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Comment by adam_arthur | Hacker News Reader

Teaching in general is best done by "peeling the onion".

By that I mean, start with the highest level explanation first, form an understanding of that, and then repeat the same lesson but at a lower level of abstraction. You have to be able to walk before you can run.

First, explain the concept like I'm 5. Use examples that provide real world context for why the concept is useful. Demonstrate examples of using the concept to solve a problem.

Then explain the concept again, with an added layer of complexity. Go through the same examples with this layer of complexity.

And so on. The more levels, the easier it will be for people to ease into learning the concept. When confronted with something that's too foreign, many people just shut off their brains, and convince themselves they're "bad at math" etc.

The hardest part about learning is really breaking through the wall of initial understanding. Once you know the language, everything becomes a bit easier to learn.

I honestly think math education can be hugely improved with this approach. This could even be done in a "high tech" way, by having an e-textbook with a slider. Adjusting the slider changes the level of abstraction and granularity that it teaches the concepts at.

Final note.

Math should be taught with well named variables and contextual examples. Why use X and Y when you could carSpeed and carAcceleration? Or whatever, you get the point. People can't relate at all to math when all the examples use generic terms without context. At that point it just becomes rote repetition. Obviously this commentary applies moreso to earlier math education.

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> Why use X and Y when you could carSpeed and carAcceleration? (...) People can't relate at all to math when all the examples use generic terms without context.

Couldn't disagree more.

The whole point is that you don't care about the meaning of the numbers. They are just numbers. A big part of math is learning to strip numbers of their context and look at them bare.

Besides, variables with several letters in them are extremely ugly. They look like products. (They should be banished from programming languages also, but I realize that this is a controversial idea.)

Ah, sounds like the common trend in the hard sciences of putting "difficulty" on a pedestal. Can't make it too easy to learn, need to make things generic and obtuse so only the most dedicated can master the material!

You can apply context to any generic abstraction/formula, otherwise the math is useless, by definition.

Teaching people effectively requires context.

I hope I don't have to read your code with single letter variables everywhere then! There's a reason that's considered a bad practice. Math gets a pass, why?

> Ah, sounds like the common trend in the hard sciences of putting "difficulty" on a pedestal. Can't make it too easy to learn, need to make things generic and obtuse so only the most dedicated can master the material!

I'm not sure that this kind of snark is helpful if you want to teach something.

My point is precisely to make things as easy as possible, not harder! Variables are just slots. How would you rewrite the following using "meaningful" variable names?

The two solutions of aXX+bX+c = 0 are X=(-b+√(bb-4ac))/2a and X=(-b-√(bb-4ac))/2a ?

Any "meaningful" names that you could assign to the terms X, a, b, c would depend heavily in the context. You would convert this equation into an unrecognizable monstrosity. But this equation appears in many different, completely unrelated contexts! The main point of math, I'd say, is that no matter what is the context, the equation is always one and the same and it is solved in exactly the same way. Thus it pays off to have variable names as meaningless as possible.

Apologies for the snark, definitely not warranted. I do see sentiments along those lines a lot, so it's a bit of a pet peeve of mine!

My suggestion is to first teach the generic concept wrapped in context that's more relatable to the student. Similar to code, it's easier to understand the purpose of a variable when you understand the surrounding context as well.

For your formula above, you could define the axes as something meaningful. Say, e.g. instead of A, B and C you have Supply, Demand and Price.

If I read a formula such as:

Demand = 100 + 50Price

Supply = 50 + 20Price

It's easier for somebody to relate to utility of the math than something like:

A = 100 + 50C

B = 50 + 20C

Of course, the math is exactly the same, but you give the student something to relate to.

Now, Supply/Demand is not something the average person can relate to. Obviously context has something of a cultural and familiarity bias. But there are examples that are likely to be close to universal. I use economics here, just because it's a discipline I'm familiar with :)

I don't advocate teaching strictly in contextual examples, but to use them first to provide ELI5 level introduction to concepts. The utility behind the math becomes more clear.

Of course, this is just my take/experience. I spent the majority of time in college skipping class and self teaching, and I discovered that the fastest approach to learning something new is to start at highest level of abstraction and work down. Usually I would learn concepts from resources other than the textbook, as they conveyed information more effectively.

Trying to learn concepts "bottom up" rather than "top down" works, but is less time efficient.

> Of course, the math is exactly the same, but you give the student something to relate to.

No, you give the student an extra problem they need to solve: Figure out what the hell supply and demand has to do with price. Understanding that isn't expected of high schoolers and teaching it isn't a part of mathematics.

History shows there are no intuitive equations outside of basic counting and summing. Even the concept of rate of change weren't explored before the 17th century, before then people had very ridiculous views on what equations physics uses, that is what you get if you try to invoke students intuition here.

> Figure out what the hell supply and demand has to do with price.

Are more people going to relate more to a problem framed that way, or framed as what A has to do with B?

If you're comparing the two, it's self evident that the real world example is easier for the average person to understand. People know what price of a good is. People know that factories make goods, so there must be some supply. Demand is maybe more foreign, but I admit this domain is not the most widely understood... just off the cuff to make a point.

The solution to both problems is the same, does not require specialized domain knowledge at all, so I don't get your point.

There are plenty of more culturally widely understood domains we could use, rather than economics. Physics is of course the easy go to. Not sure what you mean by ridiculous views on physics, clarify?

> People know what price of a good is. People know that factories make goods, so there must be some supply. Demand is maybe more foreign, but I admit this example is not the most widely recognized... just off the cuff to make a point.

Have you ever tried to argue with an average person? Those deductions aren't simple to make at all.

> Are more people going to relate more to a problem framed that way, or framed as what A has to do with B?

All students have already seen countless of examples of what A and B can be. So when they see an equation with A and B they can relate it to apples, to money, to distances, to mass, to temperature etc, at once. If you use special names for the variables most students assumes that the equation only works for those variables and either memorises it or tries hard to understand what makes those input kinds special.

Edit:

> Physics is of course the easy go to.

If you use physics then your class is no longer a math class but a physics class. It takes years for students to build intuition for even basic physics.

> Not sure what you mean by ridiculous views on physics, clarify?

Before newton people didn't understand rate of change or movement properly, instead they came up with shit explanations like this:

> Based on Aristotelian physics, Scholastic physics described things as moving according to their essential nature. Celestial objects were described as moving in circles, because perfect circular motion was considered an innate property of objects that existed in the uncorrupted realm of the celestial spheres. The theory of impetus, the ancestor to the concepts of inertia and momentum, was developed along similar lines by medieval philosophers such as John Philoponus and Jean Buridan. Motions below the lunar sphere were seen as imperfect, and thus could not be expected to exhibit consistent motion. More idealized motion in the "sublunary" realm could only be achieved through artifice, and prior to the 17th century, many did not view artificial experiments as a valid means of learning about the natural world. Physical explanations in the sublunary realm revolved around tendencies. Stones contained the element earth, and earthly objects tended to move in a straight line toward the centre of the earth (and the universe in the Aristotelian geocentric view) unless otherwise prevented from doing so.[25]

https://en.wikipedia.org/wiki/History_of_physics#Medieval_Eu...

That was what the best of humanity could come up with over 2 millennia. You can't expect an average modern high schooler to do better than that without first being taught physics.

> There are plenty of more culturally widely understood domains we could use, rather than economics. Physics is of course the easy go to.

Kids dont know basic physics naturally. They need to learn that and it takes quite a lot of effort.

Using physics examples kids who struggle with C = 40*B is not making it simple. It is complicating whole thing by making claims the kids don't believe yet.

I love your example because it shows how different we people think. It has the opposite effect on me! I find these two equations

    Demand = 100 + 50*Price
    Supply = 50 + 20*Price

extremely confusing! What are the units of demand? Is it units per day, average units per capita of a population? Is it in money? It surely must be money, since it is equated to Price with a unitless constant in-between. Then, do the two prices on each line refer to the same quantity? No, it cannot be, for one concerns demand (the price that people will pay?) and the other supply (the money that it costs to build something). Thus these two equations are unrelated on first sight. There must be some hidden, "obvious" relation between them but I'm at a loss. I'm not making things up, this is really what I see here. I'm a 42-year old who knows something about computers, physics and mathematics, but nothing about economy, and these two lines are utterly mystifying.

Now, for this:

    A = 100 + 50*C
    B = 50 + 20*C

This is just two linear equations in three variables, which has a one-dimensional space of solutions.

" The two solutions of aXX+bX+c = 0 are X=(-b+√(bb-4ac))/2a and X=(-b-√(bb-4ac))/2a ?

You would convert this equation into an unrecognizable monostrosity."

For any beginner this is a unrecognisable monstrosity.

Having real names for the variables can give them something to hold on. Something they can relate to. So it makes sense to their brain.

And later on, you can abstract it - and they are fine with it and see the benefits. (I still want programming variable names to be as descriptive as possible)

You (and probably everyone else around here) are used to abstraction - a beginner to math is not!

(and sadly, most people are even after 12 years of schoolmath still beginners in math)

And the focus on abstraction is a huge wall, making everything harder than it needs to be.

" Thus it pays off to have variable names as meaningless as possible. "

I agree that this is very helpful to achieve, when you are dealing with plain math only. But I think this thinking actually lead to lots of real world harm - when variables are not meaningless factors in math - but representing real persons. Which is forgotten quite often, if some buerocrat for example just sees them as a "meaningless variable" and treats them like this. But in reality this can mean, someones existence just got crushed.

> For any beginner this is a unrecognisable monstrosity. Having real names for the variables can give them something to hold on.

Can you rewrite this same equation using "real names" so that it becomes less monstrous? I don't think that's possible, but maybe you have something in mind.

You equation is abstract.

I recognize it exactly as this: as some abstract equations - but nothing more. (am I supposed to recognize them?)

There is of course no real point in adding names later on. At least not without context.

I would start with a real world example. And then - yes have the equations with long names.

And then after that - have the equation as abstract plain math - in the shape you posted it. But not starting with it.

Can you illustrate how you would teach inner product using a real world example without using mathy names like x, y and z.

Not on the spot - but with some thoughts very likely yes.

And like I said: nothing wrong with using x, y and z - when teaching und using abstract math.

My point was and is, that this should come later. After the real world examples.

Most math doesn't have real world examples though. Inner product doesn't. It is useful in many places, but all of those applications are more complicated than inner product itself and are therefore best taught after inner product.

"Most math doesn't have real world examples though. "

I don't think so, as you can embed allmost any math concept in a real world context. Then the numbers get meaning - then the brain accepts dealing with it easier.

The first time I had real fun with math, was when I dived into computergraphic and had to come up with equations for various vector related stuff.

Suddenly there was meaning to it - the equations provided real benefit. With great visual results. So I had fun improving my equations with pen and paper. I never had that doing abstract math. Only slight relief when some equation finally solved, but it remakned abstract and meaningless.

So this is where I would start: a real world problem - and then to solve it, show how abstraction helps. And all the various needed math concepts along the way.

But starting with abstractions or focusing too much on them, just creates more mental blocks. I certainly had some with math - and I liked science from being very young. For common persons it is much worse. Just have a look how the common people discuss numbers related to covid for example ..

> I don't think so, as you can embed allmost any math concept in a real world context.

Not without adding a lot of complexity. What I meant is that the concepts taught in maths very rarely map directly to any real world problems. GPU programming is a good example, it uses a lot of vector related maths but the leap from not knowing about vectors to GPU programming is huge and not a good way to learn the subject, since you would have to think both about the complexity of GPU's and graphics and about the vector abstraction at the same time. The reason you had fun doing GPU programming is because you already knew about the interface of vectors so you had no problems working with them.

"reason you had fun doing GPU programming is because you already knew about the interface of vectors so you had no problems working with them"

Actually no.

I had to relearn them pretty much completely, as I forgot all the abstract lessons before. But I had fun doing so. Because my brain saw real world use.

Abstraction makes the mechanics of solving a problem orthogonal to the specifics.

It's easier to do abstract reasoning (spotting deep connections between patterns) when you abstract things. I'm pretty sure this has some solid evidence behind it.

It's really a question of whether you want to teach solving problems in general, or teach how to reliably solve a specific problem. If it's the latter, sure, in other subjects (e.g. accounting, science) they use specific terms a lot more, because the problem is more important than the problem solving.

As a teaching tool, teachers start with "3 apples" rather than "3x" problems. This is usually closer to K than 12 though.

As I say it: prigramming is about manipulating abstractions. If you were good at algebra, you should not have problems with programming.

Programming is easier because people care about readability! At least to a much greater extent than in math

A lot of people are bad at math, but decent programmers. Yet, programming is basically just composing expressions the same as you would in math? Why the disparity?

Most programs you write are very concrete. Concrete programs can have concrete variable names.

Then the more you abstract things the less variable names matter. If you create a general purpose list calling its content type "item" or "x" are both equally readable since "item" doesn't say anything that "x" doesn't.

Now math, the entire subject of math is just a bunch of abstractions. Now there are a lot of concrete applications of math, but math itself isn't concrete.

How often is that applicable.

Take the math for Physically Based Rendering (PBR), for example the Cook-Torrance reflectance equation.

For me at least, it's 1000x easier to understand with descriptive variable names than with L, p, w, kd, c, ks, D, F, G, n, L, dw

No idea where else this equation is useful because it used single letter variables.

This applies to learning about applications of math, it doesn't apply to learning math. If you give the Cook-Torrance reflectance equation with proper variable names to some random linear algebra student they should be able to solve it since it is just basic maths, but they will struggle immensely trying to do anything with it since all the names will confuse them.

It's always a tradeoff.

Single-letter variables are extremely superior if you're working out math on paper or at a board. It is simply not possible to work through a long and complex calculation or derivation with variable names that are not compact.

You also end up concentrating more on structure than on identity. You recognize idioms (like the dot products in your example) and it almost doesn't matter what the things being dotted together are; the physical form of the problem guides you to what they must be, even if you lose track. There is only one way it can go.

The flip side is that if someone else does the derivation, and you just need to use the result, it's basically completely incomprehensible to you. You haven't put in the time to understand the problem yourself (if you had, you'd "just" have derived or at least worked through everything yourself), so the extra affordance of long, descriptive variable names often makes a tremendous difference to final understanding.

This, incidentally, is a major point of difference between programming code and mathematics. For any good mathematics problem, there is only one right answer, and any attempt to solve that problem will reach it. (Possibly in a different guise, but the solutions will certainly be isomorphic.) Programming is so often business-driven or real-world-constraint-driven that this is not true. Two attempts to solve the problem may diverge radically. And no one will ever derive business logic from ZFC set theory! So for mathematicians, the fundamental understanding of the problem domain necessary to do good mathematics enables the use of terse, fluid notation. Since programming just doesn't work that way, its verbose, rigid notation is much more useful.

> Abstraction makes the mechanics of solving a problem orthogonal to the specifics.

Not sure what you mean. Providing context does not change underlying math or mechanics of solving the problem at all. Just the same as if you asked somebody to solve FizzBuzz or BeepBoop.

What it does is make the rote exercise of solving more meaningful and relatable. Things that are relatable tend to be more easily internalized.

Talking about early childhood apples etc is missing the point. I'm talking about generic methodology to learn a new concept in a timeboxed way, not a multi-year development. Though thinking behind the two is the same.

It changes the problem in the minds of the student. If you call variables X and Y they wont assume that X and Y are special for this case. If instead you call them kilograms and kilometers people will assume that you teach them something that has to do with kilograms and kilometers and not some general purpose math equation.

It's easy. You say here's a general purpose math equation. Now here's an example of computing something useful with it, using real world concepts.

But often those real world applications are way harder to understand than the initial abstract problem they had. Involving other subjects only works if the students are already well versed in those subjects.

And before you say "just learn those subjects first then?", often it is easier to first learn the abstract math and then learn those concepts than the other way around, since making sense of those equations is hard even when you do know the math and close to impossible when you don't.

Letters make it easier. If you insist on everything being sentences, you will hit the ceiling fast.

The notation did not developed because people were bored. The notation developed and sticked because it made expressions easier to deal with.

That's all fine and dandy when you're teaching maths to someone interested in pursuing or studying maths, if your point is to teach someone who hasn't yet made the connection between the abstractions and their meaning then you should use relatable things - and as their knowledge increases move to remove the familiar labels and show what is the underlying abstraction.

I would think that by now that shouldn't even be up for discussion - much less for programming where your objectives, besides achieving a calculation or a side-effect, are also that your code be intelligible to other human beings. Even in assembly you have symbols for move, jump, etc.

And no, different people have different ways of thinking, not everyone has the same combination of nature and nurture to be completely comfortable with pure abstract mathematical notation - and for most part they don't need to. When you assign fancy greek letters as symbols you're also labelling them, so why not label them with something that a person learning might recognise.

Outside of teaching people that aren't familiar with it, yeah, sure, use all the numerical elegance you can muster (and I would still argue that even there it's still detrimental if the objective is sharing knowledge as widely as possible).

That sounds good in theory, but in practice it is the other way around, the worst students needs purity the most. It is well known that word problems are much harder for students to read, learn and solve than pure problems. Only really good students do well on word problems, the students who are less good at abstractions do the worst at them.

Or do people here really think that nobody has ever tried to make a math textbook with lengthier names and descriptions? It isn't used since it doesn't make things better, it makes them worse.

As someone who wasn't that good at math, I think the convention of x and y feels pretty abstract and arbitrary. Bad students have trouble connecting real world quantities to single letter variables. Lengthier and meaningful variable names will make the task slightly easier, as carSpeed is easier to reason about than x or y. Once students grasp the concept of a variable, the names can be switched. Just my 2 cents.

Why do you assume that students know what speed is? Have you seen this video?

https://www.youtube.com/watch?v=Qhm7-LEBznk

The problem here is that she don't know what speed is. She knows a lot of formulas, but not what speed actually is. That is a typical student, if you add speed in a formula you will make them confused and the problem will get harder.

My first year teaching math (12th grade, "standard" level (below "intensive" and "honors")), I had a student that could solve "equations with fractions for x" but she was struggling with a word problem. I knew she could do the math, but I was struggling to understand where she was losing it.

The word problem was basically the same as in the video. If you're moving 20 miles/hour how long will it take to go 40 miles. (honestly, this was 20 years ago ... the problem probably said "20 MPH" ... it wasn't a very good textbook)

I read the problem to her, similarly to how the guy in the video was trying to explain. "If you are going 20 miles... per hour..."

She was in drivers ed at the time, so I tried talking about her speedometer and "MPH and how that stands for Miles Per Hour"...

Eventually I wrote it down for her as a fraction "20 miles / 1 hour" and not only did she figure it out but she also said "Oh my god, that's what MPH means? I thought that was just a number I had to match with the speed limit signs!"

One of the toughest parts of teaching is figuring out where you need to start (for each student). The thing that gets a student stuck might not even be related to the material... they may have an incorrect preconceived notion that you're not aware of so they think they know what MPH is, but it's actually something completely different, so you are not effectively communicating with each other because all the words you are using have different meanings.

Well, I think the author got a point here. There is no perfect solution for this problem, yes. Maybe for this student, this way works, but not for others.

What I am saying is First, if student don't even understand what is carAccerleration or carSpeed is, then how do you expect them to understand the X and Y term?

Second, to explain an example, you may not use carSpeed or anything related to it at all, but why use X and Y while there are so many better names out there. It really depends on the context of the example trying to show. So my point is using carSpeed or carAccerlation as programming naming technique bad? Yes, it's bad. But using greek terms to show a student who are really bad at understanding things comes even worse. How we can explain them what is alpha, what is beta, etc

I am not an official teacher at any school, but I volunteered to an educational organization to teach students about science experiments. The odd is there is a huge gap in knowledge in ONE class, students range from 6th grade to 9th grade. Student of 6th grade in class don't even know what is physics and chemistry (all stuffs they know is adding and substracting). My job is explaining to them what electromagnetic field is and how the tesla coil works. That's come to my point, you can't just throw to them a bunch of equations and tell them it is it. I had to solve this by breaking the term down into smaller pieces and didn't talk about any equations at all. After that, could I expect them know how to solve physics problems? Of course, it's not. But at least they had a push, and if you want someone to move forward, a small push is what they need (Newton's first law related). That is the 'A-hah' moment.

One more point, if you have read 'A Brief History of Time' of Stephen Hawking, you'll see he barely used any complicated terms or equations in his book (he did use, but not much), he just explain things and break it down into small pieces so even non-physiscal peoples will understand.

And in conclude, if it's for generalize things, then an equation with greek terms is needed. However, to explain things to student who are bad at math and really need help, there is so many better ways to name terms in math examples instead of using greek letters.

I'm not sure that actually pans out in the real world. It seems that there's a gap between people understanding the very basics and anything slightly more complex.

We can either believe that there's just a big percentage of people that will never be able to grasp beyond arithmetic (mathematically dumb) or that perhaps we aren't teaching things in the best way possible?

When I say using descriptive/relatable labels, is in the context of a relatable problem. I also think this is common, in the very first levels of introducing maths for small children it's a normal procedure. At some point between that and more advanced topics it seems to jump off and the conclusion seems to be "Many students fail to master requisite concepts before advancing to more complex ideas, leaving them ill-prepared to succeed in higher level Science, Technology, Engineering, and Mathematics (STEM) coursework.". Of course it's the students (usually kids) that fail, not the very smart lecturers and teachers. At every point in time probably people think of the way they're doing something as the most sensible or best way of doing it.

I think there was quite a leap when programming let go of pure mathematical notation to more regular language likeness - it seems many at the time voiced their opposition to that too.

Not long ago I spent some hours chasing around a problem where had the authors I encountered tried to explain it using more descriptive problems I would have grasped it in minutes, because it was really basic. But yeah, maybe it sounds really smart when it's shrouded in jargon, symbols and meanings that only someone immersed in that field of study would understand.

Didn’t I read recently that Polya attributed Russia’s better performance in high school maths (vs the USA) to its preference for word problems?

Turns out I don’t really care about just numbers but am quite interested in applied math.

School math was hell because my brain does not memorize useless uninteresting stuff.

Same here. Solving completely artificial problems is boring.

Variable names with several letters in them should be banished from programming languages?

What do you do when you need more than 26 variables in your program?

Unicode bro!

I think teachers make the mistake of focusing on the end goal, for example: “thinking in abstract terms”.

A student starts in some state and that state may not have a direct transition to “understanding abstraction”.

A teacher needs to understand the state the student is in and find a sequence of transitions that lead to the desired state.

It’s difficult for teachers to do this if their expertise is limited to knowing what the final state should look like, but not being familiar with (or having forgotten) the state space that leads there.

> Teaching in general is best done by "peeling the onion".

My experience is that best way depends on the student, the thing you want to teach and probably even on the time of the day.

The best teachers are able to adapt to that.

> I honestly think math education can be hugely improved with this approach.

Teaching someone is more like a swiss cheese. Different students have different problems at different levels, and different holes at different levels, that they need to fill to progress more effectively.

A slider (or any completely linear progression of learning) is not, in general, going to be helpful to anyone.

Traditional textbooks can be of variable quality, but it's a benefit that you can flick through to any page. If a teacher can identify what page might be most useful to you, all the better. A point on a slider is unlikely to give you that.

> Why use X and Y when you could carSpeed and carAcceleration?

One of the purposes of math, one that makes it useful to teach to everyone universally, is that it teaches 'abstraction'.

Math questions very often are about car speed and acceleration, or some other contextual example (at least in Western education systems). Using v and a to represent those makes it easier to use algebraic tools that you learn that apply generally, rather than re-learn tools for each specific situation.

It seems you may have misinterpreted my comment.

The point is you start with context specific examples, and high level naming. Teach concepts in plain english. After which you can re-cover the same material in progressively more generic and low level forms.

It's much much easier to understand the generic form of something when you actually understand contexts where it's useful, and see practical examples.

I think you misunderstood my slider comment. You have a textbook, but slider adjusts the level of abstraction that it teaches you the concept at. In what way is this incompatible with flicking to a page?

The TLDR; version is Explain like I'm 5, then ELI10, ELI15... and so on. There's a tendency in the hard sciences to look down on presenting concepts in a simplified form, so I understand many here will disagree with this approach. You have to struggle through the seminal 1000 page textbooks from the 70's to truly learn!

I don't think I misinterpreted your comment.

You gave an example about the "slider" not being useful, and then said a textbook that you could flip to would work better. But the ideas are not opposed at all.

Feel free to clarify!

It's typical in school education to provide different textbooks for different students. 5 year olds already get an ELI5 textbook, 10 year olds get an ELI10 textbook, roughly speaking (a good ELI10 textbook summarises the ELI5 material again).

The slider is not useful - first because the goal of differentiating material by ability is already met in schools, and second because a student can't easily identify where on the slider they need to be, so it creates a further challenge for them to figure out their own curriculum. We only ask this of students when they get to Masters level, or beyond, with good reason.

If I were teaching you math, I would identify that your concept of mathematical algebra is poor, given that you haven't identified why variables x and y are useful in math as an abstraction (and why the justification for this is clearly different from computer science).

Throwing some shade, I love it!

You're missing the point. I'm talking about a strategy for learning any new concept. Just because you're taught addition at age 5, doesn't mean learning calculus at age 15 is now easy. Is it a necessary abstraction to understand how to do higher math? Of course! But math education tends to focus on bottom up approach with rote exercises rather than teaching from an intuitive perspective first.

A good example is learning integrals. Another commenter mentioned they had been solving integrals for a year before understanding at all what their potential use was. Here's a formula, solve 20 problems. That's rote learning. That's not useful at all.

It doesn't matter the concept or discipline. If I give you simple explanations first, it will be faster for you to learn, versus starting with more complex ones. Otherwise there wouldn't be an art to teaching at all.

Your last comment is funny, because it implies a misunderstanding of what I'm saying. If you name X or Y, Speed or Acceleration, the math is exactly the same. There is no change in abstraction or takeaway in the education.

And it's different from comp sci how? I can change variable names arbitrarily in a program and behavior remains the same. I can explain a data structure to you via an easy to parse drawing and high level explanation, or I can show you a formula and say go at it. Same concepts, different teaching methods.

High school level math education is notoriously hated, for good reason. The fact that materials and practices are not entirely overhauled to make it easier for the average student, is just amazing to me. The focus by and large is not in making things relatable or intuitive at all. But there will always be people like you that defend a broken system to the death!

> The fact that materials and practices are not entirely overhauled to make it easier for the average student, is just amazing to me.

I think school curriculum is just fine, and already covers the points that you are trying to make, but better.

A much bigger problem is that there are few teachers competent to teach math in this way, because there is a shortage of math teachers, because teachers are significantly underpaid for the high-level skills you are touching upon.

The math curriculum, as it is in most countries, is fine. The "Common Core" maths principles begin with "1. Make sense of problems".

Algebra is done with letter variables with good reasons: the process of solving a problem involves translating reality into a mathematical model, doing the mathematical work abstractly in a standardised way, then translate back into reality.

A main purpose of algebra is that you don't have to hold your model of reality in your head while you do the mathematical work, which would quickly lead to cognitive overload on harder problems. [In computer code, the opposite is true - since you have a computer to do the mathematical work, you want to be always focussed on the model]

> The whole point is you start with context specific examples, and high level naming

Every school already does that. For the first years math is almost only questions like: "You have 5 apples and you get 3 more Apples from Emmy, how many Apples do you have?", with associated pictures.

Then after many years of using apples and then learning numbers you start to show them that you can have variables like X and Y. Then when they already understand how variables work there are no issues using them. If they still have problems it is because they never properly understood variables, which is strange since we already spent like 10 years teaching math up to variables. You might argue that 10 years isn't enough time to learn variables, but many kids learn them way faster than that and we can't slow them down even further just so a few more kids don't feel left out.

Providing well named variables isn't slower.

It's not faster to code using one letter variable names, for obvious reasons. Because you're bounded by time to understand and make decisions, and not by how long it takes to press a few keys.

I don't advocate teaching more slowly, but teaching more effectively. Teaching "top down" is more effective, and not widely used, from what I've seen.

In your example you describe a multi-year period where basic examples are given in early childhood, and then made more generic over time. That's similar to what I'm describing, but not really the same.

I'm describing the same methodology but done repeatedly with any new domain being learned.

When you learn a new concept in computer vision, this methodology can still apply. Sure, you can throw the formula to do some image transformation at somebody and have them go at it. But showing it being used, explaining the logic behind it in natural language and so forth will convey the idea much more effectively.

The "bad" math education that turns so many people off tends to be more of a "bottom up" approach, solve 10 generic and meaningless problems using this formula.

There is no reason to repeat the process, after the first 10 years they already have enough examples of what X and Y can be that you don't need to do it for them, they already do it for themselves. If they don't then they haven't properly understood variables.

Yet I run into comp sci textbooks that use useless variable names for what are actually meaningful values all the time!

The first thing I do, and recommend anyone do, when learning new algorithms is to immediately convert the one letter var gibberish into a real program with proper variable names. Suddenly what looked like a huge, meaningless expression becomes so easy to understand!

The lack of focus on readability is rampant through math and math adjacent fields.

Example:

Another commenter mentioned the Cook-Terrance Reflectance Model. When I google this, I get a large expression with one letter variables. So I'm supposed to store in active memory which letter maps to which real world concept?

There's a reason one letter variables is considered a terrible coding practice. Why rely on human memory caching, when you can actually be descriptive?

Computer science mostly isn't math though. A small subset of it can be called math, but most stuff are just applications of math. If we are talking about teaching applications of math then the answer is very different.

> solve 10 generic and meaningless problems using this formula

I used to think this too, even when I started teaching. But I think if you look at a (modern?) textbook you'll see that those problems are not actually generic.

They are often designed problems that are trying to help you gain insights as you solve them. Each problem is probably demonstrating a different interesting pattern, if you are looking for that. Admittedly, students are not taught what to look for in those problems and if the text book as "answers to odd problems" in the back of the book, then there are probably 2x the number of problems necessary.

I teach HS Computer Science and I have to teach about division with ints and doubles. It would be nice if I could just explain the rules and give 2 or 3 examples and be done with it... but it turns out I have to have them work through lots of "generic and meaningless problems"

10 / 5 5 / 10 10 / 10 10 / 3 3 / 10 10 / 5.0 10.0 / 5 10.0 / 5.0 10 / 3.0 10.0 / 3 10.0 / 3.0 10 / 10.0 10.0 / 10 10.0 / 10.0

Each of those problems demonstrates a different "edge case". Towards the end there are some repetitive ideas I could leave off (once they know that double/int, int/double, double/double all evaluate to a double it may be overkill to keep repeating those patterns, but...) If I don't have them work through each of those problems, it is likely that at least one student will pick up the wrong pattern and run into problems in their labs when they encounter those patterns.

That being said, I think the biggest problem with math education (in the US) is that many Elementary teachers would probably rank their Mathematics skill as one of their lowest skills. I personally know an elementary school teacher (with a master's degree) that took her last Math class her Junior year in high school. I met her in her college Math credit course which was a Computer Science survey course where we listened to talks by researchers and wrote 1 page papers about what they were researching.. I know she's a great teacher, but she just doesn't have the math background to teach more than "these are the steps to do math, follow them to get the answer."

> That being said, I think the biggest problem with math education (in the US) is that many Elementary teachers would probably rank their Mathematics skill as one of their lowest skills.

Absolutely yes. I'm a secondary level math/physics teacher, and I've come to the conclusion that the problem truly stems from elementary level teachers who don't understand the math and just teach rote steps. Then, when things diverge from those rote steps (usually around fractions, negatives and variables, from what I've seen), that's where you really start to lose students.

The teachers themselves have no underlying grasp of the material and what it actually means, so of course they can't really impart that to the student (especially multiplying/dividing fractions and doing operations with negatives). Then it just compounds in middle school and by the time they get to high school it's a huge mess.

I'm not sure it makes sense to compare math presentation to software.

Unlike a math textbook, you don't show how combinations of symbols are manipulated over time in a codebase as part of a coherent narrative that builds toward a math conclusion.

You also don't generally try to teach something new in the code itself, as opposed to in a team wiki or cited reference.

The onions seem like the wrong metaphor here, since you are going from inside to outside.

Concerning the final note: that is already the case. And then one abstracts away from that for two reasons: 1. This works just as well for boats, trucks or non-vehicles, so you use v and a. 2. Have you ever tried to write and read a formula like MassOfTheCar * VelocityOfTheCar * VelocityOfTheCar /2 =....? Notation needs to be actually usable in practice, not just pretty for single line examples.

> The onions seem like the wrong metaphor here, since you are going from inside to outside.

I'm describing a "top down" approach, as opposed to "bottom up". Or outside in, rather than inside out.

> Notation needs to be actually usable in practice, not just pretty for single line examples.

Yes, math was practiced long before computers were a thing. There's a reason it's considered a bad practice to use single letter variables in code.

I can assure you it's much easier to read complicated formulas in well written code than in conventional math notation!

If you were to use all one letter variable names and cram all computations into a single large expression, that would be considered by most to be poor code. There's a reason why, and it's been proven out in industry.

Math is no different, other than cultural biases and historical reasoning.

In math what you suggest is considered bad practice and there is also a strong reason why. Handwaving that away as "bias" or "history" is extremely rude. Both approaches have their merits and optimize for different uses. This could be an interesting discussion, but you seem uninterested in having one.

I'm open to being convinced!

What is the strong reason?

Not PP, but I can guess at a few, at least:

- It easier to see the 'essence' of an equation/problem when all irrelevant detail is omitted.

- Many concepts are so general that you'd need a small essay to have a 'descriptive' name, rather than just, say, x.

- Algebraic manipulation often leads to non-intuitive concepts in intermediate calculations, e.g. meter-dollars. It's handy to be able to just call such a thing 'a' (or whatever) if it helps make calculations easier.

> I can assure you it's much easier to read complicated formulas in well written code than in conventional math notation!

I could not disagree more. The long names obstruct the shapes of the expressions, making it difficult to recognize common concepts visually. There's a higher need for intermediate variables (to avoid expressions becoming physically too long), and the intermediate variable names often need to be very long to properly describe their meaning (and may represent highly abstract concepts). Terser code can avoid trying to name these things that cannot be meaningfully named.

> If you were to use all one letter variable names and cram all computations into a single large expression, that would be considered by most to be poor code. There's a reason why, and it's been proven out in industry.

I agree that it's not a good idea to put everything in a single expression. But it's not a good idea to put every single mathematical expression in its own variable either, and in my experience the sweet spot lies much closer to the terse code.

I've worked with math-heavy code in both terse and verbose styles, and refuse to work on verbose math code again.

> Math should be taught with well named variables and contextual examples. Why use X and Y when you could carSpeed and carAcceleration?

Definitely no beyond elementary school level and very basic concepts.

If is much easier to do abstra manipulation and calculations with letters then words. And beyond very very basics, you will do a lot of that.

Might teaching be done Best by learning about the student(s) and figuring out what approach resonates?

"ELI5...N"

Exactly!

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