[jacobolus]

The current pedagogy and curriculum is nowhere close to the “most efficient path” (nor is it the most intuitive, best organized, easiest to extend, ...). It’s just an arbitrary history-dependent path people happened to come up with, mostly centuries ago, and “refactoring” any of it is almost impossible.

It’s more like our current public transit system: it gets some people where they’re going, somewhat on time, but it’s generally pretty crummy and full of historical inequity.

[robot_no_419]

I never said it was the most intuitive, just the most streamlined and most efficient. I'm having a hard time understanding why you think it's not the best organized. Most math courses are taught with a pretty straight forward approach: start with the axioms and definitions, prove the easy and auxiliary theorems that are easily derived from the axioms, prove the fundamental theorems that make the subject useful. In other words, the shortest path from the axioms to the important theorems. I don't see a way to make it more organized or compact, but I'm open to hear what you think is a more condensed or organized way to teach math.

And no, this is rarely the most intuitive or contextual way to learn math. Another analogy - a library doesn't sort their books by which ones were best reads or most influential, but by topic and author. Similarly, math curriculums are organized by a hierarchy of which theorems can prove the next theorem with no explanation of which ones are important. Organization doesn't always provide intuition.

[trasz]

What you describe is probably the least efficient way to learn math - or anything else for that matter - because you’re trying to learn things you can’t, at that point and for a while afterwards, reason about. They don’t have connection to anything else. Compare that with the opposite - learning stuff you can already mentally associate with parts of known reality.

[zozbot234]

> learning stuff you can already mentally associate with parts of known reality.

This is not always feasible or effective. Sometimes it's just better to start by doing some simple reasoning about things in isolation, and build the proper connection and context afterwards.

[WoahNoun]

> It’s just an arbitrary history-dependent path people happened to come up with, mostly centuries ago, and “refactoring” any of it is almost impossible.

This is absolutely not true. If anything, math education has a tendency to keep losing intuition over time as it's refactored for modern approaches and notation.

[jacobolus]

There are few (if any) important differences between algebra textbooks from 400 years ago, trigonometry textbooks from 300 years ago, and calculus textbooks from 200 years ago vs. their current counterparts. The way we teach vector calculus is more than a century old. Introductory statistics courses still often haven’t caught up with the existence of computers. Undergraduate level math textbooks from 60–90 years ago are still among the most popular course sources across most subjects, including abstract algebra, analysis, etc. Hot “new” material comes from the 19th–early 20th century. The curriculum (at least say 8th grade through undergrad level) is calcified and dead, like a bleached coral.

Once you get to math grad school you can find more material that uses approaches and notations that are only about 50 years old.

The most significant “recent” change to be found from the 20th century is the “Bourbaki-zation” of mathematics, especially sources intended for expert readers: cutting out pictures, intuition, and leading examples in favor of an extremely spare and formal style that alienates many newcomers and chases them out of the field. And I guess at the high school level, there’s the domination of pocket calculators (displacing slide rules) which came about in the 1970s–80s.

There is massive, massive room for improvement across the board.

If you read works by e.g. Euler, other than being in Latin they still seem pretty much modern (we did tighten up some of the details in the century or two afterward), because much less has changed in the way we approach those subjects than you would expect. By contrast, if you read Newton or his contemporaries/predecessors, the style is often completely different and almost unrecognizable/illegible to modern audiences, building on the millennia old tradition of The Elements and Conics.

For another serious transformation, look to the way computing is taught, which has changed quite dramatically in the past 50 years. Nothing remotely like that is happening in up-through-undergraduate mathematics.

[solveit]

Can you recommend a book that you think presents,say, calculus, significantly better than the books commonly in use?

[jacobolus]

http://www.science.smith.edu/~callahan/intromine.html is one idea (and read that page/book for a critique of the kind of typical ~200 year old textbook/course we still use today), but this could be a lot better with a bigger budget and more support.

Just look what you can do with high-production-value video animations: https://www.3blue1brown.com/lessons/essence-of-calculus

[abdullahkhalids]

This is a fairly well written book. But I don't see anything qualitatively better about it than the best standard math textbooks. What am I missing?

[jacobolus]

It largely dropped the "memorize this set of symbol-pushing rules then apply them to a long list of exercises" version of differential/integral calculus found in typical introductory textbooks, in favor of a "make up a model for a situation, then program a simulation into a computer and see what happens" approach. That makes for a radically different experience for students.

It’s hard to simply say this is “better”: it depends what skills and content you are trying to teach. The more computing-heavy version arguably does a lot better job quickly preparing students to engage with scientific research literature (because differential equations are a fundamental part of the language of science). But it might make it harder for students to e.g. dive into a traditional electrodynamics course intended for future physicists, full of gnarly integrals to solve.

Most of the people proposing even more significant departures (in content or style) aren’t writing introductory undergrad textbooks.

[Qem]

"Elementary Calculus: An Infinitesimal Approach", by Jerome Keisler. Learning calculus is made harder than necessary by the legacy of clumsy epsilon and delta formalism. This formalism is not the intuitive approach Newton and Leibniz used to develop Calculus, based on infinitesimals, that was shunned later because it took time until Abraham Robinson made it rigorous in the 1960s. The author made the entire book available for free online: https://people.math.wisc.edu/~keisler/calc.html See also: https://en.wikipedia.org/wiki/Nonstandard_analysis

[fmap]

Open any book on differential geometry and compare the treatment of differentiation with the needlessly index heavy treatment in any undergraduate calculus textbook.

The point is that we treat the differential of a real valued function as a function/vector/matrix for historical reasons. The simpler perspective that always works is that the differential of a function is the best linear approximation of the function at a given point. But for historical reasons most math textbooks restrict themselves to "first order functions" and avoid, e.g., functions returning functions.

This also leads to ridiculous notational problems when dealing with higher order functions, like integration and all kinds of integral transforms.

[nyokodo]

> full of historical inequity.

Granted the academic profession has historical inequality but what about the math itself displays that?

[jacobolus]

The biggest problem is that it is very unfriendly to uninitiated newcomers and makes insufficient effort to draw people in. You end up with a culture that is unfortunately insular and has trouble engaging with even engineers and scientists, much less the general public. It’s also not very friendly to people who approach problems in different ways: symbol pushing has been elevated and anyone who has difficulty with symbol pushing (for whatever reason) ends up at least partly excluded.

Students who have a lot of practice/experience by the time they get to be teenagers (often via extra-curricular help and support) are much better prepared than those without that practice. Which is of course not a problem per se, you see the same in any field and it’s great if kids want to learn ahead of their peers. But then the content, curricular design, and pedagogy of mathematics courses leave students with the impression that those differences in preparation are due to innate differences in aptitude (“I suck at math”; “she’s just a math person”; ...), toss less well prepared students into the deep end to sink without enough support, and ultimately chase a huge number of people away who might otherwise find the subject beautiful and interesting, and could meaningfully contribute.

[red75prime]

Well, we can't know that until we find (or won't find) the more effective way of teaching (or a way to do math without "symbol pushing" for that matter).

Until then it will not be wise to break what works (even for a minority of students).

[jacobolus]

Current incentives are set up to make even the most trivial attempts to run against the mainstream definitions and notations extremely difficult.

[simonh]

I don’t think it’s fair to say they are set up to do that. They weren’t conceived with that purpose. It’s just a fact of life that once we’ve invested a huge amount of effort in one set of conventions it’s very costly to change those conventions.

[jacobolus]

I don’t mean that some secret committee got together to “set up” all of the social incentives of the entire school system, university system, textbook industry, scientific publication system, engineering fields, etc.

What I mean is that there are incentives for the people involved in those systems which are extremely difficult to reform, and as long as the current incentives prevail it is all but impossible for anyone to refactor things like basic mathematical notions and notations.

Switching and retraining costs are high, gaps in inter-operability are expensive, and there is almost nobody who will achieve any career advancement through promoting changes to the high school and early undergraduate curriculum.

Mathematicians are generally most interested in pushing on the shiny boundaries of the field rather than trying to clean up the centuries-old material for novices. Teachers have their hands full enough with their students to do much new research in pedagogy. Practitioners in industry have their own problems to solve.

[le-mark]

My mind goes to various, unfortunate notational conventions.

[WoahNoun]

Such as? Many branches of mathematics have their own mutually unintelligible dialects of notation. Many longer papers or Ph.D thesis will just create notation just for the context of the paper.

[User23]

An obvious one is that traditional mathematical convention requires single glyph variable names due to the unfortunate decision to save paper by denoting the product by juxtaposition. That’s an admittedly trivial example though, even though the higher order consequences are considerable.

Computing science is when notation came into its own. Younger mathematicians have taken those lessons to heart, but as the old saying goes, progress comes one funeral at a time.

Being forced to mechanically parse and interpret a syntax has a way of really bringing out any ambiguity.

[ithinkso]

> that traditional mathematical convention requires single glyph variable names due to the unfortunate decision to save paper by denoting the product by juxtaposition.

It's not to save paper or because of the product. You don't know the solution to the problem you are working on from the beginning and most of the time is spent writing and writing and writing in a scratchpad trying to solve what you need. Anything longer than a single glyph for variables would be too tedious so everyone evolved to use single letters. And then the papers are written with the same convention since it's natural. You have variable names though with the use of subscripts with the added benefits that it can be (and is) used to elegantly group relevant variables together giving you some sort of abstraction

I once wrote a comment about it here on HN - language in maths is not a programming language used to tell a computer how to go from A to B, but a natural language used to talk about maths between peers. Every natural language have idioms, inconsistences and other quirks. Polish will not change for you so it's easier for you to learn it, it will change in the way that let's polish people communicate better with each other which also include a lot of historical and cultural happenstances. Same with maths

There are attempts like Esperanto and other artificial languages like that and I think any attempts at 'codification' of maths to use some programming language has the same chance of success of wide adoption

[darepublic]

> They are attempts like Esperanto and other artificial languages like that and I think any attempts at 'codification' of maths to use some programming language has the same chance of success of wide adoption

Aren't existing programming languages already types of codified artificial math dialects which have seen wide adoption

[User23]

> language in maths is not a programming language used to tell a computer how to go from A to B, but a natural language

Right, we're on the same page, I just think this is a bad thing and you evidently think it's a good thing. I'm well aware many mathematicians don't, because it's how they were trained and unlearning is the hardest kind of learning. The ambiguity[1] of natural language is observably ill-suited for formal reasoning, and the experience of computing science has shown this conclusively.

Do bear in mind that the pioneers in our field were virtually all trained mathematicians. They were well aware of the historic faults of the field because having to make programs actually work forced them to be.

The legacy fuzzy pencil and paper approach of traditional mathematics is going to end up being to proper formal mathematics just as what's now called philosophy is to formal logic.

[1] Let's not confuse ambiguity with generality.

[ndriscoll]

> single glyph variable names due to the unfortunate decision to save paper by denoting the product by juxtaposition.

Programmers tend to have this lack of fluency with written math that they completely miss: the concise names are not to save paper or make writing easier or anything like that. They're because they make the structure of expressions easier to visually identify and parse. The shapes of expressions are an incredibly important feature of the language and often contain implicit structural analogies. You need to be able to see those analogies to correctly read mathematics, and long variable names would obscure that part of the language.

I suppose it's similar to having enough fluency in a natural language to mechanically translate the words of a poem, but you can't properly read things like the metre, so you've unknowingly missed half of what the author originally wrote and lost it all in translation.

[bee_rider]

I haven't encountered much resistance to n_{arbitrarily complex subscripts}

[zozbot234]

> An obvious one is that traditional mathematical convention requires single glyph variable names due to the unfortunate decision to save paper by denoting the product by juxtaposition.

Generally you'd use upright text in square brackets to denote longer variable names, the notation is often seen in applied fields. But this quickly becomes clunky with longer expressions.

[worthless-trash]

> Being forced to mechanically parse and interpret a syntax has a way of really bringing out any ambiguity.

This is absolutely beautifully said user23. I as a programmer often struggle with understanding notations used in some papers.

[agumonkey]

There's a value in compact / structural notation though, to an extent of course. But o come from the world of verbose application programming :)

[goatlover]

Like APL notation being a tool of thought.

[xnyan]

Is there something you could say about these unfortunate notations or an article you you could point me to so that I can understand what they are?

[nextos]

For example, calculus uses notation and terminology that predates the modern limit-based field Weierstraß and others built. It's really confusing [1].

Statistics is even worse. A mix of old tricks developed to avoid computations when these were expensive. See [2].

[1] A Radical Approach to Real Analysis https://www.davidbressoud.org/aratra/

[2] The Introductory Statistics Course: A Ptolemaic Curriculum? https://escholarship.org/uc/item/6hb3k0nz

[aeneasmackenzie]

Limits are not an inherent part of calculus. You can do all calculus relevant for the physical world just fine with nilpotent infinitesimals if you but give up excluded middle.

[jqgatsby]

I've heard of this constructivist approach to calculus, but hadn't made the connection with nilpotents. that's really interesting, could you explain why nilpotenxy and forgoing the law of the excluded middle relate to each other?

[nextos]

My point is not that limits are an inherent part of calculus. My point is that calculus as currently taught mixes infinitesimal-like notation that predates limits with limit-based calculus.

[kazinator]

So, like, inequality against people with visual disabilities and dyslexics?

[scythe]

I would counter that the current pedagogy -- at least high school through early undergrad -- is the most efficient path, or close to it, for teaching students to become electrical engineers in the analog era. Historically, that was the most math-heavy profession that had a lot of jobs (not just professors/researchers). We just haven't updated it in a long time.

[jacobolus]

That’s probably not too far off the mark... with the proviso that we are fixing the notation, terminology, and problem solving methods for electrical engineering to what was historically used in the 1950s, and not allowing any more radical “refactoring” of those ideas or methods.

I don’t think this is actually the most effective way to train analog electrical engineers, or the most effective possible set of conceptual/notational tools for practical electrical engineering.