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If one fixes a point in the future, there are an infinite number of paths to that point. The focus therefore should be on that distant point. If the student/parent goal is to get into a top rate university program (say engineering or medicine), then that fixed point is either one or two years of calculus in high school and the ability to ace the BC calculus AP exam. The question then becomes what is the best regimen for a given kid to put said kid in position to get a 5 on that exam.
As those infinite paths converge onto the fixed point, their paths converge as well. So by first year in high school, students should be in the range of Algebra 2-Geometry-Algebra 3-Trig. Consider it like a 95% confidence interval. More than that and one is more than 95% confident the goal would be reached. Below that range, say Algebra 1 in 9th grade, and one is not very confident said goal will be reached.
There is no reason to expect that we all have the same goals. If you wish to get into Cal Tech engineering, you shoulder consider accelerating the program by one grade--your competition in Asia sure is. If you are content with a English lit and the local U, then a year or two less ambitious would likely do (with more emphasis to be placed on your writing and breadth/depth of literature).
The three errors I see are that (a) students are not on the pathway that best suits them, (b) students that state more ambitious goals are in complete denial as to the work that is required of them, and (c) that students/parents place all their demands on the curriculum far too late in the process and allow many years of non-education or mis-education.
How can it be that 13 years is not enough to take a kid from counting to calculus? Insane. My kids are on track to take two years of calculus in junior high because we have bothered not to waste their elementary school years on coloring, poster-making, and all the other inane activities that supplant real education. When they do their 60 minutes of math a day, it is a pure 60 minutes, not 15 of horseplay, 30 minutes of review and 15 minutes of watered down math. Is it really any wonder that universities offer elementary math and pre-algebra? Someone should sue the states for their debilitating school systems. Seriously.
Math is sequential. If the student hasn't memorized his or her times tables and hasn't mastered long division, he won't be able to do algebra. And if he can't pass algebra, he'll have a tough time graduating from high school. In my state we've moved away from the concrete and "drill and kill" in favor of conceptual math. While this is a noble cause, most kids can't think abstractly until late elementary school, so this is lost on them. We now have remedial math classes at the community college level since many students are unable to pass a basic math competency test.
It seems as if it's now up to the parents to drill their kids or enroll them in an outside program to supplement what schools teach. Maybe you could teach in a parochial or private school where the state doesn't dictate the math standards.
Well said. While some children can and do think abstractly, I've seen many more who don't. It's those that don't "get it" that seem to get the most lost later on. It's also obvious by how many times that they can "redo" their problems and have the highest score taken, that it's not a true measure of what they understand IMHO. Personally, I suspect that this "hooplah' of "higher test scores" can be taken with a large grain of salt. Many parents supplement their child's education. That's also a small detail that doesn't get factored into the equation (pun intended) of elevated testing scores.
Second, no one drilled multiplication tables into me. We were expected to learn them on our own. Everyone wants to be spoon fed nowadays.
We learned our tables, one at a time, beginning with 2x1, 2x2... We were told we had to stand at the front of the class and recite them, the next day -- powerful motivation. Next day, it was 3x1, 3x2... Every single one of us learned to multiply.
Also, it seems like these new, new math curriculums are more like English class--too hard for kids who struggle with language difficulties--they should be able to feel successful at something. Interestingly, they never use these curriculums with the special ed kids--not that I've ever seen anyway, unless they're in a regular classroom for math class.
There's your answer to why they don't seem to use it with special ed kids (bolded) - especially ones who struggle with language-based pedagogy.
Generally, these types of children need the structure of traditional arithmetic before they can grasp the "concept" of what's being asked of them.
Some folks just need basic steps to arrive at the right answer. Then it becomes an "Oh, I get it!" moment. Those are the kids that seem to struggle the most with the "new, new, math (which is rapidly becoming the "standard in Math Education today"). Unfortunately, the current educational "fad" is to fail to recall history IMHO.
You can't use reasoning and logic if you haven't mastered basic computation skills. Abstract ability is built on the concrete. For example, try telling a student to solve this problem: "Mr. Jones can correct 150 quizzes in 50 minutes. His student aide can correct 150 quizzes in 75 minutes. Working together, how many minutes will it take them to correct 150 quizzes?" It's difficult for a student to "problem solve" if they don't know how to determine what 150 divided by 50 is (or they never memorized 15 divided by 5), they can't write an algebraic equation from the information given, and they don't know how to solve for x.
It's like trying to drive your car on a freeway when you only vaguely know where the accelerator and brake are.
To me, the numbers were a snap; the problem for me and others was setting up the problem.
There is something to be said for the 'drill and kill'. Kids learn combinations and how to recognize the problem and know the answer without having to figure it out. Automatic thinking.
If you look at the math scores in high school, you will see that kids aren't doing too well in math. If all the wordy math in elementary is oh-so-efficient, then math scores should be way up. They're not. New math programs are failing our kids. Period.
Having learned the basic math facts, a person can apply them to what is needed for every day life (measuring, percentage, etc.). It does not make it easier to have to weed through all the wordiness to try to figure what application is needed to find an answer to a problem.
True. IMHO there also is an element of "assessment" where an instructor feels that a student is solid on a concept, only to find out later that he or she really isn't. This is particularly the case if the child has a short attention span, or has difficulty retaining concepts that they have learned in the past. I have had many assessments come home stating that my child was solid on certain concepts. I've had him solve related problems only to find the answer to be off by one or two (sometimes more). I understand there is a time limit to what teachers can teach, but I think that there should be adequate time to take a moment to recite basic math facts. The thing that gets me is that my kids can play the math games, yet cannot tell me what the "concept" is. My 3rd grader knows how to divide using manipulatives, but can't seem to manipulate actual numbers.
She was endlessly frustrated until I sat down and taught her short division. In everyday math, partial quotients are taught in 4th grade and short division is taught in 5th. I feel it's putting the cart before the horse.
I know the hopeful end-result is for a child to be able to cluster problems in his or her head - which is fine for those who can do so - but there are many people who cannot - even children who have had constructivist math. The thing is, with all of the time spent teaching the grasping of concepts, there is precious little time for rote memorization. Once I continually pressed the dreaded "drill and kill" with my son, he was able to understand how to arrive at the correct answer, and as a result of feeling that he could actually do it and get it right once and a while, he was more enthusiastic about explaining the concept to me, rather than simply rushing through it to get it the heck over with.
I agree that there's something to be said for rote memorization. I find it interesting that from the time he began Everyday Math in the first grade, to now (going into 6th - 7th if I hadn't kept him back in 2nd), the curriculum has been revised 3 times. With each new revision, I find more and more traditional math e.g. math facts and more in-depth fraction explanation.
This leads me to conclude that A. the "angry masses" are being placated, or B. The college professors are seeing that they have more and more students needing remediation in basic arithmetic before moving on to calculus.
I don't see anything wrong with that question. For second grade it seems ok to me. I don't see what two or three more questions would do. I would rather my 3rd grader have a few well selected problems than a whole worksheet of drill and kill.
Hmmm. I remember at the time thinking about just why this problem bothered me so much. I remember coming up with 5 or 6 reasons... for the sake of brevity, I'll just give you a couple:
1. "Explain your thinking" - The problem IS the explanation. For gosh sake, 2 people each ate 2 hot dogs THAT IS THE EXPLANATION. What else is there to explain? 2*2=4 because it does. There isn't anything to explain.
2. This problem was given in a unit on addition. This isn't an addition problem. The use of the word "each" makes it a multiplication problem. (Yes, I realize that all multiplication problems can be done with addition.)
Then, I take huge exception to the single problem assignment. One problem per day is not adequate. Getting the correct answer to a single problem is not an indicator of understanding, nor is it an indicator of fluency, both of which are ESSENTIAL in mathematics.
I'm sure I'll get flamed for this but, understanding is massively overrated at this age and stage. Our educational pedagogy leadership has convinced us that "understanding" is somehow the end game. At this stage of math, fluency and accuracy are of paramount importance, not "understanding". I would much rather that my kids can quickly and accurately add and multiply in their heads than have the basic concept of addition and multiplication but need to use a calculator. The FACT is that most people do NOT walk around with calculators but DO need to do sums and estimations all the time.
Somehow we've gotten it stuck in our heads that there is some mystical "understanding" to extract from math but we don't do this in other subjects.
Why is "school" spelled that way? Explain your thinking.
Why is an apostrophe used as the symbol in a contraction? What symbol would you rather use?
These are the RULES of grammar and spelling and they need to be LEARNED (dare I say MEMORIZED?) not understood. There is no deeper meaning. In fact, often other country's conventions are completely different from other country's.
I love math and am all about understanding what you are doing but, reform math has put the cart before the horse. The focus on understanding at the expense of fluency and accuracy is a joke. In math there is a right and a wrong and any system that teaches gray is just plain wrong.
Ah, Chicago School math...nothing good ever comes out of the Chicago School!
I know I'm a little late to the party but I just read another math thread, and came here. I did have the "show your working" stuff in school where you'd be asked something like:
Quote:
Kerry has 220 sheep and Jacinder has 102. They decide to put all their sheep together in one big field. How many sheep do they have? Show your working.
So I would put 220 over 102 and do long addition, and that was my "working". But I definitely had multiplication tables drilled into my head, over and over. We were also made to redo a math problem over and over again until we got it right. There is one that I still haven't solved to this day, and it's haunting me
Hmmm. I remember at the time thinking about just why this problem bothered me so much. I remember coming up with 5 or 6 reasons... for the sake of brevity, I'll just give you a couple:
1. "Explain your thinking" - The problem IS the explanation. For gosh sake, 2 people each ate 2 hot dogs THAT IS THE EXPLANATION. What else is there to explain? 2*2=4 because it does. There isn't anything to explain.
2. This problem was given in a unit on addition. This isn't an addition problem. The use of the word "each" makes it a multiplication problem. (Yes, I realize that all multiplication problems can be done with addition.)
Then, I take huge exception to the single problem assignment. One problem per day is not adequate. Getting the correct answer to a single problem is not an indicator of understanding, nor is it an indicator of fluency, both of which are ESSENTIAL in mathematics.
I'm sure I'll get flamed for this but, understanding is massively overrated at this age and stage. Our educational pedagogy leadership has convinced us that "understanding" is somehow the end game. At this stage of math, fluency and accuracy are of paramount importance, not "understanding". I would much rather that my kids can quickly and accurately add and multiply in their heads than have the basic concept of addition and multiplication but need to use a calculator. The FACT is that most people do NOT walk around with calculators but DO need to do sums and estimations all the time.
Somehow we've gotten it stuck in our heads that there is some mystical "understanding" to extract from math but we don't do this in other subjects.
Why is "school" spelled that way? Explain your thinking.
Why is an apostrophe used as the symbol in a contraction? What symbol would you rather use?
These are the RULES of grammar and spelling and they need to be LEARNED (dare I say MEMORIZED?) not understood. There is no deeper meaning. In fact, often other country's conventions are completely different from other country's.
I love math and am all about understanding what you are doing but, reform math has put the cart before the horse. The focus on understanding at the expense of fluency and accuracy is a joke. In math there is a right and a wrong and any system that teaches gray is just plain wrong.
I agree with you. There are many of us who needed to learn those rules before we could "get it." There's also the element of frustration when trying to teach a young child how to work a problem. IMHO, if they are given too many ways to arrive at an answer and then not given enough time to master one of them, the frustration of getting the wrong answer will turn them off to math in general. I equate it to kicking a guy who's already down. It builds self-confidence to be able to get the right answer in the simplest way before going on to find different ways to solve. In terms of the very young, I really think that it would be better if they focused on the easiest way to arrive at the correct answer, and then worked with the conceptual piece.
I also find it very interesting that each time they revise it (our school's on the third revision) it becomes more and more like "traditional" math. Fractions are (thankfully) being addressed much more. Is it because they are trying to placate the angry masses, or is it because college professors are now seeing the fruits of deemphasis in terms of basic arithmetic? Just what exactly are the students who end up in remedial mathematics classes having to learn or relearn?
I'd be interested in finding that out. Does anyone out there know anything about that?
Last edited by cebdark; 04-22-2009 at 06:52 PM..
Reason: added something
Just what exactly are the students who end up in remedial mathematics classes having to learn or relearn? I'd be interested in finding that out. Does anyone out there know anything about that?
These students are unable to pass a proficiency test. This could be the result of many factors: missing school when certain concepts were taught--math is sequential; moving, especially between states, since standards are different; a learning disability that went undetected; a teacher whom the student did not like or who did not like him/her, incompetent teachers, family problems, disliking the subject, etc. Many do poorly at algebra or geometry because they never learned elementary math and usually "hate" math as a result.
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While some children can and do think abstractly, I've seen many more who don't. It's those that don't "get it" that seem to get the most lost later on. It's also obvious by how many times that they can "redo" their problems and have the highest score taken, that it's not a true measure of what they understand IMHO. Personally, I suspect that this "hooplah' of "higher test scores" can be taken with a large grain of salt. Many parents supplement their child's education. That's also a small detail that doesn't get factored into the equation (pun intended) of elevated testing scores.
