Note : This post was triggered off by my answers on a quora.com question on math : http://www.quora.com/Has-US-culture-created-a-self-fulfilling-prophecy-that-math-and-science-are-hard-to-learn
Since I've a lot more thoughts on the subject will update as and when they occur to me here.
Q) Why are Math and Science so hard to learn?
"Never Let your Schooling Interfere with your Education" - Mark Twain
Seems like a typical "Nature vs. Nurture" problem.
Maybe it's time we started thinking more positively instead with "Nature AND Nurture" i.e. using the best of Natural and Environmental inputs possible together.
I don't think country alone has everything to do with it.
Teaching/Learning style match or mismatch has more to do with it.
The amount of effort the parent/student puts in is also very important.
How much learning/education is valued at home and insisted upon is also a definite factor.
Though the quote above seems to accentuate the myth that Maths is only for "mathsy-people", it does give an idea that differences exist among students. Differences in learning style, mental modelling etc.
1) 100s of years of mathematical discovery has to be crammed into a few years. That too with neither the need nor the symbolic learning style for most kids. Effects speak for themselves
2) Notation and symbols don't come naturally to most people. 2 mathematicians working on calculus came up with different notations. The "better" notation won out. So notation is not the same as knowledge or meaning. Need to concretize the problem/solution in different ways to get the necessity and the invention through to the students.
George Polya has written a book on Teaching Mathematics by "Discovery" and on how to train Math teachers.
3) Much of Mathematics was invented/discovered while trying to solve real-life problems. What students get is the dessicated extract. Time to put the juice back into Math. People always learn better when they have a need to use it in a real life situation. See : http://learnersparadise.blogspot.com/2010/11/teach-yourself-mathematics-in-intuitive.html
I had an amazing math teacher who helped me see the beauty behind the notation. He was working on his PHD thesis on facial recognition using wavelet transforms. The formulae never clicked with him until his guide advised him to learn from applications of wavelet transforms in Economics. They used Graphs etc. to explain the basis/logic of the topic. This suited the natural learning style of my teacher and he could bridge the gap between the formulae and the mental model constructed from the graphs.
"Necessity is the mother of Invention". If a student feels engaged and more motivated when it comes to school/college math the results will follow.
1) Applied mathematics and understanding the reasoning behind it will attract more students. This increases the pool of students who like mathematics.
2) Theorem proving also has it's own heuristics as captured so nicely by George Polya in his books. He talks about "discovering" the reasoning behind the math and how expert mathematicians work out the problem by a mix of intuition, trial-and-error and experience.
3) Math has huge applications in Science and Technology. Having a far larger pool of interested students will definitely be a win-win situation ANY which way you look at it.
a) Multiple Intelligence (Interacting with the world in different ways)
b) Learning Style (Input Preferences - visual, auditory, kineaesthetic)
c) Creativity in engaging the students
Basically it's about being right-handed in a left-handed world or vice-versa. If the way-you-think-and-learn is not the way-you're-taught of course you're not gonna get it!!
Garbage-In, Garbage-Out. If you enter data into a computer in an incorrect "data-format" it's no wonder if it is rejected.
Somebody may like abstract symbolic formulae and pick those up very fast. Someone else may like to hear/visualise the problem. Someone else may like to work with their hands i.e. a concrete model of the problem.
Example :
1) Watson and Crick finally solved the problem of DNA structure using a ball and stick model to match the scant x-ray diffraction data!! One was very hands-on, the other very knowledge/math oriented. BOTH people contributed to solving different parts of the problem and won a Nobel Prize!!
2) Richard Feynman "invented" his own geometry and came up with his own symbols and notation. From his book accounts he was both a Visual person and a hands-on person. He repaired radios and learnt Ham Radio with his father.
To enable the student to identify and analyse the problem, choose from multiple options and choose the best option. Also ability to adapt problem or solution as per circumstances.
Unfortunately many times the goal/"function" is assumed to be getting through syllabus by the teacher. So the "form" is that seen in the book. Teacher may be stuck in his/her own way of thinking and may not try/know enough about student's way of thinking.
So if "function" is to convey meaning to a student the "form" of the teaching must adapt to student i.e. learning-style, interests/hobbies, pre-existing knowledge from his her background, questions etc.
So teacher must really understand the subject.
Teacher must also understand the way the student thinks.
And must be able to customize the message in the form the student will most readily accept and process it.
Reading body language and testing the understanding would help the teacher ensure that the student doesn't just nod the head but is really enthused with the understanding of the topic and it's implications in the student's life.
Experts think differently when compared to Novice or Intermediate learners. They've already mapped out most things and are able to quickly match the problem characteristics with solution characteristics. They're not fooled by surface similarities and see the underlying forces at work. They hence are able to choose right/better answers. They also have more options which they can quickly scan for possible solutions.
Since I've a lot more thoughts on the subject will update as and when they occur to me here.
Q) Why are Math and Science so hard to learn?
"Never Let your Schooling Interfere with your Education" - Mark Twain
Seems like a typical "Nature vs. Nurture" problem.
Maybe it's time we started thinking more positively instead with "Nature AND Nurture" i.e. using the best of Natural and Environmental inputs possible together.
Environmental Factors
Just need to look at successful people (and unsuccessful ones) and figure out the factors involved. What made the difference? Esp. if the person was not "mathsy" at first but then somehow got to be good at maths.I don't think country alone has everything to do with it.
Teaching/Learning style match or mismatch has more to do with it.
The amount of effort the parent/student puts in is also very important.
How much learning/education is valued at home and insisted upon is also a definite factor.
One Size Fits ALL
“Everybody is a genius. But if you judge a fish by its ability to climb a tree, it will live its whole life believing that it is stupid.” ― Albert EinsteinThough the quote above seems to accentuate the myth that Maths is only for "mathsy-people", it does give an idea that differences exist among students. Differences in learning style, mental modelling etc.
1) 100s of years of mathematical discovery has to be crammed into a few years. That too with neither the need nor the symbolic learning style for most kids. Effects speak for themselves
2) Notation and symbols don't come naturally to most people. 2 mathematicians working on calculus came up with different notations. The "better" notation won out. So notation is not the same as knowledge or meaning. Need to concretize the problem/solution in different ways to get the necessity and the invention through to the students.
George Polya has written a book on Teaching Mathematics by "Discovery" and on how to train Math teachers.
3) Much of Mathematics was invented/discovered while trying to solve real-life problems. What students get is the dessicated extract. Time to put the juice back into Math. People always learn better when they have a need to use it in a real life situation. See : http://learnersparadise.blogspot.com/2010/11/teach-yourself-mathematics-in-intuitive.html
I had an amazing math teacher who helped me see the beauty behind the notation. He was working on his PHD thesis on facial recognition using wavelet transforms. The formulae never clicked with him until his guide advised him to learn from applications of wavelet transforms in Economics. They used Graphs etc. to explain the basis/logic of the topic. This suited the natural learning style of my teacher and he could bridge the gap between the formulae and the mental model constructed from the graphs.
"Necessity is the mother of Invention". If a student feels engaged and more motivated when it comes to school/college math the results will follow.
1) Applied mathematics and understanding the reasoning behind it will attract more students. This increases the pool of students who like mathematics.
2) Theorem proving also has it's own heuristics as captured so nicely by George Polya in his books. He talks about "discovering" the reasoning behind the math and how expert mathematicians work out the problem by a mix of intuition, trial-and-error and experience.
3) Math has huge applications in Science and Technology. Having a far larger pool of interested students will definitely be a win-win situation ANY which way you look at it.
Nature - I Want it My WAY
It's rather surprising that I didn't find much information being mentioned here on advances in Cognitive Science esp.a) Multiple Intelligence (Interacting with the world in different ways)
b) Learning Style (Input Preferences - visual, auditory, kineaesthetic)
c) Creativity in engaging the students
Basically it's about being right-handed in a left-handed world or vice-versa. If the way-you-think-and-learn is not the way-you're-taught of course you're not gonna get it!!
Garbage-In, Garbage-Out. If you enter data into a computer in an incorrect "data-format" it's no wonder if it is rejected.
Somebody may like abstract symbolic formulae and pick those up very fast. Someone else may like to hear/visualise the problem. Someone else may like to work with their hands i.e. a concrete model of the problem.
Example :
1) Watson and Crick finally solved the problem of DNA structure using a ball and stick model to match the scant x-ray diffraction data!! One was very hands-on, the other very knowledge/math oriented. BOTH people contributed to solving different parts of the problem and won a Nobel Prize!!
2) Richard Feynman "invented" his own geometry and came up with his own symbols and notation. From his book accounts he was both a Visual person and a hands-on person. He repaired radios and learnt Ham Radio with his father.
Nurture - "Form must follow function"
What, is the purpose of math/science education? Short answer - Problem solving.To enable the student to identify and analyse the problem, choose from multiple options and choose the best option. Also ability to adapt problem or solution as per circumstances.
Unfortunately many times the goal/"function" is assumed to be getting through syllabus by the teacher. So the "form" is that seen in the book. Teacher may be stuck in his/her own way of thinking and may not try/know enough about student's way of thinking.
So if "function" is to convey meaning to a student the "form" of the teaching must adapt to student i.e. learning-style, interests/hobbies, pre-existing knowledge from his her background, questions etc.
So teacher must really understand the subject.
Teacher must also understand the way the student thinks.
And must be able to customize the message in the form the student will most readily accept and process it.
Reading body language and testing the understanding would help the teacher ensure that the student doesn't just nod the head but is really enthused with the understanding of the topic and it's implications in the student's life.
Mastery - Learning is about Connections
The more connections you make between something new and something old the better you learn it. The more emotion is associated with learning the more firmly it is anchored in your memory.Expertise
Having breadth and depth of knowledge and skill in using that knowledge in real-life or at work is expertise.Experts think differently when compared to Novice or Intermediate learners. They've already mapped out most things and are able to quickly match the problem characteristics with solution characteristics. They're not fooled by surface similarities and see the underlying forces at work. They hence are able to choose right/better answers. They also have more options which they can quickly scan for possible solutions.
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