Why is Math so special?

[Guest]

I've never had a child in mute terror over an intellectual experience of a proof in trigonometry; I did have a child in terror over books, over ideas, over the symbolic baggage of the heritage that formed their world.

 

Accelerating math? As long as a child can function cognitively on the required level, absolutely. But for some other areas, not only cognitive but also emotional aspect should be considered - and maybe, sometimes, one really isn't to rush in all directions possible at a certain age.

 

Different people are terrorized by different things. When I read this I thought of my younger sister. We shared a room when we were kids, and she used to wake me up, grabbing my feet under the covers, screaming and sobbing in terror: "The numbers are coming! The numbers! The teacups!" To this day I'm not quite sure what it was all about.

 

Also, it's worthwhile to consider acceleration at at variety of points/ages. It was standard in previous centuries to send children of twelve and thirteen up to Cambridge or Oxford where they were expected to tackle all sorts of adult textual material, and many precocious kids from wealthy households read freely in their parents' libraries with no one to approve or disapprove of what they encountered at a tender age. This isn't to say one approach or the other is better than the other, just to point out that kids will respond differently depending not only on their particular personality but also perhaps according to the norms of the historical period in which they grow up. (Oh, no, she's doing cultural relativism again! -- this is meant nicely by the way; I'm joking about myself.)

[Ester Maria]

I do agree with you, Karen, with regards to both of your points. People indeed are terrorized by different things, respond differently if put through a same educational experience, and much of the modern fuss about acceleration was unheard of in the previous centuries.

 

What I had in mind, though, was an intellectual, conscious experience, as well as the conscious emotional reaction, more than the drawbacks of such an experience in dreams, nightmares, etc. What I meant is that most of the children of this generation (and probably also of our generation), due to many factors from prolonged childhood to greater life expectancy to changing of economic system and so on, do not mature emotionally as young as previous generations did. What is considered precocious in my daughters might, in fact, have been a regular experience of a child born into a wealthy family who had a privilege of similar education at those ages. But that's exactly the point I have in mind - it's not so much about some kind of *biological* determination (even if we can speak of a cognitive development that's more or less "sound", on average, at certain ages), but more about the mismatch (in this generation) between intellectual and life experience. My point was that math, as a formal science, isn't really life-dependent (it's too early for me to be coherent, I hope you get what I mean: the understanding of math, especially conceptually, depends not on one's life experience but rather on cognitive development), while things which directly stem from the culture, all of the symbolic baggage, are life-dependent. And as time passes and world changes, some of it becomes so remote from a typical life experience of an average child that it becomes not only difficult to comprehend, but emotionally heavy... Think of Dickens, a popular writer of his age (!) and a historical portrait (yes, I know, you're probably reading this in utter disbelief that somebody who approaches literature so formally as I do is writing these lines and taking history into account for a moment :))... for a modern child, Dickens is a heavy experience. Even better, think of Bible. I cannot think of an emotionally heavier experience, actually, even for myself - it's a world so removed from the experience, and if one attempts to understand it, it can be quite heavy, quite disturbing in fact, especially when paired with other things that make up a symbolic experience of a child that's growing up in 20th/21st centuries.

 

Math, even if on its higher levels scary as a concept (take infinity for example), doesn't have that direct emotional touch, doesn't serve as a sort of stepping stone for building a relationship with the world and an ethical/emotional/etc. stance... That's why I meant that, generally, humanities might be more difficult to accelerate and to work on an overall higher level in that sphere, being that it's more directly emotional rather than "only" conceptual experience.

 

Not sure if this makes any sense, I still haven't had my coffee. :)

[HeidiD]

Sometimes when dd and I are reading her textbook -- which I quite like actually, compared to others I've seen -- it feels like we have to actually translate it; it feels almost like it does when I'm working with a foreign language.

 

This is exactly how I've always described the process to my kids - we are "translating" this material to its most comprehensible format in order to arrive at a solution. I'm guiding them to develop this approach quickly, on their own, each in accordance with their particular thinking style. Some are spatial, some are more verbal, so we're either stripping away the "excess" language or augmenting the material in reverse.

 

Textbook styles obviously vary a great deal (eg. Saxon vs. the very "wordy" Life of Fred) but at advanced levels such as engineering, wordiness would tend to get in the way of efficiency, so a student needing a large amount of verbal description to conceptualize would have more difficulty than someone who doesn't require it, and would work at a slower pace.

 

In addition, the authors of math texts (and technical writers in general) are often more spatial thinkers who tend to be less accomplished with verbal and written communication, and also sometimes assume more knowledge on the part of the reader than necessarily exists. So the predominantly verbal thinker (more frequently female than male) is a mismatch in that milieu, but would probably not be drawn to a math-oriented career anyway.

[regentrude]

Textbook styles obviously vary a great deal (eg. Saxon vs. the very "wordy" Life of Fred) but at advanced levels such as engineering, wordiness would tend to get in the way of efficiency, so a student needing a large amount of verbal description to conceptualize would have more difficulty than someone who doesn't require it, and would work at a slower pace.

 

In addition, the authors of math texts (and technical writers in general) are often more spatial thinkers who tend to be less accomplished with verbal and written communication, and also sometimes assume more knowledge on the part of the reader than necessarily exists. So the predominantly verbal thinker (more frequently female than male) is a mismatch in that milieu, but would probably not be drawn to a math-oriented career anyway.

 

It is interesting that you bring up textbooks. There is a problem with the use of textbooks in math and science education in schools. My college students have not learned to read a textbook with continuous verbal explanations; all they want is find the colored box that gives the important equation, find the highlighted summary, find the soundbite that gives them the formula to plug and chug. Publishers cater to this trend by having colorful boxes, sidebars, pictures, tables - anything but actual text that actually explains a concept.

So, books that are actually written well and explain science through good language (they exist!) are not liked by the students who find them too wordy, because they have not learned to talk or write about concepts - to them , physics is about putting numbers in an equation.

So the bad teaching in schools closes the door to a use of those verbal ways of learning.

In contrast, my introductory texts were black and white, had mostly text and few illustrations and dealt with the subject on 400 pages instead of the 1000 pages in a "modern" book.

[Guest]

It is interesting that you bring up textbooks. There is a problem with the use of textbooks in math and science education in schools. My college students have not learned to read a textbook with continuous verbal explanations; all they want is find the colored box that gives the important equation, find the highlighted summary, find the soundbite that gives them the formula to plug and chug. Publishers cater to this trend by having colorful boxes, sidebars, pictures, tables - anything but actual text that actually explains a concept.

So, books that are actually written well and explain science through good language (they exist!) are not liked by the students who find them too wordy, because they have not learned to talk or write about concepts - to them , physics is about putting numbers in an equation.

So the bad teaching in schools closes the door to a use of those verbal ways of learning.

In contrast, my introductory texts were black and white, had mostly text and few illustrations and dealt with the subject on 400 pages instead of the 1000 pages in a "modern" book.

 

Exams have undergone this kind of change too. Dh, educated in the UK, often tells me about the fact that his science exams well into upper levels were essays rather than problem sets. And in the US we've tended to move even further from problem sets towards multiple choice exams for so many, many subjects.

[Guest]

 

In addition, the authors of math texts (and technical writers in general) are often more spatial thinkers who tend to be less accomplished with verbal and written communication, and also sometimes assume more knowledge on the part of the reader than necessarily exists. So the predominantly verbal thinker (more frequently female than male) is a mismatch in that milieu, but would probably not be drawn to a math-oriented career anyway.

 

That is what I was trying to say about how the math books we encounter are written; you said it much better.

 

However, i disagree with your conclusion, and this is precisely what Marilyn Burns is trying to confront: it's not that the verbal thinker would not be drawn to math at all events; rather, he or she has not been given verbal tools to interpret or work through spatial problems, or ways to help develop spatial thinking over a number of years.

[Bostonian]

In contrast, my introductory texts were black and white, had mostly text and few illustrations and dealt with the subject on 400 pages instead of the 1000 pages in a "modern" book.

 

I too have noticed that textbooks have become fatter and more expensive. I'd be interested to know what your favorite math and science textbooks are -- I might buy them used.

[HeidiD]

It is interesting that you bring up textbooks. There is a problem with the use of textbooks in math and science education in schools. My college students have not learned to read a textbook with continuous verbal explanations; all they want is find the colored box that gives the important equation, find the highlighted summary, find the soundbite that gives them the formula to plug and chug. Publishers cater to this trend by having colorful boxes, sidebars, pictures, tables - anything but actual text that actually explains a concept.

 

Yes, I agree. Pictures, sidebars and blurbs have overpowered the text in many cases (in order to compete with our soundbite/visually-based electronic communication habits?). Comparing an old college chemistry text I have on my shelf to one that I bought recently for homeschooling, the large photographs alone account for approximately 25% in the newer book. I can see that constant exposure to these types of texts could definitely undermine one's ability to process more verbally-based explanations efficiently. Ideally, there is somewhat of a balance between verbal explanations of the material, and the material itself (concepts supported by facts, formulas, and diagrams).

 

That being said, wouldn't a math text with "continuous verbal explanations" be just as unwieldy as one with too many pictures, etc.? The subject doesn't lend itself (in an an efficient way, at least) to a predominantly verbal presentation.

 

So, books that are actually written well and explain science through good language (they exist!) are not liked by the students who find them too wordy, because they have not learned to talk or write about concepts - to them , physics is about putting numbers in an equation.

 

So the bad teaching in schools closes the door to a use of those verbal ways of learning.

 

From what I've observed, the bad teaching is often the result of an instructor's inability to explain the concepts clearly or an inability to relate to the students' level of understanding. And this problem extends to some textbooks also. I think it's partly due to the fact that most people who are highly verbally-inclined aren't drawn to technical areas, and the people in technical fields who are writing the books and manuals aren't the best writers or communicators because their strengths lie in the opposite direction :).

 

Why would a person whose strengths lie in one direction choose a career in the opposite direction? How much effort should be expended to make scientific and technical instruction beyond the high school level as user-friendly as possible for those whose intellectual strengths lie in a different area? And how much effort should an instructor expend to accomplish this? The problem isn't always with the instructor; it can also be the result of a mismatch between the student and the course of study.

[regentrude]

From what I've observed, the bad teaching is often the result of an instructor's inability to explain the concepts clearly or an inability to relate to the students' level of understanding.

 

Let me say something very heretical here (which I have debated with colleagues numerous times): the role of the instructor is grossly overrated, at least at college level

 

My teaching experience tells me one thing: the brightest students will succeed, not matter how well or badly I teach. The students who fail now would fail no matter how I teach, too. The only area where the quality of instruction has a small impact are a few mid-range students - I may help a C student achieve his B, or a B student to scrape a low A. But at the ends of the distribution, student success is entirely determined by the student.

 

We frequently try to compare results between different instructors in the same course - everybody is hoping to prove that the own students win - and the reality is that, averaged over a large class, there is no correlation between the (common) test results and the instructor.

 

The instructor personality and quality definitely has an impact on the student's perception of the course... but I believe the impact on the educational outcome is overrated.

(Please remember, I am talking about college level instruction where the instructor acts more as a facilitator and mentor and only a third of the actual work is happening in class - the situation in an elementary school classroom would certainly be different.)

[Crimson Wife]

My teaching experience tells me one thing: the brightest students will succeed, not matter how well or badly I teach. The students who fail now would fail no matter how I teach, too. The only area where the quality of instruction has a small impact are a few mid-range students - I may help a C student achieve his B, or a B student to scrape a low A. But at the ends of the distribution, student success is entirely determined by the student.

 

:iagree:

But let's not forget that statistically, the majority of students will fall in this middle-range.

[regentrude]

But let's not forget that statistically, the majority of students will fall in this middle-range.

 

Yes - and there they will pretty much stay, regardless of the teacher. I have not yet managed to turn an average student into a brilliant one, no matter how hard I try.

[HeidiD]

Let me say something very heretical here (which I have debated with colleagues numerous times): the role of the instructor is grossly overrated, at least at college level

 

Not heretical at all - I think you're absolutely right (particularly in regard to math & science).

[kiana]

Let me say something very heretical here (which I have debated with colleagues numerous times): the role of the instructor is grossly overrated, at least at college level

 

My teaching experience tells me one thing: the brightest students will succeed, not matter how well or badly I teach. The students who fail now would fail no matter how I teach, too. The only area where the quality of instruction has a small impact are a few mid-range students - I may help a C student achieve his B, or a B student to scrape a low A. But at the ends of the distribution, student success is entirely determined by the student.

 

We frequently try to compare results between different instructors in the same course - everybody is hoping to prove that the own students win - and the reality is that, averaged over a large class, there is no correlation between the (common) test results and the instructor.

 

The instructor personality and quality definitely has an impact on the student's perception of the course... but I believe the impact on the educational outcome is overrated.

(Please remember, I am talking about college level instruction where the instructor acts more as a facilitator and mentor and only a third of the actual work is happening in class - the situation in an elementary school classroom would certainly be different.)

 

I agree that it's overrated. However, I will say that I have seen instructors at university whose averages were far below their colleagues. (Say, 15 points below, on a 100 point scale, in a large class.) There were many students who would probably have earned a C, but due to the fact that they didn't realize until midterms that it was better to skip class and read the book than attend lectures, they'd misunderstood some of the early part of the course and were left playing catchup.

 

I do also agree, however, that barring the few really, really dreadful ones out there, most instructors are more than adequate.

[EKS]

Let me say something very heretical here (which I have debated with colleagues numerous times): the role of the instructor is grossly overrated, at least at college level

 

My teaching experience tells me one thing: the brightest students will succeed, not matter how well or badly I teach. The students who fail now would fail no matter how I teach, too. The only area where the quality of instruction has a small impact are a few mid-range students - I may help a C student achieve his B, or a B student to scrape a low A. But at the ends of the distribution, student success is entirely determined by the student.

 

We frequently try to compare results between different instructors in the same course - everybody is hoping to prove that the own students win - and the reality is that, averaged over a large class, there is no correlation between the (common) test results and the instructor.

 

The instructor personality and quality definitely has an impact on the student's perception of the course... but I believe the impact on the educational outcome is overrated.

(Please remember, I am talking about college level instruction where the instructor acts more as a facilitator and mentor and only a third of the actual work is happening in class - the situation in an elementary school classroom would certainly be different.)

 

My father got his degree in physics from Caltech, and he never went to lectures (for science and math courses) because he considered them a waste of time. All of the learning happened in the process of doing the problems.

[Storm Bay]

Let me say something very heretical here (which I have debated with colleagues numerous times): the role of the instructor is grossly overrated, at least at college level

 

My teaching experience tells me one thing: the brightest students will succeed, not matter how well or badly I teach.

:iagree: for the most part, because the brightest are able to learn anyway, sometimes just from reading the text, and some are going to struggle regardless. My next comment is based on my brother's experience; he is a university instructor in physics. I do think that in some subjects, such as Physics, a gifted instructor (gifted in teaching, and I do think that a very few instructors are truly gifted there), can make a difference in another way, and that is in getting students who are only in the class for a mandatory credit so interested in the subject that they change their majors based on this. There are a few who regularly turn college students on to a subject or even a career this way. These instructors often are able to reach most of the students in the middle ability range who are willing to do the regularly.

Edited December 11, 2010 by Karin

[cdgni]

Totally guessing here... but could it be that people have been taught both directly and indirectly that acceleration means kids may be able to manipulate numbers but won't possibly understand the concepts, because they need to reach a certain age for their brain to develop in ways that allow this understanding?

 

In a way this pertains to my daughter. She "sees" patterns quickly and finds the right answer in her mind. However, she doesn't always understand how she knew the right answer. Her mind figures it out in the subconscious level, but when I ask her to show me how she accomplished it- she can't. We have to backtrack and go over some steps even though she can quickly figure out the pattern.

 

 

The other aspect of it is just plain anti-intellectualism, I think, and it's aimed at math because that is the representative nerdy field. Kids who accelerate in math are not going to be "able to be kids," they'll be socially inept, etc. I think this incredible stereotype is alive and flourishing in our society.

 

Quite unfortunate, but I agree with KarenAnne. What a shame that we don't encourage and foster a society that encourages mathematical abilities starting in the primary years.

[Storm Bay]

Quite unfortunate, but I agree with KarenAnne. What a shame that we don't encourage and foster a society that encourages mathematical abilities starting in the primary years.

:iagree: It would be an uphill battle for a while, though, but it would be very good.

[Thomas]

I think math is an important subject since it shows ones ability to abstract surrounding world into simple concepts. It also shows the problem solving abilities and how easy can an individual think about a solution when he/she needs to deal with a real problem in life. Besides all these, mathematics has a lot of application in real world as you can see in most engineering fields.

[trying my best]

Math is a foundation of many things in life like thinking LOGICALLY, finding short solution to a problem, knowing there a lot of choices but you need to pick the right one. Math was my favorite subject when i was little. I am also spending more time on math with my son wanting for him to love math and to know math. Math can be the hardest subject so I teach it young. When you know math it makes things easier :)

[HelenNotOfTroy]

My teaching experience tells me one thing: the brightest students will succeed, not matter how well or badly I teach. The students who fail now would fail no matter how I teach, too. The only area where the quality of instruction has a small impact are a few mid-range students - I may help a C student achieve his B, or a B student to scrape a low A. But at the ends of the distribution, student success is entirely determined by the student.

 

I want to comment on this. When I took geometry in high school, we had a student teacher for the second semester. I got A's on all my tests the first semester and flunked the second semester. This was due to having a different teacher, I did not do any more or less homework/studying at any point during the year (which is to say, I did none).

[GypsieFamily]

I agree with much of what has been said. I think, to go back to e original question, that maths is different because the younger accelerated child with an aptitude for maths may ceiling when the maths becomes more abstract at algebra rather than concrete. Despite always usi g problems like 5x = 25, algebra was too overwhelming for my then 8 year old and we've had to back off completely. We now practice just maths facts, and will start formal maths again when I think he is conceptually ready for the abstract.

 

Dd8.has a kindle with many classics from Gutenberg. She has read dickens. She reads Anne of green gables series, narnia, Peter Pan, and whatever else is taking her fancy. Staying with the classics, the themes are generally age appropriate. She is overly cautious wight self censuring, but willing to try almost anything. Her sentence structure and vocabulary are archaic and stand out even to a casual acquaintance. She has had discussions with me about how many blows of an ax it took to remove various historical personages' heads, and made a few comments about the many wives of Henry viii. Other than this. It poses no other difficulties in schooling her. It actually makes it easier.

 

It is easier hs an accelerated literacy kid. It poses extra challenges hs a maths accelerated kid.

[jennynd]

I don't agree with this at all. In my whole life I 've been blessed with great Math teacher. I came from another country (Taiwan). My math teacher sparks my love to math and I don't think it will be the same if not for them. Actually standard did a study that if we remove the bottom 10% bad teacher, US international score will raise dramatically.

Now, it might different when you go to high school/college. I admit I skipped class becuase I know I can self study. But not from elementry and middle school kis. they are still learning how to study.

 

Let me say something very heretical here (which I have debated with colleagues numerous times): the role of the instructor is grossly overrated, at least at college level

 

My teaching experience tells me one thing: the brightest students will succeed, not matter how well or badly I teach. The students who fail now would fail no matter how I teach, too. The only area where the quality of instruction has a small impact are a few mid-range students - I may help a C student achieve his B, or a B student to scrape a low A. But at the ends of the distribution, student success is entirely determined by the student.

 

We frequently try to compare results between different instructors in the same course - everybody is hoping to prove that the own students win - and the reality is that, averaged over a large class, there is no correlation between the (common) test results and the instructor.

 

The instructor personality and quality definitely has an impact on the student's perception of the course... but I believe the impact on the educational outcome is overrated.

(Please remember, I am talking about college level instruction where the instructor acts more as a facilitator and mentor and only a third of the actual work is happening in class - the situation in an elementary school classroom would certainly be different.)

Edited December 31, 2010 by jennynd

[Storm Bay]

I don't agree with this at all. In my whole life I 've been blessed with great Math teacher. I came from another country (Taiwan). My math teacher sparks my love to math and I don't think it will be the same if not for them. Actually standard did a study that if we remove the bottom 10% bad teacher, US international score will raise dramatically.

Now, it might different when you go to high school/college. I admit I skipped class becuase I know I can self study. But not from elementry and middle school kis. they are still learning how to study.

 

It's true that there are a few teachers who can help kids love or at least like a subject, which can make a huge difference. Also, some teachers have a gift. I have a db who has done this for students in university physics (which has a lot of math, and he originally had to choose between majoring in math or physics.) He had been able to get students so excited about Physics that they have changed their majors. They can enable more dc to understand a subject. However, I think that Regentrude's point is still valid, particularly at the university level. Even the most gifted teachers aren't going to be able to enable every student to excel in that subject because, despite PC philosophy, not everyone is going to excel in math. She also made a great point about Physics (1 in 6 are women in university Physics--was it in this thread? It's getting so long now) which makes sense given the differences in male/female brains (the ratio of 1 in 6 is very close to the 80/20 ratio of brain differences in men & women regarding spatial abilities, etc.)

Edited December 31, 2010 by Karin