Transcript ALBEL SLIDES FOR SENIA CONFERENCE 8 March

Strategies and Activities to overcome
Maths difficulties in Primary Maths
Date: Friday 26 February 2016
Venue: ISKL Melawati Campus
Time: 9.45 am
by: Mdm Aishah Binte Abdullah
Ms Anaberta Oehlers
About us
• What we do.
• Profile of our students
• Teaching Approaches at the DAS
2
What we will be covering
1) Introduction of ourselves
2) DAS Maths programme – Our
teaching approaches
3) CRA
4) A background on Singapore Maths
curriculum
5) Strategies, Activities and
Heuristics
At the end of this session...
You should be able to:
• Apply hands-on activities to understand and work
out concepts.
• Learn the short cuts of remembering times tables
• Demonstrate how to solve word problems by
drawing bar models, branching out and by drawing
tables.
Our Teaching Methodology incorporates :
OG
Orton-Gillingham Approach :
Cognitive
Direct and Explicit Instruction
C-R-A
Simultaneously Multisensory
Structured, Sequential and Cumulative
Diagnostic and Prescriptive based on the needs
of the child, with an emphasis on conceptbuilding, addressing gaps in their learning.
POLYA
Emotionally Sound, structured for success while
keeping in touch with mainstream school
mathematics syllabus.

the problem
ESSENTIAL
MATHS
PROGRAMME
I
a strategy
I’m sure I can get
the answer

that the answer
makes sense
The DAS Essential Maths programme is carried out :
Concrete
Students manipulate
with tools that enable
them to gain
understanding of the
mathematics concept
eg.
1
2
+
2
5
Representational
Students relate the
concrete objects
into drawings, bar
models, graphs or
tables
Abstract
Students use numbers,
letters and standard
algorithms to represent
Mathematics concepts
=
1 +
2
1
2
5 + 4 = 9
10
10
10
2
5
Fraction
tiles
2 =
5
Model
Representation
Working out
the fraction sum
6
FRAMEWORK FOR MATHEMATICS SUCCESS
1. Develop working memory in students - help students to
Remember, Retain and Recall --> To help children remember
Mathematics concepts :
PROMPT - Get students to :
c) play mathematics games with students to i) encourage them to
notice and focus.
ii) to provide children with the opportunities to plan, sort, count,
measure, compare, match, put together and take apart as part
of number sense and quantity preservation. (ATTRIBUTES,
CONCEPTS , SKILLS)
7
PROBE - Ask students to a) Explain the steps of arriving at their answers in precise
mathematical terms.(PROCESSES)
b) Express comparisons and similarities between similar looking
shapes, lines and objects using precise language terms.
EXPAND - ( METACOGNITION)
a) Ask students- Is there a simpler, different way of getting the
same answer?
2.Develop Reasoning and Communication for Mathematics
success in students:
8
MAIN TEACHING and LEARNING POINTS
A) Precision in Perception and Precision in ExpressionPROMPT, PROBE, EXPAND
B) Repetition for stickiness
C) Use of precise mathematical language in teaching and learning
D) Semiotics ( the science of symbols and gestures)-Gestures such as precise pointing is important. Only when the mathematical
talk is precise and the pointing is precise, there will be synchronicity
between the TEACHER, the PUPIL and the REST OF THE CLASS . Then it is
more likely that LEARNING takes place.
9
What are some topics your
students have difficulties with?
10
Common confusing topics:
•
•
•
•
•
•
•
•
•
•
•
•
•
Placeholder problems
Pattern and Grouping
Pattern Logic
Age Problem
Problems involving ‘more than’, ‘fewer than’ quantities
Finding the area of a diagonal shaded shape in a rectangle
Transfer concept involving money
‘Short of’ concept
Comparison concept
Remembering times tables
Division concept
Conversion-Improper fractions
mixed numbers
Factors and Multiples
Maths difficulties
GRASPING FUNDAMENTAL
CONCEPTS :
COM PREHENSION:
The language of Word Problems
*Placeholder problems
*Pattern and Grouping
*Pattern Logic
*Addition, Subtraction, Regrouping
*Remembering times tables
*Division concept
*Conversion-Improper fractions
mixed numbers
*Finding the area of a diagonal shape
*Age related problems
More than, fewer than
Solution:
Use concrete materials to visualise
Abstract  Visual
Too much information in the word
problems- unable to ‘see’ and make
relationships between the ‘known’
and the ‘unknown.’
Solution:
Use the basic heuristicsDraw a model / diagram
Make a table / Systematic listing
Branch out
12
Some Useful Strategies
13
Placeholder problem
Part- Whole relationship P2 / 3
(8-9yrs)
___ + 165 = 702
The number question is
represented by a model as
follows:
?
Understand:
165
?
702
702
165
100s
Work out:
702 165 = 537
6
_
10s
1s
7
9
10
1
1
6
5
5
3
7
2
Placeholder problem
Part- Whole relationship P2 / 3
(8-9yrs)
The number question is
represented by a model as
follows:
(Part)
___ + 345 = 907
?
Understand:
?
345
(Part)
907
(Whole)
(Part)
907
345
Work out:
907 345 = 562
(Whole)
(Part)
100s
8
_
10s
1s
9
10
7
3
4
5
5
6
2
Pattern + Grouping
P 2/3/4 (8 – 10yrs)
:
Find the 72nd shape in the pattern
••••
1. Understand: Identify the basic pattern and group it.
2. 72 ÷ 4 = 18.
3. The last shape (72nd) is
4. OR
5. 72 ÷ 8 = 9.
6. The last shape (72nd) is
Pattern + Grouping
P 2/3/4 (8 – 10yrs)
:
Find the 64th shape in the pattern
1
10
x 6 rows = 60
61
64
1. Understand: Identify the basic pattern and group it.
2. 64 ÷ 10 = 6 rem 4
3. The 64th shape is
•••• ?
Pattern Logic P3 (9 – 10 yrs)
=
=
=
How many
represent?
does one
Pattern Logic (P3 / 4)
Understand and work out: Substitution or working backwords
=
(3)
=
=
How many
( 4 X 3 = 12 )
( 2 X 12 = 24 )
does one
represent? Answer 24
Hundreds board- addition, subtraction
Addition
Subtraction
45 46 47 48
52 53 54 55 56
55 56 57 58
62 63 64 65 66
65 66 67 68
72 73 74 75 76
75 76 77 78
82 83 84 85 86
45 + 33 = 78
86 - 34 = 52
Teach Regrouping
100s
10s
1s
1
1
2
7
1
1
5
2
4
2
+
12
100s
10s
3
Relate Subtraction with Addition
_
1s
1
2
4
2
1
2
7
1
1
5
21
Tips to remember the 3 times tables
Row 0
1x3
3
Row 1
4x3
2
Row 2
7x3
1
2x3
3x3
6
9
5x3
6x3
5
8
8x3
4
3
6
9
2
5
8
1
4
7
9x3
7
22
Tips to remember the 7 times tables
Activity 1
Tips to remember times tables
8 times tables :
1
increase by 2
one
3
4
4
5
6
7
8
8
9
8
6
4
2
0
8
6
4
2
0
8
6
decrease by
two
9 times tables
increase by 1
one
2
3
4
5
6
7
8
9
9
10
9
8
7
6
5
4
3
2
1
0
9
8
decrease by
one
Activity 2
X
6
7
1
0
2
1
1
6 0
2 1
2
8
2
4
3
0
4
6
4
2
5
8
6
4
3
4
5
6
7
8
9
2
3
3
4
4
5
8
9
7
0
4
1
1
2
8 0
1
6
4 2
8
3
3
5
4
2
4
9
5
6
6
3
7
2
0 4
5
8
6
6
7
4
8
2
9
8
7
6
5
4
3
2
1
Napier’s
Bones for
Multiplication
67
x 7
469
79
x 4
316
Long multiplication
568
x 67
3996
+
34080
3 8076
60
7
568
568
x
7
3 996
x 60
34080
Activity 3
Column Division (Long division) – a concrete visual
experience eg. 7106 ÷ 3 =
⑦
⑩+①
THOUSANDS
⑦
HUNDREDS
⑪
⑳
TENS
⑳+⑥
ONES
⑳
©
©
©
Remainder
1 Thousand
2 Hundreds
2 Tens
Working –Representational stage
Th
3
H
T
O
2
3
6
8
7
1
0
6
Quotient
6
1
1
9
2
1
0
8
2
2
6
4
2
Remainder
Steps for division
• 7103 ÷ 3 = 2368 (quotient) 2 (remainder)
• Step 1 – divide 7 Th by 3
2x3=6 7–6=1
• Step 2 – rename the 1 Th as 10 Hun
10 + 1 = 11 Hun
• Step 3 – divide 11 Hun by 3
3 x 3 = 9 11 – 9 = 2
• Step 4 – rename the 2 Hun as 20 tens
20 + 0 = 20 tens
• Step 5 – divide 20 Tens by 3
6 x 3 = 18 20 – 18 = 2
• Step 6 – rename the 2 Tens as 20 ones
20 + 6 = 26
now
• Step 7 – divide 26 ones by 3
8 x 3 = 24
26 – 24 = 2
1 Th is left over
There are 11 Hun now
2 Hun are left over
There are 20 Tens now
2 Tens are left over
There are 26 ones
2 Ones are left over
Tens
4
Ones
1
_
1
_
Tenths Hths
3
5
7
0
2
3
0
2
8
_
5
0
Placement of the decimal
point
Sue shares $15 with 3
friends. How much is
each friend’s share?
Answer:
Each friend’s share is $3.75
2
0
2
0
0
0
Rule:
When a zero is added to the
tenths place, draw a line
between the ones and the
tenths and mark a prominent
dot on the line in the divident
and also in the quotient to
separate the wholes from the
parts.
Activity 4
Shaded area P4
Find the area of the shaded part of the
rectangle:
Use the following principle- Cut into parts and move the parts around.
12cm
7 cm
2cm
Concrete stage- 1. Cut out and remove the shaded part
Put together
2cm
2. Put the remaining parts together.
10cm
7 cm
3. Area of the new rectangle  L x B
 10cm x 7cm = 70cm²
Steps:
12cm
1. Find the area of the rectangle:
L X B = 12 X 7 = 84cm²
2. Remove 2cm from the length.
Now find the area of the rectangle:
L X B = 10cm X 7cm = 70cm²
3. Area of the shaded part =
84cm² - 70cm² = 14cm²
7cm
2cm
10cm
7
cm
Activity 5
Rounding up / down
Sum  Round off 54 to the nearest 10
Round off 109 to the nearest 10
Answer: ≈ 50
Answer : ≈ 110
Step 1  Underline the digit in the tens place.
Step 2  Look at the digit to its right in the ones place.
Step 3 If that digit is 5 or greater, add 1 to increase the
digit in the tens place. Then replace the digit in the
ones place with a zero.
Step 4  If that digit is less than 5, add zero to the digit in the tens
place. Then replace the digit in the ones place with a zero.
+
tens
ones
5
4
0
5
tens
+
0
ones
109
1
1 1
0
Rounding up / down
Sum  Round off 404 to the nearest 100
Round off 950 to the nearest 100
Answer: ≈ 400
Answer : ≈ 1000
Step 1  Underline the digit in the hundreds place.
Step 2  Look at the digit to its right in the tens place.
Step 3 If that digit is 5 or greater, add 1 to increase the digit in the
hundreds place. Then replace all digits to the right with zeros.
Step 4  If that digit is less than 5, add zero to the digit in the
hundreds place. Then replace all digits to the right with zeros.
hundreds tens
hundreds tens
9 5 0
4 0 4
0
+
4 0 0
+
1
10 0 0
Activity 6
Mixed numbers
1
5
7 =
Improper fractions
12
7
1. Identify the denominator - 7
2. 1 whole is made up of 7 parts
1
5
7
5 out of 7 parts
1
5
7 =
12
7
5
7
7
7
Factors and multiples – P4-6 (10 – 12 yrs)
Factor x Factor = Product
1 x
12
2 x
6
3 x 4 = 12 (multiple)
The factors of 12 are : 1,2,3,4,6 and 12
12
1
3 x
4
6
2
4
3
37
Heuristics Strategies
Applying Polya’s 4-step processes :
Understand
Plan
Solve
Check
38
Age problem – P3 (9 – 10 yrs)
Gopal is 15 years old.
In 3 years' time, he will be twice as old as Ravi.
How old is Ravi now?
Understand:
Now 
In 3 years’ time
Gopal
15
15 + 3 =18
Ravi
?
18 ÷ 2
Age model –Solution P3/4
Draw a model:
15
Gopal
18
Gopal
3
18
2 units  18
I unit  9
9 3=6
Answer: Ravi is 6 years old now.
Ravi
?
3
9
Check: 6 + 3 = 9
9 x 2 = 18

18 – 3 = 15
Practice Question 1
– P3 / P4
2 years ago, Lina was 11 years old and her uncle was 43 years
old. In how many years’ time will Lina’s uncle be thrice as old as
Lina.
Understand:
Lina
2 years’ ago 
11
Now 
11 + 2 = 13
Uncle
43
43 + 2 = 45
Age model –Solution P3/4
Lina
some
years
16
2 units
48
Uncle
Difference in ages 45 - 13 = 32
2 units  32
1 unit  32 ÷ 2 = 16
16 – 13 = 3
Check: In 3 years’ time:
Lina  13 + 3 = 16
Uncle  45 + 3  48

48 ÷ 16 = 3
Answer: In 3 years’ time Lina’s uncle will be thrice as old as Lina.
‘Short of’ concept (P5 / 6) 11- 12yrs -- Draw a model and work out
Stan wanted to buy 8 papayas but found that he was short of
$1.80.If he were to buy 5 papayas , he would have $6 left over.
How much money did Stan have?
$1.80
$6
?
3 units  $6 + $1.80 = $7.80
1 unit  $7.80 ÷ 3 = $2.60
5 units  5 x $2.60 = $13
$13 + $6 = $19.
Answer: Stan has $19.
Practice Question 2
Sue wanted to buy 7 pens but she was short of
$9.90. If she bought 4 pens, she would have
$ 7.20 left.
a) How much did a pen cost?
b) How much money did Sue have?
4 pens
$ 7.20
4 pens
$ 7.20
7 pens
a)
7 pens – 4 pens = 3 pens
3 pens  $7.20 + $ 9.90 = $ 17.10
1 pen  $17.10 ÷ 3 = $ 5.70
Answer: Cost of a pen is $5.70
b)
($ 5.70 x 4) + $ 7.20 = $30
Answer: Sue had $30
$ 9.90
COMPARISON CONCEPT - Draw a model and work out
Hari and Farith have $120. Peter and Farith have $230.
(6)
(1)
Peter has 6 times as much money as Farith.
How much money does Hari have?
$ 120
H
F
H
P
5 units
P
Working steps:
P
$ 230
5 units  $ 230 - $ 120 = $ 110
1 unit  $ 110 ÷ 5 = $ 22
$ 120 - $ 22 = $ 98
Answer: Hari has $ 98.
P
P
P
Practice Question 3
COMPARISON CONCEPT
Joel and Nick had
$75 altogether. Joel and Fay (1)
had $145 altogether.
(3)
If Fay had thrice as much money as Nick, how much money did Joel
have?
$ 75
J
N
J
F
2 units
F
F
$ 145
Working steps:
2 units  $ 145 - $ 75 = $ 70
1 unit  $ 70 ÷ 2 = $ 35
$ 75 - $ 35 = $ 40
Answer: Joel had $ 40.
BRANCHING
1
5
Ryan has a box of toy cars. 4 of the cars are red and 12 of
them are blue. The rest of the cars are green. There are 48
more blue cars than red cars. How many green cars does Ryan
have?
1
12
12
box
1
4 red
3
4
3
12
3
12
=
4
12
5 + 4
12
12
blue
blue
red
Working :
5
Green 1 - 12 -
48
green
2 units  48
4 units  48 x 2 = 96
Answer: Ryan has 96 green cars.
48
Practice Question 4
BRANCHING
In a class, 40% of the pupils like Soccer. The rest of the pupils prefer
Badminton and Tennis in the ratio 2 : 1. Given that 8 more pupils prefer
Badminton to Tennis, how many pupils are there in the class?
40 %
100 %
1
5
5
class
60 %
rest
2
5
3
5
Soccer
Badminton
8
Tennis
1 unit  8
5 units  5 x 8 = 40
Answer: There are 40 pupils in the class
49
MAKE A TABLE / GUESS and CHECK
Debbie bought a Smartphone for $175.
She paid the cashier in $10 and $5 notes.
She used 23 notes altogether.
How many of each type of note did she use?
Let’s try ‘half- half’. Start the
estimates in the middle.
Number of $10
notes
Number of $5 notes
Total value of the
notes
11
12
(11 x $10) + (12 x $5)
= $170
12
11
(12 x $10) + (11 x $5)
= $175
Use more
$10 notes and
fewer $5 notes
Answer: She used 12 notes of $10 and 11 notes of $5.
50
Practice Question 5
MAKE A TABLE / GUESS and CHECK
There are 24 coins in a box.
Some are 20-cent coins and the rest are 50-cent coins.
The total value of the coins is $9.30.
How many 50-cent coins are there in the box?
Let’s try ‘half- half’. Start the
estimates in the middle.
Number of
Number of
20-cent coins 50-cent coins
Total value of the coins
12
12
(12 x 20¢) + (12 x 50¢) = $8.40
11
13
(11 x 20¢) + (13 x 50¢) = $8.70
10
14
(10 x 20¢) + ( 14 x 50¢) = $9.00
9
15
( 9 x 20¢) + (15 x 50¢) = $9.30
Use more
50-cent coins and
fewer 20-cent coins
Answer: There are 15 50-cent coins in the box.
51
52
Evaluating the Progress of Dyslexic Children on a Small
Maths Group Intervention Programme in Singapore
Comparison between Pre-test and Post test scores- 13 students
DATE
Term 2 Wk 1
Term 3 Wk 10
Name of student
PRE TEST / 30
POST TEST /30
1. Agnes
4
15
2. Danial
11
17
3. Sue Lee
5
16
4. Therasa
5
16
5. Poh Wee Soon
15
16
6. Rennie
10
13
7. Kathy Loh
20
22
8. Zack Goh
23
28
9. Sunny Tan
14
withdrew
10. Kenny Toh
6
21
11. Kate
5
20
12. Thomas
4
16
13. Lian
6
18
P6 Advanced Maths
(WORD PROBLEMS)
Pilot Run  March 2015
to August 2015
Duration:6 months on
the programme
Entry Criteria: Primary 6
Participating Centres ,
• 3 EdTs
• 3 Learning Centre
•13 students
:
Hope you will try out the strategies
From: Mdm Aishah  [email protected] and
Ms Anaberta  [email protected]