I can't find the thread in which I talked (ranted?) about multiplication tables, so I'll start a new one. This is very, very bad. Bad teacher, no apple! To have memorization trump conceptual understanding is a fundamental failure to understand what mathematics is all about. Besides ... commutitative property of multiplication!!! Even Wolfram Alpha know this.
tit for tat, its "commutative." had you spent a little less time conceptualizing and a little more time rote learning you could have been a 5-tool pedant.
Extra syllable for emphasis.
Well if your kid gets it who cares what the grade is really. For the kid who doesn't conceptually get multiplication, memorization is necessary bc then you get to long division and you need to have that base. Though in this house, we're not necessarily getting the memorization either. Math is a battle and whenever I work with Ryley, one of us ends up in tears. Usually me. Mark, why you live so far away? Math tutoring at the beach this summer please.
What math is he good at? What math does he find hard? What sorts of physical/spatial challenges does he like? I know he's into music, which I would call a physical/spatial challenge. Keys, strings, etc. have certain relationships that can be described aesthetically, spatially, mathematically. (I'm just channeling Pythagorus here.)
Using leverage to move heavy things, making machines, etc. is another realm of physical/spatial knowledge. (Archimedes. Weapons, heh.)
For me, it starts with what I want to do. What problem do I need to solve, what do I want to accomplish? That motivates the math. Newton is the kind of guy who might have just gone off and invented calculus on a whim. But that's not how it happened. He had a problem to solve, and invented an entirely new branch of mathematics, and an entirely new way to describe the world, so that he could describe gravity in a beautiful, concise way.
I draw the analogy to language. We begin to comprehend and speak because of immediate, pressing need. And I think of math as a language. 2+2=4 has nouns, subject/object, a verb and a conjunction. If there is a need to know a language in order to solve a problem, to understand something at a deeper level, or to describe something, or just to think using a particular paradigm, then there's some motivation for learning the language.
I think comfort zone is important too. Start from a comfort zone, and make forays out into new territory until they become comfortable too. There was an underutilization of metaphor and analogy when I was taught math. Multiplication and division are just addition and subtraction on steroids. On a rudimentary level, they are the same damn thing, with just one little twist. I don't know why they have to make it so damn hard. Blows my mind.
When you get to array and matrix operations then multiplication is some weird shit. Matrix multiplication is hardcore. College students majoring in science struggle with it. But with plain old numbers, I don't even understand why the four basic operations aren't all taught at the same time. 4+4+4 and 3x4 can be taught as being the same thing expressed with different "words", just a small tweak to semantics and syntax. I just don't see this as a huge leap requiring a full grade level of development.
Now memorization, multiplying two digit numbers 'n the fancier stuff is a separate issue from the syntax and semantics. These are handy parlor tricks, short cuts that should be handy in the tool box. But the parlor tricks are not the language.
Our brains are wired to do syntax and semantics. Symbolic language is something our little brains get started on very, very early. I don't understand why this isn't the emphasis in math.
In fact, I find estimation to be the one "math parlor trick" I use these days. Even though I was some kind of freaking mathlete in jr high and high school, I keep Excel handy if I need precision.
How many seconds are there in a year? Quick.
4000 seconds in a hour x 20 hours in a day x 400 days in a year = 32,000,000 The hardest arithmetic is 8 x 4. And I know that one.
60 seconds in a minute x 60 minutes in an hour x 24 hours in a day x 365 and a 1/4 days in a year = 31,557,600
That example comes out closer than most similar estimations I do. But multiplying two digit numbers together? I'm an engineer, not an accountant. I just need a ball park. Knowing WHAT to do is way more important than the details of HOW. I don't want to dismiss the HOW. But the HOW is fucking pointless without the WHAT.