Exploring the structure of numerical expressions (alien)

Identity show tooltip helpexplode

Authors show tooltip help

Chiappini G., Pedemonte B., Robotti E., Viglienzone P.

Subject domains show tooltip help

  • Arithmetic

Topics show tooltip help

  • Numerical expressions
  • Tree representation

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

  • Numerical expression
  • Tree representation
  • Linear representation
  • Aplusix

Description show tooltip help

This pedagogical plan concerns an approach to the numerical expressions through an introduction of a new representation system: a tree representation. It makes use of the Aplusix software, which allows to represent expressions either in linear expression or as a tree.

This plan presents an approach to numerical expressions based on an exploration of the structure of numerical expressions. The main aim of the plan is to understand the structure of numerical expressions and the rules that etablish the hierarchical priority of its signs.

Rationale show tooltip helpexplode

Target show tooltip helpexplode

Rationale show tooltip helpexplode

The PP design is mainly based on two theoretical frameworks: the Semiotic registers of representation (R. Duval) and the Activity theory frame. These two different theoretical frameworks can be used to justify the processes that can determine the achievement of the educational goals.

Our hypothesis based on these frames is that specific tasks related to an integrated use of Aplusix tree  representations, could favour students in understanding the structure of numerical expressions. As a matter of fact the ability to represent a given mathematical concept in at least two registers and to perform conversions from one register to another should be an indicator of conceptual understanding of the notion.

Theoretical framework show tooltip help

The first theoretical frame of reference is the Semiotic registers of representation(R. Duval). Three semiotic registers are used in the PP: symbolic language (usual) representation, tree representation and natural language representation. The hypothesis is that the change of registers can be useful to understand the mathematical structure underpinning numerical expressions. The ability to represent a given mathematical structure in at least two registers and to perform conversions from one register to another can also be for student an indicator of control of the expression structure.

The second theoretical framework of reference is the Activity Theory. We use it to analyse transformations made in the development of educational activities when the new operative and representative possibilities of Aplusix are used.

Population show tooltip helpexplode

School level show tooltip help

7 grade

Age range show tooltip help

11-12 years old

Student prerequisites show tooltip help

  • Knowledge of the operations with numbers (integers and rational numbers)
  • Basic skills in solving numerical expressions
  • Familiarity with basic computer functions.

Teacher prerequisites show tooltip help

  • No specialised mathematics knowledge is necessary beyond that normally required for teaching at this school level.
  • Familiarity with basic computer functions.
  • Familiarity with the Aplusix DDA.

 

Context show tooltip helpexplode

Physical context show tooltip help

Computer suite permitting a computer-student ratio of 1:1 or 1:2. 

Institutional context show tooltip help

The contents addressed in the module are part of the Italian maths curriculum for the 7 grade school. 

Link to national maths curriculum document.

Goals show tooltip helpexplode

Content-epistemological goals show tooltip help

Understand the structure of numerical expressions
Distinguish between procedural and structural aspects of numerical expression

Cognitive goals show tooltip help

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

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

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

Theoretical framework show tooltip help

Kieran (1989) highlights that an important aspect of the students' difficulties in learning algebra is to recognise and use structure. Structure includes the "surface" structure  and the systemic structure. 

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 pedagogical plan comprises activities to be carried out in a computer suite under the active supervision of the teacher. The students can work individually or in pairs.

 

Time show tooltip help

10 hours

Actors' roles show tooltip help

 TUDENTS

  • Solving tasks
  • Problem solving

TEACHER

  • group supervision
  • cognitive structuring
  • participating in class discussion
  • leading class discussion
  • moderating class discussion
  • mediating class discussion

RESEARCHERS (discretionary)

  • Observing

What to do and how show tooltip help

This pedagogical plan comprises 3 distinct modules:

MODULE 1

  • Initial test

MODULE 2

  • Exploring expressions tree:
  1. Introduction to the tree construction in Aplusix
  2. Relationship between a linear representation and a tree representation  of expressions
  3. From the natural language to the algebraic language for expressions

MODULE 3

  • Final test

The teaching activities and the way to propose them in class are described in detail in the relevent section of the plan.

 

Initial test

Identity show tooltip helpexplode

Authors show tooltip help

Chiappini G., Pedemonte B., Robotti E., Viglienzone P.

Subject domains show tooltip help

  • Arithmetic

Topics show tooltip help

  • Algebraic resolution of a numerical expression
  • Traslation of a statement in algebraic language

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

Description show tooltip help

This sub-pedagogical plan is constituted by a test to value knowledge of students about expressions. The text is divided into two parts: a part on Aplusix and a part on paper and pencil. In the first part students have to treat numerical expressions working on the test modality of Aplusix system. In the second part they have to convert some linear expressions in natural language and into tree expressions and vice-versa some tree expressions in linear expressions and in natural language. This test should be proposed to students who previously treated expressions topics in the school.

Rationale show tooltip helpexplode

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Population show tooltip helpexplode

School level show tooltip help

7 grade

Age range show tooltip help

11-12 years old

Student prerequisites show tooltip help

  • Knowledge of the operations with numbers (integers and rational numbers)
  • Basic skills in solving numerical expressions
  • Familiarity with basic computer functions.

Teacher prerequisites show tooltip help

  • No specialised mathematics knowledge is necessary beyond that normally required for teaching at this school level.
  • Familiarity with basic computer functions.
  • Familiarity with the Aplusix DDA.

 

Context show tooltip helpexplode

Physical context show tooltip help

Computer suite permitting a computer-student ratio of 1:1 or 1:2. 

Institutional context show tooltip help

The contents addressed in the module are part of the Italian maths curriculum for the 7 grade school. 

Link to national maths curriculum document.

Goals show tooltip helpexplode

Content-epistemological goals show tooltip help

The aim of the test is to highlight students competencies about numerical expressions to compare these results with the results of the final test which will be proposed to students at the end of the pedagogical plan (final test)

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

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.

Test modality [Feature]

The Test activity provides a 30-minute test on exercises chosen from The Map or from a file. The duration of a test can be different when the exercises come from a file. A test is generated from The Map by clicking on a dot (representing a family of exercises) and choosing “Test” in the “Launch” menu or by opening a file of exercises and choosing the ‘Test’ mode. 
The remaining time is indicated in the left part of the toolbar. 
When an exercise is finished, students can modify their solutions by clicking on the “Modify the exercise” button. 
A test is finished when: 

  • All the exercises are solved or abandoned (solved means that the student says they are). 
  • The time is over. 
  • The student clicks on the « Stop the test » button. 
  • The student quits Aplusix. 

In the three first cases, Aplusix gives a score and provides students with an opportunity to correct their solution using a “Self-correction” activity.

Resources show tooltip helpexplode

Activity1: Numerical calculation [Resource for students]

This is an Aplusix file which should be solved by students in Aplusix test modality. Activities of this card ask to solve some numerical expressions. 

 

Activity 2: Natural language and algebraic language [Resource for students]

This is a communication paper-pencil activity containing two tasks asking to pass from an algebraic expression to a statement and viceversa from a statement to an algebraic expression. 

 

Activity 3: Natural language and algebraic language [Resource for students]

This is a communication paper-pencil activity containing two tasks asking to pass from an algebraic expression to a statement and viceversa from a statement to an algebraic expression. 

 

Work plan show tooltip helpexplode

Setting show tooltip help

In the first activity, 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

1 hour and half

Actors' roles show tooltip help

The first activity is performed 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 performed by students in paper-pencil environment. They work in pairs. The teacher is not supposed to intervene.

What to do and how show tooltip help

Students are disposed in pairs for each computer where Aplusix is running. 

The teacher gives some brief information about the use of Aplusix. In particular she explains how to pass from a step to another one, how select a part of expression, how replace it to solve the expression. She can eventually propose an example. Then, the teacher asks students to open the file "Activity 1" which will be opened in Aplusix in test modality and the teacher asks students to solve the test.

At the end of the test students have to save their work.

Successively the teacher delivers to each couple of students the other two activities that must be solved in paper-pencil environment. At the end of the work she picks up the filled activities.

 

Exploring the structure of expressions comparing different representations

Identity show tooltip helpexplode

Authors show tooltip help

Chiappini G., Pedemonte B., Robotti E., Viglienzone P.

Subject domains show tooltip help

  • Arithmetic

Topics show tooltip help

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

  • Numerical expression
  • Linear representation of expressions
  • Tree representation of expressions
  • Expressions in natural language
  • Aplusix

Description show tooltip help

Rationale show tooltip helpexplode

Target show tooltip helpexplode

Rationale show tooltip helpexplode

The PP design is mainly based on two theoretical frameworks: the Semiotic registers of representation (R. Duval) and the Activity theory frame. These two different theoretical frameworks can be used to justify the processes that can determine the achievement of the educational goals.

Our hypothesis based on these frames is that specific tasks related to an integrated use of Aplusix tree  representations, could favour students in understanding the structure of numerical expressions. As a matter of fact the ability to represent a given mathematical concept in at least two registers and to perform conversions from one register to another should be an indicator of conceptual understanding of the notion.

Theoretical framework show tooltip help

The first theoretical frame of reference is the Semiotic registers of representation(R. Duval). Three semiotic registers are used in the PP: symbolic language (usual) representation, tree representation and natural language representation. The hypothesis is that the change of registers can be useful to understand the mathematical structure underpinning numerical expressions. The ability to represent a given mathematical structure in at least two registers and to perform conversions from one register to another can also be for student an indicator of control of the expression structure.

The second theoretical framework of reference is the Activity Theory. We use it to analyse transformations made in the development of educational activities when the new operative and representative possibilities of Aplusix are used.

Population show tooltip helpexplode

School level show tooltip help

7 grade

Age range show tooltip help

11-12 years old

Student prerequisites show tooltip help

  • Knowledge of the operations with numbers (integers and rational numbers)
  • Basic skills in solving numerical expressions
  • Familiarity with basic computer functions.

Teacher prerequisites show tooltip help

  • No specialised mathematics knowledge is necessary beyond that normally required for teaching at this school level.
  • Familiarity with basic computer functions.
  • Familiarity with the Aplusix DDA.

 

Context show tooltip helpexplode

Physical context show tooltip help

Computer suite permitting a computer-student ratio of 1:1 or 1:2. 

Institutional context show tooltip help

The contents addressed in the module are part of the Italian maths curriculum for the 7 grade school. 

Link to national maths curriculum document.

Goals show tooltip helpexplode

Content-epistemological goals show tooltip help

  • Learn how to represent a numerical expression as a tree.
  • Learn how to “build” a tree given a numerical expression.
  • Learn that there is only a linear expression for a tree 
  • Learn that it could be different tree representations for an expression represented in linear form.
  • 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 the three representation systems: natural language, symbolic language, tree.

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

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

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 pedagogical plan comprises activities to be carried out in a computer suite under the active supervision of the teacher. The students can work individually or in pairs.

 

Introduction to the tree construction in Aplusix

Identity show tooltip helpexplode

Authors show tooltip help

Chiappini G., Pedemonte B., Robotti E., Viglienzone P.

Subject domains show tooltip help

  • Arithmetic

Topics show tooltip help

  • Numerical expressions
  • Tree representation

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

  • Numerical expressions
  • Mixed tree representation
  • Controlled tree representation

Description show tooltip help

This pedagogical plan focus on introduction of tree representations of numerical expressions in Aplusix. In the system three modalities are presented to construct a tree representation. In this pedagogical plan we propose to student the mixed representation and the controlled representation. Our hypothesis is that the integration of these modalities in Aplusix could favour students to acquire capabilities in constructing tree representation of expressions in a gradual way.

Rationale show tooltip helpexplode

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Population show tooltip helpexplode

School level show tooltip help

7 grade

Age range show tooltip help

11-12 years old

Student prerequisites show tooltip help

  • Knowledge of the operations with numbers (integers and rational numbers)
  • Basic skills in solving numerical expressions
  • Familiarity with basic computer functions.

Teacher prerequisites show tooltip help

  • No specialised mathematics knowledge is necessary beyond that normally required for teaching at this school level.
  • Familiarity with basic computer functions.
  • Familiarity with the Aplusix DDA.

 

Context show tooltip helpexplode

Physical context show tooltip help

Computer suite permitting a computer-student ratio of 1:1 or 1:2. 

Institutional context show tooltip help

The contents addressed in the module are part of the Italian maths curriculum for the 7 grade school. 

Link to national maths curriculum document.

Goals show tooltip helpexplode

Content-epistemological goals show tooltip help

Learn how to represent a numerical expression as a tree.

 

Cognitive goals show tooltip help

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

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 a tree of numerical expressions in Mixed representation mode and Controlled representation mode 

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

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.

Resources show tooltip helpexplode

Card 1: Introduction to the tree construction in Aplusix [Resource for students]

This card presents tasks to introduce students to the tree representation of a numerical expression making use of two specific functionnalities of Aplusix: the mixed tree representation and the controlled tree representation.

Work plan show tooltip helpexplode

Setting show tooltip help

This pedagogical plan comprises activities to be carried out in a computer suite under the active supervision of the teacher. The students can work individually or in pairs.

 

Time show tooltip help

1 hours

Actors' roles show tooltip help

 TUDENTS

  • Solving tasks
  • Problem solving

TEACHER

  • group supervision
  • cognitive structuring
  • participating in class discussion
  • leading class discussion
  • moderating class discussion
  • mediating class discussion

RESEARCHERS (discretionary)

  • Observing

What to do and how show tooltip help

Students are disposed in pairs for each computer where Aplusix is running. The teacher delivers to each couple of students the card containing the tasks.

Tasks have to be solved working with Aplusix. The answer to the task have to be written in the students notebooks or in the card.

At the beginning the teacher gives some information about Aplusix tree representation. In particular he/she explains that there are two ways to construct a tree: the mixed representation  and the controlled representation modes. The teacher explains that the focus of the lesson is to understand how constructing a tree from a linear representation of expression. 

At the end of the activity a brief discussion could be proposed by the teacher  to institusionalise the use of the two specific function of Aplusix which allow to construct the tree representation.

Exploring the structure of expressions comparing linear representations and tree representations

Identity show tooltip helpexplode

Authors show tooltip help

Chiappini G., Pedemonte B., Robotti E., Viglienzone P.

Subject domains show tooltip help

  • Arithmetic

Topics show tooltip help

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

  • Numerical expression
  • Linear representation of expression
  • Tree representation of expression

Description show tooltip help

This pedagogical plan focus on the relationship between a linear expression and its tree representation. The aim of the interaction between these two representations is to highlight the hierarchical structure of an expression to unserstand the priority of the elements constituent the expression (parentheses, operators, etc.). 

Rationale show tooltip helpexplode

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Population show tooltip helpexplode

School level show tooltip help

7 grade

Age range show tooltip help

11-12 years old

Student prerequisites show tooltip help

  • Knowledge of the operations with numbers (integers and rational numbers)
  • Basic skills in solving numerical expressions
  • Familiarity with basic computer functions.

Teacher prerequisites show tooltip help

  • No specialised mathematics knowledge is necessary beyond that normally required for teaching at this school level.
  • Familiarity with basic computer functions.
  • Familiarity with the Aplusix DDA.

 

Context show tooltip helpexplode

Physical context show tooltip help

Computer suite permitting a computer-student ratio of 1:1 or 1:2. 

Institutional context show tooltip help

The contents addressed in the module are part of the Italian maths curriculum for the 7 grade school. 

Link to national maths curriculum document.

Goals show tooltip helpexplode

Content-epistemological goals show tooltip help

  • Learn how to represent a numerical expression as a tree.
  • Learn how to “build” a tree given a numerical expression.
  • Learn that there is only a linear expression for a tree 
  • Learn that it could be different tree representations for an expression represented in linear form.

Cognitive goals show tooltip help

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

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

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

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.

Resources show tooltip helpexplode

Card 2: Linear and tree representations of expressions: their relationship [Resource for students]

The tasks proposed in this card concern the relationship between a linear representation of expression and its tree representation. In particular tasks propose to construct both a linear representation of expressions presented as tree and vice-versa a tree representation of a linear expression.

Card 3: How many tree representations can be found for a linear expression? [Resource for students]

The tasks proposed in this card focus on the observation that it is possible to find different tree representations of a linear expression while the opposite case (to construct different linear representations from a tree representation) it is always not possible.

 

Work plan show tooltip helpexplode

Setting show tooltip help

This pedagogical plan comprises activities to be carried out in a computer suite under the active supervision of the teacher. The students can work individually or in pairs.

 

Time show tooltip help

2 hours

Actors' roles show tooltip help

 TUDENTS

  • Solving tasks
  • Problem solving

TEACHER

  • group supervision
  • cognitive structuring
  • participating in class discussion
  • leading class discussion
  • moderating class discussion
  • mediating class discussion

RESEARCHERS (discretionary)

  • Observing

What to do and how show tooltip help

Students are disposed in pairs for each computer where Aplusix is running. The teacher delivers to each couple of students the card 2 containing the tasks.

Tasks have to be solved first with paper and pencil and then working with Aplusix. The answer to the task have to be written in the students notebooks or in the card.

At the end of this card a discussion can be planned to explicit some concepts about the relationship between the tree representation and the linear representation of an expression that have been emerged by the solution of the tasks.Some rules about the hierarchical structure of an expression should be made explicit (priority of parentheses, multiplication sign respect to addition, etc.). 

After this discussion card 3 could be delivered to students by the teacher. Another discussion can be planned when students finish the tasks involved in this card. 

Solving arithmetic task expressed in natural language using tree representations and linear representations

Identity show tooltip helpexplode

Authors show tooltip help

Chiappini G., Pedemonte B., Robotti E., Viglienzone P.

Subject domains show tooltip help

  • Arithmetic

Topics show tooltip help

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

  • Numerical expressions
  • Linear representation of expressions
  • Tree representation of expression
  • Expressions in natural language

Description show tooltip help

In this pedagogical plan, tasks focus on the relationship among three representations for a numerical expression: natural language, linear representation, tree.

Rationale show tooltip helpexplode

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Population show tooltip helpexplode

School level show tooltip help

7 grade

Age range show tooltip help

11-12 years old

Student prerequisites show tooltip help

  • Knowledge of the operations with numbers (integers and rational numbers)
  • Basic skills in solving numerical expressions
  • Familiarity with basic computer functions.

Teacher prerequisites show tooltip help

  • No specialised mathematics knowledge is necessary beyond that normally required for teaching at this school level.
  • Familiarity with basic computer functions.
  • Familiarity with the Aplusix DDA.

 

Context show tooltip helpexplode

Physical context show tooltip help

Computer suite permitting a computer-student ratio of 1:1 or 1:2. 

Institutional context show tooltip help

The contents addressed in the module are part of the Italian maths curriculum for the 7 grade school. 

Link to national maths curriculum document.

Goals show tooltip helpexplode

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 the three representation systems: natural language, symbolic language, tree.

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

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

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.

Resources show tooltip helpexplode

Card 4: From natural language to the algebraic language [Resource for students]

This card focus on different repreentation of a numerical expression. In particular tasks concern the passage from natural language to the tree or linear representation of the expression.

 

Work plan show tooltip helpexplode

Setting show tooltip help

This pedagogical plan comprises activities to be carried out in a computer suite under the active supervision of the teacher. The students can work individually or in pairs.

 

Time show tooltip help

1 hour

Actors' roles show tooltip help

 TUDENTS

  • Solving tasks
  • Problem solving

TEACHER

  • group supervision
  • cognitive structuring
  • participating in class discussion
  • leading class discussion
  • moderating class discussion
  • mediating class discussion

RESEARCHERS (discretionary)

  • Observing

What to do and how show tooltip help

Students are disposed in pairs for each computer where Aplusix is running. The teacher delivers to each couple of students the card 4 containing the tasks.

Tasks have to be solved first with paper and pencil and then working with Aplusix. The answer to the task have to be written in the students notebooks or in the card.

At the end of this card a discussion can be planned to explicit how a statement of an equation, expressed in natural language, can be transformed in algebraic language (by a  tree representation or a linear representation).

Final test

Identity show tooltip helpexplode

Authors show tooltip help

Chiappini G., Pedemonte B., Robotti E., Viglienzone P.

Subject domains show tooltip help

  • Arithmetic

Topics show tooltip help

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

Description show tooltip help

This sub-pedagogical plan is constituted by a test to value knowledge of students about expressions after having worked with tree representation of expressions. The text is divided into two parts: a part on Aplusix and a part on paper and pencil environment. In the first part students have to treat numerical expressions working on the test modality of Aplusix system. In the second part they have to convert some linear expressions in natural language and into tree expressions and vice-versa some tree expression in linear expressions and in natural language. This test should be given some results about the effectiveness of this pedagogical plan.

Rationale show tooltip helpexplode

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Population show tooltip helpexplode

School level show tooltip help

7 grade

Age range show tooltip help

11-12 years old

Student prerequisites show tooltip help

  • Knowledge of the operations with numbers (integers and rational numbers)
  • Basic skills in solving numerical expressions
  • Familiarity with basic computer functions.

Teacher prerequisites show tooltip help

  • No specialised mathematics knowledge is necessary beyond that normally required for teaching at this school level.
  • Familiarity with basic computer functions.
  • Familiarity with the Aplusix DDA.

 

Context show tooltip helpexplode

Physical context show tooltip help

Computer suite permitting a computer-student ratio of 1:1 or 1:2. 

Institutional context show tooltip help

The contents addressed in the module are part of the Italian maths curriculum for the 7 grade school. 

Link to national maths curriculum document.

Goals show tooltip helpexplode

Content-epistemological goals show tooltip help

The aim of this test is to compare the results of the initial test with the results of this test to  validate the effectiveness of this pedagogical plan.

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

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.

Test modality [Feature]

The Test activity provides a 30-minute test on exercises chosen from The Map or from a file. The duration of a test can be different when the exercises come from a file. A test is generated from The Map by clicking on a dot (representing a family of exercises) and choosing “Test” in the “Launch” menu or by opening a file of exercises and choosing the ‘Test’ mode. 

The remaining time is indicated in the left part of the toolbar. 

When an exercise is finished, students can modify their solutions by clicking on the “Modify the exercise” button. 

A test is finished when: 

  • All the exercises are solved or abandoned (solved means that the student says they are). 
  • The time is over. 
  • The student clicks on the « Stop the test » button. 
  • The student quits Aplusix. 

In the three first cases, Aplusix gives a score and provides students with an opportunity to correct their solution using a “Self-correction” activity.

Resources show tooltip helpexplode

Activity 1 [Resource for students]
Resource contents show tooltip help
Activity 2 [Resource for students]
Activity 3 [Resource for students]

Work plan show tooltip helpexplode