Generalization questions at early stages: the importance ...€¦ · September 21–26, 2014,...

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


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


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


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


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


Pupils have translated algebraically the verbal

sentence “The number of finger biscuits is 1 more

than twice the number of chocolate cookies”.


• 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”.


• 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.


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


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


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


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.


The cycle of teachers’ mathematics education


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


The cycle of teachers’ mathematics education


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


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)


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)


Piero is discussing the relational meaning of the equal sign.

23

Example 1 (8 years) – USUAL BEHAVIOR


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


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)


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.”



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


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


... 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


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


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


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



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



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)


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)


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


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


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


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


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


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


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


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


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


• 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


• 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?


Thank you

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Generalization questions at early stages: the importance ...€¦ · September 21–26, 2014,...

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