1) The document discusses 4 levels of mathematics language used in teaching: students' language using familiar objects, materials language using concrete objects, mathematics language using proper terms, and symbolic language using symbols. 2) It provides strategies and concepts for addition, subtraction, multiplication, and division, including counting on, doubles, decomposition, cover-ups, repeated addition, arrays, measurements, combinations, partitioning and quoting. 3) Visual and verbal models are recommended for introducing new concepts along with using familiar objects and materials to help students make connections.

1. Core Concepts
Skills
Strategies

2. 4 levels of Mathematics Languages
Languages Descriptions
Student’s By this level/ stage, teacher is using students’ language or
Language familiar objects for introduces and teaches students.
For example: apples, cars, books and so on.
Materials By this level/ stage, teacher is using concrete materials such
Language as counters, MAB blocks, Unifix Blocks.
Mathematics This level/ stage, teacher is using proper mathematics
Language language such as three, two, one, ….
Symbolic This level/ stage, teacher is using symbolic to teach students.
Language
For example: 1,2,3…+, -, ,

3. Addition
Strategies:
1. Counting up (count on) is most easy way of
addition. It is a way to add small number into
bigger number.
For example: 3+2= ?
Then count up two from 3; 4,5.

4. Addition Strategies
2. Use Double
Students can remember the pattern of the
doubles in sequence.
1+1=2 4+4=8 7+7=14
2+2=4 5+5=10 8+8=16
3+3=6 6+6=12 9+9=18
If 2+1+2+1= 2+2+1+1
= 4 +2
=6

5. 4 Levels of Mathematics Languages
Children Alice has 4 books and Melody has 3 books , put all
books together on the table.
Language
How many books on the table?
Materials There are 4 counters put 3 more counters make 7
counters altogether.
Language
Mathematics Four add three equal seven
Language
Symbolic 4 + 3= 7
Language

6. Subtraction
• Skill : Decomposition (Jamieson-Proctor, 2012)
3 12
Tens Ones
4 2
- 1 5
2 7
Trade 1 ten
into 10 ones.

7. Strategy for Subtraction
• Cover-ups
This strategy is just like ‘take away’ skill.
For Example:
Alice has 3 cupcakes and rabbit take away one cupcake
from her. How many cupcakes does Alice has?
3–1=?

8. Multiplication
Concept
• Multiplication is repeated addition of some equal
set or group (Jamieson-Proctor, 2012).
For example:
Alice has 4 boxes of cupcakes and each box has 2
cupcakes. How many cupcakes does she has
altogether?
Answer : 4 X 2 = 2+2+2+2
=8

9. 4 types of Multiplication Models (Jamieson-Proctor, 2012)
Set Model
2+ 2+2 = 3X2
Array Model This model is used on measure area.
For example: The length of a square is 2 cm and the width is 2 cm.
What is the area of the square?
2 cms
2 X2 = 4 cm
2 cms
Measuremen I bought 2 ribbons each 3 meters long. How many meters did I buy?
t Model 2X3 = 3+3
= 6 meters
Combination I have 2 different patterns shirts and 3 different patterns of pants. How
many outfits can I make?
Model
2 X3 = 3+3
=6

10. 4 Levels of Mathematics Languages
Language Stage Materials Stage Language Stage Recording Stage
Student Concrete Materials such Altogether, how many No symbols, children
Language as apples, balls . draw pictures only
Materials Substituted objects. Add repeatedly, how No symbols, children
Language E.g: unifix blocks, many, altogether draw pictures only
counters
Mathematics Substituted objects. Multiply, multiplication, No symbols, children
Language E.g: unifix blocks, euals draw pictures only
counters
Symbolic - multiply 2X3=
Language 4 X 3=

11. Division
Concept
• Division is separating an amount into equals groups (Jamieson-Proctor, 2012).
• There are two ways to do the separation process : partition and quotation
Teaching sequences
• Use Student’s language (1st level of Mathematics language)
• Introduce: Teacher use concrete materials students are familiar with to
demonstrate the concept, whole class activity, oral activity.
• Models: Visual and verbal
• How : Teacher use familiar object to help student to make connection about the
concept of division is to separate the total amounts into equals groups. Division
is the inverse of multiplication.

12. Partition and Quotation
Partition Quotation
Partition is sharing objects into groups Quotation is repeated subtraction
equally and see how many there are in where a small quantity. To find the
the group. number of groups needed to divide up
the total.
I have 8 flowers, give 2 flowers to each
friend. How many friends can I give to?
12 3=？
1 group of 3
2 group of 3
3 group of 3
4 group of 3