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Calculus for Seven Year Olds [pdf] (mathman.biz)
318 points by barry-cotter 4 days ago | hide | past | favorite | 128 comments

Copyright 1991. This was probably making the rounds of homeschooling circles I hung out in.

We tend to do a sucky job of teaching math to kids. A lot of elementary school teachers are women who were encouraged to go into early education because they weren't doing so well in math and this is very much gender bias.

My ex husband wasn't good at math. No one told him he should spend his days teaching little kids because of it. He was a career soldier and he just got tutored in college (by me) for his math classes. He was just expected to work at it if it didn't come naturally and he did.

So we just culturally pass around this idea that math is scary and little kids can't learn it rather than going "Well, adults just need to up their game and, by god, explain it better."

My oldest was math phobic by the time I pulled him out of school to homeschool. My one and only goal for math was to get him over his phobia.

He loves physics. I hate calculus. Calculus makes sense to him and he has read calculus books for fun.

I was good at explaining the math to him and he learned lots of solid math concepts even though he likely has dyscalculia and isn't good at crunching the numbers. But I got the concepts through to him and that's more important.

He's what gets called "calculator dependent" in some circles.

> he likely has dyscalculia

Ironically enough, "someone who likes math but hates arithmetic" isn't a self-contradictory thing. It's actually a perfect description of a pure (that is, professional) mathematician!

Evidence: look no further than the infamous "Grothendieck prime" 51....

Erdós is described in Hoffman's biography ('The Man who Loved only Numbers') - I can't recall, nor immediately find by whom - as only being able to count up three numbers: '1, 2, n'.

Reminds me of the "zero one infinity rule"[0] rule for programming. Roughly stated it says that software should be designed around 3 cases: 0/no instance of a situation, 1/single instance, or an infinite/limitless instances.

[0] https://en.m.wikipedia.org/wiki/Zero_one_infinity_rule

I have a degree in maths and I suck at arithmetic. Understanding abstract mathematical concepts has little to do with speed and accuracy of thought and more to do with visualisation. But then again you are much better off with both, I often struggled because I could see how to get to the solution but struggled moving the symbols around accurately and fast enough to get to the correct soln in the required time limit.

That would explain a lot. I am the opposite. Do well at the math that requires number crunching but terrible at things like geometry where I have to visualize stuff. Probably because of aphantasia.

I would say visualisation is great to understand those mathematical concepts that can be visualised well. There are many mathematical concepts for which this is not true. Logic comes to mind.

It's not really important, but the "Grothendieck prime" is 57 according to Wikipedia.


True, 51 is "exmadscientist prime".

<boldly updates wikipedia>

...I'll take it.

But, hey, I probably won't make that mistake again. Probably.

> Ironically enough, "someone who likes math but hates arithmetic" isn't a self-contradictory thing.

I didn't even realise anyone would expect there to be a relationship between "liking" arithmetic and actual mathematics. I worked with mathematicians and was called one myself (it was a stretch, I think) but I have no idea who in the lab was or was not proficient in doing numeric calculations by hand. It was just not relevant. There was one guy though who liked to show off doing things like multiplying long numbers in his head, which was amusing.

It would suck to lack the ability to manipulate algebraic expressions efficiently, but you could go quite far without it in many branches. My most interesting results were identifying certain topological features of certain discrete structures of relevance to biology. There was no arithmetic or algebra involved at all except to write down the results formally.

I was in a bit of a weird half-rate math competion team/club where some people who were there to get out of history class and some people who were very much the opposite and were some of the smartest people I've ever talked to. One thing I noticed is that all the power competitors had good calculation ability, but whereas some people were just really fast calculators and, like your lab friend, could calculate huge multiples in their head, others were actually subpar at that type of mental math and instead were just familiar with, for instance, all the perfect squares/cubes/powers of 5 up to 20 and beyond because they've just encountered them a lot.

When i was in university i did a math minor. I definitely remember the people who were math majors tended to be fairly bad at simple arithmatic (or at least just very average)

Which may be because you simply can't do everything good without practice and in pure math, well, there are things you don't do as much. When I go back to the 3rd and 4th degree polynoms we solved back in school, it's a hassle. Back then I was quite good at solving them but nowadays it just takes time to get accustomed to these kind of problems.

"A number less than 100 is prime if it sounds prime, and isn't 51 or 91."

(To me, 57 is too obviously 3 less than the obviously-divisible-by-3 60.)

I remember the moment in high school physics where I had an epiphany that the velocity equations the class was memorizing looked a lot like derivatives. We just happened to be covering derivatives in calculus at the time and my personal discovery meant I no longer had to memorize the velocity equations (I hated memorizing stuff)!

I brought up my discovery with my calculus prof later and he was so pleased for me (of course it was obvious to him). It's these small victories / "click moments" that grew my love of math and I hope to pass them on to my kids through their own discovery.

Maybe it's just me. But when I get the feeling like I just beat the system, even in a really small way, it's a rush.

Women has long been the "secret sacrifice zone" of domestic economics. Or, as Connelly put it, "woman is the slave of the slave."

> we just culturally pass around this idea that math is scary

There are thriving zones of anti-intellectualism in countries that have just one simple excuse for cultivating primitive thinking: it's lucrative to exploit.

Math, reasoning, debate, science, and democracy itself constitute educated civilisation. Everything else is a collection of stale bread and mournful circuses.

Wow that's a really big claim. Is there any research or reason you claim elementary teachers, specifically women, were pushed into the job (or to teach lower grade) because they aren't as good at math?!

That's a big statement.

Without research it's pretty absurd and comes off to me as sexist.

Though I think I get maybe the intention that sexist ideas and gendering subjects has created an education gap.

The teachers i know do it because it's their passion and many have post grad degrees and advanced training despite not being paid enough to cover the loans without income based repayment..

were pushed into the job (or to teach lower grade) because they aren't as good at math?!

That's not at all what I said. No one pushes them into teaching. It gets framed as "Well, if you can't hack the math, you have an easy out. You can just go teach children. That doesn't require a strong math background."

Men get discouraged from teaching small children and get stigmatized as "possibly a child molester." Women get the message, implicitly or explicitly, that caring for and teaching young kids is what we are best suited for.

It is generally true that women fairly often get told, explicitly or implicitly, that they don't need to really have career ambition because it's really a man's responsibility to provide for the family. The teaching profession is known to be family friendly and is often considered an ideal choice for a young woman who wants a career and a family.

Women are given an "easier" answer for their life than making it on their own. They can marry well (and never mind that this leaves a lot of women out in the cold these days). In contrast, men are generally told they need to be good earners to have any hope of marriage.

I don't have any sources at my fingertips and a lot of this is common knowledge, so I don't think I need to back it up. I became aware of this as a trend at universities when I was involved with The TAG Project* years ago and regularly spoke with people in the education field or who had gone to college and women got very different messaging from counselors and staff if they were struggling with certain subjects than men typically got.

We still have a huge gender disparity in who teaches what grade levels and in how comfortable most women are with STEM subjects, so I don't have any reason to believe any of this has fundamentally changed. Judging by the stats for how many men you find in early childhood education versus women, it's clear that society as a whole still shuttles women in that direction by some mechanism and men in another direction.

* http://www.tagfam.org/

Anecdotally and from my personal experience, this is unfortunately accurate. I would add two points to this depiction of scholastic life:

Teaching is not a highly regarded position in US society in general (I.e. those who cannot do, teach).

There is a trend within the US and UK to view STEM subjects negatively. The derogatory “nerd” stereotype is not so prevalent among other cultures. Growing up in the US, it was very clear that you could be either attractive/well-liked or good at math. For those that happened to possess both features, discussions would come from the angle that the person is attractive/well-liked in spite of being good at STEM. The underlying premise seemed to be that you had to extra good-looking to be labeled as attractive if you were also to do well in math.

I'm not sure I'd call it a "trend" (since there seems to be a concerted, if not obviously successful effort now to get more kids interested in STEM in the US at least), but more of a long-running thread of anti-intellectualism in general.

> Judging by the stats for how many men you find in early childhood education versus women, it's clear that society as a whole still shuttles women in that direction by some mechanism and men in another direction.

Or perhaps, women, in average, have a higher natural predisposition to take care of young children than men.

That would not be a surprise, since women get pregnant and, as far as I know, generally took care of children while men went out to hunt and fight.

Of course, nobody should be pressured to choose a profession based on a stereotype.

Or perhaps, women, in average, have a higher natural predisposition to take care of young children than men.

A few quotes from the below article:

From colonial times and into the early decades of the 19th century, most teachers were men.

1840s: Feminization Begins

The reformers argued that women were by nature nurturing and maternal, as well as of high moral character.


In contrast, early programmers were mostly women and it is now a male dominated field. There is a lot of research and we are quite confident that trends of this sort in various fields are driven by societal factors, and never mind how much people of a particular era try to come up with various justifications for how one gender is supposedly innately more predisposed to X for some reason.

I wonder if maternal instincts might factor in here too. There was a hacker news article not too long ago about "Despite social pressure, boys and girls still prefer gender-typical toys"


Please remember to assume good faith by taking a charitable interpretation of people's words.

For example, we can see reference to the broader issue of gender bias rather than so narrowly focus on a specific claim about elementary teachers. To that point, there is substantive and substantial evidence of gender bias in STEM:

Why So Few? Women in Science, Technology, Engineering, and Mathematics


Thanks for the reminder. Yeah I definitely called that out in my comment. But my point still stands a lot of this comment thread has a, charitably put it, strongly male point of view.

> We tend to do a sucky job of teaching math to kids. A lot of elementary school teachers are women who were encouraged to go into early education because they weren't doing so well in math and this is very much gender bias.

While this may be true and in general is in line with my experience, it might be better for you to reframe it in a more indirect way because sooner or later you'll find someone offended.

I appear to be the highest ranked openly female member on HN. I can suck up a few downvotes from people who don't like me or don't like what I am saying and that, in fact, seems to be par for the course here.

Thank you for otherwise validating my position, but I'm not going to soften it. If the truth about our world bothers people, they are free to make something else true instead of asking me to cater to their feelings at the expense of stating the truth.


I will add I am unwilling to back down on this point because we are actively giving young women a math phobia and crushing their dreams with it instead of telling them to get a tutor and work for what they want and we are doing so in a manner that causes them to pass their math phobia onto future generations.

This isn't just a case of "sexism sucks." We are poisoning our entire society with this garbage and it negatively impacts huge numbers of people.

Or we could not pre-emptively self-censor on the basis of imagined offense that doesn't even exist yet.

Besides, if something really is offensive, saying the same thing just indirectly is rather cowardly. Being offensive in an indirect manner is still being offensive, just less people might notice. Any opinion worth having is worth having openly.

I'm not commenting on the original quote because I'm not sure I agree with it, but worrying about offending people is destroying the fabric of our society.

By not talking openly and freely, we're keeping our thoughts and feelings inside, which brings emotions to boil. There's no release valve. We can't vent, and because we have less honest discourse with others whose opinions differ, we become internally polarized against different opinions.

By preemptively taking offense on the behalf of others, we're imagining potential victims and dulling our discourse. Over time it erodes critical thinking skills, because everything is encoded in layers of indirection. The meaning is dropped in favor of a soft tone. We begin speaking in hardcoded memes.

Taking offense to words flies in the face of our free speech liberties, individualism, and freedom. We're putting up barriers that don't exist. If we keep it up, we'll soon find ourselves living under the oppression of self-imposed thoughtcrime.

There's something to be said about having a thick skin and being open to those you disagree with.

(I'm liberal, btw.)

I completely agree with you (and with other replies to my comment). It's just saw people's lives ruined for much less than that. Heck, storms happened over not what they actually said, but what people thought they said. Even being female and holding progressive views doesn't offer much protection - I saw feminists attacking other feminists using strange invented terms like TERF or SWERF.

For example, you may well have an opinion as to whether people identifying as females but but being biologically male should compete in sports with other females, use women's restrooms and changing rooms etc., but you better be careful when you decide to express this opinion outside a circle of close friends - especially if you have a family to feed.

I was homeless for nearly six years, along with my two adult sons for whom I was financially responsible. I am absolutely not a stranger to the fact that the world is perfectly happy to watch me starve while going "La la la not listening" to my observations that my gender is a factor in my intractable poverty.

This is precisely why I am unwilling to back down on some things.

If it is any comfort, I took your initial comment as well-meaning and intended to be helpful. It's just not a position I can embrace.

There comes a point past which politely going along to get along is part of the problem, not part of some kind of solution.

I have a strong track record of leaving nuanced comments on HN asserting that "No, it is not as simple as sexism and intentional exclusion of women in X case. Here is the data and women make choices." etc.

But in this case, I am quite confident that women not only get actively encouraged to go into early childhood education, they get actively advised that this is what they should do if their initial desired major has a strong math component and they are having trouble with the math and this is not done to men.

My oldest son ended up math phobic because his teachers couldn't teach it and brought a lot of baggage to the table. So I have seen firsthand what it can do to a child and that child doesn't have to be a girl to be negatively impacted by the emotional baggage that so many elementary school teachers have over the subject of math.

It can be a boy whose relationship to math is being egregiously harmed long term because of so many elementary school teachers having had their own dreams crushed on the excuse that "math is too hard for you" rather than being told "You can audit the class. You can get a tutor. You have to pass it, but it's fine if you only get a C as long your overall GPA is high enough. etc."

It doesn't just break women to have this done to women. It is breaking our society.

So if men value math and STEM, this is a practice they should be concerned about as well. It isn't just a women's issue by any stretch of the imagination.

I'm not angry at you. I'm not offended at your attempt to look out for my welfare in a cold, cruel world.

It's just the wrong answer in this case and I've basically crawled naked across broken glass for nearly twelve years on HN to claim my right to speak my mind to some degree here and I am going to exercise that right on details of this sort.

Your account is only 4 years old ;-P

But seriously, +34k karma in four years is nuts

Doreen has been around on HN for a lot longer than this account. She's definitely one of the OGs. I would believe 12 years, that's about how long I've been here and I don't remember a time when she was not a member.

I guess the broken glass is dispersed over another account.

The irony of a tech person calling acronyms strange!

Not all of them, just these, created just to label someone whose views some people don't like. It is "strange" for me, because it's heavily loaded, it includes a false premise (that the victim is transphobic), and is used towards people who have nothing to not only with "radical" feminism (whatever this would mean), but any feminism at all.

All terms are invented.

Anything can offend someone. It's the nature of seeking truth. Conclusions have to be challenged in order to be verified and assumed true.

If I'd be in a conversation of 3 people, and one might be offended, that's 50% of people I have to be cautious with. If it would be an audience of 1000 people, 1 in 1000 could be offended... Should we stop speaking altogether?

I think the current (quite absurd) situation is a function of three factors: 1) how easily people get offended these days, 2) how easily accusations can spread thanks to network effects, e.g. on Twitter, 3) how much damage can be done by people fixed on causing you maximum damage legally, e.g. calling your employer etc.

I've taught 7-year olds, rich kids specifically in technology magnet program.

The pre-requisite information necessary for this would be way above their heads. Adding fractions, conceptualizing fractions to graphs, graphing functions... and that's just the first part!

Most 7-year olds can barely conceptualize fractions. First/second grade math curriculums are centered around basic multiplication, shape recognition and counting.

7-year olds aside, I don't even think most of the high-schooler freshmen I've taught would be able to grasp this content.

Most 7-year olds can barely conceptualize fractions

When my oldest was having trouble with this at about that age, I pulled a pie out of the fridge and began cutting it up.

"This is one pie cut into two pieces. This is half of the pie. You write this as 1/2.

This is one pie cut into four pieces. This is one fourth or one quarter of the pie. You write this as 1/4. Notice how the pieces are smaller than when it was 1/2.

This is one pie cut into eight pieces. These are eighths. You write this as 1/8. Notice how the pieces keep getting smaller as the number on the bottom gets bigger."

He never had a problem conceptualizing it again. It made complete and perfect sense why a larger number on the bottom meant it was a smaller amount.

This is how we did it, with Cuisenaire rods in the 1960s. For the simple fraction cases of the time, 1/3 1/4 1/5 2/3 &c it was perfect. Nobody seemed to struggle with the concepts here.

What freaked me out was .decimals which I totally did not get for a long time. the scale function effect of 0.001 defeated me for years. how can it be 1/1000th, when 1000 has THREE ZEROS

> the scale function effect of 0.001 defeated me for years. how can it be 1/1000th, when 1000 has THREE ZEROS

I remember getting tripped up on that all the time. Still do sometimes, having to slow down and "remember" that fewer 0s are needed on the left vs. the right (1000 <-> .001)

I wonder if—back when this notation was "invented"—the decimal or some other mark had been placed above or below the ones place, if this would be easier.

So it would be something like 1000̲ and 0̲001. Now both have the 3 zeros, and it's not optional whether to include the first 0 in the fractional case.

The inverse of 8̲ is 0̲125, the inverse of 50̲ is 0̲02, and half of 5̲ is 2̲5.

The ones place is the only position that doesn't have an inverse position. Tens is tenths, hundreds is hundreths, but one is unity. It's the "pivot point".

I like this idea, and now wonder why the radix point is placed where it is. I assume it has something to do with integers being invented first?

Haha this is painfully relatable. I still never know whether to count the starting 0 in decimals so I get confused reading numbers under 1e-3. Somehow I got a job where I get paid to do math.

Congratulations! You got a talented kid. It didn't mean the method would work if a kid was not ready, though. I did exactly the same thing to other 7-year olds, using pizza, using cereal, using number lines, and using shapes. None worked.

IIRC there's some research that suggests maths education before 8th grade doesn't really do a whole lot long-term. I doubt that'd be true in a perfect world with personalized maths education for everyone, but I'd buy that waiting until a kid's capable of conceptualizing stuff rather than trying to push them when they're not ready really cuts down on learned helplessness.

ETA: On the other hand, I'm not sure how I feel about that framing. I hated it as a kid when I asked about something and an adult wouldn't even try to explain it to me, or point me in the right direction to learn, and just say I wasn't ready to learn that yet.

This is certainly how the school my children goes to teaches it (after all the home education recently I now have a pretty good idea how they teach!) And yet, I get the impression that most of the children still find the concept of fractions hard.

Mine (8 and 10) do have that conceptual understanding but that doesn't mean that they are totally fluent with adding, dividing fractions etc. yet despite both being at the top of their years at school in maths. I reckon 99% of 7-year olds would definitely struggle with those sheets. I think my 10 year old would enjoy them (he likes maths a great deal!) but that my 8 year old wouldn't.

On the other hand I think it's good to have high expectations of what children are capable of understanding mathematically - they often surprise you!

Sometimes the problem is the notation itself(divided by symbol/fraction doesn't have any meaning to kids yet, and there are at least 4 different ways you can write divided by or fraction of). It might be easier to start off by writing "1 of 8 pieces" and then teach the shorthand 1/8 and then show the connection to division. Basically it's like learning APL for the first time

Edit: typo

It's also excessively common for a teacher with twenty to forty students to more or less expect kids to all learn the same way, pick it up from the book and written exercises, and not really need an explanation. My son ran into situations where he said he didn't understand what the math book said and asked the teacher to explain it to him and the teacher picked up the book and read to him the exact thing he wanted her to explain to him some other way because he hadn't understood it.

Another thing that happens these days is that a game of telephone has gone from "there are lots of valid ways to understand things, so don't punish kids who get the right answer with a different method than the one and only way you've been told to teach" to "teach every kid a half dozen different analogies and algorithms for every concept and punish them if they don't know which one you want them to use."

I was literally thought fractions like this in school, except the cake was drawn, I'm having trouble imagining a different solution :)

My oldest really struggled with math but really wanted to understand. I made things as physical as possible for him.

I also pulled out building blocks and demonstrated the concept of cubing a number or squaring a number. We did a lot of hands-on things to show him math in the real world.

I think I used CD cases to show him "times" and explained that "times tables" were originally people doing tile counting just like this and how if it was "4x2" it was 4 down one side and two down the other and it adds up to eight and you can count them.

I demonstrated for him that multiplication was shorthand for lots of addition and exponents were shorthand for lots of multiplication and learning this stuff would make your life easier, not harder, because it was a whole lot of shortcuts for stuffing lots of numbers into a small amount of writing.

He was so mad when I finally explained to him that "algebra is just them calling the blank space in your equation X so you can move it around." He was fuming and I don't think he has ever gotten over being mad because his response to that was "That's it! They made that seem hard?! I've been doing that in my head to calculate damage in video games for years!!!!!!" and he stomped off furiously.

He wasn't that old. He was in elementary school at the time and he's really bad at math.

Dragonbox algebra uses a shy dragon in a box for this concept: https://dragonbox.com/products/algebra-5 . I noticed also that a teacher admitting to be bad at math is seen as normal. As a parent, your really need to be careful and support your child in math more so that in other subjects.

The usual problem with this pedagogy is that kids get confused when presented with improper fractions (eg. 11/8), and so on. I guess this is probably still a good intuition to start with, but how did you tackle those extensions later on?

I don't recall that coming up as an issue for him. I don't see why that would be a problem. You can just explain that 8/8 is one whole, leaving 3/8. I don't see why that would be hard to demonstrate.

"These are two pies. Let me cut them into eight pieces each. Let's count to eleven pieces. That's one whole pie plus 3 parts of the second pie, so it's the same as 1 3/8."

"but we cut 16 slices this time so isn't it 11/16"

Three additional pizza slices automatically appear in my mind when you mention this. I found it more difficult to explain why the multiplication of two negative numbers gives a positive one.

Hm, that's a good one, I'm struggling to explain it to myself now!

My first thought is area, if you imagine four quadrants, draw a rectangle with side lengths positive from the origin, it's top right and multiplying the lengths gives you the area. If you instead take both sides negative then it's bottom left, but the area is the same.

However.. it's also the same if only one side extends negatively, so this is not at all satisfying.

(If that's made it even more confusing, the misleading error there is that side lengths are multiplied to give area, not positions on an axis relative to origin - the rectangle centred at the origin also has the same area.)

It sometimes gets easier if you first define a negative number as an "opposite positive." You can then engage in a discussion of what the opposite of an opposite is. Ultimately though, finding multiple approaches for the teaching the same concept is going to serve you better than locking in one specific analogy.

There's no one right way to teach math (or anything, really) because teaching is a dynamic relationship between teacher and student that is highly contextual and subjective. That, I think, is one of the underlying issues with standardization in schools. Something that works for 80% of kids will unfortunately fail that remaining 20%.

First, accept that a negative is the inverse, or opposite of something.

We can intuitively understand that multiplying slices by a negative inverts them - turns them from slices given into slices taken away. (Distribute the multiplication into addition if necessary to drive the point home.)

So it follows that multiplying negative slices by a negative inverts them again, turning them back into slices given.

It’s not a convenient visual, but if a negative slice represents the “taking away” of a slice, then a negative negative takes away the “taking away.”

The identical cake example is used in the first chapter of this book.

7-year olds seams a bit young but Jason Roberts from Techzing through Math Academy has lead a program that got 10-year olds to get through calculus.



He told through the years his progress in his podcast (among other interesting things, like being one of the first developers at Uber and other fascinating stuff) https://techzinglive.com/

In eastern europe the normal college-track educational path was to get a general education up until about thirteen which would cover the basics (what today we would consider a GED) and then you attend a gymnasium that focuses on a specific subject, for example music, or engineering, or foreign languages, etc. A thesis or project can be required to graduate from the subject which is then used as an entrance examination for a (three year) university, at which point you have what in the U.S. would be a strong masters degree. In the math gymnasiums, you not only learn calculus but also basics of abstract algebra, differential equations, topology, etc. Not advanced stuff, but basic stuff. But stuff that in the U.S. system is reserved for college, here it is taught in gymnasium (what used to be "prep" high school in the U.S.)

When you are young, that is the best time for you to learn these subjects. The only limitation is your ability to focus. For those who are young and are able to focus, they can do amazing things whether it is learning to play instruments, learning math, learning foreign languages, or learning any kind of trade. And historically pre-teens spent a lot of time learning, except it was with the family or another one-on-one apprenticeship scenario, so that by the time they reached teenager status they were already able to do useful stuff to support themselves and even got married. In modern societies, we tend to stretch education out farther and farther and simultaneously assume the young aren't able to learn as well as adults, when the opposite is true. Moreover we think that in the past, there was just less knowledge to absorb, but it is really the modern world in which de-skilling and mass production has made it so that a large proportion of the population is doing boring, repetitive, unskilled work, whereas in the past whether you were a baker or bricklayer or an organ player or a farmer you were doing skilled work, and you generally managed to learn the basics of that work before you became a teenager.

I'm teaching my 9yo son to take baby steps in algebra and I can totally relate to your comment. When he encountered a variable "X" for the first time he was completely flabbergasted. I tried several examples but it's just plain difficult for a human mind to go from numbers to an abstract concept of a "place holder". It's a big leap of faith to go from "a specific number" to "a place holder that you later fill up with". I asked him to divide "5 * A" by "5" and he was like "I don't know what A is how do I divide?".

A related challenge is to translate descriptions into mathematical equations. Things like "Bob ate 5 times as many berries as Sue". Funny thing is in his head he works out and comes up with the correct answer. But when I ask him to go through the process it's totally alien to him. The only way I think is to make him practice with as many such challenges as possible.

Now when I look back 30 years ago, I recall the struggle I went through to even understand the meaning of "X + 2 = 3" let alone do the magic of moving 2 to the right and change its sign.

When due to lockdown I was doing some homeschooling and giving some exercises to my 7yo, I was also anxious to introduce variables; the magic "x" that scares so many.

I ended up drawing squares or triangles instead. Maybe they scared me less, but he really seemed to grok it. When I asked and what's the value of "this", he'd just give the answer without knowing something about equivalent transformations and the stuff.

Made me wonder if math should maybe be taught more intuitively, not so much about the formal stuff, which in the end is pretty useless for our minds if we don't have any intuitive grasp of it.

I'm fairly certain that's what Common Core was trying to do... And parents around the US freaked out because it was different than how they learned math and states like Texas made it illegal...

Start with an empty box of instead of x. I have found that an empty box transfers to them the concept of placeholder that needs to be filled. it can transition to a x later

You can also compare to words like it, thing, she, he, and equations to definite descriptions.

"I don't know who she is."

"The culprit was wearing a red coat."

"The mystery number plus 3 is equal to 5."

"I'm thinking of a number, but I'm not going to tell you what it is. But I will tell you that if I multiply it by 3, and then subtract 5, I get 1."

I remember how frustrating it was to do this.

For me it helped to just be really explicit with it:

X + 2 = 3 X + 2 - 2 = 3 - 2 X + 0 = 1 X = 1

That basically unlocked it for me. I was older than 9 though :). For division I'd probably just say that you actually have (1/5)5A = (5/5)A = 1A, but that probably necessitates symbolic juggling to be something you already have as a skill.

Anyway, symbolic manipulation is interesting, even without letters.

> Most 7-year olds can barely conceptualize fractions.

I've found the most extraordinary thing with my kids. If I ask them what is a third of 12, they'll scratch their heads. But if I ask them how they would divide 12 bits of chocolate between them and me, they'll give me an answer immediately.

It even works when there is a remainder.

Just teach fractions and division with chocolate.

Yeah, there's a bunch of research showing that people learn better with concrete examples of mathematical problems first, rather than the abstract conceptualisations which tend to be more useful to us.

This was in college 15+ years ago, so I don't have any citations unfortunately.

The introductory section talks about a 7 year old working on the quadratic formula. What?? At 7!

“Calculus for 7 Year Olds (Who Are Geniuses)” should be the title.

7 might be a bit on the young side, but it's well known that there are significant differences in how math is taught in different countries; there's some interesting US vs Russia/former USSR maths discussion here:


Totally agreed with you. I also taught 7-year old. Fewer than 10% of them would truly understand fraction means ratio. Heck, many of them had a hard time understanding what the word "ratio" means. Even fewer would understand what decimal numbers are or why they exist. I also saw how Russian School of Maths taught 7-year old. They would use an entire year in their honor class to introduce the concept of variable, first using analogies, then using terms like "rules", and eventually formalizing the concept.

> Even fewer would understand what decimal numbers are or why they exist.

Heck, even I (40+ yo, software professional for ~16 years) only recently realised the importance of the decimal "dot" notation and its relation to the positional number system while I was teaching my son.

And then the importance of "zero" dawned on me. All these years I never appreciated what was all the fuss about the number zero.

I think a problem for many people with fractions is that there are many different ways to think of them. Numbers, slices of a pie, divisions, ratios, scales, quantities of non-whole units. This is a benefit: you can think of them in the most convenient way for what you are doing at a given moment. But I think it confuses people.

I wonder if it would be helpful to ban the division sign ÷. Often when I ask a young person what is 8/4 they will say 2/1 (great, they know how to simply fractions!). But 8 divided by 4 goes exactly, and you can just do that.

First present division of natural numbers with natural number answers as e.g. 18/6 = 3.

Then present 1/2 ???!!!

But wait, one object split equally between two people means each person gets half of the object. So 1/2 is a half.

There’s a wide range though. I know a six year old who can do things like prove the law of cosines or the chain rule. He’s not just regurgitating this stuff he actually understands it. Blows my mind.

Maybe they’re implying 7 year old geniuses who could read by the age of 3 and speak 4 languages by the age of 4. Anyhow, I agree that it’s not for the average 7 year old but it’s a cool book nonetheless

> Maybe they’re implying 7 year old geniuses who could read by the age of 3 and speak 4 languages by the age of 4.

Speaking four languages by the age of 4 is just ordinary development, assuming you're in an area where all four are spoken. You should be speaking them before the age of 2.

> You should be speaking them before the age of 2.

"Speaking them". 2 years old are not fluent in any language. Sure, they speak but they vocabulary is extremely limited they can't even make themselves understood most of the time which ends up in them being incredibly frustrated.

If you had nothing to do all day except play and were surrounded by adults speaking four different languages 24/7, you'd easily pick up all four to a basic child level within four years.

Languages aren't actually hard. They're only hard when you're trying to fit them into the many other demands an adult has on their time and because you're expected to have at least a minimal adult vocabulary, which is much broader than a child's.

The one element that gets harder with age is learning to control your mouth muscles to make unfamiliar phonemes. Learning to make new sounds with native fluency is harder at 40 than it is at 4.

If you want to do this properly you'll have to hire a voice coach who specialises in native intonation and can take you through a detailed physical retraining program. This isn't something most language courses/teachers can give you.

Full immersion works too, ideally with a native partner. But it takes longer and you'll usually still some have accent left.

My son speaks 2 at 3 years old. I’m amazed that is possible though, yes, it is an ordinary development in a bilingual household.

I agree that's just standard but parent said they speak 4 languages before 2. That's total and complete BS.

I have a 2 year old and I speak with her in one language the mom in another and me and the mom speak in a third language between us.

She knows words in the 3 languages and can construct phrases in 1 or 2 of those languages.

She is is also much more developed verbally than any of the kids her age we meet regularly but she definitely doesn't speak 3 languages.

Is that an indictment on the educational system and broader society or a comment that this material is only appropriate for gifted children?

My 5 year old was able to pick up fractions pretty easily. We always talk about portions of objects in real life whenever the opportunity arises, and this game helped a lot as well:


I think your ideas about what a certain age can do are heavily skewed by the overall state of your area's education.

100%. As an experiment, one could look up a sample curriculum for preschool in different areas. The divide is massive.

A quick google search using US UK and Russia shows this very nicely.

You're absolutely correct, but it proves my point.

I taught in a state that's consistently rated in the top 5 for public education. The specific institution I worked at was a private school with admissions tests well above the public curriculum.

When I worked in lower grades, the children were not particularly gifted, but they were among the top for their age/grade in the US.

I don't know how I feel about this. I've always been good at math, and was one of five kids (out of class of 120) that took Calculus as a senior in high school in the early 1990's. To even take Calculus as a senior, you had to have taken four years of math in the previous three years. I was even the HS Math Champion" for my state.

However, when I took Calculus in High School, it was taught similarly to this. Lots and lots and lots of math foundations, infinite series, areas under curves, and I was able to trudge through it but I didn't really grasp what it all meant or why I was doing it. Finally, about 2/3 of the way through the year, I was working on a problem and had an "oh, shit, this is just about how things are changing". It was a Eureka moment, and entirely based on such a simple statement that was never brought up in lectures or our crusty 1960's texbook.

> one of five kids (out of class of 120) that took Calculus

I am always puzzled by this. Calculus was 8th - 9th material in Turkey in the 80s. In classrooms with 45 - 50 kids per one over-worked teacher.

We did set theory and Boolean logic in third grade. I think that is far more useful for understanding mathematical thinking than Calculus is.

Same in Canada. ...and then I moved to the US, and the school was a full YEAR behind, and only covered the basics.

...and I feel like it's getting worse.

> It was a Eureka moment, and entirely based on such a simple statement that was never brought up in lectures

Young me picked up the basics of cryptography at seven or eight, a couple of years before classes in the first foreign language began at age nine or ten. In Finland, for the Swedish-speaking minority, the first foreign language to learn is Finnish — often mentioned as one of the world's most difficult languages to learn. Neither the textbook nor the teacher mentioned how languages work, and specifically not how they are in no way related to cryptography. Young genius me concluded that surely the teacher will soon tell us how this crypto works, so we could move on. While waiting for that to happen, I skipped more or less everything. It took a couple of months to realize the actual situation, by that time the damage had been done. Picked up Finnish seven years later, the natural way through a colleague at a summer job.

That little omission had a detrimental effect on my average grades throughout the school years though, and therefore I certainly do not judge anyone's abilities based on their paper qualifications.

Were you thinking that maybe foreign languages were substitution ciphers, and that you were eventually going to be taught the substitutions you needed?

I think I had a similar thought when I was about six or seven and came across the Cyrillic alphabet printed in the back of a dictionary. I thought "oh wow, I guess if you learn this, then you know Russian!" -- thinking of Russian as something like English transliterated into Cyrillic.

There are phenomena in language that do work a little bit this way (https://en.wikipedia.org/wiki/Relexification, https://en.wikipedia.org/wiki/Avoidance_speech) but usually at the word level and not in contexts that will really help second language learners.

Hmmm, something seems to be missing from the explanation there?

Why would the teacher be expected to know anything about cryptography, let alone know that a particular student has any interest in it?

What a great story. I dont think you are alone in this. I also think this applies to more than just math.

Can you elaborate more about that sentence? With example, perhaps?

During pandemic lockdowns I had the opportunity to teach my 7yo son maths at home. It does not work for all kids, but he showed a very strong interest in symbolic manipulation. School overemphasizes arithmetic and arithmetic algorithms ad infinitum (extremely tedious and boring!). Our work plan is simple: a problem each day (together when the concepts are new), immediate reward after he finishes, and increase difficulty once he is ready. After one year, we have just started trigonometric equations at home, while they are still working on simple multiplications at school.

In the same way that some kids are good at sports, drawing, music, foreign languages, or playing videogames, some of them are very good managing abstractions. Those would be very happy learning calculus or other advanced mathematics, just for fun!

I really wish there were something like an "executive" refresher classes on basic maths with a focus on practical applications and modeling.

I took Calculus II/III, Linear/Diff EQ and some others in college, but I haven't written a new mathematical model in almost two decades and am rusty.

Try modelling something and see where it goes?

The thing with "modelling" is that in my experience it's often justified post-facto so you don't actually learn anything when it comes to modelling as an art. That being said what you are describing is basically a physics textbook, I would say engineering, but they have no soul.

I have done data modeling for a living and I don't agree that you don't learn anything and that it's used to justify something - you just have to do it rigorously.

To that point though, most modeling with any accuracy is done by computer now, and not by hand, which is what my point is. It's impossible to gain an intuition about something if the computer just tells you the answer.

The YouTube channel 3blue1brown has a good "Essence of calculus" [1] and "Differential equations, a tourist's guide" [2] series that I enjoyed a lot as someone who took calc in college but hasn't used it since.

[1] https://www.youtube.com/playlist?list=PLZHQObOWTQDMsr9K-rj53...

[2] https://www.youtube.com/playlist?list=PLZHQObOWTQDNPOjrT6KVl...

Not sure if it’s precisely what you’re looking for, but I’m currently reading and working through Coding the Matrix, which is linear algebra by way of Python. There are few assumptions and you start coding pretty early on.

Thanks, I feel like I've run across the book a few times over the years but never was suggested it. I'll check it out

Try Khan Academy, I found it to be a great refresher for things like that.

This resource really isn't that great IMHO. For anyone (esp. children) to grasp a new concept it helps to have relatable problems that can be solved in gradual steps.

After dividing up a cookie in Ch. 2 it continues further and starts to discuss proof via harmonics!

My 6yo daughter has exceptional talent and interest for Math. She's already discovering some fundamental theorems for herself (number theory, algebra, geometry). Does anyone here know any resources to help her development?

If she wants to learn math in the classical fundamental way, Spivak's calculus is actually a fantastic resource. It begins with nearly no pre-requisite number theory and has a very axiomatic approach, which while pedantic makes things much more accessible to someone without any prior training.

Other Calculus book worth mention is "Elementary Calculus: An Infinitesimal Approach", by Jerome Keisler. One thing that makes learning Calculus harder than necessary is the clumsy epsilon and delta formalism, that became sort of a COBOL for Calculus. This is not the intuitive approach Newton and Leibniz used to develop Calculus, based on infinitesimals, but shunned later because it took time until Abraham Robinson made it rigorous in the 1960s. The book by Keisler showcases nicely how higher math education could improve, if academia had less inertia. The author made the entire book available for free online: https://people.math.wisc.edu/~keisler/calc.html See also: https://en.wikipedia.org/wiki/Nonstandard_analysis

The YouTube channel 3blue1brown has some great, accessible introductions to higher level math.

That's a truly fantastic recommendation.

Other thoughts:

* The "Exploding Dots" project came up with a creative way to present place value that leads into a lot of other concepts like number bases, algebraic structures, polynomials, and other things. (It's actually like a generalization of the abacus, with fun computer animation.) I'm not sure which presentation of this is best -- I think the original inventor has gone through like three different versions and there are several versions available online now.

* I'm a huge fan of the late math journalist Martin Gardner, who wrote a lot of really entertaining stuff about math as well as puzzles and games. His books are really brilliant and cover a very wide territory. On the other hand, they were mostly written for adult audiences (originally, mainly readers of Gardner's Scientific American column) and may contain cultural references that kids wouldn't get, not least because some of them go back to the 1960s. :-( Also in some cases there have been further discoveries in the decades since the columns were published, so learning about them only from Gardner's old work may not give a good sense of where things have gone since then (e.g., he famously originally introduced both the Game of Life and the RSA algorithm to mass audiences, which then learned a lot about both topics in the ensuing decades). But maybe have a peek because they do stress the "math is fun and you can do it for fun" and also "math is all kinds of stuff and not just arithmetic" notions.

* Gardner does have two math books for kids (the aha! series) that are very fun and introduce a lot of fascinating stuff about logic and paradoxes, among other topics.

* Two online games that involve proof and circuit synthesis (spoiler alert: these are actually often isomorphic to each other because of Curry-Howard and stuff), which I think I heard about on HN:



I totally loved both of these and I think they could conceivably be accessible for a very motivated and very talented child.

Another online logic game that I also learned about on HN and loved:


Unlike the other two, this one uses (only) text and symbols instead of blocks and arrows. It's meant for college undergraduates, but I envision highly mathematically-talented middle-schoolers possibly being able to finish it. The user interface can probably also be improved a lot to make it clearer what you're allowed to do in each context. Maybe keep this one in mind for the future? :-)

Contemporary pre-college math education is famously very weak on discrete math topics (like number theory, logic, and combinatorics). I don't know if this is due to a desire to train aerospace engineers for the Space Race, or because discrete math has radically increased its profile only in the past few decades with the rise of computer science, or as a kind of backlash against the New Math which tried (mostly unsuccessfully) to teach kids formal logical foundations through set theory. I wish I knew some more good discrete stuff for younger audiences.

I haven't ever done any of it myself, but I hear Khan Academy's explanations (especially in math) are great and very self-paced.

Here are some concept maps you could you could print for her: https://minireference.com/static/conceptmaps/math_and_physic... https://minireference.com/static/conceptmaps/linear_algebra_... The topics are more for university students, but by the sounds of it, I think she'll be able to handle ;)

I find having a map of the "terrain" is always good when exploring.

Probably too basic for your daughter, but check out Breaking Numbers Into Parts. [1] Also, there's Russian Math School, which is popular in some circles.

1: https://www.amazon.com/Breaking-Numbers-into-Parts-Second/dp...

Tell us more about how she explored and discovered those theorems. Sounds like you should be giving advice!

I was recently recommended Brainpop Jr. I signed up only 2 days ago so can't yes attest to the results.

Check out the iOS game "Slice Fractions"

Let children be children, not experiments in the aspirations of parents and adults.

Sometimes children seek this kind of stimulation. I used to love trying to solve 'advanced' problems. Making new connections, i.e. "wow, so THATS why that is the way it is" were some of my favorite memories. I did Math Olympiad as a kid not because my parents pushed me - on the contrary - I pushed on them (and my teacher) to get into the program.

Obviously it's not for everyone, but I personally I would have killed to have started learning the 'interesting' stuff earlier in my life.

How is this considered aspirations of their parents? One of the most thrilling things in life is to learn new skills and understand how the world works. My child loves to learn new things and is very excited when he does something new.

I am sorry that you feel that way and maybe you personally did not like to learn new things. Don't cast your negative judgement on the enjoyment of others learning guised as "Just let kids be kids". You assume that people doing this are pulling a Beethoven's father or something. Check your assumptions.

Don't educate people. Let them remain ignorant.

I'm also assuming here, but it does sound like you don't have children of your own.

Looks pretty out-dated (1991). A lot of the math is very poorly formatted, and a lot of code's in an ancient form of BASIC.

Doesn't look like a good fit for a primary textbook since it's too scattered and slow, though some of the content might work as mini-games, sorta like crossword puzzles.

Does math really change?

I guess the basic is less popular now that TI-89's are being replaced with smartphones, but its probably not as ancient as you think.

Yes, math changes, but it's moreso the presentation that's ancient. The formatting is so retro that it seriously damages legibility while simultaneously failing to give students a taste of actual mathematical discourse, and the coding in the text is obsolete.

That students are still required to use TI-89's in class is a travesty itself -- TI-89's are horribly obsolete in modern practice and really only serve to hamper cheating on exams. At best, teaching students with TI-89's to better prepare them for exams might be justified as a trick to inflate their scores, but it still seems like a wasted opportunity to acquire a working familiarity with actual tools that students might actually use in the future.

The BASIC code is worse yet as it can't even be argued for as an exam-score-inflation trick; it's simply obsolete. Students learning coding ought to learn modern languages.

The focus should be on what's best for the students. And this isn't it. The retro stuff belongs in a museum.

When I was 9, my next eldest brother of 16 was captivated by calculus at school and wanted to share it with me. Without algebra he managed to get across to me how it worked though I am sure I had no idea what the point of it was. Sadly I have little capacity for memorising anything so eight years later in my final school exams I was still using first principles. It didn't go too well.

I will keep this for reference. It is a good starting point that I can build on with my kids.

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