01 – VIEWS ON AND APPROACHES TO (RESEARCH IN) MATHEMATICS EDUCATION............9
1. WHAT IS MATHEMATICS?.............................................................................................9
THE NATURE OF MATHEMATICS....................................................................................9
WHERE DOES MATHEMATICS COME FROM AND WHAT IS A MATHEMATICAL TRUTH?. 11
TEACHERS’ EPISTEMOLOGICAL BELIEFS ABOUT MATHEMATICS*................................11
2. WHY SHOULD IT BE TAUGHT?....................................................................................11
THE QUADRIVIUM....................................................................................................... 12
SOME (GOOD) REASONS TO LEARN MATHEMATICS....................................................12
FUNTER M. ZIEGLER (2016) IN ICME13 PLENARY LECTURE.........................................12
STEM.......................................................................................................................... 12
INTEREST GROUPS AND THEIR AIMS FOR MATHEMATICS TEACHING (ERNEST, 2014) 13
3. HOW SHOULD IT BE TAUGHT?...................................................................................13
GENERAL EDUCATIONAL PSYCHOLOGY PERSPECTIVE (SEE VERSCHAFFEL, GREER, DE
CORTE, 2007)*............................................................................................................ 13
DOMAIN-SPECIFIC MATHEMATICS EDUCATIONAL PERSPECTIVE (VERSCHAFFEL, 2014)
................................................................................................................................... 13
THE SKILLS OR DRILL APPROACH (BAROODY, 2013)...................................................14
BINARY NUMBERS UP TO 16....................................................................................15
CUISENAIRE RODS................................................................................................... 15
STRUCTURALISTIC APPROACH (“NEW MATH”)............................................................15
CUISENAIRE RODS................................................................................................... 15
LOGI BLOCKS........................................................................................................... 15
MAB MATERIAL........................................................................................................ 16
CRITICAL COMMENTS AND CONCERNS....................................................................16
REALISTIC APPROACH................................................................................................. 16
CHARACTERISTICS AND CRITICISMS OF THE TRADITIONAL APPROACH...................16
STARTING PROBLEM................................................................................................ 16
PROF. H. FREUDENTHAL.......................................................................................... 17
REALISTIC APPROACH (E.G. RME IN THE NETHERLANDS): PRINCIPLES....................17
REAL-WORLD CONTEXTS......................................................................................... 18
CELIA HOYLES.......................................................................................................... 18
RICHARD NOSS........................................................................................................ 18
BRIAN GREER.......................................................................................................... 18
SOME CRITICAL COMMENTS AND CONCERNS..........................................................18
THREE APPROACHES: SOME GENERAL COMMENTS....................................................19
4. RESEACH IN MATHEMATICS EDUCATION....................................................................19
MATHEMATICS EDUCATION AS A RESEARCH FIELD N ITS OWN..................................19
FISCHBEIN (1990, P. 10).......................................................................................... 19
1
, WITTMANN (1995, P. 363).......................................................................................19
SHORT INTRODUCTION OF TWO OTHER TEXTS..............................................................20
VAN DEN HEUVEL, M. & DRIJVERS, P. (2014)..............................................................20
KILPATRICH, J. SWAFFORD, J. & FINDELL, B. (2001)....................................................20
02 – EARLY NUMERICAL ABILITIES.....................................................................................21
EARLY NUMERICAL ABILITIES......................................................................................... 21
A CONCEPTUAL ANALYSIS OF THE COMPONENTS OF EARLY NUMERICAL ABILITIES:
ANDREWS & SAYERS (2015).......................................................................................... 21
NUMBER SENSE: A POORLY DEFINED CONSTRUCT.....................................................21
WHAT IS NUMBER SENSE? BERCH’S RESPONSE..........................................................21
WHAT IS NUMBER SENSE? THE ANSWER BY ANDREWS & SAYERS (2015)..................21
A. PREVERBAL NUMBER SENSE: STANISLAS DEHAENE – BRIAN BUTTERWORTH......21
THE “STARTER’S KIT” FOR MAGNITUDE UNDERSTANDING (BUTTERWORTH, 2016)
............................................................................................................................. 21
MENTAL NUMBER LINE......................................................................................... 22
LOWER-ORDER NUMBER SENSE: EXAMPLE 1........................................................22
LOWER-ORDER NUMBER SENSE: EXAMPLE 2........................................................22
LOWER-ORDER NUMBER SENSE: EXAMPLE 3........................................................23
FROM NON-SYMBOLIC TO SYMBOLIC....................................................................23
B. FOUNDATIONAL NUMBER SENSE (FONS).............................................................23
FONS COMPONENTS............................................................................................. 24
FONS: CONCLUSION............................................................................................. 32
C. APPLIED NUMBER SENSE.....................................................................................32
APPLIED NUMBER SENSE: 3 RELATED AREAS.......................................................32
APPLIED NUMBER SENSE: EXAMPLES...................................................................33
AN EMPERICAL ANALYSIS OF THE COMPONENTS OF EARLY NUMERICAL ABILITIES:
CIRINO (2011)................................................................................................................ 33
CIRINO (2011)............................................................................................................. 33
QUANTITATIVE TASKS USED BY CIRINO (2011)...........................................................34
03 – EARLY MATHEMATICAL FOCUSING TENDENCIES........................................................35
INTRODUCTION.............................................................................................................. 35
MATHEMATICAL FOCUSING TENDENCIES....................................................................35
MATHEMATICAL ENTITIES ARE ALL AROUND US.........................................................35
MATHEMATICAL FOCUSING TENDENCIES: BASIS CLAIMS............................................35
MECHANISM OF SELF-INITIATED PRACTICE.................................................................35
BUT ALSO…................................................................................................................ 36
DIFFERENT TENDENCIES............................................................................................. 36
SFON.............................................................................................................................. 36
HOW CAN YOU MEASURE A SPONTANEOUS MATHEMATICAL TENDENCY?..................37
2
, METHODOLOGICAL CRITERIA FOR SFON MEASURES (HANNULA & LEHTINEN, L&I,
2005)....................................................................................................................... 37
SFON MEASURES........................................................................................................ 37
SFON IMITATION TASK: ELSI BIRD TASK (HANNULA & LEHTINEN, L&I, 2005)...........37
SFON PICTURE TASK (BATCHELOR ET AL., L&I, 2010)..............................................37
WHAT DO WE KNOW ABOUT SFON?............................................................................38
SFONS............................................................................................................................ 39
SFONS PICTURE TASK (RATHÉ ET AL., ECRQ, 2019)....................................................39
RESEARCH QUESTIONS............................................................................................... 40
MEASURES.................................................................................................................. 40
RESULTS CROSS-SECTIONAL STUDY (RATHÉ ET AL., ECRQ, 2019)..............................40
RESULTS LONGITUDINAL STUDY RATHÉ.....................................................................41
OTHER SFONS RESEARCH........................................................................................... 41
SFOR (JAKE MCMULLEN)................................................................................................. 41
SFOR TELEPORTATION TASK.......................................................................................41
SFOR TELEPORTATION TASK: CODING.....................................................................42
SFOR BREAD TASK...................................................................................................... 42
MAIN RESULTS FROM THE STUDIES OF MCMULLEN....................................................42
SFOP.............................................................................................................................. 43
INTRODUCTION........................................................................................................... 43
SFOP MEASURES (MEVARECHT ET AL., SUBMITTED)...................................................43
SFOP MEASURE: TOWER TASK (WIJNS ET AL., 2019)...................................................43
SFOP MEASURE: SCORING.......................................................................................... 44
RESEARCH QUESTIONS............................................................................................... 44
MEASURES.................................................................................................................. 44
RESULTS..................................................................................................................... 45
CONCLUSION AND DISCUSSION.....................................................................................45
1. CONCEPTUAL ISSUES.............................................................................................. 45
2. MEASUREMENT ISSUES........................................................................................... 45
3. DEVELOPMENTAL ISSUES: SEQUENCES..................................................................46
3. DEVELOPMENTAL ISSUES: INFLUENCING FACTORS.................................................46
3. DEVELOPMENTAL FACTORS: IMPACT ON GENERAL MATH DEVELOPMENT..............46
4. EDUCATIONAL ISSUES............................................................................................ 46
04 – MENTAL AND WRITTEN ARITHMETIC..........................................................................48
SOLVE THESE PROBLEMS AS QUICKLY AS POSSIBLE...................................................48
GENERAL INTRODUCTION.............................................................................................. 48
AIMS AND SCOPE........................................................................................................ 48
ADAPTIVE VS. ROUTINE EXPERTISE................................................................................48
3
, TWO TYPES OF EXPERTISE (HATANO, 2003)...............................................................48
HIGHER-ORDER NUMBER SENSE (BERCH, 2005, P. 334).............................................48
STRATEGY DEVELOPMENT IN THE DOMAIN OF SINGLE DIGIT ARITHMETIC....................49
ADDITION AND SUBTRACTION < 10 (EXAMPLE 3 + 4 = ?)..........................................49
ADDITION AND SUBTRACTION > 10 (EXAMPLE 8 + 9 = ?)..........................................49
ADDITION AND SUBTRACTION > 10 (EXAMPLE 17 – 8 = ?).........................................49
ADDITION AND SUBTRACTION: SOME MAJOR FINDINGS AND ISSUES..........................49
OVERLAPPING WAVES THEORY (SIEGLER)...............................................................49
VIGNETTE................................................................................................................ 50
RECIPROCAL RELATION BETWEEN STRATEGY USE AND MATHEMATICAL PRINCIPLES
................................................................................................................................ 50
FINGERS.................................................................................................................. 50
ARITHMETIC RACK................................................................................................... 50
MAB MATERIAL........................................................................................................ 51
NUMBER LINE.......................................................................................................... 51
STRIVING FOR ROUTINE EXPERTISE.........................................................................51
STRIVING FOR ADAPTIVE EXPERTISE.......................................................................52
MULTIPLICATION AND DIVISION: STRATEGY VARIETY AND CHANGE (E.G. 8 X 7 = ?). .52
TEACHING MULTIPLICATION: THE MECHANISTIC APPROACH....................................52
TEACHING MULTIPLICATION: THE REALISTIC APPROACH.........................................52
STRATEGY DEVELOPMENT IN THE DOMAIN OF MULTI-DIGIT ARITHMETIC......................53
TRADITIONAL VS. REFORM-BASED VIEW ON MULTIDIGIT ARITHMETIC........................53
LONG DIVISION ALGORITHM....................................................................................53
STANDARD WRITTEN ALGORITHMS.........................................................................54
3 ALGORITHMS FOR WRITTEN SUBTRACTION..........................................................54
MENTAL COMPUTATION........................................................................................... 54
MENTAL COMPUTATION STRATEGIES.......................................................................54
TEACHING FOR ROUTINE EXPERTISE IN MULTI-DIGIT MENTAL ARITHMETIC.............55
MAB MATERIAL..................................................................................................... 55
SUANPAN.............................................................................................................. 55
TEACHING FOR ROUTINE EXPERTISE IN MULTI-DIGIT MENTAL ARITHMETIC.............56
TEACHING FOR ADAPTIVE EXPERTISE IN MULTI-DIGIT MENTAL ARITHMETIC (MATHE
2000)....................................................................................................................... 56
VIGNETTE................................................................................................................ 56
MODEL OF STRATEGY CHANGE (LEMAIRE & SIEGLER, 1995)......................................57
OVERLAPPING WAVES THEORY (SIEGLER)..................................................................57
CHOICE/NO-CHOICE METHOD (LEMAIRE & SIEGLER, 1995).........................................57
CHOICE METHOD..................................................................................................... 57
PROBLEMATIC QUALITY OF STRATEGY EFFICIENCY DATA WITH CHOICE METHOD 58
4
, TASK-BASED MEASURE OF STRATEGY FLEXIBILITY...............................................58
CHOICE/NO-CHOICE METHOD (SIEGLER & LEMAIRE, 1997°........................................58
SUBJECT-BASED MEASURE OF FLEXIBILITY..............................................................59
PREVIOUS STUDIES WITH CHILDREN.......................................................................59
FLEMISH PUPILS’ EDUCATIONAL BACKGROUND.......................................................59
CONDITIONS............................................................................................................ 59
VISUAL REPRESENTATION USED IN THE NO-CHOICE DS CONDITION.......................59
VISUAL REPRESENTATION USED IN THE NO-CHOICE SBA CONDITION.....................60
VISUAL REPRESENTATION USED IN THE CHOICE CONDITION..................................60
ANALYSIS................................................................................................................. 60
RESULTS: REPERTOIRE & FREQUENCY.....................................................................60
RESULTS: EFFICIENCY.............................................................................................. 61
RESULTS: FLEXIBILITY.............................................................................................. 61
CONCLUSION........................................................................................................... 61
SBA IN MATHE 2000 TEXTBOOK..............................................................................62
05 – WORD PROBLEMS & MATHEMATICAL MODELLING.....................................................63
INTRODUCTION.............................................................................................................. 63
THE “BUSES” ITEM (NAEP, 1983)................................................................................63
THE “CAPTAIN”S PROBLEM” (IREM DE GRENOBLE, 1980)...........................................63
% COMPUTATION-BASED RESPONSES TO ‘CAPTAIN’-LIKE PROBLEMS IN RADATZ’
(1983) STUDY............................................................................................................. 63
BASIC NOTIONS............................................................................................................. 64
WORD PROBLEMS....................................................................................................... 64
EXAMPLES............................................................................................................... 64
NO WORD PROBLEMS (!)......................................................................................... 64
FUNTIONS................................................................................................................... 64
APPLICATION........................................................................................................... 64
SHARPENING THINKING........................................................................................... 64
CONCEPT FORMATION............................................................................................. 64
STARTING PROBLEM................................................................................................ 64
STEP 1: DISTRIBUTING OBJECTS...........................................................................65
STEP 2: WORKING WITH BIGGER PORTIONS AND INTRODUCING THE DIVISION
SCHEME................................................................................................................ 65
STEP 3: FURTHER ABBREVIATION OF THE DIVISION ACT AND THE DIVISION
SCHEME................................................................................................................ 65
WORD PROBLEM SOLVING.......................................................................................... 66
SOME RESEARCH RESULTS............................................................................................ 66
CONTENT SPECIFIC KNOWLEDGE (I)...........................................................................66
CONTENT SPECIFIC KNOWLEDGE (II)..........................................................................67
5
, NO USE OF HEURISTICS.............................................................................................. 67
HEURISTIC: MAKE A DRAWING....................................................................................67
METACOGNITIVE SKILLS.............................................................................................. 67
KEY WORD STRATEGY................................................................................................. 68
“AMPUTATED” PROCESS OF MATHEMATICAL MODELLING..........................................68
INADEQUATE CONCEPTIONS AND BELIEFS.................................................................68
EXAMPLES OF P-ITEMS (VERSCHAFFEL ET AL., 1994).................................................69
NON-REALISTIC (NR) AND REALISTIC REACTIONS (RR) TO P-ITEMS............................69
% OF RR’S ON P-ITEMS (VERSCHAFFEL ET AL., 1994).................................................69
% OF RR’S IN OTHER STUDIES....................................................................................69
ALERTING INSTRUCTION IN YOSHIDA ET AL.’S (1997) STUDY.....................................69
CAUSES?........................................................................................................................ 70
PROBLEM FROM A PORTUGUESE TEXTBOOK (6TH GRADE).........................................70
RULES OF THE GAME OF WORD PROBLEMS................................................................70
CAUSES...................................................................................................................... 70
TEST 2 (VERSCHAFFEL ET AL., 1997)..........................................................................70
TEST 2........................................................................................................................ 70
RESULTS FOR TEST 1.................................................................................................. 71
RESULTS FOR TEST 2.................................................................................................. 71
4 CAUSES.................................................................................................................... 71
DEWOLF ET AL. 2011.................................................................................................. 71
REMEDIES...................................................................................................................... 72
INTERVENTION STUDY RESEARCH IN LEUVEN.............................................................72
1. GOALS................................................................................................................. 73
PROBLEM SOLVING PLAN......................................................................................73
HEURISTICS.......................................................................................................... 73
2. MAJOR PILLARS.................................................................................................... 74
TASK ABOUT STEP 1............................................................................................. 74
TASK ABOUT STEP 2............................................................................................. 75
TASK ABOUT STEP 3............................................................................................. 75
TASK ABOUT STEP 4............................................................................................. 75
TASK ABOUT STEP 5............................................................................................. 76
PROJECT LESSON.................................................................................................. 76
RESEARCH DESIGN............................................................................................... 76
RESULTS............................................................................................................... 76
RESULTS (2)......................................................................................................... 77
TO CONCLUDE............................................................................................................... 77
OVERALL CONCLUSION OF DESIGN EXPERIMENTS......................................................77
6
, IMPORTANT BARRIERS................................................................................................ 77
MATHEMATICAL MODELLING.......................................................................................78
SOLVING PROCESS: AN EXTENDED MODEL................................................................78
06 – RATIONAL NUMBERS................................................................................................. 79
DEFINTIONS, CONCEPTS................................................................................................ 79
RATIONAL NUMBERS................................................................................................... 79
NUMBERS................................................................................................................... 79
“LACK OF DISCLOSURE”............................................................................................. 79
NUMBERS: CHANGES IN MEANING..............................................................................80
RATIONAL NUMBERS IN THE CURRICULUM....................................................................80
CURRICULUM IN PRIMARY SCHOOL (FLEMISH EXAMPLE)............................................80
CURRICULUM IN SECONDARY SCHOOL (FLEMISH EXAMPLE).......................................80
THE TRANSITION FROM NATURAL TO RATIONAL NUMBERS...........................................82
EXAMPLE 1C: HOW TO DIVIDE A FRACTION BY A FRACTION....................................82
SOME EXERCISES........................................................................................................ 83
A THEORY ABOUT KNOWLEDGE AND LEARNING (CONCEPTUAL CHANGE THEORY).......83
CONTINUOUS VS. DISCONTINUOUS............................................................................84
NAIVE CONCEPTIONS.................................................................................................. 84
CONCEPTUAL CHANGE............................................................................................... 84
EXAMPLARY ELABORATION IN RESEARCH ABOUT RATIONAL NUMBERS........................85
CONCEPTUAL CHANGE IN MATH.................................................................................85
EARLY NUMBER KNOWLEDGE.....................................................................................85
INITIAL INSTRUCTION.................................................................................................. 85
CONSEQUENCE........................................................................................................... 85
AND AS A CONSEQUENCE…........................................................................................ 86
INTERFERENCE OF NATURAL NUMBER KNOWLEDGE IN TASKS ABOUT RATIONAL
NUMBERS................................................................................................................... 86
“BUGGY ALGORITHMS”............................................................................................ 86
FROM A DISCRETE TO A DENSE NUMBER LINE............................................................86
RESEARCH INSTRUMENT............................................................................................. 87
RESULTS (14-15 YO)................................................................................................... 87
BETWEEN 3/8 AND 5/8............................................................................................... 87
ANOTHER EXAMPLE.................................................................................................... 88
INSTRUCTION FOR CONCEPTUAL CHANGE..................................................................88
HOWEVER…................................................................................................................ 88
AND MOREOVER…...................................................................................................... 88
FRACTIONS AS PART/WHOLE RELATIONS....................................................................88
TEXTBOOK EXAMPLES................................................................................................ 89
7
,8
,MATHEMATICS & LANGUAGE
EDUCATION – 2022/2023
01 – VIEWS ON AND APPROACHES TO (RESEARCH IN) MATHEMATICS
EDUCATION
1. WHAT IS MATHEMATICS?
THE NATURE OF MATHEMATICS
ø Definitions
o Early definition
The science of quantity (Aristotle)
Niet per se uitgedrukt met nummers
o Famous definitions given by scientist
Mathematics is the science that draws necessary conclusions
(Benjamin Peirce)
Mathematics is the art of giving the same name to different
things(Henri Poincaré)
Mathematics is the classification and study of all possible patterns
(Walter Warwich Sawyer)
o Definitions in dictionaries
The study of the measurement, properties, and relationships of
quantities and sets, using numbers and symbols
The abstract science which investigates deductively the conclusions
implicit in the elementary conceptions of spatial and numerical
relations, and which includes as its main divisions geometry,
arithmetic, and algebra (Oxford English Dictionary)
The science of structure, order, and relation that has evolved from
elemental practices of counting, measuring, and describing the
shapes of objects.
Mathematics is a broad-ranging field of study in which the
properties and interactions of idealized objects are examined.
o Definition by Wikipedia
o Definition by Courant & Robins
ø Features (De Corte et al., 1996)
o Dual nature of mathematics
“On the one hand, mathematics is rooted in the perception and
description of the ordering of events in time and the arrangement of
objects in space and the solution of practical problems in the real
world.
On the other hand, out of this activity emerge symbolically
represented structures that can become objects of reflection and
elaboration, independent of their real-world roots. (…)
This duality is acknowledged in the distinction between the pure and
applied mathematics.”
(De Corte, Greer & Verschaffel, 1996, p. 500)
o Intrinsic inaccessibility of mathematical (thinking) objects
9
, There is the intrinsic inaccessibility of mathematical (thinking)
objects (Duval, 2002), implying that their conceptual acquisition
necessarily passes through the acquisition of one or more semiotic
representations (verbal, symbolic, visual, auditory, tactile, etc.)
The development of mathematics is inextractably dependent on
systems of symbols and graphical means of envisioning information
(e.g., the Arabic place-value system compared to the Roman
system)
Multiplying in Roman vs. Arabic number system: what works
best?
LXVIII 68
x LIV x 54
Clearly, it is much easier to construct and apply
computational algorithmic procedures with the Arabic
decimal number system than with the Roman number
system.
And in the binary number system: what works best?
1000100
X 110110
o Hierarchically structured development
Both in the history of mathematics as a cultural development
(phylogeny) and an individual child’s mathematical development
(ontogeny) can be conceived in terms of various types and levels of
representations which build on one other as the mathematical ideas
become more abstract.
Example of the hierarchically structured development of math
ø Images*
o Hoe mensen nadenken over wiskunde
10
