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.

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

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".

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.

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!

Edit: typo

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.

"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."

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.)

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%.

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.”

https://mag.uchicago.edu/education-social-service/can-fifth-...

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/

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.

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.

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 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."

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.

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

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.

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

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

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 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.

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.

"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.

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.

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.

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

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

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.