7 ms·

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.

by casion 5y ago

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.

tartoran 5y ago
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
- thaumasiotes 5y ago
  > 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.
  - tartoran 5y ago
    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.
    - jdmoreira 5y ago
      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.
  - jdmoreira 5y ago
    > 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.
  - TheOtherHobbes 5y ago
    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.
AndrewKemendo 5y ago
Is that an indictment on the educational system and broader society or a comment that this material is only appropriate for gifted children?
bmy78 5y ago
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.
DoreenMichele 5y ago
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.
- riffraff 5y ago
  I was literally thought fractions like this in school, except the cake was drawn, I'm having trouble imagining a different solution :)
  - DoreenMichele 5y ago
    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.
userbinator 5y ago
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: https://news.ycombinator.com/item?id=22941144 https://news.ycombinator.com/item?id=22941144
ArcticCelt 5y ago
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. https://mag.uchicago.edu/education-social-service/can-fifth-graders-do-calculus https://mag.uchicago.edu/education-social-service/can-fifth-... https://outlooknewspapers.com/math-academy-multiplies-success-thanks-to-pusd-private-partners/ https://outlooknewspapers.com/math-academy-multiplies-succes... 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/ https://techzinglive.com/
- rsj_hn 5y ago
  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.
spoonjim 5y ago
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.
vishnugupta 5y ago
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.
- phorkyas82 5y ago
  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.
  - hdctambien 5y ago
    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...
- cowl 5y ago
  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
colordrops 5y ago
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: https://www.educationalinsights.com/fraction-formula-game https://www.educationalinsights.com/fraction-formula-game
hintymad 5y ago
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.
- vishnugupta 5y ago
  > 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.
forgotmypw17 5y ago
I think your ideas about what a certain age can do are heavily skewed by the overall state of your area's education.
- mashka 5y ago
  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.
- casion 5y ago
  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.
deleted 5y ago
[deleted]
parenthesis 5y ago
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.
forinti 5y ago
> 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.
- disgruntledphd2 5y ago
  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.

by casion 5y ago

tartoran 5y ago

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

thaumasiotes 5y ago
> 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.
- tartoran 5y ago
  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.
  - jdmoreira 5y ago
    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.
- jdmoreira 5y ago
  > 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.
- TheOtherHobbes 5y ago
  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.

AndrewKemendo 5y ago

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

bmy78 5y ago

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.

DoreenMichele 5y ago

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.

riffraff 5y ago
I was literally thought fractions like this in school, except the cake was drawn, I'm having trouble imagining a different solution :)
- DoreenMichele 5y ago
  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.

userbinator 5y ago

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: https://news.ycombinator.com/item?id=22941144 https://news.ycombinator.com/item?id=22941144

ArcticCelt 5y ago

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. https://mag.uchicago.edu/education-social-service/can-fifth-graders-do-calculus https://mag.uchicago.edu/education-social-service/can-fifth-... https://outlooknewspapers.com/math-academy-multiplies-success-thanks-to-pusd-private-partners/ https://outlooknewspapers.com/math-academy-multiplies-succes... 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/ https://techzinglive.com/

rsj_hn 5y ago
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.

spoonjim 5y ago

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.

vishnugupta 5y ago

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.

phorkyas82 5y ago
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.
- hdctambien 5y ago
  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...
cowl 5y ago
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

colordrops 5y ago

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: https://www.educationalinsights.com/fraction-formula-game https://www.educationalinsights.com/fraction-formula-game

hintymad 5y ago

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.

vishnugupta 5y ago
> 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.

forgotmypw17 5y ago

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

mashka 5y ago
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.
casion 5y ago
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.

deleted 5y ago

[deleted]

parenthesis 5y ago

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.

forinti 5y ago

> 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.

disgruntledphd2 5y ago
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.