Introduction to algebra: structural sense of expressions (alien)

Identity show tooltip helpexplode

Authors show tooltip help

L. Maffei, M.A. Mariotti, C. Sabena - UNISI

Subject domains show tooltip help

  • mathematics

Topics show tooltip help

  • arithmetic
  • algebra

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

  • Aplusix
  • didactical cycle
  • registers of representation
  • tree representation
  • equivalence
  • additive and multiplicative structure of a number
  • structure of a numerical expression
  • feedback signs

Description show tooltip help

The PP proposes an introduction to syntactic aspects of algebraic thinking through the manipulation of numerical expressions in structural terms, with the use of Aplusix software. The possibility of identifying a structure in an algebraic expression is supported by the innovative representation given by the software, the tree graph, together with the standard representation.

The types of activities involve three systems of representation: natural language (NL), standard representation of expressions (SR), and the tree representation (TR).

Rationale show tooltip helpexplode

The passage from a procedural approach typical of arithmetic to a structural one typical of algebra is fundamental to seize the meaning of algebraic manipulation. The didactical goals of the PP are twofold: the acquisition of a structure sense for algebraic expressions, and its rupture with respect to arithmetical computations. A key point in this passage is played by the equivalence between expressions, given by the use of the properties. It will be the final focus of the PP. Conversions between TR, SR and NL are exploited in order to reach such  a goal.

Theoretical framework show tooltip help

Difficulties encountered by pupils in gaining competencies in algebraic calculation are well-known, and they have been addressed many times and from different point of views (Freudenthal, 1983, Tall & Thomas 1991, Kieran 1992).

The introduction of algebra requires in fact the development of a different way of thinking, which cannot be considered as a pure generalization of arithmetics (Mariotti & Cerulli, 2003). Differently from arithmetics, whose main goal is to do calculations so as to have a result, algebra offers an operational language for representing, analyzing and manipulating relations which contain both numbers and letters.

This diversity is explained by Sfard (1991) in terms of two different perspectives from which mathematical objects can be conceived: in a structural way (as objects) or in an operational way (as procedures or processes).

"There is a deep ontological gap between operational and structural conceptions [...] Seeing a mathematical entity as an object means being capable of referring to it as if it was a real thing-static structure, existing somewhere in space and time. It also means being able to recognize the idea 'at a glance' and to manipulate it as a whole, without going into details [...] In contrast, interpreting a notion as a process implies regarding it as a potential rather than actual entity, which comes into existence upon request in a sequence of actions. Thus, whereas the structural conception is static, instantaneous, and integrative, the operational is dynamic, sequential, and detailed." (Sfard, 1991).

We address the problem of gaining a structural approach to algebraic expressions, by adopting a cognitive-semiotic view-point, provided by the theory of Semiotic Mediation. The theory draws from a Vygotskijan paradigm, and frames in particular processes of mediation carried out by means of artefacts in classroom context. In this perspective, the use of artefacts for accomplishing a task leads the individual to the construction of personal meanings which are related to the actual use of the artefact. On the other hand, mathematical meanings may be related to the artefact and its use: in particular, in classroom context under the guidance of an expert (i.e. the teacher) the personal meanings, related to the actual use of an artefact, may be generated and evolve towards mathematical meanings. Because of this potentiality, we will say that an artefact may function as a semiotic mediator. The semiotic mediation function of an artefact is not automatically activated; in fact such a function can be exploited by the expert (in particular the teacher) who has the awareness of the semiotic potential of the artefact both in terms of mathematical meanings and in terms of personal meanings. Such evolution is fostered by the teacher's action, guiding the process of production and evolution of signs centred on the use of an artefact.

The evolution of signs, central in the perspective of semiotic mediation, can be described in terms of transformations of registers of representation (Duval, 1995). Furthermore, the notion of semiotic register could reveal a useful tool both in the analysis of the semiotic potential of the artefact, as well as in the design of tasks both within Aplusix and in paper and pencil environment. Specifically, Aplusix provides two different kinds of representation systems for algebraic expressions: the “standard” representation (SR), and the “tree” representation (TR). They can be considered two different registers of representation in the sense of Duval. According to Duval (1995), a "register of representation" is a semiotic system (i.e. a system of signs) that provides for specific possibilities of transformation of the representations. In our case, natural language allows transformations such as paraphrases and reformulation, and standard and tree representation allow transformations that express operations among numbers and symbols. Beyond this kind of transformations, that pertain to a specific register and are called "treatments", there are transformations that occur between different registers: they are called "conversions". Treatments and conversions are central in mathematics:%2

Target show tooltip helpexplode

Rationale show tooltip helpexplode

The PP situates in the first year of the upper secondary school when, according to the Italian curricula, students approach algebra.

Population show tooltip helpexplode

School level show tooltip help

First Year of Secondary School

Age range show tooltip help

14-15

Population description show tooltip help

A standard mathematics classroom

Student prerequisites show tooltip help

Arithmetic computation (computation with integers, fractions).

Teacher prerequisites show tooltip help

Familiarity with Aplusix software.

Context show tooltip helpexplode

Physical context show tooltip help

Classroom equipped with computers, overhead projector.

Socio-cultural context show tooltip help

Standard classroom.

 

Goals show tooltip helpexplode

Curricular goals show tooltip help

  • Exploiting the potentialities of the TR environment to provide the students with the possibility of revising notions already encountered in the previous school level (reinforcement);
  • Supporting students in the passage from the procedural approach to the structural approach: such passage is fundamental to seize the equivalence between algebraic expressions, that is one of the main topics of their school level.

Content-epistemological goals show tooltip help

Structure of numerical and algebraic expressions.

Cognitive goals show tooltip help

From a cognitive viewpoint, the recourse to a novel representation (TR), different by the one usually adopted (SR), should break automatisms linked to the computation of numerical expressions, thus allowing the students to become conscious of the procedures they are activating. Such recourse should provide students also with control competences: they should become able to activate an instrument (the TR) that endowes them of a resolution scheme, whenever an expression or a sub-expression results difficult to be treated (meta-cognitive level).

Instrumental goals show tooltip help

Use of Aplusix to have the representation of expressions in different registers. Both treatments and conversions are embedded in the software.

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

The pedagogical plan uses Aplusix to support students in gaining a structural sense for numerical expression. This is assumed to be a basic prerequisite for approaching algebra.

 

Theoretical framework show tooltip help

The PP is organized according to the notion of Didactical cycle pointed out in the paradigm of Semiotic Mediation (Mariotti). A Didactical cycle is an iterative cycle of the following activities: activities with the artefact, individual production of signs, collective production of signs. Therefore, cycles are made of alternative phase of work within Aplusix and collective discussion.

Aplusix

Aplusix is an application for helping secondary school students to learn algebra. It lets students solve exercises and provides feedback: it verifies the correctness of the calculations and of the end of the exercises.

Aplusix has been designed to be integrated into the regular work of the class: it is close to the paper-pencil environment, it uses a very intuitive editor of algebraic expressions (in two dimensions); it contains 400 patterns of exercises organized by themes (numerical calculation, expansion, factorization, and solving equations, inequations and systems of equations) and by complexity. It also contains an exercise editor allowing teachers to build their own lists of exercises.

The application records all of the students’ actions. This allows the student and the teacher to observe them later with a “Replay system”. Teachers also have access to statistics concerning their classes indicating the amounts of exercises they worked on, amounts of well-solved exercises, amounts of incorrect calculations, and scores.

Aplusix runs on the local network of the school. An administration application allows managing classes, teachers and students (account creation, modification and suppression). Aplusix can also be installed on a personal computer in particular at home.

 

Different representations [Feature]

Aplusix allows the use of tree representations of algebraic expressions. 

There are four types of representation:

Usual representation: the “standard” (symbolic) representation of algebraic expressions.

Free tree representation: expressions can be edited as trees. In this mode, there is no constraint and no verification of the tree when it is edited (all sort of incorrect trees can be built).

Controlled tree representation: there are constraints and scaffolding when a tree is edited: internal nodes must be operators and leaves must be numbers or variables. The arity of the operators must be correct.

Mixed representation: each leaf of the tree is a usual representation of an expression. A usual representation can be expanded as a tree by clicking at the “+” button that appears when the mouse cursor is near a node; a tree, or a part of a tree, can be collapsed into a usual representation by clicking at the “-” button that appears when the mouse cursor is near a node.

In our PP we mainly refer to standard representation (SR) and tree representation in the free mode (TR).

 

 

Work plan show tooltip helpexplode

Setting show tooltip help

The activities are carried out in a classroom equipped with computers. Students will mainly work in pairs and are involved in collective discussion.

Time show tooltip help

18

Actors' roles show tooltip help

Students are requested to solve tasks and to partecipate in class discussion.

Teacher introduces the software, supervises students in their solving activities and coordinates class discussion.

 

What to do and how show tooltip help

The PP is made of four didactical cycles; each of them has specific didactical and research goals and is organized in different activities and phases. A detailed account is provided in the description of such activities and phases.

A pre-test and two post-tests complete the PP.

 

Process documentation show tooltip help

  • Students’ production both in Aplusix and in paper and pencil during the group work on the given tasks in classroom.
  • Reports individually produced by students as homework.
  • Initial and final tests.
  • Video and audio recordings.
  • Field notes.

We intend to use all these data in an integrated way, so to have relevant information on all the phases of the learning process from different perspectives.

 

Initial Test

Identity show tooltip helpexplode

Authors show tooltip help

L. Maffei, M.A. Mariotti, C. Sabena - UNISI

Subject domains show tooltip help

Topics show tooltip help

properties of operations
conversions of NL into SR

 

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

properties of operations
conversions

 

Description show tooltip help

The test aims at detecting the level of students in the topics we will address in the PP. In particular, students' competencies on conversions between natural language and standard representation is explicitly investigated.

 

Rationale show tooltip helpexplode

Providing a means to gain information on the students' competencies before their work with Aplusix and the PP.

 

Target show tooltip helpexplode

Rationale show tooltip helpexplode

The PP situates in the first year of the upper secondary school when, according to the Italian curricula, students approach algebra.

Population show tooltip helpexplode

School level show tooltip help

First Year of Secondary School

Age range show tooltip help

14-15

Population description show tooltip help

A standard mathematics classroom

Student prerequisites show tooltip help

Arithmetic computation (computation with integers, fractions).

Teacher prerequisites show tooltip help

Familiarity with Aplusix software.

Context show tooltip helpexplode

Physical context show tooltip help

Classroom equipped with computers, overhead projector.

Socio-cultural context show tooltip help

Standard classroom.

 

Goals show tooltip helpexplode

Content-epistemological goals show tooltip help

Investigating the students' competencies on conversions between natural language and standard representation.

 

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

Resources show tooltip helpexplode

Pre-test [Resource for students]

 Written sheet to be individually completed by the students.

Work plan show tooltip helpexplode

Setting show tooltip help

Classroom.

 

Time show tooltip help

30 minutes

Actors' roles show tooltip help

Students individually solve. 

Process documentation show tooltip help

Collection of the sheets.

 

Didactical cycle 1: Familiarization

Identity show tooltip helpexplode

Authors show tooltip help

L. Maffei, M.A. Mariotti, C. Sabena - UNISI

Subject domains show tooltip help

  • mathematics

Topics show tooltip help

  • arithmetics
  • equivalence
  • fraction

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

  • familiarization
  • SR
  • equivalence

Description show tooltip help

Students are introduced to the software enviroment through a partial familiarization. The first representation they meet is SR. The activity is organized around the notion of equivalence and the feedback provided in Aplusix when checking it in artithmetic context.

 

Rationale show tooltip helpexplode

This first session is the first encounter of the students with the tool, and therefore it involves a certain familiarisation with it.
Two features of the software are introduced: SR and the feedbacks provided by Aplusix when checking the equivalence between expressions.
We may speak of a partial familiarisation with the tool, in the sense that not all the features are introduced from the very beginning, but they are introduced as the PP goes on. 

 

Theoretical framework show tooltip help

Difficulties encountered by pupils in gaining competencies in algebraic calculation are well-known, and they have been addressed many times and from different point of views (Freudenthal, 1983, Tall & Thomas 1991, Kieran 1992).

The introduction of algebra requires in fact the development of a different way of thinking, which cannot be considered as a pure generalization of arithmetics (Mariotti & Cerulli, 2003). Differently from arithmetics, whose main goal is to do calculations so as to have a result, algebra offers an operational language for representing, analyzing and manipulating relations which contain both numbers and letters.

This diversity is explained by Sfard (1991) in terms of two different perspectives from which mathematical objects can be conceived: in a structural way (as objects) or in an operational way (as procedures or processes).

"There is a deep ontological gap between operational and structural conceptions [...] Seeing a mathematical entity as an object means being capable of referring to it as if it was a real thing-static structure, existing somewhere in space and time. It also means being able to recognize the idea 'at a glance' and to manipulate it as a whole, without going into details [...] In contrast, interpreting a notion as a process implies regarding it as a potential rather than actual entity, which comes into existence upon request in a sequence of actions. Thus, whereas the structural conception is static, instantaneous, and integrative, the operational is dynamic, sequential, and detailed." (Sfard, 1991).

We address the problem of gaining a structural approach to algebraic expressions, by adopting a cognitive-semiotic view-point, provided by the theory of Semiotic Mediation. The theory draws from a Vygotskijan paradigm, and frames in particular processes of mediation carried out by means of artefacts in classroom context. In this perspective, the use of artefacts for accomplishing a task leads the individual to the construction of personal meanings which are related to the actual use of the artefact. On the other hand, mathematical meanings may be related to the artefact and its use: in particular, in classroom context under the guidance of an expert (i.e. the teacher) the personal meanings, related to the actual use of an artefact, may be generated and evolve towards mathematical meanings. Because of this potentiality, we will say that an artefact may function as a semiotic mediator. The semiotic mediation function of an artefact is not automatically activated; in fact such a function can be exploited by the expert (in particular the teacher) who has the awareness of the semiotic potential of the artefact both in terms of mathematical meanings and in terms of personal meanings. Such evolution is fostered by the teacher's action, guiding the process of production and evolution of signs centred on the use of an artefact.

The evolution of signs, central in the perspective of semiotic mediation, can be described in terms of transformations of registers of representation (Duval, 1995). Furthermore, the notion of semiotic register could reveal a useful tool both in the analysis of the semiotic potential of the artefact, as well as in the design of tasks both within Aplusix and in paper and pencil environment. Specifically, Aplusix provides two different kinds of representation systems for algebraic expressions: the “standard” representation (SR), and the “tree” representation (TR). They can be considered two different registers of representation in the sense of Duval. According to Duval (1995), a "register of representation" is a semiotic system (i.e. a system of signs) that provides for specific possibilities of transformation of the representations. In our case, natural language allows transformations such as paraphrases and reformulation, and standard and tree representation allow transformations that express operations among numbers and symbols. Beyond this kind of transformations, that pertain to a specific register and are called "treatments", there are transformations that occur between different registers: they are called "conversions". Treatments and conversions are central in mathematics:%2

Target show tooltip helpexplode

Rationale show tooltip helpexplode

The PP situates in the first year of the upper secondary school when, according to the Italian curricula, students approach algebra.

Population show tooltip helpexplode

School level show tooltip help

Grade 9

Age range show tooltip help

14 - 15

Population description show tooltip help

 Students are at the beginning of the upper secondary school. 

Student prerequisites show tooltip help

Arithmetic computation (computation with integers, fractions).

Teacher prerequisites show tooltip help

Familiarity with Aplusix software.

Context show tooltip helpexplode

Physical context show tooltip help

Classroom equipped with computers, overhead projector.

Socio-cultural context show tooltip help

Standard classroom.

 

Goals show tooltip helpexplode

Curricular goals show tooltip help

  • Exploiting the potentialities of the TR environment to provide the students with the possibility of revising notions already encountered in the previous school level (reinforcement);
  • Supporting students in the passage from the procedural approach to the structural approach: such passage is fundamental to seize the equivalence between algebraic expressions, that is one of the main topics of their school level.

Content-epistemological goals show tooltip help

Equivalence between numerical expressions.

 

Cognitive goals show tooltip help

  • Interpretation of the signs linking two consecutive steps;
  • Making conjectures on these signs;
  • Understanding the meaning of the signs that check the equivalence in Aplusix.

 

Instrumental goals show tooltip help

Use of Aplusix to have the representation of expressions in different registers. Both treatments and conversions are embedded in the software.

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

Work plan show tooltip helpexplode

Setting show tooltip help

The activities are carried out in a classroom equipped with computers. Students will mainly work in pairs and are involved in collective discussion.

Time show tooltip help

A 2-hours session

Actors' roles show tooltip help

Students are requested to solve tasks and to partecipate in class discussion.

Teacher introduces the software, supervises students in their solving activities and coordinates class discussion.

 

Process documentation show tooltip help

 Video-recordings

Log files from Aplusix.

Phase 1: Introduction to Aplusix

Identity show tooltip helpexplode

Authors show tooltip help

L. Maffei, M.A. Mariotti, C. Sabena - UNISI

Subject domains show tooltip help

  • mathematics

Topics show tooltip help

  • arithmetics
  • equivalence
  • fraction

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

  • familiarization
  • SR
  • equivalence

Description show tooltip help

Teacher introduces students to Aplusix environment, that is set in SR. He shows:
- the use of the virtual keyboard for editing expressions;
- the possibility to create an empty box or a box containing the same expression when performing calculations.
No explanation on the possible feedbacks given by Aplusix are given by the teacher.

 

Rationale show tooltip helpexplode

Only few features of Aplusix are presented to the students from the beginning, so as to leave them to make sense of the signs used by Aplusix to link two consecutive steps.

 

Theoretical framework show tooltip help

This phase aims at starting the instrumental genesis (Rabardel).

 

Target show tooltip helpexplode

Rationale show tooltip helpexplode

The PP situates in the first year of the upper secondary school when, according to the Italian curricula, students approach algebra.

Population show tooltip helpexplode

School level show tooltip help

Grade 9

Age range show tooltip help

14 - 15

Population description show tooltip help

 Students are at the beginning of the upper secondary school. 

Student prerequisites show tooltip help

Arithmetic computation (computation with integers, fractions).

Teacher prerequisites show tooltip help

Familiarity with Aplusix software.

Context show tooltip helpexplode

Physical context show tooltip help

Classroom equipped with computers, overhead projector.

Socio-cultural context show tooltip help

Standard classroom.

 

Goals show tooltip helpexplode

Instrumental goals show tooltip help

 Introduction to the software and in particular to SR.

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

Aplusix

In this phase SR is presented to students.

 

Tool access show tooltip help

NIL

SR [Feature]

SR is the representation offered by the tool which is similar to the representation in paper and pencil.

 

Resources show tooltip helpexplode

DC1_Ph1 [Resource for students]
Resource contents show tooltip help
DC1_Ph1 [Resource for students]

Work plan show tooltip helpexplode

Setting show tooltip help

The activities are carried out in a classroom equipped with computers. Students will mainly work in pairs and are involved in collective discussion.

Time show tooltip help

30 minutes

Actors' roles show tooltip help

The teacher through an overhead projector introduces Aplusix to the students.

 

What to do and how show tooltip help

The teacher shows:

  • how to insert an empty line in order to carry out the solution of a task;
  • how to write fractions in SR.

 

Process documentation show tooltip help

Video recording

Log files from Aplusix

 

Phase 2: The equivalence sign

Identity show tooltip helpexplode

Authors show tooltip help

L. Maffei, M.A. Mariotti, C. Sabena - UNISI

Subject domains show tooltip help

  • mathematics

Topics show tooltip help

  • arithmetics
  • equivalence
  • fraction

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

  • familiarization
  • SR
  • equivalence

Description show tooltip help

Accomplishing simple calculations on rational numbers, students are requested to pay attention to Aplusix feedbacks. In particular the task is centred on the interpretation of the signs evolution during the computation.

Rationale show tooltip helpexplode

The immediate check on the equivalence is a specific feature of Aplusix and it is provided by different signs connecting consecutive steps. The interpretation of these sign may not be transparent to students and therefore we choose to devote the very first activity to it.

 

Theoretical framework show tooltip help

Instrumental genesis

Internalization

Target show tooltip helpexplode

Rationale show tooltip helpexplode

The PP situates in the first year of the upper secondary school when, according to the Italian curricula, students approach algebra.

Population show tooltip helpexplode

School level show tooltip help

Grade 9

Age range show tooltip help

14 - 15

Population description show tooltip help

 Students are at the beginning of the upper secondary school. 

Student prerequisites show tooltip help

Arithmetic computation (computation with integers, fractions).

Teacher prerequisites show tooltip help

Familiarity with Aplusix software.

Context show tooltip helpexplode

Physical context show tooltip help

Classroom equipped with computers, overhead projector.

Socio-cultural context show tooltip help

Standard classroom.

 

Goals show tooltip helpexplode

Curricular goals show tooltip help

 

Content-epistemological goals show tooltip help

Equivalence between numerical expressions.

 

Cognitive goals show tooltip help

  • Interpretation of the signs linking two consecutive steps;
  • Making conjectures on these signs.

Instrumental goals show tooltip help

Interpretation of the feedbacks of Aplusix.

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

Aplusix

 

Tool access show tooltip help

NIL

SR [Feature]

SR consists in the usual representation of expressions.

Equivalence signs [Feature]

The control on the equivalence is given by means of three different signs. Black lines point out that an expression is equivalent to the previous one; red crossed lines that you have not fulfilled the equivalence; blue crossed lines indicate that an expression is not well formed (i.e. you have opened a parenthesis but you have forgotten to close it).

 

Resources show tooltip helpexplode

DC1_ph2-Familiarization with Aplusix [Resource for students]
Familiarization worksheet - completed [Resource produced by students]

Students' observations and interpretations about the signs provided by Aplusix as feedback on the equivalence between steps.

 

DC1_Ph2-ExercisesSolved [Resource produced by students]

 The files contain students' productions in Aplusix.

Work plan show tooltip helpexplode

Setting show tooltip help

The activities are carried out in a classroom equipped with computers. Students will mainly work in pairs and are involved in collective discussion.

Time show tooltip help

1 hour

Actors' roles show tooltip help

Students in pairs solve the tasks. The teacher supervises giving mainly technical support.

 

What to do and how show tooltip help

 Students work both in Aplusix and in paper and pencil.

Process documentation show tooltip help

  • log files of Aplusix;
  • worksheet in paper and pencil;
  • audio and video recording;
  • field notes.

 

Phase 3: Discussion

Identity show tooltip helpexplode

Authors show tooltip help

L. Maffei, M.A. Mariotti, C. Sabena - UNISI

Subject domains show tooltip help

  • mathematics

Topics show tooltip help

  • arithmetics
  • equivalence
  • fraction

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

  • familiarization
  • SR
  • equivalence

Description show tooltip help

Starting from the interpretations given by the students to the equivalence sign, the teacher guides the discussion. If necessary, the teacher shows by means of an overhead projector the solution of an exercise which has been solved by the students in the previous phase.

Rationale show tooltip helpexplode

In this phase students discuss, under the guidance of the teacher, on the interpretation given to the different signs given by Aplusix when solving an exercise. The aim consists in attributing the right meaning to the three different signs.

 

Theoretical framework show tooltip help

 Semiotic mediation of the equivalence sign

Target show tooltip helpexplode

Rationale show tooltip helpexplode

The PP situates in the first year of the upper secondary school when, according to the Italian curricula, students approach algebra.

Population show tooltip helpexplode

School level show tooltip help

Grade 9

Age range show tooltip help

14 - 15

Population description show tooltip help

 Students are at the beginning of the upper secondary school. 

Student prerequisites show tooltip help

Arithmetic computation (computation with integers, fractions).

Teacher prerequisites show tooltip help

Familiarity with Aplusix software.

Context show tooltip helpexplode

Physical context show tooltip help

Classroom equipped with computers, overhead projector.

Socio-cultural context show tooltip help

Standard classroom.

 

Goals show tooltip helpexplode

Content-epistemological goals show tooltip help

Equivalence between numerical expressions.

 

Cognitive goals show tooltip help

Understanding the meaning of the signs that check the equivalence in Aplusix.

 

Instrumental goals show tooltip help

Sharing meanings in the classroom, with regard to the three feedback signs of Aplusix

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

Work plan show tooltip helpexplode

Setting show tooltip help

The activities are carried out in a classroom equipped with computers. Students will mainly work in pairs and are involved in collective discussion.

Time show tooltip help

30 minutes

Actors' roles show tooltip help

The teacher co-ordinates the discussion, so to guide the evolutions of  the mathematical meanings (the equivalence in particular) emerging from the activities

 

What to do and how show tooltip help

Classroom discussion using an overhead projector.

 

Process documentation show tooltip help

Video recording.

Didactical cycle 2: Structure of natural numbers

Identity show tooltip helpexplode

Authors show tooltip help

L. Maffei, M.A. Mariotti, C. Sabena - UNISI

Subject domains show tooltip help

  • arithmetic

Topics show tooltip help

  • additive and multiplicative structure of a natural number
  • introduction of a new representation (the TR) as visualization of the operations between numbers
  • introduction of parentheses and hierarchies

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

  • structure of a natural number
  • Aplusix
  • tree representation

Description show tooltip help

Aplusix will be used only for what regards the TR. The SR will be treated in the paper and pencil environment, the students are acquainted with. Conversions will then be performed between tree representations in Aplusix and the standard representations in paper and pencil.

Rationale show tooltip helpexplode

Essentially, two functionalities of the software are presented: the “tree representation” and its feedback. The conversion from the tree representation to the standard one is required for the moment in the paper and pencil environment, that is the usual one for the students.

The first activity is based on writing a natural number through the graph and a single operation. Drawing from it, the arithmetic operators are observed to be considereb by the software at least binary. Furthermore, the tree representation is explored to revise the commutative property of operations, corresponding to the commutation of leaves in the tree, and to have feedback on equivalent graphs.

The second activity is based on writing a natural number through the graph and two operations. Drawing  from  it, the arithmetic operators are observed to be  considered by the software in different way. The operators  + and * can also support three terms, whereas the operators - and / cannot support three terms. The software gives a positive feedback when one inserts three numbers in case of addition and multiplication, and a negative feedback in case one tries to put three sons of a single node in case of subtractions and divisions.

Theoretical framework show tooltip help

Difficulties encountered by pupils in gaining competencies in algebraic calculation are well-known, and they have been addressed many times and from different point of views (Freudenthal, 1983, Tall & Thomas 1991, Kieran 1992).

The introduction of algebra requires in fact the development of a different way of thinking, which cannot be considered as a pure generalization of arithmetics (Mariotti & Cerulli, 2003). Differently from arithmetics, whose main goal is to do calculations so as to have a result, algebra offers an operational language for representing, analyzing and manipulating relations which contain both numbers and letters.

This diversity is explained by Sfard (1991) in terms of two different perspectives from which mathematical objects can be conceived: in a structural way (as objects) or in an operational way (as procedures or processes).

"There is a deep ontological gap between operational and structural conceptions [...] Seeing a mathematical entity as an object means being capable of referring to it as if it was a real thing-static structure, existing somewhere in space and time. It also means being able to recognize the idea 'at a glance' and to manipulate it as a whole, without going into details [...] In contrast, interpreting a notion as a process implies regarding it as a potential rather than actual entity, which comes into existence upon request in a sequence of actions. Thus, whereas the structural conception is static, instantaneous, and integrative, the operational is dynamic, sequential, and detailed." (Sfard, 1991).

We address the problem of gaining a structural approach to algebraic expressions, by adopting a cognitive-semiotic view-point, provided by the theory of Semiotic Mediation. The theory draws from a Vygotskijan paradigm, and frames in particular processes of mediation carried out by means of artefacts in classroom context. In this perspective, the use of artefacts for accomplishing a task leads the individual to the construction of personal meanings which are related to the actual use of the artefact. On the other hand, mathematical meanings may be related to the artefact and its use: in particular, in classroom context under the guidance of an expert (i.e. the teacher) the personal meanings, related to the actual use of an artefact, may be generated and evolve towards mathematical meanings. Because of this potentiality, we will say that an artefact may function as a semiotic mediator. The semiotic mediation function of an artefact is not automatically activated; in fact such a function can be exploited by the expert (in particular the teacher) who has the awareness of the semiotic potential of the artefact both in terms of mathematical meanings and in terms of personal meanings. Such evolution is fostered by the teacher's action, guiding the process of production and evolution of signs centred on the use of an artefact.

The evolution of signs, central in the perspective of semiotic mediation, can be described in terms of transformations of registers of representation (Duval, 1995). Furthermore, the notion of semiotic register could reveal a useful tool both in the analysis of the semiotic potential of the artefact, as well as in the design of tasks both within Aplusix and in paper and pencil environment. Specifically, Aplusix provides two different kinds of representation systems for algebraic expressions: the “standard” representation (SR), and the “tree” representation (TR). They can be considered two different registers of representation in the sense of Duval. According to Duval (1995), a "register of representation" is a semiotic system (i.e. a system of signs) that provides for specific possibilities of transformation of the representations. In our case, natural language allows transformations such as paraphrases and reformulation, and standard and tree representation allow transformations that express operations among numbers and symbols. Beyond this kind of transformations, that pertain to a specific register and are called "treatments", there are transformations that occur between different registers: they are called "conversions". Treatments and conversions are central in mathematics:%2

Target show tooltip helpexplode

Rationale show tooltip helpexplode

The PP situates in the first year of the upper secondary school when, according to the Italian curricula, students approach algebra.

Population show tooltip helpexplode

School level show tooltip help

First Year of Secondary School

Population description show tooltip help

A standard mathematics classroom

Student prerequisites show tooltip help

Arithmetic computation (computation with integers, fractions).

Teacher prerequisites show tooltip help

Familiarity with Aplusix software.

basic functionalities of Aplusix software.

Context show tooltip helpexplode

Physical context show tooltip help

Classroom equipped with computers, overhead projector.

Socio-cultural context show tooltip help

Standard classroom.

 

Goals show tooltip helpexplode

Content-epistemological goals show tooltip help

Revising the decomposition of numbers through the arithmetic operations.

 

Instrumental goals show tooltip help

 Working with the TR in Aplusix

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

Resources show tooltip helpexplode

Activity_1.1 - “From the number to its expression with an operation” [Resource for students]

Phase 1: After having logged in the Aplusix DDA, students are requested to load the file Activity_1.exo.

Phase 2: A discussion in order to highlight some relevant issues coming from the work done within the software environment is guided by the teacher.

Phase 3: Students are requested to convert their findings within Aplusix in paper and pencil.

 

Resource contents show tooltip help

Contents of the file Aplusix_1.1.exo:

1 Write the number 4 as product of two natural numbers.

2 Write it as sum of two natural numbers.

3 Write it as difference of two natural numbers.

4 Write it as quotient of two natural numbers.

Work plan show tooltip helpexplode

Setting show tooltip help

laboratory equipped with computers

Time show tooltip help

2 hours

Actors' roles show tooltip help

Students work alone when interacting wiht the sofware and collaboratively when partecipating at discussions.

The teacher proposes the activities and partecipate to the discussion in order to collect all the pupils' interventions.

Activity 1: The commutative law

Identity show tooltip helpexplode

Authors show tooltip help

L. Maffei, M.A. Mariotti, C. Sabena - UNISI

Subject domains show tooltip help

  • arithmetic

Topics show tooltip help

  • decomposition of a natural number
  • commutative law
  • associative law

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

  • structure of a natural number
  • Aplusix
  • tree representation

Description show tooltip help

The activity focuses on the exploration of the applicability of the commutative law for operations with natural numbers starting from the decomposition of a number.

Firstly students are introduced to the tree representation provided by the software, and then they are asked to translate them into the SR.

The activity consists of three phases: the first one is carried out in the software environment, the second one is a discussion led by the teacher, and the third one is accomplished in paper and pencil.

Rationale show tooltip helpexplode

This activity aims at showing on the TR the commutativity of some arithmetical operations and the not-commutativity of others. The output given by the software, which gives a negative feedback in case one tries to commute the leaves of a tree in case of subtractions and divisions, is considered as a springboard for starting the discussion on the commutative law.

 

Theoretical framework show tooltip help

An outcome feedback is something showing whether or not the result, or a intermediate step, is correct (Balzer & al., 1989). This definition is suitable for describing the type of feedback given by Aplusix which doesn't orientate towards a correct solution in case of errors, but stimulates to reflect on the reason of the wrong sign.

 

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Population show tooltip helpexplode

School level show tooltip help

First Year of Secondary School

Age range show tooltip help

14-15

Population description show tooltip help

A standard mathematics classroom

Student prerequisites show tooltip help

Arithmetic computation (computation with integers, fractions).

Teacher prerequisites show tooltip help

basic functionalities of Aplusix software.

Context show tooltip helpexplode

Physical context show tooltip help

Classroom equipped with computers, overhead projector.

Socio-cultural context show tooltip help

Standard classroom.

 

Goals show tooltip helpexplode

Content-epistemological goals show tooltip help

 The commutative law and the arithmetic operations.

Cognitive goals show tooltip help

 Understanding the feedback given by the software

Instrumental goals show tooltip help

 Building a tree in Aplusix

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

Work plan show tooltip helpexplode

Setting show tooltip help

laboratory equipped with computers

Time show tooltip help

1 hour

Phase 1: Treatments in TR

Identity show tooltip helpexplode

Authors show tooltip help

L. Maffei, M.A. Mariotti, C. Sabena - UNISI

Subject domains show tooltip help

  • arithmetic

Topics show tooltip help

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

  • decomposition of a number
  • inverting leaves
  • commutative law
  • TR

Description show tooltip help

Students have to represent the decomposition of natural numbers building trees within Aplusix environment. They explore trees with different operators and in doing so they observe that the commutative law holds for addition and multiplication but not for subtraction and division.

The activity introduces the student to a new representation provided by the software, namely the tree representation. This is the first step of a gradual familiarization with the tool we aimed at.

 

Rationale show tooltip helpexplode

 The didactical goal of this phase consists in stimulating pupils to reflect on the meaning of decomposing a natural number. We hypothesize that the tree can be considered as a tool for thinking, in other words that this structure make new meanings emerge. Nodes, leaves, links between them put into evidence a hierarchical dependence among the elements constituting it.

Theoretical framework show tooltip help

According to the notion of 'structure sense' (Hoch and Dreyfus, 2006), the tree acts as a semiotic mediator of the structure of an expression.

 

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Population show tooltip helpexplode

School level show tooltip help

First Year of Secondary School

Age range show tooltip help

14-15

Population description show tooltip help

A standard mathematics classroom

Student prerequisites show tooltip help

Arithmetic computation (computation with integers, fractions).

Teacher prerequisites show tooltip help

basic functionalities of Aplusix software.

Context show tooltip helpexplode

Physical context show tooltip help

Classroom equipped with computers, overhead projector.

Goals show tooltip helpexplode

Instrumental goals show tooltip help

  1. building a tree;
  2. understanding of the feeback given by the software.

 

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

Resources show tooltip helpexplode

DC2_A1_Ph1 [Resource for students]
Exercise_1.1.alg [Resource produced by students]
Worksheet "Activity1_PHASE1" [Resource produced by students]

At the end of the activity, the worksheet contains the answer given by the students to a question based on the activity carried out within the software.

Work plan show tooltip helpexplode

Setting show tooltip help

laboratory equipped with computers

Process documentation show tooltip help

Video-recording
Collection of written sheets
Log files from Aplusix

Phase 2: Discussion

Identity show tooltip helpexplode

Authors show tooltip help

L. Maffei, M.A. Mariotti, C. Sabena - UNISI

Subject domains show tooltip help

Topics show tooltip help

  • commutativity of + and *
  • not-commutativity of - and /

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

  • mathematical discussion
  • commutative law

Description show tooltip help

 

Students are involved in a classroom discussion in which they compare and contrast the answers given in the previous phase. The teacher guides the discussion giving the possibility to each pupil to intervene. At the end of this phase it should be clear for all the students the applicability of the commutative law.

 

Rationale show tooltip helpexplode

Our aim consists in giving students the possibility to discover by means of their observations the commutativity of + and * and the not-commutativity of - and /.

Theoretical framework show tooltip help

According to the Vygotskian theory, interacting with a tool gives access to the knowledge incoroporated in it.

Target show tooltip helpexplode

Rationale show tooltip helpexplode

The PP situates in the first year of the upper secondary school when, according to the Italian curricula, students approach algebra.

Population show tooltip helpexplode

School level show tooltip help

First Year of Secondary School

Age range show tooltip help

14-15

Population description show tooltip help

A standard mathematics classroom

Student prerequisites show tooltip help

Arithmetic computation (computation with integers, fractions).

Teacher prerequisites show tooltip help

basic functionalities of Aplusix software.

Context show tooltip helpexplode

Physical context show tooltip help

Classroom equipped with computers, overhead projector.

Socio-cultural context show tooltip help

Standard classroom.

 

Goals show tooltip helpexplode

Content-epistemological goals show tooltip help

 The commutative law and the arithmetic operations.

Cognitive goals show tooltip help

Analysing the TR in Aplusix

 

Instrumental goals show tooltip help

Sharing the information on how working with TR in Aplusix

 

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

Work plan show tooltip helpexplode

Setting show tooltip help

laboratory equipped with computers

Time show tooltip help

From half an hour to an hour

Actors' roles show tooltip help

 The teacher co-ordinates the discussion, so to guide the evolutions of  the mathematical meanings emerging from the activities

Process documentation show tooltip help

Video-recording

 

Phase 3: Conversions TR-SR

Identity show tooltip helpexplode

Authors show tooltip help

L. Maffei, M.A. Mariotti, C. Sabena - UNISI

Subject domains show tooltip help

Topics show tooltip help

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

  • SR
  • paper and pencil

Description show tooltip help

Students are asked to convert the trees into standard expressions.This conversion is required for the moment in paper and pencil environment, that is the usual one for the students.

 

Rationale show tooltip helpexplode

According to Duval's assumption on the importance of the conversion between register in order to gain a deep understanding of a piece of knowledge, we request students to 'translate' from TR to SR. 

 

Theoretical framework show tooltip help

 

 

According to Duval's assumption on the importance of the conversion between register in order to gain a deep understanding of a piece of knowledge, we ask students to 'translate' from TR to SR. 

 

Target show tooltip helpexplode

Rationale show tooltip helpexplode

The PP situates in the first year of the upper secondary school when, according to the Italian curricula, students approach algebra.

Population show tooltip helpexplode

School level show tooltip help

First Year of Secondary School

Age range show tooltip help

14-15

Population description show tooltip help

A standard mathematics classroom

Student prerequisites show tooltip help

Students must have accomplished the first two phases of the Activity 1.

 

Teacher prerequisites show tooltip help

basic functionalities of Aplusix software.

Context show tooltip helpexplode

Physical context show tooltip help

Classroom equipped with computers, overhead projector.

Socio-cultural context show tooltip help

Standard classroom.

 

Goals show tooltip helpexplode

Content-epistemological goals show tooltip help

Expressing in standard representation an arithmetic relationship given in TR

Cognitive goals show tooltip help

Conversions from TR to SR.

 

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

Resources show tooltip helpexplode

Dc2_A1_Ph3-Structure of natural numbers [Resource for students]

Work plan show tooltip helpexplode

Process documentation show tooltip help

Video-recording
Collection of written sheets
Log files from Aplusix

Field notes

 

Phase 4: Report

Identity show tooltip helpexplode

Authors show tooltip help

L. Maffei, M.A. Mariotti, C. Sabena - UNISI

Subject domains show tooltip help

Topics show tooltip help

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

Description show tooltip help

Students are invited to write their impressions about the use of Aplusix, both in general, and in the specific tasks they have been involved in.

Rationale show tooltip helpexplode

The students are required to think about their activities with Aplusix, and to produce a written text.

Theoretical framework show tooltip help

In this production, the signs introduced in the previous lessons will be part of a semiotic chain that is related to the mathematical meanings that are the goals of the PP.

 

Target show tooltip helpexplode

Rationale show tooltip helpexplode

The PP situates in the first year of the upper secondary school when, according to the Italian curricula, students approach algebra.

Population show tooltip helpexplode

School level show tooltip help

First Year of Secondary School

Age range show tooltip help

14-15

Population description show tooltip help

A standard mathematics classroom

Student prerequisites show tooltip help

Arithmetic computation (computation with integers, fractions).

Teacher prerequisites show tooltip help

basic functionalities of Aplusix software.

Context show tooltip helpexplode

Physical context show tooltip help

 Classroom

Socio-cultural context show tooltip help

Standard classroom.

 

Goals show tooltip helpexplode

Cognitive goals show tooltip help

 Students are asked to reflect on their own experience with Aplusix, in the previous lessons

Instrumental goals show tooltip help

Students are asked to reflect and describe how the software helped them in solving the activities

 

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

In the first question, students have the possibility to freely write their impressions.
In the second question, they are asked to specify the contribution of Aplusix in the solving processes they have been involved in.

 

Theoretical framework show tooltip help

 The second question addresses both cognitive and meta-cognitive levels.

At cognitive level, we expect that students make references to the signs introduced by Aplusix in the previous tasks (artifact-signs), and show some other signs related to them, so to give hints of an evolution towards the mathematical meanings.
 

Resources show tooltip helpexplode

DC2_A1_Ph4-Report [Resource for students]

 Written sheet, to be individually filled.

Work plan show tooltip helpexplode

Setting show tooltip help

Either in the classroom, or as homework.

 

Actors' roles show tooltip help

Students individually answer the questions.

 

Process documentation show tooltip help

Collection of written sheets

Activity 2: The associative law

Identity show tooltip helpexplode

Authors show tooltip help

L. Maffei, M.A. Mariotti, C. Sabena - UNISI

Subject domains show tooltip help

Topics show tooltip help

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

Description show tooltip help

The activity focuses on the exploration of the binary nature of all the operations. This goal can be reached through the feedback and signs provided by Aplusix, which accepts trees with tree leaves in the case of + and * but not in the case of - and /.

This happens because the associative law does not hold for - and /.

The activity consists of three phases. Firstly students work within Aplusix environment, where they supposed to interpret the different behaviours of the software and to discover the need for sub-trees. Then they translate trees into expression and are guided to reflect on the conversion and more specifically on the role of the different signs intervening in each case. Finally a classroom discussion is set up  in order to set the point of the main topics of this activity: 

  • binary nature of all the operations;
  • associative law;
  • signs intervening (trees, sub-trees, parentheses, operators,...).

Rationale show tooltip helpexplode

Essentially, two functionalities of the software are presented: the “tree representation” and its feedback. The conversion from the tree representation to the standard one is required for the moment in the paper and pencil environment, that is the usual one for the students.

The first activity is based on writing a natural number through the graph and a single operation. Drawing from it, the arithmetic operators are observed to be considereb by the software at least binary. Furthermore, the tree representation is explored to revise the commutative property of operations, corresponding to the commutation of leaves in the tree, and to have feedback on equivalent graphs.

The second activity is based on writing a natural number through the graph and two operations. Drawing  from  it, the arithmetic operators are observed to be  considered by the software in different way. The operators  + and * can also support three terms, whereas the operators - and / cannot support three terms. The software gives a positive feedback when one inserts three numbers in case of addition and multiplication, and a negative feedback in case one tries to put three sons of a single node in case of subtractions and divisions.

Theoretical framework show tooltip help

Difficulties encountered by pupils in gaining competencies in algebraic calculation are well-known, and they have been addressed many times and from different point of views (Freudenthal, 1983, Tall & Thomas 1991, Kieran 1992).

The introduction of algebra requires in fact the development of a different way of thinking, which cannot be considered as a pure generalization of arithmetics (Mariotti & Cerulli, 2003). Differently from arithmetics, whose main goal is to do calculations so as to have a result, algebra offers an operational language for representing, analyzing and manipulating relations which contain both numbers and letters.

This diversity is explained by Sfard (1991) in terms of two different perspectives from which mathematical objects can be conceived: in a structural way (as objects) or in an operational way (as procedures or processes).

"There is a deep ontological gap between operational and structural conceptions [...] Seeing a mathematical entity as an object means being capable of referring to it as if it was a real thing-static structure, existing somewhere in space and time. It also means being able to recognize the idea 'at a glance' and to manipulate it as a whole, without going into details [...] In contrast, interpreting a notion as a process implies regarding it as a potential rather than actual entity, which comes into existence upon request in a sequence of actions. Thus, whereas the structural conception is static, instantaneous, and integrative, the operational is dynamic, sequential, and detailed." (Sfard, 1991).

We address the problem of gaining a structural approach to algebraic expressions, by adopting a cognitive-semiotic view-point, provided by the theory of Semiotic Mediation. The theory draws from a Vygotskijan paradigm, and frames in particular processes of mediation carried out by means of artefacts in classroom context. In this perspective, the use of artefacts for accomplishing a task leads the individual to the construction of personal meanings which are related to the actual use of the artefact. On the other hand, mathematical meanings may be related to the artefact and its use: in particular, in classroom context under the guidance of an expert (i.e. the teacher) the personal meanings, related to the actual use of an artefact, may be generated and evolve towards mathematical meanings. Because of this potentiality, we will say that an artefact may function as a semiotic mediator. The semiotic mediation function of an artefact is not automatically activated; in fact such a function can be exploited by the expert (in particular the teacher) who has the awareness of the semiotic potential of the artefact both in terms of mathematical meanings and in terms of personal meanings. Such evolution is fostered by the teacher's action, guiding the process of production and evolution of signs centred on the use of an artefact.

The evolution of signs, central in the perspective of semiotic mediation, can be described in terms of transformations of registers of representation (Duval, 1995). Furthermore, the notion of semiotic register could reveal a useful tool both in the analysis of the semiotic potential of the artefact, as well as in the design of tasks both within Aplusix and in paper and pencil environment. Specifically, Aplusix provides two different kinds of representation systems for algebraic expressions: the “standard” representation (SR), and the “tree” representation (TR). They can be considered two different registers of representation in the sense of Duval. According to Duval (1995), a "register of representation" is a semiotic system (i.e. a system of signs) that provides for specific possibilities of transformation of the representations. In our case, natural language allows transformations such as paraphrases and reformulation, and standard and tree representation allow transformations that express operations among numbers and symbols. Beyond this kind of transformations, that pertain to a specific register and are called "treatments", there are transformations that occur between different registers: they are called "conversions". Treatments and conversions are central in mathematics:%2

Target show tooltip helpexplode

Rationale show tooltip helpexplode

The PP situates in the first year of the upper secondary school when, according to the Italian curricula, students approach algebra.

Population show tooltip helpexplode

School level show tooltip help

First Year of Secondary School

Age range show tooltip help

14-15

Population description show tooltip help

A standard mathematics classroom

Student prerequisites show tooltip help

Arithmetic computation (computation with integers, fractions).

Teacher prerequisites show tooltip help

basic functionalities of Aplusix software.

Context show tooltip helpexplode

Physical context show tooltip help

Classroom equipped with computers, overhead projector.

Socio-cultural context show tooltip help

Standard classroom.

 

Goals show tooltip helpexplode

Content-epistemological goals show tooltip help

Associative law with respect to the different arithmetic operations

Use of parenthesis in numerical expressions

 

Cognitive goals show tooltip help

Treatments in TR in Aplusix

Conversions of sub-trees in Aplusix in expressions with parenthesis in SR.

Instrumental goals show tooltip help

  1. building a sub-tree

 

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

Work plan show tooltip helpexplode

Setting show tooltip help

laboratory equipped with computers

Time show tooltip help

1 hour

Process documentation show tooltip help

Video recordings
Log files from Aplusix
collection of written materials
field notes

 

Phase 1: Treatments in TR

Identity show tooltip helpexplode

Authors show tooltip help

L. Maffei, M.A. Mariotti, C. Sabena - UNISI

Subject domains show tooltip help

Topics show tooltip help

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

Description show tooltip help

Students are provided with an electronic sheet presenting decompositions of natural numbers with trees having three leaves under the same operator. Some of these decompositions are marked by the software as wrong. Students are asked to interpret this behaviour and to find out correct answers. 

In this way students are supposed to exploit a new component of the software, that is the sub-tree.

Rationale show tooltip helpexplode

Target show tooltip helpexplode

Rationale show tooltip helpexplode

The PP situates in the first year of the upper secondary school when, according to the Italian curricula, students approach algebra.

Population show tooltip helpexplode

School level show tooltip help

First Year of Secondary School

Age range show tooltip help

14-15

Population description show tooltip help

A standard mathematics classroom

Student prerequisites show tooltip help

Arithmetic computation (computation with integers, fractions).

Teacher prerequisites show tooltip help

basic functionalities of Aplusix software.

Context show tooltip helpexplode

Physical context show tooltip help

Classroom equipped with computers, overhead projector.

Socio-cultural context show tooltip help

Standard classroom.

 

Goals show tooltip helpexplode

Content-epistemological goals show tooltip help

Associative law with respect to the different arithmetic operations

 

Cognitive goals show tooltip help

Treatments in TR in Aplusix

 

Instrumental goals show tooltip help

 Treatments in TR in Aplusix (cognitive and instrumental goals are trictly linked, as a choice for the PP)

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

Resources show tooltip helpexplode

DC2_A2_Ph1-Structure of natural numbers [Resource for students]

Work plan show tooltip helpexplode

Process documentation show tooltip help

Video-recording
Collection of written sheets
Log files from Aplusix

Phase 2: Conversions TR-SR

Identity show tooltip helpexplode

Authors show tooltip help

L. Maffei, M.A. Mariotti, C. Sabena - UNISI

Subject domains show tooltip help

Topics show tooltip help

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

Description show tooltip help

The activity is carried out in paper and pencil. Students are asked to write the corresponding expressions of the trees they have worked with during the previous phase. They are also guided in their reflection on the conversion made; in particular the focus is centred the signs emerging in the activity.

 

Rationale show tooltip helpexplode

Target show tooltip helpexplode

Rationale show tooltip helpexplode

The PP situates in the first year of the upper secondary school when, according to the Italian curricula, students approach algebra.

Population show tooltip helpexplode

School level show tooltip help

First Year of Secondary School

Age range show tooltip help

14-15

Population description show tooltip help

A standard mathematics classroom

Student prerequisites show tooltip help

Arithmetic computation (computation with integers, fractions).

Teacher prerequisites show tooltip help

basic functionalities of Aplusix software.

Context show tooltip helpexplode

Physical context show tooltip help

Classroom equipped with computers, overhead projector.

Socio-cultural context show tooltip help

Standard classroom.

 

Goals show tooltip helpexplode

Content-epistemological goals show tooltip help

Use of parenthesis to express precendence in numerical expressions

 

Cognitive goals show tooltip help

Converting Aplusix sub-trees in expressions with parenthesis in SR

 

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

Resources show tooltip helpexplode

Dc2_A2_Ph2 [Resource for students]

Work plan show tooltip helpexplode

Process documentation show tooltip help

Video-recording
Collection of written sheets
Log files from Aplusix

 

Phase 3: Discussion

Identity show tooltip helpexplode

Authors show tooltip help

L. Maffei, M.A. Mariotti, C. Sabena - UNISI

Subject domains show tooltip help

Topics show tooltip help

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

Description show tooltip help

The students share their findings under the guidance of the teacher.
The teacher coordinates the students' interactions and leads the discussion toward the following goals:

  • reflecting on the binary nature of all the operations;
  • highlighting the validity of the associative law for + and * (and the correspondent not-validity in the case of - and / as putted into evidence by the software feedback);
  • the sub-tree component;
  • the need for parentheses in SR, corresponding to the sub-tree in TR.

Rationale show tooltip helpexplode

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Population show tooltip helpexplode

School level show tooltip help

First Year of Secondary School

Age range show tooltip help

14-15

Population description show tooltip help

A standard mathematics classroom

Student prerequisites show tooltip help

Arithmetic computation (computation with integers, fractions).

Teacher prerequisites show tooltip help

basic functionalities of Aplusix software.

Context show tooltip helpexplode

Physical context show tooltip help

Classroom

 

Socio-cultural context show tooltip help

Standard classroom.

 

Goals show tooltip helpexplode

Content-epistemological goals show tooltip help

Associative law with respect to the different arithmetic operations

Use of parenthesis in numerical expressions

 

Cognitive goals show tooltip help

Treatments in TR in Aplusix

Conversions of sub-trees in Aplusix in expressions with parenthesis in SR.

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

Work plan show tooltip helpexplode

Setting show tooltip help

 Classroom

Time show tooltip help

From half an hour to an hour

Actors' roles show tooltip help

 The teacher co-ordinates the discussion, so to guide the evolutions of  the mathematical meanings emerging from the activities

Process documentation show tooltip help

Video-recording

 

Didactical cycle 3: Syntactical skills

Identity show tooltip helpexplode

Authors show tooltip help

L. Maffei, M.A. Mariotti, C. Sabena - UNISI

Subject domains show tooltip help

Topics show tooltip help

  • Computations according to priority and hierarchical rules
  • introduction of more than operators in the same tree

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

Description show tooltip help

The cycle focuses on the syntactical skills in the artithmetical operations. Aplusix is used by the students in two different ways: translating an expression from SR to TR, and performing calculations directly on the trees.

Two activities constitute the cycle: Activity 1 and Activity 2.

Rationale show tooltip helpexplode

TR in Aplusix, by its nature, has a hiercharical structure. As a consequence, the tree forces students to compute according to specific priorities which are embedded in the tree structure itself.

Conversions aim at making students aware of the corresponding priorities rules in SR which may be expressed by specific signs, e.g. parentheses. Each environment allows specific possibilities of treatment.

Theoretical framework show tooltip help

Difficulties encountered by pupils in gaining competencies in algebraic calculation are well-known, and they have been addressed many times and from different point of views (Freudenthal, 1983, Tall & Thomas 1991, Kieran 1992).

The introduction of algebra requires in fact the development of a different way of thinking, which cannot be considered as a pure generalization of arithmetics (Mariotti & Cerulli, 2003). Differently from arithmetics, whose main goal is to do calculations so as to have a result, algebra offers an operational language for representing, analyzing and manipulating relations which contain both numbers and letters.

This diversity is explained by Sfard (1991) in terms of two different perspectives from which mathematical objects can be conceived: in a structural way (as objects) or in an operational way (as procedures or processes).

"There is a deep ontological gap between operational and structural conceptions [...] Seeing a mathematical entity as an object means being capable of referring to it as if it was a real thing-static structure, existing somewhere in space and time. It also means being able to recognize the idea 'at a glance' and to manipulate it as a whole, without going into details [...] In contrast, interpreting a notion as a process implies regarding it as a potential rather than actual entity, which comes into existence upon request in a sequence of actions. Thus, whereas the structural conception is static, instantaneous, and integrative, the operational is dynamic, sequential, and detailed." (Sfard, 1991).

We address the problem of gaining a structural approach to algebraic expressions, by adopting a cognitive-semiotic view-point, provided by the theory of Semiotic Mediation. The theory draws from a Vygotskijan paradigm, and frames in particular processes of mediation carried out by means of artefacts in classroom context. In this perspective, the use of artefacts for accomplishing a task leads the individual to the construction of personal meanings which are related to the actual use of the artefact. On the other hand, mathematical meanings may be related to the artefact and its use: in particular, in classroom context under the guidance of an expert (i.e. the teacher) the personal meanings, related to the actual use of an artefact, may be generated and evolve towards mathematical meanings. Because of this potentiality, we will say that an artefact may function as a semiotic mediator. The semiotic mediation function of an artefact is not automatically activated; in fact such a function can be exploited by the expert (in particular the teacher) who has the awareness of the semiotic potential of the artefact both in terms of mathematical meanings and in terms of personal meanings. Such evolution is fostered by the teacher's action, guiding the process of production and evolution of signs centred on the use of an artefact.

The evolution of signs, central in the perspective of semiotic mediation, can be described in terms of transformations of registers of representation (Duval, 1995). Furthermore, the notion of semiotic register could reveal a useful tool both in the analysis of the semiotic potential of the artefact, as well as in the design of tasks both within Aplusix and in paper and pencil environment. Specifically, Aplusix provides two different kinds of representation systems for algebraic expressions: the “standard” representation (SR), and the “tree” representation (TR). They can be considered two different registers of representation in the sense of Duval. According to Duval (1995), a "register of representation" is a semiotic system (i.e. a system of signs) that provides for specific possibilities of transformation of the representations. In our case, natural language allows transformations such as paraphrases and reformulation, and standard and tree representation allow transformations that express operations among numbers and symbols. Beyond this kind of transformations, that pertain to a specific register and are called "treatments", there are transformations that occur between different registers: they are called "conversions". Treatments and conversions are central in mathematics:%2

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Population show tooltip helpexplode

School level show tooltip help

First Year of Secondary School

Age range show tooltip help

14-15

Population description show tooltip help

A standard mathematics classroom

Context show tooltip helpexplode

Physical context show tooltip help

Classroom equipped with computers, overhead projector.

Socio-cultural context show tooltip help

Standard classroom.

 

Goals show tooltip helpexplode

Content-epistemological goals show tooltip help

Syntactical skills in numerical expressions

Cognitive goals show tooltip help

Conversions between TR and SR: observations of differences
Use of parenthesis and precendence rules in SR.
 

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

Work plan show tooltip helpexplode

Setting show tooltip help

The activities are carried out in a classroom equipped with computers. Students will mainly work in pairs and are involved in collective discussion.

Process documentation show tooltip help

Video recordings
Collection of written sheets
Field notes

 

Phase 1: Conversions SR - TR

Identity show tooltip helpexplode

Authors show tooltip help

L. Maffei, M.A. Mariotti, C. Sabena - UNISI

Subject domains show tooltip help

Topics show tooltip help

Language show tooltip help

English

Keywords show tooltip help

Description show tooltip help

Students are asked to convert standard expressions into trees. Since it is the first time they have to build trees (and not only interpreting them or working on a ready-made tree), the proposed expressions are quite simple from a syntactical point of view. 

Expressions are provided in paper and pencil and the students create the corresponding trees in Aplusix.

Rationale show tooltip helpexplode

Students in approaching how to build a tree learn to visualize the hierarchical structure of an expression. The sub-tree acquires the function of mediating the priority of computation.

 

Theoretical framework show tooltip help

structure sense

semiotic mediation

Target show tooltip helpexplode

Rationale show tooltip helpexplode

The PP situates in the first year of the upper secondary school when, according to the Italian curricula, students approach algebra.

Population show tooltip helpexplode

School level show tooltip help

First Year of Secondary School

Age range show tooltip help

14-15

Population description show tooltip help

A standard mathematics classroom

Student prerequisites show tooltip help

Arithmetic computation (computation with integers, fractions).

Teacher prerequisites show tooltip help

Familiarity with Aplusix software.

Context show tooltip helpexplode

Physical context show tooltip help

Classroom equipped with computers, overhead projector.

Socio-cultural context show tooltip help

Standard classroom.

 

Goals show tooltip helpexplode

Content-epistemological goals show tooltip help

Use of parenthesis and precendence rules in SR.

Cognitive goals show tooltip help

Conversions between TR and SR.
Use of parenthesis and precendence rules in SR.

 

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

Resources show tooltip helpexplode

Dc3_ph1-Syntactical aspects [Resource for students]

Work plan show tooltip helpexplode

Setting show tooltip help

The activities are carried out in a classroom equipped with computers. Students will mainly work in pairs and are involved in collective discussion.

Process documentation show tooltip help

Video-recording
Collection of written sheet

 

Phase 2: Computing on the tree

Identity show tooltip helpexplode

Authors show tooltip help

L. Maffei, M.A. Mariotti, C. Sabena - UNISI

Subject domains show tooltip help

Topics show tooltip help

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

Description show tooltip help

Trees are provided in Aplusix and the students have to calculate the result of each expression working directly on the tree.

 

Rationale show tooltip helpexplode

The activity is based on the mediation provided by the tree to the calculation procedure. In fact, the computation has to be done starting from the bottom of the tree and going upwards by substituting sub-trees by the result coming from the binary operation (expressed by the sub-tree).

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Population show tooltip helpexplode

Context show tooltip helpexplode

Goals show tooltip helpexplode

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

Resources show tooltip helpexplode

Dc3_Ph2-Syntactical aspects [Resource for students]

Work plan show tooltip helpexplode

Process documentation show tooltip help

Video-recording
Collection of written sheets
Log files from Aplusix

Phase 3: Comparision TR - SR

Identity show tooltip helpexplode

Authors show tooltip help

L. Maffei, M.A. Mariotti, C. Sabena - UNISI

Subject domains show tooltip help

Topics show tooltip help

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

Description show tooltip help

Rationale show tooltip helpexplode

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Population show tooltip helpexplode

Context show tooltip helpexplode

Goals show tooltip helpexplode

Cognitive goals show tooltip help

Reflections on conversions between TR and SR, and from paper and pencil and Aplusix environment

 

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

Resources show tooltip helpexplode

Dc3_Ph3-Syntactical aspects [Resource for students]

Work plan show tooltip helpexplode

Setting show tooltip help

The activities are carried out in a classroom equipped with computers. Students will mainly work in pairs and are involved in collective discussion.

Process documentation show tooltip help

Video-recording
Collection of written sheets

 

Phase 4: Discussion

Identity show tooltip helpexplode

Authors show tooltip help

L. Maffei, M.A. Mariotti, C. Sabena - UNISI

Subject domains show tooltip help

Topics show tooltip help

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

Description show tooltip help

Rationale show tooltip helpexplode

TR in Aplusix, by its nature, has a hiercharical structure. As a consequence, the tree forces students to compute according to specific priorities which are embedded in the tree structure itself.

Conversions aim at making students aware of the corresponding priorities rules in SR which may be expressed by specific signs, e.g. parentheses. Each environment allows specific possibilities of treatment.

Theoretical framework show tooltip help

Difficulties encountered by pupils in gaining competencies in algebraic calculation are well-known, and they have been addressed many times and from different point of views (Freudenthal, 1983, Tall & Thomas 1991, Kieran 1992).

The introduction of algebra requires in fact the development of a different way of thinking, which cannot be considered as a pure generalization of arithmetics (Mariotti & Cerulli, 2003). Differently from arithmetics, whose main goal is to do calculations so as to have a result, algebra offers an operational language for representing, analyzing and manipulating relations which contain both numbers and letters.

This diversity is explained by Sfard (1991) in terms of two different perspectives from which mathematical objects can be conceived: in a structural way (as objects) or in an operational way (as procedures or processes).

"There is a deep ontological gap between operational and structural conceptions [...] Seeing a mathematical entity as an object means being capable of referring to it as if it was a real thing-static structure, existing somewhere in space and time. It also means being able to recognize the idea 'at a glance' and to manipulate it as a whole, without going into details [...] In contrast, interpreting a notion as a process implies regarding it as a potential rather than actual entity, which comes into existence upon request in a sequence of actions. Thus, whereas the structural conception is static, instantaneous, and integrative, the operational is dynamic, sequential, and detailed." (Sfard, 1991).

We address the problem of gaining a structural approach to algebraic expressions, by adopting a cognitive-semiotic view-point, provided by the theory of Semiotic Mediation. The theory draws from a Vygotskijan paradigm, and frames in particular processes of mediation carried out by means of artefacts in classroom context. In this perspective, the use of artefacts for accomplishing a task leads the individual to the construction of personal meanings which are related to the actual use of the artefact. On the other hand, mathematical meanings may be related to the artefact and its use: in particular, in classroom context under the guidance of an expert (i.e. the teacher) the personal meanings, related to the actual use of an artefact, may be generated and evolve towards mathematical meanings. Because of this potentiality, we will say that an artefact may function as a semiotic mediator. The semiotic mediation function of an artefact is not automatically activated; in fact such a function can be exploited by the expert (in particular the teacher) who has the awareness of the semiotic potential of the artefact both in terms of mathematical meanings and in terms of personal meanings. Such evolution is fostered by the teacher's action, guiding the process of production and evolution of signs centred on the use of an artefact.

The evolution of signs, central in the perspective of semiotic mediation, can be described in terms of transformations of registers of representation (Duval, 1995). Furthermore, the notion of semiotic register could reveal a useful tool both in the analysis of the semiotic potential of the artefact, as well as in the design of tasks both within Aplusix and in paper and pencil environment. Specifically, Aplusix provides two different kinds of representation systems for algebraic expressions: the “standard” representation (SR), and the “tree” representation (TR). They can be considered two different registers of representation in the sense of Duval. According to Duval (1995), a "register of representation" is a semiotic system (i.e. a system of signs) that provides for specific possibilities of transformation of the representations. In our case, natural language allows transformations such as paraphrases and reformulation, and standard and tree representation allow transformations that express operations among numbers and symbols. Beyond this kind of transformations, that pertain to a specific register and are called "treatments", there are transformations that occur between different registers: they are called "conversions". Treatments and conversions are central in mathematics:%2

Target show tooltip helpexplode

Rationale show tooltip helpexplode

The PP situates in the first year of the upper secondary school when, according to the Italian curricula, students approach algebra.

Population show tooltip helpexplode

School level show tooltip help

First Year of Secondary School

Age range show tooltip help

14-15

Population description show tooltip help

A standard mathematics classroom

Student prerequisites show tooltip help

Arithmetic computation (computation with integers, fractions).

Teacher prerequisites show tooltip help

Familiarity with Aplusix software.

Context show tooltip helpexplode

Physical context show tooltip help

Classroom equipped with computers, overhead projector.

Socio-cultural context show tooltip help

Standard classroom.

 

Goals show tooltip helpexplode

Cognitive goals show tooltip help

 Reflections on conversions between TR and SR, and from paper and pencil and Aplusix environment

 

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

Work plan show tooltip helpexplode

Setting show tooltip help

 Classroom

Time show tooltip help

One hour

Actors' roles show tooltip help

 The teacher co-ordinates the discussion, so to guide the evolutions of  the mathematical meanings emerging from the activities

 

Process documentation show tooltip help

Video-recording

 

Didactical cycle 4: Towards a structural reading

Identity show tooltip helpexplode

Authors show tooltip help

L. Maffei, M.A. Mariotti, C. Sabena - UNISI

Subject domains show tooltip help

Topics show tooltip help

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

Description show tooltip help

Rationale show tooltip helpexplode

The natural language permeates all the activities in classroom. Of course it has been used in all the previous activities. Now, we make the students focus on it as a specific representation register (Duval).

 

Theoretical framework show tooltip help

Difficulties encountered by pupils in gaining competencies in algebraic calculation are well-known, and they have been addressed many times and from different point of views (Freudenthal, 1983, Tall & Thomas 1991, Kieran 1992).

The introduction of algebra requires in fact the development of a different way of thinking, which cannot be considered as a pure generalization of arithmetics (Mariotti & Cerulli, 2003). Differently from arithmetics, whose main goal is to do calculations so as to have a result, algebra offers an operational language for representing, analyzing and manipulating relations which contain both numbers and letters.

This diversity is explained by Sfard (1991) in terms of two different perspectives from which mathematical objects can be conceived: in a structural way (as objects) or in an operational way (as procedures or processes).

"There is a deep ontological gap between operational and structural conceptions [...] Seeing a mathematical entity as an object means being capable of referring to it as if it was a real thing-static structure, existing somewhere in space and time. It also means being able to recognize the idea 'at a glance' and to manipulate it as a whole, without going into details [...] In contrast, interpreting a notion as a process implies regarding it as a potential rather than actual entity, which comes into existence upon request in a sequence of actions. Thus, whereas the structural conception is static, instantaneous, and integrative, the operational is dynamic, sequential, and detailed." (Sfard, 1991).

We address the problem of gaining a structural approach to algebraic expressions, by adopting a cognitive-semiotic view-point, provided by the theory of Semiotic Mediation. The theory draws from a Vygotskijan paradigm, and frames in particular processes of mediation carried out by means of artefacts in classroom context. In this perspective, the use of artefacts for accomplishing a task leads the individual to the construction of personal meanings which are related to the actual use of the artefact. On the other hand, mathematical meanings may be related to the artefact and its use: in particular, in classroom context under the guidance of an expert (i.e. the teacher) the personal meanings, related to the actual use of an artefact, may be generated and evolve towards mathematical meanings. Because of this potentiality, we will say that an artefact may function as a semiotic mediator. The semiotic mediation function of an artefact is not automatically activated; in fact such a function can be exploited by the expert (in particular the teacher) who has the awareness of the semiotic potential of the artefact both in terms of mathematical meanings and in terms of personal meanings. Such evolution is fostered by the teacher's action, guiding the process of production and evolution of signs centred on the use of an artefact.

The evolution of signs, central in the perspective of semiotic mediation, can be described in terms of transformations of registers of representation (Duval, 1995). Furthermore, the notion of semiotic register could reveal a useful tool both in the analysis of the semiotic potential of the artefact, as well as in the design of tasks both within Aplusix and in paper and pencil environment. Specifically, Aplusix provides two different kinds of representation systems for algebraic expressions: the “standard” representation (SR), and the “tree” representation (TR). They can be considered two different registers of representation in the sense of Duval. According to Duval (1995), a "register of representation" is a semiotic system (i.e. a system of signs) that provides for specific possibilities of transformation of the representations. In our case, natural language allows transformations such as paraphrases and reformulation, and standard and tree representation allow transformations that express operations among numbers and symbols. Beyond this kind of transformations, that pertain to a specific register and are called "treatments", there are transformations that occur between different registers: they are called "conversions". Treatments and conversions are central in mathematics:%2

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Population show tooltip helpexplode

Context show tooltip helpexplode

Goals show tooltip helpexplode

Cognitive goals show tooltip help

Reflections on
- the different modalities in which and expression can be read in natural language;

- the relationships of this different readings with TR and SR

 

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

Work plan show tooltip helpexplode

Phase 1: Conversions SR- NL and TR- NL

Identity show tooltip helpexplode

Authors show tooltip help

L. Maffei, M.A. Mariotti, C. Sabena - UNISI

Subject domains show tooltip help

Topics show tooltip help

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

  • NL

Description show tooltip help

The activity is carried out in paper and pencil. Students are asked to translate in natural language expressions provided either in TR or  in SR.

Rationale show tooltip helpexplode

In this phase students explicitly focus on the role of natural language as a representation register. The tree is here conceived as a tool supporting the computational procedure, thanks to its hierarchical structure. According to the Vygotskian theory, the tree is meant here as an interiorized tool to structure computation.

Theoretical framework show tooltip help

Interiorization of a tool (Vygotskian theory).

 

Target show tooltip helpexplode

Rationale show tooltip helpexplode

The PP situates in the first year of the upper secondary school when, according to the Italian curricula, students approach algebra.

Population show tooltip helpexplode

School level show tooltip help

First Year of Secondary School

Age range show tooltip help

14-15

Population description show tooltip help

A standard mathematics classroom

Student prerequisites show tooltip help

Arithmetic computation (computation with integers, fractions).

Teacher prerequisites show tooltip help

Familiarity with Aplusix software.

Context show tooltip helpexplode

Physical context show tooltip help

Classroom equipped with computers, overhead projector.

Socio-cultural context show tooltip help

Standard classroom.

 

Goals show tooltip helpexplode

Cognitive goals show tooltip help

Conversions SR-NL and TR-NL.

 

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

This activitie introduces the natural language register as a specific element of reflections. It is the third register explicitly introduced (the other being TR and SR).

Students are required to convert from SR to NL and from TR to NL, and they are requested to find out more than one expression in NL. In fact, for each expressions different readings are possible. They can grouped into two categories:

- a "procedural reading": indicates the passages to be followed, in the left-right order;

- a "structural reading": indicates the operations from a more global point of view.

In our view, procedural reading is more coherent with SR, whereas structural reading with TR.

Theoretical framework show tooltip help

Since according to our hypothesis students will mainly propose procedural readings, the tree comes to have the role of semiotic mediator for a structural reading.

Resources show tooltip helpexplode

General description show tooltip help

 Activity to be solved in group work, in paper and pencil environment.

Dc4_Ph1-Lecture of expressions [Resource for students]

Work plan show tooltip helpexplode

Setting show tooltip help

The activities are carried out in a classroom equipped with computers. Students will mainly work in pairs and are involved in collective discussion.

Process documentation show tooltip help

 Video-recording

Collection of written sheets

Phase 2: Conversions NL - SR and NL-TR

Identity show tooltip helpexplode

Authors show tooltip help

L. Maffei, M.A. Mariotti, C. Sabena - UNISI

Subject domains show tooltip help

Topics show tooltip help

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

Description show tooltip help

This activity requires performing conversions from NL to SR and from NL and TR.

In NL, both the kinds of readings (structural and procedural) are proposed. Thus, this can be the first time in the PP taht students see a structural reading of expressions.

Rationale show tooltip helpexplode

Conversions (in the sense of Duval) are performed between TR and NL and between SR and NL.

 

Theoretical framework show tooltip help

Conversions between representation registers (Duval).

 

Target show tooltip helpexplode

Rationale show tooltip helpexplode

The PP situates in the first year of the upper secondary school when, according to the Italian curricula, students approach algebra.

Population show tooltip helpexplode

School level show tooltip help

First Year of Secondary School

Age range show tooltip help

14-15

Population description show tooltip help

A standard mathematics classroom

Student prerequisites show tooltip help

Arithmetic computation (computation with integers, fractions).

Teacher prerequisites show tooltip help

Familiarity with Aplusix software.

Context show tooltip helpexplode

Physical context show tooltip help

Classroom equipped with computers, overhead projector.

Socio-cultural context show tooltip help

Standard classroom.

 

Goals show tooltip helpexplode

Cognitive goals show tooltip help

Conversions NL-SR and NL-TR

 

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

Resources show tooltip helpexplode

Dc4_Ph2-Writing expressions [Resource for students]

Work plan show tooltip helpexplode

Process documentation show tooltip help

Video-recording.

Collection of written sheets

 

Phase 3: Discussion

Identity show tooltip helpexplode

Authors show tooltip help

L. Maffei, M.A. Mariotti, C. Sabena - UNISI

Subject domains show tooltip help

Topics show tooltip help

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

Description show tooltip help

Students' answers to the previous tasks are collectively discussed. The teacher guides the discussion giving the possibility to each pupil to read aloud some answers.

 

Rationale show tooltip helpexplode

Two different readings of the tree should emerge:

  1. 'top-down reading', which corresponds to a structural reading;
  2. 'bottom-up', which corresponds to an operational reading.

The 'top-down reading' is coherent with the usual reading of a written text (e.g. NL).

The 'bottom-up reading' is guided by the scheme of use of



Theoretical framework show tooltip help

Difficulties encountered by pupils in gaining competencies in algebraic calculation are well-known, and they have been addressed many times and from different point of views (Freudenthal, 1983, Tall & Thomas 1991, Kieran 1992).

The introduction of algebra requires in fact the development of a different way of thinking, which cannot be considered as a pure generalization of arithmetics (Mariotti & Cerulli, 2003). Differently from arithmetics, whose main goal is to do calculations so as to have a result, algebra offers an operational language for representing, analyzing and manipulating relations which contain both numbers and letters.

This diversity is explained by Sfard (1991) in terms of two different perspectives from which mathematical objects can be conceived: in a structural way (as objects) or in an operational way (as procedures or processes).

"There is a deep ontological gap between operational and structural conceptions [...] Seeing a mathematical entity as an object means being capable of referring to it as if it was a real thing-static structure, existing somewhere in space and time. It also means being able to recognize the idea 'at a glance' and to manipulate it as a whole, without going into details [...] In contrast, interpreting a notion as a process implies regarding it as a potential rather than actual entity, which comes into existence upon request in a sequence of actions. Thus, whereas the structural conception is static, instantaneous, and integrative, the operational is dynamic, sequential, and detailed." (Sfard, 1991).

We address the problem of gaining a structural approach to algebraic expressions, by adopting a cognitive-semiotic view-point, provided by the theory of Semiotic Mediation. The theory draws from a Vygotskijan paradigm, and frames in particular processes of mediation carried out by means of artefacts in classroom context. In this perspective, the use of artefacts for accomplishing a task leads the individual to the construction of personal meanings which are related to the actual use of the artefact. On the other hand, mathematical meanings may be related to the artefact and its use: in particular, in classroom context under the guidance of an expert (i.e. the teacher) the personal meanings, related to the actual use of an artefact, may be generated and evolve towards mathematical meanings. Because of this potentiality, we will say that an artefact may function as a semiotic mediator. The semiotic mediation function of an artefact is not automatically activated; in fact such a function can be exploited by the expert (in particular the teacher) who has the awareness of the semiotic potential of the artefact both in terms of mathematical meanings and in terms of personal meanings. Such evolution is fostered by the teacher's action, guiding the process of production and evolution of signs centred on the use of an artefact.

The evolution of signs, central in the perspective of semiotic mediation, can be described in terms of transformations of registers of representation (Duval, 1995). Furthermore, the notion of semiotic register could reveal a useful tool both in the analysis of the semiotic potential of the artefact, as well as in the design of tasks both within Aplusix and in paper and pencil environment. Specifically, Aplusix provides two different kinds of representation systems for algebraic expressions: the “standard” representation (SR), and the “tree” representation (TR). They can be considered two different registers of representation in the sense of Duval. According to Duval (1995), a "register of representation" is a semiotic system (i.e. a system of signs) that provides for specific possibilities of transformation of the representations. In our case, natural language allows transformations such as paraphrases and reformulation, and standard and tree representation allow transformations that express operations among numbers and symbols. Beyond this kind of transformations, that pertain to a specific register and are called "treatments", there are transformations that occur between different registers: they are called "conversions". Treatments and conversions are central in mathematics:%2

Target show tooltip helpexplode

Rationale show tooltip helpexplode

The PP situates in the first year of the upper secondary school when, according to the Italian curricula, students approach algebra.

Population show tooltip helpexplode

School level show tooltip help

First Year of Secondary School

Age range show tooltip help

14-15

Population description show tooltip help

A standard mathematics classroom

Student prerequisites show tooltip help

Arithmetic computation (computation with integers, fractions).

Teacher prerequisites show tooltip help

Familiarity with Aplusix software.

Context show tooltip helpexplode

Physical context show tooltip help

Classroom equipped with computers, overhead projector.

Socio-cultural context show tooltip help

Standard classroom.

 

Goals show tooltip helpexplode

Cognitive goals show tooltip help

Reflections on
- the different modalities in which and expression can be read in natural language;

- the relationships of this different readings with TR and SR

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

The discussion is situated after two phases in which natural language has been explicitly addressed, with conversions NL-TR and NL-SR in both directions.

In the discussion, the terms "procedural readings" and "structural readings" will be introduced. The difficulty and importance of a structural reading will be underlined by the teacher.

Work plan show tooltip helpexplode

Actors' roles show tooltip help

  The teacher co-ordinates the discussion, so to guide the evolutions of  the mathematical meanings emerging from the activities

Process documentation show tooltip help

Video-recording.

 

Exercises

Identity show tooltip helpexplode

Authors show tooltip help

L. Maffei, M.A. Mariotti, C. Sabena - UNISI

Subject domains show tooltip help

Topics show tooltip help

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

Description show tooltip help

Rationale show tooltip helpexplode

The natural language permeates all the activities in classroom. Of course it has been used in all the previous activities. Now, we make the students focus on it as a specific representation register (Duval).

 

Theoretical framework show tooltip help

Difficulties encountered by pupils in gaining competencies in algebraic calculation are well-known, and they have been addressed many times and from different point of views (Freudenthal, 1983, Tall & Thomas 1991, Kieran 1992).

The introduction of algebra requires in fact the development of a different way of thinking, which cannot be considered as a pure generalization of arithmetics (Mariotti & Cerulli, 2003). Differently from arithmetics, whose main goal is to do calculations so as to have a result, algebra offers an operational language for representing, analyzing and manipulating relations which contain both numbers and letters.

This diversity is explained by Sfard (1991) in terms of two different perspectives from which mathematical objects can be conceived: in a structural way (as objects) or in an operational way (as procedures or processes).

"There is a deep ontological gap between operational and structural conceptions [...] Seeing a mathematical entity as an object means being capable of referring to it as if it was a real thing-static structure, existing somewhere in space and time. It also means being able to recognize the idea 'at a glance' and to manipulate it as a whole, without going into details [...] In contrast, interpreting a notion as a process implies regarding it as a potential rather than actual entity, which comes into existence upon request in a sequence of actions. Thus, whereas the structural conception is static, instantaneous, and integrative, the operational is dynamic, sequential, and detailed." (Sfard, 1991).

We address the problem of gaining a structural approach to algebraic expressions, by adopting a cognitive-semiotic view-point, provided by the theory of Semiotic Mediation. The theory draws from a Vygotskijan paradigm, and frames in particular processes of mediation carried out by means of artefacts in classroom context. In this perspective, the use of artefacts for accomplishing a task leads the individual to the construction of personal meanings which are related to the actual use of the artefact. On the other hand, mathematical meanings may be related to the artefact and its use: in particular, in classroom context under the guidance of an expert (i.e. the teacher) the personal meanings, related to the actual use of an artefact, may be generated and evolve towards mathematical meanings. Because of this potentiality, we will say that an artefact may function as a semiotic mediator. The semiotic mediation function of an artefact is not automatically activated; in fact such a function can be exploited by the expert (in particular the teacher) who has the awareness of the semiotic potential of the artefact both in terms of mathematical meanings and in terms of personal meanings. Such evolution is fostered by the teacher's action, guiding the process of production and evolution of signs centred on the use of an artefact.

The evolution of signs, central in the perspective of semiotic mediation, can be described in terms of transformations of registers of representation (Duval, 1995). Furthermore, the notion of semiotic register could reveal a useful tool both in the analysis of the semiotic potential of the artefact, as well as in the design of tasks both within Aplusix and in paper and pencil environment. Specifically, Aplusix provides two different kinds of representation systems for algebraic expressions: the “standard” representation (SR), and the “tree” representation (TR). They can be considered two different registers of representation in the sense of Duval. According to Duval (1995), a "register of representation" is a semiotic system (i.e. a system of signs) that provides for specific possibilities of transformation of the representations. In our case, natural language allows transformations such as paraphrases and reformulation, and standard and tree representation allow transformations that express operations among numbers and symbols. Beyond this kind of transformations, that pertain to a specific register and are called "treatments", there are transformations that occur between different registers: they are called "conversions". Treatments and conversions are central in mathematics:%2

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Population show tooltip helpexplode

Context show tooltip helpexplode

Goals show tooltip helpexplode

Cognitive goals show tooltip help

Reinforcing

 

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

Resources show tooltip helpexplode

Homework [Resource for students]
Exercises post DC4 [Resource for students]
Resource contents show tooltip help

Work plan show tooltip helpexplode

Setting show tooltip help

 Homework

Actors' roles show tooltip help

Students solve the exercises as homework.

 

Process documentation show tooltip help

 No documentation

Final test

Identity show tooltip helpexplode

Authors show tooltip help

L. Maffei, M.A. Mariotti, C. Sabena - UNISI

Subject domains show tooltip help

  • mathematics

Topics show tooltip help

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

Conversions

Structural and procedural readings

 

Description show tooltip help

The test aims at gaining information on the competencies acquired by students during their activities with Aplusix following the PP.
In particular, the test focuses on:
conversions between SR and NL. and between TR and NL
structural and procedural readings
identification of structures in different registers

 

Rationale show tooltip helpexplode

The test aims at gaining information on the competences the students have acquired during the PP.

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Population show tooltip helpexplode

Context show tooltip helpexplode

Goals show tooltip helpexplode

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

Resources show tooltip helpexplode

Final test [Resource for students]

Work plan show tooltip helpexplode

Setting show tooltip help

Classroom

 

Time show tooltip help

1 hour

Actors' roles show tooltip help

 Students individually answer to the questions of the test

Process documentation show tooltip help

Collectin of written sheet.

 

Final report

Identity show tooltip helpexplode

Authors show tooltip help

L. Maffei, M.A. Mariotti, C. Sabena - UNISI

Subject domains show tooltip help

Topics show tooltip help

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

Description show tooltip help

Rationale show tooltip helpexplode

Theoretical framework show tooltip help

Difficulties encountered by pupils in gaining competencies in algebraic calculation are well-known, and they have been addressed many times and from different point of views (Freudenthal, 1983, Tall & Thomas 1991, Kieran 1992).

The introduction of algebra requires in fact the development of a different way of thinking, which cannot be considered as a pure generalization of arithmetics (Mariotti & Cerulli, 2003). Differently from arithmetics, whose main goal is to do calculations so as to have a result, algebra offers an operational language for representing, analyzing and manipulating relations which contain both numbers and letters.

This diversity is explained by Sfard (1991) in terms of two different perspectives from which mathematical objects can be conceived: in a structural way (as objects) or in an operational way (as procedures or processes).

"There is a deep ontological gap between operational and structural conceptions [...] Seeing a mathematical entity as an object means being capable of referring to it as if it was a real thing-static structure, existing somewhere in space and time. It also means being able to recognize the idea 'at a glance' and to manipulate it as a whole, without going into details [...] In contrast, interpreting a notion as a process implies regarding it as a potential rather than actual entity, which comes into existence upon request in a sequence of actions. Thus, whereas the structural conception is static, instantaneous, and integrative, the operational is dynamic, sequential, and detailed." (Sfard, 1991).

We address the problem of gaining a structural approach to algebraic expressions, by adopting a cognitive-semiotic view-point, provided by the theory of Semiotic Mediation. The theory draws from a Vygotskijan paradigm, and frames in particular processes of mediation carried out by means of artefacts in classroom context. In this perspective, the use of artefacts for accomplishing a task leads the individual to the construction of personal meanings which are related to the actual use of the artefact. On the other hand, mathematical meanings may be related to the artefact and its use: in particular, in classroom context under the guidance of an expert (i.e. the teacher) the personal meanings, related to the actual use of an artefact, may be generated and evolve towards mathematical meanings. Because of this potentiality, we will say that an artefact may function as a semiotic mediator. The semiotic mediation function of an artefact is not automatically activated; in fact such a function can be exploited by the expert (in particular the teacher) who has the awareness of the semiotic potential of the artefact both in terms of mathematical meanings and in terms of personal meanings. Such evolution is fostered by the teacher's action, guiding the process of production and evolution of signs centred on the use of an artefact.

The evolution of signs, central in the perspective of semiotic mediation, can be described in terms of transformations of registers of representation (Duval, 1995). Furthermore, the notion of semiotic register could reveal a useful tool both in the analysis of the semiotic potential of the artefact, as well as in the design of tasks both within Aplusix and in paper and pencil environment. Specifically, Aplusix provides two different kinds of representation systems for algebraic expressions: the “standard” representation (SR), and the “tree” representation (TR). They can be considered two different registers of representation in the sense of Duval. According to Duval (1995), a "register of representation" is a semiotic system (i.e. a system of signs) that provides for specific possibilities of transformation of the representations. In our case, natural language allows transformations such as paraphrases and reformulation, and standard and tree representation allow transformations that express operations among numbers and symbols. Beyond this kind of transformations, that pertain to a specific register and are called "treatments", there are transformations that occur between different registers: they are called "conversions". Treatments and conversions are central in mathematics:%2

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Population show tooltip helpexplode

Context show tooltip helpexplode

Goals show tooltip helpexplode

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

Resources show tooltip helpexplode

Final report [Resource for students]

Work plan show tooltip helpexplode

Setting show tooltip help

 Classroom

Time show tooltip help

1 hour

Actors' roles show tooltip help

Students answer open questions in paper and pencil environment.

 

Process documentation show tooltip help

 Collection of the written sheets.

Delayed Test

Identity show tooltip helpexplode

Authors show tooltip help

L. Maffei, M.A. Mariotti, C. Sabena - UNISI

Subject domains show tooltip help

Topics show tooltip help

  • arithmetic
  • algebra

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

Conversions

Strategies

 

Description show tooltip help

The test requires conversions SR. TR, and NL into the two others registers.
Students are also required to explain how they proceed to perform such conversions, and to suggest strategies for mates in difficulty

 

Rationale show tooltip helpexplode

The test is proposed after some weeks the activies with Aplusix have ended. It aims at monitoring students' abilities in conversions tasks, and above all their choices in such activities. In fact, students are left free to choose which conversion to perform before the other.

Furthermore, some questions address the meta-cognitive level.

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Population show tooltip helpexplode

Context show tooltip helpexplode

Goals show tooltip helpexplode

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

 

Resources show tooltip helpexplode

General description show tooltip help

Written sheet to be individually completed.

 

Delayed test [Resource for students]

Work plan show tooltip helpexplode

Setting show tooltip help

Classroom

 

Time show tooltip help

1 hour

Actors' roles show tooltip help

Students' individually answer the questions

 

Process documentation show tooltip help

Collection of sheets