Imagine I walked into your house right now, stuck you into a car, drove you to a building, and sat you down with a bunch of other people. You look up at the board at the front of the room and you see a word written on it. "Concurrency".
What's that? You wonder. Why am I here? Who are these people?
The person at the front of the room proceeds to explain the nuances of concurrent software design in distributed computing environments. They frequently remind you that you will be tested on this, and that the test is very important, and that if you fail - that is bad.
You look around this room and notice everyone else carefully studying. Some people seem to be putting their hand up and getting correct answers - how do they know? Why do they find this so easy? You take a stab at the questions, you get them wrong. It's not making any sense. Why do I have to do this anyway?
Why
This is the question that inevitably pops up when doing something difficult - and it's a perfectly reasonable question. When you are suffering, when things are hard, you want to know why. You want to be reassured that the suffering is both temporary, and going to be worth it in the long run.
I've been tutoring maths (and physics, economics and computer science) for many, many years now, and I've taught students of all ages and capabilities for a range of different examinations in multiple countries and I've always seen the same thing. Students that really struggle with the subject will always have this resistance, and will always ask the same question - what's the point of this? Why do I need to know it? When am I going to use it?
And for the first few years of teaching maths I used to try and convince them that while they probably never would use trigonometry in their day to day life, that they were learning transferable skills that would help them in other areas and so forth. But a moment of reflection probably about 5 years into my teaching career made me realise this wasn't true. And I don't like telling people things that aren't true. For one I think they can sense it anyway. Secondly even if they don't pick up on it they will later discover the truth anyway.
I have a masters in mathematics, and I learned all sorts of exotic ideas - abstract algebraic constructions - that while to me, are of great beauty and aesthetic, and make a very interesting study, aren't actually needed for any of the problems of day to day living.
There are other skills and knowledge I've accumulated since leaving university that I know to be far, far more useful in solving every day problems and getting along in life. So I think we need to stop pretending that there is some actual point in learning all this mathematics.
The history
The actual reason this is still stressed so much in the syllabus has a historical origin. The old medieval education system was founded upon 3 pillars - grammar, logic, rhetoric.
The idea was "grammar" was how you learned to ingest knowledge and understand other people's ideas. They are the rules of language and are needed to receive communication.
Then "logic" was your ability to process this information and draw conclusions. For this they taught Greek Geometry, as this was seen to be the pinnacle of logical thought.
Finally "rhetoric" was your ability to articulate yourself and persuade others of the conclusions you had come to.
If you really think about it, it makes a lot of sense. The intention behind this form of education was to create people that could understand the world about them, other peoples ideas, could process them with logic, and could then go forth and share their discoveries with the world.
This thinking has sort of decayed and we're now just left with a relic in there that everyone must learn higher level mathematics. But nobody really knows why, we just sort of do it, because we always have.
The cost
So the reality is, there is no good reason for children to be learning algebra, trigonometry etc. etc. In fact if I was to design a maths syllabus from the ground up now, I'd probably just do basic numeracy, then quite a bit of probability, statistics and game theory, and round it off with a smidge of set theory and formal logic.
These are the areas of maths that are most applicable to daily living, and forming an understanding of the world around us. Perhaps even ideas like non-linearity and coupled systems would help people think about their lives in a more useful way than human intuition normally gravitates towards.
So not only have we kept this relic of an ancient educational philosophy without understanding why, we've removed the pillars around it that really made it make sense, then we've thrown in a bunch of useless stuff and left out all the good stuff.
And if you thought that was a bad idea - the worst part is yet to come. I used to play a game with all my students, and with many of the parents as well (and I still do whenever I get the chance!), and I would ask them one simple question.
What is the definition of the word "mathematics".
And in all the last 15 years I've been asking that question, guess how many people could come up with a satisfactory answer? Absolutely zero. Not one person. Most people have never even thought about it. Some people will give it a good go but always come up short. They sort of reflect on their studies and try and grasp at what it meant for them. But they always give definitions that are easily proved wrong.
Like they'll say things like "it's about numbers", to which I ask what triangles have to do with that. They'll say it's about "spotting patterns", which you have to do in all the sciences as well so that's not a good definition for mathematics either. Or recently I got told it was "how to make logical arguments", which is also what lawyers have to do.
So we have an education system that not only does a complete botch job of teaching the subject (as if that wasn't a big enough crime in itself), they never even bother to tell students what it is they are studying in the first place.
What is the credibility of an organisation that issues certificates, certifying competence in a subject, that none of the certificate holders can explain what that subject is. It boggles the mind.
What about the good students
So to finish my story, I also noticed something about the very good students in maths. While they did not really push back on the whole "why do I need to learn this?" line, I would still ask them questions about what they were doing. Questions like, "why is this useful?", or "what is this used for?". And it would slowly dawn on them they were quite proficient in doing these things like, solving various equations, or answering these problems, yet had no idea why they were doing it or what the knowledge was even for.
And for them, it still greatly increased their ability and understanding of the subject when they found out why mathematicians cared about these things, and what types of problems they are used to solve. And no, they are not every day problems. So while I'm not saying the knowledge has NO value, it is just highly specific to certain types of work.
Like concurrent distributed computed systems, there are only a handful of people on the planet that actually need to know how that works. Specifically - software engineers working on large-scale distributed systems. Think, the guys that work at google and microsoft. That's it, nobody else needs to know. And I think you could argue that is the case for much of the maths syllabus. If you're not going to become a scientist, you probably don't need to know it.
Still, the students that were good at maths, still found it interesting what all these egg-heads did with this kind of maths, and it made them better at using it and solving problems that were a bit outside their comfort zone. I.e. it allowed them to think for themselves and apply the knowledge because they had context and understanding that wasn't just the memorisation of a process.
Summary
So given how spectacularly the modern education system fails, and on so many levels at once, it's not difficult (I think) to do a better job yourself when home educating.
You just have to first of all think of the reasons why someone would want to do the thing you're about to teach your child. And there's always a reason, and if you don't know it, it will take you about 5 minutes of googling and reading around the subject yourself to find the answer. And you'll probably find it quite interesting yourself to round off your own lacklustre education with real knowledge and not just stuff you were forced to memorise in order to pass an exam.