2015-2016MAED 6161Teaching and Learning Mathematics in the Changing Curriculum

Ng Ka Wai 1155073128Chang Hei Man Kate 1155073105Eugene Sze 1155073134Tse Tung Ho 1155006557Tam Wai Man 1010091880

Agenda

Introduction

Summary of the article

Examples

Reflection based on the local context

Conclusion

Reference

Introduction2 Primary School Teachers

2 Secondary School Teachers

1 International School Teacher

●Article:

[C4]Watson, A. & Mason. J. (2006). Seeing an exercise as a single mathematical object: Using variation to structure sense-making, Mathematical Thinking and Learning, 8(2), 91-111.

Theory of Variation (Marton & Booth, 1997)Three important factors for learning

1.Discernment

2.Variation

3.Simultaneity

Theory of variation (Con’t)

❖Some aspects of the object of learning vary;

❖simultaneously keeping other aspects invariant or constant;

❖allowing the discernment of the appropriate critical aspects by the students.

Hypothetical learning trajectory (Simon 2005)Three main components:

Advice to Teachers (By Watson and Mason 2006)

1.Anticipate responses of students and hence have an influence on students’ experiences to the mathematical phenomena.

2.Directing the discernment towards the planned direction by using constrained variation of exercises.

Design of constrained variation of exercises

1.The number and the nature of the differences presented to learners should be constrained to let learners focus on important variables.

2.Both combination of several similar examples and further not-quite similar examples should be provided to let learners self-correct their errors and make further refinement on their conjectures.

Reflection based on the local context

Private through-trained School

Primary School

Secondary School

Reflection based on the local context

Private through-trained School

Discovery and Guided Independent Thinking

Parents: Academic Achievement → Tutorial Sessions

Practicing the “surface syntactic structure”

Result:

Calculation Skill & Method

Not Mathematical Concept

Reflection based on the local context

Primary School & Secondary School

Abundant Examinations

Territory-wide System Assessment (TSA)

Pre-Secondary One Hong Kong Attainment Test (Pre-S1)

The Hong Kong Diploma of Secondary Education Examination (HKDSE)

Teachers: Preparing students for Exam

Reflection based on the local context

Reality: Limited Lessons & wide syllabus

Solution: Remembering the pattern → “surface syntactic structure” AGAIN

Result: Cannot solve when the structure of question is changed

Solution: Providing learning experiences

Reflection based on the local context

“Variation is a tool to scaffold the construction of

different tasks that are conceptually related”

(Watson & Mason, 2006)

Example: Different among sphere, pyramid and

prism

Conclusion

Variation:

❖enhance students understanding and the underlying theoretical framework behind this notion

❖textbooks are unlikely to be able to have the resources to design such variation-based exercises for every mathematical concept given their need to work towards tight timeframes

Teacher's role

1.make sure learners' focus is towards the hypothesised learning outcome

2.guide learners to wider and more abstract mathematical concepts.

Some advice for teachers1. Revise and review the mathematical concept to be

learnt by the learners

2. Identify opportunities for learners to generalise

3. Plan and identify the variations required to highlight the learning outcome

4. Provide learning activities that provide controlled and relevant variation

5. Micro-modelling opportunities should be offered in sequences

References

Marton, F.; Booth, S. (1997). Learning and Awareness. New Jersey: Lawrence Eribaum Associates.

Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26, pp. 114-145

Watson, A.; Mason, J. (2006). Seeing an exercise as a single mathematical object: Using variation to structure sense-making. Mathematical Thinking and Learning, 8 (2), pp. 91-111