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

Stage Feature/Activity What is Concerned

0 Intuitive/Mathematics Formal Definition OrientedDichotomous Evaluation

1 Exploring/Developing Test Situations & InvarianceTextbook based

2 Local Model (topic-wise) / Ist Order Analysis

Learning CharacteristicsStudents Centered

3 Local Theory (topic-wise 、societal)/2nd Order Analysis

Mechanism of CDCausal Model

4 General Theory/ Theorizing

Theory of CL

Conceptualizing

Views about C.D at Stage 1 (Data 1) vs. Performances at Stage 2 (Data 2)

Qualitative analysisDriving ForceLearning ProcessLearning Product

Hypothetical Learning Routes for Maths. Eds on C.D.

H.L.R R1:stage 1, 2, 3, 1, 2, 3,...4, R2:stage 1, 2, 3, 4 R3:stage 1, 2, 1, 2, 3,...4,

Reflection Learning shows the insufficient

學然後知不足 Knowing the insufficient

知所不足

(Confucisous)

The End~

Thank You

Views about C.D. at Stage 1

Remainder ( degeneration) eg.T11

Linear eg.T4

Horizontal ; eg.T7 ,T12 Navigating Expanding Inductive

Vertical ; eg.T8 , T10, T13, T14 ,T15 Hierarchy Internerization

Sfard’s theory Accommodation

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.

( Specific, Accurate, Economic ; Accommodation cognitive strategy ; )

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

Comment: H , R

T12’ Performance on C.D

Strategy Situation Diagnostic assessment Representation

Comment: V

T13’ Performance on C.D

Attribute and Strategy Difficulty

Comment: H

T14’ Performance on C.D

Hierarchy Strategy Misconception

Comment: V

T15’ Performance on C.D

H.L.T Effect Hierarchy

Comment :V

Activity and Driving Force By Stage

Stage Activity Driving Force

0 Mathematics Motives of joining the program

1 Developing Test

Two-Dimensional SpecificationPilot Study

2 First Order Analysis

National SurveyDiscourse

3 Second Order Analysis

Explaining whyDesigning Learning Activities

4 Theorizing Practice in Teaching

Process and Product by Stage

Stage Process Product

0

1 ChaosClarifyingSynthesizing

Rich InformationItem FormatWhat has been Learned: Part 1

2 ConjecturingTestingStructuring

National Education PhenomenaWhat has been Learned: Part 2

3

4

Driving Force at Stage 0 Why the learning community organized?

Motives of joining the program ：Atmosphere of the Campus in 2000 ： Responsibility of university faculty.

Teaching, Research, Service Mission of the Institutions. Self-Regulation

Driving Force at Stage 1

Two Dimensional Specification.

Pilot Study

Working on a two Dimensional Specification.

Dimension X:

Features and Attributes of the Topic.

Dimension Y:

Context of Concept Development

and Cognitive Strategy.

Two Dimensional Specification

Y X Topic a Topic b

Features Attributes ............

Everyday Life ............

School life ............

Natural phenomena ............

Social-cultural ............

Historical ............

Virtual ............

Pilot Study

Interview

Developing coding

Conjecturing

Processes at Stage 1

1. Chaos exploring

2. Clarified with 2-dim specification.

3. Synthesizing as a booklet

Products at Stage 1

Ⅰ.Rich Information on the 2-dim

specification table.

Ⅱ.Test Booklet

Ⅲ.Hypothetical Local Understanding Route.

Driving Force at Stage 2

1. National Survey

2. Discourse within the community Leadership Narrative Reflexion

Process and Product at Stage 2

Conjecturing-Testing Strategy Attribute HLT testing Errors Facility (whole picture)

Process and Product at Stage 2

Structuring National Education Phenomena. Different Topics

Understanding level Development process Growth of different grades

Cross Topics

Everyday life:

小中和同學在回家的路上邊走邊聊天，大家都說自己做的事情大約要花1 小時的時間，你認為哪一個說法最合理？

1.one episode of soap opera 2.train-ride from Taipei to Taichung 3.run 100 meters 4.one aerobics lesson 理由 : □這件事結束的時間扣掉開始的時間，就是1 小時。 □每次做這件事的時間都是1 小時。 □因為做這件事感覺時間經過了很久，所以是1 小時。 □做這件事時，時鐘上的指針都指在數字1 的地方，所以是1 小時。 □其他

Which is one hour activity?（ Time; Zhong, J. , 2002 ）

School life

325個小朋友表演大會操，每一排有 13個小朋友，那麼可以排成幾排？

① 325×13 ② 325−13③ 13×25④ 325 13⑤ 13+13⑥ 13 325

Performing group exercise in school(Beginning Algebra; Horng, Y.C , 2002)

遠方有一隻蝴蝶正向這九朵高低不同的花飛過來，猜想它會停留在那一朵花 ( 或相同高度的花 ) 上面？

Ⓐ Ⓑ Ⓒ

Ⓓ

承上題 35你選擇該選項的原因是 ( 可複選 ) Ⓐ 最高或最矮Ⓑ同高度的花最多Ⓒ平均的高度Ⓓ中間的高度Ⓔ和蝴蝶最近的Ⓕ其他 ____________________

Natural phenomena

The butterfly shall stop at which flower?（ Statistics; Chang, S.T, 2002 ）

Historical(space; Horng, W.S ,2002)

考慮以下各立體的關係：將立方（即長方體ABCDEFGH）沿 ABHG的方向斜切可得到兩個壍( ㄑㄧㄢˋ ) 堵，將壍堵（ prism, qian-du, ABEFGH）沿 BEH的方向斜切會得到陽馬 (pyramid, yang-ma, BEFGH）及鼈( ㄅㄧㄝ ) 臑 ( ㄋㄠˋ ) （ tetrahedron, bie-nau, ABEH）各一。（ Jiu Zhang Suan Shu ）

F

E

A

B

H

G

立 （ 橢 ） 方E

A

H

G

CB

Which figure is a triangle ？

. .

.

a b c d

Mathematician 4 0 1 0

Elementary school student 4 2 3 2

(a) ( b) (c) (d)

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

Two-Tier Multiple Choice and explanation

Probability, Argumentation,

Multiple-Tier multiple choice Item

Statistics, Measure, Space, Geometry Shape, Argumentation, Algebraic Operations, Infinity

Completion Item Argumentation, Geometry Shape,

Matching Argumentation,

Virtual Linear Equation

Grade

7

85.7 83.6 79.5 71.7 43.9

48.4

~30

Grade

8

90.9 88.4 81.8 76.4 57.0

61.6

~35

Grade

9

92.2 91.4 88.2 84.9 66.164.5

~50

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 ) 。

常見之函數概念錯誤 (或困難 )

(9) 認為函數都應該是線性函數。(10) 數字之間必須要有“固定的關係”才是函數， 但物件之間只要有表格即是函數。(11) 只要有圖形，就是函數。(12) 不會直接從圖形判斷 , 仍使用代數計算。(13) 認為函數應有與其配合的公式，否則就不是函數。(14) 對鉛直線判別法與水平線判別法認識不清。(15) 對不熟悉的圖形不會判斷是否為函數。(16) 認為長條圖也應該是線性的。

常見之函數概念錯誤 (或困難 )

(17) 不了解平移的觀念 ( 合成函數的觀念 ) ， 且公式記憶不清，雖學過平移的方法 ( 公式 ) ， 但不會使用。(18) 對離散型函數及其圖形表示法不清楚。(19) 知道函數定義域範圍的意義 , 但答題時無法應用。(20) 對受測學生而言，將表格轉化成關係式， 比將圖形轉化為關係式來的困難。 （就線性函數而言）

View about Concept Development

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

proto-type effect ; eg. Geometric shape, Symmetry, Space, Elementary Geometry, Measurement

abuse analogy effect ; eg. Argumentation

H.L.T. in Reasoning of Number Patterns

Comprehending Generalizing Symbolizing

Checking 1 Checking 2