[gnodar]

It is hard, but only if you try to find the general solution. If instead you think in the way the problem designer (who had a target audience of 3rd graders in mind) intended the solution to be found, then it becomes easy to find a solution: one of each toy from the yo-yo to the car (inclusive). That gets you $5.18 from the goal of $43.94, so get another car (the problem doesn't state you can't have repeats).

[exporectomy]

Then it's a matter of luck whether you tried that pattern. I started with the highest prices items and didn't get far. It's mostly a lottery if it depends on people guessing that way.

I would start several approaches like writing a formula, bounding the number of toys (6 to 50?), looking for easy multiples, etc. the declare them all too hard and quit.

[bottled_poe]

Unfortunately that’s exactly the issue here. That justification puts more focus on training a (simplified and flawed) process rather than understanding the fundamental nature of the problem being presented.

[Aerroon]

But that's how you build to to the learning about the eventual general solution.

First graders here get fill in the blank questions:

__ + 3 = 10

Years later they'll learn

x + 3 = 10

Filling in the blanks isn't that hard for most kids, but dealing with x can be.

[bottled_poe]

I would argue that arithmetic is overvalued in early mathematics education. Category theory, as an example, has many more practical applications and leads into arithmetic. In my experience, early education in mathematics skips many of the prerequisites and you end up learning these things through rote-learning instead of building that knowledge from the fundamentals.

[tsimionescu]

> Category theory, as an example, has many more practical applications and leads into arithmetic.

Now this is a sentence I never expected to read.

I aheb thought before about the fact that derivatives are a much simpler and more useful concept than exponentials and logarithms, but I very much fail to see how category theory is useful for anything other than researching the foundations of math.

[myWindoonn]

Arithmetic is just decategorified set theory. The main advantage of teaching arithmetic via set theory is that it allows for building intuitions about counting.

It shouldn't be surprising that two sheep plus two sheep is four sheep. We can go further. Suppose that we have lots of longhair sheep and shorthair sheep. If we choose some sheep, how many ways can we have some shorthair and some longhair? This gives exponentiation intuitively, so that we could ask how many ways we could choose zero sheep from a collection of zero sheep, or in other words, why zero to the zeroth power is one.

The parts of category theory that you're imagining, with the morphisms and natural transformations, doesn't have to be taught before arithmetic. It can be taught when lambdas are first introduced, when we write "f(x)" on the board for the first time.

[tsimionescu]

But none of this modern theoretical understanding helps me see what is 131 + 121, or 2^10. Expressing arithmetic operations as set operations and counting elements is only useful for really small numbers - if you actually want to know the answer, you ignore the set-based or category based foundations and use an algorithm.

This is the major problem I have always had with examples of category theory use: they always sound nice and give very illuminating intuitions for certain mathematical structures, which is very useful for doing mathematics and furthering your understanding. But they never directly answer any practical questions - for those, you always abandon the abstractions and start getting into the nitty gritty of the specific domain. At best, they help you take a specific algorithm from domain 1 and apply it in domain 2.

Am I wrong? Is there some way to actually get the answer to how many ways you can combine longhair and shorthair sheep in sets of 10 sheep, other than simply counting all combinations, or using traditional arithmetic/geometry etc. (e.g. repeated multiplication, angle measurements)?

[bottled_poe]

I specifically mentioned arithmetic, which is often confused for mathematics.

[tsimionescu]

Yes, which is most bizarre, since school-level arithmetic is the first kind of mathematics ever invented, far before writing, specifically for its practical uses.

Are you really claiming that it's of more practical use to understand the properties of monoids and rings than it is to understand that 2 + 2 = 4? Or are you actually talking about arithmetic beyond what is normally taught in school?

[TchoBeer]

Category theory has many more practical applications than arithmetic?

[iso8859-1]

Certainly, see Haskell. It enables reasoning about interfaces, easy as algebra.

[amw-zero]

Can you explain how category theory can be used to figure out how much to tip a pizza delivery person?

[iso8859-1]

It shows that tipping is not consistent with mathematics and pizza people should be formally compensated as to achieve this consistency. Perfectly congruent.

[jameshart]

The question doesn't ask you to find a solution - it says "what combinations" can you find - plural.

You're not done when you've found one. Are you sure you can't buy ten yoyos, three cars and a pinwheel and hit the number too?

[novia]

If you keep reading it asks if your solution was the only possible solution. This implies that you only have to give one solution and prove that there are no other solutions (very difficult) or find two solutions to show that there could be many more than just one.

When the son said that some people in the class got the problem exactly right, I doubt they spit out all 279 combinations.

[jameshart]

All of which is just more evidence in the column of 'this is a terrible question, ill-posed, and pedagogically invalid'