Structural aspect of algebraic expressions (familiar)

Identity show tooltip helpexplode

Authors show tooltip help

J. Trgalova, H. Chaachoua - MeTAH, Grenoble (France)

Subject domains show tooltip help

  • mathematics

Topics show tooltip help

  • algebra

Language show tooltip help

English

Country show tooltip help

France

Keywords show tooltip help

  • numerical expression
  • algebraic expression
  • representation system
  • tree
  • Aplusix

Description show tooltip help

This scenario describes an innovative approach to the algebraic expressions through an introduction of a new representation system: a tree representation. It makes use of the Aplusix software, which allows to represent algebraic expressions either in symbolic algebraic language (usual representation), or as a tree. Both representations can be displayed on the screen at the same time as well.

The scenario is organized in two parts: (1) learning, (2) assessment.

The learning part is divided itself into 4 units: (1) introduction to the tree representation system (RT), (2) interplay between natural language representational system (RNL) and RT, (3) interplay between symbolic language representation (RSL) and RT, and (4) treatment tasks in RT.

The activities are designed to be done either in the traditional paper - pencil environment or in the computer-based environment provided by Aplusix software.

Rationale show tooltip helpexplode

A given algebraic expression can be considered from two points of view:

- as a computation programme: it indicates the sequence of computations to do in order to obtain the number sent by the programme when numerical values are assigned to letters involved in the expressions (procedural aspect);

- as an object whose form can be described and with which operations, such as simplifying, factoring, substituting into another expression, etc., can be done (structural aspect).

Considering the structural aspect of algebraic expressions is required for instance when determining of the equivalence of two expressions is at stake. However, in the teaching of algebra at the secondary level of French school system, the structural aspect of algebraic expressions is less “visible” for the students and is often suppressed by the procedural aspect.

A description of an algebraic expression in natural language leads to considering its structural aspect: for example the expression (2x + 1) (x - 4) is described as a product of a sum and a difference, a sum of the product of 2 by x and 1, and the difference of x and 4. The first word of the sentence gives the form of the expression. A tree representation of an algebraic expression presents a certain “proximity” to the natural language representation. It allows highlighting the form of an algebraic expression, which is given by the highest level assembler. In this scenario, the activities are designed to allow the students to explore the structure of algebraic expressions using first the natural language and the tree representation of algebraic expressions. These two representation systems will then be linked to the usual symbolic language representation.

The innovative aspect of this scenario lies in the use of Aplusix educational software, which provides a microworld where students can use both tree and symbolic language representations of algebraic expressions.

Theoretical framework show tooltip help

Semiotic registers of representation (R. Duval)

• Three semiotic registers: symbolic language (usual) representation, tree representation and natural language representation

• Hypothesis: the ability to represent a given mathematical concept in at least two registers and to perform conversions from one register to another is an indicator of conceptual understanding of the notion

• For each register:

–Internal rules

–Specific tasks (formation, treatment)

• Between registers:

–Conversion rules

–Congruence or not

–Specific tasks (conversion)

Anthropological theory of didactics (Y. Chevallard)

• Viability of the tasks in the Aplusix environment

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Population show tooltip helpexplode

School level show tooltip help

Grade 10

Age range show tooltip help

15-16 years

Population description show tooltip help

The student population that the scenario is designed to address does not include students with serious visual and/or motor impairments. This is because the tools used are unsuitable for meeting the specific needs of those students.

Student prerequisites show tooltip help

Meaning of letters in algebra (unknown, general number)

Algebraic expressions    

Familiarity with basic Aplusix functionalities

Teacher prerequisites show tooltip help

Mathematics knowledge required for teaching algebra at this school level

Familiarity with Aplusix software

Context show tooltip helpexplode

Physical context show tooltip help

 Computer lab with a computer-student ratio of 1:1 or 1:2

Institutional context show tooltip help

The contents addressed in the scenario are part of the French mathematics curriculum for the Grade 10 (classe de seconde).

Link to the mathematics curriculum

Goals show tooltip helpexplode

Curricular goals show tooltip help

The students will be able to:

  • identify the form of an algebraic expression given in either of the following representation systems: tree, natural language, symbolic language;
  • "translate" an algebraic expression given in one representation system into another one;
  • solve problems involving algebraic expressions given in either of the three representation systems.

Content-epistemological goals show tooltip help

Understand the structure of algebraic expressions

Distinguish between procedural and structural aspects of algebraic expressions

Cognitive goals show tooltip help

Articulate the three representation systems: tree, natural language and symbolic language

Social-affective goals show tooltip help

Develop the willingness and capacity to:

  • work collaboratively;
  • participate in class discussion;
  • question one's own work through critical evaluation of the work of others.

Instrumental goals show tooltip help

Use Aplusix to construct tree representations of algebraic expressions.

Use Aplusix to solve algebra problems involving algebraic expressions given in either of the three representation systems.

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

A given algebraic expression can be considered from two points of view:

- as a computation program: it indicates the sequence of computations to do in order to obtain the number sent by the program when numerical values are assigned to letters involved in the expressions (procedural aspect);

- as an object whose form can be described and with which operations, such as simplifying, factoring, substituting into another expression, etc., can be done (structural aspect).

Considering the structural aspect of algebraic expressions is required for instance when determining of the equivalence of two expressions is at stake. However, in the teaching of algebra at the secondary level of French school system, the structural aspect of algebraic expressions is less “visible” for the students and is often suppressed by the procedural aspect.

A description of an algebraic expression in natural language leads to considering its structural aspect: for example the expression (2x + 1)(x - 4) is described as a product of a sum and a difference, a sum of the product of 2 by x and 1, and the difference of x and 4. The first word of the sentence gives the form of the expression. A tree representation of an algebraic expression presents a certain “proximity” to the natural language representation. It allows highlighting the form of an algebraic expression, which is given by the highest level assembler. In this scenario, the activities are designed to allow the students to explore the structure of algebraic expressions using first the natural language and the tree representation of algebraic expressions. These two representation systems will then be linked to the usual symbolic representation.

The innovative aspect of this scenario lies in the use of Aplusix educational software, which provides a microworld where students can use both tree and symbolic language representations of algebraic expressions.

Theoretical framework show tooltip help

 

Semiotic registers of representation (R. Duval)

• Three semiotic registers: symbolic language (usual) representation, tree representation and natural language representation

• Hypothesis: the ability to represent a given mathematical concept in at least two registers and to perform conversions from one register to another is an indicator of conceptual understanding of the notion

• For each register:

    –Internal rules

    –Specific tasks (formation, treatment)

• Between registers:

    –Conversion rules

    –Congruence or not

    –Specific tasks (conversion)

Anthropological theory of didactics (Y. Chevallard)

• Viability of the tasks in the Aplusix environment

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.

Tree representation mode [Feature]

Aplusix-Tree allows the use of tree representations of algebraic expressions. It also includes two new types of exercises: “Transform a usual (symbolic) representation into a tree representation” and “Transform a tree representation into a usual representation”.

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.

 

Work plan show tooltip helpexplode

Setting show tooltip help

This scenario alternates sessions in a usual classroom, in a classroom with a videoprojector and in a computer lab, under the active supervision of the teacher. The students work individually, in pairs or collectively.

 

Time show tooltip help

5,5 hours

Actors' roles show tooltip help

STUDENTS

•solving tasks

•participating in class discussion

TEACHER

•group supervision

•cognitive structuring

•leading and moderating class discussion

RESEARCHERS (discretionary)

•observing

What to do and how show tooltip help

 

This scenario has been refined into two distinct scenarios, Learning and Assessment, which are designed to be dealt with in strict order. Activities are to be carried out individually, in pairs or collectively, depending on the specific situation.

Pretest

Identity show tooltip helpexplode

Authors show tooltip help

J. Trgalova, H. Chaachoua - MeTAH, Grenoble (France)

Subject domains show tooltip help

  • mathematics

Topics show tooltip help

  • algebra

Language show tooltip help

English

Country show tooltip help

France

Keywords show tooltip help

  • numerical expression
  • algebraic expression
  • Aplusix

Description show tooltip help

The goal of the scenario being helping students understand the structure of algebraic expressions they deal with usually in symbolic language register, the pretest aims at revealing their difficulties in mastering the tasks within this register (calculate, develop and simplify, factor). We suppose that the interaction between natural language and symbolic language registers will also contribute to revealing students' difficulties related to the structural aspect of algebraic expressions. The table below summarizes goals, types of activities, environments and duration of activities proposed in the pretest:

Goals

Description

Environment

Time

Treatment in RSL

Calculate

Develop and simplify

Factor

Aplusix

50 min

Conversion RNL - RSL

Communication games

Paper/pencil

30 min

Rationale show tooltip helpexplode

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

The aim of the pretest is to identify students'difficulties in algbera, mainly those that are related to the structural aspect of algebraic expressions. 

A series of exercises are proposed to the students in the first phase of the pretest where they are asked to perform numerical computation and transformations of algebraic expressions written in a usual, symbolic language. The exercises are to be done with Aplusix in test mode, i.e., the students do not benefit of any feedback from the system.

In the second phase, we focus on how students read algebraic expressions written in symbols: do they pay attention to parentheses? do they read from left to right, like a text? do they respect the structure of the expressions? The activities proposed are communication games between pairs of students.

Aplusix

Tool access show tooltip help

NIL

Resources show tooltip helpexplode

General description show tooltip help

The ressources are of two kinds:

  • Aplusix exercises (numerical computation, expand and simplify, factor)
  • printable worksheet with paper-pencil activities for students
Activity 1 : numerical calculation [Resource for students]

aplusix file 

 

Activity 2: Expand and simplify [Resource for students]

 aplusix file

Activity 3: Factor [Resource for students]

aplusix file

 

Activity 4: Conversion RNL-RSL [Resource for students]

communication paper-pencil activity 

Activity 5: Conversion RNL-RSL [Resource for students]

communication paper-pencil activity 

Work plan show tooltip helpexplode

Setting show tooltip help

In the first 3 activities, the students work individully with Aplusix software in test mode.

In the last 2 activities, the students work in pairs in paper-pencil environment.

 

Time show tooltip help

80 minutes

Actors' roles show tooltip help

The first 3 activities are done by students with Aplusix. The students solve exercises in the proposed order.
The teacher is not supposed to intervene, only in the case of technical problem.

The last 2 activities are done by students in paper-pencil environment. They work in pairs. The teacher is supposed to manage the classroom discussion at the end of the session.

Process documentation show tooltip help

The students' actions are recorded by Aplusix. It will then be possible to reply students' solving processes.

 

Learning

Identity show tooltip helpexplode

Authors show tooltip help

J. Trgalova, H. Chaachoua - MeTAH, Grenoble (France)

Subject domains show tooltip help

  • mathematics

Topics show tooltip help

  • algebra

Language show tooltip help

English

Country show tooltip help

France

Keywords show tooltip help

  • numerical expression
  • algebraic expression
  • representation system
  • tree
  • Aplusix

Description show tooltip help

This scenario describes an innovative approach to algebraic expressions through an introduction of a new representation system: a tree representation. It makes use of the Aplusix software, which allows to represent algebraic expressions either in symbolic algebraic language (usual representation), or as a tree. Both representations can be displayed on the screen at the same time as well.

The learning scenario is divided into 4 units: (1) introduction to the tree representation system (RT), (2) interplay between natural language representational system (RNL) and RT, (3) interplay between symbolic language representation (RSL) and RT, and (4) treatment tasks in RT.

The activities are designed to be done either in the traditional paper - pencil environment or in the computer-based environment provided by Aplusix software.

Rationale show tooltip helpexplode

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Theoretical framework show tooltip help

Population show tooltip helpexplode

School level show tooltip help

Grade 10

Age range show tooltip help

15-16 years

Population description show tooltip help

The student population that the scenario is designed to address does not include students with serious visual and/or motor impairments. This is because the tools used are unsuitable for meeting the specific needs of those students.

Student prerequisites show tooltip help

Meaning of letters in algebra (unknown, general number)

Algebraic expressions    

Familiarity with basic Aplusix functionalities

Teacher prerequisites show tooltip help

Mathematics knowledge required for teaching algebra at this school level

Familiarity with Aplusix software

Context show tooltip helpexplode

Physical context show tooltip help

 Computer lab with a computer-student ratio of 1:1 or 1:2

Institutional context show tooltip help

The contents addressed in the scenario are part of the French mathematics curriculum for the Grade 10 (classe de seconde).

Link to the mathematics curriculum

Socio-cultural context show tooltip help

Goals show tooltip helpexplode

Curricular goals show tooltip help

The students will be able to:

  • identify the form of an algebraic expression given in either of the following representation systems: tree, natural language, symbolic language;
  • "translate" an algebraic expression given in one representation system into another one;
  • solve problems involving algebraic expressions given in either of the three representation systems.

Content-epistemological goals show tooltip help

Understand the structure of algebraic expressions

Distinguish between procedural and structural aspects of algebraic expressions

Cognitive goals show tooltip help

Articulate the three representation systems: tree, natural language and symbolic language

Social-affective goals show tooltip help

Develop the willingness and capacity to:

  • work collaboratively;
  • participate in class discussion;
  • question one's own work through critical evaluation of the work of others.

Instrumental goals show tooltip help

Use Aplusix to construct tree representations of algebraic expressions.

Use Aplusix to solve algebra problems involving algebraic expressions given in either of the three representation systems.

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

A given algebraic expression can be considered from two points of view:

- as a computation program: it indicates the sequence of computations to do in order to obtain the number sent by the program when numerical values are assigned to letters involved in the expressions (procedural aspect);

- as an object whose form can be described and with which operations, such as simplifying, factoring, substituting into another expression, etc., can be done (structural aspect).

Considering the structural aspect of algebraic expressions is required for instance when determining of the equivalence of two expressions is at stake. However, in the teaching of algebra at the secondary level of French school system, the structural aspect of algebraic expressions is less “visible” for the students and is often suppressed by the procedural aspect.

A description of an algebraic expression in natural language leads to considering its structural aspect: for example the expression (2x + 1)(x - 4) is described as a product of a sum and a difference, a sum of the product of 2 by x and 1, and the difference of x and 4. The first word of the sentence gives the form of the expression. A tree representation of an algebraic expression presents a certain “proximity” to the natural language representation. It allows highlighting the form of an algebraic expression, which is given by the highest level assembler. In this scenario, the activities are designed to allow the students to explore the structure of algebraic expressions using first the natural language and the tree representation of algebraic expressions. These two representation systems will then be linked to the usual symbolic representation.

The innovative aspect of this scenario lies in the use of Aplusix educational software, which provides a microworld where students can use both tree and symbolic language representations of algebraic expressions.

Theoretical framework show tooltip help

 

Semiotic registers of representation (R. Duval)

• Three semiotic registers: symbolic language (usual) representation, tree representation and natural language representation

• Hypothesis: the ability to represent a given mathematical concept in at least two registers and to perform conversions from one register to another is an indicator of conceptual understanding of the notion

• For each register:

    –Internal rules

    –Specific tasks (formation, treatment)

• Between registers:

    –Conversion rules

    –Congruence or not

    –Specific tasks (conversion)

Anthropological theory of didactics (Y. Chevallard)

• Viability of the tasks in the Aplusix environment

Aplusix

Tool access show tooltip help

NIL

Work plan show tooltip helpexplode

Setting show tooltip help

This scenario alternates sessions in a usual classroom, in a classroom with a videoprojector and in a computer lab, under the active supervision of the teacher. The students work individually, in pairs or collectively.

 

Time show tooltip help

4 hours

Actors' roles show tooltip help

STUDENTS

•solving tasks

•participating in class discussion

TEACHER

•group supervision

•cognitive structuring

•leading and moderating class discussion

RESEARCHERS (discretionary)

•observing

What to do and how show tooltip help

 

This scenario is refined into 4 elementary scenarios:

• introduction to tree representation of algebraic expressions

• interplay between natural language and tree representations

• interplay between symbolic language and tree representations

• treatment tasks with tree representation

These scenarios are designed to be dealt with in this order.

Introduction to tree representation

Identity show tooltip helpexplode

Authors show tooltip help

J. Trgalova, H. Chaachoua - MeTAH, Grenoble (France)

Subject domains show tooltip help

  • mathematics

Topics show tooltip help

  • algebra

Language show tooltip help

English

Country show tooltip help

France

Keywords show tooltip help

  • numerical expression
  • algebraic expression
  • representation system
  • tree
  • Aplusix

Description show tooltip help

 Introduction to tree representation of algebraic expressions with Aplusix

Rationale show tooltip helpexplode

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Population show tooltip helpexplode

School level show tooltip help

Grade 10

Age range show tooltip help

15-16 years

Population description show tooltip help

The student population that the scenario is designed to address does not include students with serious visual and/or motor impairments. This is because the tools used are unsuitable for meeting the specific needs of those students.

Student prerequisites show tooltip help

Meaning of letters in algebra (unknown, general number)

Algebraic expressions    

Familiarity with basic Aplusix functionalities

Teacher prerequisites show tooltip help

Mathematics knowledge required for teaching algebra at this school level

Familiarity with Aplusix software

Context show tooltip helpexplode

Physical context show tooltip help

 Computer lab with a computer-student ratio of 1:1 or 1:2

Institutional context show tooltip help

The contents addressed in the scenario are part of the French mathematics curriculum for the Grade 10 (classe de seconde).

Link to the mathematics curriculum

Socio-cultural context show tooltip help

Goals show tooltip helpexplode

Curricular goals show tooltip help

The students will be able to:

  • identify the form of an algebraic expression given in either of the following representation systems: tree, natural language, symbolic language;
  • "translate" an algebraic expression given in one representation system into another one;
  • solve problems involving algebraic expressions given in either of the three representation systems.

Content-epistemological goals show tooltip help

Learn how to represent an expression (numerical or algebraic) as a tree.

 

Cognitive goals show tooltip help

Articulate the three representation systems: tree, natural language and symbolic language

Social-affective goals show tooltip help

Develop the willingness and capacity to:

  • work collaboratively;
  • participate in class discussion;
  • question one's own work through critical evaluation of the work of others.

Instrumental goals show tooltip help

Use Aplusix to construct tree representations of algebraic expressions.

Use Aplusix to solve algebra problems involving algebraic expressions given in either of the three representation systems.

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

Aplusix

 See general description of Aplusix software at the top scenario level.

Tool access show tooltip help

NIL

mixed representation mode [Feature]

In a mixed representation mode, 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. This mode seems to be particularly adapted to the introduction of the tree representation: the students observe how a given expression is progressively developed into a tree.

Resources show tooltip helpexplode

General description show tooltip help

Usage scenario for the teacher. 

Printable worksheet with instructions for the teacher how to carry out the activity.

 

Usage scenrio for the teacher [Resource for students]

Ressource for the teacher

 

Work plan show tooltip helpexplode

Setting show tooltip help

In an ordinary classroom equipped by a computer with Aplusix-tree software and a videoprojector, the teacher manipulates the computer and manages class interactions.

 

Time show tooltip help

55 minutes

Actors' roles show tooltip help

 

Teacher:

- manipulating the computer

- questioning the students

- managing classroom discussions

- institutionalising

Students:

- participating to classroom discussions

Researchers:

- observing

What to do and how show tooltip help

 The teacher follows the instructions from the usage scenario.

Conversion RNL-RT

Identity show tooltip helpexplode

Authors show tooltip help

J. Trgalova, H. Chaachoua - MeTAH, Grenoble (France)

Subject domains show tooltip help

  • mathematics

Topics show tooltip help

  • algebra

Language show tooltip help

English

Country show tooltip help

France

Keywords show tooltip help

  • numerical expression
  • algebraic expression
  • representation system
  • tree
  • Aplusix

Description show tooltip help

 Conversion tasks between natural language and tree representation

Rationale show tooltip helpexplode

A given algebraic expression can be considered from two points of view:

- as a computation programme: it indicates the sequence of computations to do in order to obtain the number sent by the programme when numerical values are assigned to letters involved in the expressions (procedural aspect);

- as an object whose form can be described and with which operations, such as simplifying, factoring, substituting into another expression, etc., can be done (structural aspect).

Considering the structural aspect of algebraic expressions is required for instance when determining of the equivalence of two expressions is at stake. However, in the teaching of algebra at the secondary level of French school system, the structural aspect of algebraic expressions is less “visible” for the students and is often suppressed by the procedural aspect.

A description of an algebraic expression in natural language leads to considering its structural aspect: for example the expression (2x + 1) (x - 4) is described as a product of a sum and a difference, a sum of the product of 2 by x and 1, and the difference of x and 4. The first word of the sentence gives the form of the expression. A tree representation of an algebraic expression presents a certain “proximity” to the natural language representation. It allows highlighting the form of an algebraic expression, which is given by the highest level assembler. In this scenario, the activities are designed to allow the students to explore the structure of algebraic expressions using first the natural language and the tree representation of algebraic expressions. These two representation systems will then be linked to the usual symbolic language representation.

The innovative aspect of this scenario lies in the use of Aplusix educational software, which provides a microworld where students can use both tree and symbolic language representations of algebraic expressions.

Theoretical framework show tooltip help

Semiotic registers of representation (R. Duval)

• Three semiotic registers: symbolic language (usual) representation, tree representation and natural language representation

• Hypothesis: the ability to represent a given mathematical concept in at least two registers and to perform conversions from one register to another is an indicator of conceptual understanding of the notion

• For each register:

–Internal rules

–Specific tasks (formation, treatment)

• Between registers:

–Conversion rules

–Congruence or not

–Specific tasks (conversion)

Anthropological theory of didactics (Y. Chevallard)

• Viability of the tasks in the Aplusix environment

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Theoretical framework show tooltip help

Population show tooltip helpexplode

School level show tooltip help

Grade 10

Age range show tooltip help

15-16 years

Population description show tooltip help

The student population that the scenario is designed to address does not include students with serious visual and/or motor impairments. This is because the tools used are unsuitable for meeting the specific needs of those students.

Student prerequisites show tooltip help

Meaning of letters in algebra (unknown, general number)

Algebraic expressions    

Familiarity with basic Aplusix functionalities

Teacher prerequisites show tooltip help

Mathematics knowledge required for teaching algebra at this school level

Familiarity with Aplusix software

Context show tooltip helpexplode

Physical context show tooltip help

 Computer lab with a computer-student ratio of 1:1 or 1:2

Institutional context show tooltip help

The contents addressed in the scenario are part of the French mathematics curriculum for the Grade 10 (classe de seconde).

Link to the mathematics curriculum

Socio-cultural context show tooltip help

Goals show tooltip helpexplode

Curricular goals show tooltip help

The students will be able to:

  • identify the form of an algebraic expression given in either of the following representation systems: tree, natural language, symbolic language;
  • "translate" an algebraic expression given in one representation system into another one;
  • solve problems involving algebraic expressions given in either of the three representation systems.

Content-epistemological goals show tooltip help

 

• Learn how to represent an expression described in natural language as a tree.

• Learn how to “read” an expression represented by a tree.

Cognitive goals show tooltip help

• Articulate natural language and tree representations of algebraic expressions.

• Get an idea of a semantic congruence between these two registers.

 

Social-affective goals show tooltip help

Develop the willingness and capacity to:

  • work collaboratively;
  • participate in class discussion;
  • question one's own work through critical evaluation of the work of others.

Instrumental goals show tooltip help

Use Aplusix to construct tree representations of algebraic expressions.

Use Aplusix to solve algebra problems involving algebraic expressions given in either of the three representation systems.

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

Aplusix

 See general description of Aplusix software at the top scenario level. 

Tool access show tooltip help

NIL

controlled representation mode [Feature]
free tree mode [Feature]

Resources show tooltip helpexplode

Exercise 1 [Resource for students]

 

Aplusix file with a series of exercices (conversion RNL -> tree) to be done in a controlled tree mode.

Exercise 2 [Resource for students]

Aplusix file with a series of exercices (conversion RNL -> tree) to be done in a free tree mode.

 

Exercise 3 [Resource for students]

A printable worksheet with a series of exercises (conversion tree -> RNL) to be done in paper - pencil

 

Work plan show tooltip helpexplode

Setting show tooltip help

Three activities are proposed, two of which are done in a computer lab and the third one can be done in an ordinary classroom (without computer). In the computer lab, the students work individually or in pairs. The third activity is done individually.

 

Time show tooltip help

90 minutes

Actors' roles show tooltip help

 

Students:

- solving the exercises

- participating to classroom discussions

Teacher:

- orchestrating the classroom

- questioning the students

- managing classroom discussions

- institutionalising

What to do and how show tooltip help

In a computer lab, the teacher orchestrates the classroom activity. The students work individually or in pairs on exercises indicated by the teacher. The teacher interrupts the students’ work when necessary in order to organise a short whole class discussion.

The activity to be done without computer is worked out individually. The teacher distributes a worksheet to every student. He supervises the students’ work.

After each activity, a whole class discussion and a synthesis phase are foreseen.

 

Conversion RSL-RT

Identity show tooltip helpexplode

Authors show tooltip help

J. Trgalova, H. Chaachoua - MeTAH, Grenoble (France)

Subject domains show tooltip help

  • mathematics

Topics show tooltip help

  • algebra

Language show tooltip help

English

Country show tooltip help

France

Keywords show tooltip help

  • numerical expression
  • algebraic expression
  • representation system
  • tree
  • Aplusix

Description show tooltip help

Conversion tasks between symbolic language and tree representation

 

Rationale show tooltip helpexplode

A given algebraic expression can be considered from two points of view:

- as a computation programme: it indicates the sequence of computations to do in order to obtain the number sent by the programme when numerical values are assigned to letters involved in the expressions (procedural aspect);

- as an object whose form can be described and with which operations, such as simplifying, factoring, substituting into another expression, etc., can be done (structural aspect).

Considering the structural aspect of algebraic expressions is required for instance when determining of the equivalence of two expressions is at stake. However, in the teaching of algebra at the secondary level of French school system, the structural aspect of algebraic expressions is less “visible” for the students and is often suppressed by the procedural aspect.

A description of an algebraic expression in natural language leads to considering its structural aspect: for example the expression (2x + 1) (x - 4) is described as a product of a sum and a difference, a sum of the product of 2 by x and 1, and the difference of x and 4. The first word of the sentence gives the form of the expression. A tree representation of an algebraic expression presents a certain “proximity” to the natural language representation. It allows highlighting the form of an algebraic expression, which is given by the highest level assembler. In this scenario, the activities are designed to allow the students to explore the structure of algebraic expressions using first the natural language and the tree representation of algebraic expressions. These two representation systems will then be linked to the usual symbolic language representation.

The innovative aspect of this scenario lies in the use of Aplusix educational software, which provides a microworld where students can use both tree and symbolic language representations of algebraic expressions.

Theoretical framework show tooltip help

Semiotic registers of representation (R. Duval)

• Three semiotic registers: symbolic language (usual) representation, tree representation and natural language representation

• Hypothesis: the ability to represent a given mathematical concept in at least two registers and to perform conversions from one register to another is an indicator of conceptual understanding of the notion

• For each register:

–Internal rules

–Specific tasks (formation, treatment)

• Between registers:

–Conversion rules

–Congruence or not

–Specific tasks (conversion)

Anthropological theory of didactics (Y. Chevallard)

• Viability of the tasks in the Aplusix environment

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Theoretical framework show tooltip help

Population show tooltip helpexplode

School level show tooltip help

Grade 10

Age range show tooltip help

15-16 years

Population description show tooltip help

The student population that the scenario is designed to address does not include students with serious visual and/or motor impairments. This is because the tools used are unsuitable for meeting the specific needs of those students.

Student prerequisites show tooltip help

Meaning of letters in algebra (unknown, general number)

Algebraic expressions    

Familiarity with basic Aplusix functionalities

Teacher prerequisites show tooltip help

Mathematics knowledge required for teaching algebra at this school level

Familiarity with Aplusix software

Context show tooltip helpexplode

Physical context show tooltip help

 Computer lab with a computer-student ratio of 1:1 or 1:2

Institutional context show tooltip help

The contents addressed in the scenario are part of the French mathematics curriculum for the Grade 10 (classe de seconde).

Link to the mathematics curriculum

Socio-cultural context show tooltip help

Goals show tooltip helpexplode

Curricular goals show tooltip help

The students will be able to:

  • identify the form of an algebraic expression given in either of the following representation systems: tree, natural language, symbolic language;
  • "translate" an algebraic expression given in one representation system into another one;
  • solve problems involving algebraic expressions given in either of the three representation systems.

Content-epistemological goals show tooltip help

 

•Learn how to construct a tree representation of an expression given in symbolic language as a tree. 

• Learn how to “build” a symbolic languagerepresentation of an expression  represented by a tree.

Cognitive goals show tooltip help

 

• Articulate symbolic language and tree representations of algebraic expressions.

• Get an idea of the fact that these two registers are not semantically congruent.

Social-affective goals show tooltip help

Develop the willingness and capacity to:

  • work collaboratively;
  • participate in class discussion;
  • question one's own work through critical evaluation of the work of others.

Instrumental goals show tooltip help

Use Aplusix to construct tree representations of algebraic expressions.

Use Aplusix to solve algebra problems involving algebraic expressions given in either of the three representation systems.

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

A given algebraic expression can be considered from two points of view:

- as a computation program: it indicates the sequence of computations to do in order to obtain the number sent by the program when numerical values are assigned to letters involved in the expressions (procedural aspect);

- as an object whose form can be described and with which operations, such as simplifying, factoring, substituting into another expression, etc., can be done (structural aspect).

Considering the structural aspect of algebraic expressions is required for instance when determining of the equivalence of two expressions is at stake. However, in the teaching of algebra at the secondary level of French school system, the structural aspect of algebraic expressions is less “visible” for the students and is often suppressed by the procedural aspect.

A description of an algebraic expression in natural language leads to considering its structural aspect: for example the expression (2x + 1)(x - 4) is described as a product of a sum and a difference, a sum of the product of 2 by x and 1, and the difference of x and 4. The first word of the sentence gives the form of the expression. A tree representation of an algebraic expression presents a certain “proximity” to the natural language representation. It allows highlighting the form of an algebraic expression, which is given by the highest level assembler. In this scenario, the activities are designed to allow the students to explore the structure of algebraic expressions using first the natural language and the tree representation of algebraic expressions. These two representation systems will then be linked to the usual symbolic representation.

The innovative aspect of this scenario lies in the use of Aplusix educational software, which provides a microworld where students can use both tree and symbolic language representations of algebraic expressions.

Theoretical framework show tooltip help

 

Semiotic registers of representation (R. Duval)

• Three semiotic registers: symbolic language (usual) representation, tree representation and natural language representation

• Hypothesis: the ability to represent a given mathematical concept in at least two registers and to perform conversions from one register to another is an indicator of conceptual understanding of the notion

• For each register:

    –Internal rules

    –Specific tasks (formation, treatment)

• Between registers:

    –Conversion rules

    –Congruence or not

    –Specific tasks (conversion)

Anthropological theory of didactics (Y. Chevallard)

• Viability of the tasks in the Aplusix environment

Aplusix

  See general description of Aplusix software at the top scenario level.

Tool access show tooltip help

NIL

second view [Feature]

Resources show tooltip helpexplode

General description show tooltip help

Three types of resource are provided:

–Students’ worksheets including task texts (for tasks to be done in paper – pencil environment);

–Lists of Aplusix exercises (for exercises to be done with the software);

–Teacher’s document (for an activity to be orchestrated by the teacher using Aplusix in a videoprojection mode).

 

Exercise 4 [Resource for students]

Aplusix file with a series of exercices (conversion RSL -> tree) to be done in a controlled tree mode.

 

Exercise 5 [Resource for students]

Aplusix file with a series of exercices (conversion RSL -> tree) to be done in a free tree mode

Exercise 6 [Resource for students]

Aplusix file with a series of exercices (conversion tree -> RSL) to be done in a free tree mode.

Work plan show tooltip helpexplode

Setting show tooltip help

The activities are to be done in a computer lab. The students work individually or in pairs.

 

Time show tooltip help

80 minutes

Actors' roles show tooltip help

Students:

- solving the exercises

- participating to classroom discussions

Teacher:

- orchestrating the classroom

- questioning the students

- managing classroom discussions

- institutionalising

 

What to do and how show tooltip help

The teacher orchestrates the activity in the computer lab. The students work individually or in pairs on exercises indicated by the teacher. The teacher interrupts the students’ work when necessary in order to organise a short whole class discussion.

Treatment in RT

Identity show tooltip helpexplode

Authors show tooltip help

J. Trgalova, H. Chaachoua - MeTAH, Grenoble (France)

Subject domains show tooltip help

  • mathematics

Topics show tooltip help

  • algebra

Language show tooltip help

English

Country show tooltip help

France

Keywords show tooltip help

  • numerical expression
  • algebraic expression
  • representation system
  • tree
  • Aplusix

Description show tooltip help

Treatment tasks in tree register

 

Rationale show tooltip helpexplode

A given algebraic expression can be considered from two points of view:

- as a computation programme: it indicates the sequence of computations to do in order to obtain the number sent by the programme when numerical values are assigned to letters involved in the expressions (procedural aspect);

- as an object whose form can be described and with which operations, such as simplifying, factoring, substituting into another expression, etc., can be done (structural aspect).

Considering the structural aspect of algebraic expressions is required for instance when determining of the equivalence of two expressions is at stake. However, in the teaching of algebra at the secondary level of French school system, the structural aspect of algebraic expressions is less “visible” for the students and is often suppressed by the procedural aspect.

A description of an algebraic expression in natural language leads to considering its structural aspect: for example the expression (2x + 1) (x - 4) is described as a product of a sum and a difference, a sum of the product of 2 by x and 1, and the difference of x and 4. The first word of the sentence gives the form of the expression. A tree representation of an algebraic expression presents a certain “proximity” to the natural language representation. It allows highlighting the form of an algebraic expression, which is given by the highest level assembler. In this scenario, the activities are designed to allow the students to explore the structure of algebraic expressions using first the natural language and the tree representation of algebraic expressions. These two representation systems will then be linked to the usual symbolic language representation.

The innovative aspect of this scenario lies in the use of Aplusix educational software, which provides a microworld where students can use both tree and symbolic language representations of algebraic expressions.

Theoretical framework show tooltip help

Semiotic registers of representation (R. Duval)

• Three semiotic registers: symbolic language (usual) representation, tree representation and natural language representation

• Hypothesis: the ability to represent a given mathematical concept in at least two registers and to perform conversions from one register to another is an indicator of conceptual understanding of the notion

• For each register:

–Internal rules

–Specific tasks (formation, treatment)

• Between registers:

–Conversion rules

–Congruence or not

–Specific tasks (conversion)

Anthropological theory of didactics (Y. Chevallard)

• Viability of the tasks in the Aplusix environment

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Theoretical framework show tooltip help

Population show tooltip helpexplode

School level show tooltip help

Grade 10

Age range show tooltip help

15-16 years

Population description show tooltip help

The student population that the scenario is designed to address does not include students with serious visual and/or motor impairments. This is because the tools used are unsuitable for meeting the specific needs of those students.

Student prerequisites show tooltip help

Meaning of letters in algebra (unknown, general number)

Algebraic expressions    

Familiarity with basic Aplusix functionalities

Teacher prerequisites show tooltip help

Mathematics knowledge required for teaching algebra at this school level

Familiarity with Aplusix software

Context show tooltip helpexplode

Physical context show tooltip help

 Computer lab with a computer-student ratio of 1:1 or 1:2

Institutional context show tooltip help

The contents addressed in the scenario are part of the French mathematics curriculum for the Grade 10 (classe de seconde).

Link to the mathematics curriculum

Socio-cultural context show tooltip help

Goals show tooltip helpexplode

Curricular goals show tooltip help

The students will be able to:

  • identify the form of an algebraic expression given in either of the following representation systems: tree, natural language, symbolic language;
  • "translate" an algebraic expression given in one representation system into another one;
  • solve problems involving algebraic expressions given in either of the three representation systems.

Content-epistemological goals show tooltip help

•Calculate a numerical expression given as a tree.

• “Operate” with an algebraic expression represented by a tree (in this case, develop and simplify).

 

Cognitive goals show tooltip help

• “Experiment” the difference in dealing with numerical and algebraic expressions expressed in tree and in symbolic language registers respectively (using second view functionality of Aplusix).

Social-affective goals show tooltip help

Develop the willingness and capacity to:

  • work collaboratively;
  • participate in class discussion;
  • question one's own work through critical evaluation of the work of others.

Instrumental goals show tooltip help

Use Aplusix to construct tree representations of algebraic expressions.

Use Aplusix to solve algebra problems involving algebraic expressions given in either of the three representation systems.

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

Aplusix

 See general description of Aplusix software at the top scenario level. 

Tool access show tooltip help

NIL

second view [Feature]

Resources show tooltip helpexplode

General description show tooltip help

Three types of resource are provided:

–Students’ worksheets including task texts (for tasks to be done in paper – pencil environment);

–Lists of Aplusix exercises (for exercises to be done with the software);

–Teacher’s document (for an activity to be orchestrated by the teacher using Aplusix in a videoprojection mode).

 

Exercise 7 [Resource for students]

 

Aplusix file with a series of exercices (Calculate) to be done in a controlled tree mode

Exercise 8 [Resource for students]

 

Aplusix file with a series of exercices (Develop and simplify) to be done in a controlled tree mode.

Work plan show tooltip helpexplode

Setting show tooltip help

The activities are to be done in a computer lab. The students work individually or in pairs.

 

Time show tooltip help

20 minutes

Actors' roles show tooltip help

 

Students:

- solving the exercises

- participating to classroom discussions

Teacher:

- orchestrating the classroom

- questioning the students

- managing classroom discussions

- institutionalising

What to do and how show tooltip help

 

The teacher orchestrates the activity in the computer lab. The students work individually or in pairs on exercises indicated by the teacher. The teacher interrupts the students’ work when necessary in order to organise a short whole class discussion.

Assessing

Identity show tooltip helpexplode

Authors show tooltip help

J. Trgalova, H. Chaachoua - MeTAH, Grenoble (France)

Subject domains show tooltip help

  • mathematics

Topics show tooltip help

  • algebra

Language show tooltip help

English

Country show tooltip help

France

Keywords show tooltip help

  • numerical expression
  • algebraic expression
  • representation system
  • tree
  • Aplusix

Description show tooltip help

 

Treatment tasks in tree register,
conversion tasks between RNL and RT and between tree RSL and RT

Rationale show tooltip helpexplode

A given algebraic expression can be considered from two points of view:

- as a computation programme: it indicates the sequence of computations to do in order to obtain the number sent by the programme when numerical values are assigned to letters involved in the expressions (procedural aspect);

- as an object whose form can be described and with which operations, such as simplifying, factoring, substituting into another expression, etc., can be done (structural aspect).

Considering the structural aspect of algebraic expressions is required for instance when determining of the equivalence of two expressions is at stake. However, in the teaching of algebra at the secondary level of French school system, the structural aspect of algebraic expressions is less “visible” for the students and is often suppressed by the procedural aspect.

A description of an algebraic expression in natural language leads to considering its structural aspect: for example the expression (2x + 1) (x - 4) is described as a product of a sum and a difference, a sum of the product of 2 by x and 1, and the difference of x and 4. The first word of the sentence gives the form of the expression. A tree representation of an algebraic expression presents a certain “proximity” to the natural language representation. It allows highlighting the form of an algebraic expression, which is given by the highest level assembler. In this scenario, the activities are designed to allow the students to explore the structure of algebraic expressions using first the natural language and the tree representation of algebraic expressions. These two representation systems will then be linked to the usual symbolic language representation.

The innovative aspect of this scenario lies in the use of Aplusix educational software, which provides a microworld where students can use both tree and symbolic language representations of algebraic expressions.

Theoretical framework show tooltip help

Semiotic registers of representation (R. Duval)

• Three semiotic registers: symbolic language (usual) representation, tree representation and natural language representation

• Hypothesis: the ability to represent a given mathematical concept in at least two registers and to perform conversions from one register to another is an indicator of conceptual understanding of the notion

• For each register:

–Internal rules

–Specific tasks (formation, treatment)

• Between registers:

–Conversion rules

–Congruence or not

–Specific tasks (conversion)

Anthropological theory of didactics (Y. Chevallard)

• Viability of the tasks in the Aplusix environment

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Theoretical framework show tooltip help

Population show tooltip helpexplode

School level show tooltip help

Grade 10

Age range show tooltip help

15-16 years

Population description show tooltip help

The student population that the scenario is designed to address does not include students with serious visual and/or motor impairments. This is because the tools used are unsuitable for meeting the specific needs of those students.

Student prerequisites show tooltip help

Meaning of letters in algebra (unknown, general number)

Algebraic expressions    

Familiarity with basic Aplusix functionalities

Teacher prerequisites show tooltip help

Mathematics knowledge required for teaching algebra at this school level

Familiarity with Aplusix software

Context show tooltip helpexplode

Physical context show tooltip help

 Computer lab with a computer-student ratio of 1:1 or 1:2

Institutional context show tooltip help

The contents addressed in the scenario are part of the French mathematics curriculum for the Grade 10 (classe de seconde).

Link to the mathematics curriculum

Socio-cultural context show tooltip help

Goals show tooltip helpexplode

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

Aplusix

  See general description of Aplusix software at the top scenario level.

Tool access show tooltip help

NIL

free tree mode [Feature]

Resources show tooltip helpexplode

General description show tooltip help

Printable worksheet with instructions for students

 

Exercise 9 [Resource for students]

Printable worksheet with instructions for students

Work plan show tooltip helpexplode

Setting show tooltip help

The activities are to be done in a computer lab. The students work in pairs.

 

Time show tooltip help

55 minutes

Actors' roles show tooltip help

Students:

- solving the exercises

- participating to classroom discussions

Teacher:

- orchestrating the classroom

- questioning the students

- managing classroom discussions

- institutionalising

 

What to do and how show tooltip help

The teacher orchestrates the activity in the computer lab. The students work in pairs. They create their own expressions represented as trees and they describe them in natural language or write them in symbolical language. The other student has then construct a tree representing the expressions given either by a tree or written in symbols. The teacher supervises the students’ work and interrupts it when necessary in order to organise a short whole class discussion.

 

Posttest

Identity show tooltip helpexplode

Authors show tooltip help

J. Trgalova, H. Chaachoua - MeTAH, Grenoble (France)

Subject domains show tooltip help

  • mathematics

Topics show tooltip help

  • algebra

Language show tooltip help

English

Country show tooltip help

France

Keywords show tooltip help

  • numerical expression
  • algebraic expression
  • representation system
  • tree
  • Aplusix

Description show tooltip help

 The posttest activities will be similar to pretest activities, in order to be able to compare the outcomes. Their design will take into account the results of the pretest.

Rationale show tooltip helpexplode

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

Work plan show tooltip helpexplode