The variant of CPA Approach

As Mathematics teachers in Singapore, we have always heard about CPA (Concrete – Pictorial-Abstract) approach in teaching of mathematics. With the increasing popularity and the easily availability of virtual manipulative online, there is a need to include the manipulative approach. In the last sharing in TSG 18, Dr Lee shares about the CVPA approach (Concrete Virtual Pictorial Abstract). In his study, he has worked with some schools. Virtual manipulative can help to improve both the efficiency and effectiveness of the classroom instruction.

For more information, you can download his article from eJMT website

Questioning and Mathematics

It is important for Mathematics teacher to ask questions in class. They should spent about 30% of the time answering questions. The questions should be able to provoke deeper understanding the lessons.  Teachers should move from knowledge dispenser to knowledge constructor. It is important to think about the possible questions to be asked so as to bring about deeper learning in the students.  When doing the analysis of the conversation in class, it is possible to consider the following coding schemes:

(1) Understanding Students’ Thinking
Telling Answer, Asked closed questions, request explanation, pressing for vigour
Best Practices
Constructs knowledge from conjectures, Facilitates reasoning

(2) Yes / No, Name/ state, Describe/ explain questions (Hiebert et al, 1999)




Emotions and problem solving

“Belief about mathematics and problem solving (orientation) may influence the choices made by the solver” (Schoenfeld. 1985). It is not doubt that emotions do influence positive solving process. Surprisingly, a review of meta-analysis of non-logitutinal surveys and longitudinal reviews and PISA data reveals mixed results. It might be influenced due to age and culture.

Emotions are essential part of the problem solver’s regulation. It has been suggested that basic emotions (anger, sadness, fear, respect surprise and disgust) rarely occurs in basic problem solving process Emotions has physiological, psychological and social function in self-regulation process. Physiological adaptions makes the body ready to face the challenge. In psychological self-regulation, emotions biases cognitive process and direct attention.Emotions also play the adapting function in social coordination of the group. Negative emotions are part of productive process. Students must be taught to cope with such emotions by joking. It is important that teacher should acknowledge such emotions in problem solving and plan task that is manageable for the students. There is a need to teach teach the students explicitly how to handle such emotions. For example, 5eachers could model the appropriate  enthusiasm and self-regulation behavior in problem solving process.

East Asian Verus Western Asian Mathematics Teaching

In East Asia, the mathematics education is more inclined towards product-oriented approach in which the students have to be grounded in their content knowledge. There is strong belief that the students must have “product” in which they can develop the process. In East Asia, teachers usually adopt didactic teaching in which students are passive receivers of information. East Asian countries are heavily influenced by CHC (Confucian Heritage Culture). The East Asians are more compliant, obedient and respect for superior.  On the other hand, the western countries see the value of Mathematics in its process (of analyzing the reality). Thus, the style of teaching is more process-oriented and individualized learning. The students are seen as active construction of their knowledge and mathematical communications and reasoning.
In Japan and Korea, parents spend more than USD$10 billion on tutions. In Japan, it is called Juku schools while in Korea, is is known as tutorial school. Usually, such tution schools advocate the memorizing of the mathematical facts.
So, how do I see myself as a Mathematics teacher in East Asia? I am the truly “East Asian product” type as I believe that students must have the Mathematical content knowledge (“product”) before the process developed. That does not mean I advocate root learning. As what Confusion has said “Seeking knowledge without thinking is labour lost; thinking without seeking knowledge is perilous” (?????????????), I strike a balance between focusing on the process and the product. As a ordinary teacher who wants nothing but the best for my extraordinary students (who just have the knack to make me both angry and happy at all times), I used varied teaching strategies based on the different learning theories (behavourism, cognition and construction) to make mathematics both meaningful and fun for the students. For example, I designed varied activies for my students from getting the students to practice their numeracy skills using web resources to using Scratch to explore the geometric properties.

TEDS-M Teacher Education and Study in Mathematics

TEDS-M 2008 is a comparative study of teacher education with a focus on the preparation of teachers of mathematics at the primary and lower secondary levels.It provides a cross-national data-based study about initial mathematics teacher education. TEDS-M is both relevant to educators and policy makers.In this study, the MCK (Mathematical Content Knowledge) and MPCK (Mathematical Pedagogical Content knowledge) of the would-be teachers are investigated. MPCK consists of (1) Mathematical curricular knowledge (Big idea), (2) Knowledge of planning for mathematics teaching and learning (Activities, possible misconceptions) and (3) Enacting mathematics for teaching and learning (Explaining mathematics concepts). For more info, please visit:


The Chaos Theory

Chaos theory is  popularized by Lorenz’s butterfly effect: “does the flap of a butterfly’s
wings in Brazil set off a tornado in Texas?” Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions (From wikipedia).

It is actually not new and this ideal can be found in some famous Mathematician quotes:

“We ought then to consider the present state of the universe as the effect of its previous
state and as the cause of that which is to follow. An intelligence that, at a given instant,
could comprehend all the forces by which nature is animated and the respective situation of the beings that make it up, if moreover it were vast enough to submit these data to analysis, would encompass in the same formula the movements of the greatest bodies of the universe and those of the lightest atoms. For such an intelligence nothing would be
uncertain, and the future, like the past, would be open to its eyes.” (Laplace, 1814)

“There is a maxim which is often quoted, that ‘The same causes will always produce the
same effects.’ To make this maxim intelligible we must define what we mean by the same
causes and the same effects, since it is manifest that no event ever happens more that once,so that the causes and effects cannot be the same in all respects. […]
There is another maxim which must not be confounded with that quoted at the beginning of this article, which asserts ‘That like causes produce like effects’. This is only true when
small variations in the initial circumstances produce only small variations in the final state
of the system. In a great many physical phenomena this condition is satisfied; but there are other cases in which a small initial variation may produce a great change in the final state of the system, as when the displacement of the ‘points’ causes a railway train to run into another instead of keeping its proper course.” (Maxewell, 1876)

India National Presentation

(1) Introducing history would make mathematics education more complete and help develop social cultural perspective.

(2) Traditional Pathshala (Hindi world of school) culture of learning numbers : Local orientation and a strong sense of functionality; recollective memory was the primary mode of learning.

(3) India has a strong Mathematical tradition. The major challenge in Mathematics education is the divisions in India society. Schools are stratified (poor schools for ppor students)

(4) Critical perspective : Problem posing pedagogues, folk tales, stories

(5) Face challenge in development of Maths teachers (viewed as “2nd rate” profession)

(6) A group of passionate mathematicians has come together to from National Initiative on Mathematics Education . I applaud their enthusiasm and their passion for mathematics education in their home country.

(7) Mathematics Assessment : Lots of questions and you have only 2 to 3 minute to solve each question.Extremely high stakes (Gang steals papers. Students commit suicide) . Emphasize on procedures and manipulative skills , heavy dependance on memorization
(8) Mathematics Training and Talent Search (MTTS) . Highly interactive. Think along with teacher. Do not allow the students to take notes

(9) More information on India Mathematics Education can be found at :

Constructionism : theory of learning or theory of design?

By Chronis Kynigos
(Director of Educational Technology Lab, National and Kapodistrian University of Athens School of Philosophy, Faculty of Philosophy, Pedagogy and Psychology Department of Pedagogy)

(1) More info about the projects in Educational Technology Lab, can be found in :

(2) Constructionism is first coined by Seymour Papert “”The word constructionism is a mnemonic for two aspects of the theory of science education underlying this project. From constructivist theories of psychology we take a view of learning as a reconstruction rather than as a transmission of knowledge. Then we extend the idea of manipulative materials to the idea that learning is most effective when part of an activity the learner experiences as constructing a meaningful product.”

(3) Constructionism can be addressed as an epistemology of learning associated with a theory of learning and design

(4) Constructionism is relevant since digital society is full of objects to be tinkered with and tools for collective mathematical activity and communication.

(5) Constructionist media : A digital media as an expressive medium for students. Designed for students and teacher

(6) Useful Links:

TSG Gp 18 : Session 2

(1) Use of Internet In a Mathematics Assessment System
– SMAPP (Singapore Mathematics Assessment and Pedagogy Project)
– Assessment for Learning
– More information can be found at SMAPP
– Positive response from teachers and students

(3) Written math exams with Internet Access : A teacher’s perspective on the impact on day to day Math Teaching

– Denmark conducted exams in 2008 to 2010 in which students can use Internet
– For Mathematics, the use of Internet exam is not relevant as students can solve problems easily without using Internet
– However, the pupils are now currently equipped with laptops with CAS (Computer Algebra System) . Teachers use the laptop for classroom teaching.

(4) Identifying Cognitive Appropriate Technological Tasks based on Students’ level of Geometric Thinking
– There are different geometry applets/animations. Teachers need to use them properly
– The teachers can use geometry-related technology tasks for(1) Developing Analysis Dynamic Activities , (2) Developing Abstraction Dynamic Activities (3) Developing Deduction Dynamic Activities. These tasks are based on the the geometry learning as proposed by van Hiele (

(5) How to use video in a class?
–    As an informative tool
–    trigger a formative process about a given mathematical content?
– Teacher must know when to use in class
– (Website in Brazil)
–    Video-least used resource (lack of training for teachers to use video)