You can download all the important posts as PDF book "Unlocking the Secrets of Quantitative Thinking".
You can read more "How math can be applied to so many different fields and how we can use math in real life".
I am quite sure kids are more scared of strange mathematical names than of ideas, sequence of math operations, or numbers they represent.
Motivation is first step to make a decision to even consider making new assumptions! Hence, motivation is like turning a light switch ON.
Browsing web is similar to day dreaming. But, you should use your own BrainBrowser to explore worlds, ideas you already have in your mind.
You may decide to add 2+3, just 'cause. Or you measure distance with laser to get 2+3. Math will not care where the numbers came from.
Social and emotional contexts, and not only physics or engineering, frequently dictate which math calculations you are about to perform.
Math and emotions? If you get raise of $150 you will feel differently than if the raise was $15000. Countable objects matter in social contexts.
He made an emotional decision to think logically. :-)
Axioms is just a FANCY word for initial assumptions, truths you are dealing with, inside any field (not only math) where logic is used.
Mathematics is defined not by objects it counts, nor by reasons or logic why those objects are counted, but with concepts used to define matheatics axioms and to define proofs of mathematical theorems.
Your motivation and decision making process have their own axiomatic system you are using, perhaps, unconsciously, not calling them axioms..
Motivation (p) has nothing to do with the assumption (q) you are about to make. Yet both are necessary for s to happen, p AND q => s.
There's a rich world of ideas right behind math axioms. Axioms deny you access to them, yet it's from these ideas axioms came into being.
That "IF..." in mathematics, can open so many new doors in mathematics for you, but at the same time closes or other doors other IFs, from other fields, can show you if you don't keep open mind and try to combine different ideas.
Mathematics is about COUNTS. The question that many ask is do we really need to know so much about counts during the schooling?
Mathematics is about counts, and counts only. The way we think about them is logic. Axioms, theorems, lemmas are labels for logical results.
School may teach you how to use logical thinking, but it can not teach you to chose correct initial assumptions.
If math is derived from 9 axioms where teachers found all those apples, pears, giving, getting, having, and claiming these words are math.
You can use calculator to deal with numbers. But what device you will use to deal with logic for objects' relationship the counts represent?
Of course, even using calculator, you can explore the properties of numbers, discover number patterns.. Only have to know what to look for.
Whatever is outside the calculator, or more precise, abacus, is outside pure math.
Math is like a calculator! Only YOU know what numbers will be typed in, what they represent, what sequence of operations will be performed.
The confusion for kids learning math is constant mix of items that are counted and counts, numbers obtained. Separate the logic for two.
Moreover, WHAT are you counting IS NOT part of mathematics. Only counts and what you do with counts is a part of mathematics!
YOU keep tab where the numbers, counts come from. Mathematics will not do that for you. Units in physics are not part of mathematics.
Math has to start calculation with SOME numbers. Where those numbers come from is up to YOU. Is it from a guess, a measurement, a counting?
Students should be shown that math doesn't care if numbers came from measurements or from guessing. Math will equally use both numbers.
Where numbers come from? How about from love? http://yarzabek.wikispaces.com/, "Let me Count the Ways I Love You"..
Art is also about exploring human values, moral, ethical questions, like, how do you apply laws of physics or mathematical calculations.
Structured, logical thinking doesn't guarantee that your results will be correct. Initial assumptions, axioms is what matters the most. With wrong assumptions you can make structurally sound mistakes.
Art, as physics or math, has its own axiomatic system. The subjects of art axioms are emotions, moral, feelings, human experience.
In art, as long as the emotion, feeling, moral message is REAL, it does not matter whether the art work follows laws of physics.
In art, fantasy worlds are allowed, breaking laws of physics is possible, because art is about showing REAL emotions, feelings, moral messages.
In science, truth is about physical laws, while in art it is about true emotions, feelings, moral messages.
With word 'IF' you can build so many assumptions, and, if you hold to them, you can build a lot of fantastic worlds imaginary or not.
Track to creativity. Chose world of interest. Axiomatize it. Chose world where you interpret results. Axiomatize it. Play with both worlds.
You could develop calculus from axioms of course, but dealing with real world situations helped to pursue that direction of thinking sooner.
Interesting thing with invention is you don't have to prove nor show how you got it. But, again, same can be said for a mathematical proof.
If you want to understand calculus ask your teacher, on first lecture, to demonstrate limit in probability and stochastic integral.
Puncturing the Math Axiomatic Bubble Frontier to access inner mathematics functionality- http://goo.gl/ykvCC
The thrust, in jet engines, is obtained, in its essence, from fuel and oxidant molecules accelerated by mutual ELECTRICAL repulsive forces.
To me, the most interesting examples in math teaching will be in aviation, pirates and their treasure islands, and sculpture. Try it.
Education should not only show what other people have done. Education should encourage, lead students to make discoveries by themselves.
The idea of differential and integral does not follow from real numbers knowledge. These ideas can be thought early in primary school.
The concept of a mathematical function should not be first introduced as a formula, but as an arbitrary pairs of numbers. Pupils are conditioned to think of a function as a continuous line. Later there are issues with statistics: function is a set of distinct dots. There is no formula at all. Hence, the rule how you pair one number with another can be a formula, but also can be completely random assignment. Math function is about pairing two numbers. You can also pair random chosen numbers, you do not need to calculate second number from first. The rule can be input or output, but that restricts the function in the way that you have to know input to get the other paired number, the output. Because, function can have a pairing rule "pick first number, then, don't ask another person to pick another number without looking at the first number, then pair two numbers". I want to emphasize that function need not to be defined in a restrictive way by using words \"inputs\" and outputs\" which is more related to computer science. Function is first and foremost a pair of ordered numbers. My examples shows why the \"input\" \"output\" definition is restrictive and possibly misleading. I my view, the word pair best describes the function. Then we can use word map, association of two numbers etc. Input and output really leads someone to think that there need to be formula or some dependency between output and input. But, it is not so. It can be, but that's too restrictive for function definition. As in my example, a function can be "pick an output that in no way depends on input". Or, pick one number, then cover it (hide it) then ask another person to pick another number. Pair these tow numbers. Here, output in no way depends on input, yet this is a function.
While kids may like airplane models for many reasons, demonstrating to them role of AIRFOILS for lift and performance will be very important.
Once student knows how to calculate rectangle area right away he/she should be thought to calculate surface area under the curve (integral).
If I were to teach kids math, I would let THEM chose the numbers and chose what to do with them. Add, subtract, multiply, divide. #math
Mathematics does not care, at all, if the numbers it deals with are coming from dogmatic religious thoughts or from scientific analysis.
Want to know what math is all about?? Artists, sculptors, writers, movie makers, students, read on! http://goo.gl/C6vrL
Everything is in accordance to the Laws of Physics. NOT TRUE! It matters how you use them! Decisions have important place.
During schooling (don't mix that with education!) best thing you can do is to follow your own ideas and ask, then answer your own questions.
Combustion is, in it’s essence, an ELECTRICAL reaction between the fuel and oxygen molecules. From my blog http://tiny.cc/fboud
Education should not stop at achieving literacy in students only, it should encourage use of their minds too. Students are not fax machines.
There is a physicist and there is a composer. Both focused on two different domains of human thought, both very valuable for our experience.
There is a physicist and there is a composer. Both dealing with air waves but from completely different directions of thinking.
Physics is aware only of SOUNDS. But our interpretation in brain, whatever that means, will call sounds music, noise, speech, singing.
You apply math only AFTER you chose WHAT to count. Conversely, choosing WHAT to count and WHAT is counted has nothing to do with math!
Primary/Secondary education is not market driven, It's a monopoly. "Customers" are forced to be in school no matter how useless "service" is.
One of the most important application of mathematics: counting the beats of your heart.
What would you like to do, what you have a talent for, and what economy, i.e.market is looking for can hardly be all found in one job.
One of the most important application of mathematics: counting the beats of ones heart.
While applied mathematics is used to count heart beats, pure mathematics is not interested whose heart it is.
One of the most important application of mathematics: counting the beats of your heart.
To get a number you use logic, intuition, feeling, estimate, assumption, measure, imagination. To add two numbers you have to forget all that.
Anyone can call himself a "thinker". But, who is "Correct Assumption Maker"?. Crappy assumption in -> brilliant logic --> Crap out.
You can guess coffee strength by how bad was the caffeine withdrawal..
After all the transfer, give, receive, produce, supply, send, one would think that ENERGY is an object. It's not. It's a calculated value.
At one point you may ask yourself who needs an energy balance calculation of brain biochemical reactions during decision making.
Some parts of mathematics relationships are developed by quantifying completely non mathematical relationships - http://tiny.cc/5r5hk
Axiom Frontier, World #1, World #2, what is it? http://tiny.cc/mm38c
Women Mathematicians. Danica McKellar, Hollywood star (IMDB: http://tiny.cc/lwhrh) and published math author (J. Phys.) http://tiny.cc/cwrk8
Where math comes from and origins of fundamental math concepts for better understanding mathematics.Blog http://tiny.cc/lyah8
Word "IF" is a Magical Wand of Math. Using "IF" you can generate vast quantity of numbers, functions, patterns with no reference to reality.
Proof in math is like a proof in any other field. Logic of proof is same. The only difference is math has well defined starting assumptions.
You can read more "How math can be applied to so many different fields and how we can use math in real life".
I am quite sure kids are more scared of strange mathematical names than of ideas, sequence of math operations, or numbers they represent.
Motivation is first step to make a decision to even consider making new assumptions! Hence, motivation is like turning a light switch ON.
Browsing web is similar to day dreaming. But, you should use your own BrainBrowser to explore worlds, ideas you already have in your mind.
You may decide to add 2+3, just 'cause. Or you measure distance with laser to get 2+3. Math will not care where the numbers came from.
Social and emotional contexts, and not only physics or engineering, frequently dictate which math calculations you are about to perform.
Math and emotions? If you get raise of $150 you will feel differently than if the raise was $15000. Countable objects matter in social contexts.
He made an emotional decision to think logically. :-)
Axioms is just a FANCY word for initial assumptions, truths you are dealing with, inside any field (not only math) where logic is used.
Mathematics is defined not by objects it counts, nor by reasons or logic why those objects are counted, but with concepts used to define matheatics axioms and to define proofs of mathematical theorems.
Your motivation and decision making process have their own axiomatic system you are using, perhaps, unconsciously, not calling them axioms..
Motivation (p) has nothing to do with the assumption (q) you are about to make. Yet both are necessary for s to happen, p AND q => s.
There's a rich world of ideas right behind math axioms. Axioms deny you access to them, yet it's from these ideas axioms came into being.
That "IF..." in mathematics, can open so many new doors in mathematics for you, but at the same time closes or other doors other IFs, from other fields, can show you if you don't keep open mind and try to combine different ideas.
Mathematics is about COUNTS. The question that many ask is do we really need to know so much about counts during the schooling?
Mathematics is about counts, and counts only. The way we think about them is logic. Axioms, theorems, lemmas are labels for logical results.
School may teach you how to use logical thinking, but it can not teach you to chose correct initial assumptions.
If math is derived from 9 axioms where teachers found all those apples, pears, giving, getting, having, and claiming these words are math.
You can use calculator to deal with numbers. But what device you will use to deal with logic for objects' relationship the counts represent?
Of course, even using calculator, you can explore the properties of numbers, discover number patterns.. Only have to know what to look for.
Whatever is outside the calculator, or more precise, abacus, is outside pure math.
Math is like a calculator! Only YOU know what numbers will be typed in, what they represent, what sequence of operations will be performed.
The confusion for kids learning math is constant mix of items that are counted and counts, numbers obtained. Separate the logic for two.
Moreover, WHAT are you counting IS NOT part of mathematics. Only counts and what you do with counts is a part of mathematics!
YOU keep tab where the numbers, counts come from. Mathematics will not do that for you. Units in physics are not part of mathematics.
Math has to start calculation with SOME numbers. Where those numbers come from is up to YOU. Is it from a guess, a measurement, a counting?
Students should be shown that math doesn't care if numbers came from measurements or from guessing. Math will equally use both numbers.
Where numbers come from? How about from love? http://yarzabek.wikispaces.com/, "Let me Count the Ways I Love You"..
Art is also about exploring human values, moral, ethical questions, like, how do you apply laws of physics or mathematical calculations.
Structured, logical thinking doesn't guarantee that your results will be correct. Initial assumptions, axioms is what matters the most. With wrong assumptions you can make structurally sound mistakes.
Art, as physics or math, has its own axiomatic system. The subjects of art axioms are emotions, moral, feelings, human experience.
In art, as long as the emotion, feeling, moral message is REAL, it does not matter whether the art work follows laws of physics.
In art, fantasy worlds are allowed, breaking laws of physics is possible, because art is about showing REAL emotions, feelings, moral messages.
In science, truth is about physical laws, while in art it is about true emotions, feelings, moral messages.
With word 'IF' you can build so many assumptions, and, if you hold to them, you can build a lot of fantastic worlds imaginary or not.
Track to creativity. Chose world of interest. Axiomatize it. Chose world where you interpret results. Axiomatize it. Play with both worlds.
You could develop calculus from axioms of course, but dealing with real world situations helped to pursue that direction of thinking sooner.
Interesting thing with invention is you don't have to prove nor show how you got it. But, again, same can be said for a mathematical proof.
If you want to understand calculus ask your teacher, on first lecture, to demonstrate limit in probability and stochastic integral.
Puncturing the Math Axiomatic Bubble Frontier to access inner mathematics functionality- http://goo.gl/ykvCC
The thrust, in jet engines, is obtained, in its essence, from fuel and oxidant molecules accelerated by mutual ELECTRICAL repulsive forces.
To me, the most interesting examples in math teaching will be in aviation, pirates and their treasure islands, and sculpture. Try it.
Education should not only show what other people have done. Education should encourage, lead students to make discoveries by themselves.
The idea of differential and integral does not follow from real numbers knowledge. These ideas can be thought early in primary school.
The concept of a mathematical function should not be first introduced as a formula, but as an arbitrary pairs of numbers. Pupils are conditioned to think of a function as a continuous line. Later there are issues with statistics: function is a set of distinct dots. There is no formula at all. Hence, the rule how you pair one number with another can be a formula, but also can be completely random assignment. Math function is about pairing two numbers. You can also pair random chosen numbers, you do not need to calculate second number from first. The rule can be input or output, but that restricts the function in the way that you have to know input to get the other paired number, the output. Because, function can have a pairing rule "pick first number, then, don't ask another person to pick another number without looking at the first number, then pair two numbers". I want to emphasize that function need not to be defined in a restrictive way by using words \"inputs\" and outputs\" which is more related to computer science. Function is first and foremost a pair of ordered numbers. My examples shows why the \"input\" \"output\" definition is restrictive and possibly misleading. I my view, the word pair best describes the function. Then we can use word map, association of two numbers etc. Input and output really leads someone to think that there need to be formula or some dependency between output and input. But, it is not so. It can be, but that's too restrictive for function definition. As in my example, a function can be "pick an output that in no way depends on input". Or, pick one number, then cover it (hide it) then ask another person to pick another number. Pair these tow numbers. Here, output in no way depends on input, yet this is a function.
While kids may like airplane models for many reasons, demonstrating to them role of AIRFOILS for lift and performance will be very important.
Once student knows how to calculate rectangle area right away he/she should be thought to calculate surface area under the curve (integral).
If I were to teach kids math, I would let THEM chose the numbers and chose what to do with them. Add, subtract, multiply, divide. #math
Mathematics does not care, at all, if the numbers it deals with are coming from dogmatic religious thoughts or from scientific analysis.
Want to know what math is all about?? Artists, sculptors, writers, movie makers, students, read on! http://goo.gl/C6vrL
Everything is in accordance to the Laws of Physics. NOT TRUE! It matters how you use them! Decisions have important place.
During schooling (don't mix that with education!) best thing you can do is to follow your own ideas and ask, then answer your own questions.
Combustion is, in it’s essence, an ELECTRICAL reaction between the fuel and oxygen molecules. From my blog http://tiny.cc/fboud
Education should not stop at achieving literacy in students only, it should encourage use of their minds too. Students are not fax machines.
There is a physicist and there is a composer. Both focused on two different domains of human thought, both very valuable for our experience.
There is a physicist and there is a composer. Both dealing with air waves but from completely different directions of thinking.
Physics is aware only of SOUNDS. But our interpretation in brain, whatever that means, will call sounds music, noise, speech, singing.
You apply math only AFTER you chose WHAT to count. Conversely, choosing WHAT to count and WHAT is counted has nothing to do with math!
Primary/Secondary education is not market driven, It's a monopoly. "Customers" are forced to be in school no matter how useless "service" is.
One of the most important application of mathematics: counting the beats of your heart.
What would you like to do, what you have a talent for, and what economy, i.e.market is looking for can hardly be all found in one job.
One of the most important application of mathematics: counting the beats of ones heart.
While applied mathematics is used to count heart beats, pure mathematics is not interested whose heart it is.
One of the most important application of mathematics: counting the beats of your heart.
To get a number you use logic, intuition, feeling, estimate, assumption, measure, imagination. To add two numbers you have to forget all that.
Anyone can call himself a "thinker". But, who is "Correct Assumption Maker"?. Crappy assumption in -> brilliant logic --> Crap out.
You can guess coffee strength by how bad was the caffeine withdrawal..
After all the transfer, give, receive, produce, supply, send, one would think that ENERGY is an object. It's not. It's a calculated value.
At one point you may ask yourself who needs an energy balance calculation of brain biochemical reactions during decision making.
Some parts of mathematics relationships are developed by quantifying completely non mathematical relationships - http://tiny.cc/5r5hk
Axiom Frontier, World #1, World #2, what is it? http://tiny.cc/mm38c
Women Mathematicians. Danica McKellar, Hollywood star (IMDB: http://tiny.cc/lwhrh) and published math author (J. Phys.) http://tiny.cc/cwrk8
Where math comes from and origins of fundamental math concepts for better understanding mathematics.Blog http://tiny.cc/lyah8
Word "IF" is a Magical Wand of Math. Using "IF" you can generate vast quantity of numbers, functions, patterns with no reference to reality.
Proof in math is like a proof in any other field. Logic of proof is same. The only difference is math has well defined starting assumptions.
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