apply-maths-in-a-useful-real-life-way mentality, but because of mathematical interest.
Oh, and screw Andrew Hacker man:
http://www.slate.com/articles/technology/technology/2013/02/should_algebra_be_in_curriculum_why_math_protects_us_from_the_unscrupulous.html
Yeah, with Hacker’s work I’m trying to draw a distinction – I think the specific lessons he presents and distils are worthwhile (i.e., he’s a good teacher, by my definition of the term) but his vision for math education is (like any good vision) specific and idiosyncratic, and therefore I’m not particularly persuaded by his advocacy to make all of math education suit this vision. So yay on Hacker the teacher, nay on Hacker the policy-maker.
Put another way: I think he has thoughtful, sensible goals, and seems effective in reaching them. That’s good teaching. But his goals are not the only possible ones, and I’m not persuaded by his arguments to the contrary.
Fair enough! My only objection is that although Hacker is as you say, a good teacher, his policies are very counterproductive to education in general, yet are attractive to most people as they appeal to that popular idea that knowledge has to be useful in the real world.
He is in a way, the Donald Trump of education.
He is far too focused and precise to be the Donald Trump of education. At least he’s advocating for something more important than himself and actually seems to mean it.
“Instead, what education needs are many architects, scattered across the mountainside, building shelters for the students under their care.”
As opposed to a Stalin, Mao or E.D. Hirsch dictating a common core to be applied, by force as necessary, to all students at all altitudes and inclinations on the mountain?
If you’re talking about Common Core, it wasn’t by any stretch of imagination dictated by a single person: they were in fact created by “many architects, scattered across the mountainside, building shelters for the students under their care” in the metaphor. See the website:
http://www.corestandards.org/about-the-standards/frequently-asked-questions/
There is also the slight issue that a top-down imposed math curriculum might also be better than what much of the US currently uses… (e.g., http://www.edweek.org/ew/articles/2011/03/17/25math.h30.html). The USSR did basically do a top-down curriculum+pedagogy design, and the results weren’t that bad (e.g., plenty of excellent mathematicians, Russia still doing well on standardized tests despite overall worse SES than the US). Not to say the US needs a one-size-fits-all curriculum, but we’re obviously too far on the opposite side: so little common practices that we have almost no ability to build upon successful innovations.
And that even ignores the fact that in an age of highly-effective analytics (e.g., we can assess a students’ detailed basic Algebra skills in <= 35 questions using adaptive testing), most teaching is stuck using methods from the 1800's and 1990's. If aviation was like this, we'd still be flying in custom planes that range from biplanes to jets, crashing frequently, and autopilot would still be getting tested effectively at universities but used by only a small fraction of real pilots.
Which does bring up one issue: your tools partly cause your standardization in your pedagogy. If all you have is a blackboard and a slide deck, your class starts looking an awful lot like, well, a typical class. If, by comparison, all teachers had was a copy of Matlab or Maple, you'd see a lot more simulation and graphing. If they all were practiced at using a semi-autonomous system like CMU's spin-off the Cognitive Tutor, you'd see teachers doing a lot more targeted one-on-one remediation (by using the data that tells them in real time what each student is struggling on). And so on. Just like how water flows downhill, different tools and familiarity with strategies to use those tools will vastly impact what strategies teachers can easily apply. If a critical mass of tools end up eventually dominating the space, we will likely see emerging pedagogical standards of various sorts. The quality of such standards may vary greatly upon which tools succeed, and unfortunately success is highly unlikely to depend much on their actual quality (marketing to districts and saving teachers' time will likely be the two greatest factors, rather than efficacy data).
” If, by comparison, all teachers had was a copy of Matlab or Maple…”
As a high school math teacher, I haven’t been impressed with my colleagues’ general comfort level with software. Perhaps it’s the schools I’ve been in. Granted, there’s also the issue that teachers in general are having their time consumed by administrative hoops and hence don’t have the time to delve into this sort of thing, but WolframAlpha and Desmos are both free, and don’t appear to be widespread (at least, again, at the schools I’ve been at). Khan Academy (prepared material) is far more popular than WolframAlpha and Desmos (make-your-own). You might as well ask them to write their tests in LaTeX. 😉
I agree with you that we need more standard practices; often, practices aren’t even standard within the same building, let alone the county or state. But I’m not sure that software-based solutions would get general traction.
I have an analogy.
The school puts a class on the schedule called “Painting.” They hire a man who paints nice paintings of sail boats to teach the class. The teacher would like to hand out brushes and paint and canvas and for the most part let the kids paint what they want, while offering some tips on technique.
The textbook gives great detail of the Renaissance and Rembrandt, gives a passing mention of the impressionists and ignores the moderns.
Someone asks, “why are we teaching painting”?
“Because these kids may need to paint their house some day.”
And the school says, more rollers fewer brushes.
Industry says, “We assume that these kids know nothing when they get to us, Painting is mostly prep, applying roller to the wall is just the last 10%. Besides, we are all going to sprayers.”
The University says, “we are looking for the next breakthrough genius. Big thinkers and and high-concept is more important that technique.”
Some kids will one day paint their living room. A few will make their own nice paintings of sail boats in their spare time. Many will have fond memories of painting class although they will never paint again. Others will say that it was a big waste of time,
As usual, fantastic post. I would add one thing: there isn’t one body we can call “the” math curriculum. Despite standardizing influences, the mathematical experiences of any two students will still be wildly different. In what I’ve seen, that’s true even for students sitting in the same classroom, to say nothing of those thousands of miles apart. So, math could be many different mountains (some of which are even plains or valleys).
In my view, that’s a good thing, since the demands on math education are so varied. However, I hope we can find a way to make this more clear to students, if nothing else so that those who don’t like the view from their current mountain will know to try other math mountains.
Sticking with the architecture metaphor…. what is important to just about any building, be it skyscraper, shed, palace or pagoda? You need to have a solid foundation or the whole thing is going to collapse. Most of the discussion of “what should math education look like” revolves around high school math. Algebra for all vs statistics vs home-ec math vs whatever, I agree having many different architects and many options for students is a great idea… except the key to succeeding in any of these is a solid k-8 foundation. And heaven forbid a student in the teamwork, conformity centric path wants or needs to switch to the independent, creative path. Without the foundation jumping from one style to another (especially a “harder” path) is prohibitively difficult. We already see this with the numbers of remedial students in college courses. If your high school was algebra in name only and group project based and the college class is college level and independent work oriented, expect to need remediation.
Bravo! Once you notice how wrong our current math education is and how unfit for the world today, you made the first step. That step is already impossible for most educators. They keep claiming that the kids off today are to blame. Times they are changing. But too slow.
Kids these days are visual learners. They learn faster through interactive ways. Long gone are the days of just book and paper. They have shorted attention span, so it’s important to keep lessons interesting. Zap-Zap Math offer my kids that.
ok, does every student need every concept, basic and foundation of math? would a person rather actually perform a useful task such as taking the trash out for a university campus as a livelihood, and not busy them self with haggling the employer over the pay rate or manouvering the work schedules to get the laziest tour of the trash bucket flipping routes? where some deem an all out affront to existence on a subject they excel in, it may be entirely useless to some to waste valuable survival energy pursuing. with earths billions of available bodies attached to brains, now being casually replaced with machine effort, no actual brains are required (and less math too). this badmathwdrwngs is very entertaining but of no actual remedy outside the land of teaching and communicating a specialized skill set to newborn blobs of life. aspire to greatness, yes, but keep the feet on the ground!
Reblogged this on Crazy Pasta Child.
Great post. Found it via Joanne Jacobs.
The 19th century mathematical analyst was a “calculator”.
In the 20th century, the best selling author, Neville Shute, was the calculator for aircraft designer Barnes Wallis, who later went on to design the bouncing bomb which destroyed the Ruhr Dams in 1943.
That’s why “calculator” ends with an “…or” – because it was originally a person. On the other hand the “computer” has always been a machine, so it ends in “…er”.
I think “computers” were also people, before electronic computers were invented. (Have you seen the movie Hidden Figures yet?)
I love your posts. 🙂
Reblogged this on math – update and commented:
We finally start – motivated by – this article, our so long waited EDU Posts Series…
I have a quibble about your examples. For instance, memorizing the derivative of the arcsec(x) is pointless. However, understanding inverse functions and learning how to derive their derivatives from the more common inverse is quite useful over a broad range of scientific disciplines including basic probability, random processes, and statistics. I don’t know how you are supposed to develop a deep understanding of statistics and p-values without a deep understanding of calculus, and nobody ever learned a dollop of math without working specific examples.
Hence, learning how to derive the derivative of the arcsec(x) can be quite fundamental to understanding p-values even if you forget the formula within minutes. It’s a tool and no more the goal than knowing what a wrench is for the sake of knowing what a wrench is. Good understandings of basic disciplines like calculus, linear algebra, probability, etc. has allowed me to be an autodidact in a wide range of inquiries. I’m thankful that I was exposed to them early.