Thursday, May 2, 2013

Teaching ‘Time’ – the fun way

One of the objectives we (as Teachers) may want to teach 3rd standard students to learn about Time is ‘to tell time to the nearest 5 minutes.’ Most of the children by age 7-8 are able to tell time as 5’o clock, 8’o clock, etc where the short hand is at the particular number while the long hand is almost taken for granted to be at number 12.
So, for the children to learn the skill of telling time close to 5 mins, the concept of 1 hour is equal to 60 mins needs to be introduced.
Mr.Rajendran in his book, ‘Inikkum Kanakku, part-2 for Class 3 and 4’ writes about an interesting way to present this concept by introducing a sequel to the ‘The Hare and the Tortoise’ story.
The tortoise won the race. But rabbit wants another one. Both start to practice hard. Just a day before the race, rabbit meets tortoise and boasts that I can finish the race in 1 hour. For that tortoise replies, 'Oh it takes 60 minutes for me.' Considering himself as being at an advantage, he tells the same to his friend peacock. The more sensible peacock says to rabbit, 'You are a fool.' Why do you think peacock calls the rabbit a fool." And that starts the whole new discussion on introducing minutes, hours in the subject on time.. …..”
The subsequent discussion on why the peacock called the hare stupid will definitely yield to a point where at least one of the students will say that both means the same. Thus, the children arrive at an abstract concept like 1 hour = 60 minutes in an exciting way. From this, we could build on the concept how the multiplication table 5 is used to denote the minutes in clock and thereby making sense of the minutes on the clock.

Attaining the Concept

Let’s play a game. I am going to introduce names of animals in two categories.

Category 1
Category 2
Now you got the first set of animals. Let me introduce second and the third one.
Category 1
Category 2
Can you guess a pattern in Category 1 and 2? If you can, hold on, if you can’t its okay. Now, I am going to ask you some questions. Tell me under which category ‘goat’ comes.
Yes you are right - Category 1.
Category 1
Category 2

Now next one - ‘tiger’?
Yes! Category 2
Category 1
Category 2
Now tell me where does,
  • Horse comes?
  • Deer?
  • Giraffe
  • Hen

Do this one by one.
Category 1
Category 2

Now, are you able to identify the category? Not yet, don’t answer. Let’s go on further. Can you give me an example for category-1? An example for category-2? Elicit at least three sets of examples.
Category 1
Category 2

Can you now tell me what the category means?
Yes! Category 1 is ‘Domestic animals’ and Category 2 is ‘Wild animals’.
In the game, you must have got your ‘aha’ moment earlier or later, but it was to your thinking that helped you get there. Even if you didn’t get it and got the answer externally you were able to take in the concept inside so easily and naturally.
This technique called ‘Concept Attainment Model’ was developed by Jerome Bruner, a pioneer in Constructivist Pedagogy.
Information processing and pattern recognition, it seems, one of the fundamentals of human cognitive ability. CAM builds on that. Bruner uses an ‘Yes/No’ model instead of two categories model I used to introduce one concept by giving an example against a non-example.
I used it to introduce ‘Punctuation’ and ‘Capitalisation’ by presenting a positive example of a correct sentence against the negative example of a wrong sentence. Then we discussed on deducing what made each sentence right or wrong, thus developing a list of grammar rules for writing a correct sentence.

What is Mathematical Ability?

[from the book 'Inikkum Kanakku' by Mr.Rajendran]
We think that ability to solve a problem quickly is M.A. But a computer/ a calculator can do the same thing much faster.
So, we think that following criteria marks as Math Ability
  • Thinking different ways to solve one problem
  • Estimate/ Guessing an answer before actually doing the problem [Estimation]
  • Finding an apt way to solve the problem
  • Ability to make similar such questions and solve it
  • Finding/ Discovering/ Constructing a theory by solving a current problem and testing the same by creating more such problems
  • Trying to find relationship between existing and previous concepts

Math and Abstract Ideas
One way to teach Math is to employ the following model
Experience à Language à Pictures à Symbols/ Abstraction
This is very similar to other models like
Concrete à Pictorial à Abstract
Manipulative à Visual à Abstract

What can be called as Numeracy [Ennarivu]? []
The knowledge about various concepts of a number and the ability to form more such concepts can be defined as Numeracy.
E.g. Take a number 12. What can you tell about this number?
  • 10+2 = 12
  • 12 ones make 12 or 1 tens and 2 ones make 12
  • Less than 13
  • Etc..

This knowledge/ ability will increase as per the class of the student.

 Concept of Game/ Gamification [Box item,]
Why do children play or watch a game like cricket, football again and again?
In a game, when a bowler bowls the ball, nobody can actually predict what is going to happen.
In that case, the audience might speculate what will happen, while same thing else altogether could happen or the exact same thing could happen as expected. This uncertainty is the nature of any sports.
This characteristic is present in almost every sport either less or more depending on the nature of the game. There is no such game where one can predict the outcome, i.e. who the winner is going to be.
When a classroom activity has that nature of uncertainty, the students are hooked to live through to see the end. Just mere reciting a poem, imitating a procedure or finding answers to five similar problems cannot be considered exciting.

The Surprise Element [Box item –]
Presenting a concept, teaching children how to solve problems and then asking them to solve 10-15 similar such problems to solve doesn’t build any Mathematical ability.
There should be an element of surprise in it and a deep thirst to solve a problem, discover the concept behind it, build a mental model, test that model by creating similar such problems and in the process to discover more such surprises will definitely improve the Mathematical ability. Any Math activity should have this surprise element to completely involve the children.

Inikkum Kanakku – Part 1 [A Maths Teaching Guide for Class 1 and 2] By Mr.G.Rajendran

Certain fundamentals []
  • To create an interest in a concept/ subject of discussion should be the main aim of a teacher.
  • Use concrete and existing materials
  • The students should be able to build upon their already existing prior knowledge and build upon their mental models
  • If I cannot learn the way you teach, can you teach me the way I learn
  • Learning to learn is education
10 Commandments for Teaching Maths
1. Ask open-ended questions where there is no single answer for it.
E.g. Tell me about # 5
  • It has only one digit
  • It is an odd number
  • It is between 4 and 6
  • Etc..
 E.g. Tell me about # 5 and 8
  • They both are one digit numbers
  • 8 is the largest among the two numbers
  • Etc..

2. Ask continuous multiple questions:
  • This helps to elicit curiosity
  • Helps navigate and introduce new concepts easily
  • Can give an interesting problem and/ or a situation for class work or home work
  • Children have the joy of finding an answer for themselves
  • Posing a puzzle always hooks them.
  • Asking questions like
    • Why?
    • How?
    • What will happen if?

3. Maths as Language
  • What is addition?
  • What does 5+3 means?
We should treat and translate abstract and symbolic concepts in Maths into Language. The above examples try to do that.

4. Multiple meanings in one statement
We should make the children derive multiple meanings in one statement just like in a language statement.
E.g. “I came from Delhi only today morning.”
E.g. 5+3 = 8, also means
  • 3+5 = 8
  • 8-5 = 3
  • 8-3 = 5

5. Multiple ways of getting to an answer
E.g. 674+308
To achieve this, use the following steps:
Step 1: Pose an interesting problem before the students to solve
Step 2: Let the students employ their own ways to try solving it
Step 3: Allow them to explain how they got their answers
Step 4: Discuss which method could be the best or introduce the most apt step saying that ‘This is how I would solve it. If you find it useful you can also use it.’
Step 5: Give more interesting puzzles to solve using the currently discovered methods.

6. Develop theorems/ axioms from solving problems
Let the students ‘deduce’ the theorem or a rule instead of giving it to them in the first place.
E.g. Introduce the following problem saying ‘There is a magic in this. Can you find it?’
5+3 =                                                               6+5 =
3+5 =                                                               5+6 =
The concluding theorem is “Distributive property of Addition”
Note: This method of teaching is more formally called ‘Inductive Teaching’. This works on the idea of ‘allowing students to construct theories internally.’

7. Reverse theorems
E.g. ‘In a triangle, the sum of its angles is 180 degrees.’ A student should also be able to state ‘A shape with the sum of all its angles 180 degree is a triangle.’

8. Find out the derivates as well
If in a number statement 5+3 = 8, a student should be able to answer
3+5 = ?
8-3 = ?
8-5 = ?
E.g. 20% of 50 is 10, then
What percentage of 50 makes 10?
20% of what gives 10?

9. Make your own questions
The best Mathematician is the one who is able to make his/ her own questions.
10. The beauty of Maths
Introduce various Math puzzles, Number games, Interesting Mathematical patterns, Maths stories, Number songs to generate interest in the subject

Characteristics of a good Math Activity
Based on the 10 Commandments, we could conclude that a good Math Activity will have the following characteristics.
  • There should be an unquenchable thirst to find the answer
  • Will have the qualities of a sport/ game, the uncertainty factor in the activity.
  • Will give the children a chance to think, estimate, hypothesise and test one’s own hypothesis
  • A question that is open so that children could try multiple ways to do it [instead of asking one question – one answer kind of problems]
  • Will have the possibility of asking continuous questions
  • The students will be able to construct a theorem/ axiom at the end of the Math activity.