MATHEMATICS & LANGUAGE
EDUCATION – 2022/2023
01 – VIEWS ON AND APPROACHES TO (RESEARCH IN) MATHEMATICS EDUCATION
1. VERSCHAFFEL (2016) ‘Views on and approaches to (research in) mathematics education’
(1) What is (a) THE NATURE OF MATHEMATICS
mathematics? ø Evolution: “the science of number and space” → “the science of patterns”
Answers from ø Aspects that could be mentioned to characterize mathematical knowledge:
the i. The dual nature of mathematics: descriptions of perceived reality and autonomous
philosophy of abstract constructions
mathematics • Universal utilitarian subject ↔ specialized, esoteric and esthetic side
• Dual aspect of mathematics: practical/applied side vs. theoretical side
ii. The intrinsic inaccessibility of mathematical (thinking) objects: conceptual acquisition
necessarily passes through one/more semiotic representations
iii. Its hierarchically structured development
• Both the cultural/individual planes: various types/levels of
representations which build on one another → more abstract
mathematical ideas (= 1e major element of development)
(b) THE NATURE OF MATHEMATICAL OBJECTS AND TRUTH
ø 2e major element of development = shift away from view of mathematics as the epitome of
certain knowledge; shift toward its conceptualization as a fallibilistic activity grounded in
human practices
i. Shift: absolutist views (mathematics = knowledge that is absolutely secure and
objective; apply “foundationalism” strategy) → failed to secure the certainty of
mathematical knowledge (Ernest, 2014)
ii. Absolutist views + Platonistic views (mathematical objects are real and exist in an
objective and superhuman realm) → belief that mathematical truths are discovered
and not invented
iii. Alternative for absolutist and Platonist view: fallibilism
• Mathematics = the outcome of social processes
• Mathematical knowledge = fallible and open to revision
• Mathematics = associated with sets of social practices
• Academic mathematics, school mathematics and ethnomathematics →
distinct, but intimately interrelated
ø 4 principal orientations beliefs concerning the nature of mathematics (related to distinction
between an absolutist and fallibilist view)
i. Absolutist view:
• Formalism-related orientation: mathematics = exact science, developed
by deduction
• Scheme-related orientation: mathematics = collection of terms, rules
and formulae
ii. Fallibilist view:
• Process-related orientation: mathematics = science which mainly
consists of problem solving processes and discovery of structure and
regularities
• Application-related orientation: mathematics = science which is
relevant for society and life
ø These beliefs = dynamic and flexible, and linked to certain socio-cultural and educational
contexts
(2) Views on the (a) VARIOUS COMPETING INTEREST GROUPS WITH DISTINCT AIMS FOR MATHEMATICS EDUCATION –
purposes of DIFFERENT VIEWS
mathematics ø Industrial trainers (right conservative side of the social spectrum): emphasize back-to-basics
teaching and numeracy as the main mathematical aim (combined with absolutist set of decontextualized
learning: but utilitarian truths and rules)
What are ø Old humanistic mathematicians: conservative mathematicians preserving rigor of proof and
mathematics purity of mathematics; transmit pure mathematical knowledge (combine this aim with view of
for? mathematics as an absolutist bdy of structured pure knowledge)
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, ø Public educators: emphasize development of critical awareness and democratic citizenship,
combined with fallible knowledge that is socially constructed in diverse practices
(3) Views on and (a) Intro:
approaches ø Various views and approaches → 3 major views
to learning ø Mechanism: links to behaviorist/empiricist perspectives
and teaching i. Knowing = associations between stimuli and reactions which are established through
(elementary) repetition and reinforcement
mathematics ø Structuralism: associated with Piaget’s theory of cognitive development
ø Constructivist/realist: reminiscences to constructivist, situative and socio-historical theories
of cognition and learning
ø 3 descriptions are prototypes → real mathematics = mixtures of elements of the different
views/approaches
(b) MECHANISTIC APPROACHES
ø Dominated the (elementary) mathematics educational scene, up to the 1950s
ø Focus of instruction = factual and procedural content
i. Through basic learning principles: association, memorization and automatization;
through repeated practice and reinforced behavior
ø Little attention to conceptual understanding
ø Instruction = heavily teacher-directed, teacher = dispenser of pieces of knowledge and skills
to be learnt
ø Baroody (2003), skills- or drill-based approach: focus on memorization of basic skills by rote
(= basisvaardigheden uit het hoofd leren)
i. Mathematical knowledge = collection of socially useful information
ii. Children = “empty vessels”
iii. Aim of math instruction = inform them about how to do math; transmit information by
means of direct instruction and practice (no understanding) – transmittal and mastery
iv. Skills approach: not the teacher’s goal to develop conceptual understanding in the
children → “ours is not to reason why, just invert and multiply” (popular
mnemotechnic device)
ø Criticism: mechanistic approach = problematic → new approaches
ø Still many computer-based ‘drill-and-practice’ programs for learning basic arithmetical skills
(c) STRUCTURAL APPROACHES
ø Economic and scientific developments → revise curricula
ø US “New Mathematics”: aim = to make rational and critical citizens
i. Emphasize insight into mathematical structures, through the study of abstract
concepts (sets, relations, graphs, algebraic structures, number bases other than 10,
etc.)
ø Criticism: a lot of controversy; overemphasis on abstract concepts and neglect of practical
applications
i. Europe: implementation depended on national cultures and local educational systems
→ many divergences (Servais)
ii. Elementary schools: situation = quite diverse
iii. Starting too much from the structure of the discipline and too little from the
perspective of the learning child who is discovering
ø Main features of New Math curriculum (p. 8)
ø Belgium curriculum pays more attention to structural aspects and tends to establish stronger
strive for abstraction and formalization (= result of involvement in New Math movement)
(d) CONSTRUCTIVIST/REALISTIC APPROACHES
ø Aim at the integrated mastery of mathematical proficiency that goes beyond the learning of
procedures, but also goes broader than the strongly conceptually oriented approach of the
new math
ø Notion of mathematical proficiency:
i. Five interwoven strands: conceptual understanding, procedural fluency, strategic
competence, adaptive reasoning, and a productive disposition to see mathematics as
sensible/useful/worthwhile
ø Meaning-oriented, process-oriented, learner-oriented and/or problem-oriented forms of
teaching and learning
ø Ample attention to the facilitation of personal exploration → rich environment to explore;
building on children’s intuitive knowledge and informal skills and their inherent need to
understand, to their capability of inventing and flexibly applying their own solution strategies
ø Teacher = “mentor”: teacher poses worthwhile task (complex and challenging, frequently also
whether or not there was formal instruction on the content)
ø Example: RME (Realistic Mathematics Education; Dutch realist approach)
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, i. “Mathematics as a human activity”, mathematics education = “putting pupils in touch
with phenomena”
ii. Guided re-invention
iii. Focal point = learner’s own activity, on his/her process of “mathematization”
• Treffers (1987): two types of mathematization which should receive proper
attention in an RME program
a. ‘Vertical’ mathematization: process of reorganization within the
mathematical system itself
b. ‘Horizontal’ mathematization: come up with mathematical tools
which help to organize/solve a problem in a real-life situation
iv. Core principles RME:
• Realistic contexts (“realistic” does NOT necessarily refer to a connection with
the real world; implies that the situation is imaginable for the student)
• Progressing towards higher levels of abstraction and internalization (from
concrete, intuitive, informal level → general, explicit, formal level)
• Learners are considered as active participants: construct and develop
mathematical tools & insights
• Learners have opportunities to interact with each other and challenged to
share their experiences with others + reflect
• Instruction should take into account/contribute to the hierarchical organization
and interconnectedness of mathematical knowledge components and skills
v. Criticism:
• Positive results → validity of conclusions have been questioned
• Math war
(4) Mathematics (a) BRIEF DESCRIPTION OF HOW MATHEMATICS EDUCATION HAS GROWN AND ESTABLISHED ITSELF
education as AS A RESEARCH FIELD ON ITS OWN
a research ø 1970: different identifiable traditions reflecting the state of research in mathematics
field on its education
own i. Empirical-scientist tradition: psychological/instructional theories and methods
ii. Disciplinary-oriented and more pedagogical-didactical in nature
ø Interactions scholars/researchers → idea that mathematics can be delineated
ø Process involved new theoretical views & establishment of a methodological stance
ø Designed context: subject to test and revision; involve interdisciplinary teamwork
ø Design research has 2 goals:
i. Contributing to the innovation and improvement and innovation of classroom
practices
ii. Advancement of theory building about learning from instruction
ø Criticism:
i. Teaching to the test = most powerful untested component of the “treatment”
2. VAN DEN HEUVEL & DRIJVERS (2014) ‘Realistic Mathematics Education’
(1) What is Realistic (a) RME = domain-specific instruction theory for mathematics
Mathematics Education? ø Rich, “realistic” situations = prominent position in the learning process → source for initiating
development of mathematical concepts, tools, procedures
ø Broader connotation of “realistic”: students are offered problem situations which they can
imagine; ‘making something real in your mind’
i. → Problems can come from the real world but also from the fantasy
world/fantasies, or the formal word of mathematics
(2) The onset of RME (a) Start of RME: founding of Wiskobas (basis for RME) → 1973: decisive impulse to reform the prevailing
approach to mathematics education
(b) 1960’s: mathematics education in the Netherlands dominated by a mechanistic teaching approach
(mathematics as a scientific discipline)
ø Inflexible and reproduction-based knowledge (taught at a formal level, in an atomized manner)
ø Alternative = ‘New Math-movement, it was Freudenthal’s merit (verdienste) that Dutch
mathematics was not affected by the formal approach of the New Math movement
(3) Freudenthal’s Guiding (a) Hans Freudenthal, 1946: professor of pure and applied mathematics
Ideas About Mathematics ø Later argued for teaching mathematics that is relevant for students + thought experiments →
and Mathematics how can student be offered opportunities for guided re-invention of mathematics?
Education (b) Empirical sources + method of the didactical phenomenology:
ø Describing mathematical objects/structures/… in their relation to the phenomena for which
they’re created + taking into account students’ learning process → theoretical reflections on
the constitution of mental mathematical objects
ø Contributed to the development of RME
ø Freudenthal about the then dominant approach: ‘anti-didactic inversion’
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, (c) Students should be active participants in the educational process, mathematics = human activity →
should not be learned as a closed system, but as an activity of mathematizing reality/mathematics
(d) Took over Treffers’ distinction of horizontal and vertical mathematization
ø Horizontal: using mathematical tools to organize and solve problems situated in real-life
situations
ø Vertical mathematization: process of reorganization within the mathematical system →
shortcuts by using connections between concepts and strategies
i. Stressing RME’s “real world” perspective too much → neglecting vertical
mathematization (!)
(4) The Core Teaching (a) RME has much in common with current approaches; involves core principles for teaching mathematics
Principles of RME (originally from Treffers → reformulated)
a. Activity principle: students = active participants; mathematics = human activity which is best
learned by doing mathematics (idea of mathematization)
b. Reality principle: (1) importance of the goal attached to mathematics education, (2)
mathematics education should start from problem situations which are meaningful to students
→ teaching starts with problems in rich contexts, puts students on track of informal context-
related solution strategies
c. Level principle: learning mathematics = students pass various levels of understanding; models
are important for bridging the gap
i. Models have to shift: ‘model of’ to ‘model for’
ii. Level principle = reflected in the didactical method of “progressive schematization”
d. Intertwinement principle: mathematical content domains are not isolated, but heavily
integrated in the curriculum (also applies within domains)
i. E.g. number sense, mental arithmetic, estimation and algorithms are taught in close
connection
e. Interactivity principle: learning mathematics is not an individual activity, but also a social
activity → RME favors whole-class discussions and group work; interaction = reflection, which
helps reach a higher level of understanding
f. Guidance principle: RME teachers should have a proactive role in students’ learning; should
include scenario’s which have the potential to work as a lever to reach shifts in students’
understanding → teaching/programs should be based on long-term teaching trajectories
(5) Various Local Instruction (a) Core teaching principles → local instruction theories and paradigmatic teaching sequences focusing on
Theories specific mathematical topics
ø Van den Brink: new approaches to addition/subtraction up to 20
ø Streefland: prototype for teaching fractions intertwined with ratios and proportions
ø De Lange: new approach to teaching matrices and discrete calculus
(b) RME → new approaches to assessment in mathematics education
(6) Implementation and (a) RME → more reform-oriented textbooks and mechanistic ones disappeared
Impact (b) Also influential worldwide!
(7) A Long-Term and Ongoing (a) RME is still a work in progress; not a unified approach → lots of various accents in RME
Process of Development (b) This diversity was not a barrier, but stimulated reflection and revision – supported the maturation of the
RME theory
(c) Current debate: going back to mechanistic approach?
ø Made proponents (voorstanders) of RME more alert to keep deep understanding and basic
skills more in balance in future developments of RME + enhance the methodological
robustness
3. KILPATRICK & SWAFFORD (2001): ‘Adding It Up’
(1) Executive Summary (a) Mathematics epitomizes the power of deductive reasoning → participation in society
(b) Article is about mathematics from pre-kindergarten to eight grade, addresses this concern:
too few students are successfully acquiring the mathematical knowledge, the skill & the
confidence they need to use the mathematics they have learned
(c) Curriculum has many components: heart of math = concepts of number and operations with
numbers
ø Much of the rest of the mathematics curriculum is intertwined with number concepts
(d) Proficiency with numbers and numerical operations is an important foundation for further
education in mathematics
REPORT PERSPECTIVE
(e) (!) Report perspective is broader than just computation
1. Numbers and operations are abstractions → communication about numbers requires some form
of external representation (graph, system of notation,…)
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