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Math for kids outside of the Calculus Sequence
- remich 3y agoDoes anyone have resources for learning some of these topics as an adult? Some of them I've never looked into, some I have but it's been a while, and I'd love to have a way to revisit them.
- mindcrime 3y agoI don't mean to sound glib, but for basically every subject on that list (and then some) there are multiple instances of video lecture series on the topic on Youtube. I could recommend some specific ones if I had my list handy, but it's a Google Doc and our work network blocks access to Google Docs / Google Drive / etc. Just try searching Youtube though, for "<subject name> lectures" and then sample what you find to see if you like the instructor's style. Along with that, there are those "Schaum's Guides" for all (or nearly all) of these subjects as well. Go through a lecture series on the topic and follow along in the corresponding Schaum's Guide and you should be able to make quite a bit of progress. Brilliant might also have some useful stuff. And lastly, check Alibris for used copies of older textbooks. A texbook that costs $150.00 for the "current" (in use) edition can often be found for like $10.00 for the previous or earlier edition, since there is no real demand for those anymore. And if you're not in a formal class, you don't need to care about the specifics of the edition. So save some money and find a nice cheap copy to use.
- kqr 3y agoRead lots of books. Reading just one book you'll walk away with gaps in your understanding you don't even know are there. A mentor can point them out for you and give you directions on how to close them, but absent that you have to approach everything from as many angles as you can and hope that you will fill in the gaps by sheer luck. (That's a coupon collector's problem for you!) I recently re-read and took extensive notes on de Finetti's Theory of Probability which I always recommend. But don't stick to just that.
- wheelinsupial 3y agoI've found using the search bar at the bottom of HN (or https://hn.algolia.com/ https://hn.algolia.com/) to usually give quite a few questions and random comments with resources.
- ivansavz 3y agoYou might want to check out my linear algebra book. It has standard LA topics, but also goes into more advanced topics in the last three chapters (e.g. probability theory and modular arithmetic). See extended preview here: https://minireference.com/static/excerpts/noBSLA_v2_preview.pdf https://minireference.com/static/excerpts/noBSLA_v2_preview.... If you think you might need a more basic book that focuses on a review of high school math topics for adults, then this book would be better: https://nobsmath.com https://nobsmath.com See extended preview here: https://minireference.com/static/excerpts/noBSmath_v5_preview.pdf https://minireference.com/static/excerpts/noBSmath_v5_previe... Both books written specifically with adult learners in mind.
- rahimnathwani 3y agoI recently started working through 'An Introduction to Statistical Learning with Applications in Python'. It's free to download here: https://www.statlearning.com/ https://www.statlearning.com/ Read the first couple of chapters, and see if it works for you. It even has a nice minimal intro to a Python, Numpy, Pandas and Matplotlib.
- masfuerte 3y agoI always liked the Martin Gardner books from his Scientific American columns. Some suitable for children are listed here: https://nrich.maths.org/books https://nrich.maths.org/books
- vmilner 3y agoThankfully when I mentioned some topics from a Gardner book when at school in the 80s my maths teacher was encouraging and didn’t squash me for “reading ahead”. Also the Uk libraries near me at the time seemed to have enlightened people putting Gardner books in them.
- armchairhacker 3y agoOne more anecdote: I took Discrete Math in high school because I was already taking Calculus, had space in my schedule, and didn’t want to take another elective like French or video production. Discrete Math was the “bottom-tier” class for people who weren’t interested in higher-level math and just needed to fulfill math requirements. Which is ironic because the stuff I learned there I’m using today much more than calculus or statistics.
- gumby 3y agoThis is totally true and there is so much more math that is accessible and fun to teens. The list in the blog post is great. We had fun doing some of this and I'm doing it now with my step kid who really likes math but also likes to be in class with his friend. I did encourage my own kid to take calculus in year 11 rather than the usual year 12 so that he would enjoy physics more. His physics teacher told me she changed a little so she would call out calculus applicability or offer some extra problems for him, not so much for him, she said (I think she didn't like him, actually) but because it was just more fun to have someone who understood the math. Interestingly he said he was the only calc student who actually wanted to be in the class (a dozen kids). He said the rest were there because their parents had been pushing them.
- sobriquet9 3y agoI don't get why calculus is considered to be so hard that a new education path around it should be built. Sure, if you do rigorous development from axioms up it might be, but that's not the way one should start. After all, we teach kids to count without number theory and Peano axioms. After skipping calculus, the article recommends to do statistics, probability, game theory, and mathematical finance. Yeah, right. You may Monte Carlo your way through some problems, but it would be really hard to get understanding of those subjects.
- iraqmtpizza 3y agoIt's considered hard because you are expected to intuit the theorem or process you need to use instead of being given an algorithm for choosing one which works for all cases. Integration by parts (and optionally integration by parts again) is a good example. I guess geometry and all math is like that. Or at least it's taught that way. But at least geometry is visual.
- sobriquet9 3y agoIt's only true for integration, and not even all of it. Differentiation can be done mechanically, and killer app of calculus (finding maximum) only requires differentiation.
- iraqmtpizza 3y agoI dropped out before vector calculus and differential equations but i'm confident they have literally nothing to do with making money in software development. in fact I would wager money that the more math you take during your computer science education the less money you make. say goodbye to startups. say hello to academia and top ramen
- sobriquet9 3y agoIf you are talking about CRUD, sure. But there is software in self-driving cars, 3D printers, drones, etc. that uses vector calculus and differential equations heavily. Same with applied statistics. For example, if your software has anything to do with quantitative models used in trading, it would be pretty hard to work on it without calculus. Even reading a paper on effectiveness of COVID-19 vaccines requires one to know what p value is and isn't, which in turn requires PDF and CDF (calculus again).
- sheepybloke 3y agoI think the context of mathematics is more important than what is taught. I took AP Calc with AP Physics, and I found I understood calc much better when it was in the context of basic physics. Instead of just teaching math, we need to put it in contexts that are more interesting to students that helps encourage them to learn.
- simscitizen 3y agoI ran into this issue. I took Calculus BC as a sophomore in high school and then ran out of math classes to take. So I basically didn't do math for two years until I took classes in college. By then my math skills had definitely atrophied and it definitely showed in various math and engineering courses that I took in college.
- injb 3y agoThis is a really good idea. I'd emphasize combinatorics and probability because these are actually the most useful. But I would add group theory too! For me this was the first time I realized that mathematics doesn't have to be all about numbers or geometry. Although, come to think of it, add geometry to the list too.
- threatofrain 3y agoGeometry is already in the Common Core. The article is complaining about how getting ahead can result in child boredom in class. The "synthetic" approach to Geometry is also something that undergrad largely abandons. Part of the problem of the bureaucracy of current schooling is that your child has to either be way ahead or way behind for people to consider an out-of-band adjustment.
- JJMcJ 3y agoThe very best schools can get kids into upper level or even college classes. One reason some families are willing to pay $35000/year to send their kids to the high end private schools, either boarding schools or elite day schools, e.g., https://en.wikipedia.org/wiki/Dalton_School https://en.wikipedia.org/wiki/Dalton_School.
- guidoism 3y agoI don’t think you necessarily need to be in an “elite” school. My high school allowed us to take classes at the local university.
- jjav 3y ago> One reason some families are willing to pay $35000/year to send their kids to the high end private schools Sadly that's the low end of private high schools. Many are over 50K and most are over 40K in silicon valley.
- JJMcJ 3y agoThat's a ten year old number. Not surprised it's way higher now.
- MissingAFew 3y agoI like this, enrichment not acceleration. Brilliant. Not everything is a race, there are plenty of topics to learn that are "off the beaten path".
- JJMcJ 3y ago> you as an adult get asked to “stop helping them get ahead of the class” Whether by parental training or self study, some teachers treat it almost as a crime to understand the material before the class starts. In college you can sometimes talk your way out of the Intro To XYZ classes and go directly to the upper level stuff. I had friends who did that, so taking senior/graduate seminars in history instead of sleeping through U. S. History 101, where you could get a B if you remembered Washington was president before Lincoln.
- emidln 3y ago>> you as an adult get asked to “stop helping them get ahead of the class” > Whether by parental training or self study, some teachers treat it almost as a crime to understand the material before the class starts. I remember a 7th grade math teacher confiscating my workbook when she found out I had done the entire workbook in the first week. I never really understood why. I spent most of that year telling the teacher she already had my homework and refused to do it again on principle. I ended up taking a 0 on every homework assignment and had a C in the class instead of an A+. Fuck that teacher. > In college you can sometimes talk your way out of the Intro To XYZ classes and go directly to the upper level stuff. I did this for first year history, computer science, and calc classes. Highly recommend giving it a try.
- JJMcJ 3y agoThe famous story of young C. F. Gauss adding 1 + 2 + . . . + 100 by noting the familiar formula. Brought it up to his teacher. Lucky for him the teacher recognized this was no ordinary pupil and contacted the Duke of Brunswick, who paid for his university education later. Today, he'd be lucky not to end up handcuffed in a squad car. That's another reason to pay 50K for a private school for your kids if you can afford it. Also, for getting into upper division classes, I wish I'd had the sense to do it, looking back.
- IIsi50MHz 3y agoWhereas I, not understanding the high school textbook or teacher explanations on some problem, but having reversed a procedure from the problems which had answers in the back of the book, was told one day: - It probably doesn't work for all problems in the set (but was given two chances to prove it from the unassigned problems and some the teacher made up on the spot). - Okay, but you can only use it on homework and still have to write every step; can't use it on quizes, tests, or extra credit because it takes up too much space and takes too long to grade. - If I could find a math symbol for the key operation within one day, I could use it ONE take-home extra credit that had been due that day. The key operation? For all positive N, f(N) = N+N-1...until N=zero. I couldn't find the requested symbo l. When I took my first uni maths class, I asked my prof. He answered and wrote it before I even finished describing: "Oh, sigma."
- mempko 3y agoWhat's great about this is that there is SO MUCH MORE MATH to learn outside the calculus pipeline. Reading this post and seeing the list made me realize, 'wow, I've heard and know a lot of this but there is so much here to learn'. Math enrichment is a lifelong process. Never stop!
- whoomp12342 3y agolove it
- rahimnathwani 3y ago"The key here is to look for problems that a kid can do without having to understand too much mathematical machinery." This is exactly what math competition problems are for, no? There are books available with past competition problems, e.g. from Math Kangaroo.
- zeroonetwothree 3y agoSort of. They kind of assume a different specific set of knowledge.
- chongli 3y agoMath competitions are fun for recreation but the topics they cover tend not to be very productive for educational purposes. Enrichment topics actually broaden a student’s mathematical foundation, preparing them for a career in pure mathematics much better than the “calculus path”, which is focused on applied math for science and engineering.
- zitterbewegung 3y agoList of items should have Linear Algebra. It is by far the most useful topic that could be done outside of the Calculus sequence because it can help you with Machine Learning / computer science in general and even quantum computing. (Quantum computing is not really as important now ).
- 1980phipsi 3y agoI would guess only a small fraction of students will do something in real life where they need linear algebra beyond the simple stuff taught in an Algebra 2 course.
- zitterbewegung 3y agoTrue though but the article does say alternative math classes and I don’t see why to not have an understanding of Linear Algebra when things like number theory and graph theory are on the list.
- davesque 3y agoAnother great topic is infinite set theory (not just naive set theory as the author mentioned) touching on the transfinite cardinals. Maybe a little advanced, but if you can just get to the part where you see how there are different "sizes" of infinity, it can help a lot to understand how infinity is not a number but instead more a way of describing trends in a mathematical process over time. Sort of adjacent to Calculus in the sense of wrangling with infinity, but decidedly different since it's dealing with discrete mathematics.
- kccqzy 3y agoCardinals are either boringly easy or impressively hard. Ordinals are more fun. But yes, let's get the kids started with the basics of cardinals.
- zeroonetwothree 3y agoI’ve always found this area of math a bit navel gazey but if you like it then you might like the book Surreal Numbers by Knuth
- davesque 3y agoI suppose it is. But, to me, that's actually kind of the point. I think a meta lesson of infinite set theory is that logic can lead to strange places. So a practitioner should think of logic as a way to get from point A to point B in conceptual space and not so much as a way of exploring what conclusions are reachable from point A. Actually maybe it's fine to do the latter, but just take what you discover with a grain of salt.
- gnicholas 3y agoWhen I was in high school, I chose to take standard-level IB math instead of higher-level IB math. The standard-level math focused on probability and combinatorics, whereas higher-level was focused on calculus (you had to take calculus first). I appreciate the opportunity to have learned about these other subjects, which are arguably more useful in everyday life (and I went on to be a lawyer and a startup founder, where advanced calculus would not have been especially useful). It also meant my schedule was somewhat easier, since the standard-level class was not as challenging, and I was more mature by the time I took calc AB as a senior. I'm glad my parents realized that getting as far down the calculus path as possible was not the only goal, and suggested this path for me.
- kqr 3y agoI've always been surprised that probability and statistics are not getting more stage time in school. It's one of the most practically useful areas of mathematics, and also one of the least intuitive so people with little training get it wrong all the time.
- goeiedaggoeie 3y agoMy 13 year old is doing some basic probability already at school, something I did not get exposure to until I was considerably older.
- lern_too_spel 3y agoTo learn many topics in statistics, you need calculus. The reverse is not true.
- kqr 3y agoYou mean to prove many theorems in statistics, right? You can learn and use plenty of statistics with very few weeks of calculus.
- lern_too_spel 3y ago
- esalman 3y agoOur 2 years (and 2 months) old is ahead of his class at the daycare because he knows his ABC's and 123s very well by now, and is able to form full sentences. We cannot just let him join the 3 year old class because he'll not fit physically. I am looking for ways to "enrich" his learning in some other ways.
- gnicholas 3y agoI've heard good things about Khan Academy's app for little kids. We didn't use it since our kid was older by the time we started, but we've gotten a lot of mileage out of the regular Khan Academy app, starting around age 5. Both are free!
- esalman 3y agoThe khan kids app is great! I basically steered him away from Cocomelon and towards khan kids and a few other learning apps.
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- carabiner 3y agoHave him play outside in the woods and streams and with other kids.
- esalman 3y agoCurrently sheltering in place due to Texas heat.
- zeroonetwothree 3y agoI wouldn’t worry about maximizing learning at that age, it’s much more about physical and social development. Being “ahead” at age 3 is basically irrelevant long term
- twiddling 3y ago
- ecshafer 3y agoSchools should honestly have two parallel tracks of math (which students should take at the same time). There should be the standard "analysis" mathematics of algebra, geometry, trigonometry, calculus. But there should also be a track of (what I will shorthand to) Discrete mathematics, which focuses on proofs, combinatorics/ probability, number theory, graphs, etc. Discrete Mathematics used to very much be a niche mathematical field, who needs to know about number theory!? But computers has really made these other fields incredibly important to modern society. Plus writing proofs is a good exercise for anyone.
- hdctambien 3y agoI don't think you'll find too many Math teachers that disagree with you. There are a lot of things that schools should do. I'm curious to think through what a second track like this would look like. Assuming the "normal" 8-12 track is: Algebra 1 -> Geometry -> Algebra 2 -> Trig/PreCalc -> Calculus I think you need Algebra 1... maybe I'm too stuck in the old ways.. but at some point you need to understand what a variable is and how to "solve for x". How to plot points, read and interpret a graph. Identify patterns in series of numbers, etc.. Call it what you want, but without the content of Algebra 1 you're going to have a hard time communicating ideas in the language of Mathematics. And these kids also have a Physics graduation requirement where they will need to at least solve f=ma. Geometry is usually the "proofs" class. You're only really learning geometry so you can write proofs. You could plug&play that with a Discrete Math/Sets/Boolean/Logic class. I think Geometry is conceptually easier to understand as a 14/15 year old because you can "see" that the proofs work. Truth tables are kind of visual, but still a little more abstract than triangles and rectangles. Combinatorics/Probability is already a half year course that's usually combined with the half year course of Statistics. I can see non-AP versions of this class split into two full year classes. I imagine this would be something like what you're thinking of: Algebra 1 -> Discrete Math -> Probability -> Statistics The only thing standing in the way of something like this is politicians (state boards of education) and startup costs. For example, the graduation requirements in Texas are "4 credits of Math including Algebra, Geometry, and Algebra 2" (and the content of those classes are explicitly laid out in the TEKS). And you would also need to buy new textbooks/curriculum... which is money that schools don't really have to spend.
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- a-dub 3y agocomputability proofs could be a fun one for kids.
- zoomablemind 3y agoWe are teaching math to kids "because it's important and is everywhere". Mostly kids believe us and do learn it in school... Yet do we use that much of the math in our lives ourselves? Do we even have that much chance to apply the approaches and solutions we learned? Most of the math is outsorced to devices and ready-made solutions. A lot of those internal details are too complex to EL5. So if a kid did really crack the math at his grade, then instead of expanding it further, I'd rather try to find ways to apply that knowledge in life. Spot the uses around, make uses in projects. See how that math is fused in physics, well, really around us.