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.