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Summary Mathematics & Language Education (P0R44a)
Summary of the lectures Mathematics & Language Education in the academic year .
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Mathematics & Language Education – P0L36A Prof. Lieven Verschaffel & Wim van DoorenCHAPTER 1
VIEWS ON AND APPROACHES TO
(RESEARCH IN) MATHEMATICS EDUCATION
1. WHAT IS MATHEMATICS ?
1.1 THE NATURE OF MATHEMATICS
DEFINITION(S)
Famous definitions given by scientists
- Aristotle The science of quantity
- Benjamin Peirce (1870) Mathematics is the science that draws necessary conclusions
- Henri Poincaré (1850-1912) Mathematics is the art of giving the same name to different
things
- Walter Warwick Sawyer (1955) Mathematics is the classification and study of all possible
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.
- 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.
FEATURES
D UAL NATURE OF MATHEMATICS
- “On the one hand, mathematics is rooted in the perception of the ordering of events in time
and 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.”
I NTRINSIC INACCESSIBILITY OF MATHEMATICAL ( THINKING ) OBJECTS
- 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 inextricably dependant on systems of symbols and
graphical means of envisioning information (e.g.: the Arabic place-value system compared to
the Roman system)
o In geography, biology, music etc. you have direct access to the object of study; this is
not the case for mathematics where the object of study is not visible, audable,
touchable… It is only accessible through representations.
H IERACHICALLY STRUCTURED DEVELOPMENT
Both 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.
E.g.: From natural to whole (negative) to rational to more irrational or real numbers etc.
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,Mathematics & Language Education – P0L36A Prof. Lieven Verschaffel & Wim van Dooren
1.2 WHERE DOES MATHEMATICS COME FROM AND WHAT IS A
MATHEMATICAL TRUTH ?
- Is mathematics discovered or invented?
- Is a mathematical truth absolutely and unconditionally true?
MATHEMATICS : DISCOVERED OR INVENTED?
Discovered:
- Plato Math exists somewhere in a perfect world which we need to unravel (like the story of
the cave and the shadows)
- Galilei Mathematics is something that exists (like our material world) in a superhuman,
objective realm, and, thus is discovered by mankind.
- There is something called math that is in the world, that dictates how the machinery of nature
works, and mathematicians have to discover and describe and systematize that externally
existing reality.
Invented:
- Mathematical truths are not discovered but invented; they are creations of mankind situated in
human practices.
IS A MATHEMATICAL TRUTH ABSOLUTELY AND UNCONDITIONALLY TRUE ?
- Mathematics knowledge is absolutely, eternally and unconditionally true = Absolutism
- Mathematical knowledge is fallible, i.e. subject to failure, refutation, change, development,
debate, .. = Fallibilism
- Both (related) questions continue to elicit hot debates among philosophers and
mathematicians
- The position one takes on the first question tends to be related to the position towards the
second one.
- The position one takes in this debate affects one’s answer to the next questions (why teaching
math? How teaching math?)
1.3 EPISTEMOLOGICAL BELIEFS ABOUT MATHEMATICS
Teachers’ epistemological beliefs (Felbrich et al., 2008)
Absolutist Fallibilist
1. Formalism-related 3. Process-related
An exact science that has an axiomatic basis A science which mainly consists of problem
and is developed by deduction and logic. solving processes and discovery of structure
E.g.: ‘Mathematical thinking is determined by and regularities.
abstraction and logic.” E.g.: ‘If one comes to grip with mathematical
problems, he/she can often discover something
new (connections, rules and terms)’
2. Scheme-related 4. Application-related
A collection of terms, rules and formulae A science which is relevant for society and life.
E.g.: ‘Mathematics is a collection of procedures E.g.: ‘Mathematics helps to solve daily tasks and
and rules which precisely determine how a task problems’
is solved.’
These are 4 extreme positions they distinguished to position teachers with their epistemological beliefs
(those aren’t very well articulated, but they affect how a teacher thinks, behaves etc. in their
mathematics classes).
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,Mathematics & Language Education – P0L36A Prof. Lieven Verschaffel & Wim van Dooren
2 WHY SHOULD IT BE TAUGHT?
- There are many (good) reasons for teaching and learning mathematics
- Throughout the history of mathematics education, different arguments have dominated the
debate.
- Different groups in society emphasize distinct arguments
- The answer to the ‘why’ question may have a great impact on the answer to the ‘what’
question
- The answer to the ‘why’ question is associated with the answer to the ‘how’ question.
REASONS TO LEARN MATHEMATICS
- It’s the tool and language of commerce, engineering and other sciences – physics, computing,
biology etc.
- Mathematics plays a vital, often unseen, role in many aspects of modern daily life.
- As society becomes more technically dependent, there will be an increasing requirement for
people with a higher level of mathematical training.
- Analytical and quantitative skills are sought by a wide range of employers.
- A degree in mathematics provides you with a broad range of skills in problem solving, logical
reasoning and flexible thinking.
- This leads to careers that are exciting, challenging and diverse in nature (so, good job
prospects)
- Mathematics is a universal part of human culture. Good citizenship requires a minimal contact
with this part of human culture.
- Mathematics in nice and fun (some say)
Günter M. Ziegler (2016) in ICME13 plenary lecture
1. Toolbox for everyday life
2. Part of culture, 6000 years old
3. Essential basis for high tech
Should we teach all these in the same school subject, by the same teacher, for the same groups of
students? Or in three different subjects, taught by different kinds of teachers, for different groups of
students?
STEM
- STEM (Science, Technology, Engineering and Mathematics) is a term used to group together
these academic disciplines.
- This term is typically used when addressing education policy and curriculum choices in
schools to improve competitiveness in science and technology development.
- The term typically also refers to attempts to connect and integrate the four STEM disciplines.
- Puts serious pressure to the identity and superiority of mathematics as an ‘untouchable’
curricular domain.
- Mathematicians and math teachers don’t like the STEM They risk losing their identity
INTEREST GROUPS AND THEIR AIMS FOR MATHEMATICS TEACHING
(ERNEST, 2014)
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, Mathematics & Language Education – P0L36A Prof. Lieven Verschaffel & Wim van Dooren
3 HOW TO LEARN AND TO TEACH MATHEMATICS?
3.1 EDUCATIONAL PSYCHOLOGY PERSPECTIVE (VERSCHAFFEL ET AL,
2007)
1. Behaviorist/Empericist
2. Cognitive/Rationalist
3. Situative/Pragmatist-Sociohistoric
3.2 MATHEMATICS EDUCATION PERSPECTIVE
MECHANISTIC OR SKILLS APPROACH
Example 1a: How to do the multiplication algorithm
Example 1b: How to divide a fraction by a fraction
• Earlier, we learnt how to multiply fractions:
– E.g. 1 x 2 = 2
8 3 24
• Today I will show you how to divide fractions: step 1: change
the operator; step 2: invert nominator and denominator; step
3: do the multiplication
– E.g. 1 ÷ 2 = 1 x 3 = 3
8 3 8 2 16
• Let’s now practice this newly learnt skill:
– Exercice: 1 ÷ 1 =
4 6
• Summary: recapitulation of the rule; drawing attention to the
difference with the rule for multiplication; stimulating students
to memorize the rules well
T HE SKILLS OR DRILL APPROACH (B AROODY , 2003)
“The skills approach, with its focus on memorization of basic skills by rote (…) is based on the
assumption that mathematical knowledge is simply a collection of socially useful information (facts,
rules, formulas, and procedures). Except for the mathematically gifted, children are viewed as ‘empty
vessels’, largely helpless, and incapable of understanding many aspects of school math. The aim of
math instruction is to inform them about how to do math. Proponents of the skills approach believe that
the most efficient way to transmit information is by means of direct instruction and practice – without
taking the time to promote understanding about the whys of procedures. A teacher directs all aspects
of the learning process to ensure efficient instruction (transmittal and mastery) of basic skills.
This method was very dominant worldwide until the 1950’s.
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