From "How Math Became an Object of the Culture Wars/As was true in the nineties, today’s fights about math are not entirely about what kids actually learn in their classrooms" by Jay Caspian Kang (The New Yorker).
November 15, 2022
"William Heard Kilpatrick, one of the most influential pedagogical figures of the early twentieth century, would have felt right at home in today’s educational culture wars."
"Back then, as now, the traditionalist defense of math education came from the idea that the subject created order and discipline in the minds of young students. The child who could solve a geometric proof, for example, would carry that logic and work ethic into his professional life, even if it did not entail any numbers at all. Kilpatrick, a popular reformer... dismissed that idea. Algebra and geometry, he believed, should not be widely taught in high schools because they were an 'intellectual luxury,' and 'harmful rather than helpful to the kind of thinking necessary for ordinary living.' Not everyone was going to need or even have the intelligence to complete an algebra course, Kilpatrick reasoned. Why bother teaching it to them? In 1915, Kilpatrick chaired an influential National Education Association committee tasked with looking into the reform of math instruction in high school. He amplified his attack on the place of math in schools, as the committee’s report declared that nothing in mathematics should be taught unless 'its (probable) value can be shown,' and recommended the traditional high-school-mathematics curriculum for only a select few...."
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Algebra and geometry, he believed, should not be widely taught in high schools because they were an 'intellectual luxury,
All those cheering should take the idea to its logical conclusion:
Other intellectual luxuries:
- Greek and Latin, and by extension the origin and etymology of words.
- Literature, outside of the Bible or other spiritual texts.
- Science and The Scientific Method
- History, except the very recent which may come up in conversation
- Languages, except for those bordering on your immediate area (reachable in a day or two by horseback)
When people say, "I was told there'd be no math," I guess they are referring to this guy.
1) Today's battles are mostly about how the teaching will occur, not what is to be taught.
2) However I agree with the basic premise, we are teaching material to the students that they are not interested in, and will never use.
3) The biggest problem is the college for all movement. My high school no longer offers general ed classes. All classes are college prep or higher. We have gutted our vocational tech offerings (no more wood shop or metal shop, no more auto shop, no more home ec, no more ROTC, no more masonry) Every student is required to fill out a financial aid request form as a junior, and take the SAT.
4) Meanwhile, there is no accountability for the students. This year I have ten "Juniors" who have less than a year's worth of credits. (Two of them have literally zero credits) I have at least a half dozen students who have not turned in a single assignment all year. (one of them also manages to be absent on the days I give tests, so literally has a zero in my class. This student has been selected to attend leadership conferences at the local college. It may have something to do with her being Black) I have students with forty or fifty tardies. I have a student who skips my fourth period class and double lunches almost every day. The admin has told us not to fight with the students over cell phone usage in class, and the program that used to allow me to monitor what sites the students were going to on their laptops no longer works. A third of my students are failing my class (which they have to pass to graduate) and at least one other. This is in a district that mandates that we give the students a minimum of 50% on any assignment and test they turn in, regardless of their actual score.
5) Despite everything I wrote in #4, I have been contacted by exactly one parent all year.
If proficiency with algebra is linked with general intelligence, then learning it matters. If, for no other reason, to differentiate students on their intellectual ability.
More useful than all the social studies, history, literature and PE classes I took in high school were statistics, algebra, and differential equations. Especially differential equations.
Has anybody recently looked at the rate of change of US government spending relative to the rate of change of US productivity?
Trading Economics
This hits home. As someone who had to retake (after failing) geometry, algebra 1, remedial math for business majors and calculus I agree 100%. I went on to become a successful trial lawyer and yet I have never had to use geometry, algebra or calculus. I still remember the shame and embarrassment of having to hide this fact from my friends in HS and college. Students would be better served with a course in economics.
Kilpatrick's assumption was probably that half the pupils would grow up to be homemakers and most of the rest would become the industrial proletariat or the salesforce or the burgeoning teacher/social worker class. He didn't see that finance, management, science and technology would become so central to the economy. Geometry and algebra might be more useful in work like the building trades than Kilpatrick thought -- he apparently thought of his workers as brainless drudges -- and in any case, thinking of numbers as interesting and having to study algebra and geometry would reinforce the basic arithmetical skills.
It looked for a minute like the New Yorker was calling someone on the progressive side a "culture warrior" -- a hopeful sign -- but no, for them it's always the other side who are the culture warriors.
Mathematics Made Difficult, Carl Linderholm, "Guess the Next Number"
"A great deal of what we learn at school is of little use in later life. This is especially true of mathematics. Beyond the most basic arithmetic, which does have a use in checking the bill in a restaurant, there is very little that is ever used again except by specialists. A knowledge of probability theory is handy for an undertaker, so that he can work out when his customers are likely to need him; a little topological group representation theory is not amiss if you happen to end up a quantum mechanic, repairing other peoples' quanta when they begin to wear out. But for most of us, most of our mathematics moulders away slowly as the brain cells blink out, cell after cell, in our heads. It never gets used.
Number guessing is an exception. Why is it not taught in the schools? ..."
He means what is the next number in this series stuff, that turns up on tests that determine your future. He winds up recommending lagrange extrapolation, which covers all cases.
You can do it without math almost.
What is the next number?
1 2 4 8 16 _?
form the differences
1 2 4 8
and those differences
1 2 4
and those
1 2
and those
1.
Now just repeat that difference
1 1
and build the series back up starting from 1 with those differences
1 2 3
1 2 4 7
1 2 4 8 15
1 2 4 8 16 31
and 31 is your answer. It's the value of the lowest degree polynomial going through the known points at the next position.
More useful than all the social studies, history, literature and PE classes I took in high school were statistics, algebra, and differential equations. Especially differential equations.
Has anybody recently looked at the rate of change of US government spending relative to the rate of change of US productivity?
Trading Economics
In my work, I use algebra several times a week. I write equations and then "solve for x".
Algebra is skill that few people would teach to themselves but that can be useful through ordinary adult life.
In high school, I took trigonometry and calculus -- neither of which I ever have used afterwards. I do not, however, regret the time and effort I spent on them.
Back in the 60s you could graduate high school with just a secondary math class. Maybe just basic math, I don't know. I took algebra because it was required for UC admission.
And I encountered algebra in chemistry, physics, and genetics. Good luck avoiding it.
Only recently was algebra 1 made mandatory. Like 1990s. They were even talking about making algebra II mando in California!
I assume that didn't work out? Lol
If we taught kids what they will use in everyday adult life, what would we teach? That economists are right half the time, just like a coin flip? That most of what they read in the news is not accurate or slanted? That a lot of people don’t get married and of those who do half get divorced? That climate change worry is overblown but the national debt is a looming crisis?
High school math isn’t high-level requirements. Let’s stick with it. It’s dead lifts for the brain.
Now do English.
What's the point of 'teaching' it when we already fucking speak it...
National Education Association: A hundred years of dumbing down math class.
I didn't know the NEA was around that long, and looking it up, found it started before the Civil War.
In 1915, William Heard Kilpatrick, an influential education thinker, pushed for eliminating Algebra and Geometry class for most students because they probably won't need it.
Of course he was correct, and the same could be said for most of the high school curriculum. The problem is knowing which students can afford to have their career and life options severely limited before they even have a driver's license.
That would require much more honesty and cruelty than we could muster. I prefer a system that would intellectually challenge each student as much as possible, and let them choose a path later on.
We've allowed a group of math-phobic elementary school teachers to dominate this discussion. They bred math hatred for generations. But, math is the core need for the highest income STEM careers worldwide. Take your pick: easy and 'necessary' math and a low income, or 'hard' math and a high income.
The need for math has been proven many times over and all across the world. Resistance is ideological and reveals fear of being outcompeted or fear of failure. Nothing more.
Why is it that the UK is home to all sorts of math superstars (going back to Sir Isaac Newtown for sure, Alan Turing, etc.), while other similar European countries aren't that great? What happens in the UK's math teaching culture that works so well?
So the NEA was opposed to education even in 1915. China runs rings around us in education while Biden struts around in a Mao jacket. Easy to predict our future.
This is rather silly. Before WW I, USA had no computers, and very few electrical devices. Nor a massive Defense establishment dependent on R&D and high tech weapons. Today, we live in a society where a large amount of our economy invovles knowing basic math and applying science of various sorts.
In 1915, there was almost no air force, radio was in its infancy, and the US army was using horse drawn artillery. 25% of the USA were farmers, and as late as 1940, only 10 percent of students went to college, and 50% didn't graduate from HS.
You can forgive Mr. Kilpatrick in 1915 for thinking math was a waste of time. Today, its the starting point for massive numbers of computer scientists and programmers, engineers, and technicans and researchers needed to run a society and defend the country.
"the committee’s report declared that nothing in mathematics should be taught unless 'its (probable) value can be shown,'"
If we applied that rule to the entire K-12 curriculum, the school day would be much shorter than 6 1/2 hours. Which would create a daycare crisis.
Math taught as an end to itself is a failure except for the subset of people who thrive on pure math. Math taught as a tool that furthers other interests -- computer programming, real-world money counting, home repair and measurement -- is a success. During my schooling years math was taught backwards, as a cause rather than as an effect. It's why a lot of people hate math.
I do object to the obsession with Calculus and higher forms of math. As a finance/business major I was forced to take calculus classes. My roommate was a computer science guy and had to take at even more advanced calculus.
Neither of us ever had to use in our jobs.
General of the Army Omar Bradley would do algebraic equations during his spare time as a method of relaxation. True story.
Why is it that the UK is home to all sorts of math superstars (going back to Sir Isaac Newtown for sure, Alan Turing, etc.), while other similar European countries aren't that great? What happens in the UK's math teaching culture that works so well?
Rene Descartes and Pythagoras would like a word with you.
My granddaughter (now 11) was struggling in math, so I started tutoring her. She was very timid at first, reticent to volunteer answers. However, as we worked together she found her confidence and is now performing well in both math and her other subjects.
I believe math is unique in its ability to instill confidence in the young student. The key is, there is a correct answer. The student can know with certitude that they got it right.
gahrie's comments illuminate the reality in most of our public schools. It matters little what is taught if nothing is learned.
A friend of mine was a 'counselor' in the state prison system. He said the most pathetic thing he saw was a good-hearted retired lady teacher coming in to teach the prisoners alegbra, calc, and trig. I understood what he meant.
I have always been happy to leave STEM subjects to those who love them, same as athletics and almost everything else our schools are tasked to teach. I retook Alg I (my only summer school experience), did well in Geometry, and stopped paying attention in Alg II after the first few weeks.
College required no straight math--I took jock physics (OK call it historian's physics if you want), did well in symbolic logic, and took some math-using Geography lab courses.
I did well enough, and better, on the math portions of the standardized tests, but math simply doesn't interest me much and I've never used more than basic algebra in my private and work life.
Enigma, "math phobic elementary teachers". Too right. I tutored in-class math, fourth grade, at a top private school. The teacher stopped by my table where four kids were learning ultra basic algebra, equation balancing and the like. All well within their grasp. Her parting remark was "I couldn't do that. It looks complicated." Fortunately the school and parent population believed firmly in helping kids advance as fast as they were interested, so her inability was ultimately not a problem. But I was astounded that any teacher within shouting distance of Silicon Valley could have been so blithely comfortable with that level of math ignorance.
Here is the hard truth that I believe- most children don't really understand math beyond the simple operations of addition/subtraction and multiplication/division in getting results. If you asked a normal 18 year old American high school student of even IQ 110 or so to explain why you offset rows left when doing multiplication by hand, they couldn't tell what that added space each row signified. If you asked them to explain why we need 10 digits for our dominant base ten numbering system, you would certainly draw a blank stare (I know these things because I have spent some significant time in the past tutoring high school students for pay and for free in math, physics, and chemistry, and you quickly learn their limitations, and these were bright students who just needed more tutoring for placement tests). At first, I thought it probably was a failure at the earlier levels of education- I know for a fact that the teachers I had in first-third grade didn't explain such things- you were expected to just remember by rote to do it, and not think about it any deeper than that. However, I now think that such details are just beyond the grasp of the large majority of people. I imagine the ability to understand a geometric proof, for example, is probably limited to the top 10% of people in intelligence. Lots of students can memorize the quadratic or the Pythagorean, for example, but almost none of the children graduating from high school could derive either one after having passed HS algebra and geometry (and both are really easy derivations as far as mathematics go).
So, I am not necessarily sure that Kilpatrick was wrong. Such classes, of course, need to be offered to those students who can grasp and use the logical tools mathematics teaches you, but maybe it shouldn't be mandated for all.
Geometry was almost fun, and like a game until trigonometry entered the picture. Physics and statistics were more of a problem.
If you understand why the political left hates math, you can see where the concept of "right answer" became "white supremacy."
...today’s fights about math are not entirely about what kids actually learn in their classrooms
FIFY: "Math is not what kids actually learn in their classrooms."
There that now matches reality more than his statement.
Enigma asked what it is about England.
I just ran across an article last week that addresses this:
https://www.popsci.com/excerpt-loonshots/
Part of the answer was the Royal Society.
I'm a former fellow of the Royal Society. Unfortunately not that one. I was in The Royal Society for the Arts and Manufactures. I thought having "John R Henry, FRSA" on my business cards looked pretty cool. Until I found that nobody had ever heard of either in the US and I stopped renewing.
John Henry
yet I have never had to use geometry
Bullshit. If you can define a line or a point or a shape like rectangle then you have "used geometry." If you are aware that a square has four equal sides then you "know geometry." If you ever said "acute angle" then you "used geometry." If you are aware a circle is 360 degrees then you "know geometry." If I magically removed all geometric concepts and vocabulary from you then you would be at a loss to describe the world around. Square that circle, man.
Math concepts are so closely intertwined with modern living we lose touch with where these terms integral to the technology we've adopted came from. Maybe there's a German word for "we know so much nowadays we don't where the knowledge came from."
Calculus is the first time math is fun.
RideSpaceMountain, Ha! My father used to do Taylor series expansions while swimming laps.
"Math is the language of the universe, the lifeblood of technology and the skill that most helps one avoid being ripped off."
It has been ages ago, but I moved through the Algebra, Geometry, Trig, College Algebra, and Calculus sequence in high school. I was required to take Calculus in college, and found it to be "easy" for two reasons -- I had good math teachers in high school who prepared me well for higher level math, and I was a Social Sciences major and my calculus course was not as intense as the one the engineers are taking.
I believe some level of math is important to all students. I disagree with those who believe they "never use algebra" in their adult lives -- I think it is something that comes up frequently in my life, unrelated to my career. As others have mentioned, knowledge of geometry is relevant not only to the STEM majors, but also to plumbers, carpenters, and countless blue collar professions.
I do think we should rethink the inclusion of calculus for high school students, even those who are ready to tackle advanced math concepts. In my opinion, Statistics is a far more valuable course of study for nearly all students. It has application not only for those who will use it in their career, but also in everyday life . . . such as understanding the economic blather that presidents and politicians use to mislead voters, and to make themselves look good.
I'm so tired of the argument " advanced math is a waste because it's never used in real life "
Pick any academic subject other than the purely artistic. I claim that you can't understand it without understanding the math. Economics. Chemistry. History. Linguistics. Psychology. Business. Engineering. Medicine. Anything involving data, analysis, and statistics. Anything requiring the scientific method.
If you can't use appropriate math to prove to yourself that what you are being taught is correct, then you are just taking someone else's word for it.
Sure, in the work world you may be using tools to bypass having to do all that again, but you will have a better reason to trust them, and a better chance to detect when they break down.
Math concepts are so closely intertwined with modern living we lose touch with where these terms integral to the technology we've adopted came from.
Ask anyone under the age of 18 why we "dial" a telephone number or what it means when someone says "It's at your two o'clock".
is there really any additional discussion needed on this topic. The answer is well known to be provision of more than one educational track. Arizona is moving in this direction. Students should have choice. That's the American way.
The one argument that can be made for mandating students study these in HS is that it allows those who have the talent and interest to find this fact out.
For what it's worth, the vast majority of what I studied in law school had been useless to my professional career, and most of the law that I now use was not part of the curriculum.
I've never used art or music in my life. Why did I have to take it?
Vidal used to say that HS should NOT have arts programs--those with talent and drive would find their ways to those areas, and you wouldn't have a bunch of half-assed half-talented wannabes taking up space.
I agree that simple statistical reasoning and logical thinking are scarce, and scarcely taught.
The old “educator” is funny. We are 1500 years past the ancient Greeks and he wants us to be dumber than they were. This is an argument for having no schools at all. Everyone would be insufferable. By human nature people would think they were as smart as everyone else, but they would know nothing. What would they base their votes on?
At every step of the way kids would take the easy way out, and once out they could never go back. There would be stem people, but fewer of them, and many who could have been would never know it.
Perhaps it would be like the small town where everyone made a living being a social influencer. We just need enough people who took math to sell bonds to China.
Even at the entry level, math reveals that some things are concrete and immutable.
How quaint.
One of my cousins has a son who graduated HS in 2018. He's not that academically gifted and struggled in school, but he WANTED a NY Regents diploma (now the only option in NY) and not just a GED. To get there he had to get thru the Algebra Regents exam. He was tutored and worked so hard, and ended up with the same score on the test that I received many years ago, an 85. It was so important to him to have the diploma, walk across the stage at graduation - the whole thing. He's an exceptionally hard-working young man, and now works one job full time and apprentices with an electrician in addition. The character and determination he showed in getting himself thru Algebra is the same he continues to show today. I don't think we need to give everyone a true "college prep" education, but there is a lot to be said for challenging students to do something tough.
Blogger Critter said...
is there really any additional discussion needed on this topic. The answer is well known to be provision of more than one educational track. Arizona is moving in this direction. Students should have choice. That's the American way.
If Hobbs is elected, as seems likely now, I think the education reform Ducey pushed through will not get very far. Th Democrats tried to run an initiative to reverse the legislation. A Governor indebted to the teachers' union will cover this with a pillow.
Joe Smith said...
Now do English.
What's the point of 'teaching' it when we already f-----g speak it...
I cannot vouch for this being true as it's something I've only read, but apparently we're the only nation that teaches it's language in a language course. I recall being told in other classes whre we had to write esssays that we wouln't be graded on our English... In other nations, if you write an essay in, let's say, your history class, the paper is graded on both history AND grammar.
And on math- I use geometry and algebra near every day. It's just not called algebra and geometry. There are a LOT of jobs that require use of both but they're not posed as math problems. Carpentry is one of those jobs. Carpet laying, roofing, etc. They're both problem solving techniques.
And for a lot of us, basic science like (P1*V1)/T1 = (P2*V2)/T2 comes into practice every day. It's why diesel engine work without a spark plug, and why you don't wing open air valves to pressurize a line, but do it slowly.
I think part of the problem teaching math is that while most math, even calculus, comes into play in many professions, mathematice departments and teachers are interested in pure math, not application of math, and the students are never taught where in real life math matters.
I don't actually use it, but knowing basic calculus aids in undersanding some of the world around us. If you care about that sort of thing. I took it in HS and have never actually used it. I know it has real life uses, but not in my job.
To my mind, the point of school is teaching kids how to think. Math teaches us to follow logical steps and rules to get a correct answer. English/literature teaches us how to read progressively more complicated texts and grapple with higher ideas beyond sex/drugs/rock'n'roll. History teaches us that nothing under the sun is new. Science teaches logic and applied math.
The whole point is so that we can then go into the real world and recognize BS when we see it, really. "Say, that guy says Republicans were all for Jim Crow, but I know the South was solidly Democratic. Hmmmm. Maybe he's BSing me. . . ."
In other nations, if you write an essay in, let's say, your history class, the paper is graded on both history AND grammar.
I teach high school history. I had a consultant report me to the principal, and the principal write me up, because I insisted it was part of my job as a history teacher to grade grammar and spelling.
Joe Smith said...
"Now do English.
What's the point of 'teaching' it when we already f-----g speak it..."
(I'm guessing this is tongue in cheek, but maybe not.)
I remember taking this attitude explicitly about 6th grade, or thereabouts. I blew it off and that was fine; I had done a good job picking up proper English grammar through osmosis as I grew up. Then I took a foreign language and I was lost. All those grammatical terms and principles I didn't know came back and bit me in the butt.
Gahrie,
Re vocational ed, I am in full agreement. But there are still some programs. I am getting involved in one with the Packaging Machine Manufacturers Institute (PMMI) called Packaging Challenge. They just finished their first on a couple weeks ago.
6 teams at 6 high schools were each given a crate of parts and components, some vague instructions and and told to build a fully automated machine to fill 20 marbles into bottles.
I had a chance to see the results at the recent PackExpo trade show and spent about an hour with the teams and instructors. Very sophisticated machines, lots of skill and creativity on display. Mostly juniors and seniors. Some 9th & 10 graders.
Any one of those kids could start at $50-60m the day after they graduate.
Sure beats spending 4 years and scores of thousands getting a degree in fuck-all just so you can get a job as a barista in a Starbucks.
Here is a press release on the program https://www.pmmi.org/show-news/pack-expo-international-challenges-students-creativity-new-machine-building-competition
I also talked with the teachers. Highly competent and interesting. One of them gave me a virtual tour of their facilities. Complete machine shop with multiple Bridgeports, lathes and 6 CNC machines. 8-10 3D printers, sheet metal shop, wood shop and more. Any of my clients with manufacturing plants would kill to have a maintenance shop like that. The school is in a Chicago suburb but I don't remember which one. All were public schools.
So I agree with you generally about the vocational programs but there do seem to be plenty of exceptions. This being a pretty spectacular one.
John Henry
I think part of the problem teaching math is that while most math, even calculus, comes into play in many professions, mathematice departments and teachers are interested in pure math, not application of math, and the students are never taught where in real life math matters.
I remember how much I hated word problems in grade and high school. Yet those were almost the only ones that are directly useful in later life. I think there should be the basic math to make sure of understanding but then word problems all day long. We do math word problems every day and don't even notice.
I also think that estimation should be taught every year K-12. How much is $7.18 minus $2.53 without calculating. Or 432 X 927. Any answer within +/-15% being counted as correct.
Or estimating measurements. I tried to do this with my grandkids. Make them guess how long, high, heavy, many gallons and so on then measure and see how well they did. I was not that successful at getting them to do that because it was not school.
And some basic statistics would be helpful. Mean/Median/Mode might have come in handy a week or so back when Brandon was talking about "the most common price of gasoline". Not the mean or average price. Not the median with half of all gas being more and half being less, but the mode or most common price. He was trying to hoodwink us and nobody even noticed.
Or the concept of "normal" as in a normal/gaussian/bell curve distribution. Is transgender "normal"? Not if only 1-2% of the population is transgender.
Or standard deviations. I can't remember the last time I actually calculated a standard deviation. Perhaps a year or two ago. But I use the concept of 6 sigmas most every day.
John Henry
Or a lot of other really basic stuff that we need to know for daily life to make sense.
Math or any subject taught in K-12 is more than a set of facts one learns, or it should be. Those pesky word problems teach more than math, they teach how to analyze language to extract meaningful information. The concepts taught in geometry are not all used in everyday life, but the fundamentals: shapes and relationships between different shapes are. If you study music you are also teaching your brain to analyze sound and rythm. And then there's the fact that music relies on fractions... English... should be about teaching language and the vocabulary of language (parts of speech, types of sounds used in a language, etc.) and nuances of language as in how punctuation can change the meaning of a sentence or how many different things a phrase can mean. I'm certain that our host can elaborate even further on the language aspects of what could and should be taught in K-12.
My late father was a math professor at a university in Canada. He lamented how poorly the students were in basic math - he once had me do a test he gave his students (I was in the 10th grade at the time). There were many things Ididn't understand in the test, but I still answwered more than half of the questions correctly. Yes, I was considered pretty good at math in high school. And this was more than 50 years ago.
If you get nothing else out of K-12 you should at least get some skills at critical thinking - how to spot BS when you see or hear it.
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