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I told you my experience; your experiences may obviously differ. I would not worry so much about finding the perfect text; just start with something and see if it works for you. Time spent searching for the perfect text would be better spent on the GRE review books. By the way, my experience with GRE review books has been terrific!
Understood. It may be a myth that somewhere a great book has been written that can solve the mystery of mathematics for a particular individual, but it is a tantalizing dream. Humans like to dream such dreams. The same thing goes for a "magic bullet" path to math mastery. That's why the stories of those who intuitively grasp math are fascinating. We know math savants and prodigies exist. That is scientifically proven fact. Knowing that, we can hope that their abilities can be emulated in ourselves given the right technique or teaching to show the way. Maybe it is a curse to think that way. As you suggest, just wade in and get our feet wet. Begin the journey with a first step and do what we can.
Understood. It may be a myth that somewhere a great book has been written that can solve the mystery of mathematics for a particular individual, but it is a tantalizing dream. Humans like to dream such dreams. The same thing goes for a "magic bullet" path to math mastery. That's why the stories of those who intuitively grasp math are fascinating. We know math savants and prodigies exist. That is scientifically proven fact. Knowing that, we can hope that their abilities can be emulated in ourselves given the right technique or teaching to show the way. Maybe it is a curse to think that way. As you suggest, just wade in and get our feet wet. Begin the journey with a first step and do what we can.
This is about as likely as finding the "magic" that will make me an artist!
It's time the OP stops dreaming about finding the right magic formula, recognizes that different people have different neural skills (and learn differently) and gets to work to find what works for the OP. There's no magic here, just mental exploration and work.
For people like me who tend to think visually, some of the visual tools (mostly not found in textbooks) are the most exciting developments in recent decades. For others, story problems might work. For others, who knows? I sure don't.
For me, and people get mad at me for all the useless information that floats around my brain, I understand hard concrete facts. 1+1=2. Where I get lost is when it becomes abstract and therefore interpretable. such as 1+1=2 UNLESS yada yada yada and then there are margins of error. Algebra and Geometry are super east for me but the higher I went, the less I was able to conceptualize what I learned.
What I have learned as Ive gotten older is that I know Im able to learn things when they are shown in a material way so if teachers in school taught higher level maths in 3D lessons and actually showed me how things worked and why, I would get it. Same with my higher level sciences, I aced my way through the biologies because I could visualize what was being taught. But chemistry was my downfall. I learned from the labs but the long equations screwed me up.
I could have probably done a lot better if my math wasnt just going through the same math as everyone else in small town high school and the math teacher actually cared whether we understood or not or were just there for the math requirement to graduate.
I also have almost zero ability to understand the vastness of space and quantum theory and string theory and all that, even when its dumbed down. I just cant visualize it, I have no frame of reference
Your story can be generalized to some extent. For example, there is a book on the art of thinking whose reviewer puts it this way.
"What style of thinker are you?
The Synthesist: Sees likeness in apparent opposites, interested in change
The Idealist: Welcomes broad range of views, seeks ideal solutions
The Pragmatist: Seeks shortest route to payoff: “Whatever works,”
The Analyst: Seeks “one best way,” interested in scientific solutions
The Realist: Relies on facts and expert opinions, interested in concrete results"
The Art of Thinking: The Classic Guide to Increasing Brain Power
by Allen F. Harrison (Author), Robert M. Bramson (Author)
Maybe if the art of thinking were taught to children as a fundamental skill in addition to the 3 R's, they would do better in school and life. Their teachers might be more fulfilled as well.
This is about as likely as finding the "magic" that will make me an artist!
It's time the OP stops dreaming about finding the right magic formula, recognizes that different people have different neural skills (and learn differently) and gets to work to find what works for the OP. There's no magic here, just mental exploration and work.
For people like me who tend to think visually, some of the visual tools (mostly not found in textbooks) are the most exciting developments in recent decades. For others, story problems might work. For others, who knows? I sure don't.
I absolutely agree. I am exploring and working on parallel tracks full-time to do the work. This thread is one of the tracks and it is rewarding for me (maybe for others). The other tracks are going well too. I believe mastering math is more than just following the standard sequence of subjects. I could be wrong.
Here is a quote on the nature of mathematics, from Duke University. I have not received specific permission to reprint it. I found it on their website and hope I may be forgiven.
"In addition to theorems and theories, mathematics offers distinctive modes of thought which are both versatile and powerful, including modeling, abstraction, optimization, logical analysis, inference from data, and use of symbols. Experience with mathematical modes of thought builds mathematical power--a capacity of mind of increasing value in this technological age that enables one to read critically, to identify fallacies, to detect bias, to assess risk, and to suggest alternatives. Mathematics empowers us to understand better the information-laden world in which we live."
1. Math is, to some degree, something that either comes naturally or not. If it doesn't come naturally to you, putting in more time isn't really going to help much. Math came very naturally to me, and I'd get straight A's in math without studying or any effort at all. Other subjects were more about putting in time, and I was usually too lazy for them.
2. As others have said, most elementary school teachers are liberal arts focused, and are very poor at teaching math.
3. Math tends to be more cumulative than other subjects, so somebody who falls behind, either due to a poor teacher, or do to relocating from one school district to another, may have a hard time catching up, especially if they never get a teacher that is willing to help them catch up. If somebody moves from a school district that teaches long division in 3rd grade, and moves to a district that teaches it in 2nd grade, they may always be at a disadvantage. I know that Katy Perry said that her parents frequently relocated, and that is one reason why she always struggled with math, even though she seems to be at least a somewhat intelligent person.
On other hand, in social studies, every year is mostly independent of each other. In my high school, in 9th grade, we learned about the non-Western world, 10th grade was European history, 11th grade was US history, and 12th grade was half a year of participation in government, and half a year of economics. I had a very poor teacher in 9th grade who taught us almost literally nothing. But since there was no overlap between years, that gap in knowledge had no real impact on later years of social studies. Similarly, in science, if you have a poor teacher for biology, it doesn't impact your ability to learn chemistry or physics in a later year.
4. Many math teachers are more about discipline than teaching math. I've often mentioned having a personality conflict with my 8th grade math teacher. She expected everything done one way, and only one way, and would give a 0 for anything done any other legitimate way. It basically meant unlearning everything that I had learned the previous year. She also expected us to show every step, no matter how trivial, including things as trivial as 1+1=2, of you'd get a 0 for that problem. We didn't learn math from her. What we learned was that she was the boss and we weren't, and that we had to obey everything she said and do everything her way. Maybe that's a useful skill for the real world, but it doesn't help you learn any math.
1. Math is, to some degree, something that either comes naturally or not. If it doesn't come naturally to you, putting in more time isn't really going to help much.
I have the complete opposite view. If you want to excel at higher level math, then you must put in the hours and beat your brains out if you have to.
The biggest reason for failure is that people give up too easily. This goes for any subject or pursuit.
I have the complete opposite view. If you want to excel at higher level math, then you must put in the hours and beat your brains out if you have to.
The biggest reason for failure is that people give up too easily. This goes for any subject or pursuit.
What are you defining as higher level math? I suspect that you are somebody for whom math didn't come naturally, so you had to spend the time. I literally never studied in math and got all A's.
1. Math is, to some degree, something that either comes naturally or not. If it doesn't come naturally to you, putting in more time isn't really going to help much. Math came very naturally to me, and I'd get straight A's in math without studying or any effort at all. Other subjects were more about putting in time, and I was usually too lazy for them.
2. As others have said, most elementary school teachers are liberal arts focused, and are very poor at teaching math.
3. Math tends to be more cumulative than other subjects, so somebody who falls behind, either due to a poor teacher, or do to relocating from one school district to another, may have a hard time catching up, especially if they never get a teacher that is willing to help them catch up. If somebody moves from a school district that teaches long division in 3rd grade, and moves to a district that teaches it in 2nd grade, they may always be at a disadvantage. I know that Katy Perry said that her parents frequently relocated, and that is one reason why she always struggled with math, even though she seems to be at least a somewhat intelligent person.
On other hand, in social studies, every year is mostly independent of each other. In my high school, in 9th grade, we learned about the non-Western world, 10th grade was European history, 11th grade was US history, and 12th grade was half a year of participation in government, and half a year of economics. I had a very poor teacher in 9th grade who taught us almost literally nothing. But since there was no overlap between years, that gap in knowledge had no real impact on later years of social studies. Similarly, in science, if you have a poor teacher for biology, it doesn't impact your ability to learn chemistry or physics in a later year.
4. Many math teachers are more about discipline than teaching math. I've often mentioned having a personality conflict with my 8th grade math teacher. She expected everything done one way, and only one way, and would give a 0 for anything done any other legitimate way. It basically meant unlearning everything that I had learned the previous year. She also expected us to show every step, no matter how trivial, including things as trivial as 1+1=2, of you'd get a 0 for that problem. We didn't learn math from her. What we learned was that she was the boss and we weren't, and that we had to obey everything she said and do everything her way. Maybe that's a useful skill for the real world, but it doesn't help you learn any math.
I'm going to merge your first three:
Math being cumulative very true and what many people miss. If you don't get, say, multiplication then division won't come to you. If you don't understand fractions then percentages will be speaking Martian. And so on up through the classes.
You are correct up to a point about History. You can easily fail the chapter on the Mexican War and pass the one on the Civil War. But what happens is that for History taught in a linear fashion, which it usually is, then you start losing context and the "why" something happened later on.
I taught both US and World History and I'd always start the US year with a short (40 questions or so) quiz on what those 9th Graders had (supposedly) learned in US the year before in 8th (Exploration through Civil War). Most couldn't even name the American Revolution or the beginnings of the country (the latter impacts in 10th Grade when they take Local, State and National Government which is a High School Assessment class).
World was a whole different set of problems due to the bouncing around between regions and civilizations.
Hence why I always preferred teaching electives.
For your fourth item I had a 7th Grade Math teacher much the same, Pre-Algebra would be what it's called today. He assigned two pages from the text every day, no more and no less, all were word problems. Never explained the problems and it didn't matter if you got the answer correct if you didn't set up the answer in his format the question was marked wrong. I handed in innumerable assignments with every answer correct but zero for the grade. It took me awhile to catch on but also impacted my desire to do math.
Another personal anecdote:
I was a local elected official for over 20 years and in all that time I was one of the few council members who understood the budget.
This is an article about mathematically gifted children. They can not always be identified by academic and testing success. I'll try to find something on the mathematically un-gifted.
Discovering Mathematical Talent.
ERIC Digest #E482.ERIC Identifier: ED321487
Publication Date: 1990-00-00
Author: Miller, Richard C.
Source: ERIC Clearinghouse on Handicapped and Gifted Children Reston VA.
One topic that interests me - what defines a mathematician? At what point in the sequence of study does one become a mathematician? PhD? earning a salary doing mathematics? publishing a mathematics book? When did Einstein become a mathematician or was he never one? Am I barred from the title because i am retired and do it for "fun" or because my interest is in its philosophy? I know - it doesn't really matter.
The wikipedia entry defines it as "A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems." https://en.wikipedia.org/wiki/Mathematician
It's genetic. You either have the aptitude or you don't--although most people fall somewhere in between. But there are plenty of people who just cannot do math. They are not dumb and they can be very smart at languages. It's how the brain is wired. No amount of studying or tutoring is going to ram math into their heads.
They can get really good at geometry because that's visual and it's logical but get into algebra and...wha?????
My physics teacher was great at math but could he spell the words he was trying to write? No. And could he learn to spell? No. He couldn't help it because that's how his brain was wired. He couldn't draw either.
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