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transcript
Nicolina A. Malara, Giancarlo Navarra University of Modena and Reggio Emilia, Italy
September 21–26, 2014, Herceg Novi, Montenegro
12th International Conference of The Mathematics Education into the 21st Century Project
The Future of Mathematics Education in a Connected World
Generalization questions at early stages:
the importance of the theory
of mathematics education
for teachers and students
In the socio-constructive teaching, maths
teachers have the responsibility to:
• create an environment that allows pupils to
build up mathematical understanding;
• make hypotheses on the pupils' conceptual
constructs and on possible didactical
strategies, in order to possibly modify such
constructs.
This implies that teachers must not only acquire
pedagogical content knowledge but also
knowledge of interactive and discursive patterns
of teaching.
The socio-constructivist approach
2 September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)
In a constructivist teaching perspective the
teachers need:
• to be offered chances, through individual study
and suitable experimental activities,
• to revise their knowledge and beliefs about the
discipline and its teaching, in order to overcome
possible stereotypes and misconceptions,
• to become aware that their main task is to
make students able to give sense and
substance to their experience and to construct
a meaningful knowledge by interrelating new
situations and familiar concepts.
The teachers’ needs
3 September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)
The actual answers to these needs are extremely
complex in the case of classical thematic areas,
such as arithmetic and algebra, which suffer from
their antiquity, and the teaching of which is
affected by the way they historically developed.
In the traditional teaching and learning of
algebra the study of rules is generally privileged,
as if formal manipulation could precede the
understanding of meanings.
The general tendency is to teach the syntax of
algebra and leave its semantics behind.
The state of the art
4 September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)
We believe that the mental framework of
algebraic thought should be built right from the
earliest years of primary school when the child
starts to approach arithmetic, by teaching him or
her to think of arithmetic in algebraic terms.
In other words, constructing their algebraic
thought progressively, as a tool for reasoning,
working in parallel with arithmetic. This means
starting with its meanings, through the
construction of an environment which informally
stimulates the autonomous processing of that we
call algebraic babbling.
Our hypothesis on the approach to algebra
5 September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)
Algebraic babbling can be seen as the
experimental and continuously redefined
mastering of a new language, in which the rules
may find their place just as gradually, within a
teaching situation which:
is tolerant of initial, syntactically “shaky” moments,
stimulates a sensitive awareness of the formal
aspects of the mathematical language.
Algebraic Babbling
6 September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)
While learning a language, the child gradually
appropriates its meanings and rules, developing
them through imitation and adjustments up to
school age, when he/she will learn to read and
reflect on morphological and syntactical
aspects of language.
We believe that a similar process has to be
followed in order to make pupils approach the
algebraic language, because it allows them to
understand the meaning and the value of the
formal language and the roots of the algebraic
objects.
An example of algebraic babbling
The babbling metaphor
7 September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)
Pupils have translated algebraically the verbal
sentence “The number of finger biscuits is 1 more
than twice the number of chocolate cookies”.
8 September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)
• 1×2 • a+1×2 (a = number of finger biscuits) • fb+1×2 • a×2+1 • fb+1×2=a • fb=ch+1×2 • a=b×2+1 • a×2+1=b (a = number of chocolate cookies) • (a–1)×2
Please, reflect on the pupils’ sentences.
An example of algebraic babbling (10 years)
Pupils have translated algebraically the verbal
sentence “The number of finger biscuits is 1 more
than twice the number of chocolate cookies”.
9 September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)
• 1×2 • a+1×2 (a = number of finger biscuits) • fb+1×2 • a×2+1 • fb+1×2=a • fb=ch+1×2 • a=b×2+1 • a×2+1=b (a = number of chocolate cookies) • (a–1)×2
Pupils are going to discuss on the correctness of
the paraphrases expressing in many different
ways the same sentence.
An example of algebraic babbling (10 years)
10
Example – USUAL BEHAVIOR
September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)
It is rare that a teacher devotes significant
attention to the linguistic aspects of the
language of mathematics both for semantic and
syntactical aspects.
Usually, he/she does not promote students'
reflection on the interpretation of formulas as
linguistic objects in themselves and as
representations that objectify processes of
solving problem situations.
The teacher does not encourage the meta-
cognitive and meta-linguistic aspects in the
teaching of mathematics.
Example – A PRE-ALGEBRAIC PERSPECTIVE
The teacher has to:
• interpret each pupils’ writing and understand
their underlying ideas,
• Make the pupils interpret the writings and assess
their efficacy, reflecting on their correcteness
and fitness to the verbal sentence,
• discuss on the equivalences or differences
among the writings and select the appropriate
ones.
11 September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)
Example – A PRE-ALGEBRAIC PERSPECTIVE
The teacher must be able to:
• act as a participant-observer, i.e. keep his/her
own decisions under control during the
discussion, trying to be neutral and proposing
hypotheses, reasoning paths and deductions
produced by either individuals or small groups;
• predict pupils’ reactions to the proposed
situations and capture significant unpredicted
interventions to open up new perspectives in
the development of the ongoing construction.
This is a hard-to-achieve baggage of skills
12 September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)
Our studies make us aware of the difficulties that
teachers meet as to the design and
management of whole classroom discussions.
They highlight how, in the development of
discussions, teachers:
• do not make pupils be in charge of the
conclusions to be reached
• tend to ratify the validity of productive
interventions without involving pupils.
Our beliefs
13 September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)
We believe that a careful reflective analysis of
class processes is needed if one wants to lead a
teacher to get to a productive management
with pupils.
The project promotes
• a revision of the teaching of arithmetic in
relational sense
• an early use of letters to generalize and to
codify relationships and properties
• a reshaping of teachers’ professionalism
(knowledge, beliefs, behaviors, attitudes,
awareness) through sharing processes of
theoretical questions connected to practice.
1998 - 2014
14 September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)
The scientific setup of ArAl Project is illustrated in
• the Theoretical Framework
• the Glossary
• the Units
The Glossary constitutes in many aspects the
theoretical heart of the project.
It was conceived with the aim of aiding teachers
in their approach to theory, through the
clarification of specific conceptual or linguistic
constructs in mathematics and in maths
education. 15 September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)
Methodology
The development of the project is based on a net
of relationships involving:
• the university researchers as maths educators
• the teachers-researchers as tutors
• the teachers
• the pupils.
16 September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)
The cycle of teachers’ mathematics education
17 September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)
Classroom activity
Joint reflection among
teachers, tutors,
maths educators
Development of
theoretical framework,
methodologies,
materials
leads to
leads to
influences
Transcript
Other commentators
Teacher
E-tutor
A teacher • records a lesson,
• sends its Commented
Transcript (CT) to an E-tutor.
The E-tutor • comments the CT,
• sends the new version to
other members of the team.
The other members • write their comments too.
The CT so reached becomes
a powerful tool for teschers’
reflection and learning.
The Multicommented Transcripts Methodology
18 September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)
The cycle of teachers’ mathematics education
19 September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)
Classroom activity
Joint reflection among
teachers, tutors,
maths educators
Development of
theoretical framework,
methodologies,
materials
leads to
leads to
influences
MTM Class Episodes
influence
We show now a set of classroom episodes which
testify:
• the effects of the joint work among teachers,
tutors and maths educators which make
teachers embody theoretical results addressing
a new classroom practice;
• the achievement of new believes, of a new
language and of new ways of acting in the
classroom;
• the pupils’ conceptualizations and attitudes
towards a relational and pre-algebraic vision of
arithmetics.
The classroom episodes
20 September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)
21
The pupils are reflecting on:
5+6=11 11=5+6
Piero observes:
“It is correct to say that 5 plus 6 makes 11, but
you cannot say that 11 'makes' 5 plus 6, so it is
better to say that 5 plus 6 'is equal to’ 11,
because in this case the other way round is also
true”.
Example 1 (8 years)
September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)
What can we say about Piero’s sentence?
22
The pupils are reflecting on:
5+6=11 11=5+6
Piero observes:
“It is correct to say that 5 plus 6 makes 11, but
you cannot say that 11 'makes' 5 plus 6, so it is
better to say that 5 plus 6 'is equal to’ 11,
because in this case the other way round is also
true”.
Example 1 (8 years)
September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)
Piero is discussing the relational meaning of the equal sign.
23
Example 1 (8 years) – USUAL BEHAVIOR
September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)
Teachers and pupils 'see' the operations to the
left of the sign ‘=‘ and a result to the right of it. In
this perspective, the ‘equal’ sign expresses the
procedural meaning of directional operator and has a mainly space-time connotation (leftright,
beforeafter).
The task “Write 14 plus 23” often gets the reaction
‘14+23=’ in which ‘=‘ is considered a necessary
signal of conclusion and expresses the belief that
a conclusion is sooner or later required by the
teacher. '14+23' is seen as incomplete.
The pupils suffer here from lacking or poor control
over meanings.
24
Example 1 (8 years) – A PRE-ALGEBRAIC PERSPECTIVE
September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)
When shifting to algebra, this sign acquires a
different relational meaning, since it indicates the
equality between two representations of the
same quantity.
Piero is learning to move in a conceptual
universe in which he is going beyond the familiar space-time connotation.
To do this, pupils must ‘see’ the numbers on the
two sides of the equal sign in a different way;
the concept of representation of a number
becomes crucial.
25
Miriam represents the number of sweets: (3+4)×6.
Alessandro writes: 7×6.
Lea writes: 42.
Miriam observes: "What I wrote is more transparent,
Alessandro’s and Lea’s writings are opaque.
Opaque means that it is not clear, whereas
transparent means clear, that you understand.”
Example 2A (9 years)
September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)
What can we say about Miriam’s sentence?
26
Miriam represents the number of sweets: (3+4)×6.
Alessandro writes: 7×6.
Lea writes: 42.
Miriam observes: "What I wrote is more transparent,
Alessandro’s and Lea’s writings are opaque.
Opaque means that it is not clear, whereas
transparent means clear, that you understand.”
Example 2A (9 years)
September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)
Miriam reflects on how the non-canonical
representation of a number helps to interpret and illustrate the structure of a problematic situation.
27
Example 2A (9 years) – USUAL BEHAVIOR
September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)
Traditionally, in the Italian primary school, pupils
become accustomed to seeing numbers as
terms of an operation or as results.
This leads, inter alia, to see the solution of a
problem as a search for operations to be
performed. The prevailing view is that of a
procedural nature: the numbers are entities to
be manipulated.
The pupils are not guided towards reflection,
through the analysis of the representation of the
number, on its structure.
Actually teachers rarely explain that…
28
Example 2A (9 years) – A PRE-ALGEBRAIC PERSPECTIVE
September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)
... each number can be represented in different
ways, through any expression equivalent to it:
one (e.g. 12) is its name, the so called canonical
form, all other ways of naming it (3×4, (2+2)×3,
36/3, 10+2, 3×2×2, ...) are non canonical forms,
and each of them will make sense in relation to
the context and the underlying process.
As Miriam observes, canonical form, which
represents a product, is opaque in terms of
meanings. Non canonical form represents a
process and is transparent in terms of meanings.
29
Example 2A (9 years) – A PRE-ALGEBRAIC PERSPECTIVE
September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)
Knowing how to recognize and interpret these
forms creates in the pupils the semantic basis for
the acceptance and the understanding, in the
following years, of algebraic writings such a-4p,
ab, x2y, k/3.
The complex process that accompanies the
construction of these skills should be developed
throughout the early years of school.
The concept of canonical/non-canonical form
has for pupils (and teachers) implications that
are essential to reflect on the possible meanings
attributed to the sign of equality.
Let us see example of these skills:
30
Example 2B (11 years) - A PRE-ALGEBRAIC PERSPECTIVE
September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)
The pupils have the task of representing in
mathematical language the sentence:
“Twice the sum of 5 and its next number.”
When the pupils’ proposals are displayed on the
whiteboard, Diana indicates the phrase of Philip and justifies her writing: “Philip wrote 2×(5+6), and
it is right. But I have written 2×(5+5+1) because in
this way it is clear that the number next to 5 is a
larger unit. My sentence is more transparent”.
What can we say about Diana’s sentence?
31
Example 2B (11 years) - A PRE-ALGEBRAIC PERSPECTIVE
September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)
The pupils have the task of representing in
mathematical language the sentence:
“Twice the sum of 5 and its next number.”
When the pupils’ proposals are displayed on the
whiteboard, Diana indicates the phrase of Philip and justifies her writing: “Philip wrote 2×(5+6), and
it is right. But I have written 2×(5+5+1) because in
this way it is clear that the number next to 5 is a
larger unit. My sentence is more transparent”.
Diana is emphasizing the relational aspects of the number made evident by its non-canonical form.
32
Example 3A (10 years)
September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)
The task for the pupils is: ‘Translate the sentence 3×b×h into natural
language’.
Lorenzo reads what he has written: “I multiply 3
by an unknown number and then I multiply it by
another unknown number.”
Rita proposes: “The triple of the product of two
numbers that you don’t know” Lorenzo observes: “Rita explained what 3×b×h is,
whereas I have told what you do.”
What can we say about Lorenzo’s sentence?
33
Example 3A (10 years)
September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)
The task for the pupils is: ‘Translate the sentence 3×b×h into natural
language’.
Lorenzo reads what he has written: “I multiply 3
by an unknown number and then I multiply it by
another unknown number.”
Rita proposes: “The triple of the product of two
numbers that you don’t know” Lorenzo observes: “Rita explained what 3×b×h is,
whereas I have told what you do.”
Lorenzo captures the dichotomy process-product.
Another example:
34
Example 3B (two teachers)
September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)
Rosa and Viviana are two teachers of one of our
groups. They are discussing on a problem
concerning the approach to equations using the
scales: “There are 2 parcels of salt in the pot on
the left, and 800 grams in the pot on the right”.
Rosa explays her task: “How heavy is the salt?”
Viviana observes: “It would be better to write:
Represent the situation in mathematical language
in order to find the weight of a packet of salt”.
Please, comment Rosa’s and Viviana’s sentences.
35
Example 3B (two teachers)
September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)
Rosa and Viviana are two teachers of one of our
groups. They are discussing on a problem
concerning the approach to equations using the
scales: “There are 2 parcels of salt in the pot on
the left, and 800 grams in the pot on the right”.
Rosa explays her task: “How heavy is the salt?”
Viviana observes: “It would be better to write:
Represent the situation in mathematical language
in order to find the weight of a packet of salt”.
Rosa and Viviana are reflecting on the dialectics
representing/solving.
36
Examples 3A-3B – USUAL BEHAVIOR
September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)
Rosa’s task is set in a ‘classical’ arithmetic
perspective: she looks for the solution and
emphasizes the search for the product.
This way, the pupils learn that the solution of a
problem coincides with the detection of its result
and with the search of operations.
The consequence of this attitude is that the
information of the problem are seen as
ontologically different entities and separated into
two distinct categories: the data and what one
needs to find.
Pupils solve the problem by operating on the
former and finding the latter.
37
Examples 3A-3B – A PRE-ALGEBRAIC PERSPECTIVE
September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)
Viviana’s task is set in an algebraic perspective: it
induces a shift of attention from elements in play
towards the representation of the relationships
between them and on the process.
She draws the pupils from the cognitive level
towards the meta-cognitive one, at which the
solver interprets the structure of the problem and
represents it through the language of
mathematics.
38
Examples 3A-3B – A PRE-ALGEBRAIC PERSPECTIVE
September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)
This difference between the attitude that favours
solving (Rosa) and that which favours
representing (Viviana) is connected to one of the
most important aspects of the epistemological
gap between arithmetic and algebra: while
arithmetic implies the search for solution, algebra
delays it and begins with a formal transposition of
the problem situation from the domain of natural
language to a specific system of representation.
39
Example 4 (12 years) - A PRE-ALGEBRAIC PERSPECTIVE
September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)
Thomas has represented the relationship between two variables this way: a=b+1×4 and he
explains: “The number of the oranges (a) is four
times the number of the apples (b) plus 1”.
Katia replies “It's not right: your sentence would
mean that the number of oranges (a) is the number of apples (b) plus 4 (1×4 is 4). You have
to put the brackets: a=(b+1)×4”.
Reflect on Thomas’s and Katia’s sentences.
40
Example 4 (12 years) - A PRE-ALGEBRAIC PERSPECTIVE
September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)
Thomas has represented the relationship between two variables this way: a=b+1×4 and he
explains: “The number of the oranges (a) is four
times the number of the apples (b) plus 1”.
Katia replies “It's not right: your sentence would
mean that the number of oranges (a) is the number of apples (b) plus 4 (1×4 is 4). You have
to put the brackets: a=(b+1)×4”.
Thomas and Katia are discussing the translation
between natural and algebraic language and
the semantic and syntactic aspects of mathematical writings.
41
Example 4 (12 years) - USUAL BEHAVIOR
September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)
Sentences in the mathematical language as,
e.g., a=(b+1)×4, are generally seen from an
operational point of view rather than an
interpretative one.
Students unaccustomed to reflecting on the
meanings of the sentences written in algebraic
language, in this case merely observe that “a=b+1×4 is wrong because there are no
brackets”.
42
Examples 4 - A PRE-ALGEBRAIC PERSPECTIVE
September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)
Translating from the natural language to the
mathematical one (and vice-versa) favors
reflecting on the language of mathematics, that
is interpreting and representing a problematic
situation by means of a formalized language or,
on the contrary, recognizing the problematic
situation that it describes in a symbolic writing.
Closely related to the act of representing is the
issue of respecting the rules in the use of a
(natural or formalized) language.
In teaching mathematics, rules are generally
‘delivered’ to pupils, thus losing their social value
of a support to the understanding and sharing of
a language as a communication tool.
43
Examples 4- A PRE-ALGEBRAIC PERSPECTIVE
September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)
Pupils should be guided to understanding that
they are acquiring a new language which has a
syntax system and semantics.
They internalize from birth that compliance to the
rules allows communication, but it is highly unlikely
that they will transfer this peculiarity to the
mathematical language. In order to overcome
this obstacle, we ask pupils to exchange
messages in arithmetic-algebraic language with
Brioshi, a fictitious Japanese pupil who speaks
only in his mother tongue. This trick works as a
powerful didactical mediator to highlight the
importance of respecting the rules while using the
mathematical language.
44
Open questions
September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)
• When and how does the curtain on algebra
begin to open?
• Which teachers’ attitudes can favour pre-
algebraic thinking?
• Do you agree with the idea that algebra
doesn’t follow arithmetic, but rather develops
by intermingling along with it right from the first
years of primary school?
• Which mathematics education should future
teachers receive in order to improve their
sensitiveness towards those micro-situations that
allow to ‘see algebra within arithmetic’?
45
Open questions
September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)
• Which is your position about these topics?
• Which kind of difficulties (cultural, social
political difficulties) do you see about the
spreading of this type of teaching in the
classes? Which constraints?
• Which is the status of Early algebra in your
country?
• Which are the teachers' dominant beliefs about
algebra and early algebra in your country?
• Which importance do these questions have in
the pre-service mathematics education of
teachers?
46 September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)
Thank you