Conceptualizing Mathematics Concept Development : Mathematicians and Maths Educators as Co-learners Fou-Lai Lin and Chuang-Yih Chen Department of Mathematics National Taiwan Normal University Plenary Speech on “International Conference on Science & Mathematics Learning” December 16-18, 2003 , Taiwan
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Fou-Lai Lin and Chuang-Yih Chen Department of Mathematics National Taiwan Normal University
Conceptualizing Mathematics Concept Development : Mathematicians and Maths Educators as Co-learners. Fou-Lai Lin and Chuang-Yih Chen Department of Mathematics National Taiwan Normal University. - PowerPoint PPT Presentation
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ConceptualizingMathematics Concept Development : Mathematicians and Maths Educators as Co-learners
Fou-Lai Lin and Chuang-Yih Chen
Department of Mathematics
National Taiwan Normal University
Plenary Speech on “International Conference on Science & Mathematics Learning” December 16-18, 2003 , Taiwan
An Integrated program
Concept Development:
Mathematics in Taiwan (CD-MIT)
(08,’00~07,’03)
The Context
The Topics Studied
Grade 7-9 P.IInfinity *Wang, W.C
Probability Wang, C. D.
Symmetry Tso, T. Y.
Algebraic Operations *Horng, Y.C.
3-D Horng, W. C.
Geometric Shape *Chen, C.Y.
Statistic *Chang, S.T.
Function *Chang, Y.C.
Measurement *Huang, W.D.
Variables & Linear Function Tsao, B. C.
Grade 7-9 P.ILinear Equation *Wu, B.K.Absolute Value and Inequality *Horng, B.F.
Mathematics Argument Lin, F. L.(Pattern and Shapes)Grade 1-6Fractions Leu, Y. C.Measurement Cheng, K.HTime Chung, C.Elemantary Geometry Chang, Y. C.
*8/17 P.I are Novice researchers in Math. Ed
The Topics Studied
Schedule
Aug,’00 Nov.’01 Jan~May ’02 Nov.’02 Aug.’03
Prog. Organizing interview pilot study national final
with P.I survey report
Stage 0 Stage 1 Stage 2
Data1 Data2
Stages of Understanding Children’s Mathematics Concept Development
Social Interactive ; eg. T1,2,3 (Theoritically),T6 Language & representation
V
S
R
L
H
Views (Stage 1) v.s. Performance(Stage2)
R L H V S
R
L 3 1
H 1 2 3
V 1 2 1
S
Views
Performances
Educating Mathematics Educators
Degree of agreement 4 3 2 1
Frequency 12 2 2 1
Non-maths educator 4 2 1 1
Q1:Four years ago, what were your research areas? Maths., statistics, computer, edu., maths. Edu.Q2:Would you continue to research mathematics education?
N=17 P.I (Dec. ‘03)
Data 1 Interview:
Would everyone explain your ideas about concept development? (05,11,’01)
Subjects:
-Project directors (17)
-Graduated students (4)
-Teachers (2)
Basically they are at the certain degree of Vygosky
Informal vs. Formal
Spontaneous vs. Scientific
From daily life vs. From school
T1,2,3
(Ph.D Students)
(Group
Discussion)
( Dual ; Social interaction ; )
Concerning the changes of the concept development :
(1)Qualitative presentation:—a better control of the complexity of concepts
(2)Strategy—more systematized when solving problems with concepts
(3)Quantitative presentation—the facility of getting the right answers
(4)The path of the concept development is not linear, is recursive.
T1,2,3
(Ph.D Students)
(Group
Discussion)
Presenting with a view of ‘process concept’
(1)Goal of study is to match the most appropriate time for learning the concept with students’ growth.
(2)For examples, statistic diagrams:
-Containing the activities of reading the diagrams, getting information to compose diagrams and explaining diagrams.
T4
( Process concept ; Readiness ; )
It’s a computerizing model.
-Cultivating everyday experiences to gather small units together in mind as a database, via mental organization, and then express the concepts in various ways
T5:
( ; CD , Inductive ; Computer model )
(1)The final destination of concept development is to know how to define the concept.
(2) The development of concept like the process of cooperating by a midwife and a sculptor. The former produce something and the latter get rid of the improper.
(3)Be able to distinguish examples and counterexamples under different circumstances, then can be assessed in expressing the concepts in different forms.
(4) And during the development towards the goal, we need language and symbols
T6:
( Def., S.I.R ; Functional ; Midwife, Sculptor)
T7:
(1) It’s navigating between certainty and uncertainty.(2)The sequence of growth like the process as Teacher demo.(certain) Student learning(certain) Teacher give counter example(uncertain) Student adjusted(certain) Teacher give improper example(uncertain) Student learn (certain)
(Exp,Non-exp. ; Navigating ; )
T8: (1)The intuitive understanding is from semantic interpretation of the words.
(2)Then accommodation by the conflicts and counterexamples towards the final concepts.
(3)For example, independent events is intuitively viewed as disjoint events…
( Misconception, Language ; Accommodation ; )
T9:(1)The process of growing is like a
concentric circles model.
(2)It is a genetic process.
(3)The acceptance of different degree of inaccuracy reveals the level of growth.
( Acceptance ; ; )
T10: The content of knowledge is formed by lots of
subconcepts. For example, about linear function, elementary school children have experiences of the covariance of two variables, but without the words; junior high school students begin to learn the term, but they might view y=f(x)=8 is and y=8 is not a linear function, after they adapt both examples as linear function; they then come to learn the quadratic function… Through the process of interiorization and condensation, then abstract to the generalized concept.
( Set of subconcepts; Text-book based ; )
T11: Assume one is deported to a barren island,
one starts to forget one used to know, concept is the last bit of knowledge that still keep in one’s mind, it is not easy to forget.
This metaphor could be used to build up the hierarchy among concepts.
( Remainder of cramming ; Remaining process ; Barred Island )
T12(1): Use the concept ‘Development’ as an example
of concept. Interpretation the concept ‘Development’ for
example Chicken is growing Duck is growing Dog is growingChicken and duck both are oviparous.Dog is viviparous.They are changing from samll, hairless to big and
with hair.
( Manifold ; Change, Inductive ; City development)
Changes are the essence of concept development-different in volume, different in forms and growing. Change can be revealed by concept map, transition, association, evolution and degeneration.
Development then can be used to say city development.
T12(2): ( Manifold ; Change, Inductive ; City development)
T13:(1) The inner characteristic of concepts have to be emphasized and bounded. The character is more like an inner language, not language for communication.
About concept, I still don’t have its definition.
It can be explained by the envelop model( envelop of curve, surface)
( Manifold, S.I.R ; Smoothing ; Envelop model)
T13:(2)
A concept is enveloped by Features of the concept Situations terminology Symbolic representation Examples
Under the suitable circumstances, the correct usage of the thinking can be reached. Symbols, special terms and plenty of examples are needed. Then it will reach the completion of concept
( Manifold, S.I.R ; Smoothing ; Envelop model)
T14: (1)From informal to formal
(2)The sequence of teaching material will affect students’ concept development.
(3)The development of concept could be interpreted with the aspect of one dimensional hierarchical levels, but sometimes also with the aspect of multi-dimensions model.
( Informal vs. Formal ; Informal to formal hierarchy ;
understanding level )
T15:(1)
(1)Development is a process towards the status that one is able to express ‘the concept’ in a specific, accurate and economic way.
(2)The process usually is carrying with certain misconceptions.
(3)For example, the concept similarity is linked with’ looks like’ ‘like photo copy’ ‘enlargement’ ‘look the same’ ‘proximate’ ‘akin’, etc.
One important feature of similarity is the directional position of the figures. The final stage is one can define the concept of similarity of two figures as the distance between any two corresponding points of the two figures are proportional.
Analogy is used prevalently in recognizing triangles and quadrilateral.
T15:(2)
T4’ Performance on C.D
Intuition Formal learning Hierarchy
Comment: L
T6’ Performance on C.D
Hierarchy (situations, features) Architecture of development
(attributes, features)
Comment :V
T7’ Performance on C.D
2-dim. Specification table Strategy H.L.U.R
Comment: H
T8’ Performance on C.D
Hierarchy (situation) Formal learning
Comment: L
T10’ Performance on C.D
Ambiguity of Tolerance Condensation ( representations, examples)
Comment: H
T11’ Performance on C.D
Attributes Proto-type Over-generalization Strategy of proportional reasoning Back to semantic
Sample:6 Mathematicians and 5 elementary school students
Stratified Systematic Sampling and the Sample Size Roughly Proportion to its Population on each Stratum.
Sampling
The Six Stratums
1.Taipei city2.Kaohsiung city3. Northern area4. Central and Southern areas5. Eastern area6. Small school ( N < 1400)Note : Total population is 956823 in 882 junior high
schools. (statistics in Education 2002)
The SampleP=956,823 S=45,633 S/P=4.77%
Stratum T K N C-S E S-school
Population (%)
102442
(10.7%)
64858
(6.8%)
306255
(32.0%)
350841
(36.6%)
29194
(3.1%)
103233
(10.8%)
Sample size
(Schools)
4764
2
23291
17838
6
14690
5
3950
2
2062
2
S/P (%) 4.65% 3.59% 5.82% 4.19% 13.53% 2.00%
A sample size of 1067 shall ensure that with 95% confidence the maximum error of the true proportion is within 3%.
The sample size of each grade population in CD-MIT study is about 1150.
Geographical Distribution of the Sample in CD-MIT
How Students Recognize a Triangle?
By :(1) Perception : look like, essentially, feel,…(2) Name : triangular board, sandwich, triangular cylinder,…(3) Appearance : whole, solid, global,…(4) Subfigure : the cylinder has two triangles on the top and
bottom,…(5) Completion : connect the line then it is, non-collinear three points decide a triangle,…(6)Component : vertex 、 angle 、 edge 、 face(7)Attribute : closed 、 line must attach 、 solid or dotted line 、 the width of line, the altitude of face,…(8)Signifier : just a representation, nothing to do with its widthness, trickiness
Test Booklet
Item Format of topics Topics
Multiple Choice All
Two-Tier multiple choice Item
Statistics, Measurement ,Space, Argumentation, Geometry Shape, Time
Hypothetical Understanding Routes of Parallelograms (Chen; C.Y. , 2002)
Facility
H.U.R of Defining a Rectangle(Lin & Yang ; 2001)
A reinvention Process of defining a rectangle(1) Analyzing properties of a rectangle(2) Conjecturing the cheapest way through establishing the
logical relationships of properties actively(3) Oversimplifying the necessary conditions for being a
rectangle(4) Conflicting(5) Re-conjecturing the cheapest way (6) Re-establishing the logical relationships of properties (7) Formulating an Aspect of Definitions
Strategies Used by Students
Diagonal strategy 、 Central line strategy
Tabulation
Strategies of reasoning of number pattern. Concrete Proportional Recursive Functional
Attributes of Infinity
1.無限大 Infinite in size
2.無限多 Infinite in amount
3.無窮遠 Infinite in distance
4.無限逼近 Infinite in proximity
5.無限重複 Infinite in recursion
HLT testing
Task-Comprehending
Generalization Symbolization
Errors ( function)
(1) No function concept.(2) 對函數的定義模糊 ( 自變數和應變數之間的關係倒置 ) 。(3) 認為“比”和函數無關。(4) 可寫出關係式者就是函數。(5) 表示函數的數學式中必須要有 x 和 y 。(6) 對常數函數的數學慣用表示法不熟悉。(7) 將常數函數視為數線。(8) 將函數概念與生活情境連結有困難( 無法將情境問題轉換為數學問題 ) 。
1. Hierarchical level; eg. Fraction2. Ambiguity of Tolerance; eg. Geometric shape; linear function 3. Recognizing the possible relations of knowing & doing; Argumentation4. Variety of L.T ; eg. Shape, Argumentation5. Effects