When I'm working with engineers of my generation or older, we routinely do basic math to a couple of significant digits in our heads while working out problems. I just watch the younger engineers struggle with the basic math. It's not that they don't know how to do it, just that they can't do it as second nature and it becomes a stumbling block. I remember one extrememly bright engineer who had to pause the discussion to open a spreadsheet to do a very basic math problem. I was stunned.
First, I have problem with your terminology. Mastering the mechanics is a broad term that is not synonymous with memorizing tables. I certainly think that being able to add two numbers together is important and would qualify as "mechanics".
Let me start a principle: Education should be focused on giving the learner what she needs to be productive and successful. (I will leave the definition of "productivity" and "success", but hopefully you agree with the basic point... education is to benefit the learner).
My personal experience says that rote memorization of multiplication tables is an exercise that can be skipped on the path to mastery of mathematics. I believe there is plenty research that backs this up (although I don't have the energy to dig it up right now).
I will agree that some tedious exercises are useful for mastery. I feel strongly that not all tedious exercises lead to mastery.
It is clear that goal-centered, learner driven education is often quite effective (I am not saying that it is always effective, or that other forms of education aren't appropriate at times).
But, you have certainly learned things based on your own goals and driven only by the pure desire for knowledge. You learned your first language this way. You learned an incredible amount of stuff, driven only by curiosity, before we stuffed you in a classroom and told you you had to perform based on grades.
And, for that matter, what have you learned lately? I assume (at least I hope so) that you are still learning things, without being graded or told how to learning them.
Learner driven education doesn't mean you don't end up memorizing math values, or verb-tenses. It does mean that you start with the end in mind. I have certainly had flash cards to learn Spanish... and I did memorize verb tenses, but I did this because I understood the need for them-- and I didn't start until I understood the need for them.
The said myth of our education system is that learning isn't natural. This is why we push so much meaningless stuff on students, and this is why we feel the need to give grades, a completely artificial measure of how much we know.
My point is not that you can't pull out a calculator when confronted with 9x6, it is that your ability to work on more complicated problems is hampered because you have to take out a calculator to do it.
he problem is that the educator doesn't know which tedious exercises will be the ones that work for each individual child and the child doesn't have the experience to decide for himself.
I certainly support allowing children to leap ahead on things they have a strong interest it, but it is also useful (or even necessary) to expose them to things they have no initial interest in if only to broaden their education. If you love math, you still need to be able to write. You might never know if you like music without trying it. Exposure to a sport in PE might be the start of a life-long love affair. I think you must balance "learner driven" with adult experience to get that "goal-centered" part correct.
She can tell me answers, orally, like how the blood flows through the body and heart if I ask the right questions. She even made unusual connections on a high level sometimes. She just couldn't pass those tests.
I kinda agree with on this. The only thing I don't like is the curve thing where you do a percentage gets A, B, C, etc. My issue with it is you could potentially have more than 5% students that are incredible and deserve the A in one class and the opposite result in another. So you because you have a class of "dummies" you need to award one with an A to meet a quota when most deserve a D for instance.
Right - but in that class I took where I got the highest grade with a B+, the professor thought no one earned an A..
My impression is that curves changed in the late sixties, and a bigger percentage got Bs and As than before. But... that's just what I was tuned in to re discussion going on at my school; don't know if that happened across the board. People were very agitated about the whole grading thing being over emphasized, right about '65.
I do agree that the standards should be higher, where in theory a small amount get As and so forth - more the work should be more difficult. My kids' school have a slightly different grading system than what I've seen typical in other schools - you have to get a 94 average or higher in order to an "A"; 84 - 93 is a "B", etc. This boosts up the expections at each grade level.
She can tell me answers, orally, like how the blood flows through the body and heart if I ask the right questions. She even made unusual connections on a high level sometimes. She just couldn't pass those tests.
What I disagree with is this.... if you want to give me an introductory course on music, you don't sit me down and start with scales. If I am going to master music, then you are first going to have me listen to great music... and then you are going to put an instrument in my hand.
When I develop the appreciation of music... and get the desire to play like Dizzy Gillespie, then sure, I am going to have to learn scales... but at this point I will have already understood what I am doing and I will know exactly what the scales are for.
I feel strongly that this is how humans tick... we are driven by purpose and motivated by desire. Let me understand why I am working hard and then I will do what it takes to master it.
In math, we do this completely backwards... Kids aren't shown the beauty of math, they aren't shown the power of math. What we do is give them a bunch of the rote actions of math and tell them it will be important later.
I know this from experience... by the time kids reach high school, we have basically squashed their mathematical curiosity and have taken away the joy of discovery. Math is taught as a bunch of things to memorize... and for most people, this is all it ever is (people who go on to careers in math learn in spite of, not because of, the crap they are given to do in school).
Because, especially with math - you need the basic tools. If you can't remember your multiplication tables, going on to division, fractions, ratios and percentages, geometry and algebra, is going to be slow and tedious torture.
Yes, I understand that different people learn differently... I have met many very mathematical people who tell very similar stories. To say that it will always be "slow and tedious" is clearly wrong.
But my real conviction on this topic comes from my experience teaching physics in high school.
There was a group of students with very high GPA's who had a very troubling (to me) trait-- they had memorized a set of problems-- kind of like Mad Libs. As long as I gave them a problem they had seen before, and gave them clues, they would fill in the blanks and turn the crank perfectly.
The trouble with this is that they were completely unable to think mathematically. What they were doing wasn't anything like what mathematicians do-- they had a set of recipes that they could not deviate from.
These were very hard working students who had worked so hard that they had never had to think or to understand what they were doing.
This was very frustrating for me as a teacher
And that's what the goal is supposed to be- that they become educated. That they know more than they started out knowing.
And for some parents and children - the only important goal is an A on a piece of paper- what they know or don't know is secondary.
That's why I like the grading scale as it is over here.
First... I assume you agree that grades are supposed to motivate. Whether you see them as a carrot or a cudgel, the primary goal of grades is to pressure kids to perform the way we want them to perform.
