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I had a similar experience where math went from being easy and fun to an intimidating and painful slog. My problem was just how focused most courses are on learning techniques for solving problems. I found all those endless substitutions that you learn in calculus to be infinitely dull and so it was difficult to do a good job. Ditto for the solution techniques for differential equations. Don’t get me started on matrix inverting. I think I had to do a 5x5 matrix once for a homework assignment. What a colossal waste of time.
Proof-based math classes came like a revelation to me. When I took Real Analysis, for the first time in over a decade, math was fun. You weren’t just memorizing and reapplying recipes. You were seriously thinking about unique problems and devising solutions. And all the while, you were learning where all these techniques actually came from and how everything connected together.
I don’t understand why we can’t have more proof heavy math in high school. Who cares whether you remember the arctan substitution or whatever in an integral; I’d always just use a solver for that anyway. I’d rather be learning about what an integral is in the first place.
I'm a private tutor who works with adults on proof-based math. I've often had a similar thought to the one you're expressing here --- I also found proofs pretty revelatory when I first exposed to them and wondered where this magical tool had been all my life --- but I wonder how well this experience would scale to the mass of students in high school math classes.
After teaching proof-writing to my students for several years now, I've seen a lot of variation in how quickly students take to the skill. Some of them have the same experience that it sounds like you and I had, where it "clicks" right away, some of them struggle for a while to figure out what the whole enterprise is even about, and everything in between. Basically everyone gets better at it over time, but for some that can mean spending a decent amount of time feeling kind of lost and frustrated.
And this is a very self-selected group of students: they're all grown-ups who decided to spend their money and spare time learning this stuff in addition to their jobs! For the kind of high school student who just doesn't really think of themselves as a "math person", who isn't already intrinsically motivated by the joy of discovering what makes integrals tick, I think it would be an even harder sell. High school math teachers have a hard job: they have to try to reach students at a pretty wide range of interest and ability levels, and sadly that often leads to a sort of lowest-common-denominator curriculum that doesn't involve a lot of risk-taking.
A wonder no one uses this for programming.
Uses what?
Proofs
A more rigourous approach was tried after WW2, when Americans feared the Soviets were edging ahead mathematically/scientifically. It was called "New Math" [0]. For an example of the type of textbook high school students were taught from, check out Dolciani's Modern Introductory Analysis (the 1960s and 1970s editions only; the later editions were dumbed down, especially when Dolciani died) [1], which starts with set theory, logic, field axioms, and proof writing techniques.
> I don’t understand why we can’t have more proof heavy math in high school.
proof based math requires critical thinking and its a lot harder to scale the teaching of critical thinking. We dont' pay enough for teachers of quality to be able to do this at the public school level. Its also much harder to test for in standardized tests.
> Its also much harder to test for in standardized tests.
You could test it using interactive proof verifiers. This would also make it a lot easier to teach, since proof verifiers can handle even very complex mathematical proof via the repeated application of a mere handful of rules. (The rules are also surprisingly similar to the familiar "plug and chug" workflow of school-level math, only with different underlying objects - lemmas and theorems as opposed to variables and expressions.)
I completely disagree. Proofs are very abstract. Learning to read them is a skill you have develop before you can learn anything from them.
Great post! It's always interesting to see the experiences of fellow peers going through Math Academy.
It took myself 2 1/2 months to complete Mathematics for Machine Learning on Math Academy last year (2024) working through reading material, taking notes, and completing all the exercises took all day everyday I loved it, this was after I completed Khan Academy (starting from the beginning of mathematics negative numbers, to the end differential equations) because I kept putting it off for years when I got to busy.
The main thing for me was learning not to get too frustrated when getting an answer wrong. If I made a mistake, I focused on understanding what went wrong, looking up youtube videos on the topic if it was confusing, and then trying again.
At the end of a lesson I wish I had someone to bounce questions off of but thats when I used chatGPT.
Congrats!
Thanks for reading my blog! M4ML is the next course for me after I complete MF3. What are you doing now on math academy!
Yeah! Unfortunately at the time I had gotten laid of from work so I had extra time just not extra $ to keep paying monthly subscription, also some courses still said coming soon at least the ones I wanted to take when I was taking M4ML.
Good Luck with M4ML its a great course! Covers a lot, I was impressed, wish there was some videos or more visuals but it doesn't hurt to use youtube. I took maybe 7 pages of notes on my github and each over 4000-8000 lines (I used the notes to do the step by step exercises it was easier for me to type notes and do the exercises on computer than pen and paper this is what I used to do).
I take the notes because I will probably forget, but I think its key to always be learning and keep practicing even when your done the course.
Once I get hired again I will def take Discrete Mathematics. In the mean time I've just read books on ML and LLMs, free online courses, youtube videos etc.
I understand the feeling. You'll get another job soon and be back to crushing math problems on math academy. Thanks I'm excited for M4ML a lot. I feel like I'll experience math in a way I never have before. Would you be willing to share your github of notes?
Could you have completed the entire curriculum on your own using ChatGPT without needing Math Academy?
Not at all. ChatGPT can't even reliably do arithmetic.
I really want to do Math Academy and even briefly tried it a year ago. It's absolutely great but it's also very expensive. I know that math skills are invaluable, it's far cheaper than schooling, and that long term the investment is likely to pay for itself but when you're skint $49/month is still a pretty hefty sum, especially if you live outside of America. For context in the UK, a basic gym membership (£17/month) and a SIM only phone plan with unlimited data (£22/month on a two year contract) only costs £1 more in total than Math Academy (£38/month). I can't help but feel that the people who would benefit from it the most are also the people least likely to be able to afford it.
Go on eBay and buy the following Open University book sets. They go for around £30-50 a pop: MU123 (basics), MST124 (more complex). 6 months worth of study in each book set. If you like it do MST125 (even more complex) and M140 (stats) after. That's the first year of a mathematics degree literally from the ground up through GCSE and A-level stuff. If you really like it, get a student loan and do the associated accredited degree.
£30 for 6 months is pretty damn cheap and you get to keep it forever!
ebay example of the latest edition for sale: https://www.ebay.co.uk/itm/197011707080
On archive.org too if you are happy with PDFs: https://archive.org/search?query=creator%3A%22The+MU123+Cour...
First MU123 book A: https://archive.org/details/BookAMU1232ndedOU2014/MU123-Book...
This is a proper accredited course developed over 50 years or so with its own textbooks and material from a respectable university, not a gamified subscription portal experiment put together by god knows who that can disappear in a puff of smoke at no notice.
I'm studying the Q31 (BSc Maths) on Open University.
I can second this recommendation. The maths books are _excellent_.
It's hard to explain how, but let me try: most of the maths textbooks I possess (plenty of them) are written with the assumption that you attend lectures at a classroom and use them for extra material/exercises/reference.
The OU books are written with the assumption that you learn from them as the primary material, so they go a lot further with regard to explaining things as well as producing them from first principles.
I just skimmed through MU123. The content and level is somewhere around the level of these two courses on Math Academy:
- 5th grade math
- prealgebra
The book does look high quality. But I'm surprised it covers these fundamentals, given it's for a university course.
Yeah, MU123 is basic (high school) mathematics. The reason for that is that the way the Open University works, they have no formal requirements to join, so they cannot assume that students know even the basic stuff (because they might have left school earlier, or they might be out of formal education for years, or be from a different domain, etc), so the aim of that course is to quickly help you catchup regardless your background.
If you are above this level, you would start with the "intensive start", which skips MU123 (allowing you to pick another module in its place) and then starts with MST124 (precalculus, trigonometry and single-var calculus, roughly), moving you on to MST125 (intro to proofs, number theory, more calculus, linear algebra, etc), in a faster pace.
Yeah I get the part about filling in some missing foundation. But it's the level that surprised me. Some of the MU123 content is covered in 5th grade in the US, i.e. around age 10 or 11.
But I guess there are people who will have forgotten that stuff, if they're not using much math day to day.
The OU maths books are indeed very good. This is the way to go.
The way I come to look on such offers (monthly unlimited subscriptions) is not the net price itself, and not future supposed returns to it (who knows what they be, and they for sure will depend on many other things), but how many hours a week I am willing to spend on that service.
If you can and willing dedicate on average 2 hours a day (a big commitment but I think I was able to hold it for several month with them) the cost of mastering, say, Linear Algebra will be ~4 less then if you subscribe and will be spending ~30 minutes a day.
> very expensive
I guess it depends on where you are at in the world, but in our neck of the woods $50/month is an absolute bargain compared to using a tutor. Not to mention you get to work at your own pace and to practice spaced repetition consistently.
It is. But I don't think there is an alternative way to make it sustainable. There are just not that many people who are serious about self-education, and you won't like it to cater to the less dedicated customers.
> I can't help but feel that the people who would benefit from it the most are also the people least likely to be able to afford it.
Even if it were $10/mo, the people who would benefit from it the most (around the world) still can't afford it.
Did you try contacting them and asking for a discount? Sometimes all you have to do is ask.
my read as a US person is that math academy is optimized towards students who would otherwise be well served by an in-person supplemental math program. at the earlier grades for math academy (grades 4-5 etc) the main competition i've encountered are in person programs like AoPS, Russian Math, or Kumon. The prices for those range between $450-$100/mo and for a student or student and parent combo that may be looking to supplement their math classes or for somebody who needs to home school for a period of time, mathacademy at $50/mo is a steal.
I wonder if they could charge lower rates for people who live in poorer parts of the world.
$49/month is almost nothing to me now, but it would be prohibitively expensive for a 15 y.o. me in freshly independent Czechia.
I suspect it would also be prohibitively expensive for most 15 y.o.s in the developing world today, and these are the guys and gals who stand to gain the most.
It's not just the price. You'll find a number of 15 year olds have no ability to spend money online.
+1
I wish there was PPP for the subscription, i tried for a few months but stopped the subscription recently.
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I use it.
the format basicly consists of a list of modules (lessons or reviews) to work though. each lesson module starts with a short test on how to solve the problem it teaches, 1-2 worked problems and 2-3 problesm then more short text on how to solve some variation of the problem coupled with 2 solved problems and 2-3 for you to solve. review modules just give you a few problems to solve. if you can solve 3 without making any mistakes, it will finish the module and you can go tot he next one. every few days you are presented with a quiz that tests what you know and genearates review modules for you to go though before retesting you.
overall, it seems spartan but its effective. you spend most of your time actuvly engaing with the problems with just enough information to solve them. the structure of the courses are setup such that you learn and master prerequisites before you are presented a lesson so there isn't much of the frustration you might find where a problem implicitly assumes you know something you don't. That a big reason they start you with a placement test.
overall I'd say the $50 price tag is worth it. its very efficient vs reading and working though problem sets where you don't necessarily know what you need to know to solve them. you're not spending time figuring out what your knowledge gaps are as it fulls them in as you go for you.
you could check the website slightly less quickly and see you can cancel at any time within the first 30 days and get a full refund
https://www.mathacademy.com/terms-of-service#cancellation
It's what I did 10 days ago before deciding to try it out
I don’t trust those plans - it’s easy to forget to cancel it, and most products are simply hoping that you will forget. If I actually think I will want to cancel it, I will not sign up for services with this pattern. It’s a simple rule that saves a lot of mental overhead.
That is not the case with MA they will refund you with a click of a button and it will cost them money since stripe keep their cut and don’t refund it to MA.
and that is fine, it's your choice but has nothing to do with the other party not offering an option to test, change your mind and not spend money.
I personally just put a reminder in my calendar for all such things and be done with it.
Please don't post insinuations about astroturfing, shilling, brigading, foreign agents, and the like. It degrades discussion and is usually mistaken. If you're worried about abuse, email hn@ycombinator.com and we'll look at the data.
I'll call out 3b1b and khan academy for me. Especially over covid. Made math fun again.
My middleschool principal thought it'd be a good idea to skip me over pre-algebra into alg 1.
Turns out that doesn't work great, and I still have confidence issues because I have a hard time remembering the properties of addition & multiplication by name. I know the rules.
A noun that only refers to one thing isn't a real word, so if you want to cure yourself of "the associative property" being meaningless, you could study other algebras where the rules are different.
this was actually one of the things that really turned things around for me actually. it took probably 25 years in between.
My middleschool principal thought it'd be a good idea to skip me over pre-algebra into alg 1.
1. Read each sentence at half your normal reading pace
2. Skip every other chapter.
Sounds ridiculous, right?
That's my reaction when people propose grade skipping as the only solution for a child whose natural pace is 2x the 'standard' pace at which math is taught in school.
Yes, it's ridiculous. They should really only grade-skip in math after giving the student take-home exercises during the current year that will serve as a replacement for the skipped grade. It's irresponsible to do otherwise, imo.
Agreed, but I'd rank the choices in this order. #3 is still better than #4.
1) Allow the child to go at their natural pace.
2) Grade skip every 2 years, with take-home exercises.
3) Grade skip every 2 years, without take-home exercises.
4) Force the child to go at the same pace as the rest of their same-age peers.
I a missing component is a plan for recourse if the student doesn't take to the new pace/material.
That was my biggest problem, that and I wasn't actually on-board for the skip. Educators need to learn how to admit when they fucked up and learn how to improvise a new strategy.
a lot of school is redundant, and the courses are often non-sequential
skipping chapters of a novel doesn't work very well, but it works great for the encyclopedia, and pretty well for a lot of textbooks
it's also not that hard to use khan academy or wikipedia to fill in the gaps, if you did miss something
The best solution is to offer accelerated math classes in public schools, in both elementary and middle. Mostly in middle school because elementary math can usually be handled through differentiated instruction by the teacher, unless the child is exceptionally advanced.
I really like the way my kid's middle school does it: accelerated 6th grade math covers the entirety of the 6th, 7th, and 8th grade standard math curriculum, which sets the kids up for algebra in 7th grade and geometry in 8th. Because the standard middle school math curriculum is essentially just advanced arithmetic, it's pretty straightforward to bundle this way. It also makes it easy to inject 7th graders who missed 6th grade accelerated math into the accelerated track if they pass the algebra qualifying test before 7th grade.
When I was growing up the G&T program started in 4th grade and cohorts from multiple schools were pulled into a school that ran the "gifted" program. Essentially all the kids were tracked from 4th grade through high school graduation and there was no real possibility for non-G&T kids to get into the "gifted" classes in middle school. In HS that just transitioned into APs and college dual-enrollment; by the time I graduated HS in '99, I had 22 credit hours of college classes banked, including dual-enrollment bio and calc 1 + 2, plus a bunch of humanities APs.
Today -- at least in our bay area public high school -- there's no tracking outside of math and the vast majority of classes can contain students in multiple grades. That absolutely was not the case when I was in school, and imho it's an improvement.
I'm thinking specifically about the USA math curriculum. It's pretty sequential until 8th grade or so.
Filling in gaps is fine for people with good study skills, but that excludes the vast majority of elementary school students.
the "smart" kids do seem to have those skills, though. either that, or they're being tutored on the side, or they just require fewer examples to get it
whatever the case is, i think the idea behind skipping grades is that the kid isn't learning much in the classes they're in. they may not learn much in the next level either, but it allows the school to test that they've learned what they were supposed to (from class or elsewhere), while wasting less of the student and teacher's time
that said, testing out seems like it'd be better than forcing the kids to sit through yet another math class, even if it's one level higher. more time to touch grass, or read in the library, etc.
Yeah the smart kids may need fewer examples or fewer practice reps, but very few kids can skip entirely, say, 4th grade math, and not struggle to catch up. It seems unnecessarily painful, when instead they could be taught smoothly at double the pace.
Congrats on your progress!
Over the past few years, while homeschooling my daughters, I've come to see the way math is usually taught as horribly pathological. In the US, where we live now, it's often seen as a competitive activity -- almost like a sport. In the UK, where I grew up, that wasn't the case but still it was taught as this huge body of knowledge and skills with almost no motivation.
My daughters are so advanced in math and I really don't believe it's even mostly due to innate ability. It's because, just to take an easy random example, when we studied geometry our very first lesson was me pointing out that the word "geometry" just means "earth measuring", and it was useful for farmers to be able to do that. Or, when we proved the irrationally of sqrt(2), of course I entertained them with the tale of Hippasus being thrown into the sea by the Pythagoreans. For basically everything we've learned there are so many fun stories. It makes me sad that most students of math never get to hear them.
As a b and c grade student, who messed about, stumbled through a not very good info technology degree at university I definitely agree with this. The stories and lore are what makes me now so interested in programming and software engineering. I've pretty much taught myself everything programming related and that's what I work as too. I desperately want to learn math up to and including calculus as I feel like it's a hidden shame that I'm a programmer with not much math ability. I'm actually considering signing up for math academy.
I am currently studying for our country's version of the SAT and, having tried Math Academy — having been convinced there is nothing anywhere as polished and developed on the market — I still had to cancel my subscription after the first month. The price just wasn't worth it; over a single year, it translates to a cost greater than one-on-one tutoring.
Small companies have to understand the value of local pricing — nobody is willing to pay above h percent of their salary for a service X, and there's only so much that rule can be bent. I understand that, at the end of the day, the company still has all their expenses in USA prices, but for digital services with no manufacturing or logistic costs, it can be better to make a modest profit than none at all.
I decided to try it 10 days ago exactly because of the pricing.
It would be impossible for me to have one-on-one tutoring for a year at only €465 ($499 but I'm in EU). And that's regardless of the tutoring quality
Wow that’s a pretty glowing review of the service. Sucks about the pricing though.
I haven’t really looked at math academy, but I was in school (including college) I probably learned 40% of math from khan academy, 40% from textbooks, and maybe 10% from lectures.
How does math academy compare to Khan academy?
Math Academy uses spaced repetition for skills with tiny, to-the-point interactive lessons (typically following "theory, some exercises, theory, some more exercises" formula) based on an initial diagnostic test, where the skills are structured as a graph of dependencies.
I didn’t, at the time, appreciate how challenging a problem it was until I started researching Bayesian Knowledge Tracing. While their definition of a skill can be a bit narrow, thus putting more time into reviewing things I'd rather move on from, it does work from what I've observed.
I recall they had a course on Abstract Algebra and other more advanced subjects, so if you're really interested, the great thing about subscriptions is that you can afford to try it.
Always great to hear from people on the far side of the valley of despair. I don't think it is pointed out enough that people who fall off of "mount stupidity" can sometimes get really really stuck. In my experience when they do that at work it is quite traumatic.
Another good book for the author and others is "5 Elements of Effective Thinking" by Burger & Starbird. It thinks about thinking which can sometimes side step the depression of suddenly not thinking you know anything about anything that accompanies that big drop off mount stupid.
Thanks for reading my blog post! I'm going to pickup that book today and make sure to start reading it!
what do you mean by falling off mount stupidity, especially at work?
It was discussed in the article, but to be more explicit, sometimes a person who is sure of their understanding of things learns a new thing and that new thing opens their eyes to a huge amount of complexity they were missing. They go from feeling like they knew everything there was to know about a thing to feeling like they know little to nothing about the thing. (this is the "Falling off Mount Stupidity")
Depending on how senior they are at work, that can be quite traumatic. A lot of people in tech sort of base their self image on how smart they perceive themselves to be with respect to their peers. When that perception inverts their own world model makes them feel worthless.
In the two cases where people I was managing this occurred (that I knew of) their productivity dropped like a rock and they became seriously depressed. One I managed to get back on track, the other left tech and I have lost track of where they ended up.
Comment was deleted :(
Mount Stupidity is the peak of overconfidence greatly outpacing your competency level. So, falling off is essentially being humbled by expreiences that make you realize you do not know as much as you think you do, and your confidence takes a major dive as a result.
Mount Stupidity relates to section two of the blog post where it references a concept related to the Dunning-Kruger effect.
Math Academy is awesome, I'm fully hooked, but, repeating something I wrote elsewhere: it is a bleak existential confrontation with your ineptitude with fractions.
you and me both buddy
I'm signing up like, oh, I have a lot of gaps I can fill in with calculus, and it's like, no, you got a lot of gaps you need to fill in with simplifying cube root expressions. The best is every once in awhile it double checks to make sure I still know what multiplication is, with like Dick and Jane bought 10 apples problems. I have given it no reason to believe otherwise! But I trust the algorithm.
Also, I go too fast through them and do stuff in my head that I should write down and make dumb mistakes, and when I get the "Incorrect" I'm like, yeah, I see exactly the dumb thing I did, let's move on, and it's like, no, let's do a next problem that's real nice and easy to make sure you get this and I'm like "stop patronizing me motherfucker".
I’m doing math academy and I have two comments. First, the value of this site comes from its content which is very thorough. I think it’s a great way to learn math.
Second - I love the website. It reminds me of what I think of as the golden age of web design where sites were mostly server side rendered with a little jquery / Ajax sprinkled in, and more information density was preferred.
Hey everyone! This my blog. Just made an account on here so I could comment. Thank you all for the support and reading my story! :)
I have enjoyed the challenge of relearning mathematics with Math Academy as well. I find the format and reviews extremely helpful- it is so refreshing to end a lesson or review early if you are getting all the answers right compared to the drudgery of my schooling experience where you are getting question after question that isn't introducing a new mental challenge.
My only desire is that their site worked on my phone- it would be nice to do a lesson when I have some free time and some paper.
That is known as "drill," and is vital to math success.
I'd wager there are not many skills or occupations were drill isn't vital to success.
I read this article because I wanted to learn more about their Math Academy experience, but I found the preamble and backstory a little long, which caused me to skim.
Re: Math Academy, I used the service for ~3 weeks last year from a post here on HN by the guy responsible for the AI/ML knowledge graph behind the platform (I believe his first name is Justin). I was "only" doing about 30-60 minutes a day (a little bit higher than their guidance, but low for someone not doing math otherwise IMO).
N.B. Due to substandard early instruction combined with being "gifted and talented", I was placed by the test into Math Foundations 1 (or 2?). For example, I still don't have an active/working mastery of the unit circle. So if you're a real whiz, YMMV.
I found Math Academy effective at showing me my weaknesses and sharpening those skills in the short term, but I probably didn't do it for long enough to benefit from the spaced repetition effects. I found the UI/UX better than Khan Academy (sans AI), and much less tedious (when I demonstrated understanding, the questions moved on or increased the complexity vs. doing the full problem set no matter what).
When I cancelled within the first month to receive my refund (see other commenters mentioning the high price), I was surprised to see my support email and refund request email both went to one of the founders (or owner?), Sandy Roberts, who was emailing me while also attending her daughter's college orientation (or helping her move, can't recall right now).
Cancelling was painless once I realized I was getting a response from someone at the platform --- so if you're interested in trying it, I can recommend giving it a shot. Maybe there's some sort of economy for them if more (adult) people sign up, because 50 USD still feels a bit steep.
"other commenters mentioning the high price"
I understand that everybody has different financial circumstances, but personally I find it so odd how people prioritize their spending. $50/mo to level up your math game? Too much. 8x $6 lattes per month - totally worth it. $200k+ for a university education after which you STILL won't know basic math (or much else useful for most majors) - super totally worth it.
For me I'm just willing to pay a lot more than other folks are to learn interesting skills. Math, sailing, music, leatherworking, perfume making, whatever - to me that's such a good use of money.
Thanks for the reply, and I know I'm replying ~a day late to your comment.
The near-term costs vs. the long-term payoffs of learning the math skills I was learning were pretty clear (immediate costs, opaque long-term benefits other than being better at math in a general sense). I didn't have anywhere to apply the math, so I decided to spend my time learning more from sources like "Structure and Interpretation of Computer Programs", which is zero-cost to me.
I don't drink any $6 drinks of any sort; I also can't go back to change how much I did or didn't pay for a university education that did or didn't involve learning math for my major.
Other commenters I read who shared price concerns also didn't mention their latte habits or cost of their advanced degrees, so I didn't find this comment very helpful for making that comparison in my specific case.
I agree with your overall point but I don't think those comparisons are very useful. Regardless of my monthly latte consumption, an extra 50/month is 50/month... the only real comparison imo is how much you'd be saving vs hiring a tutor or simply going through books yourself for free. I think it comes down to whether you have the drive to learn from books. If so then that's clearly the best move. But I'm willing to pay the 50 because this is the only approach that's worked for me so far. It's worth it but it still stings.
Fair point. IOW, what you're saying is: If learning math is your goal, is this the best (or most cost effective) way to do it? Sure, for some it would be, but for others they'd be better off with books or some other alternative.
This may be a great article.
As an aside, all Dunning and Krueger showed is that everybody thinks they're in the top 1/3 to 1/4. (At least everybody in undergrad school at Cornell.)
Oh it’s much worse than that, and it wasn’t everybody in Cornell undergrad, it was a grand total of like 45 undergrads at Cornell who signed up for extra credit (which I imagine is a pretty big confounding factor - the A students don’t need it, and the F students don’t bother, on top of the massive confounding factor of only measuring undergrads from Cornell - they didn’t measure any people who are truly incompetent or unskilled).
The DK experiment depends on people ranking themselves against the cohort, other people who they don’t know. The DK effect probably doesn’t exist, it has been argued compellingly that the paper does not demonstrate what it claims to.
“To establish the Dunning-Kruger effect is an artifact of research design, not human thinking, my colleagues and I showed it can be produced using randomly generated data.”(!!!) https://www.scientificamerican.com/article/the-dunning-kruge...
“the asymmetry reported by Kruger and Dunning actually goes away, and even reverses, when the ability tests given to participants are very difficult.” https://talyarkoni.org/blog/2010/07/07/what-the-dunning-krug...
“The Dunning Kruger Effect is probably not real” https://www.mcgill.ca/oss/article/critical-thinking/dunning-...
If you are downvoting, I hope you've read at least one of D&K's papers. Or at least looked at the plots. Not meta-stuff pro or con, but what Dunning and Krueger wrote.
I didn’t downvote anyone, in fact I upvoted your comment. Why do you ask? I am getting downvoted here, but I don’t know why. Yes I have read D&K’s paper and studied the plots and methodology multiple times. I thought I was mostly agreeing with you here. What part of their plots or paper are you thinking of? In my opinion, the paper dramatically overstates the claims of what it actually demonstrates experimentally. The title alone primes the reader to jump to a conclusion that isn’t supporter by their data; they did not show people who are unskilled or “incompetent” on an absolute scale, they did not test all skills in general (they only tested 4 relatively easy “skills” and one of those was subjective), they did not test the general population. Looking at the plots again won’t address the confounding issues I mentioned, even if I don’t dispute their data at all.
Every time I read the DK paper, even now, I’m struck by how wild it is that so many people refer to this as a human-wide cognitive bias, when there was so little evidence to begin with, the study was so small and biased, so many people have called out problematic issues with their data and experiment design. If randomly generated data shows the same effect, then what, exactly, did D&K demonstrate?? https://digitalcommons.usf.edu/numeracy/vol9/iss1/art4/
Oh, no, I'm asking people who downvoted your comment to please get their heads in the game. I don't understand their point of view.
Oh haha, thanks!
Math Academy and a SQLite course pulled me out of despair as well. I grew up thinking I was relatively smart and being able to learn on my own, but only randomly and sporadically. After not learning much the past few years and feeling very stupid, I decided to look for some paid courses to take on Hacker News.
The SQLite course was in a very different video format and took roughly 20 hours, but I learned a lot and immediately used that knowledge in two software projects that would've seemed insurmountable to me.
As for MA, it's taken a lot longer and has been difficult. I'm now at 6000XP and halfway through Fundamentals II. I have a lot of thoughts, but I (kind of) plan to (probably not) write a review after having completed Foundations I-III, since I haven't talked to anyone else who's done so so far.
Any chance you remember the name of the SQLite course?
It was High Performance SQLite:
https://highperformancesqlite.com/
The course content could all be learned from the excellent SQLite docs, and in greater detail. However, I think paying the $200 was worth it for me, since the course led me through a structured learning path, this is my "year of structured learning", after all. I had some minor complaints, but in the end I learned a ton and feel empowered because of it. I'd like to take his Postgres course too but the price is too high for me.
which sqlite course ?
I started reading the article without paying much attention to the word Academy in the title, and was attracted by the single word 'Math'. In the first part, I found it interesting when the OP talked about their learning experience. In the final part, my interest faded when their writing turned towards a business entity.
Reeks of a publicity stunt.
I am going through Math Academy and I like it very much. I have done advanced technical work in my field but my math background had weaknesses from my public schooling in a large urban area and some experimental math instruction in high school. The ability to do it over is oddly exhilarating.
Who is building the Math Academy equivalent for other subjects (especially other sciences)?
From listening to interviews, it seems like after completing a full undergrad's worth of math courses, they plan to expand to CS and math-adjacent fields (probably physics at least?).
I’ve used Khan Academy with the kid and always been impressed, but Math Academy sounds very promising indeed. Anyone have an informed view on the relative merits? I don’t mind paying the monthly fee if there’s genuine value.
I tried math academy about a year ago since I wanted to finally get a strong grasp on linear algebra.
But I gave up during the diagnostic test. It was very, very long, and didn't seem to be adjusting in difficulty, and asked similar questions. I'm normally a fast test-taker, but after about a third, I figured it would take me an hour and a half or two hours more.
I hope they've updated it by now.
You can do the diagnostic test in multiple sittings: I think I split it up into 3 parts?
To some extent, though, I don't think you're meant to spend lots of time on each question struggling to figure it out as it's trying to determine what bits of the course you can skip, so if you can't answer a question relatively easily it's best to just say "Don't know" and move on so you get some revision during the course.
It's still the same and I agree it's a splash of cold water to sign up for the platform and immediately be put through the gauntlet. I would say it's worth getting through it as none of the actual course material is like that, it's much more bite-sized. But I hope they figure out a way to smooth out that onboarding process, maybe split the test into segments and weave them into the rest of the material somehow.
def try again. I personally absolutely bombed it. But you know its been 20 years since I've been in a math class. Its all to figure out where to start you off. It doesn't matter where that is, it matters where you go from there.
Inspiring and well written. It resonated with me for I find myself in a similar position. I wonder how much time did the author commit on weekly basis. Nonetheless, I wanted to signup on Math Academy immediately but doesn't look cheap.
Are there any other recommended websites for learning math (apart from Khan Academy, Math Academy)?
Hey! This is my blog post thanks for reading! At my peak I spent roughly 4h a day on math academy because I wanted to get 100+ XP. I've brought it down to about 2h a day since I'm also teaching myself python for my goal of being a MLE in the future.
A rare, insightful post from someone who didn’t do well in maths in high school. A true HN treat!
If anyone from Math Academy is reading: the price point just doesn't work for me. I'm happy to pay ~$100/year for our family's Duolingo subscription. $49/month is too much.
Thank you for sharing your wonderful story. The prelude triggered a lot of emotions and memories from my own past.
Math Academy looks very enticing!
Dunning-Kruger effect does not look like that!
Compare and contrast with actual one: https://en.wikipedia.org/wiki/Dunning%E2%80%93Kruger_effect
Sure, novices overestimate their skills - but the relationship is monotonous, with no peaks or valleys at all.
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$50/month? I know people with children who can't afford the NYC mandated trash cans which cost the same...
Tried it for one of my kids, mini-review:
From Parent: Expensive Alpha quality - missing absolute value signs, auto-marks "1./2" as wrong compared to "1/2", parental controls have bugs, eg you want to pause, then change the date won't update the date you're pausing to. Uses XP (experience points) as gamified motivator and then doesn't respect their value by docking unnecessarily or due to bugs. Emailed on sign up to be "personally welcomed and invited to respond with feedback, or any questions" So I did, because I want this kind of effort to succeed. Response totally ignored all content of the email and suggested he could delete the account for me(!)
From student, who is quite a way ahead of his peers in math. "boring, annoying and stupid"
Account is paused due to the above, if we cancel it I'll be pretty annoyed given the unused portion and expense.
Khan Academy has very similar merit. You can donate a lot less than USD$50 / mth.
Maybe we got unlucky and the good intentions of "personally welcoming and requesting feedback, inviting questions" was lost due to getting snowed under so response LLM happened and totally ignoring follow up rather than being deliberately rude? Could well be.
Lotta hype about it, justified? I didn't see it.
This kind of started to feel like a Math Academy promo at the end.
Side note: An absolute pet peeve of mine is how the Dunning-Kruger paper is being misrepresented, to the point of abused, like the graph included in this blog post.
In what specific way did this post misrepresent or abuse the Dunning-Kruger concept? (Btw, the graph used is the same one used on the Wikipedia page for DK.) If you’re able to explain what you understand to be misrepresented, you can clear up the misconception for others — like me.
You can find the original paper here: https://www.researchgate.net/publication/12688660_Unskilled_...
It's a mere 15.5 pages of actual text.
So, instead of engaging in a discussion, and sharing your knowledge, with someone genuinely interested in learning from you— to improve upon the seeming misconception that bothers you — you link to paper and do nothing to correct your own pet peeve. Maybe consider that human life is finite, no person will ever be able to read or analyze everything, so you can help others when you have a piece of knowledge. Relevant - https://xkcd.com/1053/.
I don’t see that graph anywhere on the Wikipedia page
https://en.m.wikipedia.org/wiki/Dunning%E2%80%93Kruger_effec...
I do see other graphs that tell a different story. Namely, that confidence is a monotonically increasing function of competence. If the data supports the idea that there is a valley of despair where confidence decreases as competence increases, I must be missing it.
Here, from Wikipedia:
https://commons.wikimedia.org/wiki/File:Dunning%E2%80%93Krug...
[edit: yes, it isn't currently on the Wiki page. On the other hand, I've seen that graph associated with that work before]
This is Commons, not a Wikipedia article. This image is incorrect, has been removed from the enwiki article, and is in fact explicitly tagged with a disputed factual accuracy notice.
Dunning-Kruger described a relationship between people's subjective opinion of their skill, and their performance on a test. They find the subjective curve is less steep than the objective one (low performers believe they are closer to the center than they really are, and so do top performers). There's no "peak of stupid", or anything else on that graph.
Repeating vague associations you've seen on the Internet before is how misinformation spreads.
I dispute nothing you write. Looking at the paper that graph is not within it.
Either my eyes skipped past it or that dispute notice was added after I linked the image. Regardless it belongs there.
I have previously seen a similarly shaped graph with Dunning-Kruger effect discussions many times, including on Wikipedia I believe. Now I'm curious what the source of the misrepresentation is since it does not appear quite derivable without artistic interpretation from the paper's data.
Regardless, I'm glad to update and add to my beliefs.
Please note that despite the implication that seems to be in your final statement, I did not mean to say the graph was correct, only that it is a graph commonly associated with the paper's message and thus understandable for the author to have used. From that, the use of it doesn't quite come from nowhere. I'm fact, I didn't really say much at all. While Wikipedia is the first search result, the Decision Lab is next which has a similar, even more distorted graph on their page [0] and yet is a fairly well esteemed organization.
Glad to improve my knowledge but that the graph is in common use is not misinformation even if the graph itself misinforms and isn't from the paper.
I am likewise baffled by this. The entire "Mount Stupid" theory of Dunning-Kruger is wrong, and the blog shows that same wrong graph for me.
Maybe the author is running some kind of A/B test between the actual Dunning-Kruger paper graph and the fake one?
Nice story - I’ve seen you on the leaderboard a few times. Good luck through the rest of Foundations III
For child, being precocious in a subject is usually a curse. Being bright and a generally fast learner is also a trap. Hitting the wall is inevitable for almost everyone, but until that point your self-image is built on forward velocity, and especially relative velocity — you’re just faster than your peers. Turns out there are faster kids, they just aren’t at your school.
Parents can make this worse but it’s pretty hard to prevent it.
> Turns out there are faster kids, they just aren’t at your school
Moving from Slovenia to SFBA in my mid 20's (~2015) was ... super fun like that. Sooo many people here are that most brilliant super talented engineer/founder/whatever from their home locale. But here we are just the norm.
We've got an idiom for that: being a big fish in a small pond.
Then you move or have some experience that opens your eyes and you see that there are so many people out there that you're not actually as special / smart / talented / athletic etc as you thought.
I've had the experience a few times myself, and it's always a bit of an existential wakeup call.
My first job at Microsoft fresh out of college, my office mate had 2 PhDs, one in chemistry and one in physics.
Related - About 10 seconds into my first job I decided that staying quiet for a bit and listening to people around me would be a very good strategy throughout life.
Yeah, i was the best physicist the tiny impoverished state school had seen in years. I'm_significantly_ behind everyone else in my PhD program. But then, i tell myself if i can't be the best prepared, i can be the hardest worker. But realistically... nah. I'd rather stay _inside_ the 5th story window of my office. It's not a race, unless you're losing
Erdos could learn as much in five minutes as the average person can in five years, but there are more things to study than he had five-minute intervals.
What
Erdos learn a million time faster than Grug. But Erdos still not learn everything.
If topics infinite: Knowledge of Erdos bigger than knowledge of Grug, but ignorance of Erdos as big as ignorance of Grug: both infinite :)
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Being in the 99.9th percentile for intelligence just means that there are 8 million people in the world who are smarter than you. And a few more every day.
> Moving from Slovenia to SFBA in my mid 20's (~2015) was ... super fun like that. Sooo many people here are that most brilliant super talented engineer/founder/whatever from their home locale. But here we are just the norm.
Very similar story here! Grew up in the Netherlands, joined an SV company in my late twenties. Huge mix of imposter syndrome and sadness that I hadn't been able to experience such brilliant people earlier.
Very much regret not choosing different educational paths that could've let me surrounded by them a decade earlier. I would've enjoyed life much more. On the other hand, I don't think either of us could've realized it before experiencing it first-hand, so no reason to beat oneself up over it.
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The really bad thing about being precocious or a fast learner is it allows terrible working habits to develop in school - if you can max out the standardised tests by doing almost no work, this develops awful habits for life in general.
Yeah, this is why IMO there should be more project-based work and just move faster and harder for skilled students in general.
Get students to build their own drone with MCUs - so they're forced to overcome challenges and practical trade-offs.
Introduce algebra in primary school, and calculus much earlier - so Green's functions, etc. can be taught in high school to those who want to study mathematics, and they feel what it's like to struggle to master concepts early on.
Hopefully with more of a shift to online courses and AI, this will be possible. Unfortunately the majority of schools just act more like a daycare centre / prison.
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"Introduce algebra in primary school, and calculus much earlier"
algebra and calculus is college level century ago, we already bring it on HS level in some part of the world (Asian country) we already learn it earlier than most of the world but that's not sustainable
Calculus is 16-18 (A level, years 12-13) in the UK, 15-16 for kids who do advanced maths.
Very true. Before working for a company I expected it would be OK or didn't give much thought about it. Meanwhile school was easy and I didn't put anything close to 40 hours per week. Plus the school was close. The first months working for a company were OK but then I was thinking oh damn I'm going to have to work 40 years like this?
And then after a few years you learn that you can accept the workload but it's not enough to guarantee everything will be OK. In school everything is handed to us, really.
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Yep, we all have to hit the wall and that’s where we find out what we’re made of. It can be a valuable experience with the right people around to help.
Can you give more detail on what you mean by it can be a valuable experience with the right people around to help.
My son (7 years old) is gifted in Math and as a parent I find it extremely hard to decide how much I should push him (register him to math competition, weekend math club ...) and how much I should just let him get 100% on exam and not accelerate the learning.
He needs to learn grit and how to ask for help. He needs to learn some things are hard and that he can’t always lean on his intelligence.
The best way to guarantee a gifted kid wastes a lot of their potential is to be in an environment that is too easy. It creates a devastating mental habit that won’t trigger until later in life, like college. Whenever they try to do something that doesn’t come easy, their brain will try to shut down out of a kind of frustration. They won’t know how to overpower it. It will cause depression, anxiety, shame and low self worth later on. Because the gifted kid will know they are wasting their potential, but blame themself for not being good enough to deal with it. It feels like being broken.
All of this is created by being rewarded for maxing out the rewards of a trivial environment. Someone needs to patiently and compassionately teach them to value overcoming appropriately sized challenges. To find and operate on the edge of their potential and ask for help to operate beyond those limits.
So yeah, grit and asking for help. Intelligence is mostly wasted without it.
It's just like exercise - if you just stay lifting weights you can manage easily, you never really progress, even though it still feels easy.
Hey, stop describing me accurately, that's mean.
I faced the same problem. Some analysis in retrospect, having kids who have now graduated college:
- your child has a wall. At 7 he is not hitting that wall.
- that wall is probably mostly related to the pure math concepts, and probably less to his actual age when he encounters them. This is my assertion and I cannot prove it but let’s assume it is true. Precalc or calc is a typical wall moment, but for others it might be geometry or trig.
- one response to an eager math learner is to move them through the curriculum faster. They are happy, because everything is fun prior to the wall! You get to be the parent of that kid who is great at math! Let’s put the pedal to metal!
- what acceleration means is that your kid will hit the wall at 13 instead of 15, or 14 instead of 16, etc.
- those two years can make a big difference. Accelerating might be positive, in that they hit that at an age where you can support them better. It might be negative, in that they now have a crisis that their peers can’t relate to. Not accelerating might mean that they respond to the wall by pouring their energies into age-appropriate activities instead, like listening to loud music or being grumpy.
So no easy answers here. We did not think ahead clearly, and pushed forward, and had some decisions to make later. In retrospect I think it turned out fine, but I wish I had known that I was pulling the wall forward in time.
If you're hitting a hard "wall" either some concepts are not being taught effectively, or there are some undetected gaps in your previous learning that make some things difficult to understand for you. There's nothing specifically about precalc that makes it inherently harder than, e.g. Algebra II or whatever if the teaching is effective. So being able to access alternate sources of understanding, such as Khan Academy or the Math Academy OP talks about, can be especially important.
Moving through the curriculum faster is a common approach but it's also risky, because that's how the gaps are created that can then hinder your understanding later. Of course if you have reached true mastery of a given topic, moving forward is preferable to being bored to death, but assessing whether that applies can also be difficult at times.
In my experience as a parent, you can provide the resource but don’t need to push. Love of math will happen if it has the right environment. For a 7yo I might suggest looking onto Epsilon camp, and Art of Problem Solving (which is on line).
My own kid went to MathPath (middle school camp by same people as Epsilon Camp). Loved it. “Yes, dad really, I want to spent a whole month of my summer doing math.” The social experience is great for kids to be with other kids that like math.
rich people stuff is so fascinating to me my family went on one vacation my whole life i wonder why jesus made us poor because i loved school so much
If you're 'good' enough/identified a certain way as a kid, they'll bend over backwards to get you in things like that even if you're not well off. I wasn't from a well-off family, but test scores in the top 0.1% meant somehow there were scholarships to make camps and programs accessible once/if I expressed an interest. Whatever amount was required to make it affordable.
I'm a thoroughly useless adult, so it was a waste of money on their part, but it does happen. Or at least it used to.
I got put into some “smart kid” activities in grade school, but as a poor kid with zero advice from parents, I really had no idea what to do with it.
No one told me that math is really 90% about writing proofs, all those homework problems I did were just the weed-out stuff, the academic equivalent of Leetcode.
So when I got put into some “real” academic math as a teen, I crashed and burned hard. I didn’t have a tutor and it never would have occurred to me to ask for one, so that was that.
When I was 18 years old in my first year of college, after my first semester grades came in, a guidance counselor set up a 1-on-1 with me to talk about the Rhodes Scholarship process and what my research interests were.
My response was: 1) what the heck is a Rhodes Scholarship and 2) how could I possibly have “research interests” as an 18 year old college freshman.
That was the final chapter of society considering me “gifted”, but it was just as well, I couldn’t imagine any greater success beyond getting a job and being able to afford my own apartment.
I'm curious how old you are?
Mostly because a lot of my personal interests/ability to self-develop was related to Internet access. (My parents made VERY QUESTIONABLE financial choices and opted to pay for Internet access instead of food or clothing so I might have been freezing and my clothes all had holes in them but I could go online to talk to other smart kids.)
Also because I remember me + my parents being sat down when I was in elementary school and having my options talked about. In middle school once I was proven to have programming and math aptitude during the dot com boom, educational experts came to us and discussed specific gifted learning options (including things like private schools, skipping grades, or even pulling me out of school altogether for private instruction). None of this was initiated by my parents - it was brought to us. This was in the 90s.
I was born in 1985, we got dialup around 1996 I think?
I did teach myself programming in the 90s, after my friend loaned me his floppy disk with all his QBASIC stuff. Then dabbled in PHP, MySQL, etc.
We had one computer programming class in high school and I never got to take it because I had too many other electives. I don’t think it would have done much for me by the time I could have taken it.
It never really occurred to me as a teen that I could use the internet for getting really good at academics or broader “self-development” - I guess I just cared about video games and making money. Parents’ attitude was as long as I was getting As and going to college they didn’t need to do anything.
This might be the first time in my life I’ve seen someone with a similar experience. As a big fish in a small pond, opportunities just present themself to you. Free summer camp that provides college credits? Going to national/state competitions just because? It’s all second nature once you’re ’that kid’. Even bullying goes away because everyone knows you have the ear of the teachers and administrators and/or wants your help on homework.
Of course you still hit the wall later. But I see all the reports of how terrible it is to be gifted and am so grateful that my experience was different.
You get away with so much, it's a terrible adjustment to be 'normal' after that. I still struggle frequently, and have to take a lot of steps not to come off as an arrogant prick. Luckily, I have a fair amount of charisma, and I used to be an attractive young woman, which conceal a lot of social sins, but it's still one hell of an adjustment.
If I'm honest, I never ran into an intellectual wall. I did choose a comparatively 'easier' path, but that was more because I had a wide breadth of interests and choosing something easier meant I'd have more time to indulge my various interests. I was still getting interviews for tenure track positions out of grad school and when I did try to work post-graduate school, my first position was at an Ivy where I was the only one on staff who didn't come from an Ivy League school. (I was too lazy/too absorbed in my own things to do what was required to go to one.)
I ended up disabled in my last semester of graduate school - the 'wall' in my case is my body being unable to accommodate the social/networking demands of an academic or high powered private research career rather than my running into a topic I felt was beyond me. Particularly combined with being on my own in a HCOL area as that lifestyle required: Doing all your life management on your own with no safety net along with running at that high of an intellectual level is near impossible when you have a severe disability. (I have MS.)
I've been 'stuck' intellectually once in my life, and it was the result of a medication we tried for symptom management, and I found the feeling horrifying, if I'm honest. It was the first time I'd run into a problem where I had to sit there and think and still couldn't come up with a way to proceed, versus running into a problem and just being too damn lazy to bother. (Being able to see what I would do to solve the problem is very different from being motiviated to do so.) Apparently, most people feel that way fairly often? It made me way more sympathetic to people who didn't like school or who don't like learning.
You…have a PhD and you never encountered a research problem you didn’t immediately know how to solve?
They mean a problem they didn't immediately think of something to try for.
Yes, this. And I don't have a PhD, I have a Master's. I'm not saying the wall doesn't exist - that's why I specified I chose an 'easy' path. I'm just saying in my case the wall wasn't intellectual.
The gifted programs we have liked most focus on depth over acceleration. Finding someone who can open the deeper views of things might be more supportive of his joy and longevity in the subject.
I guess how easy it is to do depends heavily on the district, but why not have him skip some math courses and leave extracurriculars for if he's really interested in it rather than just good at it? I ended up skipping three years of math by the end of high school, though I never did any club or competitions.
My youngest is not gifted in math. She's still in the top 1/3 of her class, through diligent study, repetition, and review. Over the last year she's gone from dreading to loving math. Please keep an open mind about your kiddo's interests and don't push too hard.
It really depends on how much your son wants to do math.
As you can imagine, there is a whole world of kids like your kid who love math and want to do nothing more than math.
If you're interested I can chat with you or recommend resources here if you decide to help your kid do more math.
You shouldn’t push him. You should encourage him.
If he likes to do math you make it available, if he would rather play with legos instead of doing math you let him do that in his free time.
You can encourage learning and problem solving without it having to be math, or pushing.
Accelerated advanced math at Purdue as a freshman flung me into that wall at high velocity. Nothing like a competitively graded class to make you hate a subject for life.
The 'experimental' accelerated calculus program run by Stephen Wolfram had a similar effect and is why I decry trying to learn CS using AI and also why I don't like Mathematica or Wolfram. Fuck that guy. I loved math more than almost anybody I knew before that.
As someone who tried to learn CS properly, as far as deep fundamentals, I was told by advisors that me being terrible at math would stop my career. I switched to a CIS degree, which at my local university was learning networking and Microsoft Office mostly. I dropped out of school and went into sales, while still having an interest in software. I started to pick up software development on my own, and found that I loved it even more without worrying about math. I ended up going to a "career" school, which would have turned out terribly, but I had a professor that taught all of the important programming and CS classes (there weren't too many CS classes, just fundamentals of OOP/data structures and algorithms).
All of this to say I have been writing software professionally since 2006, and while I do struggle with the thinking behind functional programming and math-heavy subjects like graphics programming, I have written lots of business software that has brought me personal satisfaction. I would really like to understand calculus better, but I'm not sure if it would actually do anything for my skills in programming. If math is holding you back, think about whether you need the full breadth of CS knowledge, or if you just enjoy writing software.
I became better at code organization, making code maintainable and simple enough to understand unless performance was an issue, and general people skills. I can understand why math and software are so close to each other, but at the same time, I don't think it needs to hold you back unless you really want to go into a topic that is deeply intertwined with math. It took me four times to get past pre-Calculus, and once I did, I realized that I just did not enjoy that type of math and didn't need it to build useful software (as in makes people's lives easier and/or generates profit for business) that I also find fun to create.
You think that's bad? :) I went to the University of Minnesota Talented Youth Mathematics Program and they assigned a fourth grader (me) about 50 hours of math homework a week.
It genuinely wasn't until I was in my mid twenties that I wanted to look at anything mathematical again :)
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