Our Brain Is Tuned For Addition Versus Subtraction

Do you prefer adding numbers or subtracting them?

Do you make more mistakes doing subtraction than doing addition?

If the answers to the above questions are all YES, you are not alone.

We are "programmed" since young to add more than subtract.

Do you count things starting from 1 up or down from 100?

Does the page number of books starts from 1 up or down from the last number?

Does the date on a calendar month starts from 1 up ?

Day in day out, we are bombarded by addition of numbers through counting up. It is not accidental though. It is due to the fact that we do not know the maximum in those counts.

Therefore, we are accustomed to addition and will do better with this maths operation than with subtraction.

Try doing this:

1. y = 44 + 36


2. y = 44 - 36


3. y = 60 + 53


4. y = 60 - ( - 53)

We can do the above simple maths questions easily.

But, honestly, did you take a longer time to compute the subtraction?

There is a slight mental block compared to addition, right? Mine does.

There is nothing wrong with this. What I like to bring to your attention is the fact that knowing addition is the brain's preference, we can minimise maths error by aiming for addition than subtraction, if possible.

If we are to do a maths operation y = 55 - (3 + 6), do the (3 + 6) first, followed by the subtraction.

Avoid doing y = 55 - 3 - 6, with 2 subtractions involved. Although mathematically it is correct, why take the risk?

It's the same for calculating y = 53x + 16x than doing y = 53x - (- 16x).

Did you get the message? Understand how your brain works and tweet the mathematical process for a less risky computation.

;)

Principles of Learning (Mathematics)

Learning? Yes, a simple word, but the implication is far-reaching than the word itself.

Before we start learning anything useful, we need to question ourselves over its purpose.

If the answer to it is sufficient enough for you to strike on, you are in the correct path to a better future.

However, what is the mental approach to learning, before actual content acquisition?

Is the ownership of learning clear?

These are the questions you have to answer before you set out on the bright journey of learning.

The key issue here is OWNERSHIP of learning.

Besides learning the essential principles of mathematics, the crucial principles of learning has to be handled well too. Do note that both go hand-in-hand.

Principle of learning takes into account the idea that learning is the sole ownership of the learner.

The teacher is, just after all, a catalyst to speed up the learning process. He is there to address any questions that may be harder to solve or missing links that may unknowing been left out during the course of lesson delivery.

But to have a fruitful learning outcome, you must know the fact that you are the ultimate target of the process. You have to constantly remind yourself that "I am the one learning".

Whether can the teacher teach well or explain properly, the learning still goes back to you. (Do not blame anyone for failure to learn!). It becomes an excuse to deviate from the proper.

So, the principles of learning has to be clearly understood before principles of mathematics can be effectively captured.

Mathematics learning will then be a wonderful learning journey.

Wonderful here does not mean easy-going though.

Without decent struggle in learning and thinking, retention of knowledge will not last long. This is a well-known fact! (Struggle here means the mental processing of knowledge, past and current).

Study hard (and smart), and know that you are the final gem that any teacher would like to polish, provided you start off with the correct mindset.

Cheers to learning and cheers to you, the mathematics gem.

Maths Is Interesting!

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Analysis of Maths Error

Making mistakes during maths learning is a common step to mastery of its concept.

This is possible, provided the learners pick up the error made and understand them through detailed analysis.

I came across an error that puzzled me for quite some time.

The error made: Inverting y = 2x + (1/x) became 1/y = (1/2x) - x

What actually happen in the thinking during the math manipulation?

Understanding the thinking will definitely help in resolving the issue and prevent further mistakes made.

Here, I believe the student:

- confused "flipping" cards with the basic concept of maths (flipping the left y does not equate to individually flipping the right mathematical terms)

- confused the property of indices ( 1/a = a^-1) with this operation, resulting in the last term having a shocking "-" appearing after the inversion of denominator to numerator placement.

- lack practices to enhance the algebraic manipulation during the learning phase. Knowledge is not retained.

Analysing maths error is not an easy task if done alone, and without guidance.

The thinking part is an abstract art of the mind. If there is no communication to reveal the mental procedure in solving the math question, a skilled guess has to be made especially by those teaching maths.

If guidance is not available, textbook will be the next best alternative to answer for the mistake encountered. Go back to basic to dig out more understanding.

The approach is to question and question till satisfied. It will cover most aspect of the learning objectives.

However, do note that identifying and analysing maths error is a great learning process.

It serves to filter out current bad learning habits and replaces the gap with proper systematic steps. This is of much importance especially in maths where every steps count.

Therefore, do enjoy "debugging" your maths error. It is fun and not meaningless!

Errors do tell many stories. Just be friendly to them.

:-)

Linear | Angular Speed

Speed has two definitions:

1. Linear


2. Angular

Linear speed is defined as the distance travelled for a given time.

Angular speed is defined as the angle covered for a given time.

Are they proportional ?

Or are they related in any sense?

Let us take 2 pendulums hung on a slim rotating rod for analysis.




If the 2 pendulums (A and B) rotate one full cycle, the time taken by them is the same.

They covered the same amount of angular distance (360 degree) within the same amount of time.

This showed that they have exactly the SAME angular speed.

But is the Linear speed similar?

The length of the 2 circumferences travelled by the individual pendulums are not the same.

The linear length or distance is therefore NOT the same.
Length = 2 x (pi) x radius.

They took the same time to complete one full cycle, though.

The linear speed is thus DIFFERENT, having travelled different length for the same amount of time.

In this case, the angular speed is the SAME whereas the linear speed is different. Pendulum A has a higher linear speed compared with pendulum B.

Can the angular speed be different and linear speed made to be the same?

If the same slim rod is used, the answer is NO.

But if they are held by different rods, like that of the traditional analogue clock, the answer is YES.

For the linear speeds to be the same, pendulum A has to take a longer time to complete one cycle compared to pendulum B timing.

In this case, their linear speeds are the SAME while the angular speed will be DIFFERENT. This is so, since, same angular coverage but different time taken.

Why are we talking so much about this?

This is an "old" principle that was applied to the famous grandfather's clock. The setting of the pendulum position along the swinging rod (or string) is the key to the time accuracy of the clock. Ancient people has used this understanding to produce something useful for their daily needs, and this is mathematics!

Wonderful isn't it? :)

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How To Estimate Numerical Answer

Estimation in maths is not just about rounding up or down numbers.

It called for logically thinking and is done on a case-by-case basis.

Take for example, 302 x 11.

What is the estimated product of the 2 numbers?

We can choose for it to be 300 x 10, which gives 3000.

But how about being more accurate in our estimation?

300 x 11 gives 3300 !

We can use "11" instead of closing in to "10" of the first instance.

Logical thinking at play here...

But why choose "11" instead of the "10"?

The answer is simple.

Since the first number is 300, multiplying it with "11" is still within any person's ability. Therefore using "11" as part of the estimation is closer to the actual answer.

Now, how about 314 x 11?

Here, the answer estimated will be 310 x 10 = 3100, to be practical.

310 x 11 will be harder to estimate. Therefore "10" is chosen in preference to "11" as in the case of the previous example. (We are not talking about mental maths, but maths for "normal" people.)

Estimation, thus, calls for a fast but accurate production of numerical answer. It involves thinking and choosing numbers that are easy to handle.

Estimating answers can be fun as shown. It can be a challenging game where students aim to be first to give the most accurate answer.

Have fun estimating! :-) It gets more interesting as you go on.....
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Volume and Surface Area | Geometrical Relationship ?

Using geometry, we can determine the volume and surface area of any object.

However, have you wondered what is the relationship between them?

If we want to maintain the volume of an box but reduce its surface area, is it possible?

Or does the volume ALWAYS increases with increase in its surface area?

How about reduction in surface area? Will the volume also reduce?

Let us throw some numbers into an example to figure out the answers.

For simplicity, let us use a simple box. (Diagram 1)



<== Diagram 1 Box of diagram 1 has a : Volume = (2 x 8 ) x 4 = 64 cubic units

Surface area = 2 (2 x 8 ) + 2(4 x 2) + 2(8 x 4) = 112 square units


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Let modify the box (Diagram 2) to another dimension and see its geometric parameters.



<== Diagram 2 Volume = ( 4 x 4 ) x 4 = 64 cubic units

Surface area = (4 x 4) x 6 sides = 96 square units

Note, the volume remained but....
the surface area has reduced!

This saves material, right?

How about another dimension (Diagram 3)?



<== Diagram 3 ( A tall box!) Volume = (2 x 2 ) x 16 = 64 cubic units

Surface area = 2(2x2) + 4(2x 16) = 136 square units

What now?

The volume again remained,
but the surface area has increased!

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What is the conclusion from here?

It is observed that althought the volume of the box has not changed, their surface areas has changed. The direction of change( increase or decrease), however, is on a case by case basis.

This concluded that there is no relationship between volume and surface area of any object.

Therefore don't be tricked into reckoning that surface area increase will cause a definite increase in volume. Likewise for reduction also.

Hope you gain much from here.

Geometry is exciting and mysterious at the same time, right?

For relationship comparsion between another set of geometrical parameters, perimeter and area, click here.

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