Math for kids outside of the Calculus Sequence | Hacker News Reader

Math for kids outside of the Calculus Sequence

kidswholovemath.substack.com

111 pointssebg93 comments

When 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.

I'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.

My 13 year old is doing some basic probability already at school, something I did not get exposure to until I was considerably older.

To learn many topics in statistics, you need calculus. The reverse is not true.

You mean to prove many theorems in statistics, right? You can learn and use plenty of statistics with very few weeks of calculus.

If you can't prove something, even informally, you aren't doing math. Memorizing and applying magic formulas (especially in "math" and physics classes) is a big problem with the curriculum.

I agree. Algebra based Statistics is one of the things that makes people say "Math is a religion". There is a lot of "just trust me" in statistics when you don't know calculus.

I suppose you could switch Probability & Statistics to "Just Enough Calculus" & Statistics to make it a less religious experience... but that also sort of defeats the "nobody really uses Calculus" mentality.

How do you define a PDF without an integral? I guess you can work exclusively with the discrete PMF, but then you have no Guassians, which are central to probability and statistics. How do talk about non-linear transformations of random variables, for which you need a Jacobian?

You can explain density as mass with arbitrarily small steps, yes. The Gaussian is the limit of many small binomials. With some graphical demonstrations to go along with it, I've had much success getting people to fruitfully use these concepts.

You also don't need a Jacobian, just the general concept of "How much does this change when we adjust this parameter in this direction".

I'm not saying it's easier without calculus. By virtue of being more knowledge, the ability to use calculus will always be easier than not having it, all else equal.

My argument is about opportunity cost. In my experience, people can get really good intuitions and tools around probability and statistics in the same time that they would otherwise learn the fine details around integrals and Jacobians -- and they'll find much more use for it.

Obviously you’re right. But we are talking kids here. There is so much to learn in statistics at basic to intermediate level without algebra. Many social scientists or in the modern age data scientist go through life with cookbook statistics and some code examples. Sure, it lacks rigour, but intuition especially for kids breeds interest and then the subset that is mathematically inclined can go on to do statistics with all rigour.

I think it all depends on where you went to school and possibly who noticed you while you were there.

I took 2 years of Probability and Statistics in Highschool in 1998-99 and 1999-00 at a school with 200 students. At the 4000 students school I taught at a few years ago we offered AP Probability and Statistics (had have been for at least 10 years, but probably much longer than that). In both situations, you could (and many did) take Stats without Calc.

Most times when people say "schools should teach X", many schools are (and have been) doing it (taxes, car maintenance, carpentry, gardening). Just maybe not your school ... or maybe nobody told you that it was a possibility at your school... Or maybe it's not at your school, but it is offered at another school in your district...

Or maybe it's just not offered at your school. Because there is an AP exam associated with Stats, it is fairly easy to get the class made as long as there are students that want to take the class and enough teacher slots to accommodate that. If a school is understaffed in the math department and class sizes are nearing 40, then you probably won't find a Stats class there.

I followed the same path - it served me well in life, but man did my university assume when you were taking Calc 1 that you had already taken calculus in high school mess me up.

Ah… IB must’ve been why I found the discrete mathematics subject at university so much easier than my peers did - I’d already been exposed to a lot of it.

IB?

International Baccalaureate. It's like AP, but internationally accepted. To get the full diploma, you have to take three standard-level tests and three higher-level tests.

Schools 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.

I 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.

I agree with most of what you wrote, but it’s puzzling to me that you think that proofs are more of a feature of discrete math than continuous (analysis). My first introduction to proof-writing was in geometry, and a majority of proofs that I have read or written have been in analysis. But I was a math major in college.

My first introduction to proofs was also in high school (middle school?, It's been a while) geometry.

I also did proofs for the first time in Geometry. But the reason I think of proofs as more in Discrete, is that you usually get proofs immediately in Discrete via number theory, graphs, or things like induction. But you usually don't really get proofs in analysis until you take, what real analysis or complex analysis? So you typically have trig, precalc, calc 1/2/3, differential equation and partial differential equations that is just calculations. That's my thinking anyways.

List 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 ).

I 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.

True 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.

> 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.

>> 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.

I ended up taking a 0 on every homework assignment and had a C in the class instead of an A+. Fuck that teacher.

I'm curious why/if your parents didn't step in to correct the injustice.

I did something similar and caught flak for being dumb enough to intentionally get a bad grade

I would have done the same. The teacher being WrongTM does not mean I should have to redo all the coursework to get the already earned grade for It. In the described situation I know I would have been able to recruit support from multiple people to get the teacher to behave correctly. Fortunately, no teacher in my life would have been able to defend refusing to give credit for work done within the written academic standards.

Although, I did once transfer into a school whose headmaster refused to accept transcripts from another school. He claimed I couldn't have begun grammar school when I did, because my birthday was after his school's deadline (despite him acknowledging the other school's later deadline!), which to him meant that I 'must' have started a year later and therefore 'must' either go back one form or if we didn't like that, have my entire transcript rejected and repeat all forms. (Madness!)

Good read on this topic: https://www.amazon.com/Underground-History-American-Educatio...

The 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.

Whereas 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."

> In college you can sometimes talk your way out of the Intro To XYZ classes and go directly to the upper level stuff.

There's a formalized way to do this:

https://en.wikipedia.org/wiki/College_Level_Examination_Prog...

> The College Level Examination Program is a group of standardized tests created and administered by the College Board.[3] These tests assess college-level knowledge in thirty-six subject areas and provide a mechanism for earning college credits without taking college courses. They are administered at more than 1,700 sites (colleges, universities, and military installations) across the United States. There are about 2,900 colleges which grant CLEP credit.[4] Each institution awards credit to students who meet the college's minimum qualifying score for that exam, which is typically 50 to 60 out of a possible 80, but varies by site and exam.[5] These tests are useful for individuals who have obtained knowledge outside the classroom, such as through independent study, homeschooling, job experience, or cultural interaction; and for students schooled outside the United States.[6] They provide an opportunity to demonstrate proficiency in specific subject areas and bypass undergraduate coursework. Many take CLEP exams because of their convenience and lower cost (price varies by institution, though typically $89) compared to a semester of coursework for comparable credit.

CLEP is great! But it allows you to skip requirements.

My friend in the 400 level seminars was frighteningly ambitious and very very fast learner. I think he would have had a nervous breakdown sitting through History 101, Biology 101, etc etc.

My high school math ended with algebra. I placed out of geometry and trig and went into calculus my first semester in college. I was not prepared. Do not recommend.

I ended up going the opposite direction. My 11th grade trigonometry teacher forced me to repeat trig despite my having gotten a B. I was angry at the time! She did me a favor because by the time I got to Calculus in University it was very intuitive and I didn’t have much of a problem with it.

I had something similar at uni. Guidance plan had me goto Calculus without doing Trig, despite not having studied Trig in years. I had to withdraw from Calc twice.

Another 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.

I’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

I 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.

Cardinals are either boringly easy or impressively hard. Ordinals are more fun. But yes, let's get the kids started with the basics of cardinals.

I 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.

One 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.

"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.

Sort of. They kind of assume a different specific set of knowledge.

Math 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.

This 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.

This 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.

Geometry 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.

The 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.

Public schools can also grant children credits for classes taken in a local college.

And in CA, they are generally required to do so (though sometimes have to be sued to comply — see PAUSD's recent experience: https://padailypost.com/2023/03/25/lawsuit-sparks-debate-ove...).

Definitely do not need to pay thousands of dollars to get advanced instruction on math.

Many public schools allow students to partially enroll in a community college and take classes there. At the very least, if you take a modest amount of initiative, you can negotiate a way to allow for a mild deviation form the standard curriculum.

My daughter is a student in one of those elite schools. You are allowed to take one of those advanced classes in any year (i.e. Calculus as a freshman) but that doesn't mean that the school will teach it to you. If you take the class you either need to be a genius or use outside tutoring.

I don’t think you necessarily need to be in an “elite” school. My high school allowed us to take classes at the local university.

> 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.

That's a ten year old number.

Not surprised it's way higher now.

I 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.

It'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.

It'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.

I 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

If 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).

>self-driving cars, 3D printers, drones

The chance of working on one of these projects is small. The chance of working on the advanced algorithms themselves is even smaller. The chance of getting paid well to do so is 50/50 at best. On average it's a bad bet to go math heavy in computer science. But if it's what you love then go for it.

>quantitative models used in trading

Probably better off with a degree other than computer science here.

Calculus doesn't teach about p-hacking so it's probably not as useful as you think.

Calculus is not heavy math. It describes everyday physics - coordinate, velocity, acceleration, curvature. Stuff you can touch. And it describes unrelated abstract concepts like PDF and CDF, in a way you can intuitively understand by rearranging stacks of Lego bricks.

I agree that a degree in a quantitative field other than computer science is better. A CS degree is like saying "I can drive a car and use Microsoft Excel". You need to either be a really good driver, or be ready to work harder than many others to get ahead.

I don't care what it describes. Six advanced math courses is math heavy for computer science insofar as private sector employment is concerned. This is a descriptive fact about the world. Go nuts with it. We're talking statistics, differential calc, integral calc, vector calc, differential equations, discrete math. Especially when only one of the six (discrete math) is useful for making money, which is the yardstick I selected.

Curvature has literally nothing to do with making money, on average. Being able to describe physical reality at a fundamental level with specificity is not adaptive on the level of the individual. If you want to donate your time to humanity and eschew money, go ahead. But don't be sour grapes about your educational choices.

I always liked the Martin Gardner books from his Scientific American columns. Some suitable for children are listed here: https://nrich.maths.org/books

Thankfully 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.

Does 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.

I 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.

Read 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.

I've found using the search bar at the bottom of HN (or https://hn.algolia.com/) to usually give quite a few questions and random comments with resources.

You 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....

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 See extended preview here: https://minireference.com/static/excerpts/noBSmath_v5_previe...

Both books written specifically with adult learners in mind.

I recently started working through 'An Introduction to Statistical Learning with Applications in Python'.

It's free to download here: 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.

I like this, enrichment not acceleration. Brilliant. Not everything is a race, there are plenty of topics to learn that are "off the beaten path".

I 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.

Our 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.

I 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

< Being “ahead” at age 3 is basically irrelevant long term

Not if you want them to be FIRE at age 10

>> I am looking for ways to "enrich" his learning in some other ways.

You parent however you want, but instead of looking for an AP daycare how about let them expand as a toddler and then a child? Must be your first, because I got news for you, no two year old is "ahead of his class", except maybe in how big they are, or how much they eat, sleep and shit.

TBH his daycare teachers brought this up first, based on his interaction with kids older than him.

A comment above suggests infinite set theory and surreal numbers. ;)

I'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!

The khan kids app is great! I basically steered him away from Cocomelon and towards khan kids and a few other learning apps.

Have him play outside in the woods and streams and with other kids.

Currently sheltering in place due to Texas heat.

What'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!

We 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.

computability proofs could be a fun one for kids.

love it

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