Saturday, September 28, 2013

Day 6- Last Day of Class!

Day 6- Last Day of Class!

Here's a review of what the 20 + problems that we did over the last few days taught us to do...

1. Promote Visualisation

2. Spot patterns

3. Gain number sense

4. Think about communication 

( explaining the reasons behind the methods used, how teachers choose appropriate words and terms to use when teaching math)

5. Promote metacognition ( higher order thinking)

We discussed about how important it is for teachers to choose appropriate words and terms when teaching math!

This example of division is too confusing, having too many symbols present!

This is how it should be done!

We also talked about graphs and how we can extend learning from a simple graph.

A possible progression

1. Simple bar graph in proper arrangement ( as shown)
2. Have specific counters to represent a certain number in the bar graph 
(eg. Yellow= 10 people)
3. Plot them in a pie chart format or a line graph format

Using poker cards and our foreheads, we played a game of multiplication and division with our friends!

The SALUTE! game is to have us take a card each, place it on our foreheads and have us guess the number on our card, using our friends' cards as clues.

The more number of players, the more times u need to multiply and the bigger the number is for you to divide!

I truly learnt new methods of teaching and thinking over the span of these few days. 

It was mind-boggling at times, forcing my brain to think of various ways to deal with a single problem and thinking deeper into a problem and how to explain it succintly.

Nevertheless, it was a fruitful journey for me!

Thank you Dr Yeap for your time and effort!

Till next time!

Friday, September 27, 2013

Day 5- Differentiated Instruction , Measurement and Geometry

Day 5- Differentiated Instruction , Measurement and Geometry

Today we discussed quite a bit on being a Teacher of Mathematics.

: to ENRICH learning through different methods with regards to a single concept.

This promotes and develops higher order thinking in the process.

Great take-home points for me to think about too:

Do we compromise ACCELERATION
 with higher order thinking?

Do we rob children of opportunities to think deeper as we push them 
to learning one new concept after another?

Do we overteach our children that we stifle their learning unknowingly?

Also, we talked about 3 ways we can help a child understand a concept when cannot do so...

1. By Scaffolding 
2. By role modelling
3. By changing the task to a less complicated situation

Many a times, we have to make sure the terms and words 
we use are clear and specific to avoid confusion!

Problem 18

Find Out: How many different sizes of squares can we make with a set of 7 tangrams?

These are some ways our group explored with and because of the sizes of the tangrams, 
the relation of the original square to the largest square is as such:

Problem 19

Find Out: Can all types of triangles form 180 degrees?

Method 1: Through folding, we can clearly see that 3corners: 45+ 45+ 90 = 180 degrees

Method 2: By cutting and shifting the parts,  3 corners form a straight line= 180 degrees

There are probably many more different ways to solve this problem through cutting and folding!

Problem 20

Find Out: Can congruent triangles make rectangles?
  Also, find the area of a triangle.

By using 3 different folding methods, the triangles can be made in rectangles.

The formula of area of a triangle : 1/2 X Base X Height 

is explicitly explained when I saw how 

2 congruent triangles form a rectangle!

1/2 a rectangle X Base X Height!

I really feel happy that a mere memorized formula I know 
actually makes sense and  is understood today! :D

Thursday, September 26, 2013

Day 4- Fractions, Subtraction, Polygons

Day 4- Fractions, Subtraction, Polygons

We continued on with Problem 13: Mind Reading
An interesting problem that had me clueless for the first 10 minutes!

Step 1: Think of 2 digits, make known the 1st digit only.
Step 2: Put the 2 digits together, and use that number to minus off the sum of the two digits.
              (eg: 2 1 - 3 = 18)

Have your friend find out: Given the 1st digit only, find out the 2nd digit.

Dr Yeap had us try it out and was able to guess the final answer while being given the 1st digit only.

That baffled us for a while!


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


1st digit     Final Ans

  3                     27

  6                     54

  4                     36

  2                     18

Using this pattern, 2 more methods are found....

Method 2: Take 1st digit and make it a Tens and minus the 1st digit off to get the final answer.

Eg: 30 - 3 -27

Method 3: Have any 1st digits x 9 and you get the final answer.

Eg: 3 x 9 = 27

Personally, I find Method 3 the best and most straight forward method to do!

IMPORTANTLY, this problem allowed 3 levels of teaching here:

1. Learning subtraction
2. Identifying the patterns present
3. Ability to articulate the reasons of the patterns present

This is an example of a problem displaying Differentiation.

Problem 14: Fractions and Subtraction

Using a fairly simple problem, we saw how 2 methods can be used to help children understand ways to calculate the sum 
3 1/4 - 1/2 = ?

Problem 15

Find Out: How do 3 piggies share 4 pizzas equally??

Method 1: 4 divided by 3 = 3/3 + 1/3 = 1 1/3

Method 2: 4 divided by 3 = 12 thirds = 4/3 = 1 1/3

Problem 16 had us thinking about multiplication.

Find Out: How do we get answers for multiple sets of 7 birds?

Method 1: By doubling certain sets 
 Eg:  2x7= 14 , so 4x7= 14+ 14 = 28

Method 2: 2 sets of 7 added up together 
Eg: 2x7= 14 & 3x7= 21,
Therefore, 5x7= 14 + 21 = 35 

Method 3: Subtract 7 from a larger set of 7 
Eg: 10x7= 70 , so 9x7= 70-7 = 63

These provided more efficient ways of using multiplication, addition and subtraction together to arrive at answers at a faster speed!

Problem 17 of exploring Polygons made us think deep into the concept as we...

Find Out: Forming Possible polygons that has 1 dot left in the center

The Geoboard app was a good medium to explore polygons!

It was a fruitful time of consolidating our ideas as we found out the smallest to biggest polygons that can be formed!

From a few ideas...

to a complete set of the best ideas we could come up with!

Ending off, 

Whether you are a good mathematician or not, 

Here's a good Quote to advocate Math as A Math Teacher...

It's not about the answers, but the Concepts of Math that can enrich our everyday lives!

Wednesday, September 25, 2013

Day 3- Fractions and 1st Quiz!

Day 3- Fractions and 1st Quiz!

Class started with doing Problem 9, using digits 0-9.

Find out: How many ways can we form a 2 digit equation without repeating any digits?

( _ _ + _ _ = _ _)

My partners and I discussed mostly about how to arrive to getting the 
BIGGEST and SMALLEST  equations that we can come up with.

We started with the biggest and smallest number sum possible and did trial and error with the digits till we got the best answers.

BIGGEST Equation: 98 ( number sum) = 25 + 73

Smallest Equation : 39 (number sum) = 15 + 24

Then, we had our quiz! The moment that I was afraid of! Thank goodness it was not that tough for me and I learnt some good pointers during our discussion of the 2nd question in the Quiz.

Meaning of ENRICHMENT sessions: 
Providing more challenging practices for children without introducing new concepts.

An illustration that I found useful to remember what Enrichment is:
To weave more strings with the single string I have, strengthening the bond, making my string a stronger one!

Unlike ACCELERATION sessions: where children learn a new topic or concept of a higher level. 

It sure is a great difference between these two terms!

Problems 11 and 12 were on Fractions and we thought of a variety of ways to share parts of a rectangle equally among 3 and 4 persons.

Especially in Problem 11, we really cracked our brains and thought of out-of-the-box ideas to share a rectangle equally amongst 4 people! It was a good way to use creativity to solve a math problem!

Problem 11

Problem 12

Here's one point that preschool educators should be aware of when they teach Math!

Start teaching counting using

Concrete and Proportionate materials

Instead of using.....

Concrete but Non-proportionate materials

This way, we don't end up creating confusion in children as they learn and they can understand the concept of Tens and Ones clearly too! 

Day 2- on Whole Number

Day 2- On Whole Numbers

It was insightful as we learnt about the "type" of numbers we have been teaching and encountering with daily. The terms we talked about today can be confusing! But once we have examples coupled with the terms, we are less likely to mix them up!

1. Ordinal Numbers ( eg. 6th in place)

2. Nominal Numbers (eg. Player 6 won the race!)

3. Cardinal Numbers (eg. There are 6 sweets in the jar)

4. Numbers for measurement purposes (eg. The boy is 6 years old.)

Problem 5

The video of "As I was going to St Ives" - Sesame Street helped to emphasize a strong and important point in Math. The people, sacks, cats and kittens mentioned have viewers confused about what the question posted was really about.

In actual fact, they only wanted to know how many people were going to St Ives!
Answer: 1 man and his 7 wives = 8 people.

Just like how this picture shows how dogs and cats can't be put together, we should NEVER add things of different categories together! 

1 dog + 1 cat = No possible!
Unless you are looking for the number of animals....
Then 1 dog + 1 cat = 2 animals 

Also, preschoolers should start exploring with counting of the same nouns so they will understand the concept of Adding the same kind of things together.

As we did Problem 6, I simply started playing with no pre-calculations for winning. When we discussed how the number that we decided to count down by affects the winner and loser of the game, i began to see a pattern emerging.

There were a set of "bad numbers" for each count down number chosen and if it were to be my turn when there were the amount of "bad number" beans, then I would SURELY LOSE!

As our group of three ( Clare, Michelle, Lydia) explored with the beans, we also tried playing with 3 players and were able to predict the loser of the game better too!

Problem 7 had us thinking about counting in different ways using 10 frames. We experimented with 3 sets of 10 frames and thought of different number compositions to make 18.

In a more creative manner for the use of preschool settings, we can use objects such as egg trays as 10 frames for children to explore with.

Lastly,  knowing these are very useful information for us to understand how children learn! :

Pre-requisites to learning how to count

1. Knowing how to classify
2. Knowing how to rote count
3. Understanding 1 to 1 correspondence

Giving children a variation of number sets helps them to grasp the conceptual understanding of cardinal numbers too!
 Example: Sets of 3 in variation.

I am glad I learnt a whole lot of new things about whole numbers! :)

Monday, September 23, 2013

Day 1- Creating a Mathematical Climate in the Classroom

Today's our first lesson of the module and i thought it was rather light hearted and engaging!

Problem 1 showed me how there can be a number of methods used to solving a problem and that it was all about identifying the patterns in the methods used. As long as your method and pattern stays consistent, your answer can surely be found!

It was helpful when Dr Yeap have us to think back and reflect upon how we solved the problem and spend time discussing about it with our classmates. As a not-so-ardent-fan of mathematics, I feel more interested and enthusiastic in trying to solve this math problem when I do it alongside my classmates.The exchange of ideas with others also keep each other going and having fun together!

Problem 2 had the class hyped up and excited to find the solution to the 'Magic Trick'!

Our group literally REJOICED! when we had Hui Min who found the solution for us!
So here is the answer of the sequence of cards to the "Magic Trick" we learnt today!

It is an excellent game we can adapt and use for learning numbers 1-10 and spelling them too. 
Changing the numbers to shapes or other themes can do the trick too, as long as we arrange the cards in the correct pattern.

Problem 3 was about the trucks, paper and shredders. It was this evening's most "serious" problem that we discussed about. For me,  I took a while to rationalise how to solve the problem. I am glad I managed to get the answer together with my classmate!

Problem 4 was a fun hands-on activity which demonstrated many possibilities to forming shapes. 

Having us explore with different restrictions of forming shapes challenged the way we think about the positionings of our tangrams. 

For example, 2 of my classmates and I had equal number of tangrams, but we had different compositions and combinations of tangrams used to form congruent rectangles. 

As adults, we were actively engaged in exploring the tangrams in a concrete manner. Even more so for children, they will learn effectively with concrete materials given. They may even discover ideas and solutions which adults have not!

As Dr Yeap mentioned about Diene's theory; I think we were all involved in the "Play" stage at class today! :)

Saturday, September 21, 2013

Note to Parents (Pre-course reading on Chapter 1 and 2)

Dear Parents,

“How should children begin to learn and enjoy Math?”

Often, mathematics is seen as a subject that involves a lot of rote learning and the memorizing of formulas. Contrastingly, the mathematical concepts learnt should make sense to children as they learn to reason and solve math problems logically.

Learning mathematics in the 21st Century is about understanding how mathematical concepts are interconnected with one another and how mathematics is related to the real world. All these can be progressively learnt in conducive classroom environments that allow children to explore, problem-solve, make mistakes, ask questions and learn from teachers and peers.

When children are in an environment where they can exchange ideas and discuss about mathematical concepts freely, they have opportunities to reflect upon their findings, listen to their peers and learn much more than they would as individuals.

Richard Skemp, an educational psychologist, developed five aspects of mathematical proficiency, which can serve as guidelines for teachers to facilitate math learning more effectively. They are; conceptual understanding, strategic competence, adaptive reasoning, procedural fluency and productive disposition. (pp. 26-28)

With the awareness of these five aspects, teachers can intentionally plan purposeful math activities for children with the goal of having them attain these gradually. Beneficially, children have a richer understanding of concepts when they understand the relations between concepts, allowing them to remember and recall strands of information with more ease. Furthermore, the relational understanding in children will allow them to apply problem-solving skills in different situations that they encounter too.

Without a doubt, children will grow to have a positive attitude towards learning mathematics. They will develop persistence in attempting math problems and confidence in their ability to understand mathematics.