Summary ?Summary ?

DiscourseDiscourse

BASICSBASICS聯結學說

E.L. Thorndike(1874-1949)完形心理學德國 20’s

M. Wertheimer (1880-1943)W. Kohler (1887-1967)K. Koffka (1886-1941)

行為主義 (美國 )20’s – 30’s

J. Watson (1878-1958)「在一個行為主義者看來的心理學」 (

1913)I. Pavlov (1849-1936)

新行為主義30’s – 50’s

B.F. Skinner (1904-1990)

認知心理學20’s – 50’s

J.Piaget (1896-1980)J.S. Bruner (1915-)

D.P.Ausubel

折衷學派60’s – 80’s

R.M. Gagné (1916-)R. Glaser (1915-)

※Reinforcement & classroom/ social setting

※ S-R-K ※ Intrinsic & external motivation ※ Locus of control ※ Attribution

※“cause” and “effect”

The following day, when the little hoodlums came to jeer at him, he came to the door and said to them, “From today on any boy who calls me ‘Jew’ will get a dime from me.” Then he put his hand in his pocket and gave each boy a dime.Delighted with their booty, the boys came back the following day and began to shrill, “Jew! Jew!” The tailor came out smiling. He put his hand in his picket and gave each of the boys a nickel, saying, “A dime is too much-I can only afford a nickel today.” The boys went away satisfied because, after all, a nickel was money, too. However, when they returned the next day to hoot at him, the tailor gave them only a penny each.“Why do we get only a penny today?” they yelled.“That’s all I can afford.”“But two days ago you gave us a dime, and yesterday we got a nickel. It’s not fair, mister. ”“Take it or leave it, That’s all you’re going to get!”“Do you think we’re going to call you ‘Jew’ for one lousy penny?”“So don’t”And they didn’t.

Human Motivation (Bernard Weiner)

A fable reported by Ausubel (1948)

NCONCEPTION VIEW

PRIMITIVE BELIEF

cognition(100%)

affect(100%)

conscious

unconscious

CONCEPTION OF MATH: DEFINITIONCONCEPTION OF MATH: DEFINITION

CONSCIOUS

UNCONSCIOUS

affect.(100%)

view

expectation

stereotype

image

preconception

conception

primitive beliefs cognitive(100%)

Teachers’ belief on mathematics teaching: ghosts

unconscious or repressed

personal experience

practice(beliefs in action)

ghosts

process of adaptation to the context(type of school, location of the school, principal, parents, …)

conception of mathematics teaching(set of conscious and beliefs)

subject matterknowledge

educational theories

Expert judgment

COMMON BELIEFS (Lim, 1999)- Mathematics is a difficult subject- Mathematics is only for the clever ones- Mathematics as a male domain

Other Myths: Kogelman & Warren, 1978; Paulos, 1992)

Instrumental viewPlatonic viewProblem solving view (Ernest, 1991)

Vies of mathematics teaching- learner-focused- content-focused, emphasis on conceptual understanding- content-focused, emphasis on performance- classroom-focused(Thompson, 1992)

- Epistemological (how can math be acquired)- Ontological (what is math ?)

(Furinghetti, 1998)

DIMENSIONS

What ismathematics?

What islearning/teachingmathematics?

What are the roles of theteacher and pupils

What are criteria forthe judgingcorrectness?

What is problem solving?

LEVEL0

common uses ofarithmetic skillsin daily situations

mathematicalknowledgemeans rote,proceduralproficiency

memorization ofcollections of facts,rules, formulas andprocedures

teaching sequencesof topics and skillsspecified in a book

the teacher is ademonstrator of well-established procedures

pupils imitate

the teacher is anauthority forcorrectness

accurate answers asthe goal ofmathematicsinstruction

getting answers to"story problems"

helping pupils toidentify the rightprocedure ("rules ofthumb")

LEVEL1

rules continue togovern all workin mathematics

appreciation forunderstandingthe concepts andprinciples behindrules

an emergingawareness of the useof instructionalrepresentations

use of manipulativesin instruction

promote the viewthat “math is fun”

much as in Level 0 the teacher attends to the

"reasons behind therules"

pupils include someunderstand

authority forcorrectness still lieswith experts

view as a separatecurricular strand

taught separately problems unrelated to

mathematical topicsbeing studied

teaching "about"problem solving

LEVEL2

understandingmathematics as acomplex systemof differentinterconnectedconcepts,procedures andrepresentations

teach forunderstanding

understanding growsout of engagement inthe process of doingmathematics

the teacher steers pupils'thinking inmathematicallyproductive ways

he listens to pupils' ideas pupils express their ideas

the process of doingmathematics is thegoal of teaching

pupils themselvescheck their answersfor correctness

problem solving isused as a teachingmethod

teaching "via"problem solving

“What do you think mathematics is ?”FRAGMENTED COHESIVE

“How do you usually go about learning math ?”Those who hold a fragmented view incline to go about surface learning and those who hold a cohesive view incline to go about deep learning

Crawford et al (1994)

Five common views of the UK sample- Utilitarian- Symbolic- Problem solving- Enigmatic (mystic)- Absolutist- Dualistic view

Categories and subcategories of images of mathematics and frequency of corresponding responses

Images of mathematics (F=797)

Attitude Beliefs Process of learning Nature of mathematics Values/goals

(feelings) (about own mathematical ability) (teaching/learning) (in mathematics/education)(f=346 or 43.4%) (f=39 or 4.9%) (f=110 or 13.8%) (f=237 or 29.7%) (f=65 or 8.2%)

difficult(f=70)boring (f=58)anxiety(f=44)enjoyable(f=38)necessary/important(f=40)interesting (f=35)useful(f=21)exciting (f=18)rewarding (f=10)irrelevant (f=9)easy (f=3)

confusion(f=20)incomprehensible(f=12)difficult butpossible(f=4)inability(f=3)

logical thinking(f=22)mental work(f=20)problem solving(f=18)hard work(f=14)effortfulendeavour(f=11)repetitive process(f=8)getting difficult(f=4)skill(f=3)hierarchical(f=3)exploration(f=3)discovery(f=3)organising(f=1)

numbers and symbols(f=46)practical tools(f=25)formulae,equation/algebra(f=25)patterns/structure(f=20)complexity(f=19)a discipline/subject(f=26)games/puzzles(f=14)visualrepresentations/geometry(f=12)rules and procedures(f=19)theoretical(f=8)a model(f=7)a language(f=5)exactness andprecision(f=5)abstraction(f=5)proofs(f=1)

challenge(f=26)beauty ofmaths(f=20)objective(f=6)mystery(f=4)strange/foreign(f=3)orderly andtidiness(f=3)not creative/imaginative(f=2)dangerous butattractive(f=1)

Frequency (f) = number of responses corresponding to the categories or subcategories.

Categories and subcategories of images of mathematics and frequency of corresponding responses

Images of mathematics (F=727)

Attitude Beliefs Process of learning Nature of mathematics Values/goals

(feelings) (about own mathematical ability) (teaching/learning) (in mathematics/education)( f=272 or 37.4%) (f=62 or 8.5%) (f=227 or 31.2%) (f=93 or 12.8%) (f=73 or 10.1%)

Frequency (f) = number of responses corresponding to the categories or subcategories.

anxiety(f=87)boring (f=38)enjoyable(f=27)necessary/important(f=24)difficult(f=24)uncertain/conditional(f=17)exciting (f=15)rewarding (f=12)interesting (f=11)irrelevant (f=7)easy (f=6)useful(f=4)

difficult butpossible(f=21)confusion(f=13)difficult but rewarding(f=11)inability(f=10)incomprehensible(f=7)

problem solving(f=32)skill(f=30)exploration(f=20)hard work(f=19)slow and pointless(f=19)discovery(f=16)repetitive process(f=15)mental work(f=14)getting easier(f=14)hierarchical(f=13)effortfulendeavour(f=12)logical thinking(f=7)getting difficult(f=5)memorisation(f=4)investigation(f=4)organising(f=3)

games/puzzles(f=33)a language(f=12)patterns/structure(f=10)numbers andsymbols(f=6)rules andprocedures(f=7)practical tools(f=6)a discipline/subject(f=8)formulae,equation/algebra(f=3)a model(f=2)complexity(f=2)theoretical(f=2)exactness andprecision(f=1)visual representations/geometry(f=1)

mystery(f=24)strange/foreign(f=21)challenge(f=13)beauty of maths(f=5)objective(f=5)dangerous butattractive(f=3)orderly and tidiness(f=2)

A comparison between those who claimed to like mathematics and those whoclaimed to dislike mathematics

Those who claimed to likemathematics(n=36)

Those who claimed to dislikemathematics(n=26)

Reasons forliking/dislike

good at it (14) enjoy problem solving (11) sense of satisfaction (8) power of certainty/right

answer (9) logical nature (8) can use it (6) as a challenge ( 5)

beauty of maths (3)

not good at it (9) associate with strong

negative feelings (9) believe mathematics

is only for the cleverones (7)

find it difficult tounderstand (7)

blame teacher - nottaught properly (7)

Image ofmathematics as

related to nature ofmathematics :

certainty in answer (13) a practical tool (12) numbers, equations and

formulae (symbolism) (12) logical thinking (11)

certainty in answer (5) .a practical tool (12) .numbers, equations

and formulae(symbolism) (6)

logical thinking (4)

Image ofmathematics as

related to values inmathematics oreducation:

power of certainty andobjectivity(13)

creativity(4) element of mystery(3)

power of certainty andobjectivity (1)

creativity (2) element of mystery(2)

Note: number in parentheses indicates the frequency of occurrence (in the interview data).

A comparison of characteristics of mathematics teachers for those who claimed to likeand those who claimed to dislike mathematics

Characteristics of mathematics teacher ofthose who claimed to like mathematics

Characteristics of mathematics teacher ofthose who claimed to dislikemathematics

explain well give poor explanation make learning interesting and enjoyable have authoritarian or teacher-centred

teaching style are inspiring and encouraging are discouraging or humiliating have a lot of patience are lack of patience give individual attention or time to

pupils

give more time to the clever ones

get along well with pupils do not get along well with pupils

List of reasons for liking and disliking mathematics and the possible causal attribution for each reason

Reasons Possible causal attribution

For liking mathematics “good at it” or competence Positive beliefs about own ability

enjoy solving problems Problem solving view

power of certainty or right or wronganswer

Absolutist or dualistic view ofmathematics

like the logical nature ofmathematics

Problem solving view

like the challenge Problem solving view or “moreeffort”

“can use it” or like the practicalvalue of mathematics

Utilitarian view

satisfaction or rewarding Positive experience leading to positivefeelings

For disliking mathematics “not good at it” or incompetence Negative beliefs about ownability

elicit strong negative feelings suchas “hate”, “cold”

Negative experiences leading tonegative attitudes

belief that mathematics is only for“clever ones”

Negative beliefs about ownability

difficult to understand Negative beliefs about ownability

blame on teachers, believed thatthey were “not taught properly”

Others

A comparison between Malaysian (UK) teachers and students on their beliefs about thenature of mathematics

Malaysian teachers and students (UK teachers and students)

Statements Mathsteachers

n=36(n=29)

Otherteachers

(non-maths)n=64

(n=38)

Mathsstudentsn=174(n=47)

Otherstudents

(non-maths)n=133

(n=117)

TotalN= 407(N=231)

Maths is a collectionof rules andprocedures

22.2%(51.7%)

59.4%(68.4%)

9.8%(38.3%)

34.6%(79.5%)

26.8%(65.4%)

Mathematics is exactand certain

83.3%(34.5%)

85.9%(47.4%)

74.1%(40.4%)

82.0%(65.0%)

79.4%(53.7%)

A mathematicalproblem can besolved in differentways

94.4%(93.1%)

87.5%(92.1%)

90.2%(85.1%)

84.2%(84.6%)

88.2%(87.0%)

Mathematicalknowledge willchange rapidly in thenear future

58.3%(34.5%)

51.6%(15.8%)

64.4%(23.4%)

54.9%(28.2%)

58.7%(25.5%)

There are constantlynew discoveries inmaths

44.4%(75.9%)

50.0%(60.5%)

52.9%(61.7%)

45.1%(50.4%)

49.1%(58.0%)

Puzzles and gamesare proper maths

58.3%(93.1%)

54.7%(89.5%)

50.3%(72.3%)

42.1%(78.6%)

49.0%(79.7%)

Note: Figures in parentheses indicate results from the UK samples

A comparison between Malaysian (UK) teachers and students on the characteristics of a mathematician

Malaysian teacher and student (UK teachers and students)

Characteristics Maths teachersN=36 (n=29)

Other teachers(non-maths)n=64 (n=38)

Maths studentsn=174 (n=47)

Other students(non-maths)n=133 (n=117)

TotalN= 407(N=231)

Intelligent 69.4% (75.9%) 68.8% (65.8%) 78.2% (42.6%) 78.0% (53.8%) 75.9% (56.3%)

Serious 50.0% (13.8%) 62.5% (26.3%) 58.6% (27.7%) 59.1% (42.7%) 58.6% (33.3%)

Confident 83.3% (48.3%) 76.6% (26.3%) 83.3% (27.7%) 75.0% (17.9%) 79.6% (25.1%)

Male 58.3% (13.8%) 45.3% (28.9%) 61.5% (17.0%) 59.8% (27.4%) 58.1% (23.8%)

Wear glasses 44.4% (13.8%) 57.8% (18.4%) 60.9% (12.8%) 59.8% (38.5%) 58.6% (26.8%)

Odd/funny 13.9% (37.9%) 21.9% (23.7%) 43.1% (36.2%) 31.1% (23.9%) 33.3% (28.1%)

Strict 55.6% (20.7%) 48.4% (10.5%) 52.9%(10.6%) 57.6% (21.4%) 53.9%(17.3%)

Introvert 19.4%(17.2%) 26.6% (13.2%) 27.0% (8.5%) 32.6% (23.9%) 28.1% (18.2%)

Tidy/neat 33.3% (20.7%) 32.8% (7.9%) 43.7% (17.0%) 40.2% (23.9%) 39.9% (19.5%)

Bald 13.9% (6.9%) 26.6% (5.3%) 30.5% (10.6%) 31.1% (22.2%) 28.6% (15.2%)

Smart looking 8.3% (10.3%) 14.1% (0.0%) 8.0% (10.6%) 15.2% (12.0%) 11.3% (9.5%)

Female 5.6% (10.3%) 9.4% (5.3%) 20.1% (2.1%) 25.8% (4.3%) 19.0% (4.8%)

Note: Figures in parentheses indicate results from the UK samples

A comparison between Malaysian (UK) teachers and students on images of how mathematicians find new knowledge

Malaysian teacher and student groupings (UK sample)

Nine suggested ways Maths teachersn=36 (n=29)

Other teachers(non-maths) n=64(n=38)

Maths studentsn=174 (n=47)

Other students(non-maths)n=133 (n=117)

TotalN= 407(N=231)

Carrying out complicatedcalculations

60.0%(58.6%)

64.1%(63.2%)

75.9%(48.9%)

71.2%(68.4%)

71.1%(61.9%)

Guessing mathematicalrules

40.0%(72.4%)

29.7%(39.5%)

28.9%(23.4%)

28.8%(31.6%)

30.0%(36.8%)

Testing examples withformulae

91.4%(75.9%)

90.6%(86.8%)

94.8%(76.6%)

90.2%(83.8%)

92.3%(81.8%)

Trying to prove otherpeople’s formula is wrong

31.4%(93.1%)

17.2%(47.4%)

27.6%(70.2%)

22.7%(59.0%)

24.7%(63.6%)

Using their intuitions 62.9%(86.2%)

54.7%(73.7%)

48.9%(59.6%)

52.3%(61.5%)

52.1%(66.1%)

Solving real world problems 54.3%(79.3%)

53.1%(76.3%)

43.1%(68.1%)

50.0%(47.9%)

47.9%(60.5%

Discovering new formulae 77.1%(72.4%)

78.1%(71.1%)

83.3%(59.6%)

81.1%(75.2%)

81.2%(70.6%)

Creating new formula andrules

80.0%(79.3%)

73.6%(68.4%)

71.9%(55.3%)

76.5%(77.8%)

74.8%(71.5%)

Inventing new mathematicalideas

77.1%(75.9%)

81.3%(59.6%)

77.6%(60.5%)

77.3%(67.5%)

78.0%(65.4%)

Note: Figures in parentheses indicate results from the UK samples

Outcome space & critical aspects

“Can you tell me something you’ve learned?”- learning to do- learning to know- learning to understandPramling (1983)

CONCEPTION & PHENOMENOGRAPHY

Learning- increasing one’s knowledge- memorising and reproducing- applying what has been learned- understanding- seeing something in a different way- changing as a person

Marton, Dall’Alba, & Beaty (1993)

Understanding- existential understanding (that something is the case, what really is the case, the meaning of something)- hermeneutic understanding (what things look like for another, what an expression means)- phenomenological understanding (how something works, why something is the case, inherent regularity or structure)

Helmstad & Marton (1991)

Deep and surface learning

Quantitative Surface Multistructural and lower

Qualitative Deep Relational and higher

Pedagogy of variation- Discernment- Awareness- Simultaneity

Lived space

SOURCECollective Anschauuang Lim’s model

Images ofmaths Metaphors

Poetic, narrative,visual representations

Beliefs of-nature of maths,-in learning maths,-values in mathseducation

Attitudes to maths-liking, enjoyment,confidence, anxiety,perceived utility

CONSEQUENCES

※ Student ※ Intended belief implemented belief

attained belief achieved belief ※ Teacher ※ Self-monitoring

METHODOLOGY

“Mathematics is …”“Mathematics learning is …”

“※ Math is a game played according to certain simple rules with meaningless marks on paper” (D. Hilbert)

“※ Mathematics may be defined as the subject in which we never know what are talking about, nor whether what we are saying is true” (B. Russell)

“※ Mathematics has nothing do with logic” (K. Kodaira) “※ The moving power of mathematical invention is not r

easoning but imagination” (A. DeMorgan)

1. doing calculations mentally2. the idea that getting the right answer is always more important than the way of solving the problem3. doing computations with paper and pencil4. the idea that the pupil can sometimes make guesses and use trial and error5. the idea that everything ought to be expressed always as exactly as possible6. drawing figures7. the idea that one ought to get always the right answer very quickly8. strict discipline9. doing word problems10. the idea that there is always some procedure which one ought to exactly follow in order to get the result11. the idea that all pupils understand12. the idea that much will be learned by memorising rules13. the idea that pupils can put forward their own questions and problems for the class to consider14. the use of calculators15. the idea that the teacher helps as soon as possible when there are difficulties16. the idea that everything will always be reasoned exactly

Zimmermann Pehkonen questionnaireGood mathematics teaching includes:

17. the idea that different topics, such as calculation of percentages, geometry, algebra, will taught and learned separately; they have nothing to do with each other18. the idea that there will be as much repetition as possible19. the idea that studying mathematics has practical benefits20. the idea that only the mathematical talented pupils can solve most of the problem21. the idea that studying mathematics can only always be fun22. calculations of areas and volumes (e.g. the area of a rectangular and the volume of a cube)23. the idea that studying mathematics requires a lot of effort by pupils24. the idea that there are usually more than one way to solve problem25. the idea that games can be used to help pupils learn mathematics26. the idea that when solving problems, the teacher explains every stage exactly27. the idea that pupils are led to solve problems on their own without help from the teacher28. the constructing of different concrete objects (e.g. a box or a prism) and working with them29. the idea that there will be as much practice as possible30. the idea that all or as much as the pupil is capable of will be understood31. the idea that also sometimes pupils are working in small group32. the idea that the teacher always tells the pupils exactly what they ought to do.

1. For me, mathematics is the study of numbers2. Mathematics is a lot of rules and equations3. By using mathematics we can generate new knowledge4. Mathematics is imply an over-complication of addition and subtraction5. Mathematics is about calculation6. Mathematics is a set of logical systems which have been developed to explain the world and relationships in it.7. What mathematics is about finding answers through the use of numbers and formulae8. I think mathematics provides an insight into the complexities of our reality.9. Mathematics is figuring out problems involving numbers10.Mathematics is a theoretical framework describing reality with the aim of helping us understand the world

Conceptions of Mathematics Questionnaire

Fragmented: 1, 2, 4, 5, 7, 9, 12, 13, 16, 18Cohesive: 3, 6, 8, 10, 11, 14, 15, 17, 19

11. Mathematics is like a universal language which allows people to communicate and understand the universe12. The subject of mathematics deals with numbers, figures and formulae13. Mathematics is about playing around with numbers and working out numerical problems14. Mathematics uses logical structures to solve and explain real life problems15. What mathematics is about is formulae and applying them to everyday life and situations16. Mathematics is a subject where you manipulate numbers to solve problems17. Mathematics is logical system which helps explain the things around us18. Mathematics is the study of the number system and solving numerical problems19. Mathematics is models which have been devised over years to help explain, answer and investigate matters in the world

Conception of Mathematics Learning Scale 1. Mathematics is computation2. Mathematics problems given to students should be quickly solvable in a few steps3. Mathematics is the dynamic searching for order and pattern in the learner’s environment4. Mathematics is a beautiful, creative and useful human endeavour that is both a way of knowing and a way of thinking5. Right answers are much more important in mathematics than the ways in which you get them6. Mathematics knowledge is the result of the learner interpreting and organising the information gained from experiences7. Students are rational decision makers capable of determining for themselves what is right and wrong8. Mathematics learning is being able to get the right answers quickly9. Periods of uncertainty, conflict, confusion, surprise are a significant part of the mathematics learning process10. Young students are capable of much higher levels of mathematical thought than has been suggested traditionally11. Being able to memorise facts is critical in mathematics learning12. Mathematics learning is enhanced by activities which build upon and respect students’ experiences13. Mathematics learning is enhanced by challenge within a supportive environment14. Teachers should provide instructional activities which result in problematic situations for learners15. Teachers or the textbook – not the student - are the authorities for what is right or wrong16. The role of the mathematics teacher is to transmit mathematical knowledge and to verify that learners have received this knowledge17. Teachers should recognise that what seem like errors and confusions from an adult point of view are students’ expressions of their current understanding18. Teachers should negotiate social norms with the students in order to develop a co-operative learning environment in which students can construct their knowledge

Conceptions of professors Mathematicians’ way of knowing What do we do when we do mathImage of mathematician View of adults

OTHERS