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Conceptual Models (CM) for Learning Algebra in Hong Kong Secondary School by Mind Map: Conceptual Models (CM) for
Learning Algebra in Hong Kong
Secondary School
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Conceptual Models (CM) for Learning Algebra in Hong Kong Secondary School

Conceptual Learning

facilitate learning by







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


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


Mayer 2001, Paivio 1986, Rieber  

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


churchill, mayer, learning and instruction      

allow to change parameters, variable

control curve

control table


adobe multimedia learning

placed navigation element

familiar look


adobe mutlimedia learning  

moderate colour

simple design, no decoration

divide the related concepts into different sections

situative persepctives


explanation for control interface

pre training


no redundant information

suitable amount of concept

New node

importan concept

signal principal (Mayer)

alert (audio)

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

support the new learning enviroment

support meaningful learning

improve learning and teaching processes

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

the conceptual is not well understood

it is difficult to define Mathematical concepts

Background (the construction of mathematical concepts)

Ferdinando Arzarello, Ornella Robutti and Luciana Bazzini  

traditional teaching

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

mental model

concept map

Hasemann & Mansfield 1995, Doerr & Browers 1999  

Types of Learners

experience learners

novice learner

Number of Conceptual Models