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Dr. Yeap Ban Har Making Connections through Multiplication and Division | Grades 3 – 5 SYNOPSIS |Using the CPA Approach which is fundamental to Singapore schools, we will explore the learning of multiplication facts as well as multi-digit multiplication and division. The extension between multiplication and division in whole numbers and in factions will be discussed. We will also look at anchor tasks for lessons where the emphasis is on practicing a previously-learnt skill.

The CPA (Concrete Pictorial Abstract) Approach is based on Bruner’s ideas of enactive, iconic and symbolic representations. New concepts are learnt through concrete experiences. Stages in Learning Multiplication Facts

1. Counting 2. Figuring Out 3. Remembering

3 x 4 =

6 x 4 =

7 x 4 =

9 x 4 =

Let’s figure out!

Anchor Tasks for Multi-Digit Multiplication and Division

Source | Maths – No Problem! Year 4

A farmer sells his crops to 3 restaurants. He divides 525 heads of lettuce equally among the 3 restaurants. How many heads of lettuce does each restaurant receive?

Source | Math in Focus 4

Hundreds tens Ones

Note | In Partitive Division, you know the number of groups. Example – Four children shared 12 cookies equally. In Quotitive Division, you know the number of items in each group. Example – Twelve cookies are put in packs of 4 cookies.

Anchor Tasks for Practice Lessons

Source | Every Child Can Do Math

Fraction Connections

1. How many halves are there in 2

12 ?

Source | Math in Focus 4

2. Find the value of 4

3 of 20.

Source | Primary Mathematics 4


Dr. Yeap Ban Har Developing Number Sense through Counting, Adding and Subtracting | Grades K – 2 SYNOPSIS |Using the CPA Approach which is fundamental in Singapore schools, we will explore what students need to be able to do to be proficient in counting, adding and subtracting. How can teachers of younger children develop deep understanding using small numbers? The use of ten frames, number bonds and bar models will be emphasized. We will also learn several mental computation strategies.

The CPA (Concrete Pictorial Abstract) Approach is based on Bruner’s ideas of enactive, iconic and symbolic representations. New concepts are learnt through concrete experiences.

Teaching kids to count is fine, but teaching them what counts is best.

Bob Talbert

A fundamental idea in counting is that you can only count things in the same set i.e.

having the same ‘noun’. It is obvious that four oranges and an apples does not make

five apples. You don’t count different nouns!

In the same sense, 1 ten and 4 ones do not make 5 tens. Or, 1 half and 4 fifths do

not make 5 fifths. Or, worse, 5 sevenths!

Implications – integrate sorting with counting rather than teaching those as isolated

skills.

1. Sort the fruits.

2. Count how many in each basket.

Source | Earlybird Kindergarten Mathematics Reader

Let’s look at the use of ten frames for counting numbers less than 10 and numbers

greater than ten.

Implication – Help students progress from counting in ones to counting in tens and

ones.

Count, “1, 2, 3, 4, 5, 6, 7, 8, 9, 10. 1, 2. 10 and 2 is twelve.”

Instead of “1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.”

Anchor Tasks for Counting, Adding and Subtracting

1. Count 4 square tiles. Make a shape using 4 square tiles.

Source | Earlybird Kindergarten Mathematics

2. 8 + 6 Source | Maths in Focus 1

3. Subtract 28 from 300.

Source | Primary Mathematics 2 Third Edition

Number Bonds

Two children starts with a small pile of beans. They take turn to remove either one

or two beans until all the beans have been removed from the pile. The winner is the

child who takes the last bean(s).

Bar Model

Source | Primary Mathematics 2 Third Edition


Dr. Yeap Ban Har Key Note | The Big Fives in Mathematics Classrooms SYNOPSIS | What are five things that students who are good in math do? What are five key competencies that students need to have by the time they start middle school? Who are the five wise guys who can answer most of the questions we have in mathematics education? What did they say about mathematics learning?

In his engaging and thought-provoking keynote, Ban Har will offer a simple framework that guides good mathematics instruction. You will learn how students develop ability in visualization and generalization, number sense, metacognition and communication. We will see that "it is fine to teach children to count, but it is best to teach them what counts!" (Bob Talbert)

visualization

generalization

number sense

metacognition

communication

I can show.

I can draw.

I can explain.

I can write about it.

I can find ways to challenge myself.

Piaget

Dienes

Vygotsky

Skemp

Bruner

explore

structure

journal

reflect

practice

core competencies

good at math

elements of

effective lessons

learning

theories

model

scaffold

let go

explain

counsel

Yulin wants to make 14 small identical stars and 20 large identical stars using wire.

She has made 12 small stars and 7 large ones using 960 cm of wire. The length of

the wire she used for 3 large stars is the same as that for 4 small stars.

What is the length of the wire she needs to make the remaining stars?

Source | Primary School Leaving Examination

tools pedagogical


Dr. Yeap Ban Har Developing Foundations for Multiplication and Division | Grades K – 2 SYNOPSIS | Using the Problem-Solving Approach which is fundamental in Singapore schools, we will explore the acquisition of multiplication facts. What are some things that young children do that prepare them for formal learning of multiplication and division. The role of visuals such as dot diagrams and language will be emphasized.

Source | Maths - No Problem!

Stages in Learning Multiplication Facts

1. Counting 2. Figuring Out 3. Remembering

Stories for Multiplication

Equal Groups

Array

Comparison

Rate

Combination Stories for Division

Sharing

Grouping

Note | In Partitive Division, you know the number of groups. Example – Four children shared 12 cookies equally. In Quotitive Division, you know the number of items in each group. Example – Twelve cookies are put in packs of 4 cookies.

3 x 3 =

6 x 3=

7 x 3 =

9 x 3 =

Let’s figure out!

Anchor Tasks

1. Find the value of 18 x 3.

2. Find the value of 54 3.


Dr. Yeap Ban Har Fractions | Grades 3 – 5 SYNOPSIS | In this session we will look at two key ideas in learning fractions and consider strategies to teach addition, subtraction, multiplication and division of fractions.

Example 1

Share among 4 friends. They all get the same amount.

Example 2

Example 3

Grace arranged 2

1,

3

2,

4

1 in order, from the greatest to the smallest.

Is she correct?

4

1

2

1

9

8

3

13

63

2

4

3

3

1

23

1

3

13

3

1

Example 9

Find the value of 125

4 .

Give your answer as a fraction in the simplest form.

Source | PSLE Singapore 2013

Example 10

The total cost of a pen and a book is $32. The cost of the pen is 5

3 the cost of the

book. What is the cost of the book?


Example 11

Kai Li spent 3

1 of her money on 5 magnets and 11 postcards.

The cost of each magnet is 3 times the cost of each postcard.

She bought some more magnets with 4

3 of her remaining money.

How many magnets did Kai Li buy altogether?


Example 12

Mr Lim had a total of 540 long and short rulers. After selling an equal number of

both types, he had 3

1of the long rulers left and

6

1of the short ones left.

What was the total number of rulers left?



Dr. Yeap Ban Har Math Journals| Grades K – 2 SYNOPSIS | In this session we look at a framework for teaching to mastery and examine the role of journaling in the process. How does journal entries look like? What do we do with them? How does it help in assessment and differentiation?

Teaching to Mastery

To master counting, Kindergarteners should be able to tell the number of a set of objects. Fluent at counting means able to do it with accuracy, confidence and some degree of flexibility. It does not mean count fast!

To master number facts, First Graders should be able to recall 1 + 4 = 5 and, in cases, they cannot recall (8 + 9), they should be able to figure out (8 + 9 = 10 + 7). Fluency is not mere speed. When one is fluent in a language, it does not mean one speaks fast, does it?

Anchor Task o Explore o Structure o Journal o Reflect

Guided Practice

Independent Practice

The framework for teaching to master involves several stages and journaling is one of them.

In an open lesson for students who had just completed Kindergarten, the anchor task was to use unifix cubes to explore the number bonds of 5. Later, they extend their investigation to the cases of 3 and 7.

The journal writing was to look for and write down two numbers that add up to a given number. Owen was given 6, for instance.

Source | Owen’s Journal

Source | Alex’s Journal

Owen looked for two numbers that add up to 6.

He tried to record his results in his journal.

Owen was able to make number bonds for 6.

He was also able to see possibilities – 5 and 1,

3 and 3 as well as 4 and 2. Is recording was

less successful as his ability to form numerals was

still not developed. He also attempted to use

symbols (not used in this lesson) that he had little

understanding of.

Although not required, Alex had chosen to record the results of the

investigations in his journal. It can be seen that he went from random

recording to making a systematic list. It is interesting to observe the

case of 0 continued to elude many students.

Journal allows advanced learner to experience differentiated

lesson. In this case, Alex went to explore cases involving negative

numbers while waiting for the class to complete their investigation..

In an open lesson for students who had just completed Grade 1, the anchor task was to add three single digit numbers such that the vertical sum equals the horizontal sum.

After working on it with their friends and discussing it, students document how it is done.

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Math Plus Kansas City

Education