Do You Have to Learn Math Either Early or Never?

[Marat]

The psychologist Piaget says that unless you learn certain things at the proper age, you have forever passed the developmental stage at which you brain is capable of learning them well, so it is too late to acquire that knowledge later. However, his theory applies only to early childhood development.

 

But I wonder if this applies to the learning of advanced mathetmatics? Every person I knew in my first year at university who was majoring in math said that he had learned calculus when he was 12 and some relative had, for some reason never clear to me, given him a calculus text which he simply devoured. Ever since then, these math majors would say, they had found math 'easy,' and generally regarded their program of studies such a breeze that they could spend most of the day reading through the entire New York Times or listening to the radio, while everyone else majoring in other subjects had to work feverishly hard just to keep up. I kept in touch with one of these people who got a Ph.D. in math at M.I.T. and literally did all his doctoral work in a period of about three weeks.

 

So is that the secret: if you learn calculus at the proper time, somewhere around 12, math is easy, and otherwise it is dauntingly difficult? If that is true it seems profoundly unfair to aspiring mathematicians that no one explains to them that they have lost their chance to make it in the field once they are 13.

[Ringer]

Piaget only studied his own children for his developmental theories. There is no study that I've seen that age has a lot to with inability to learn math except a study that said it is very difficult for children under a certain age, I think around 11, to learn algebra or any other abstract mathematics. This is thought to be due to that their abstract reasoning abilities aren't developed enough to truly understand the methodology of the math. That was about 4 years ago I read that so I'm not really sure.

 

But there are plenty of people who are older and gain mathematical knowledge easily, I would assume more than there are children who learn it easily. The reason you hear about the children learning these things at a young age and becoming amazingly proficient is the same reason you hear about people jumping out of planes and surviving, because it's amazing not because they are the norm and we should all be able to do this.

[Marat]

But in my experience it was not especially rare that the people who felt able to become university math majors had already completely mastered calculus by age 13. When I first arrived at university I thought I was doing all right by way of mathematical preparation by having passed the advanced placement exam in calculus in my last year of secondary school, but then I was shocked to find that everyone intending to be a math major had already reached that stage five years previously. What chance would a student have just starting second year university calculus when everyone else was doing Lee Groups, Goedel's Incompleteness Theorem, Linear Algebra, and Field Theory?

[Ringer]

What do you mean especially rare, it could be you only remember those cases because they surprised you. It's sort of a confirmation bias, you remember the special exceptions and ignore those who had not made an impact. If you look at the average child most can't even do algebra when they are going into middle school. I don't believe that our school systems are THAT messed up that they can't teach algebra in the prime of mathematical learning.

 

Of course I could be wrong I just haven't seen any studies or personally experienced anything that would make me believe that excelling at calculus, or any higher math, should be simple for children in that early stage of development.

[hypervalent_iodine]

But in my experience it was not especially rare that the people who felt able to become university math majors had already completely mastered calculus by age 13. When I first arrived at university I thought I was doing all right by way of mathematical preparation by having passed the advanced placement exam in calculus in my last year of secondary school, but then I was shocked to find that everyone intending to be a math major had already reached that stage five years previously. What chance would a student have just starting second year university calculus when everyone else was doing Lee Groups, Goedel's Incompleteness Theorem, Linear Algebra, and Field Theory?

 

I did a chemistry major at university and didn't touch a single maths subject until I was in my second year, where I picked up a second year linear algebra course. I of course had to study a little harder, having not done maths since high school, yet I still managed to get full marks in all my weekly assignments and did quite well in both the mid and end of semester exams. At one point I was even tutoring a friend doing the course with me - he who had done all of the pre requisite first year maths courses whereas I had not. I think the key for some people is just the motivation to do extra work when you don't quite understand enough about a concept to apply it. I spent hours going over tutorial sheets and lecture notes until I could fully visualise everything.

 

Another thing that was pointed out to me by a friend who is doing Electrical engineering is that as our pool of knowledge expands, we have to push more and more of the concepts that we would traditionally learn at a university level to lower secondary levels to make space for all the new 'stuff' (for lack of a better term). He does a lot of tutoring for high school students in maths and was quite taken aback himself to come across a student needing help with an area in physics he himself had only just covered in a second year fields course. When you think about it too, some of the basic areas that we have learnt about in high school would have, at some point in the past, been exclusively university territory.

[HamsterPower]

It's more like he finished learning calculus at 12 so he had that much more time to review and practice before university,

Math is a matter of practicing

[Genecks]

...When you think about it too, some of the basic areas that we have learnt about in high school would have, at some point in the past, been exclusively university territory.

 

College algebra...

 

The psychologist Piaget says that unless you learn certain things at the proper age, you have forever passed the developmental stage at which you brain is capable of learning them well...

 

 

I find it hard to believe a 12-year-old acted as an autodidact to learn the topic of calculus. It must have been a good book. I'm not sure about any of you, but in my experience, most mathematics books in the 1990s were cryptic. They were still pretty cryptic into the 2000s. It wasn't until there was a medium (the Internet), where people made it very obvious through book criticism sites, such as Amazon, that many math books are f'ing cryptic. Go back even further in history and you'll notice that they are even more and more cryptic. Luckily, some libraries actually have calculus problem set books that are around from the 1970s, so I guess it's practical for a kid to pull a Matilda and just start teaching himself. But then again, many cities, towns, and the such don't have good libraries that provide people with access to the materials to be autodidacts.

 

And, to my knowledge, not many grade schools actually possess calculus books. Truth be, I haven't walked in a grade school library or middle school library in many, many years; but I'm pretty sure they don't have those college-level resources there.

 

What am I getting at?

I'm pretty sure the kid didn't do it all by himself.

 

Whenever I hear or read about someone learning about a particularly advanced topic at an early age, I can't help but consider they had someone there to hold their hand. Otherwise, they had an excellent mentor who explained concepts quite easily so they were quickly understood.

 

In terms of neuronal development, I'm skeptical of the quick need to generate associative networks for greater information retention, understanding, and episemantic database building.

 

I think much of society has been led by a variety of early-age development theories that it's somehow damaged their self-esteem and belief that they can learn an advanced topic even in old age. As such, people don't dare to attempt learning the topic. That's my view.

 

There are a lot of theories about language development and whatnot. If there is perhaps some truth to all of the neuronal development, it might be hormone-based. I believe it's about the age of 12 where people have greater difficulty learning a new language. Before that age, people don't seem to have many problems. But surely, that's what I read over and over in the university. I don't have the data nor understanding of the methods used to collect the data.

 

So, it could be just as bogus as me reading something by Daniel Dennett where he describes some guy named Walters using a slide project to discuss free-will. I did read such an article by Dennett, and never once sourced nor cited was what Walters did.

 

I don't think I can deny, however, that some people are neuronally specialized for obtaining and digesting particular aspects of their environment. In simpler terms, I believe there are surely gifted people out there. I supposedly am gifted at math. Nonetheless, I don't really care for an intense study of the field. I find mathematics classes to be particularly boring. However, set me in front of a spreadsheet, and I seem to have fun.

 

One could say that I've generated a hidden layer that causes unconscious cognitive dissonance, thus causing me to have a lack of care for picking up particular mathematics materials. As such, when given a mathematics book, I don't learn as easily anymore. This goes back to any particular age-related learning issue that might come around: Some cognitive scientists believe that accumulated life experience unconsciously plays a role in making things harder to learn, similar to how a hard drive fills up. People can only hope to have a type of Zen enlightenment in order to pick up new topics.

 

As an aside, I do find that some people have an interesting affinity for mathematics. For instance, I have two nephews who I often discuss mathematics and statistics with. One nephew had trouble understanding the concept of tying his shoe, so I explain the process in mathematical terms. It made a lot more sense to him, and he quickly grasped how to tie his shoes. My other nephews finds amusement, as I do, in trying to determine and estimate the probability of events using mathematics, more in an actuary sense.

 

Also, finally, I think there is a lot of unfairness in academia. This is for sure. There are guilds, old boys' networks, and the such. Collecting the data and information is difficult, as people don't want to be exposed. As I'm not familiar with mathematics on a graduate level, I can't tell if sophisticated equipment is required to generate many new theories; but if it is, I could see not having access to such equipment to be a problem for mathematicians.

 

I often consider that we are still in the stone age of education, but then again, we have Internet communities who are often willing to discuss academic topics. So, it's not as if people are stranded in caves these days. Maybe we're in a renaissance.

Edited January 15, 2011 by Genecks

[HamsterPower]

wow, you wrote an essay on a topic so simple as that

College algebra...

 

 

 

 

I find it hard to believe a 12-year-old acted as an autodidact to learn the topic of calculus. It must have been a good book. I'm not sure about any of you, but in my experience, most mathematics books in the 1990s were cryptic. They were still pretty cryptic into the 2000s. It wasn't until there was a medium (the Internet), where people made it very obvious through book criticism sites, such as Amazon, that many math books are f'ing cryptic. Go back even further in history and you'll notice that they are even more and more cryptic. Luckily, some libraries actually have calculus problem set books that are around from the 1970s, so I guess it's practical for a kid to pull a Matilda and just start teaching himself. But then again, many cities, towns, and the such don't have good libraries that provide people with access to the materials to be autodidacts.

 

And, to my knowledge, not many grade schools actually possess calculus books. Truth be, I haven't walked in a grade school library or middle school library in many, many years; but I'm pretty sure they don't have those college-level resources there.

 

What am I getting at?

I'm pretty sure the kid didn't do it all by himself.

 

Whenever I hear or read about someone learning about a particularly advanced topic at an early age, I can't help but consider they had someone there to hold their hand. Otherwise, they had an excellent mentor who explained concepts quite easily so they were quickly understood.

 

In terms of neuronal development, I'm skeptical of the quick need to generate associative networks for greater information retention, understanding, and episemantic database building.

 

I think much of society has been led by a variety of early-age development theories that it's somehow damaged their self-esteem and belief that they can learn an advanced topic even in old age. As such, people don't dare to attempt learning the topic. That's my view.

 

There are a lot of theories about language development and whatnot. If there is perhaps some truth to all of the neuronal development, it might be hormone-based. I believe it's about the age of 12 where people have greater difficulty learning a new language. Before that age, people don't seem to have many problems. But surely, that's what I read over and over in the university. I don't have the data nor understanding of the methods used to collect the data.

 

So, it could be just as bogus as me reading something by Daniel Dennett where he describes some guy named Walters using a slide project to discuss free-will. I did read such an article by Dennett, and never once sourced nor cited was what Walters did.

 

I don't think I can deny, however, that some people are neuronally specialized for obtaining and digesting particular aspects of their environment. In simpler terms, I believe there are surely gifted people out there. I supposedly am gifted at math. Nonetheless, I don't really care for an intense study of the field. I find mathematics classes to be particularly boring. However, set me in front of a spreadsheet, and I seem to have fun.

 

One could say that I've generated a hidden layer that causes unconscious cognitive dissonance, thus causing me to have a lack of care for picking up particular mathematics materials. As such, when given a mathematics book, I don't learn as easily anymore. This goes back to any particular age-related learning issue that might come around: Some cognitive scientists believe that accumulated life experience unconsciously plays a role in making things harder to learn, similar to how a hard drive fills up. People can only hope to have a type of Zen enlightenment in order to pick up new topics.

 

As an aside, I do find that some people have an interesting affinity for mathematics. For instance, I have two nephews who I often discuss mathematics and statistics with. One nephew had trouble understanding the concept of tying his shoe, so I explain the process in mathematical terms. It made a lot more sense to him, and he quickly grasped how to tie his shoes. My other nephews finds amusement, as I do, in trying to determine and estimate the probability of events using mathematics, more in an actuary sense.

 

Also, finally, I think there is a lot of unfairness in academia. This is for sure. There are guilds, old boys' networks, and the such. Collecting the data and information is difficult, as people don't want to be exposed. As I'm not familiar with mathematics on a graduate level, I can't tell if sophisticated equipment is required to generate many new theories; but if it is, I could see not having access to such equipment to be a problem for mathematicians.

 

I often consider that we are still in the stone age of education, but then again, we have Internet communities who are often willing to discuss academic topics. So, it's not as if people are stranded in caves these days. Maybe we're in a renaissance.

 

 

 

[MR. Science]

This is not true, people vary, some people will never be good at math like others, it really is a gift of favoratism, since math requires a lot of energy and time to study properly.

[lemur]

I think advanced math mixes various skills. Part of it is coding. People have to be able to decode variables and symbols in a meaningful way while interpreting and processing them cognitively. It's like learning to read and think about what you're reading at the same time. It can be hard and requires a lot of practice. Math also requires simultaneous multi-stage processing. For example, it can be necessary to read and interpret numbers and variables with exponents while simultaneously processing their operators. Finally, a lot of math (such as calculus) builds on successive foundations. So even though there is abstract conceptual processing going on, this processing is aided by familiarity with related procedures. In other words, there are patterns of cognitive-logic that are translated through different types of math so the more types of math you've worked on to build familiarity, the better you get at learning new, more complex/advanced techniques.

 

I'm not very good at math but I have noticed similar patterns with written ideas as well as with music. With music, for example, becoming familiarity with note-selection makes it possible to learn a new instrument fairly quickly once you have established a "feel" for how notes are keyed/selected and voiced. To a non-musician (or even someone who is narrowly specialized in a certain style of playing a particular instrument), it probably seems quite daunting to learn to play multiple instruments, just as to a person with little math experience, the prospect of learning the many forms of advanced math would seem daunting when in fact they build-on and reinforce each other, no?

 

edit: I forgot to mention my own experience with statistics. I found that learning statistics was difficult because I already had advanced qualitative methods of working with theory directly. So many of the statistical procedures are complex quantitative procedures for performing analytical tasks. So when you are using math to test validity or model a simple yes or no question in terms of data-analysis, it is harder to swallow that the results are supremely valid when you've already learned qualitative methods for addressing the same questions with comparable accuracy. Btw, I'm talking about social science here, so all the quantitative data and variable are derived from subjective or interpretive observations of theoretically interpolated observations, so while the statistics produce accurate results, the results connect fuzzily with the source of the data. It would be like using an extremely accurate calculator to count breezes observed in a forest. Every time you observe a breeze you tally it and the calculator gives you an accurate sum of the tallies, but you can't eliminate the problem of when you count air-movement as a breeze and when it is a gust or just doesn't show up on your "radar" because it is too weak or escapes your gaze. So I think it would have been easier to learn the statistical math if I would have had faith that they would give me more accurate results than the data that was input in the first place - but statistics are only as good as the data.

Edited January 18, 2011 by lemur

[7Mick7]

While it's unlikely that many people can grasp and master calculus (or even algebra) at 12, it isn't impossible. I went through my dad's statistics and calculus books as a kid, and, though I wasn't completely self taught (dad answered some questions as they arose), all of the learning was outside of my school system, as my district was not equipped to teach calculus, even at the high school level. In fact, none of my teachers through my high school years had even taken a calculus course themselves.

 

If this was the situation facing this student (and others in mathematics programs), I am surprised that this student didn't avoid math or act out during class for many years after such an experience. Sitting through classes on fractions and division after calculus isn't the least bit fun (not to mention how demeaning it is!).

 

This being said, it is certainly possible to be an autodidact in higher level mathematics. I am self-taught from Calc III through enough mathematics to understand many papers in this field and related fields and to have had the pre-recs for an advanced degree waived after spending time talking with professors in my program. Time and motivation are essential, as well as a good grasp of research skills and cross-referencing what one reads. It is definitely easier to obtain material these days than it was when I was a child/teen in an area without Internet or a good public library, and I wouldn't be surprised to find a larger amount of students these days teaching themselves from such aforementioned resources to occupy their minds and expand their knowledge outside of school.

 

 

[Anura]

• i did not understand algabre concepts 25 years ago . but im capable of understanding them today

[Athena]

I am a math idiot, and thought it impossible for me to change this. But I am fascinated by math, and read books about it and watch DVD's about math. W.W. Sawyer's book "Mathematician's Delight" is very helpful. He stresses the importance of being able to visualize the problem. Like there is all this mind preparation stuff to do, before trying to learn math concepts. I think those who natural do well in math, learned this pre math thinking from someone close to them, or through early experiences, such as using construction blocks or doing origami. The ideas are already in their heads, and learning the math, just becomes away of expressing the idea they already have.

 

A good teacher can make a huge difference. I think if someone is really struggling with math, s/he might consider finding another teacher. Just because a person really knows something, that doesn't mean this person is a good teacher. W.W. Sayer wrote of public schools failing to teach math, when children are to parrot back what the teacher is directed to teach. This parroting back, may look great on those test the government is obbessed with, but they are not proof of real learning. Real learning is on a deeper level of understanding. A child with a good memory may parrot back a lot, only to hit a wall when trying to learn more advanced concepts. That is because, this memory and parroting is not equal to grasping the necessary math concepts. When we hit this wall, we need to begin working backwards to discover that concept we did not fully grasp, because we can not move ahead without a good comprehension of the basics. In his book he puts this delightful diagram of the math concepts in each chapter. Each chapter is represented by a block and the blocks are stacked in such away a person can see how the concepts are built on the one before. We can use this chart to see which chapter we need to reread, if we don't understand something in chapter 6 or chapter 11. How many teachers work with a child to help the child discover which concept s/he is having trouble with, and then being creative with an explanation that the child can relate to?

 

PS The Teaching Company sells complete college math courses on DVD's. really cheap, compared to the price of actually attending college. They frequently have 70% off sales and that is when to buy. Then you can come here and ask questions when you need help understanding something. How cool is that!

Edited January 31, 2011 by Athena