It wasn't just the USA. Even in the conservative Portugal of the 1960s/70s a small wave of "modern math" (as it was called here) washed over the high-schools.
I believe it was more of an experimental thing, with partial adoption - i.e. only one of the math teachers would use the new syllabus on their students, with classes six days a week (yep, back then there was Saturday morning school).
All I remember now is a bit of bafflement at the set theory paraphernalia, rings and groups etc.
It seems like if you only load the HTML then use Reader mode that it does OK. I use μMatrix and have it only load HTML for sites where I have not explicitly allowed additional content.
Doesn't work for me. Firefox Reader Mode just shows the first page. (I have uBlock Origin installed, but I suspect it doesn't matter.) red_admiral's suggestion worked for me though.
> There's an endless search for an effortless way to learn math.
Nonsense. There is a search for more effective ways to teach math. And there most definitely are differences in effectiveness, so no reason to believe that any given way is optimal.
> Just like there is no way to get strong without hard work.
There are, however, lots of ways to spend a lot of hard work for little gain.
I don't understand how someone can learn math, yet be unable to solve math problems on a test.
In my 16 years of education, I saw a consistent 1:1 correlation with learning the material and doing correspondingly well on the tests. That consistency included myself - when I learned the material, I'd ace the test. When I didn't learn it, I failed the test. There wasn't any magical "learned but didn't test well" going on.
You’re speaking only of yourself and while you should be very happy you don’t suffer from any of a multitude of disorders, it’s highly classist and disrespectful to assume that nobody else else.
Consider anxiety disorders. There’s a crystal clear, fully scientifically proven reason that some can know the material but fail to perform on a test. Think more. It helps.
What do "disorders" have to do with class? At best, being from a lower socio economic class could put you at risk for whichever disorders you have in mind, but they aren't synonymous. At least, not in my mind.
> There’s a crystal clear, fully scientifically proven reason that some can know the material but fail to perform on a test.
I don't assume such people don't exist. But I'm not buying there are enough people to significantly alter overall student test results. There'd have to be a couple in every class, and I've never encountered one.
You're assuming that "math problems" is a well-defined set of problems. But depending on what you want to test/teach, there's also going to be different problems that students will do well on.
As an example: do you consider solving 579×46 quickly a "math problem"? I will solve it (relatively) quickly, since I was thought how to solve that exact problem algorithmicaly.
But if you hadn't thaught that algorithm (you could argue that nowadays we have calculators for this) then it would probably take you a bit longer to derive some form of this algorithm. But it doesn't necessarily imply that you're bad at math, just that you have not practiced this exact task before.
In my experience, doing really well in tests requires more adaption to the tasks of the tests than actual understanding. Most tests are limited on time, and stress does not exactly help you to deal creatively with completely new tasks.
Note that I don't want to say that teaching algorithms is bad. It surely helps to do some basic calculation without having to think much about it. I just want to point out that the tests are often testing a specific understanding of math, and different teaching methods often also differ in their goal of what they want students to learn well.
Being able to do arithmetic like 579x46 is foundational arithmetic, not some esoteric thing. Furthermore, teaching arithmetic while relying on a calculator is not teaching arithmetic.
It's second or third grade material. It's reasonable to expect teachers to teach it and kids to learn it. It's easy to create tests for it.
Of course they're not improving. The US education sector is like the US health sector - a device for increasing shareholder value among parasitic corporates, not for providing an effective public service.
As a result it provides vanishingly poor outcomes which keep the US below the OECD average, in spite of some of the highest per-student spending in the world.
Education is highly politicised, managed by school boards which seem to attract religious kooks and other low-quality authoritarians with eccentric ideas, and which does almost everything it can to discourage good teachers and other suspiciously well-educated and competent individuals from joining or staying in the profession.
Widespread poverty in the US population doesn't help.
Changing how math is taught won't address the essential dysfunctional broken futility of the system as a whole.
All the attempts to gamify learning math, or teach it by passively watching TV, suggest otherwise.
> more effective
Sesame Street is probably the most inefficient way to teach arithmetic ever devised. A kid can watch 1000 hours of it, and barely be able to count to 10.
We know how to effectively teach math. Present the concepts, follow up with the student working through progressively harder problems.
> And there most definitely are differences in effectiveness
Since the needle hasn't moved on results, we've likely found the optimal method.
People love math. Look at youtube channels like 3Blue1Brown - 3 million subscribers, 160 million views, or Numberphile with 3.4 million subscribers and 511 million views. Millions of people of their own free will go out of their way to at least get an introduction to math concepts. Most of this isn't practical knowledge that they need, but rather abstract concepts they learn purely for fun. Think of what fraction of the population who would rather do puzzles instead of what they actually are supposed to be doing at the time. Strip away the tedium of rote arithmetic drills and people can't get enough of real math.
Saying people simply don't like math because their eyes glaze over in math class is like saying people don't like stories because they're bored by Great Expectations.
The problem (at least one of the problems) is, kids are different. You see these kids are struggling with the traditional approach. You devise a non-traditional approach to help them. And it actually works! For them...
But some kids who would have done fine on the traditional approach now are struggling. Is that a net win? Maybe, maybe not. Depends on how many kids are in each group and how bad they're struggling.
Yep. Had exactly that experience this week. They're trying to teach my youngest how to subtract with carry. Now I taught her the column method about a year ago but she's not allowed to use that suddenly due to a new idea they have. Not a single one of the children understood it so the teacher went back to column method a couple of days later having pissed the time on that attempt out of the window. This sort of stuff happens very regularly.
Fashion is exactly it. I feel like they experiment on them rather than use established methods.
I had a talk about that with a relative (soon to be retired teacher, got a decoration for his job). His point of view was that teachers should be viewed as technicians: they are here a apply a method, not to invent new ones.
This doesn’t need to be robotic, it still give a lot of room for personalization. I find he’s really right, especially since in my country everyone want to do their own stuff in their own way. The ministry is guilty as well, because it produces an endless stream of contradictory directives, each based on theories made by pedagogists that never taught in the field.
His point of view was that teachers should be viewed as technicians: they are here a apply a method, not to invent new ones.
I think this applies to most jobs really. Research and practice are different things, and even when the same person does both it's good to keep them at least somewhat separate.
I (almost always) kinda want my doctor or mechanic to follow what's known to work, rather than try out new things. On my own software projects, experimental techniques seem to cause issues a lot more often than standard "best practice" sort of techniques and I try to keep them away from important production system. Etc etc.
Within home schooling communities, Singaporean style math has quite a following. Singapore also tends to score quite high on international exams. Basically, they focus on fewer topic more deeply. They also teach from concrete to abstract. I wish American schools would adopt this.
https://en.wikipedia.org/wiki/Singapore_math
There is another project called JUMP math which has shown some great results, and is now getting adopted widely in Canada. Essentially the approach is to present concepts as bite-sized chunks (more scaffolded) and make students gradually work their way up to tougher problems.
If you think about it, it makes sense: if the problem in math is people get stuck at some foundational step where they are required to "get" something, and start to think they "suck at math" because they are missing this piece, then making the steps VERY small will make sure everyone can make those steps, thus making everyone "good at math."
Similar to Singapore math, it's not based on a "textbook" that you read but an "exercise book" that you write and solve exercises in. I don't have personal experience with teaching using this approach, but I have heard many good things.
Stephen Boyd is somewhat famous for his class in Linear Dynamical Systems that begins with an introduction to what DiffEq really is, and how to use computation rather than abstract rules of symbolic manipulation. I felt cheated that my classes had been nearly exclusive to the abstract presentation, which was mostly useless except for rough conceptual understanding in the rest of my engineering life.
The biggest problem seems to be that they spend a helluva lot of time on math instruction, but not very much time actually doing the math.
For anything through AP calculus, you'd be better served going down the volume and spaced repetition route that math programs like Saxon use. Do things over, and over, and over, and over again until they are second nature. There's no magic bullet to get around that work that you have to do to master something.
My grade two teacher demonstrated "venn diagrams" one day, it absolutely blew my mind. She had a bunch of coloured shapes scattered on the floor, then put one hoop over the triangles, another hoop over the red shapes, and then, what to do with the red triangles? My six-year-old brain loved it when she put one hoop overlapping the other one. And yet to this day I refuse to memorize the multiplication table.
Rote learning is easier for the teacher: easier to teach, easier to mark. It's not just mathematics, it's all the subjects in school. Many of the "geniuses" merely survive school, nothing more.
I found myself thinking about multiplication tables in the shower today. I think you can get away with knowing the tables for 2, 3, and 7 really well. The rest is just tricks:
4: 2x twice
5: add a zero and divide by 2
6: a 3x followed by a 2x
8: 2x three times
9: 3x twice
Beyond that, it’s repetition of those tricks, plus the distributive property of multiplication over addition.
Another: for X from 1 to 10, the digits of 9x will add to 9, and the leading digit will be x-1. You can do this visually by putting up all your fingers and putting down the Xth one - this splits your fingers into two groups, the first is the tens place digit while the second is the ones place digit.
99 on its own is also an example where it breaks down, which is why I stopped at 10, but yes all multiples of 9 in base 10 have digits summing to a multiple of 9.
> And yet to this day I refuse to memorize the multiplication table.
> Rote learning is easier for the teacher: easier to teach, easier to mark.
You're right, but there's still a good reason to memorize the multiplication table: more advanced math builds upon lower level arithmetic, and so the less time you have to spend working on the low level details, the more time you can spend focusing on the higher level understanding. So you memorize the multiplication table so that you don't have to think about it anymore.
I'd agree that knowing basic arithmetic is important but the issue is deeper in America. The rules and problem type issue is a direct result of uninspired math teaching. Teaching arithmetic for the sake of arithmetic is boring and uninspired and leaves a person incapable of doing both the arithmetic and math. You don't need arithmetic if you have nothing to apply it too. Arithmetic is a tool but a tool without a problem to apply it to is easily forgotten.
Take this from someone who could get through advanced complex analysis but who is also bad at arithmetic. Time plus checking solves all arithmetic issues but arithmetic never solves a realistic problem. Even balancing checks(not that is useful today) if you only know how to computer you'll be at a lost until you know what to compute.
Any math higher level than arithmetic doesn't actually require doing arithmetic.
If you want to find the hypotenuse of a triangle with side lengths A and B, the answer is sqrt(A^2 + B^2). It shouldn't matter how long it takes you to multiply A with itself because you shouldn't actually be doing that. Even if one insists on getting a numerical answer for say a word problem, you can use a calculator.
If knowing your times tables provides any benefit to learning higher level math, it's merely in mitigating the inefficiency of bad instruction.
The mathematics itself might not require doing arithmetic, but human psychology forces us to approach the abstract via the concrete. A facility with arithmetic helps with building a pile of concrete examples as a springboard for the leap to the abstract.
For example, the usual example of a cyclic group is {0,1,2,3,4,5} with addition modulo 6. Maybe we start with 12 rather than 6 because it is familiar to old people used to analogue clocks. We end up with one example for each size. But we could also give {1,2,3,4,5,6} with multiplication modulo 7. Now the abstract idea of an isomorphism has a concrete example. [0->1, 1->3, 2->2, 3->6, 4->4, 5->5]. And we get a new perspective on generators. {0,1,2,3,4,5} has an obvious generator: 1. We are tempted to view 5 as an extra nuisance generator. But we notice that 6x6 = 36 = 5x7+1 means that 6x6 = 1 in the multiplicative group so 6 isn't a generator. And 2x2x2 = 8 = 7+1 == 1 mod 7, so 2 isn't a generator either.
Viewed abstractly, product groups look dull and easy. The Klein Four group is a product C2 x C2, and has four elements {(0,0), (0,1), (1,0), (1,1)}. Obviously the only set of generators is {(0,1),(1,0)} and there are two automorphisms, the identity and the map that swaps (0,1) and (1,0).
Those with a facility with arithmetic might play with {1,3,5,7} and multiplication modulo 8. Or {1,5,7,11} and multiplication modulo 12. But beware, you are venturing into dangerous territory. Which one of 3,5, and 7 is (0,1) and which is (1,0)? You are going to discover that the previous paragraph is wrong. (1,1) looks different from (0,1) and (1,0). It looks inferior and not a generator. But actually there are three minimal sets of generators {3,5}, {5,7}. and {3,7}.
Playing with similar examples gives a second insight into what mathematicians mean by isomorphism. The first insight is that names don't matter: an isomorphism is what a ConLanger would call a relex = relexification. You invent a new language by inventing new words for the same old stuff. The second insight is that we build compound names, with internal structure. We have primitive names, 0 and 1. We build compound names (0,0), (0,1), (1,0), and (1,1). Then the mathematical concept of isomorphism treats our set of compound names as a mere dust of points, discarding the structure. An isomorphism can as easily map (0,1) to (1,1) as to (1,0) because they are just names, like 3,5,7. A loose notion of isomorphism as "same shape" can trip you up, because the structure of the names isn't part of the shape.
Knowing your times tables well enough to play happily with simple examples can help learners appreciate the subtleties of advanced math.
I don't know what kind of mind finds it easy to go from such a "concrete" example to the abstract, but I don't have one.
Now let's consider taking just the first sentence on wikipedia for the Klein Four group:
> In mathematics, the Klein four-group is a group with four elements, in which each element is self-inverse (composing it with itself produces the identity) and in which composing any two of the three non-identity elements produces the third one.
Starting with the abstract definition, now it's clear what we're talking about.
The problem with math education generally tends to be that the teacher knows the concept that the students are meant to learn, but the students don't, and thus logical jumps which appear obvious to the teacher are not at all so to the student. Concrete examples which the teacher thinks clearly illustrate the concept just add noise that the student must cut through.
Consider for example the pattern 1, 2, 3, 5, 7... what is the sequence I am trying to convey? I bet you're thinking the next number is 11, but you would be wrong. It's 9. The sequence is n+1 = floor((n + n+2)/2). Of course there are an infinite number of patterns that would fit the data. Expecting you to pick the specific one of those infinite I had in mind would be absurd. So to is the number of potential truths in mathematics infinite - expecting a student to get the one a teacher has in mind from a finite number of concrete examples is likewise woefully impractical.
> Any math higher level than arithmetic doesn't actually require doing arithmetic.
That's not really true. If you're doing calculus, solving some integral, you end up doing a bunch of addition and multiplication along the way. That's not the part you're trying to learn, which is my point—by having your arithmetic down pat, you can focus on the important things, rather then having to think about the low level details at all.
This is controversial (and shouldn’t be) but what if instead of spending so much time wondering why ‘Johnny’ can’t add, we spent that money and effort on making sure that Johnny was fed properly??
Countries with as much child poverty as Canada and the United States shouldn’t be surprised when their testing results don’t improve. This is pretty fucking basic science yet I’ve been fighting this battle for over twenty years and quite literally nobody in power gives enough of a fuck to fix it.
Of course I can't find it now no matter how hard I look, but there was an excellent blog post on here many months ago explaining how the pyramid of needs has become somewhat of a lazy shortcut people like to see on PowerPoint slides, which doesn't actually communicate useful information. It made its way into the wider discourse via educators, who were actually using the model incorrectly in the first place according to a few quotes from Maslow himself in the blog. It's a shame I couldn't find the article. Basically, it's an abstraction of needs that is just too good/simple to be true.
There’s hope yet: “The reduction in child poverty would be larger still, from 13.7 percent to 3.6 percent. Only about a quarter as many children would remain in poverty after the policy package’s adoption.”
I’m very hopeful about that program, but if you’re interested, I’d love to show you a myriad of Canadian plans to end child poverty by certain dates in the past.
I have a lot of hope for the United States now but if my own country’s difficulty is a guide, this will be a tough slog!
My cousin and her late husband are dentists. They spent years helping the Native Americans with dental work. During frequent discussions, she mentioned that it's sad that they have bad dental hygiene. Ultimately what she said was, "It's really difficult to convince someone they should focus on brushing their teeth, when really the sole idea occupying their mind is if they are going to get to eat the next day." So many things could be solved so effortlessly, by us spending so little money on basic things.
That’s a wicked program but as of 2018, children were still the poorest age group in the United States. We have an incredibly long ways to go and while that program is a start, it’s barely a drop in the bucket.
I see how someone who cannot (legally) work yet can be poorer than other age groups, but I am sure that’s not what is meant. But is a child somehow deemed poorer than its caregivers? Or perhaps is the child as poor as its poorest caregiver? Or how does that fact appear?
I'm curious about this too. My best guess is that they're measuring household income and there are a lot of kids with low household incomes (more than adults with low household incomes, especially when you have more kids than adults in a household).
Poor households tend to have many kids, and the parent/s are often very young, sometimes even under 18. It's not unseen to have 32-year old parents living with their 16-year old daughter and her two babies, a 14-year old child and another ~12-year old child. Of course these are extremely poor compared to even most of low income households. Perhaps that's what was meant by the comment.
With all due respect, there are quite literally hundreds of well written papers that explain this. The number isn’t controversial in the slightest and it’s based on one of the most basic statistical observations in economics. Honestly, I can’t understand why HN is having so much trouble with this.
The answer is that in child poverty research, a child is considered to live in poverty if that child lives in a household with an income below the statistical low income cutoff.
That does not answer how the child is poorer than its parents, though.
I don’t think people have difficulty with the concept of child poverty, but your original assertion sounds super weird (and yet it sounds official, not a personal opinion, so a paper detailing that would be a better answer).
The child is probably not poorer than its parents, but instead, I'm guessing households with children are poorer on average, and even more so the more children they have. This would then mathematically result in children being poorer on average if this is determined by the household they live in. If poor people have more children, then children are going to be poor overall.
So a much less confusing way to say the same thing would be that people in households with children are poorer than people in households without children? Or even simply "poor people have more children than average"?
It’s a measurement based on the statistical low income cutoff (the “poverty line”). In child poverty research, a child is considered to live in poverty if they live in a family whose income falls below the low income cutoff.
Using that measure, in the United States 1/6 children live in poverty. Half of those live in statistically defined ‘extreme poverty’. That 1/6 has some interesting geographical and race based components that worth a read. There is a metric shit tonne of research on this and if you’re interested in a reading list, my email is in my profile! :)
In anti-poverty research, a child is considered to live in poverty if that child lives in a home whose income falls below the statistical low income cutoff point. That’s ‘the poverty line’ that you’ve hopefully heard of.
Finally, we are talking about child poverty - a little bit of respect and perhaps some research would be a far better complement than lowbrow snark.
Child poverty isn't a laughing matter, but weird statistics are.
How would we make children not the poorest age group? First poor families are more likely to have more kids than well to do ones, and kids raise the income you need to be above the poverty line.
Better statistics revolve around specific needs that children have and what percentage of those are satisfied.
Your last paragraph is completely true but better statistics are expensive. And you might be shocked by how little money there is in the anti-poverty racket...:)
Consequently, we’re stuck with statistical measures based on deviations from income and cost of living.
I think food programs are part of a solution, but a more realistic solution will require investments in mental health, addictions services, adult education, housing and child care.
I didn't notice if "participation" meant meals available or actually picked up by kids. Kids, particularly younger ones, who are in unstable households may not get to school in time to pick up what's available. Some districts have adapted by doing grab-and-go available anytime or serving meals in the classroom.
Schools in parts (or perhaps all?) of California remained open just to serve meals even while everything else was on lockdown last spring.
Unfortunately the infrastructure of school meals in the US is, like so much else, political. A sensible approach would be free (both removing the complex billing infrastructure and removing stigma of "free" vs "paying") nutritious meals for all students.
Incredibly, both "free" and "nutritious" are contentious.
Sometimes, welfare can cause more problems than it creates. So what should "people in power" do? Setup welfare programs, and in turn welfare dependence?
Education was seen a cure for poverty, which in turn is why Johnny isn't fed.
The shortest answer is that it’s far cheaper to feed two families than to incarcerate one adult.
The longer answer is that you don’t necessarily need welfare programs to help people out of poverty. Rather, a more efficient system would have ‘welfare’ programs for people in crisis that lead into a series of help up services that eventually lead to full independence.
Edit - Also, fuck no, it’s not a teach a man to fish argument. We’re talking about child poverty.
Sure, but only short-term. And if you only care short-term, because you are a politician with a short term, then fine (for you).
But that's a different plan from "the government" in general.
> a more efficient system would have ‘welfare’ programs for people in crisis that lead into a series of help up services that eventually lead to full independence.
You are describing "safety-net" welfare, but how do you ensure any form of welfare remains a safety-net, and doesn't become a norm? welfare for those not in poverty is much more likely to work this way. If you specifically set out to tackle poverty, the most important thing are educational programs, and limited welfare programs.
I'm not against welfare in any case, but I see it as a double-edged sword that is too often seen as a "throw money at the problem" solution.
> Also, fuck no
It's a metaphor, the difference between men and children aren't relevant. The abstract meaning is dependence vs self-dependence. Giving a child food solves their food, but not their poverty.
The way it was originally stated:
what if instead of spending so much time wondering why ‘Johnny’ can’t add, we spent that money and effort on making sure that Johnny was fed properly
I could just as well say "what if instead of spending so much time wondering why a man can't fish, we spent that money and effort on making sure that man is fed properly", and the answer is the same: because then they will be dependant on us (the government) to provide food. This doesn't mean we shouldn't provide food, but that both are required, since the original post post specifically proposed providing food should displace education i.e. we spend money elsewhere. In fact, what we spend on food is a problem in itself, but unrelated to what we spend on education, we should not cannibalise one in favour of the other as if they are competing solutions - they are both solutions to the same problem, but do not compete because they have different scope: one long term (poverty) the other short (hunger).
I think I outlined this pretty well in the previous post.
Children can be fed by their parents, or by the government. long-term, it is better that it be the former, short-term it might need to be the latter.
It's not sustainable or desirable to have multiple generations dependant on the government. If children become "stable" adults, but their children still require the government to feed them then nothing has been solved. The adults need to be both "stable" and educated to escape government dependence.
The main problem was that most teachers didn't get it, and so couldn't teach it. And parents were angry their kids were learning math that they couldn't understand.
So it was an implementation issue. The concept was sound.
The problem with abstract thinking is that there’s often more than one way to represent an answer, so that means the teacher has to actually understand the problems, not just match to an answer sheet.
I’m reminded of a widely circulated common core math problem to draw a series of dots showing 4 x 8 or something. The kid drew 4 rows of 8 dots, and the teacher marked it “NO!” and drew 8 rows of 4.
Maybe a dumb question, but if a non-insignificant number of teachers are simply matching answer sheets why do you need a degree and certification to do it. Doesn’t that indicate that a number are incompetent at at least one very large aspect of the job.
I think that I had New Math in sixth grade, and I remember nothing in particular of it but learning about number bases (as Feynman remarked, kind of pointless), and learning the terms commutative and associative--though it was a long time before I grasped the latter.
> learning the terms commutative and associative--though it was a long time before I grasped the latter
I think it's kinda interesting—commutativity is so much a more clearer and approachable property than associativity, and it sounds more useful, but the later is far more important.
That we get any STEM talent at all is a quirk of selection and survivor bias, which very likely happens in spite of public education methods, not because of it. I have had some truly incredible teachers, but their ability was pareto distributed. Great teachers have nothing in common with the long tail of inadequate ones. The problem they are trying to solve is how to produce citizens with math skills while preserving the system of one generic teacher talking at 30+ kids with increasingly heterogeneous experiences confined in a room and sitting at desks. It's a stupid problem.
The concrete math and geometry you learn from things like music, cooking, carpentry, navigation, engine repair, gambling, and building a basic physical experimental apparatus could take a child "up" to differential calculus in less than a couple of years. We have to look at who becomes a teacher and whether they are transmitting their tangible skills to students, or if they are themselves physically helpless talking heads for a received curriculum.
Today, I meet ostensibly educated people and their defining characteristic is they have been institutionalized. Whether it was in grad school or prison is just a matter of taste.
I’d being institutionalized bad? PhD is nearly a requirement now to progress beyond mid-level in all top STEM fields, sometimes an entry requirement. I think this is because institutionalized people all think the same way, never too much creativity or questioning what is going on. I would say that this is useful for large organizations only, but the startup community is just as enamored with it, most especially venture capitalists as a group. People that don’t submit fully to some system end up not fitting into any system after enough years.
I remember learning about sets in elementary school in the 60's - subsets, unions, intersections (joins) - year after year, and thinking that I would never need this. Now I've been doing data engineering since 1989 (when it was called systems integration) and all I do is work with sets. So glad I had that schooling.
Why would we expect teachers to understand math when they're generally not mathematicians? Why would we expect teachers to change the system that created them? Why would we expect a system that was created for sorting people to educate people? Students are trained to serve, not to think.
Anecdote: I was an ADD kid in the 90s Undiagnosed, but intelligent and bored and distracted easily. So much so, that during math, one other kid and I were sat outside the classroom in the hallway, and with very little assistance from the teachers, given worksheets on base-5 math. I essentially taught myself what different bases are, what digits represent, and so on.
I thought of this the other day as I listened to Radiolab do a thoroughly piss poor job of explaining binary, as part of a larger discussion about computer memory and the relevance of 4096 to a computer error.
I mean, they spent a lot of time talking about light bulbs going on and off, and then how, if you represent 2, you flick one bulb off and another on, but if you represent 3, you leave the first bulb on, etc. They did finally mention powers, but not in a generic sense. They didn't ever mention that binary is "just" base 2 math like normal math is base 10.
It seemed so contrived. I know they were trying to explain it to the layman, but it seems they actually made it more conceptually difficult.
I haven't been able to rant about this and here seems like the best spot.
Standout passage for me: "The villain, if there is one, might be the country’s penchant for the “quick fix.” Had Sputnik not flown, UICSM, SMSG, the Madison Project, and the other experimental programs might have evolved slowly and carefully into a national curriculum; as it was, they were shoved to center stage, lavishly financed, and told to perform a miracle overnight. They couldn’t, so the country passed on to the next educational fad (“back to basics”), labeled the previous one a failure, and blamed it for low test scores and a decline in skills."
What happened? Well, it was taught to the kids by all those Old Math teachers, using all of their perfectly-honed and time-proven Old Math techniques, of course.
Naturally this was a disaster for both sides, so as soon the correct scapegoats were identified and dispatched, they could safely return to their Old, safe, familiar system which is a disaster for only one.
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[ 3.5 ms ] story [ 173 ms ] threadI believe it was more of an experimental thing, with partial adoption - i.e. only one of the math teachers would use the new syllabus on their students, with classes six days a week (yep, back then there was Saturday morning school).
All I remember now is a bit of bafflement at the set theory paraphernalia, rings and groups etc.
It's actually kinda funny. There's an endless search for an effortless way to learn math. There is no such thing. Learning math requires work.
Just like there is no way to get strong without hard work.
Nonsense. There is a search for more effective ways to teach math. And there most definitely are differences in effectiveness, so no reason to believe that any given way is optimal.
> Just like there is no way to get strong without hard work.
There are, however, lots of ways to spend a lot of hard work for little gain.
- The scores don't effectively measure learning.
- The need to prepare students for tests undermines the ability to try different teaching approaches.
- Scores don't change overall but the distribution of scores change.
- The distribution of scores don't change but the demographics of the students in a given part of the distribution change.
- There are confounding factors such as budget or classroom size.
In my 16 years of education, I saw a consistent 1:1 correlation with learning the material and doing correspondingly well on the tests. That consistency included myself - when I learned the material, I'd ace the test. When I didn't learn it, I failed the test. There wasn't any magical "learned but didn't test well" going on.
Consider anxiety disorders. There’s a crystal clear, fully scientifically proven reason that some can know the material but fail to perform on a test. Think more. It helps.
What do "disorders" have to do with class? At best, being from a lower socio economic class could put you at risk for whichever disorders you have in mind, but they aren't synonymous. At least, not in my mind.
I don't assume such people don't exist. But I'm not buying there are enough people to significantly alter overall student test results. There'd have to be a couple in every class, and I've never encountered one.
As an example: do you consider solving 579×46 quickly a "math problem"? I will solve it (relatively) quickly, since I was thought how to solve that exact problem algorithmicaly.
But if you hadn't thaught that algorithm (you could argue that nowadays we have calculators for this) then it would probably take you a bit longer to derive some form of this algorithm. But it doesn't necessarily imply that you're bad at math, just that you have not practiced this exact task before.
In my experience, doing really well in tests requires more adaption to the tasks of the tests than actual understanding. Most tests are limited on time, and stress does not exactly help you to deal creatively with completely new tasks.
Note that I don't want to say that teaching algorithms is bad. It surely helps to do some basic calculation without having to think much about it. I just want to point out that the tests are often testing a specific understanding of math, and different teaching methods often also differ in their goal of what they want students to learn well.
It's second or third grade material. It's reasonable to expect teachers to teach it and kids to learn it. It's easy to create tests for it.
I don't think that ability necessarily has a great deal in common with understanding mathematics or the ability to do higher level math.
As a result it provides vanishingly poor outcomes which keep the US below the OECD average, in spite of some of the highest per-student spending in the world.
Education is highly politicised, managed by school boards which seem to attract religious kooks and other low-quality authoritarians with eccentric ideas, and which does almost everything it can to discourage good teachers and other suspiciously well-educated and competent individuals from joining or staying in the profession.
Widespread poverty in the US population doesn't help.
Changing how math is taught won't address the essential dysfunctional broken futility of the system as a whole.
All the attempts to gamify learning math, or teach it by passively watching TV, suggest otherwise.
> more effective
Sesame Street is probably the most inefficient way to teach arithmetic ever devised. A kid can watch 1000 hours of it, and barely be able to count to 10.
We know how to effectively teach math. Present the concepts, follow up with the student working through progressively harder problems.
> And there most definitely are differences in effectiveness
Since the needle hasn't moved on results, we've likely found the optimal method.
Learn playing! Gamify!
But people DO NOT LIKE MATHS despite them needing it. I am a professor of the thing and am not ashamed to own it. It is what it is.
Like cleanin your room, doing your bed or... shaving!
Saying people simply don't like math because their eyes glaze over in math class is like saying people don't like stories because they're bored by Great Expectations.
Sounds about right!
The problem (at least one of the problems) is, kids are different. You see these kids are struggling with the traditional approach. You devise a non-traditional approach to help them. And it actually works! For them...
But some kids who would have done fine on the traditional approach now are struggling. Is that a net win? Maybe, maybe not. Depends on how many kids are in each group and how bad they're struggling.
Fashion is exactly it. I feel like they experiment on them rather than use established methods.
This doesn’t need to be robotic, it still give a lot of room for personalization. I find he’s really right, especially since in my country everyone want to do their own stuff in their own way. The ministry is guilty as well, because it produces an endless stream of contradictory directives, each based on theories made by pedagogists that never taught in the field.
I think this applies to most jobs really. Research and practice are different things, and even when the same person does both it's good to keep them at least somewhat separate.
I (almost always) kinda want my doctor or mechanic to follow what's known to work, rather than try out new things. On my own software projects, experimental techniques seem to cause issues a lot more often than standard "best practice" sort of techniques and I try to keep them away from important production system. Etc etc.
It seemed fine to me but I wasn't really thinking about alternative teaching methods at age 4.
And we used Cuisenaire Rods instead of bar charts.
If you think about it, it makes sense: if the problem in math is people get stuck at some foundational step where they are required to "get" something, and start to think they "suck at math" because they are missing this piece, then making the steps VERY small will make sure everyone can make those steps, thus making everyone "good at math."
Similar to Singapore math, it's not based on a "textbook" that you read but an "exercise book" that you write and solve exercises in. I don't have personal experience with teaching using this approach, but I have heard many good things.
books: https://www.amazon.com/s?k=jump+math (non free) samples: https://jumpmath.org/jump/en/learn podcast: https://www.cbc.ca/listen/live-radio/1-63-the-current/clip/1...
http://ee263.stanford.edu/archive/
dy = dx + 3dz + 2
For anything through AP calculus, you'd be better served going down the volume and spaced repetition route that math programs like Saxon use. Do things over, and over, and over, and over again until they are second nature. There's no magic bullet to get around that work that you have to do to master something.
Rote learning is easier for the teacher: easier to teach, easier to mark. It's not just mathematics, it's all the subjects in school. Many of the "geniuses" merely survive school, nothing more.
4: 2x twice
5: add a zero and divide by 2
6: a 3x followed by a 2x
8: 2x three times
9: 3x twice
Beyond that, it’s repetition of those tricks, plus the distributive property of multiplication over addition.
9: 10x, then subtract once.
> Rote learning is easier for the teacher: easier to teach, easier to mark.
You're right, but there's still a good reason to memorize the multiplication table: more advanced math builds upon lower level arithmetic, and so the less time you have to spend working on the low level details, the more time you can spend focusing on the higher level understanding. So you memorize the multiplication table so that you don't have to think about it anymore.
Take this from someone who could get through advanced complex analysis but who is also bad at arithmetic. Time plus checking solves all arithmetic issues but arithmetic never solves a realistic problem. Even balancing checks(not that is useful today) if you only know how to computer you'll be at a lost until you know what to compute.
If you want to find the hypotenuse of a triangle with side lengths A and B, the answer is sqrt(A^2 + B^2). It shouldn't matter how long it takes you to multiply A with itself because you shouldn't actually be doing that. Even if one insists on getting a numerical answer for say a word problem, you can use a calculator.
If knowing your times tables provides any benefit to learning higher level math, it's merely in mitigating the inefficiency of bad instruction.
For example, the usual example of a cyclic group is {0,1,2,3,4,5} with addition modulo 6. Maybe we start with 12 rather than 6 because it is familiar to old people used to analogue clocks. We end up with one example for each size. But we could also give {1,2,3,4,5,6} with multiplication modulo 7. Now the abstract idea of an isomorphism has a concrete example. [0->1, 1->3, 2->2, 3->6, 4->4, 5->5]. And we get a new perspective on generators. {0,1,2,3,4,5} has an obvious generator: 1. We are tempted to view 5 as an extra nuisance generator. But we notice that 6x6 = 36 = 5x7+1 means that 6x6 = 1 in the multiplicative group so 6 isn't a generator. And 2x2x2 = 8 = 7+1 == 1 mod 7, so 2 isn't a generator either.
Viewed abstractly, product groups look dull and easy. The Klein Four group is a product C2 x C2, and has four elements {(0,0), (0,1), (1,0), (1,1)}. Obviously the only set of generators is {(0,1),(1,0)} and there are two automorphisms, the identity and the map that swaps (0,1) and (1,0).
Those with a facility with arithmetic might play with {1,3,5,7} and multiplication modulo 8. Or {1,5,7,11} and multiplication modulo 12. But beware, you are venturing into dangerous territory. Which one of 3,5, and 7 is (0,1) and which is (1,0)? You are going to discover that the previous paragraph is wrong. (1,1) looks different from (0,1) and (1,0). It looks inferior and not a generator. But actually there are three minimal sets of generators {3,5}, {5,7}. and {3,7}.
Playing with similar examples gives a second insight into what mathematicians mean by isomorphism. The first insight is that names don't matter: an isomorphism is what a ConLanger would call a relex = relexification. You invent a new language by inventing new words for the same old stuff. The second insight is that we build compound names, with internal structure. We have primitive names, 0 and 1. We build compound names (0,0), (0,1), (1,0), and (1,1). Then the mathematical concept of isomorphism treats our set of compound names as a mere dust of points, discarding the structure. An isomorphism can as easily map (0,1) to (1,1) as to (1,0) because they are just names, like 3,5,7. A loose notion of isomorphism as "same shape" can trip you up, because the structure of the names isn't part of the shape.
Knowing your times tables well enough to play happily with simple examples can help learners appreciate the subtleties of advanced math.
Now let's consider taking just the first sentence on wikipedia for the Klein Four group:
> In mathematics, the Klein four-group is a group with four elements, in which each element is self-inverse (composing it with itself produces the identity) and in which composing any two of the three non-identity elements produces the third one.
Starting with the abstract definition, now it's clear what we're talking about.
The problem with math education generally tends to be that the teacher knows the concept that the students are meant to learn, but the students don't, and thus logical jumps which appear obvious to the teacher are not at all so to the student. Concrete examples which the teacher thinks clearly illustrate the concept just add noise that the student must cut through.
Consider for example the pattern 1, 2, 3, 5, 7... what is the sequence I am trying to convey? I bet you're thinking the next number is 11, but you would be wrong. It's 9. The sequence is n+1 = floor((n + n+2)/2). Of course there are an infinite number of patterns that would fit the data. Expecting you to pick the specific one of those infinite I had in mind would be absurd. So to is the number of potential truths in mathematics infinite - expecting a student to get the one a teacher has in mind from a finite number of concrete examples is likewise woefully impractical.
That's not really true. If you're doing calculus, solving some integral, you end up doing a bunch of addition and multiplication along the way. That's not the part you're trying to learn, which is my point—by having your arithmetic down pat, you can focus on the important things, rather then having to think about the low level details at all.
Countries with as much child poverty as Canada and the United States shouldn’t be surprised when their testing results don’t improve. This is pretty fucking basic science yet I’ve been fighting this battle for over twenty years and quite literally nobody in power gives enough of a fuck to fix it.
If a child is struggling at the basic needs level of this pyramid then hitting the self fulfillment bit at the top is excruciatingly difficult
https://www.vox.com/future-perfect/21456242/joe-biden-povert...
I have a lot of hope for the United States now but if my own country’s difficulty is a guide, this will be a tough slog!
What... does that mean?
I see how someone who cannot (legally) work yet can be poorer than other age groups, but I am sure that’s not what is meant. But is a child somehow deemed poorer than its caregivers? Or perhaps is the child as poor as its poorest caregiver? Or how does that fact appear?
/shrug
The answer is that in child poverty research, a child is considered to live in poverty if that child lives in a household with an income below the statistical low income cutoff.
I don’t think people have difficulty with the concept of child poverty, but your original assertion sounds super weird (and yet it sounds official, not a personal opinion, so a paper detailing that would be a better answer).
I agree it's very confusing.
> there are a lot of kids with low household income
I think maybe you meant to reply to someone else.
Using that measure, in the United States 1/6 children live in poverty. Half of those live in statistically defined ‘extreme poverty’. That 1/6 has some interesting geographical and race based components that worth a read. There is a metric shit tonne of research on this and if you’re interested in a reading list, my email is in my profile! :)
I really don't expect my kids to have more disposable income than I do.
Finally, we are talking about child poverty - a little bit of respect and perhaps some research would be a far better complement than lowbrow snark.
How would we make children not the poorest age group? First poor families are more likely to have more kids than well to do ones, and kids raise the income you need to be above the poverty line.
Better statistics revolve around specific needs that children have and what percentage of those are satisfied.
Consequently, we’re stuck with statistical measures based on deviations from income and cost of living.
The black panthers used to serve free breakfast to schoolkids for a reason.
Unfortunately the infrastructure of school meals in the US is, like so much else, political. A sensible approach would be free (both removing the complex billing infrastructure and removing stigma of "free" vs "paying") nutritious meals for all students.
Incredibly, both "free" and "nutritious" are contentious.
Sometimes, welfare can cause more problems than it creates. So what should "people in power" do? Setup welfare programs, and in turn welfare dependence?
Education was seen a cure for poverty, which in turn is why Johnny isn't fed.
The longer answer is that you don’t necessarily need welfare programs to help people out of poverty. Rather, a more efficient system would have ‘welfare’ programs for people in crisis that lead into a series of help up services that eventually lead to full independence.
Edit - Also, fuck no, it’s not a teach a man to fish argument. We’re talking about child poverty.
But that's a different plan from "the government" in general.
> a more efficient system would have ‘welfare’ programs for people in crisis that lead into a series of help up services that eventually lead to full independence.
You are describing "safety-net" welfare, but how do you ensure any form of welfare remains a safety-net, and doesn't become a norm? welfare for those not in poverty is much more likely to work this way. If you specifically set out to tackle poverty, the most important thing are educational programs, and limited welfare programs.
I'm not against welfare in any case, but I see it as a double-edged sword that is too often seen as a "throw money at the problem" solution.
> Also, fuck no
It's a metaphor, the difference between men and children aren't relevant. The abstract meaning is dependence vs self-dependence. Giving a child food solves their food, but not their poverty.
The way it was originally stated:
I could just as well say "what if instead of spending so much time wondering why a man can't fish, we spent that money and effort on making sure that man is fed properly", and the answer is the same: because then they will be dependant on us (the government) to provide food. This doesn't mean we shouldn't provide food, but that both are required, since the original post post specifically proposed providing food should displace education i.e. we spend money elsewhere. In fact, what we spend on food is a problem in itself, but unrelated to what we spend on education, we should not cannibalise one in favour of the other as if they are competing solutions - they are both solutions to the same problem, but do not compete because they have different scope: one long term (poverty) the other short (hunger).Feeding children so that they can be stable adults is one of the most long-term policies I can think of.
Children can be fed by their parents, or by the government. long-term, it is better that it be the former, short-term it might need to be the latter.
It's not sustainable or desirable to have multiple generations dependant on the government. If children become "stable" adults, but their children still require the government to feed them then nothing has been solved. The adults need to be both "stable" and educated to escape government dependence.
(and remember kids, base 8 is just like base 10, really — if you’re missing two fingers)
https://www.youtube.com/watch?v=UIKGV2cTgqA
The main problem was that most teachers didn't get it, and so couldn't teach it. And parents were angry their kids were learning math that they couldn't understand.
So it was an implementation issue. The concept was sound.
I’m reminded of a widely circulated common core math problem to draw a series of dots showing 4 x 8 or something. The kid drew 4 rows of 8 dots, and the teacher marked it “NO!” and drew 8 rows of 4.
Or perhaps, that’s as intended?
I think it's kinda interesting—commutativity is so much a more clearer and approachable property than associativity, and it sounds more useful, but the later is far more important.
The concrete math and geometry you learn from things like music, cooking, carpentry, navigation, engine repair, gambling, and building a basic physical experimental apparatus could take a child "up" to differential calculus in less than a couple of years. We have to look at who becomes a teacher and whether they are transmitting their tangible skills to students, or if they are themselves physically helpless talking heads for a received curriculum.
Today, I meet ostensibly educated people and their defining characteristic is they have been institutionalized. Whether it was in grad school or prison is just a matter of taste.
https://poibella.org/papers/indefenseofmath.pdf
I thought of this the other day as I listened to Radiolab do a thoroughly piss poor job of explaining binary, as part of a larger discussion about computer memory and the relevance of 4096 to a computer error.
I mean, they spent a lot of time talking about light bulbs going on and off, and then how, if you represent 2, you flick one bulb off and another on, but if you represent 3, you leave the first bulb on, etc. They did finally mention powers, but not in a generic sense. They didn't ever mention that binary is "just" base 2 math like normal math is base 10.
It seemed so contrived. I know they were trying to explain it to the layman, but it seems they actually made it more conceptually difficult.
I haven't been able to rant about this and here seems like the best spot.
Naturally this was a disaster for both sides, so as soon the correct scapegoats were identified and dispatched, they could safely return to their Old, safe, familiar system which is a disaster for only one.
https://fs.blog/2016/07/richard-feynman-teaching-math-kids/
Adding together all the temperatures of red, yellow, green, and blue suns? Kill me now.