I am a guy who works with complex DSP algorithms all the time. I know how to get to things thinking in arrays with actual numbers. I can't say how often I decyphered some obscure mathematical hyroglyphs only to then be: "Oh that is what that means? I have been doing this for a while now, they could have described that with a sentence or two."
I get why mathematical notion looks like it does, but it sucks at doing a lot of things. That would be as if I tried to teach people a non-trivial piece of code by showing them the zeros and ones that make up the code and naming all important variables with ancient banylonian letters. Now sure, after a while the tiny fraction of them who wouldn't get scared away by this might also get fluid in a mad system like that — but if my goal is to make people understand the code why not reduce the tripping stones instead of putting some there for once?
I think mathematicians — similar to the people who didn't let themselves be scared away by zeroes, ones and babylonian letters — have become fluid at a system that they now defend, because they are the initiated magicians who understand it.
As a programmer I am aware that there is a difference between what a piece of code expresses and the syntax/representation it uses. To me mathematicians are like programmers who learned only brainfuck and refuse to consider trying out more appropriate notations for teaching purposes, because that's what they had to learn historically.
> I think mathematicians — similar to the people who didn't let themselves be scared away by zeroes, ones and babylonian letters — have become fluid at a system that they now defend, because they are the initiated magicians who understand it.
No, there is no elitism behind. Mathematical notation exists because it is the most compact form that also contains no ambiguity.
There is plenty of plain english explanation of all of mathematics, that is used to understand the concepts, but to represent them, you use math notation.
> Mathematical notation exists because it is the most compact form that also contains no ambiguity.
Citation needed. My experience thus far trying to read pretty much anything written by a mathematician involves googling a LOT of symbols, all of which have many many many many meanings depending on context.
If any of that were actually true, postfix/revers polish notation would be more popular. Math has it's own languages with all the same problems every language has.
I had a math teacher who claimed the opposite anecdote: he found that kids can do equations when there is x (obviously ones who are past their algebra fear): x + 5 = 10 is no problem. But then when you use some unfamiliar symbol, they freeze: △ + 5 = 10.
At that level, people tend to resemble trained models. I was trained on x, so I don't get it any other way.
I teached myself Basic on our C64 when I was 11 with a manual my father had. A year later we had a few hours of introduction to programming in our math class. When I read the first very abstract lesson, I thought: My god, this so hard, it’s so difficult to understand. But when I saw the code examples on the next page I had to laugh, because they very incredibly simple. It’s just the way the math book talked about them was too abstract for me. I then got for the First Time the idea: Could it be that all the other math I have problems with could be much easier if it was explained differently to me?
I have had different versions of this exact problem throughout my learning career. It is common that the very concepts of convention, representation, terminology, and symbology are more abstract/complex/unintuitive than the actual topic they describe, and the person doing the teaching usually glosses over the explanation of that part because it is so second nature to them.
I think the fundamental thing that many refuse to accept is that some people are, in fact, smarter than others. Until we can accept that fact in society, we're always going to end up reducing to the lowest common denominator.
Yet, as someone teaching electronics and programming to art students, many of whom "hate maths" there are many people who would have understood it in school if it was explained right and/or if someone gave them a good motivation why they should bother understanding it in the first place.
The truth is that many math teachers suck. I had a math teacher that managed to do integrals with us for a year without ever explaining to us (in a school specialized on mechanical engineering) what real world problems would be impossible to solve without it.
> I had a math teacher that managed to do integrals with us for a year without ever explaining to us (in a school specialized on mechanical engineering) what real world problems would be impossible to solve without it.
I sure did. She said it doesn't matter, or we don't need to understand right now. As I said, shw was not a good teacher.
As an educator myself the point I am trying to make here is that the why is crucial in complicated topics and if someone who was mentally there even feels the need to ask the why you should probably reconsider the way you chose to teach things.
I vividly remember the moment when I researched integeals for an exam. I found the typical simple examples and wondered why the hell that teacher didn't mention this once. In fact today I am convinced the right way to teach such topics is to always bring up the examples even (or especially) as you dive deeper into the material. This reminds people what they are actually dealing with.
> if someone gave them a good motivation why they should bother understanding it in the first place
Being bad at math means you'll likely not manage your finances very well, and that will hurt.
I saw a documentary recently on Netflix (or Amazon?) on what happens to athletes who became overnight millionaires. Very few managed to hold on to that wealth very long. Some pretty sad stories.
Do you have kids? Telling them their finances are a reason for learning maths is bullshit, because not every person that age has to worry about their finances.
And even if you choose that as a story you will have a hard time explaining to them why they need to learn trigonometry or something like the aforementioned integral.
As an educator I have found most people don't really need a hard rational incentive to learn a thing. All they need is a story that convinces them to learn that. E.g. when I helped unruly teens with their homework and they had an abstract geometric problem to solve, it costed me literally nothing to come up with some story of how they comanding the knights in a castle and to prevent their enemey from entering they have to figure out how much material they need to chop in the woods.
The same boring problem suddenly was tied to a heroic story of them saving the day and oh wonder, they were more interested in solving the issue.
I've seen 'em right up through high school saying that there was no purpose to learning math, and that math wasn't "relevant". And then they go into the workforce and mismanage their finances into perpetual poverty.
I also see them choose college majors apparently on the basis of avoiding all contact with math, and wind with degrees that employers aren't interest in.
Your hero story worked, good on ya! I would have related the true story of hiring a contractor to build an elliptical patio. He charged by the square foot. He laid it out, and had the bricks delivered. Something seemed off. I measured the L+W+H of the brick pallet, and the volume of one brick, and computed the area the bricks would cover. Then, measured the L+W of the patio layout, and used the area for an ellipse formula. The bricks was the right number for the area, but the area was only 2/3 of the area he quoted me and charged me for. I nicely inquired about the discrepancy, and he hemmed a bit and said he'd made a mistake, and adjusted the bill accordingly.
Of course, we both knew he tried to cheat me, quite egregiously. I figure he did this regularly and got away with it with his math challenged customers.
I recall a documentary about a famous entrepreneur who made a lot of money in high school by gambling with other teens who did not comprehend the odds.
Sure, finance doesn't interest a 6 year old. But I've never heard of a teen who didn't want to have money in their pocket, and who looked forward to being taken advantage of. Even the unruly ones.
I don't agree that finances are bullshit reasons for learning math.
P.S. In high school, I joined a poker game with some peers who I knew had engaged in fencing stolen goods. They cleaned me out. I knew enough about poker odds to realize they were card cheats, but I didn't see how they were doing the cheat. It was a cheap lesson for me :-/ The most effective scams are the ones where the mark doesn't realize he was cheated, and will come back for more. I didn't return to poker.
I just talked about this with a friend of mine who related a story about a teacher friend of his in a low-income school. Lots of gang violence, standard rough neighborhood. A kid asked the usual "why do I need to learn algebra?" question. The teacher replied "If you want to stay poor, you don't have to." The kid got the message.
I had these "though talking" teachers, and to be honest they didn't impress anybody. The best teachers I ever met all had a burning enthusiasm for their subject and managed to spark interest in their students as well. So much so that nobody ever had to ask them why a thing was needed. It was obvious that it was either needed or the lesson was so interesting that nobody gave a damn whether there was any use to it. I mean people watch documentaries about aliens and pyramids all day — teenagers don't need things to be useful to be motivated, but they need to be motivated to learn things that turn out to be useful.
There are studies on whether negative or positive drivers are more effective at getting people to learn. I won't spoil the findings, but you can probably guess by my line of reasoning already.
According to the teacher, he did. Even if he didn't change his behavior, he now knew there was indeed a reason.
I started my first small business when I was 8, and was able to do the accounting for it (not that there was much to it). I never thought about the connection between math and business at the time, because it just seemed natural and obvious. I did several businesses, earning enough that I could buy a car by the time I was 16.
I bet you knew kids during school who had small enterprises on the side.
The eternal problem with this idea the some people are smarter than others is that never finishes the statement *smarter at what?*
An overly simplistic view of intelligence and human capabilities seems to think that you’re just “smart” or at least smarter than those around you in general and that’s it, you’ve won the lottery. When in reality, people have aptitudes and gifts across spectrum of endeavors and where you may not seem smart in maths, you can be an incredibly smart engineer or artist or communicator.
It’s hard especially to have this conversation with traditionally “smart” people because you find very quickly that their entire identity is wound up in this idea of being universally intellectually superior than others through no effort of your own. Gods own gift. You challenge that, and the teeth come right out.
I'd argue this is an overly pedantic way of looking at the situation. Yes, everyone has different aptitudes and things they're good at. That fundamentally doesn't matter to the core of my argument, which is that some people are just worse at everything.
> many refuse to accept is that some people are, in fact, smarter than others.
... I'll caveat that "some students are smarter than others _at the same age_."
It's bizarre and frustrating because we already have the solution, we just need a way to assign students to a grade level that's based on how smart they are, rather than how old they are.
yah, I can't see any issues that will evolve from having a wide range of physical development level of approximately the same mental prowess in a classroom... I mean I certainly never got into any mutual violence situations with any of my classmates, nor did we ever choose each other as sexual partners.. (I can keep going with sarcastically listing things that definitely routinely happened in my k-12 classrooms that you almost certainly don't want happening between people more than a couple years apart that your policy choice would heavily facilitate).
There are valid social(ization) reasons to keep kids approximately the same age in a classroom. The lucky thing is, if you live in a city you usually have a big enough cohort of kids that you can do some of this sorting within grade, although I am personally skeptical of that as a solution. I remember ~10% of the kids being highly disruptive and it would have been nice to have them gone but there was this huge, probably majority of the class, that was dumber than me but trying hard to get ahead and keep up that I think was probably beneficial to society keeping there because they got pulled ahead more than I got held back.
>There are valid social(ization) reasons to keep kids approximately the same age in a classroom
Fair enough. I'm already of the opinion that k-12 is primarily a daycare and that smart kids will prevail no matter the education model.
> I think was probably beneficial to society..
Yes, there will always be a tension between what is best for the individual and what is best for society. A difference of opinion likely stems from a difference of personal philosophy.
While it's hard to say what's best, I think most would agree that there is a lot of room for improvement in the current education system.
> White supremacy culture shows up in math classrooms when... There is a greater focus on getting the "right" answer than understanding concepts and reasoning.
When designing airplane parts, there are definitely right and wrong answers.
Getting the right answer is important in many situations, such as that; however, understanding concepts and reasoning is important so that you can know how to get the right answer (and so that you can understand math, in general). However, there may be more than one valid way to do so (and the answer will be the same). For example, there is many way to prove Pythagorean theorem, factorial duplication formula, and other stuff; and shortcuts for making some kinds of calculations, etc.
(However, this does not seem to have to do with white supremacy; it is something else.)
To make an analogy, your idea is that a runner needs to learn stride techniques, physics and biology first and doing base physical training without understanding these is pointless. This does not work in athletics, nor does it work in math. Somebody who cannot even run 100m is not going to benefit from understanding how carbs convert to ATP and how that ATP is consumed to fire muscles or how muscles and ligaments in the legs can store and release energy or how drag slows you down etc. These are all good to know for an athlete to improve further, but not for somebody who just started running. And people had been evolving to run for millions of years!
People have not evolved for math. Their innate math ability is similar to falling after making few steps in regards of running analogy. They need a whole lot of boring and painful base training to be any good. What we get with "but they need to understand!" is people who can barely walk 1km without losing their breath insisting that when children get fun carbon shoes and tasty carb gels they will run much better than children who are made to run laps until they drop.
Sure but do we want kids to just learn what's the correct sum or do we want them to learn the process to arrived at the correct sum? From my school years as a kid who enjoyed math I had a lot of fun trying to arrive at the correct answer while most of my peers just wanted to get the "correct" answer because they were being tested on it, anything else that wasn't the correct answer had no value to them.
I believe that's what you quoted refers to:
>> White supremacy culture shows up in math classrooms when... There is a greater focus on getting the "right" answer than understanding concepts and reasoning.
> When designing airplane parts, there are definitely right and wrong answers.
Focus on getting the right answer makes kids not value the process to get to the right answers, getting the right answer is a result of a process they need to learn but if they are only tested on if the answer is right or wrong they won't learn the concepts and reasoning, they will memorise a formula and apply it at wrong times/places because they don't know why the formula is what it is.
You can set up the equations many ways, but then there's "turning the crank", aka "plug and grind" and there's the correct answer, and everything else is wrong.
Yes, I know that singularities are special cases, as well as two roots for the square root, etc.
> they won't learn the concepts and reasoning
Memorizing comes first for children. Concepts and reasoning comes later. In college, concepts and reasoning come first, and memorization is unimportant.
I remember memorizing the times tables when I was 5 or 6. I'd draw out the 10*10 matrix and fill in all the boxes, using addition. Eventually I noticed that "times" meant add the number that many times, and that I didn't have to memorize it anymore. But the work memorizing it laid the groundwork for the epiphany.
Steelmanning "There is a greater focus on getting the "right" answer than understanding concepts and reasoning."
Correct understanding and reasoning leads to correct answers. I want my airplane engineers to understand the concepts and reason through them and not have a shallow understanding of memorized solutions.
White reflects sunlight better and leads to less heating of the airframe's skin, shows up damage and leaks well, and the expensive paint job doesn't fade in the high-UV troposphere, so it's the supreme paint colour for most civil aircraft both in practical and economic terms.
> So do I, and I had some disrespect for the engineers who relied on memorization, but memorization is the first step towards understanding.
I disagree that it is always the first step towards understanding. It is for some people but it is not the only route. At the end of the day some memorization is needed in most disciplines not just math.
> "There is a greater focus on getting the "right" answer than understanding concepts and reasoning."
Approximate rewording to what should/can be done rather what should't be done ~"focus first on understand concepts and reasoning, then getting the correct answer"
> It is for some people but it is not the only route
Teaching 5 different routes to a classroom means focus is lost and little progress is made.
Besides, monkey-see monkey-do is an innate skill of humans, and it works. Especially with children, who have a very undeveloped ability to reason. Reason comes later.
Learning starting from basic principles is introduced in high school, but is foundational at the university level. By then the students can understand it.
>> "There is a greater focus on getting the "right" answer than understanding concepts and reasoning."
> Approximate rewording to what should/can be done rather what should't be done ~"focus first on understand concepts and reasoning, then getting the correct answer"
How about my rewording here does that address your main concerns from the quoted approach?
If you get any answer, isn't the only way to know whether it is right or wrong to understand it?
What is critiqued here isn't that people manage to add numbers the right way. What is critiqued here is in a metaphorical sense that students learn the results for any given input by hard, and don't understand what addition does and what happens when you add things.
As someone who had mechanical engineering classes if I had to pick someone to build my plane I'd pick someone who understands that the bend on a 5m aluminum rod is not going to be in the order of 2km over a person that accepts that result because they think they applied the rules they learned.
You only get the former kind of thinking by making people understand what they are doing. And a good way to avoid developing that understanding is by constantly focus on things being right instead of only doing so in special situations (e.g. a test).
> isn't the only way to know whether it is right or wrong to understand it?
The other way to know is to test it. Things are tested anyway, just to ensure the math was done correctly.
I designed the stabilizer trim jackscrew so it could withstand 151% of the maximum load. (The extra 1% was for tolerances.) The test people grinned and told me they were going to bust my jackscrew. Off I went to Plant 2 where they had the medieval torture rig all set up. It was two I-beams, hinged together at one end. There was a hydraulic ram connected to form a triangle. My poor little jackscrew was fixed to both I-beams.
And, with a grin, the testers started cranking up the pressure in the ram. I watched with my safety googles on. They ran it up to 150%, and the I-beam bent like limp noodle, and my jackscrew stood strong and proud! No creaks, no cracks, no spalling, no buckling. The testers were chagrined, and I simply went back to the office.
There were two takeaways here:
1. It was too strong. Evidently Saginaw Gear forged a masterpiece. I've always liked Saginaw Gear since. They're some bad mofos.
Believe me I would have loved to forklift a load onto a steel beam during mechanical engineering exams just to test my calculations, but somehow none of us were forklift certified and the school failed to provide steel beams to toy around with.
Looking back at that education the important lessons there were to acknowledge that even if multiple people try for hours under high stress to get things right, chances are more than 50% will still be wrong. And to only way around that is to really know your shit inside out.
I had many colleagues who I met uears later which wouldn't even be able to explain basic mechanical laws as all they did back then is learn for exams like other people learn poems. That is what I meant when I said understanding is more important than being right in school. If you construct a plane you better check it ten times and then test it a hundred times.
Indeed, and you correctly used a plural thus reinforcing the notion that methods with an undue focus on singular right answers a particular teacher is thinking of aren't all that great.
That's not what I meant, but I'll go with it for the moment. You can design and build a functioning automobile without math. You cannot design a functioning airplane without math. You cannot design a functioning spacecraft without a very, very strong math background.
With aircraft parts, you can make mistakes up to a point, but those mistakes will make the airplane cost more, such as being overweight. There is always an optimum answer, and I took it as a point of pride to get things exactly right.
I knew some engineers who couldn't do arithmetic very well. The results were parts that were overweight and often would not even fit together, causing expensive redesign and rework. Engineering management knew who they were, and assigned them tasks where their deficient work was not going to cause much trouble.
> You cannot design a functioning airplane without math.
Did the Wright brothers use a lot of math in their wind tunnel experiments or was it more empirical .. given that "correct" math for flight dynamics took a while longer to come about.
> With aircraft parts,
there are a lot of "right* answers; three pairs of wings, two pairs, a single pair, a single wing, push from the back, pull from the front, control surfaces here or forward canards there.
> There is always an optimum answer
There are multiple optimums in the field of avionics.
> Did the Wright brothers use a lot of math in their wind tunnel experiments
Yes, they did. They did a lot of math in their design (their notebooks survive). That's why they were successful and their contemporaries all failed.
They developed the first mathematical theory of propellers, for example, and used it to create a propeller that was twice as efficient as that of other experimenters.
See "The Wright Brothers as Engineers, An Appraisal" by Quentin Wand
> there are a lot of "right* answers
None of them flew if they got thrust/lift/drag/weight/cg/strength/stability wrong answers. Lots of people died as the result of wrong answers.
This is school-level mathematics. The 'wrong' answer isn't a different optimization of cost, strength, weight and endurance, which Boeing might consider.
It's an arithmetic or procedural error giving an incorrect result.
There's great value in understanding the method, discussing ways to make errors and so on — but I think it's nonsense to suggest this has anything to do with race.
As is true with algorithms. To piggyback on your other comment about testing, it's entirely possible (and widely experienced in the software world) to come up with something that seems correct but does not withstand real-world behavior. The premise that the "right answer" is somehow racist or exclusionary is a ridiculous statement to any engineer.
I assume the misspellings "agnecy" and "sutdents" are intentional, as showing perfectionism would be a sign of white supremacy.
Except the reference they give for these characteristics of white supremacy [1] says:
> Organizations that are people of color led or a majority people of color can also demonstrate many damaging characteristics of white supremacy culture.
The problem math education has is that 99% of it is thought by people who are still at step 1, while it should be thought by people who are at step 2 or 3.
I thought it was about either Montessori 'do not grade' or the California weird new math laws but this is simply a rant about a teacher who never learned how to teach.
Yes, people in the same grade don't have the same level. It's actually good: some of them will be great and have high grades, other not and will have low grade. Your job is to have both learn new stuff. You're paid for that. Good teachers will create class cohesion and will make pupils learn from each other (that's what happened in my class in highschool, we ended up very close to each other, and the whole class partied with each other ever other month or so, and we ended up all 22 in my math teacher's garden after we all got the final results (all passed).
(source: I have current experience with understanding and navigating special educational needs in the UK)
Our current model of teaching, loosely based on compulsory education defined in the Victorian era, does not scale. It may work for the interquartile range, but hopelessly fails the upper and lower quartiles. Kids that could achieve more academically get 'held back'. Kids that need help 'fall behind'. The median kids get along fine and hopefully cruise through school.
There are no easy solutions because, as a society, we do not want to (or cannot) pay for smaller classes and restructured teaching at a scale that works. We cannot imagine alternatives, such as not bundling kids the same age together (kids at the same age do not have the same academic capability across all subjects). We insist, for our own children, that they get a good degree so that they can get a proper job. We cannot accept, as parents, employers, or buyers of services, that there are alternative paths. So, we put all kids in the same sausage machine until the machine is worn out and the sausages are all the same.
There is nothing fixable with education because the current model can't be fixed. With the exception of the privileged who can afford it. It is luck as to whether a particular child lands up with the right school, the right teacher, the right peers, at the right stage of their academic progression. There is a high probability that they will do okay, and we seem happy with that.
There should be a revolution with learning. Most people of HN have self-taught a lot of what they know, even just by trying to keep up. Many people are able to work and learn remotely. Our ability to interact and learn at our own pace is common post-grad, and should trickle down to college and secondary school. The platforms and tools need to get better, and society needs to view it as acceptable. The only scalable solution to 'inclusion', at both ends of the distribution, is more home learning.
My school had streams for several subjects which is perhaps unfashionably pragmatic, but at least it allows the top, bottom and inner quartiles to go at their own paces with more tailored tuition.
The lower streams also took the lower exam tiers. Which limited their maximum grades, but a C (or whatever the number grade is these days) from a decent showing in Foundation Maths is better than an fail from a disastrous attempt at the Higher tier.
It won't help when children are multiple entire years behind their peers, though. Short of actually repeating years with extra tuition and major parental involvement until they can catch up, I don't know what you're supposed to do at that point. Obviously just chucking them into the next year and marking it down as a "pass" might make look good for the paperwork, but it's just compounding the misery for the poor kid who might as well be being taught quantum chemistry in Ancient Sumerian for all the good it will do them.
> It is luck as to whether a particular child lands up with the right school, the right teacher, the right peers, at the right stage of their academic progression.
We happened to get our offspring to a good school. Now we're "stuck" living in a rather expensive area until they graduate, if we move we'll most likely have to switch schools and it's a complete crapshoot.
The title is editorialized and wrong. This is the opinion of a single anonymous poster who claims to be a teacher in Canada.
I saw much of the same thing when I was in school. Everyone was in the same classroom because there was no money for anything else. I was limited by what the weakest student in the room could do, because everyone had to be able to pass.
A colleague of mine taught remedial algebra to freshmen at the University of Washington back in the 80's. She'd show the students:
and ask what 'x' is. They'd all fall over in a quivering heap saying they can't do algebra.So she tried:
and asked them to "fill in the blank".They could do that, no problem.
The problem they were having was the "x". "x" meant "algebra", and they couldn't do "algebra".
I know this is no magic solution to teaching algebra, but it is interesting how some people are triggered by it.