> Joel: The big picture there is, the school system should have given up on trigonometry and calculus a long time ago and started teaching statistics and probability. The discrete math is a lot more useful.
I could probably be convinced to jump on that bandwagon.
I disagree with the impulse to discard calculus + trig.
You need more statistics than calculus to understand the current state of neural nets in Machine Learning, but you if you don't at least know the chain-rule, you're just using library code and not understanding even a basic perceptron.
And from a personal perspective: I've used so much trigonometry in my career (plenty of graphics work) that I'm not sure where I'd be without the time spent on trig and geometric reasoning in school.
I mean, I agree with their basic thesis that stats is important because of ML and the increasing practicality in using statistics and AI to solve problems. But I don't thin what they'd discard is well thought out in terms of what is going on. Nor is it meant to be, it's just a casual interview.
Oh, on a personal level I totally agree about trig. I've used it a lot in my programming as well. But I think for the average person a basic understanding of probability is way more valuable, not because of the hard skills but because it changes the way you think.
For the average person a basic understanding of arithmetic would be nice.
People are always complaining that schools don't teach them how to do their taxes, but for most people doing taxes is a combination of addition, subtraction, and following a page of instructions about which numbers to copy from one piece of paper to another.
Imagine trying to teach 16 and 17 year olds that are going to grow into adults that struggle to copy numbers from Box A to Line 3 and then add them correctly the difference between t-value and a t-value.
(I also believe the average person should have a better understanding of probability and statistics ... but we're still working on getting the average high schooler an understanding of why 1/2 + 1/3 is not 1/6 and why you don't need a calculator to calculate 7 * 0)
Taxes are more about having the right training/mindset when approaching a potentially complex set of instructions. It's more about how daunting it feels. Same way I've talked myself out of putting in the work to think long and hard about some proof I had to complete for homework ages ago: "I don't know how much work I'll have to do and how many sub-problems I'll encounter and that stresses me out".
100% agree about probability and stats, that stuff is not intuitive nor easy to pick up outside of school.
I think the argument with taxes is not just the simple math (addition, substraction) that people complain about, it's that taxes become more confusing every year (at least in the US). And not sure how school can help with that.
They don't even need to become "more confusing" -- the simple act of changing, even if they get simpler, forces one to 1) be on the lookout for changes, 2) when/if they land, understand the impact on one's particular requirements, 3) retrain one's processes accordingly. I pay an accountant to tell me how much VAT/sales tax I'm supposed to charge and what tax relief I can ask for, rather than to calculate the actual sums.
This is absolutely not a US problem alone, btw -- every modern country has a yearly cycle for fiscal policies at the very minimum, and often you have additional changes landing throughout the year.
I identify with your view completely. My undergrad degree is in accounting and I know enough not to do my own taxes. I don't trust my ability to guess what are valid deductions and what are not when the laws are frequently updated and I'm not focused on staying current.
Please don't give Intuit (makers of TurboTax) or H&R Block your money. They both actively lobby to keep taxes complicated / non-automatic and are a primary reason why it is so easy to screw up your taxes.
Happy to protest against them but any time you give me the opportunity to choose between paying $50 or manually preparing my taxes from scratch on paper, the answer is a no-brainer.
There is a official service called "free fillable forms" that you can use to file your taxes digitally. The UX is... a bit rough... but it works and is an alternative to preparing your taxes from scratch on paper.
Alternatively, there a number of other tax software companies that do not actively lobby against tax simplification and consumer interests. TaxAct is one example I've seen mentioned on HN, but there are others.
The problem is that "no-brainer" decisions like yours help keep us stuck in the same quagmire we are in now. I think it is worth a bit of time and/or money to push back against these kinds of shady business practices.
School doesn't teach you to do your taxes because your taxes are abstruse and incomprehensible on purpose. If numeracy rose to the level it became easy, taxes would become harder.
It's really true that once you get probability, you see the world, read the news, and solve problems differently. If everyone had a good basis in stats, Vegas would be a Ghost town.
But if you spend time thinking about it, trig likely changed your perception of the world (and history) too. I think most mathematics do that to your brain, by virtue of expanding your internal language.
I can only imagine how crazy these conversations must get for the people actually trying to plan curricula.
I think you need calc to deeply understand neural networks, but there's a lot of other fields that would do well to have a more formal background in statistics starting early. Pretty much most of society would benefit, IMO.
> I could probably be convinced to jump on that bandwagon [dump calc, replace with stats]
I think, really, that the opposite needs to be done.
Math curriculums would do better to focus more on getting "the foundations" right. Starting with algebra, geometry, trigonometry and real analysis (calculus), and pursuing it rigorously for college-bound students. There should be a focus on mastery rather than covering a wide variety of stuff quickly.
Too many students get to college with a mathematical foundation that looks like Swiss cheese-- with many holes in basic skills: like fractions, exponents, trig (yes, even if they took calc in high school).
Why should geometry and trig be "the foundations" while statistics is not? Because that's the way it's always been? Sounds like a horrible reason to keep things the same.
Ever tried to teach an introductory Physics class without using geometry, trigonometry or calculus?
(Edit: for that matter, teaching statistics to students with limited exposure to calculus is not a very successful endeavour, either...)
There are a lot of things that mathematics is fundamental to besides data science, and schools need to address the requirements of all children, not just those who want to write (a very limited range of) software when they grow up.
I hated calculus until learning Physics. It was so abstract and seemed useless. I'd much rather be presented with a problem, like how far did this ball travel, then learn calculus as a tool to solve that problem.
I agree with you though that attempts to teach Physics without calculus are sorely misguided. It's ridiculous to force students to memorize kinematic equations that can be trivially derived from each other.
I got lucky enough to be able to take both Calculus and Physics concurrently from the local University rather than taking Physics without calculus from my high school. It almost seems like they should be taught as a single interdisciplinary class.
Not really an argument. Spolsky didn't suggest leaving out geometry. Trigonometry and calculus can be shortened and taught to prepare a student for statistical reasoning, which includes plenty of trig and calc. And finally, what makes the way physics is taught so sacred? Introductory physics doesn't bear much resemblance to what physicists have been talking about for the past 100 years or so, precisely because the student lacks the statistical reasoning that modern physics requires.
Having gone through both without (high school) and with (college) calculus physics, yeah, it's actually easier with. I think I agree with a sibling comment though, building the two together but taking up the time of two classes (learning the physics by learning the calculus to do it) makes more sense IMO.
Geometry is really used to introduce proofs. Turns out its easier to get teenagers to understand a proof if you can actually see its results.
I think that's a pretty good foundational skill, but I admit that it isn't obvious from the students point of view why that would be or that that is what they are even learning.
In Texas the Geometry class is designed to "strengthen their mathematical reasoning skills in geometric contexts. Within the course, students will begin to focus on more precise terminology, symbolic representations, and the development of proofs" [1]
In Texas Trig is taught in Precalc, which makes sense if you are going to take Calc which makes sense if you are going to go on to study engineering or physics or mathematics. But neither of those classes are required in Texas. You could just as well take Algebra 1, Geometry, and AP Stats.
Disagree. I find that there is a lot of supposed value in everyone learning all the foundations but I often wonder how much money, time and potential for other outcomes has been wasted on everyone being forced to learn things they never use. This is totally verboten to say I suppose but languages are very much like this, and so is much of maths. Not saying the options should not be there or the classes should not be full of kids, just that the sense that its "of course" a good idea for everyone to get these same foundations is overly prescriptive and ignores all the other possibilities and the lack of value most people actually gain from them.
I studied many years of calculus. Guess how many times I've used calculus in my electrical engineer career. 0. When needed, I use a math package instead.
Statistics, on the other hand, I encounter daily, but don't have the expertise to handle well. I know, I could study, but easier said than done.
I know this discussion is focused on Math, but I think most students get to college with a general foundation that looks like Swiss cheese. I think it's just a result of good education being hard to apply generally, and students lack of interest in certain topics.
I don't see any reason why they could not teach all of those. It's a bit like saying that you should learn English instead of your mother tongue (assuming that's not English). There is utility in knowing more than just a few things because they have commercial value or immediate use.
The reality is that we are teaching as much math as we can.
My school teaches Algebra 1, Geometry, Calc, and Stats. We sometimes are able to offer Multivariate Calculus if there are enough Juniors in the BC Calc class.
It would be nice if we could offer an Abstract Algebra or Complex Numbers course occasionally, but there just aren't enough students that get to senior year ready to take those kinds of courses. Keep in mind that there are 2 AP Calculus tests, so no student that could take Abstract Algebra would choose to not take AP Calculus due to getting college credit for it.
Since there is an AP Stats class, students will often take both AP Calc and AP Stats. Which means they are lined up to take all the fun math classes when they get to college!
Of course, these aren't the average students. These are the students that are taking full boats of AP classes and coming out of high school with 15 - 30 college credits.
What too many people are missing is basic statistical and probabilistic concepts that allow one to evaluate and contextualize data they are constantly exposed to, and calculus isn't needed for this. Calculating the area under a distribution curve isn't nearly as important.
In my rural US curriculum, these concepts were 100% part of the (very vague) "Algebra II".
Granted it seems like it didn't stick for a large segment of folks.
(Which might be a valid thing to examine in itself: for math and science we tend to act like it's just the fact that a course is missing from the curriculum that will answer why the general population is inexperienced with it, but nearly everyone also takes composition and literature classes, and I don't feel like we can say their associated skills are really at "saturated" levels.)
Maybe I am biased because I'm an Engineer but most of the stats I do leans heavily on Calculus and Linear Algebra
A good example is regressions this is quite a common technique I rely on daily and at it certainly helps to have an understanding of slope, intercept etc.
Unfortunately my "Stats for Engineers" course at uni was not great it was pretty much focused on "this is how to interpret ANOVA output from Excel".
I've found looking at some of the more complex stats I've had to wrap my head around like Principle Component Analysis - it makes a lot more sense when you can grasp the linear algebra going on behind it.
Yes, this was how my college-level class was structured. We focused on using Matlab to plot things and understand what we were plotting as we moved across the range of topics. We never did any deep dives and no calculus and I think the class was solid (and approachable in high school with a competent instructor).
For probability, you can focus on discrete-valued random variables (this is what is covered in high schools in Poland). Don't need any calculus for that.
Good old frequentist Bernoulli and conditional probability.
The problem with it is applying it to life, which is more often Bayesian.
Sadly the high school level math can never be reasonably complete.
They should mention statistical tests and normal distribution at some point, as concepts.
Calculus seems like a clear requirement for most science and engineering discipline outside of computing.
I can't imagine I'd want to do physics or mechanical engineering without calculus or trigonometry. And computer graphics would be a lot harder without trigonometry.
And for statistics and probability... how can you have a Central Limit Theorem if you don't even know what a limit is? ;-)
Being able to deal with continuous functions is certainly useful for statistics and probability.
When I was in college, prob & stats were taught in one of two ways: 1) Memorization of formulas and rules, in the course for psychology majors; and 2) Proofs, which were taught in an upper level math course.
Neither would be better than trig and calculus. Nobody remembers the formulas and rules unless they use them regularly, and a course that is heavy on proofs would be too hard for most students.
In my view there might be a third way, which is to let the teaching of prob & stats revolve around simulation, rather than by algebraic derivation (proofs) or memorization. What can you learn from a giant bag of random numbers? What if that bag is divided down the middle and given to two groups of students? What can you deduce, when you know something about a set, but not everything about it? Can you make random numbers produce outcomes that are sometimes as convincing as the outcome of an experiment?
And so on. Even to this day, if I think that I've used the correct statistical formula, I test my recollection by feeding simulated data through it.
This kind of exploration could also gently lead students into... coding.
While learning by simulation is going to garner deep understanding, it takes a lot longer. And in the end you end up discovering the same "formulae" (but of course having done the experimentation to "derive" means you understand it much better than if it had just been told to you).
I would take out only classic Euclidean geometry. Proofs in the geometry space are in the education system because they've been there for over two thousand years. Yet few people do those. Even people who do lots of 3D work don't do those. It was at one time considered important because it was the only system of formal logic around, but we're well past that.
I had a related discussion the other day with an acquaintance. We were explaining the different maths to my daughter, who is thinking about doubling up courses to get to calculus.
My acquaintance (having recently graduated college for maths) was explaining that in the real world, statistics and probability have far more use in the regular world. From my perspective perspective as a programmer, I confirmed that I'd used statistics far more often than trigonometry but with one notable exception: game engine programming.
Never have I felt more amateur in my understanding of mathematics during my programming career than when I had to (re)learn about matrix math, dot products, quaternions, Euler transformations, and other concepts I hadn't touched or even heard of in years. Maybe those concepts are less applicable as general math education now that everything's a pre-built framework with its own IDE.
Yeah or several months we've spent on solving quadratic equations and inequalities which was very repetitive and something I've never needed afterwards.
I am afraid it's tough to teach statistics and probability though. People seem to struggle with it and it's difficult to find competent teachers and time necessary for less gifted children to have a chance to understand it. It's much easier to do repetitive trigonometry drills over and over again sadly.
Both the school I went to 20 years ago and the school I teach at now offer Probability&Statistics and AP Prob&Stats and at neither school is Calculus a required course. In fact, you can take any of our 4 Computer Science courses as your 4th year math credit, but usually students choose between Stats or Calculus (or just take both)
I'd wager it's been a while since Joel was in a school and the course offerings across the board are likely quite different than he remembers.
And as a counterpoint my daughter's school had 'high ability' math students skip geometry (over my objections) to hurry them to calc, which surprise, surprise she did miserably at.
Not a whole lot of meaningful statistics you can do without at least a little calculus. The integral is an essential part of nearly all inference, even in very trivial problems such as estimating a bias coin.
Statistics without any calc is basically applying ad hoc tests without understanding what they are actually testing. This is the kind of statistics we should be getting rid of, not expanding.
You mean Shannon-Hartley theorem is defined in terms of trigonometry not statistics or calculus?
Please do go back to school. It's mostly calculus, statistics and discrete math.
Trigonometry, besides geometry, essentially only appears again as part of complex numbers and calculus. When circles or balls are involved. Phase as well, but it's offshoot of calculus, as it can be represented as complex exponentials.
You realize gains are measures of amplitudes? You realize Shannon’s information is predicated on flip flop circuits of electrical signal memory which follows superposition linearity? You realize the mathematical completeness of Euler’s formula is trigonometric?
No, I do not need think I need to go back to school.
While I understand his argument, there are people who use that trig/calc base and move forward with their math careers - NASA scientists, physicists, engineers, and if we don't provide that math education in high school, will we be hurting those careers by not continually pushing their math skills in those directions.
I don't think anyone is arguing that trig or calc don't benefit anyone, but the opportunity cost for a standard curriculum in high school is extremely high. What about the other careers/professions/fields that are hurt by teaching calculus in place of _______? Clearly we need a higher standard than that.
You need calculus to understand probability and statistics, since probability is fundamentally a question about measures and statistics about integrals of variables on measure spaces. Even if you don't know the details of measure theory, certainly you need some notion of limits. Calculus is not much more than a systematic application of limits to algebra and geometry.
Even for discrete math you need to understand ideas like convergence, otherwise probability is filled with paradoxes.
"Pick a number!"
"Any number?"
"Well, numbers that are big are less likely to be picked."
"How much less likely do they need to be in order for your question to have a sensible answer?"
"Let's talk about convergence"
You also need algebra and trig. How else would you know what an inner product is without trig? Or that correlation is just an inner product? You need some linear algebra as well to understand cross products and inner products, otherwise they will be these magical things.
Many ideas in discrete math should be understood as the discrete versions of ideas in continuous math, with the continuous math ideas often being more natural and easy to understand. Sometimes one of the best ways to estimate an infinite sum is via an integral that is much easier to solve. A lot of complicated sums seem magical until you see that they are just the discrete versions of integrals -- sometimes path integrals -- with various weights that come from measures on the spaces of interest. So if your interest is blind computation, you don't need a lot of discrete math at all, you just need to learn what the symbols mean and how to look recipes up in books. But if your focus is on understanding what you are doing, then there is not a magic barrier between discrete and continuous math, most discrete math can be viewed as a discretization of continuous ideas, and in some cases ideas in continuous math can be better understood as limiting cases of a discrete result.
The sad thing is that this is all stuff you will get in a good gymnasium in Europe but you may need to wait until college to get it in the U.S. Absolutely no reason why a teenager can't learn calculus, limits, trig, probability and statistics, as well as basic ideas in group theory. Really algebra, geometry and trig you should get in elementary school so that you can get calculus, probability and statistics, basics of differential equations and some topology in high school. Then you can take specialized courses in algorithms if you are into CS or say PDEs if you are into engineering -- this is all stuff that should be taught in high school.
College tracked teenagers in Europe often learn that and more, and not just in math centered gymnasiums, but the price of that is tracking, otherwise you have to go as the slowest learner who has no interest in a specific major in college.
I have a very bleak outlook on coding after moving to the bay. The companies that win and get big do not put stock in writing good code. They want to move fast so they can beat people in the market. So you end up valuing people who can do whatever it takes to scale and keep things working while ignoring your tech debt until you’re so big you can just hire 20k engineers to clean the mess up over the next two decades.
With this approach young engineers learn that clean code and everything people like Martin Fowler have been saying isn’t going to get them promoted. So people learn bad practices and we all get collectively worse at coding.
I’ve spent so much energy trying to get people to care about code quality and qualitative improvements, but at the end of the day the people who got these companies were they are can just say there method works and there’s no proof it would have gone better with better code or that they would have survived at all.
"I know it’s kind of a cliché but it comes back to worse is better. If you spend the time to build the perfect framework that’s going to do what you want and that’s going to carry you from release 1.0 through release 5.0 and everything’s going to be great; well guess what: release 1.0 is going to take you three years to ship and your competitor is going to ship their 1.0 in six months and now you’re out of the game. You never shipped your 1.0 because someone else ate your lunch."
You're commenting about this on Hacker News, a site that--from a technical perspective--does almost everything "wrong", or using methods that have been obsolete for 10+ years. It uses tables for layout. It does "AJAX" by creating an <img> tag and setting the src= as the url to request. It has an awful url structure (e.g. item?id=20996837). And on and on, I could list so much more.
But you (and I, and thousands of others) still use the site and probably didn't even notice or care about any of that. This is one of the largest, most heavily concentrated technical audiences on the internet, and even though those are the exact people you think would be most discerning about it, obviously nobody truly cares about how much isn't being done "properly". People will argue all day in the comments here about the right way to do things, but since they're here they clearly don't think it's important when deciding what they use themselves.
It's the same for everyone else, with every other site and product. As long as it works, the reality is that almost nobody cares how much of a mess anything is internally.
You've picked a poor example to prove your point because web development collectively jumped the shark and websites using technology from around 2006 are the most reliable and easy to use.
Back then usability, graceful degradation and progressive enhancement were appreciated by developers. Nowadays we have the complexity of Enterprise Java except with less reliability and in JS. And ads and tracking and virality everywhere.
The best code is what gets the job done, and on schedule. Customers don't care what design patterns you are implementing or what your unit test coverage is. I don't think of it as a "bleak outlook", just reality. It was true 15 years ago and is true today, and IMO overall coding standards are only getting better, not worse.
There is an optimization that needs to be done between manageable code and deliverables. If your code is poorly structured it may be very difficult to expand or iterate upon. This is not ideal, and will slow down your goals.
However you could spend forever rewriting and refactoring. What is the value in that?
Excellent point. Companies need to make money to keep the doors open. There is no money made in fixing something that isn't broke on the off chance you might end up needing to utilize that code later. Tech debt should be addressed when necessary, not just on a whim because someone is uncomfortable with less-than-optimal code. Refactored code requires testing just like any code update, sometimes more depending on how significant the change is.
Totally unrelated to this topic, but I absolutely love Joel's "You suck at excel" youtube video, done in the same vein as the old "You suck at photoshop" tutorials someone made.
He and his team created Stackoverflow, a site used by every developer every day. He's a prolific writer around software engineering & managing software engineering. He's got an incredibly blunt writing style w/o any fluff and his writing on topics such as interviewing, refactors, and technical debt are shared around often.
He has been industry famous since before Stack Overflow, mainly through his blog (joelonsoftware.com). Some of his writing from the early 2000s is still amazingly relevant.
Obviously the generalization is wrong, but I bet it's not too far off the mark.
All* developers google things. Google often places at the top StackOverflow hits for programming related questions. I bet that's how most "non-nerd" programmers arrive at SO.
*again a generalization, but even less off the mark.
He cane up with the Joel Test (https://www.joelonsoftware.com/2000/08/09/the-joel-test-12-s...). It's still incredibly relevant. A few years ago job vacancies would include things like 'we score 11 on the Joel test, come work for us'. I haven't seen it for a while, but I still do it mentally when I go to a new gig
He was very skilful at telling developers things they wanted to hear in his blog. But until Stack Overflow I didn’t know a single person who used any of his products (FogBugz, CityDesk) and it didn’t seem plausible that the hottest developers in the world would be beating down his door to work on them.
The current stupidity of the hiring process, the inane whiteboarding, the take-home tests and all that, is all down to him.
He was famous before stackoverflow; the success of stackoverflow was attributed to it's creators having a large audience to seed it with questions and answers.
I could probably be convinced to jump on that bandwagon.