I believe that math teaches logicalness and helps w/ all functions of life. W/out math, I am condemning my child to poor wages. No matter what I end up choosing for school, I will hire a math teacher.
Unschoolers know that logic exists in life. Logic isn't an entity found only in math textbooks. It's a real thing people use everyday for solving problems.
Unschoolers know that math exists in life. And that a better foundation than years of teaching is using math and understanding it conceptually. It's far easier to absorb what percentages are and what and how they're used and then puzzle out how they're represented in formal math than to try to understand abstract formal math while also trying to figure out what percentages are divorced from actually using them for real personally meaningful reasons.
(In other words, a word problem about buying a $5.74 toy with $10 isn't the same as actually having $10 and weighing all the factors involved in what to spend it on or even maybe to save it for something more costly.)
Major breakthroughs in reading are occurring now but this math thingy still has me on pretty shaky ground.
It took me a looong time to get unschooling math. I think because school leads us to believe that math is all about the rote learning of arcane formulas to manipulate numbers and the only way to learn them is by precise teaching and hours of practice.
That's necessary to get school math but not real math. There are loads of kids who can muddle through the percentage problems in the percentage chapter of the math book but haven't a clue on how to figure out 5% sales tax or what 20% off means.
Which doesn't mean schools need to teach more. It means they need to teach less and help kids explore how numbers work in their lives more. Teaching math the way schools do without giving kids the opportunity to be immersed in math that's personally meaningful is like trying to teach Spanish by teaching grammar and vocabulary without ever hearing Spanish spoken or using it for personally meaningful reasons. It's the difference between learning English as a side effect of using it versus "learning" (though it won't stick) a foreign language step by tedious step.
The problem is that if schools allow kids to absorb math the way they absorbed English the schools run up against their biggest road block which providing feedback to parents and administrators and the state that the kids are learning. Unfortunately the best methods that allow schools to provide the best feedback are also the methods that are the most difficult for kids to learn through.
I think learning to "speak" math can't be emphasized enough. That means talk through the process you're going through to solve a problem in your head. (This is musing out loud, not a lecture. It's okay if they aren't listening.) Avoid pencil and paper. If the problem seems too unwieldy, break it up into easier problems that you can solve in your head. That way kids can see -- even if they aren't "getting it" or paying close attention -- how numbers work.
Pencil and paper math is something entirely different. Some kids love it, like puzzles. But they (and those who are forced to do it) won't necessarily learn how numbers work because writing down the numbers on paper makes them permanent. The numbers can't be altered and manipulated and pushed and pulled to make them tell you the answer.
A child who has manipulated how much allowance she has to figure out how much more she needs to buy something, and figured out the scores in games -- and has heard her parents do these things so she's comfortable with the process -- can "get" what borrowing and carrying means (sometime before she takes the SATs ;-) because she's done that without realizing it. Whereas it can take hours and hours to teach it to a child whose major experience with numbers is problems in a book.
People tend to think of the addition and multiplication tables as synonymous with being able to do addition and multiplication. But memorized tables only mean a child can do the problems faster not that the child has a clue what the process means. In fact a child counting on her fingers is showing she knows what addition means.
I've tried explaining to my husband that they only need higher level maths if they wish to take an exam in the subject or want to become scientists or mathematicians.
Or engineers. Or architects. Or economists.
Why is it necessary to learn something that you are only learning to get a bit of paper that is irrelevant to everyday life and who they are?
Being a math-head and engineer myself ;-) I'd say your husband's concern isn't that they learn math for the bit of paper, so that's why your arguments aren't reaching him. His concern is that he knows to the core of his being that it takes 10+ years at 5 hours per week to acquire all the math necessary for the exams. So, even if they don't use the higher math, it's important in terms of time that they start learning now so they'll have math knowledge in case they do want to use it.
He's wrong, but there isn't an easy route to help him see that he's wrong. I could not picture unschooling math until I could see it happening with my daughter. Now I can see that natural acquisition of math is a totally different process than how math gets taught in school. Learning math naturally is to school math as learning English as a toddler is to learning Spanish in school. The first are effortless and work without formal understanding. The second are often a source of frustration and frequently result in people who never want to see either math or Spanish again.
Marilyn Burns has written a few books on children acquiring math naturally. She talks about them discovering their own algorithms. (That's math speak for figuring out a way to do something themselves ;-) So rather than teaching kids the process of borrowing and carrying, kids figure out various ways the numbers can be manipulated in order to get the answer.
(As an example, in school they'd teach a child that to subtract 4.99 from 6.00 you need to borrow a 1 from the 6 to subtract 99 from 100. Which makes no real sense. But a child can more easily picture temporarily bumping 4.99 up to 5, subtracting 5 from 6 to get 1. And then, since 4.99 is further from 6 than 5 (or however they picture it) knowing that the extra penny we borrowed has to be added to the 1 for 1.01. It sounds complex written out, but it makes a lot of intuitive sense when you have to do arithmetic in your head.)
I liked Elaine's comparison to haute cuisine :-)
And Brenda's idea of him showing them the joy in math so they can have experience with it being fun and intriguing. The problem is that your husband might not care if they like math or not. He's only concerned that they get the practice hours in that he's certain is necessary. But you could suggest that if their experiences with math are the really cool fun stuff like fractals and Fibonacci numbers then they'll be more likely to want to know more.
Perhaps if he looks at it this way: if his primary goal is to get math into them, enjoyment will get sacrificed for practice and they'll try to spend as little time as possible on math. But if his goal is for them to enjoy the time spent with him and math, then they all win because the number of hours they spend willingly on math will voluntarily increase. And they'll enjoy spending time with him too :-)
[Algebra is] "a generalization of arithmetic in which letters representing numbers are combined according to the rules of arithmetic"
That just shows that dictionary definitions can be boring ;-)
Linda Wyatt came up with the definition that Algebra is figuring out what you don't know from what you do know. Which anyone can do whether their eyes glaze over at the sight of x + 2 = y or not :-)
Wish I had similarly pithy definitions for calculus, trigonometry and differential equations. I could manipulate all the equations like a champ since they were all just pattern identification to me. But I never really got a grasp of what they were, even if I did have to actually use them in engineering ;-)
My Math Experience
My math experience is probably the opposite of most people's. I enjoyed math so much that I had the impression I'd always been good at it. Then I stumbled on my grade school report cards and it reminded me I hadn't liked grade school math and was a pretty solid C student.
The numbers and the tediousness of early math got to me. The one thing that got drilled into me was being precise was the most important part of math. If there were patterns to arithmetic, they got lost in making sure all the numbers were doing exactly what they were supposed to.
(That was a tough lesson to unlearn, especially since it got reinforced in college engineering classes. I didn't unlearn it until I was an adult when doing math in my head was easier than hunting up a calculator and realized I didn't need precision to the 6th decimal place. Being close was good enough. In fact estimation is way more useful than precision. But estimation -- though we were told it was an important skill -- was merely a chapter in the math book and then we went back to getting answers right to the umpteenth decimal point. In fact estimation made no sense at all in the context of math class. Why would anyone want to do the problem twice? Once to get a rough answer and then again to get the right answer? Just double the tediousness.)
Then I hit algebra and the emphasis shifted to pattern recognition and I soared :-) All the way through from Algebra to Trig it was nothing but recognize the problem type, apply the appropriate formula and voile! Elegance :-) I absolutely adored Geometry proofs. (Which apparently they don't even do any more.) The proofs were like games to me. In fact I see a lot of parallels between geometry proofs and answering homeschooling questions. The procedure is to take someone's thinking step by logical step from where it is to where it needs to be to understand how children learn naturally. QED :-)
I adored college math, especially calculus and differential equations. More pattern recognition and apply the formula.
But when it came time to use the math for practical purposes in engineering courses I was befuddled. I didn't really understand why I was doing what I was doing. The problems weren't elegant in engineering. They were messy with bits and pieces of different problem types mixed together. I ended up graduating near the bottom of my class (the worst of the best ;-) and the only thing that got me as high as I was was math, art, psychology and programming. Rather crude programming with computer cards, but programming nonetheless and it saved me from a career in befuddlement because programming is just geometry proofs :-)
It wasn't until I started helping my daughter figure out real life problems that I "got" arithmetic. I could finally see the patterns. Since we were doing all the math in our heads, we couldn't keep track of all the intermediate answers so I had to tear the numbers apart and rebuild them out loud for her into numbers that would add or subtract easily. And from listening to me do that she "got" how numbers worked. She can see the patterns that it took me until adulthood to see.
My husband teaches algebra and modern math courses and occasionally brings problems home that are puzzling him. He'll walk through them using the concepts he's trying to teach and words that I've lost the meaning for. He uses the same techniques I used all through school of trying to figure out how the formula fits the pattern of the problem. But I can see the problems differently now. I can see essential patterns in the problem itself without being clouded by the formulas.