Sunday, September 29, 2013

How to Teach Math as a Social Activity | Edutopia

a truly inspiring video... an Alaskan version of Dr Yeap.

How to Teach Math as a Social Activity | Edutopia

Day 6: Recap & Reflect

After 5 evenings n a Saturday morning
of solving mathematical problems,
I did some reflection ...
am I ready to return to my classroom
and be a better teacher of mathematics?

Do I have what it takes
to create a mathematical climate
in my classroom:
allowing students to ask questions,
challenging them to think deeper,
providing concrete materials,
facilitating their learning journeys?
allowing children to collect items from the outdoors,
and to work with them to solve problems.
providing a wide array of concrete materials
for sorting, counting, patterning etc. 

student working alone
in small groups

teacher observing and facilitating
what should I do
to ensure that my students
benefit from being in my class?
what do I hope to achieve?
 Dr Yeap did a quick recap:
1. teach so that children build up intellectual capital, not just so that they can apply
2. teach children to visualize
3. teach children to look for patterns
4. teach number sense
5. teach metacognition (management of thinking)
6. teach metacognition (sharing ideas/strategies/solutions with others)

Re-acquaint with the theories of:


More reminders:
1. Teach number sense but not computational skills
2. Don't let conventions be stumbling blocks, over-riding conceptual knowledge
3. Do not teach key words strategy e.g. "fewer" means that you minus--> this is not always
    the case:
Clare has 3 cookies fewer than Adeline.
Clare has 8 cookies.
How many cookies does Adeline have?
In this instance you add and not minus.

I have found the following video to be very useful and inspiring in understanding how to establish a mathematical environment and how to be a teacher of mathematics.
Video: "How to teach Maths as a Social activity"

Khob khun ka Dr Yeap!
Thank you for challenging me and
 facilitating my learning journey
for the past 6 sessions.
The fear that used to grip me is gone;
the embarrassment is no longer an issue.
I now view maths
and the teaching of maths
from an enlightened perspective.
Terima kasih!
Toh Xia!
from a very grateful heart,
Oh, this is not over yet ....
there's group assignment and individual assignment to complete.   

Friday, September 27, 2013

Day 5: Geometry & Measurement

Inspiration @ the Singapore Art Museum

Kim Yang, Sharon and I toured the Learning Gallery where we viewed, commented and pondered over Southeast Asian contemporary artworks of Eko Nugroho, Yuree Kensaku, Justin Lee, Zsa Zsa Zsu, Terra Bajraghosa among others. Walter the colossal bunny was there too.

Why were we there? Following in Greg Tang's footsteps to artfully build number sense, strengthen problem-solving skills and develop deep thinking skills in young children through contemporary artworks.

We'll be meeting again to think of creative ways to help young children sort, count, estimate and measure .....yup, it's our group assignment.

Found myself paying closer attention to the way Prof Yeap modelled how we should teach asking lots of questions:
"I wonder which container
can hold more water?      
How can you tell?             
Are you saying you can use
        2 small triangles          
to make a bigger square?   
Are there other ways
to make this square?
     Why not?              
It's a surprise, isn't it?                  the more pieces you use,
   the larger the square         
is that true?           
Can someone explain
how you can make
a smaller square with
more pieces of triangles?
Problem 18: Make Squares (using tangrams)
Priscillia, Kim Yang and I formed these squares with different number of tangram pieces:
2 pieces
                  3 pieces

               4 pieces                                                  5 pieces
                                                                  7 pieces                
We couldn't figure out how to use 6 pieces to create a square. 
I wonder why? 
Prof Yeap reminded us that
  learning is not
knowing more
         but tehinking more deeply.               

Next, we traced these squares on a white piece of paper and began taking a closer look at the squares  to answer these questions:
How many different sizes?
How many ways to create the square of one size?
How many pieces can you uese to create a bigger square?
Are there squares of other sizes?

Before we attempted to solve problem 19, we discovered that multiplication
has 4 meanings:
1. the equal groups (e.g. 6 groups of 4)
2. comparison (4 times as many girls) but not 4 times "more than"
3. area model (using square tiles e,g, 7 x4)
    as compared with array model (rows of birds or soldiers)
4. combinations (counting the possible pairings that can be made between 2 or more sets) 
5. rate (e.g. 3 km per hour
                   2 hr --> 6km (2x3)
                   3hr  --> 9km (3x3)
(refer to pages 168-169 of textbook)
Problem 19: Making squares from triangles
Method 1
Cut out a triangle (in this case an isosceles right angle triangle = 90 degrees).
How do we know that the smaller angles are 45 degrees each?
Fold the two corners from the base to the top (right angle) (45 + 45 =90).

 Method 2 (as suggested by Rowena our mathematician)
Tear off the three corners of the triangle and arrange them to form a straight line.
Does that make sense?
What is making sense?
It is making connections (connecting prior knowledge about 180 degrees on a straight line) to the present task. 
Next, we were challenged to convert a triangle into a rectangle.
Method 1:
Similar to folding triangle into square (above).
Can you calculate the area of the rectangle if you know the lengths of two sides of the triangle (one is Bcm and the other is H cm)?
If you can, you have arrived at the formula for area of triangle:
1/2 base x height

Method 2 (as shared by Rowena)
Cut off the top portion of the triangle, rotate it and place it next to the bottom half to get a rectangle.

An important reminder:
Teachers need to apply Howard Gardner's Multiple Intelligence theory in their classrooms. The more we allow our students to use their five senses plus their interpersonal and intrapersonal intelligences, the better they can assimilate and accommodate new information (Jean Piaget's theory).
As Confucius said...

in other words ....

A fruitful evening indeed!

Thursday, September 26, 2013

Day 4: Fractions, Subtraction, Multiplication & Polygons

Problem 13: Mind Reading

Step 1: Think of 2 digits & reveal only the 1st e.g. 3
Step 2: Make a number by putting the 2 digits together e.g. 35
Step 3: Add the 2 digits together 3+5=8

Step 4: Subtract the smaller number from the bigger 35 - 8 =27 
Prof Yeap guessed the answer correctly on all 4 examples.
How did he do it?
By now, we know it is not magic.
Activity: Challenge your partner: Reveal the 1st digit. Ask her to guess the final answer using the steps above. What was the 2nd digit?
Her initial response ....

0 to many digits...which one to choose???

She was baffled!#+?!

And so was I:-((
To solve the problem, look for patterns!

Method 1: The 2nd digit can be any digit from 0-9. The final answer will fit into a pattern that will provide the answers for any 1st digits used! WOW!!
     1st digit     Final Answer

  3                     27

  6                     54

  4                     36

  2                     18
 Method 2: Take 1st digit and make it a Tens. Next minus the 1st digit and voila, you'll get the final answer!
40 - 4 = 36
60 - 6 = 54 

Method 3: Taking a good look at the pattern, you will see that the answers are multiples of 9! So take any 1st digit and multiply by 99 and ta-da you'll have the final answer.
1st digit     Final Answer

  3                     27x9 =27

  6                     54x9=54

  4                     36x9=36

  2                     18x9=18
 3 x 9 = 27
Being the SloMo (someone who takes forever to understand anything mathematical), I will have to admit that the 3rd method is the easiest for me to understand and share with another person. Just follow the pattern!
So what are the Learning Outcomes for the problem solving activity?
1. practice subtractions
2. look for patterns
3. explain/communicate to others about how you arrived at the solutions

Problem 14: Fractions and Subtraction

If 1 green bar = 1,
1) show 3 1/4.
2) show 3 /14 - 1/2
Method 1
Divide all the bars into 1/4s
13 - 2
 4    4 
 4    which is the same as 2 3/4
Method 2
3 1/4 - 1/2
   /   \
11/4     2
from 5
        4 take out 2/4 (1/4+1/4=1/2) remaining 3/4
so 31/4 - 1/2 = 2 3/4

Problem 15: 3 little Pigs
Scenario: The 3 little pigs are famished after their encounter with the wolf. They decided to "pig out" and have some pizzas, only that they are now faced with a problem...
             ... they bought 4 whole pizzas.

Challenge: How can 3 pigs share 4 pizzas equally?
Answer: logically, they can't!!
              mathematically .......   
Method 1:
4 divide by 3 = 3/3 + 1/3 = 1 1/3

Method 2:
4 divide by 3
12 thirds (12/3) divide by 3 = 4/3 = 1 1/3
so each pig gets 11/3 pizza.....I'd loved to see pigs doing maths at the farm:-)
Problem 16: Multiplication (with the help of the cutest birds) 
Challenge: How do you arrive at answers for multiple rows of bird?

Method 1: (by doubling) 
If 2 x 7= 14 
then 4 x 7= 14+ 14 = 28

Method 2: (by combining) 
If 2 x 7= 14 &
   3 x 7= 21
then 5  (2+3) x 7= 14 + 21 = 35 

Method 3: (subtracting)  
If 10 x 7= 70 
then 9 x 7= 70 - 7 = 63
before this module ends, here's something to reflect upon ....

Children must learn
to figure out and also
acquire the belief that
they can figure out.
(the same goes for teachers too!)

The T-E-A-C-H-E-R factor
A child is better off with a good teacher in a lousy school
than with a lousy teacher in a good school
A child is better off with a good teacher in a large group
than a lousy teacher in a small group.
(can anyone remember who said this?)


Wednesday, September 25, 2013

Day 3: Teaching Fractions


A quick review:

Q: According to Richard Skemp, what is the difference between procedural knowledge and
    conceptual knowledge?

Simply put...
Procedural knowledge is knowing what to do.

Conceptual knowledge is understanding why you're doing it.

 e.g. Dayna  might know that in order to add fractions, she needs to find a common
       denominator, even though she has no idea why it's important. That's procedural

But Marcel realizes that a common denominator gives each increment of the fraction equal weight, thus enabling him to add them. He has conceptual knowledge

It is important for children to acquire procedural and conceptual knowledge hand-in-hand.

Nuggets from Quiz 1

DO NOT confuse "enrichment" with "acceleration".

In enrichment, we provide more challenge so that the child can be more fluent
e.g. move from concrete to pictorial (draw AB repeat patterns)
       explore other possibilities (introducing a 3rd colour)
       looking for AB repeating pattern in the environment

DO NOT introduce concept of a higher level (teaching tomorrow's lesson)...that'll be acceleration!

Problem 11: Share equally among 4 persons

Q: Just how many ways can you divide a piece of chocolate equally among 4 persons?
A: Kim Yang and I folded our rectangular pieces of paper into 16 parts and found various

Clare shared her idea with the was a smart way of using 1/16 to create 4 different shapes yet the same value: 4/16 = 1/4 
And now. to figure more ways to share that bar of chocolate equally among 4 persons...
 Dr Yeap's advice:
better to use concrete and proportionate materials such as ten base blocks
then abacus beads which are concrete but disproportionate
beginning to enjoy thinking like a mathematician ...







Day 2: Teaching Whole Numbers

Methods of Instruction

Teachers and parents alike must be mindful of the ways we teach Maths to young children:

1. Modelling (NOT explaining)                       I DO
2. Scaffolding (but not being child's clutch)   WE DO
3. Allow child to work independently             YOU DO

Maria Montessori reminds us that....

Q: Why can't my K2 daughter count?

There are 4 reasons why children cannot count:

1. they cannot classify or sort (pre-numeracy skill)
2. they cannot rote count
3. they cannot do one-to-one correspondence
4. they do not have the conceptual understanding of cardinal numbers i.e. the last number
    uttered represent the size of the objects e.g. child counts 1-2-3-4-5 pens. When asked 
    how many pens there are, the child should be able to say "5" (the last number uttered)

What these children need is opportunities to "play" (count) with concrete materials, supported by adults before they are left to work independently.

Problem 7: Making 10 Method

Every young child should use the 10-frame.

How many ways are there to make 10?
Counting in 10s, in 5s, counting on ...


Suggested Links:

Jack and the Beanstalk by Debbie & Friends

Problem 8: How do you share 51 golden eggs among 3 persons?