,

WHY DO WE MULTIPLY FROM RIGHT?

vocabulary-science-blog-multiply-from-right

Why do we multiply from right to left?

Multiplication is one of the arithmetic operations that we all use to study in our early classes. It makes our calculations easy and fast and helps us in finding patterns hidden in the number system. With the help of multiplication, we are able to perform our daily calculations very conveniently.

Going into the method of multiplication and as the title of says: -“Why do we multiply from right to left”? Is this the only way of doing multiplication? Is it predefined in the rulebooks of mathematics? Ahh! a lot of confusions.

Let us first try to figure out the meaning of multiplication. Multiplication as we know it, is repeated addition of the same number into one another. In order to get rid of adding, again and again, the same number or to make the calculations easy, we devised one more arithmetic operation known as multiplication. Through this operation, we can find the sum when a number is added to itself tens of thousands or even millions of times, in a very small unit of time and doing very fewer calculations. (Again the mathematics is fulfilling its need of making calculations easy. Here mathematics is making mathematics easy).

The approach that we follow to carry out multiplication of numbers, is what we have learned from our childhood. Let us understand by an example:

Say we want to multiply the number 32 by 4 :
We first use to write this down in the form like:
Then first multiply 4 by 2  to get 8 and then multiply 4 by 30 to get 120 . And write the answer as .

Now in doing this calculation, what we are performing is multiplying the unit numbers first and then the tens number and then add them together.

On the hidden part, we are using the distributive law.
i.e.,

4×(2+30)=4(2) + 4(30) .

\ \ \ \ \ \ \ \ \ \ =8+120 .

\ \ \ \ \ \ \ \ =128 .

This method now seems natural to us, as it’s been inculcated into deep our minds and understanding.

But we never argued, why do we always use this method of multiplication? Is there any other method also by which we can multiply any two numbers?

Let’s now take the title of the article seriously and raise the question “Why do we multiply from right? Why not from left? Why not from the middle or why not from anywhere?

When we multiply from right we used to decompose the multiplier into units, tens, hundreds, thousands and so on and so forth in increasing order, and then use to the multiply the multiplicand by all of the decomposed numbers and add the results to get the final result.

Multiplication from the left would be the same as multiplication from right, just the difference will be in the order of multiplication from the decomposed numbers and the addition.

For example, for the same question done above, the multiplication from the left would be like this:
Say we want to multiply the number 32 by 4:

In this we first multiply 4 by 30 to get 120 and then multiply 4 by 2 to get 8. And write the answer after addition them together as: 128. .

Now in doing this calculation, what we are performing is multiplying the tens number first and then the unit number and then add them together.

On the hidden part, we are using the distributive law. .

i.e.,

4×(30+2)=4(30)+4(2) .

\ \ \ \ \ \ \ \ \ \ =120+8 .

\ \ \ \ \ \ \ \ =128 .

The approach would be the same, just the order has been changed in this approach. We are using here a property that addition holds, which is a commutative property of addition.

According to this property the order doesn’t matter for doing addition:

i.e., for any given numbers a or b: .

a+b=b+a .

For example: .

2+30=30+2 .

32=32 .

So, there is no difference in performing the multiplication from the left or from the right.

Same kind of approach would be followed in performing multiplication from anywhere in between in the number, and there will be no difference in the solution.

Let’s understand by an example:

Say we want to multiply the number 425 by 3:

In order to perform multiplication from anywhere, we first decompose the number 425 into its units, tens, and hundreds:

i.e.,

425=400+20+5 .

First we can multiply the number 20 by 3 to get: 60 .

Then multiply the number 400 by 3 to get: 1200 .

Then multiply the number 5 by 3 to get: 15 .

Now, adding the results of all we get: .

425 \times 3 =1200+60+15 .

425 \times 3=1275 .

And, in this way, we can perform the multiplication from anywhere in the number.

Mathematics and Nature

vocabulary-science-nature-of-mathematics

NATURE! Who doesn’t love and admire the beauty of nature all around us? I even ignored the word mathematics when I see the word nature… just joking!

There is a reason why we usually feel more connected to nature than mathematics; it’s because nature is something which we can feel and see but mathematics is an abstract subject. When we even talk about numbers they are abstract, we cannot see two (2) but we can only understand it with reference to some objects or things in nature. For example, two bottles, two trees, two pens, etc. Mathematics is abstract in nature but that doesn’t make it boring, uninteresting, unenjoyable, or even difficult, it’s the way it is taught in schools that makes it so. There are many interventions which can make mathematics interesting, enjoyable and at the same time easy to understand. One of them is connecting mathematics with nature or we can say, showing the beauty of mathematics in nature.

Mathematics is visible in nature like in sunflower, pine-cones, honeycombs, trees, fishes, faces in form of different shapes or numbers, etc. If such aspects are highlighted and students could figure out such beauty, then it would not be that difficult for students to relate to mathematics and understand it better.

To be precise let’s take an example of  π(pi). While teaching circumference and area of circle, π is introduced to students as a constant having a value of 3.14 or 22/7. But why the value of π is 3.14 or 22/7 is rarely told to them. To help students understand this, students could be given freedom to explore. Like, they could be given something circular and asked to wrap a rope around the object to measure its circumference, then measure diameter of that object and divide circumference with diameter. After that, observations could be shared and students could analyze and construct their knowledge while connecting with nature.

And then they would get sense of what Galileo Galilei said,

“Nature is written in mathematical language.”

 

,

Assessment through a three facilitator model

Assessment is one of the most challenging task in any form of facilitation and it becomes even more difficult in a workshop model. Workshop offers its own unique challenges and opportunities. Till the interaction with the learner is continuous and comprehensive, even the remedies or steps taken by facilitator will be fruitless. Hence I will discuss a unique model by which, the concept of CCE (Comprehensive and Continuous Evaluation) can be utilised to its optimum.

The approach towards this problem is actually utilising the strength of workshop format of learning. Classroom learning in schools is generally constrained by a single teacher burden with many tasks of facilitating the learning process, evaluating the learner, create a comfortable learning environment and many others. But in workshop model, there is no constraint on the single facilitator. We believe in the 3 facilitator workshop (Lead Facilitator, Support Facilitator, Impact Assessor)  in which there is a division of various roles and responsibilities :

  1. The Lead facilitator is one who is directly involved in content delivery and conceptual understanding of the learner. The complete design, methodology, and delivery is the major responsibility of the Lead Facilitator. 
  2. The Support Facilitator is one who is the walk around person during the session. They are part of the learner surrounding and usually sit with the crowd. Based on the preassessment, they involve with the various group (For more on How to Group Learners) of learners to help them in their specific area of improvements and cognitive development. The support facilitator work as the backbone of the learning process.
     
  3. The Impact Assessor continuously evaluates and assesses learners behaviour, social and cultural responses. They act as a direct source of information for the support facilitator.

Such a model would solve the challenges of execution of CCE and help in the decision making process as the workshop proceeds from session to session.

 

How to Think like a Mathematician?

how-to-think-like-a-mathematician

[dropcap]H[/dropcap]ow do great ideas take form? How does one think and develop great concepts? And above all, how do some people are able to think in an abstract manner while others are not? We have a perception that one needs to be a genius, great scientist or a mathematician to develop great ideas. What we fail to realize is that these geniuses are born and nurtured among us just taking a different path in their growing years. They develop certain ability or a strategy that helps them master any concept. I feel any strategy in a right direction is fruitful given it is practiced for a long enough time. I will try to list here few of them that a learner can adopt and practice to be better at Mathematics. Though the context here is Mathematics but a learner can apply this in any field of understanding and knowledge discovery. .

DOUBT

When we start learning anything or try to grasp a concept, it is important to start using the first principle (The first principle approach is to start at the beginning of the concept. You clear your basics; understand it and build subsequent ideas upon it.) As soon as the bas concept is clear the next important thing in learning anything especially Mathematics is to doubt. You should not just accept anything as given rather doubt it to your level of understand by putting forth contrary examples or situations. Test it yourself approach is the best for clearing your own concepts and understanding. It is a two-way approach; one it tests the truth and falsity of the statement given and also helps you grasp the concept thoroughly. .

THINK

Thinking about the problem is necessary inorder to make new connections. Try to spend as much time as possible with the problem. As soon as you start to doubt; give your mind some time to absorb the content and ideas. When we think too hard on a problem and then give our mind some space to make connections among discreet ideas it results in a much better understanding of the problem at hand. .

REACH CONCLUSIONS AND ORGANISE

When we reach conclusions, it is important to start writing. Many of us believe that we can do this or achieve that but fail to give it a concrete form. The best is to start writing down all the ideas and try to make a mind map. Once we have written all the ideas and conclusion; the next step is to organize it and try to make logical connections among the ideas that we have written down. You will realize that seemingly discreet ideas suddenly start making wonderful connections and relationship. This might not lead you to ultimate answer but will surely take you forward with the work. .

TEST WITH EXAMPLES

It is easy to reach conclusions but difficult to verify it and say in absolute terms. So test it out with examples and relate it with other subject areas. It will help you to understand a concept thoroughly not just broadly. You will be yourself surpirsed to discover the constituent of your own thories and conclusions. Such a process will help to keep the undesired results out and further improve our results. If your conclusions seem correct, you’ll move forward and using your own theory, you can build your own examples and problems. .

THEORETIZE AND PROVE

All the vague ideas and conclusions that have been verified; tested with examples and improved must be written in general ideas annd theories. It is important to build a theory so that a logical sequence to your understanding of the problem can be identified. It will also give other readers a clear path to follow which ultimately they should also doubt to keep up the spirit of mathematics. Once the theory is ready; the hardest and the most critical step is to formulate your theory and prove it.  It requires tremendous amount of effort and some prior mathematical knowledge to formulate a proof. Once you have formulated a proof the work is done and you can repeat the process to DOUBT. .

 

 

Is Mathematics innate?

is-mathematics-innate

I gave choice to my niece of around 1 year that she could select one of the 2 packets of chocolates. One hand had 2 and the other one had 4. She was able to observe and choose the packet with more chocolates. Also, whenever given a choice she always chose toys or objects which were relatively bigger from each other. The case of my niece is not unique and the root to our perception of something big or more has been critical to our evolution. Humans were a hunter-gatherer society and it was necessary for survival to be able to identify trees bearing more fruit than the ones that bore less. The situations like this and many other throughout our evolution helped us develop a sense of number.

We never taught my niece counting or numbers or any other mathematical concept. How could we? She was only 1 year of age. These experiences show that there is something innate about the way we perceive and view numbers. And what does innate mean in mathematical understanding? Innate is the ability that the child is born with or which is present in the child from birth to understand the world and make a choice using the mathematics senses(he) is born with.

Noam Chomsky, a linguist put forth the idea of Language Acquisition Device which is hypothetically present in human beings when they are born. And research has shown that when human beings are born they have this device which helps them acquire the language that is spoken around them.  Similar to this, there is some research that shows that children also do respond to mathematical clues given to them in earlier ages. For instance, when a toddler is shown some objects say; 2 pencils and then after that, a pencil is added, toddlers are more likely to respond differently and this shows that there is some innate knowledge that children have when they are born.

To better understand, if we talk about animals, they also have some mathematical sense when they are born. Like a horse cub who is lost in the forest can easily identify the appropriate way if there is 1 tiger on his left and 4-5 on his right. He would either find another way or choose the way with one tiger as the horse cub has this innate ability to differentiate 4 or more from 1.

Mathematics is innate in humans as well as in animals, but that doesn’t mean that they are fully equipped with mathematics. They just have a sense of mathematics and that too is a part of their unconscious self.