2de
bach
Mathematics education
SUMMARY
SERENA DE LEO
1
,Table of contents
TABLE OF CONTENTS ..................................................................................................................... 2
CHAPTER 1: VIEWS ON AND APPROACHES TO (RESEARCH IN) MATHEMATICS EDUCATION .............. 5
1. MATHEMATICS: VIEWS, DEFINITIONS ............................................................................................... 5
MATHEMATICS: DIFFERENT VIEWS .................................................................................................................. 5
MATHEMATICS: DEFINING CHARACTERISTICS .................................................................................................... 5
DIFFERENT VIEWS AND DEFINITIONS ............................................................................................................... 6
2. MATHEMATICS EDUCATION: VIEWS, APPROACHES............................................................................... 6
LEARNING AND TEACHING: THREE PERSPECTIVES ............................................................................................... 6
LEARNING AND TEACHING MATHEMATICS: THREE VIEWS ..................................................................................... 7
1. MECHANISTIC APPROACH ...................................................................................................................... 7
2. STRUCTURAL APPROACH........................................................................................................................ 7
3. REALISTIC APPROACH ............................................................................................................................ 8
MATHEMATICAL PROFICIENCY........................................................................................................................ 9
THREE APPROACHES: SOME GENERAL COMMENTS ............................................................................................. 9
3. MATHEMATICS EDUCATION AS A RESEARCH DISCIPLINE....................................................................... 10
MATHEMATICS EDUCATION: RESEARCH DISCIPLINE ........................................................................................... 10
RESEARCH METHODS: RICH DIVERSITY ........................................................................................................... 10
CHAPTER 2: EARLY MATHEMATICAL ABILITIES .............................................................................. 10
1. EARLY MATHEMATICAL ABILITIES .................................................................................................. 10
MULTI-DIMENSIONAL ................................................................................................................................ 11
CONCLUSION ........................................................................................................................................... 14
2. DEVELOPMENT OF EARLY MATHEMATICAL ABILITIES ........................................................................... 14
DEVELOPMENTAL TRAJECTORIES .................................................................................................................. 14
NUMBER AND PATTERNING: EMPIRICAL EVIDENCE ........................................................................................... 15
CONCLUSION ........................................................................................................................................... 16
3. EARLY MATHEMATICS EDUCATION ................................................................................................. 17
LEARNING TRAJECTORIES APPROACH ............................................................................................................. 17
MATHEMATICAL INTERACTION AND DISCUSSION ............................................................................................. 19
CHAPTER 3: EARLY MATHEMATICAL FOCUSSING TENDENCIES ...................................................... 19
1. EARLY MATHEMATICAL FOCUSSING TENDENCIES: THEORETICAL FRAME WORK .......................................... 19
BASIS CLAIMS ........................................................................................................................................... 19
MECHANISM OF SELF-INITIATED PRACTICE ..................................................................................................... 20
2. EARLY MATHEMATICAL FOCUSSING TENDENCIES: COMPONENTS............................................................ 20
DIFFERENT TENDENCIES ............................................................................................................................. 20
SFON..................................................................................................................................................... 20
SFONS ................................................................................................................................................... 23
SFOR ..................................................................................................................................................... 24
SFOP ..................................................................................................................................................... 26
2
,3. CONCLUSION AND DISCUSSION .................................................................................................... 27
CONCEPTUAL ISSUES.................................................................................................................................. 27
MEASUREMENT ISSUES .............................................................................................................................. 28
DEVELOPMENT ISSUES ............................................................................................................................... 28
EDUCATIONAL ISSUES................................................................................................................................. 29
CHAPTER 4: MENTAL AND WRITTEN ARITHMETIC ........................................................................ 29
MENTAL AND WRITTEN ARITHMETIC: DEFINITION, TYPES OF STRATEGIES ....................................................... 29
DEFINITION.............................................................................................................................................. 29
PROCEDURAL STRATEGIES: MENTAL .............................................................................................................. 29
PROCEDURAL STRATEGIES: WRITTEN ............................................................................................................. 30
MULTI-DIGIT ADDITION AND SUBTRACTION .................................................................................................... 30
ALSO SINGLE-DIGIT OPERATIONS (MENTAL) ................................................................................................... 31
STRATEGY COMPETENCE: THEORETICAL, METHODOLOGICAL AND INSTRUCTIONAL RATIONALE ............................. 31
THEORETICAL RATIONALE ............................................................................................................................ 31
METHODOLOGICAL RATIONALE .................................................................................................................... 32
INSTRUCTIONAL RATIONALE ........................................................................................................................ 33
ROUTINE EXPERTISE (SINGLE-DIGIT ADDITION) ................................................................................................ 33
ADAPTIVE EXPERTISE (SINGLE-DIGIT ADDITION) ............................................................................................... 33
ROUTINE EXPERTISE (SINGLE-DIGIT MULTIPLICATION) ....................................................................................... 33
ADAPTIVE EXPERTISE (SINGLE-DIGIT MULTIPLICATION) ...................................................................................... 34
MULTI-DIGIT ADDITION AND SUBTRACTION ........................................................................................... 35
WHAT DO WE KNOW ABOUT STRATEGY COMPETENCE (EMPIRICAL STUDIES) ......................................................... 35
MULTI-DIGIT ADDITION AND SUBTRACTION OP TO 100 .................................................................................... 37
MULTI-DIGIT SUBTRACTION OP TO 1000 ....................................................................................................... 38
MULTI-DIGIT ADDITION AND SUBTRACTION: CONCLUSION ................................................................................. 39
MULTI-DIGIT DIVISION ..................................................................................................................... 40
WHAT DO WE KNOW ABOUT STRATEGY COMPETENCE? .................................................................................... 40
MULTI-DIGIT DIVISION (490:5) ................................................................................................................... 41
STUDY DESIGN .......................................................................................................................................... 41
RESULTS: STRATEGY USE ............................................................................................................................. 41
RESULTS: STRATEGY USE PROFILES ................................................................................................................ 42
RESULTS: COMPENSATION, REMAINDER ......................................................................................................... 43
MULTI-DIGIT DIVISION: CONCLUSION ............................................................................................................ 43
TAKE HOME................................................................................................................................... 43
MULTI-DIGIT ARITHMETIC: TAKE HOME ......................................................................................................... 43
MULTI-DIGIT ARITHMETIC: TAKE HOME ......................................................................................................... 43
CHAPTER 5: MATHEMATICAL MODELLING ................................................................................... 44
MATHEMATICAL MODELLING: DEFINITION, PROCESS, EDUCATION ................................................................ 44
MATHEMATICAL MODELLING: DEFINITION ...................................................................................................... 44
MATHEMATICAL MODELLING – EDUCATION .................................................................................................... 45
WORD PROBLEMS: FUNCTIONS.................................................................................................................... 45
APPLICATION............................................................................................................................................ 46
CONCEPT FORMATION – STARTING PROBLEM .................................................................................................. 46
3
, MATHEMATICAL MODELLING: EMPIRICAL EVIDENCE ................................................................................. 47
WHAT DO WE KNOW ABOUT MATHEMATICAL MODELLING IN ELEMENTARY SCHOOL CHILDREN? ............................... 47
WEAKNESSES IN MATHEMATICAL CONTENT KNOWLEDGE .................................................................................. 47
NO USE OF VALUABLE HEURISTICS ................................................................................................................ 48
NO USE OF VALUABLE METACOGNITIVE SKILLS ................................................................................................. 48
INADEQUATE CONCEPTIONS AND BELIEFS ....................................................................................................... 49
WORD PROBLEMS HAVE NOTHING TO DO WITH REALITY ................................................................................... 49
“AMPUTATED” PROCESS OF MATHEMATICAL MODELLING .................................................................................. 50
MATHEMATICAL MODELLING: INSTRUCTION .......................................................................................... 51
AMPUTATED PROCESS OF MATHEMATICAL MODELLING… INSTRUCTION? .............................................................. 51
NATURE OF THE PROBLEMS ......................................................................................................................... 51
PRACTISE AND CULTURE IN CURRENT INSTRUCTION .......................................................................................... 52
CF. VERSCHAFFEL ET AL., 1997 ................................................................................................................... 52
AMPUTATED PROCESS OF MATHEMATICAL MODELLING … INSTRUCTION ............................................................... 53
STRATEGY FOR WORD PROBLEM SOLVING....................................................................................................... 53
USEFUL HEURISTICS ................................................................................................................................... 53
APPROPRIATE BELIEFS AND ATTITUDES........................................................................................................... 54
TASKS ..................................................................................................................................................... 54
INSTRUCTIONAL TECHNIQUES ...................................................................................................................... 55
RESEARCH DESIGN..................................................................................................................................... 55
WORD PROBLEM TEST ............................................................................................................................... 55
CONCLUSION: INSTRUCTION ........................................................................................................................ 57
TAKE HOME................................................................................................................................... 57
MATHEMATICAL MODELLING: RELEVANNT BUT COMPLEX .................................................................................. 57
INSTRUCTION? ......................................................................................................................................... 57
CHAPTER 6 RATIONAL NUMBERS ................................................................................................ 58
RATIONAL NUMBERS AND THE “NATURAL NUMBER BIAS” ......................................................................... 58
RATIONAL NUMBERS: DEFINITION................................................................................................................. 58
RATIONAL NUMBERS: DIFFICULTIES ............................................................................................................... 59
NATURAL NUMBER BIAS ............................................................................................................................. 60
CONCEPTUAL CHANGE THEORY ........................................................................................................... 61
HOW TO UNDERSTAND NATURAL NUMBER BIAS?............................................................................................. 61
INSTRUCTION ........................................................................................................................................... 63
DUAL PROCESS THEORY .................................................................................................................... 64
PROCESSES UNDERLYING DIFFICULTIES ........................................................................................................... 64
NATURAL NUMBER BIAS IN ADULTS ............................................................................................................... 64
INSTRUCTION ........................................................................................................................................... 65
4
bach
Mathematics education
SUMMARY
SERENA DE LEO
1
,Table of contents
TABLE OF CONTENTS ..................................................................................................................... 2
CHAPTER 1: VIEWS ON AND APPROACHES TO (RESEARCH IN) MATHEMATICS EDUCATION .............. 5
1. MATHEMATICS: VIEWS, DEFINITIONS ............................................................................................... 5
MATHEMATICS: DIFFERENT VIEWS .................................................................................................................. 5
MATHEMATICS: DEFINING CHARACTERISTICS .................................................................................................... 5
DIFFERENT VIEWS AND DEFINITIONS ............................................................................................................... 6
2. MATHEMATICS EDUCATION: VIEWS, APPROACHES............................................................................... 6
LEARNING AND TEACHING: THREE PERSPECTIVES ............................................................................................... 6
LEARNING AND TEACHING MATHEMATICS: THREE VIEWS ..................................................................................... 7
1. MECHANISTIC APPROACH ...................................................................................................................... 7
2. STRUCTURAL APPROACH........................................................................................................................ 7
3. REALISTIC APPROACH ............................................................................................................................ 8
MATHEMATICAL PROFICIENCY........................................................................................................................ 9
THREE APPROACHES: SOME GENERAL COMMENTS ............................................................................................. 9
3. MATHEMATICS EDUCATION AS A RESEARCH DISCIPLINE....................................................................... 10
MATHEMATICS EDUCATION: RESEARCH DISCIPLINE ........................................................................................... 10
RESEARCH METHODS: RICH DIVERSITY ........................................................................................................... 10
CHAPTER 2: EARLY MATHEMATICAL ABILITIES .............................................................................. 10
1. EARLY MATHEMATICAL ABILITIES .................................................................................................. 10
MULTI-DIMENSIONAL ................................................................................................................................ 11
CONCLUSION ........................................................................................................................................... 14
2. DEVELOPMENT OF EARLY MATHEMATICAL ABILITIES ........................................................................... 14
DEVELOPMENTAL TRAJECTORIES .................................................................................................................. 14
NUMBER AND PATTERNING: EMPIRICAL EVIDENCE ........................................................................................... 15
CONCLUSION ........................................................................................................................................... 16
3. EARLY MATHEMATICS EDUCATION ................................................................................................. 17
LEARNING TRAJECTORIES APPROACH ............................................................................................................. 17
MATHEMATICAL INTERACTION AND DISCUSSION ............................................................................................. 19
CHAPTER 3: EARLY MATHEMATICAL FOCUSSING TENDENCIES ...................................................... 19
1. EARLY MATHEMATICAL FOCUSSING TENDENCIES: THEORETICAL FRAME WORK .......................................... 19
BASIS CLAIMS ........................................................................................................................................... 19
MECHANISM OF SELF-INITIATED PRACTICE ..................................................................................................... 20
2. EARLY MATHEMATICAL FOCUSSING TENDENCIES: COMPONENTS............................................................ 20
DIFFERENT TENDENCIES ............................................................................................................................. 20
SFON..................................................................................................................................................... 20
SFONS ................................................................................................................................................... 23
SFOR ..................................................................................................................................................... 24
SFOP ..................................................................................................................................................... 26
2
,3. CONCLUSION AND DISCUSSION .................................................................................................... 27
CONCEPTUAL ISSUES.................................................................................................................................. 27
MEASUREMENT ISSUES .............................................................................................................................. 28
DEVELOPMENT ISSUES ............................................................................................................................... 28
EDUCATIONAL ISSUES................................................................................................................................. 29
CHAPTER 4: MENTAL AND WRITTEN ARITHMETIC ........................................................................ 29
MENTAL AND WRITTEN ARITHMETIC: DEFINITION, TYPES OF STRATEGIES ....................................................... 29
DEFINITION.............................................................................................................................................. 29
PROCEDURAL STRATEGIES: MENTAL .............................................................................................................. 29
PROCEDURAL STRATEGIES: WRITTEN ............................................................................................................. 30
MULTI-DIGIT ADDITION AND SUBTRACTION .................................................................................................... 30
ALSO SINGLE-DIGIT OPERATIONS (MENTAL) ................................................................................................... 31
STRATEGY COMPETENCE: THEORETICAL, METHODOLOGICAL AND INSTRUCTIONAL RATIONALE ............................. 31
THEORETICAL RATIONALE ............................................................................................................................ 31
METHODOLOGICAL RATIONALE .................................................................................................................... 32
INSTRUCTIONAL RATIONALE ........................................................................................................................ 33
ROUTINE EXPERTISE (SINGLE-DIGIT ADDITION) ................................................................................................ 33
ADAPTIVE EXPERTISE (SINGLE-DIGIT ADDITION) ............................................................................................... 33
ROUTINE EXPERTISE (SINGLE-DIGIT MULTIPLICATION) ....................................................................................... 33
ADAPTIVE EXPERTISE (SINGLE-DIGIT MULTIPLICATION) ...................................................................................... 34
MULTI-DIGIT ADDITION AND SUBTRACTION ........................................................................................... 35
WHAT DO WE KNOW ABOUT STRATEGY COMPETENCE (EMPIRICAL STUDIES) ......................................................... 35
MULTI-DIGIT ADDITION AND SUBTRACTION OP TO 100 .................................................................................... 37
MULTI-DIGIT SUBTRACTION OP TO 1000 ....................................................................................................... 38
MULTI-DIGIT ADDITION AND SUBTRACTION: CONCLUSION ................................................................................. 39
MULTI-DIGIT DIVISION ..................................................................................................................... 40
WHAT DO WE KNOW ABOUT STRATEGY COMPETENCE? .................................................................................... 40
MULTI-DIGIT DIVISION (490:5) ................................................................................................................... 41
STUDY DESIGN .......................................................................................................................................... 41
RESULTS: STRATEGY USE ............................................................................................................................. 41
RESULTS: STRATEGY USE PROFILES ................................................................................................................ 42
RESULTS: COMPENSATION, REMAINDER ......................................................................................................... 43
MULTI-DIGIT DIVISION: CONCLUSION ............................................................................................................ 43
TAKE HOME................................................................................................................................... 43
MULTI-DIGIT ARITHMETIC: TAKE HOME ......................................................................................................... 43
MULTI-DIGIT ARITHMETIC: TAKE HOME ......................................................................................................... 43
CHAPTER 5: MATHEMATICAL MODELLING ................................................................................... 44
MATHEMATICAL MODELLING: DEFINITION, PROCESS, EDUCATION ................................................................ 44
MATHEMATICAL MODELLING: DEFINITION ...................................................................................................... 44
MATHEMATICAL MODELLING – EDUCATION .................................................................................................... 45
WORD PROBLEMS: FUNCTIONS.................................................................................................................... 45
APPLICATION............................................................................................................................................ 46
CONCEPT FORMATION – STARTING PROBLEM .................................................................................................. 46
3
, MATHEMATICAL MODELLING: EMPIRICAL EVIDENCE ................................................................................. 47
WHAT DO WE KNOW ABOUT MATHEMATICAL MODELLING IN ELEMENTARY SCHOOL CHILDREN? ............................... 47
WEAKNESSES IN MATHEMATICAL CONTENT KNOWLEDGE .................................................................................. 47
NO USE OF VALUABLE HEURISTICS ................................................................................................................ 48
NO USE OF VALUABLE METACOGNITIVE SKILLS ................................................................................................. 48
INADEQUATE CONCEPTIONS AND BELIEFS ....................................................................................................... 49
WORD PROBLEMS HAVE NOTHING TO DO WITH REALITY ................................................................................... 49
“AMPUTATED” PROCESS OF MATHEMATICAL MODELLING .................................................................................. 50
MATHEMATICAL MODELLING: INSTRUCTION .......................................................................................... 51
AMPUTATED PROCESS OF MATHEMATICAL MODELLING… INSTRUCTION? .............................................................. 51
NATURE OF THE PROBLEMS ......................................................................................................................... 51
PRACTISE AND CULTURE IN CURRENT INSTRUCTION .......................................................................................... 52
CF. VERSCHAFFEL ET AL., 1997 ................................................................................................................... 52
AMPUTATED PROCESS OF MATHEMATICAL MODELLING … INSTRUCTION ............................................................... 53
STRATEGY FOR WORD PROBLEM SOLVING....................................................................................................... 53
USEFUL HEURISTICS ................................................................................................................................... 53
APPROPRIATE BELIEFS AND ATTITUDES........................................................................................................... 54
TASKS ..................................................................................................................................................... 54
INSTRUCTIONAL TECHNIQUES ...................................................................................................................... 55
RESEARCH DESIGN..................................................................................................................................... 55
WORD PROBLEM TEST ............................................................................................................................... 55
CONCLUSION: INSTRUCTION ........................................................................................................................ 57
TAKE HOME................................................................................................................................... 57
MATHEMATICAL MODELLING: RELEVANNT BUT COMPLEX .................................................................................. 57
INSTRUCTION? ......................................................................................................................................... 57
CHAPTER 6 RATIONAL NUMBERS ................................................................................................ 58
RATIONAL NUMBERS AND THE “NATURAL NUMBER BIAS” ......................................................................... 58
RATIONAL NUMBERS: DEFINITION................................................................................................................. 58
RATIONAL NUMBERS: DIFFICULTIES ............................................................................................................... 59
NATURAL NUMBER BIAS ............................................................................................................................. 60
CONCEPTUAL CHANGE THEORY ........................................................................................................... 61
HOW TO UNDERSTAND NATURAL NUMBER BIAS?............................................................................................. 61
INSTRUCTION ........................................................................................................................................... 63
DUAL PROCESS THEORY .................................................................................................................... 64
PROCESSES UNDERLYING DIFFICULTIES ........................................................................................................... 64
NATURAL NUMBER BIAS IN ADULTS ............................................................................................................... 64
INSTRUCTION ........................................................................................................................................... 65
4
