Thursday, April 28, 2011

New, Fresh, and Fun Way to Learn Math and Real Life Math Applications

Here are my posts about mathematics and real life mathematics applications:
More articles to come. Topics will include "From Sets to Numbers to Mathematical Functions" and "What Does That 'Let's assume we have...' Mean in Mathematics?", and more.

In combination with vivid real world examples, I will try to show you how to think mathematically, when to think mathematically, and what are the general rules for you to solve mathematical problems and to define new ones, your own, from scratch. My goal is to show you the methods to use with mathematics and to reveal the motivation behind mathematical directions of development.

You can find elsewhere how to solve equations and work with algebra, how to do the integration, calculate matrices. But, at one point you may ask yourself, where all that mathematics comes from? Why there are algebra and quadratic equations, and so many other equations, at the first place? What would be the way to understand them better? How you can apply concepts in math in real world, the one you see immediately when you close the mathematics textbook and look around?

I will demonstrate you the methods, the way of approach, the directions of thinking that will help answering all these questions.

How math can be a part of physics, economics, engineering, chemistry, finance, commerce, trading, accounting, and yet, mathematics can exists as a separate and independent discipline, from all of the fields it is applied to? What is the point at which mathematics took off as a separate discipline? I am answering these  questions in my blog here.

[ applied math, applied mathematics, learn math, math, math education, rational numbers, teach math, whole numbers ]

Tuesday, April 19, 2011

Math, Real World Examples, and How You Can Understand Math Even Better

You can read more "How math can be applied to so many different fields and how we can use math in real life".

As you can see, from the picture, 2 + 3 counts can come from many different fields.


You can then abstract counting and numbers 2 and 3 as common to all those fields. Moreover, you can deal with counts alone, not even thinking where they come from. You conclude that 2 + 3 = 5 no matter what you have been counting. That's why math can take off as a separate discipline, called pure math (not that applied math is dirty!). 

Many students are not puzzled so much with real world math applications. Students know it's there! Students appear to be puzzled how math can exist as a separate discipline. When and how did it take off from all those apples, pears, and other counting objects? How math can be a part of physics, economics, engineering, chemistry, finance, commerce, trading, accounting, and yet, mathematics can exists as a separate and independent discipline, from all of the fields it is applied to? I am answering this question in my blog here, and in this article as well.

When you see a person writing down 2 + 3 =...   you don't know what is she adding! But you know the result will be 5. That's pure math!.  You see, a person is writing down 2 + 3 =… Now,  let's say it again, you don’t know what that person has in mind, which objects she was counting. But, you know that the result will be 5! That’s, so called, pure math. You did a great job abstracting math from its real world applications. You may now want to continue to develop math! Make more examples! 5 + 3 = 8. 3 x 4 = 12. 8 - 5 = 3. You can say now, that if you have 7 and you add 8 you will get 15, just by dealing with numbers, no matter what objects and rules about them were involved. You realize that you can work with numbers only! And operations you have at your disposal, addition, subtraction, division, multiplication. be sure, there are no other operations in math. All other, so called, operations are just different sequences of these basic four, +, -, x, /.  That’s pure mathematics, as the mathematicians want to say. This is probably the most important step you have made in learning math.

As I mentioned, we can separate counts 2 and 3 into different area and play with them exclusively.

These "pure" math means playing only with numbers. It's just about making new counts, i.e. numbers,  (like 2 + 9 = 11 or 7 x 5 = 35) and doing operations on them, without worrying where they come from. Motivation to do operations is up to you since real world at this point is out of our interest! This play with numbers only, is very similar to playing with marbles and counting how many marbles you have after some operations, like, addition, subtractions etc. However, if you are interested WHAT you are counting, then you can associate objects names (the ones you are counting) to the actual numbers, like 5apples, 12cars, 15$, 7CDs, songs, steps, kilometers. You keep track what you are counting on one paper, but on another paper you do actual addition, subtraction etc only with numbers! Don't forget, the object names do not influence actual numerical operations. Object names are for you only, to keep track what you have been counting. It can be, again, compared with marbles. If you say, these 5 marbles represent 5 CDs you have counted, and then you add 3 more you can say that you have 8 marbles, or, since they represent CDs that you have 8 CDs. The trick here is that, when counting marbles, YOU can decide that they represent various things you are interested to count, like apples, cars, CDs, balls, etc. Marbles are in some sense very similar to numbers, they are universal method of representing the counts of different objects.

You can play with other numbers and math operations (addition, subtraction, division, multiplication) on them and be sure they will be common and be in intersection of other fields (outside math) too.

At one point you may want to play with numbers 3 and 7, like 3 + 7 = 10, and perhaps think what are the fields where these counts can come from. Note that you used those numbers, 3 and 7, without knowing what you have counted! That's pure math! Once you are aware that you can deal with numbers by themselves, as separate entities, as we just did, you enter the area of pure math research :-) Congratulations!

Now, let's take a look at that intersecting area where the numbers are. You can enlarge that area and add more numbers. Moreover, you can add any counts you want and investigate any operations on them as you wish. You can add, 1, 5, 7, 10, 25, 26, 27, ...102, 1237. They are all in that central area. But you can, now, look at that area independently from what intersects it. You can add more numbers, like 2/3, 4/5, 6/7, 1/3, 1/129. You see how you can create new numbers (which are called rational numbers, because they are ratios of integers). That is all math, pure math. Other fields that use math and counts can use your results instead of rediscovering them each time when they need to calculate something. These fields can be any of academic fields and the math they use is called "applied math". But, you see that mathematical results can be motivated by real world examples, but also they can be devices by you, when you just play with numbers. Essentially, you do not need real world examples to develop mathematics. One example is geometry of Lobachevsky that has been developed first and only then discovered and used by Einstein in his Theory of Relativity. On the other hand, Newton's problems in Physics, like calculating speed, led to development of calculus. But be aware, calculus could be developed even without looking at physical problems.

At this point I would like to show how mathematics can be an independent discipline and also can be, and is, used in real life. I will make a comparison between lines and some shapes and numbers. When you look at the building, and you want to draw it, you  will draw some lines, squares, rectangles, trying to mimic the shape of the building as truthfully as possible. Look at the lines you draw. You draw vertical, horizontal, and lines at any arbitrary angle. It appears that which line and where you will draw it will depend on the shape of the building. And that's true. Now, look away from the  building and get a new, blank sheet of paper. Draw a line on it. Draw horizontal lines, vertical, and at arbitrary angles. You are not required to draw a building. Just draw lines. You see how you abstracted lines, and, moreover you can play with them without taking care whether they represent anything in real world. Moreover, you can draw completely new building with your lines and call a company to actually build a new object from your drawings!

Similar things is with numbers. You can count real objects and get their numbers and deal with them. You can add, subtract, multiply numbers following the real world examples, like, "how many liters of gas I will use if I travel 125 km with the car that consumes 8.9L/km..." etc. When you calculate this or any other example, you deal only with numbers. You keep track of units, like what you have counted, aside. But, then, you can notice that 7 x 5 = 35 regardless what is counted! Similar thing happened with lines! You could draw lines without worrying if they represent any object in real world. Now, you can play with numbers, any number, and use ANY operations on them without worrying do they currently represent anything in real world. That's the essence of math. When you do applied math you still do pure calculations while keeping aside the units, what you have counted. But, in pure math you start with some numbers, it's up to you with which ones, without providing reasons why they are there, and do calculations on them. Try both scenarios and you will see the point! 


Here are more links you might like as well:
[ applied math, applied mathematics, education, math concepts, math education ]

    Sunday, April 17, 2011

    How math can be applied to so many different fields and how we can use math in real life

    How to differentiate between pure math and applied math? How we can use math in real life? How math can be applied to so many different fields? Where all those volumes of math come from? Can I revisit my math from primary and secondary school and finally understand what is it about? Can I learn math now? How I can teach my kids or students effectively primary and secondary school math?

    I will be trying to answer these questions in the next several articles.

    I encourage you to read the whole article, especially the action with index cards, where I explain, essentially, how you can use math in real world situations, and two steps in real world math solving method.

    In order to “apply” math, we have to decide we want to quantify something. Quantification is the essence of math application. Note that nonmathematical relationships between objects exist long before we decided  to quantify. If the quantification is applicable to the field of our interest then we can say we can “apply” math to it, i.e. pure mathematical results may be applicable to that field. When we say mathematics can be applied to many fields, it really means, counting, quantification can be applied and done to those fields. However, quantitative relations are not the only one of importance in the field we chose to study. Knowledge of all mathematics will not help you a bit to discover a single physics law unless you first think in terms of non-math concepts.

    As you can see, from the picture, 2 + 3 counts can come from many different fields.



    You can then abstract counting and numbers 2 and 3 as common to all those fields. Moreover, once you’ve done counting,  you can deal with counts alone, not even thinking where they come from. You conclude that 2 + 3 = 5 no matter what you have been counting. That's why math can take off as a separate discipline, called pure math (not that applied math is dirty!). Pure means that suddenly we are not interested where the quantities, counts, numbers come from, we are specifically and only interested in the numbers, counts properties.

    When you see a person writing down 2 + 3 =...   you don't know what objects she has on her mind, what objects she is adding! But you know the result will be 5! That's, actually, pure math!  You see, a person is writing down 2 + 3 =… Now,  let's say it again, you don’t know what that person has in mind, which objects she was counting. But, you know that the result will be 5! You abstracted the counts and counting from objects being counted. That’s, so called, pure math. Even, probably, without knowing it, you did a great job abstracting numbers, and operations on them, from real world situations. It is important to note how you dealt here with counts only, and not with particular objects. But, if you keep, on a separate sheet, what objects you are counting, you can transfer the result back to the real world.

    Once you are aware that you can work with numbers, counts only, you are in position to investigate properties of these counts, without considering at all where they come from. You may now want to continue to develop math as a separate discipline! Make more examples, think of more numbers, operations on them! 5 + 3 = 8. 3 x 4 = 12. 8 - 5 = 3. You can say now, that if you have 7 and you add 8 you will get 15, just by dealing with numbers, no matter what objects and rules about them were involved. You realize that you can work with numbers only! And operations you have at your disposal, addition, subtraction, division, multiplication. Now, be sure, there are no other operations in math. These four are sufficient to define all "kinds" of numbers and all mathematics. Other, so called, more complex operations (although they are not so complex, in many cases) are just different sequences of these basic four, +, -, x, /, it’s just that mathematicians like give them exotic names, that's all.

    This is probably the most important step you have made in learning math so far.

    You can play with other numbers and math operations (addition, subtraction, division, multiplication) on them and be sure they will be common and be in intersection of other fields (outside math). Mathematics could have been also called Countology, a science about counts. Or even Setology, a science about sets, because sets are fundamental objects in mathematics. Calling math Setology or Countology is emphasizing the fact that no matter which "fancy" names mathematicians use inside math, those names always represent sets, or counts, or sets of counts, or sets of pairs of counts, sums, or some other relationships between them!

    At one point you may want to play with numbers 3 and 7, like 3 + 7 = 10, and perhaps think what are the fields where these counts can come from. Or, in other words, what can be counted to get those numbers. Once you find what can be counted, you then investigate the relationships between objects counted. These relationships need not to be mathematical, but can influence counts.  Note though that you used those numbers, 3 and 7, without knowing what you have counted! That's pure math! Once you are aware that you can deal with numbers by themselves, as separate entities, as we just did, you enter the area of pure math research :-) Congratulations!

    One important note about math and applied math. Looking at 2 + 3 it is impossible to say to which objects these numbers are referring to. Of course, you can try to find examples from real life for the counts 2 and 3 and their addition, but, again, going the other way, from math to real world,  just looking at 2 and 3 you can not say what you have counted. That' pure math. You can add 2 and 3 and then you can return the result to the real world, to the objects you have been counted. So, if you counted apples, it will be 5 apples, if you counted cars, it will be 5 cars. Note how math result can be reused for many different objects you have counted. In this "re-usability" lies the value of math. But, on the other hand, since you can deal with numbers only, you can investigate properties of pure numbers, their relationships, their magnitudes, you can investigate operations on them, different sets of counts, i.e. numbers. Result may be applicable to some real world scenario, and frequently is.
    Let’s start with clarifying even more and explaining in more detail the difference between pure and applied math. Let’s say you have a task at hand and some mathematical calculations to complete. Say, you want to calculate how much money you will have to pay for 5 CDs if each one costs $3.00. Now, I know it is easy to calculate, it’s 5 x 3 = 15 dollars. But, I will show you how you can tell apart what is pure and what is applied math, and from these “rules”, which are more than rules, how you can apply math to many different fields.
    Take a set of blank, white index cards, size 5" x 8" (inches), say, 50 of then. You might not use them all for our investigation, but I want you to have enough for the examples that will come.
    Take two cards now. On one card we will write down the task what we have talked about and a required calculation, on another we will write down only actual mathematical calculation. For instance, you can write, on the first card: “How many dollars I will pay for 5CDs if one CD costs $3?”. Then, on this first card, which I will call Field Card, write down the calculation requirements together with “units” you are dealing with, i.e. objects you are counting. Like this,
                               5 CDs x 3 $ = ? on Field Card
    Now, take another card. I will call this card Math Card. On this card you only write down the actual mathematics you have used, without any units, or objects counted, or explanation what the task was, like this
                               5 x 3 = 15 on Math Card

    Now when you have the result, you go back to first, Field Card, and write down 15 with “units” i.e. objects counted to it, in this case dollars: 
                               5 CDs x 3 $ = 15$ on Field Card
    Take another blank card. It will be our new Field Card. Let’s do another calculation. You want to buy 3 model airplanes and each model costs $10. How much you will have to pay for these airplanes? Again, it is easy to calculate the correct number (3x10=30 airplanes), but we still want to separate the task description and actual calculation. Hence, Field Card, the new one will have
                              3 models x 10$ = ?$ on Field Card
    On our Math Card we will write the calculation we are performing, without mentioning objects we have been dealing with:
                              3 x 10 = 30 on Math Card
    Our Math Card, at this point, will have two lines on it:
                              5 x 3 = 15
                              3 x 10 = 30
                             
    How about a new task? There are five cars and each car can Have 3 people inside. Whatis the total number of peoplethat can use the cars? 5 cars x 3 people = ? people
    Let’s take our Math Card to write down only the calculation required. But wait! We already have 5 x 3 = 15 written on it. We can use that, we don’t have to write it again! It, apparently, doesn’t matter if we multiplied 5 CDs and 3 dollars or 5 cars and 3 people. In both cases the result was count of 15 or number 15. We can conclude that no matter what objects we have counted, if we have 5 things and 3 other things, the result of multiplication will always be 15! As we have just done on Math Card, we have written only numbers without objects counted. Whether we count CDs, dollars, cars, or people, 5 x 3 will always be 15. We can reuse that calculation for next and any other example requiring to multiply counts of 5 objects with counts of 3 objects! At this point you can guess that if we have another example where we have to calculate say, 3 things x 10 things, and if we note what we have written on our Math Card, we can see that we have already written there 3 x 10 = 30. And we can reuse this result as well, for any new examples involving 3 objects and 10 objects.
    We have now three Field cards, with three tasks outlined on them. Let’s put a title on each card, in the top centre area. First card will be called “Music CDs”, second card will have title “Airplane Models”, and the third card will have the title “Transport”. You may sense that there can be similar calculations say in biology, chemistry, sales reports, your grocery list, your monthly spending, airport schedules etc. And, all these new tasks may reuse the same calculations we have written on the Math Card. In other words, we have one Math Card and many Field cards.
    We have two reusable mathematical results which we can use again and again. Let’s try! Let’s talk Environment! If I have 3 trees and each one needs 10 minutes to plant, how much time I will need to plant these 3 trees? Take a new card, write the title “Environment” at the top and write down the task I have just described. Now, take a look at the Math Card? Is there a line 3 x 10? Yes there is! We can reuse the calculation there, 3 x 10 = 30, we can go back to “Environment” card and write down calculation in units,
    3 trees x 10 min = 30 min
    We will need 30 minutes to plant 3 trees.


    Now, stack away the Field Cards, and keep in front of you only Math Card. There is one interesting thing we can say about it. We can predict, that perhaps, in future, in real life, we can have not only 3 or 5 or 10 objects. We can sense there can be calculations required for other numbers as well. Do we really need to wait for these “real world” examples to pop up so we can fill in another row in Math card!? No! We can play with numbers and fill in some new rows on Math card by ourselves! Let’s do it! You can chose any combination of numbers you like. Let’s say, we chose 7x5. It will be 35. New row on Math Card will be 7 x 5 = 35. How about 3 x 4? Sure, let’s add another row, 3 x 4 = 12. We are certain these calculations can be reused for many different real time tasks. Even we can modify slightly our own examples with CDs. Say, now, we want to calculate how much we will pay if we want to buy 7CDs and each one costs $5. We can look in Math Card and find the row with actual calculation we just entered, 7 x 5 = 35. Hence, the answer is we will need to pay $35 for 7 CDs.
     

    So, it’s clear that Math Card is reusable. The more numerical entries we put on it, the more chance is that some real life example will be a match for them.
    Now, let’s look again to Math Card. It looks nice. It looks neat, clean. We deal only with numbers there while our example cards are stacked away. Then, we look closer to the actual multiplications. I can see now, that I can deal and play only with numbers! I can generate another row on Math Card without knowing which real life example will be a match. Let’s add one more row, say, 2 x 9 = 18. Now, let’s say there are several people outside our room waiting for calculations to be done. One is in “Transport”, but another one is in “Movie Business” (want to calculate how many tickets she sells each month), and yet another person is in another field. Field!? Yes! And all of them have their index cards, Field Cards with field title written on them. We can lend, to each of these people our ready-to-use Math Card so, they can find the actual calculation that is relevant to their requirements.
    Our Math Card is right here. We look at the rows we have added. Again, since we have concluded we can add more rows, let’s examine what is the next step. For sure we have to get more blank cards from the stack. Let’s take two more. Now we have three Math Cards. I know that I can play only with numbers now. I can add more numbers. I can add more operations, like subtraction, addition, knowing for sure there are plenty of examples in real life that these calculations can relate. Moreover, I can start to begin to notice some common properties of these numbers. I can group them which one is even which one is odd. I can add new Math Card to which I will write only multiplications, and another card which will contain only subtractions. Hmm, interesting. My index cards start too look like a math book! Moreover, since they deal with pure numbers only, it can be called pure math book. And you are right. This is a point when we can see how pure math books can exists without any “real” life examples in it. It’s because we can investigate numbers properties just looking at numbers. If we want a “real world” example we can ask people with Field Cards to come forward.
    Our Math Cards can be a sole focus of our studies. Since we know we can add any number we wish, and do any sequence of math operations we wish, we might as well add to Math Card all what we have been discovering as properties of these numbers and operations. Our results will be written on new math cards, and we will detail which numbers we have used, what operations have been performed and what results we have obtained. If you open any pure math book, you will see that it is exactly the case. They are like index cards notes collected together and given Table of Contents!
    Now back to Field Cards. The important point I want to make here is that we separate and collect index cards for each field. No mix up. If we want to do “Movie Business” calculations, we will fill in index cards with calculations requests related only to movie business. For instance, we will write on index cards tasks like how many tickets is sold during summer in three theatres downtown. What is an average number of visits in the Theatre number 3? Which movie has the biggest number of tickets sold? As we have seen, for all these requirements we can chose related Math Card with ready to use pure calculations, and get the calculated results back to “Movies Business” card. This is the reason mathematics can be “reused” in many fields. As the time passes, people dealing with Field Cards may want to record their logic related to their particular field in a book and publish it! Those books usually will have titles “Mathematics for Transport Studies”, “Mathematical Biology”, “Quantitative Physiology”, “Applied Mathematics in Food Industry”, “Financial Mathematics”, “Math in Economics”. But, they are, in essence, nothing more than summarized Field Index Cards, enriched by many examples in the particular field, and with hidden “pure” Math Cards. They will rarely show that common math is reused here. But, if you have these books at hand, and compare mathematics used in them, you will see that math is one and the same, it’s just objects counted that were different.

    Real world reasoning will tell you what to count and with which mathematical operations you will start with. Mathematics will accept these starting points, but will see only numbers and operations, and not what is counted or measured. An example can be any physical law. In each formula, which is within physics, is important what you measure or count, and relationships between the objects or concepts, like mass, speed, electrical charge, distance. But when you start calculations, you will deal with pure math, counts, numbers, keeping aside what is counted (kilograms, meters, seconds, temperature, etc). For instance, in finance, buying or selling is not part of the math. That's part of financial domain. It's separate logic. When you count how much you have sold or bought, those numbers, or counts are actual mathematics. It is YOU who will have to keep track who "buys" and who "sells", while math will keep track of counts and give you results in them.

    There are two parts of an applied math problem. 1) Correctly establish relations between non-mathematical objects 2) Solve newly defined numerical, i.e. mathematical problem. The first action will require the knowledge of the real world domain, which is, in most cases, extraneous to mathematics, and, in many cases you will deal with relations and definitions that has little to do with math. For instance, when you "buy" or "sell" something and in certain quantity, buying and selling are not mathematical objects. In Physics, object moving through space, or water going through turbine, are not mathematical relations nor objects. They are non mathematical things whose actions can be quantified, that's all. But these things themselves or their actions are not mathematical objects, they are physical objects. In economics, you will have demand for 500 cars, and supply of 470 cars. But, demand and supply are not mathematical objects, and their relations are not mathematical. Their relations are defined by human will, decision making, sociological laws, and specific logic associated with it. Only when we start quantifying them, keeping track which numbers are "demand" and which numbers are "supply" we enter mathematical world. Note that without keeping track what we have counted, we would not know to which objects the numbers 500 and 470 belong too.

    Once we established what needs to be calculated, we can deal with numbers only, with mathematics only. And that's the second part of problem solving. Given these initial conditions from the world of economics or physics or trading, we now deal with calculations, perhaps solving some equations, which is pure math, and knowledge of pure math can be very beneficial at this point. Once the calculations are done (of course, we should keep, all along, track of units) we go back to the real world domain and use the newly obtained solution and results.


    One answer to the question how math can be applied to so many different fields is because so  many things can be quantified. But even before that, we have to make decision to quantify something. The very fact that we can distinguish different objects through our psychological, cognitive, abstraction, categorization capacities and abilities gives a rise to an important choice of action - to count what we can distinguish. The very fact that we can distinguish so many objects means that we can put them in sets, and count them is the core reason why mathematics can be applicable to so many real world situations.

    How Mathematics Can Be Used in Real Life

    How to distinguish between pure math and applied math? How math can be applied to many different fields? How we can use math in real life? Why math looks so complicated? Where all those volumes of math come from? Can I revisit my math from primary and secondary school and finally understand what is it about? Can I learn math now? How I can teach my kids or students effectively primary and secondary school math?

    I will be trying to answer these questions in the next several posts.

    Here are more links you might like as well:

    [ applied math, applied mathematics, axioms, math and real life, math education, math, mathematics, physics concepts, tutoring, calculus, school, education ]

      Friday, April 15, 2011

      More tweets about math, math education, math and emotions, art and physics, inventions, mind and other amazingly interesting stuff!

      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.