This document contains information about a mathematics course taught by Dr. Yeap Ban Har in Singapore, including:
- Contact information for Dr. Yeap Ban Har and background on his experience teaching mathematics.
- An introduction to the Singapore approach to teaching mathematics, which uses Bruner's constructivist theory of a spiral curriculum and the Concrete-Pictorial-Abstract approach.
- Details and case studies for several sessions on topics like early numeracy, addition/subtraction, multiplication/division, fractions, ratio, proportion, and algebra.
- Information on differentiated instruction, assessment, and the use of games, journaling and modeling in mathematics lessons.
Connect with Maths ~Maths leadership series- Session 3- the right knowledgeRenee Hoareau
Connect with Maths ~Maths leadership series- Session 3- the right knowledge presented by Rob Proffitt-White
The right knowledge – A clear valuing and understanding of mathematical content, the connections and a working knowledge of the proficiency strands underpins successful teaching
This workshop targets teachers and school leaders and aims to upskill their assessment literacy by:
• Creating cognitive activation tasks that promote critical thinking in all students
• Ensuring a consistent and shared responsibility for numeracy transfer
• Differentiating tasks through a focus on the proficiency strands
• Classifying the different problem solving types.
Connect with Maths ~ supporting the teaching of mathematics ONLINE
Engaging All Students community ~ http://connectwith.engaging.aamt.edu.au
Connect with Maths ~Maths leadership series- Session 3- the right knowledgeRenee Hoareau
Connect with Maths ~Maths leadership series- Session 3- the right knowledge presented by Rob Proffitt-White
The right knowledge – A clear valuing and understanding of mathematical content, the connections and a working knowledge of the proficiency strands underpins successful teaching
This workshop targets teachers and school leaders and aims to upskill their assessment literacy by:
• Creating cognitive activation tasks that promote critical thinking in all students
• Ensuring a consistent and shared responsibility for numeracy transfer
• Differentiating tasks through a focus on the proficiency strands
• Classifying the different problem solving types.
Connect with Maths ~ supporting the teaching of mathematics ONLINE
Engaging All Students community ~ http://connectwith.engaging.aamt.edu.au
Connect with Maths~ Teaching maths through problem solvingRenee Hoareau
Connect with Maths Early Years Learning in Mathematics community
Teaching Maths Through Problem Solving: Facilitating Student Reasoning
Presenter: Louise Hodgson
This session will focus on teacher actions, which promote problem solving and reasoning in early years classrooms. We will workshop some tasks and have opportunities for discussion.
Connect with Maths ~ supporting the teaching of maths ONLINE
Join a Connect with Maths community today http://www.aamt.edu.au/Communities
AAMT website: http://www.aamt.edu.au
These were the materials covered in last year's professional development. This year's session is a follow-up with revisiting of core ideas and extension of others.
Singapore Math Institute First AnnouncementJimmy Keng
This institute will be held in Singapore in November 2012. Ministry of Education (Singapore) teachers will register through their schools. International participants, please contact geraldynsng@sg.marshallcavendish.com for registration details.
Connect with Maths~ Maths Leadership Series-session 2-the right pedagogyRenee Hoareau
Connect with Maths ~ Maths Leadership Series
Session 2 - The right pedagogies
Presented by Rob Proffitt-White
Implementing curriculum intent requires a repertoire of pedagogies
Effective teaching of mathematics and numeracy capabilities require a range of pedagogical practices . This workshop is for teachers and school leaders who want to look at the processes involved in creating a common language around effective delivery of all mathematical proficiencies. It will focus heavily around
• Valuing teacher voice and building supportive and trusting culture for all
• Enacting the growth mindset in all classrooms
• Designing protocols and routines to support coaching/mentoring and reflecting.
Connect with Maths ~ supporting the teaching of Maths ONLINE
Connect with Maths Engaging All Students community ~ join at http://connectwith.engaging.aamt.edu.au
Connect with Maths Leadership Series: Session 1- the right teamRenee Hoareau
Building culture and capacity to enact the Australian Curriculum: Mathematics presented by Rob Proffitt-White for the Engaging All Students community. The first session will communicate the key factors and pre requisites common to schools successfully implementing elements of the initiative. This session has been designed for school leaders and Mathematics HODs wanting to prioritise numeracy and problem solving.
• Identification and remediation of common resistors
• Strategies for selecting a core key team and setting an agenda
• Valid and rigorous data professional learning communities
To view the accompanying webinar recording and resources please go to the Connect with Maths Engaging All Students community: http://connectwith.engaging.aamt.edu.au
Connect with Maths ~ supporting the teaching of mathematics ONLINE
Presentation Math Workshop#May 25th New Help our teachers understa...guest80c0981
This is presented by a Math teacher,in Army Burn Hall College For Girls ,Abbottabad.
The target group was the teachers of school section. There were certain activities also performed an demonstrated in order to introduce new teaching methodologies and to prepare our teachers to meet the need of the day.
Umber
Connect with Maths~ Teaching maths through problem solvingRenee Hoareau
Connect with Maths Early Years Learning in Mathematics community
Teaching Maths Through Problem Solving: Facilitating Student Reasoning
Presenter: Louise Hodgson
This session will focus on teacher actions, which promote problem solving and reasoning in early years classrooms. We will workshop some tasks and have opportunities for discussion.
Connect with Maths ~ supporting the teaching of maths ONLINE
Join a Connect with Maths community today http://www.aamt.edu.au/Communities
AAMT website: http://www.aamt.edu.au
These were the materials covered in last year's professional development. This year's session is a follow-up with revisiting of core ideas and extension of others.
Singapore Math Institute First AnnouncementJimmy Keng
This institute will be held in Singapore in November 2012. Ministry of Education (Singapore) teachers will register through their schools. International participants, please contact geraldynsng@sg.marshallcavendish.com for registration details.
Connect with Maths~ Maths Leadership Series-session 2-the right pedagogyRenee Hoareau
Connect with Maths ~ Maths Leadership Series
Session 2 - The right pedagogies
Presented by Rob Proffitt-White
Implementing curriculum intent requires a repertoire of pedagogies
Effective teaching of mathematics and numeracy capabilities require a range of pedagogical practices . This workshop is for teachers and school leaders who want to look at the processes involved in creating a common language around effective delivery of all mathematical proficiencies. It will focus heavily around
• Valuing teacher voice and building supportive and trusting culture for all
• Enacting the growth mindset in all classrooms
• Designing protocols and routines to support coaching/mentoring and reflecting.
Connect with Maths ~ supporting the teaching of Maths ONLINE
Connect with Maths Engaging All Students community ~ join at http://connectwith.engaging.aamt.edu.au
Connect with Maths Leadership Series: Session 1- the right teamRenee Hoareau
Building culture and capacity to enact the Australian Curriculum: Mathematics presented by Rob Proffitt-White for the Engaging All Students community. The first session will communicate the key factors and pre requisites common to schools successfully implementing elements of the initiative. This session has been designed for school leaders and Mathematics HODs wanting to prioritise numeracy and problem solving.
• Identification and remediation of common resistors
• Strategies for selecting a core key team and setting an agenda
• Valid and rigorous data professional learning communities
To view the accompanying webinar recording and resources please go to the Connect with Maths Engaging All Students community: http://connectwith.engaging.aamt.edu.au
Connect with Maths ~ supporting the teaching of mathematics ONLINE
Presentation Math Workshop#May 25th New Help our teachers understa...guest80c0981
This is presented by a Math teacher,in Army Burn Hall College For Girls ,Abbottabad.
The target group was the teachers of school section. There were certain activities also performed an demonstrated in order to introduce new teaching methodologies and to prepare our teachers to meet the need of the day.
Umber
Tom Hutchinson "Practical Intellectual Property"Jane Lambert
This is the second presentation in IP North West's seminar on IP law on 12 Oct 2011. This talk was presented by patent agent, Tom Hutchinson, principal of Hutchinson IP. In his talk, Tom considers "What is IP", "Why it is important?", "Types of IP", "Patent Attorneys" and "Tom's Top Tips". Tom is particularly well qualified to talk to FabLab because he researched additive manufacturing technology before he became a patent agent.
Becoming the Kind of Leader You Admire - The Endless Journey
The process of becoming a great leader is perpetual. It is an endless journey of self-discovery. There will be successes along the way, but no failure, only feedback from which you can choose to learn and grow. Sometimes the challenges you face will seem too enormous for you, but you will benefit more from the difficult parts of your travels than the easy roads. There will never be a convenient time for you to invest in developing yourself as a leader. You may be fortunate enough to have help – a mentor, coach or guide who provides valuable advice or support in your quest to become a great leader – but no one can give you what you deny yourself. And do not wait until you are given a position of leadership. Commit today to becoming the kind of leader you admire regardless of your role or title in your organization. This module will challenge you to set goals for your leadership development that extend far into the future, clearly imagining yourself as the leader you admire and then taking steps to become more like that leader every day. As you look back on your journey from the far future you will be amazed at your progress!
This is part of the professional development for the team that translate My Pals Are Here into Dutch and also people who are going to provide professionald evelopment for teachers using Singapore textbooks in the future.
Today we focused on the spiral approach and enrichment activities. The three-day programme covers the fundamentals of Singapore Math as well as it theoretical underpinnings and participants get to do a bit of model drawing.
An Intelligent Microworld as an Alternative Way to Learn Algebraic ThinkingCITE
4 March 2010 (Thursday) | 11:00 - 12:30 | http://citers2010.cite.hku.hk/abstract/4 | Prof. Richard NOSS, Professor of Mathematics Education & Co-director and Director of TLRP-TEL Research Programme, London Knowledge Lab
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2. 2 | P a g e
Contact Information
yeapbanhar@gmail.com
www.banhar.blogspot.com
about yeap ban har
Dr Yeap Ban Har spent ten years at Singapore's National Institute
of Education training pre-service and in-service teachers and
graduate students. Ban Har has authored dozens of textbooks,
math readers and assorted titles for teachers. He has been a
keynote speaker at international conferences, and is currently
the Principal of a professional development institute for
teachers based in Singapore. He is also Director of Curriculum
and Professional Development at Pathlight School, a primary
and secondary school in Singapore for students with autism. In
the last month, he was a keynote speaker at World Bank’s READ
Conference in St Petersburg, Russia where policy makers from
eight countries met to discuss classroom assessment. He was
also a visiting professor at Khon Kaen University, Thailand. He
was also in Brunei to work with the Ministry of Education Brunei
on a long-term project to provide comprehensive professional
development for all teachers in the country.
3. 3 | P a g e
introduction
The Singapore approach to teaching and learning mathematics was the result of
trying to find a way to help Singapore students who were mostly not performing
well in the 1970’s.
The CPA Approach as well as the Spiral Approach are fundamental to teaching
mathematics in Singapore schools. The national standards, called syllabus in
Singapore, is designed based on Bruner’s idea of spiral curriculum. Textbooks are
written based on and teachers are trained to use the CPA Approach, based on
Bruner’s ideas of representations.
“A curriculum as it develops should revisit this basic ideas repeatedly, building
upon them until the student has grasped the full formal apparatus that goes with
them”.
| Bruner 1960
“I was struck by the fact that successful efforts to teach highly structured bodies
of knowledge like mathematics, physical sciences, and even the field of history
often took the form of metaphoric spiral in which at some simple level a set of
ideas or operations were introduced in a rather intuitive way and, once mastered
in that spirit, were then revisited and reconstrued in a more formal or operational
way, then being connected with other knowledge, the mastery at this stage then
being carried one step higher to a new level of formal or operational rigour and
to a broader level of abstraction and comprehensiveness. The end stage of this
process was eventual mastery of the connexity and structure of a large body of
knowledge.”
| Bruner 1975
Bruner's constructivist theory suggests it is effective when faced with new material
to follow a progression from enactive to iconic to symbolic representation; this
holds true even for adult learners.
| Bruner 1966
4. 4 | P a g e
Open Lesson |
What do we want the students to learn?
Lesson Segment Observation / Question
How can we tell if students are
learning?
What help students who
struggle?
What are for students who
already know what we want
them to learn?
Summary
5. 5 | P a g e
Day 1 | Early Numeracy |Session 1
Rational Counting
Number Bonds
Lesson Sequence
Use of Literature
Lesson Sequence
Anchor Task
Guided Practice
(Independent Practice)
Case Study 1 |
Show 5 beans on a ten frame.
Do it in another way.
6. 6 | P a g e
Case Study 2 |
Show the teacher five pieces of square tiles.
Make a shape using five square tiles.
There are some rules that we have to follow.
7. 7 | P a g e
Whole Number Addition and Subtraction |Session 2
Materials
Strategies
Semantics
Variation
Semantics
Part-Whole
Change
Comparison
Case Study 2 |
Together, Jon and Kim have 32 coins.
Jon has 19 coins.
Find the number of coins that Kim has.
8. 8 | P a g e
Case Study 3 |
Lance has 10 coins more than Ming.
Together, they have 34 coins.
How many coins does Lance have?
9. 9 | P a g e
Case Study 4 |
At first, Lance had 10 coins more than Ming.
Then Ming gave Lance 6 coins.
Who had more coins in the end? How many more?
10. 10 | P a g e
Use of Activities for Math Learning |Session 4
Types of Lessons
To develop basic ideas, concepts and skills
To consolidate basic ideas, concepts and skills
To extend basic ideas, concepts and skills
Case Study 5 |
Use the digits 0 to 9 not more than once to make an addition equation.
11. 11 | P a g e
Holistic Assessment for Young Learners |Session 5
Assessment Benchmarks
Approaching Expectations
Meeting Expectations
Exceeding Expectations
Students should be able to perform rational counting.
Approaching Expectations The student is unable to count a plate of not more than ten
cookies.
Can the student perform one to one correspondence?
Can the student classify?
Can the student rote count?
Has the student grasp the principle of cardinality?
Meeting Expectations The student is able to count a plate of not more than ten
cookies.
Also able to read the correct numeral
Also able to read the correct number word
Also able to write the correct numeral
Also able to write the correct number word
Exceeding Expectations The student is able to count a plate of not more than ten
cookies. The student is also able to read and write the correct
numeral and number word.
12. 12 | P a g e
Day 2 | Differentiated Instruction |Session 1 and Session 2
Remediation
Enrichment
Four Critical Questions
Four Critical Questions (DuFour)
What do I want the students to learn?
How do I know they have learnt it?
What if they cannot learn it?
What if they already learnt it?
Differentiated Instruction (Tomlinson)
Content
Process
Product
Affect
13. 13 | P a g e
Case Study 1 | Basic Idea Lesson
Draw any triangle.
How are the three angles in a triangle related?
Answer the four critical questions.
DI for Struggling Learners DI for Advanced Learners
14. 14 | P a g e
Case Study 2 | Basic Idea Lesson
Anchor Task | Mom baked two cakes.
After giving half of a cake to our neigbors, we ate
5
4
of a cake.
Answer the four critical questions.
DI for Struggling Learners DI for Advanced Learners
15. 15 | P a g e
Case Study 3 | Practice Lesson
Draw triangles and find the area of each.
Answer the four critical questions.
DI for Struggling Learners DI for Advanced Learners
16. 16 | P a g e
Use of Games in Math Learning |Session 4
Types of Lessons
To develop basic ideas, concepts and skills
To consolidate basic ideas, concepts and skills
To extend basic ideas, concepts and skills
Case Study 4 |
Write expressions that include fractions and one of the four basic operations, one on each side
of the square such that the value of adjacent expressions are equal in value. Cut out the pieces,
mix them up and ask another group to arrange the pieces back again such that values of adjacent
expressions are equal.
17. 17 | P a g e
Journal Writing |Session 5
Case Study 5 | Problem-Solving Lesson
Let’s have a go at writing a math journal using this diagram as a stimulus.
18. 18 | P a g e
Day 3 | Whole Number Multiplication and Division |Session 1
Strategies
Semantics
Multiplication
Group
Array
Area
Rate
Combination
Division
Sharing
Grouping
Case Study 1 |
Compare these three lessons on division of whole numbers
Anchor Task A | Try putting 14 children into 3 equal groups.
19. 19 | P a g e
Anchor Task B | Try putting 41 children into groups of threes.
Anchor Task C | Try putting 41 liters of water into 3 containers. Is it possible to
make sure each container contains the same amount of water?
20. 20 | P a g e
Case Study 2 |
X =
Given three digits, make two numbers, a 1-digit number and a 2-digit number,
so that the product has the largest possible value.
21. 21 | P a g e
Factors and Multiples |Session 2
Jerome Bruner
Zoltan Dienes
Richard Skemp
Case Study 3 |
Use 12 square tiles to make a rectangle or square.
22. 22 | P a g e
Model Drawing |Session 4
Case Study 4 |
There are 40 children is Primary 3 Honesty.
19 of them are boys.
How many girls are there in Primary 3 Honesty?
Case Study 5 |
There are three times as many boys as there are girls in the soccer club.
There are 96 children in the soccer club.
Is this possible?
23. 23 | P a g e
Case Study 6 |
There is a group of people in a room.
A third of them are children.
A third of the children are boys.
There are 9 or 10 children in the room.
Which situation is possible?
For that situation, suggest questions that can be answered using the given
information.
24. 24 | P a g e
Holistic Assessment |Session 5
Newman’s Procedure
o Read
o Comprehend
o Know Strategies
o Transform
o Compute
o Interpret
Approaching Expectations Unable to solve word problems that is required at the current
grade level. However, the student is able to handle single-step
word problems.
Meeting Expectations Able to handle typical word problem required at the current
grade level.
Exceeding Expectations Able to handle unusual word problems and / or complex word
problems.
Case Study 7 |
At first, the ratio of the number of students in Basketball to the number of
students in Soccer was 3 : 1.
When 18 students moved from Basketball to Soccer, the there were equal number
of students in both sports.
Find the number of students in these two sports.
What if the ratio is 4 : 1?
25. 25 | P a g e
Case Study 8 |
In a group of 96 students, a third of the boys and a fifth of the girls do not have
pets at home while 70 students have pets at home.
How many boys have pets at home?
26. 26 | P a g e
Fractions, Fractions, Fractions!
an in-depth study of the teaching of fractions
Day 4 | Two Fundamentals |Session 1
Idea of ‘Nouns’
Idea of Equal Parts
Case Study 1 |
27. 27 | P a g e
Show 2 equal parts.
What do you mean by equal parts?
Show 4 equal parts.
In lesson study, we might discuss why use squares. Why not circles? Why not
rectangles?
28. 28 | P a g e
Case Study 2 |
5
2
5
1
5
1
5
3
Equivalent Fractions |Session 2
Case Study 3 |
8
?
4
1
?
9
4
3
29. 29 | P a g e
A cake is cut into 6 equal slices.
Aaron and Ben share four slices.
Case Study 4 |
What fraction of the rectangle is shaded?
30. 30 | P a g e
Basic Operations |Session 4
Case Study 5 |
Mary has a blue ribbon that is
3
2
1 m long. She has a red ribbon that is
4
3
1 m long.
Source | Primary Mathematics (Standards Edition) 6A
31. 31 | P a g e
Case Study 6 |
There are 12 cupcakes left over. Alex takes
4
3
of them home.
How many cupcakes does Alex take?
There are
2
1
3 pies left over. Ali takes
4
3
of them home.
How many pies does Ali take?
Source | Primary Mathematics (Standards Edition) 6A
32. 32 | P a g e
Case Study 7 |
The longest side of a triangle is
4
3
2 times as long as the shortest side. The
shortest side is
3
2
in. Find the length of the longest side.
Source | Primary Mathematics (Standards Edition) 6A
33. 33 | P a g e
Practice and Problem Solving |Session 5
Instructional Models
o Teaching through Problem Solving
o Teaching for Problem Solving
o Teaching of Problem Solving
Case Study 8 |
A total of 325 boys and girls attended a performance in the school hall.
5
4
of the
boys and
4
3
of the girls left the hall after the performance ended. There were 29
more boys than girls who remained in the hall. How many girls attended the
performace?
Source | Catholic High School (Primary) Primary 6 Examination
35. 35 | P a g e
Day 5 | Ratio and Proportion |Session 1
Problem-Solving Approach
Three-Part Lesson Format
A total of 325 boys and girls attended a performance in the school hall.
5
4
of
the boys and
4
3
of the girls left the hall after the performance ended. There
were 29 more boys than girls who remained in the hall. How many girls
attended the performace?
Source | Catholic High School (Primary) Primary 6 Examination
36. 36 | P a g e
Case Study 1 |
Find the area of a polygon with one dot inside it.
How does the area vary with the number of dots on the perimeter of the polygon?
37. 37 | P a g e
Find the area of a polygon with four dots on the perimeter.
How does the area vary with the number of dots inside the polygon?
38. 38 | P a g e
Advanced Bar Model Method |Session 2
Case Study 2 |
Four friends, Ravi, Johan, Meng and Emma, shared the cost of a present.
Ravi paid 50% of the total amount paid by the other three friends. Meng paid
60% of the total amount paid by Johan and Emma. Johan paid ½ of what Emma
paid. Ravi paid $24 more than Emma.
How much did the present cost?
Source | Primary Six Examination in a Singapore School
39. 39 | P a g e
Case Study 3 |
At a swimming meet, School A had 18 more swimmers than School B and 6 fewer
swimmers than School C. The ratio of the number of boys to the number of girls
from the three schools was 1 : 3.
The ratio of the number of boys to the number of girls in School A, School B and
School C were 1 : 3, 1 : 5 and 2 : 5, respectively.
Find the total number of swimmers from the three schools.
Source | Primary Six Examination in a Singapore School
40. 40 | P a g e
Teaching Algebra |Session 4
Ideas Development
o Variable
o Expression
Simplify
Expand
Factor
o Equation
Linear
Quadratic
Others
Case Study 4 |
Solve 7 – x = 4.
Source | Primary Mathematics (Standards Edition) 6A
Case Study 5 |
There are three times as many boys as there are girls in the soccer club.
There are 96 children in the soccer club.
Number of boys
Number of girls
41. 41 | P a g e
Case Study 6 |
(a) Find the value of 3s – 1 when s = 4.
(b) Solve 3s – 1 = 11.
Source | Primary Mathematics (Standards Edition) 6A
Case Study 7 |
Is it possible to factor 252 2
xx into linear factors?
42. 42 | P a g e
Is it possible for 252 2
xx = 0?
Case Study 8 |
Use algebra tiles to show 522
xx and 142
xx .
In each case try to rearrange the tiles to form a square.
43. 43 | P a g e
Holistic Assessment |Session 5
Skemp’s Types of Understanding
o Instrumental
o Relational
o Conventional
Approaching Expectations Student is unable to solve typical systems of linear equations.
The source of difficulty is likely to be
knowing the meaning of ‘solve’ (conventional)
knowing how to read algebraic expressions (conventional)
knowing how to do arithmetic manipulation (instrumental)
…
Meeting Expectations Student is able to solve typical systems of linear equations.
Exceeding Expectations Student is able to solve typical systems of linear equations.
There is also evidence that the student is able to extend his/her
understanding to less common situations.
Case Study 9 |
Solve 171
2
1
3
1
3
1
2
1
yxyx .