Principles of Mathematics

Ways To Improve Logical Learning Part Of Maths

In learning of maths, as with learning of other subjects, we need to understand our learning styles.  The styles are namely, visual, auditory and kinesthetic.

Each of us has one particular dominant style. Knowing which one will thus serve us good. However, the learning style itself is still not enough, we need to also know the types of intelligence we possess.

The learning styles are used to gather information and ideas, through the senses, for the brain. But how do we process the information captured afterwards depends much on the intelligences that we also have.

In maths learning, what we need is the logical / mathematical intelligence.

This lets us do all sort of computations and comparisons that maths requires.

An example is the conversion of an expression in logarithmic form to its index form (and vice versa).

log_aY = X <==> a^X = Y

Though, this conversion seems simple enough, it is found that many students still have difficulties in converting the above.  Do they lack this logical intelligence?

The answer is “maybe” and “maybe not”.

They may have possess this logical intelligence, but has not yet enhanced it.

One way to offset this weakness in the logical comparison part of intelligence is to introduce the visual intelligence into maths learning.

What is this visual intelligence?

This is the “appearance” aspect of information processing. Here, colour, shapes, and the likes are linked up to the information.

Why introduce the visual intelligence?

The main reason is to bridge up the right and left brain, a theory that is now well-known and practiced throughout the learning and teaching communities.

Now, let’s see how we can solve the logarithmic and index conversion through the use of the visual intelligence.

The examples above showed two very interesting ways to enhance the learning. 

The first one is to replace and “beautify” the symbols with pictures or graphics to arouse retention ability.

The second one is to make use of colours to strengthen the symbols and their placements.

By seeing pictures and colours, although they are no way close to any maths topics, the learning of maths is greatly improved especially for the logical comparison portion.

Therefore, if possible, introduce as many “visuals” into maths as you can. They will make maths learning a totally new experience. Take care of intelligences and intelligences will take care of you.

Inspired ? 

Posted in Learning Maths, Teaching Maths, Indices, Logarithm, Self-improvement, Inspiration, Psychology | No Comments »

Benefits Of Knowing How To Solve Quadratic Equations

In elementary school, we are exposed to quadratic equations and their solving. We focus alot on this topics which is deemed the basic of any algebraic studies.

The generic quadratic equation is  y = a x² + bx + c.

We know there are various methods to handle the quadratic equations.

There are:

• Factoring
• Completing the Square
• Quadratic Formula
• Graphical

Mastering all these techniques allow anyone studying maths to have the flexibility of choosing a better or suitable method that fits the nature of the question.

However, do note that if there is problem learning all these techniques at one go, click here to get some pointers.

What is the benefit?

Many maths questions are actually quadratic in expression. They may not appear so, but, on closer look, they are.

Examples:

• 3 cos² A + 2cosA + 4 = 0
• 2 (log Y)² + 2(logY) + 3 = 0
• 4^x + 3(2^x) - 5 = 0
• 5x^-2 - 7x^-1 - 6 = 0

Being able to handle the generic quadratic equation solving means having the potential to solve numerous other types of quadratic equations as listed above.

What is the obstacle if you still cannot map the quadratic solving method to the other types of expressions?

Tips: 

1. Stare at the given expression
2. Identify the terms that matches the x² format.
3. Identify the other two terms through the “x” format and pure number format.
4. After re-writting the questions in the generic quadratic form, apply any of the method to solve this quadratic equation.

And that’s all.

Simple isn’t it?

Thus, mastering any one method of handling quadratic equation allows anyone to solve many other types of quadratic equations. Therefore, it is worth the time and effort to know solving these type of mathematical expression.

Bonus information:

Let’s look into this example  

The first term can be modified to 5(x^-1)².
The second term can be modified to 3(x^-1).
The last term will be obviously the pure number “2″.

Selecting the use of quadractic formula, we can say that a = 5, b= -3 and c = -2.

Next, just apply the quadratic formula and you are close to the two answers (roots) of the equation,

x^-1 = -4/10 or 1.   Clear?  

If not, read again… Our brain needs some mental exercise at times.

Posted in Learning Maths, Self-improvement | 1 Comment »

Indices & Logarithm - Less Mistakes with Proper Writing

Every symbols in mathematics serve certain purpose. The size of the symbols and numbers also represent unique meaning.

Many mistakes and confusions were made due to improper writing of the above two items that are very much key to mathematics.

Two of the topics that are commonly populated with writing errors are Indices and Logarithm.

Indices:  a^x = Y

Logarithm: log_nA = K

From the above two examples, it is apparent that writing is crucial to give correct meaning to mathematics.

Writing therefore is an important skill while doing these two maths topics.

Discipline is needed here to maintain consistency.

What do I mean?

Let me give an example.

log₂ xy = log₂x + log₂y

The above is a correct expression written with proper size and positioning.

If writing discipline is not maintained, it may end up as:
log₂ xy = log₂x + log2y      (size and position error)

Do you notice the change in meaning when size changes?

The subscript “2″ representing “base 2″ is coveyed as “2 times y”!

This is the result of a simple “slip” of the hand in writing the size of the “2″.

Can this mistake be allowed?

I guess NO!

Why run the risk of marks being deducted or effort wasted in getting wrong answers?

Write properly is the only answer to less mistakes, if not at all.

If a^x = Y is written as    ax = Y,    how do we expect to get correct answers?

Here, you realise that maths learning is not just about mathematical concepts, human discipline and strength of concentration matters also.

Being able to write clearly and presentable is a noble capability that we should be proud to demonstrate to anyone. Strive to be focused and discipline when dealing with maths.

Happy writing! 

Posted in Logarithm, Inspiration | No Comments »

Principles of Learning Mathematics - Know Your Limits

While learning mathematics, we are always exposed to a few methods of solving a particular maths questions. The methods are taught in order to give us flexibility to select an apropriate technique to ”attack” any maths problems.

Is it good then to master all the techniques taught?

The answer is to know your limits.

If you are stressed up learning so many methods, let go of the ones that you find uncomfortable with. Master the one that seems to be the best and easiest to you.

Stay on with this chosen technique of solving problems, and apply it to similar maths questions. This is the first step in the principles of correct mathematics learning.

However, do not be compacent and stay stuck! Having more methods to solve a certain maths problems is always a better and sensible course of action any maths learners should aim for.

After mastering the first selected method, move on and try using another technique that was taught. Practice till it becomes comfortable and an easy tool to use.

In mathematics, flexibility is the norm. Questions are always varied, and thus, solution has to follow suit. It is this nature of solving mathematics that makes a good maths learners achieve much in his later life.

Knowing one’s limit is thus important especially when dealing with mathematics. Focus on one method first as too many at a go will only mess up the curious and greedy mind.

Understand that maths problem can be solved through the use of any one method. Even if the steps are more, it is still a way to obtain the answer. Slowly after mastering the selected method, you can explore another method that can shorten the solving process.

At least you are now more relax as you can fall back to the first method if the new technique cannot be comprehended finally.

However, having said that, do review the maths syllabus. Some syllabus do spell out that the students have to master a few methods to solve a certain type of maths questions. For that matter, you are left with no choice but to deal with them accordingly. Seek for help if necessary, and do not simply give up!

Finally a message that I like to share.  “Although practice makes perfect, good practice is the one that makes perfect ultimately!”

Choose the correct principles of learning and you will not go wrong. Know your mental limits.

Posted in Learning Maths, Self-improvement | 2 Comments »

Solving Maths Develops Plan Foward Capability

As in any case of solving problems, solving maths also requires certain strategy and procedures. Performing the mathematical steps to realise the result need certain skills.

One of this skill is the ability to see the “path” to the result. This ability, however, is obtained when we are able to plan what to do and reveal the intermediate goals.

Solving maths is like a mini-warfare where the enemy is the result that has to be obtained. Every steps that we take to achieve our goals has to be planned for.

We need to think ahead in each steps of computation and be capable of using whatever tools available to clear the obstacles lying in front of us.

Solving maths, thus, is a good platform for anyone to gear themselves for a future of planning, or to have better planning capability.

The steps required to solve a maths problem develops one to be able to plan forward. The steps taken at every maths operation serves a certain purpose to simplify mathematical expressions or eliminate unknowns.

Studying and doing maths is therefore a necessary part of human development in that it allows the learners to develop their mind to solve real-life problems through proper planning.

So to developing minds, cheers to maths!

Posted in Learning Maths | 1 Comment »

Why Do We Study Quadratic Equation?

In maths class, we are hammered with expressions after expressions of quadratic equations. We are taught how to solve for its roots. We are taught all the necessary methods or mathematical techniques to handle quadratic equations.

But after all these, what is the purpose?

This is the question many students of maths studies ask.

Do we need this “quadratic” knowledge in working life?

See the diagrams and photos below. They will enlighten you.

The communication dish is parabolic in shape. Parabolic is the equivalent to quadratic mathematically. Engineers need to understand quadratic equation to dsign this beautiful profile.

This wok is designed using quadratic expression. With this, food can be fried to our liking!

Without quadratic equation, who knows how a wok would look like.

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Here you see that eye-glass lens are constructed with curves matching that of the quadratic equation.

Light is thus controlled to give good image to our eyes.

Quadratic equations to the rescue, right?

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Other examples are:

1)  Distance travelled given by the quadratic equation  s = ut + (1/2) a t²

2) Electrical characterisitcs of a MOSFET (Transistor device) 
                         i = k [(Vg - Vt)VD - (1/2)Vd²]

So now do you still wonder why you study quadratic equations?

Maths do have a purpose in our daily life. Rest assure that you are studying maths for a good cause.

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