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Conceptual Models (CM) for Learning Algebra in Hong Kong Secondary School by
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# Conceptual Models (CM) for Learning Algebra in Hong Kong Secondary School

## Conceptual Learning

### facilitate learning by

Bodemer

interacting

experiencing

exploring

involving

decoding

integrating text and graphics with existing knowledge

### provided an learning environment for

developing multi-modal thinking and reasoning

re-capturing knowledge

re-discoverinfg knowledge

re-developing knowledge

having different knowledge

having different experience

having different interpretation

having mixed knowledge

having holistic knowledge

constructing new knowledge

leaners' reflection

### support higher order thinking

deep understanding

new knowledge

own interpretation and expression

## Design

Ainsworth (2006), Lowe 1999, Lewalter 2003, Mayor 2003, Patti Shank, adobe book: Multimedia learning Book: Cambridge handbookof Multimedia learning

### Represention

Mayer 2001, Paivio 1986, Rieber

Visual, Pictures, graphical representation, diagram, tables, lines, Words, equation, expression, number and symbol, theorems, notation, symbolic expressions, formula, figures

### interaction

churchill, mayer, learning and instruction

allow to change parameters, variable

control curve

control table

familiar look

### layout

moderate colour

simple design, no decoration

divide the related concepts into different sections

### help

explanation for control interface

pre training

### content

no redundant information

suitable amount of concept

### importan concept

signal principal (Mayer)

colour change (optional)

## Content (Algebra)

### procedural knowledge

baykul 1999, Ma 1999, Bali 1998

fact, values and numbers

symbol, identification between different parts, delta, symbols for root

principle, component in the theory, theory for solving quadratic equations, formula for delta, two different real numbers, two same real numbers, no real number, nautre of roots, sum of root, product of root

rules, arithmetic, indices

graphical representation, its nature

procedure, Multiplying and Dividing Monomials’, factorization

equations, Polynomials, different look of quadratic equation, (ax+b)^2=c, (ax+b)^2-c=0, ax^2+bx+c=0

functions, idea of input-processing-output to the meaning of dependent and independent variables

mathematical concepts

### conceptual Knowledge (mathematical concept)

baykul 1999, Ma 1999, Bali 1998baykul 1999, Ma 1999, Bali 1998

relationships between procedural knowledge, identify the most appropriate strategy to solve quadratic equation, further exploration on properties of quadatic graphs, identify the best formula to solve the mathematical problems, identify the best method to solve the real life problems, explore the effects of transformation on the functions from tabular, symbolic and graphical perspectives, visualize the effect of transformation on the graphs of functions when giving symbolic relations

conceptual in conceptual knowledge

## Its importance

### Learning algebra issue in Hong Kong (age 12 to 18)

existing supporting learning material, low opinion from teachers on the existing supporting learning material, low rate of usage of the existing supporting learning material

handling the variable items

traditional teaching, focus on distributive peoperty, focus on manipulative skills, cognitive obstacles and differences in levels, rich in content and skill-based, teacher-centred, directed students' thought processes, narrow the focus of the discussion, lead role, Transmission approach, examination-oriented

difficulties students faced, weaknesses in algebratic thinking, alarming in HK mathematical issue, misunderstanding: treating letters as as genealized numbers of variables, difficulty in producing relational and extended abstract responses, demonstrated a coherent argument and capacity of hypothetical thinking, student has little or no opportunity for students have the need of the development of conceputal understanding

text book, procedural paradigm, emphasis on learning computational skills, pupil -centred

## Use of CM for instruction

### Support for aquistion of meaningful learning

Book: Learning and Instruction (Mayer 2008)

Semantic Knowledge

Conceptual knowledge

Schematic knowledge

Prodcedural knowledge

Strategic knowledge

decalarative knowledge

### Design for Task for algebra learning in lesson or self learning(package)

With help of other types of learning object, approach 1, presentation object, instruction learning and teaching(before), information object, instrucion learning and teaching(before), simulation object, instruction learning and teaching (before), practice object, drill and practice (after), contextual representation, situation learning (after), approach 2, self learning with CM, practice object, drill and practice (after), contextual representation, situation learning (after)

## Problems of conceptual Learning

CM is one of the solutions for the existing problem

## Background (the construction of mathematical concepts)

Ferdinando Arzarello, Ornella Robutti and Luciana Bazzini

symbolic reconstructive approach, tranmission, listening, observing

can create obstacles to learning

### conceptual learning

construction of a rich experiential base

perceptuo-motor approach

## Conceptual Change

### ontological perspective

I prefer this perspective as it is more suitable for secondary school students

conceptual change learning model

### concept map

Hasemann & Mansfield 1995, Doerr & Browers 1999