Equality and equivalent notions in treating algebraic expressions and equations (familiar)

Identity show tooltip helpexplode

Authors show tooltip help

G. P. Chiappini, B. Pedemonte, E. Robotti, F. Vannucci

Subject domains show tooltip help

  • Algebra

Topics show tooltip help

  • Algebraic expressions
  • Polynomials
  • Equations

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

  • Algebraic Line
  • Algebraic Manipulator
  • Activity theory
  • Algebra of quantities
  • Algebra of operations
  • Algebraic expressions
  • Polynomials
  • Equations
  • Alnuset

Description show tooltip help

This pedagogical plan describes an innovative approach for introducing students to the notions of equivalence and equality when they deal with algebraic and polynomial expressions and equations. It is based on the use of a digital artefact called ALNUSET. In particular, two components of ALNUSET are used: the algebraic line component and the algebraic manipulator component.
The plan is conceptually divided into two modules.
The first focuses on the construction of the notion of equivalent algebraic expressions and on the notion of the root of a polynomial expression.
The second deals with the construction of meaning for the solution of equations considered as conditioned equality determined by the insertion of the equality sign between two algebraic expressions.

Rationale show tooltip helpexplode

Educational research has highlighted different kinds of difficulties arising from the issues addressed in this pedagogical plan, particularly difficulties concerning:

  • conceptualisation of what an expression denotes
  • comprehension of what characterizes two equivalent expressions or two opposite expressions or two reciprocal expressions
  • comprehension of the notion of polynomial roots and the capability to use them to factorize polynomial expression or to solve problems
  • comprehension of what the solution of an equation is and of the conditions that make two equations equivalent.

We believe that in order to overcome the identified difficulties and to develop genuine algebraic thought in students, it may be useful to integrate an algebra of quantities with an algebra of formal operations. An algebra of quantities is an algebra that focuses attention on quantities denoted by a literal symbol and on quantities that are expressed in an indeterminate way in the sequence of operations of a literal expression. An algebra of operations focuses on the operations which preserve formal properties of expressions within a numeric domain. This need for integration between these two types of algebra can be justified from an historical and phylogenetic standpoint as well as from a cognitive and didactical perspective. As a matter of fact, in order to give algebra the status of an autonomous discipline, around 1830 Peacock distinguished between arithmetical and symbolic algebra: in the former the attention is focused on quantities denoted by literal symbols and expressions, while in the latter there is a separation between what the literal symbols and expressions denote and attention is paid   exclusively to operations.

In this framework ALNUSET can be a very useful tool for integrating these two different types of algebra into didactical practice. Indeed ALNUSET has two main components, the Algebraic Line component and the Algebraic Manipulator component, that respectively are oriented to the development of an algebra of quantities and to the development of an algebra of operations.

Theoretical framework show tooltip help

This pedagogical plan can be defined and justified by referring to two theoretical frameworks. The first is Peirce’s semantic frame.It is used to frame the ways mathematical objects and their interaction are represented in Alnuset and the ways representations of Alnuset can be acted on.

The second theoretical framework of reference is the Activity Theory. It is used to frame the pedagogical strategy used in the PP, to analyze the contradictions that can emerge in  the development of the didactical activities and the mediating role of Alnuset representations to overcome them. 

The use of the Peirce's semiotic frame

This pedagogical plan is based on the assumption that mathematical knowledge is the result of a social construction. This social construction is realized by a continuous dialectic between the development on the operational/procedural level of mathematical techniques, and their conceptualisation on a structural/relational level.
A mathematical technique is, first of all, a way to solve a problem. Moreover, every technique is composed of mechanic and automatic reasoning and operations (Chevallard, 1992; Artigue, 2002). Mathematical techniques are realized by the use of signs; they are characterized by operative and representative orders which are shared socially.
The operative order concerns the rules of use of signs; it orients the development of the technique on the procedural level.
The representative order concerns the relationship among signs (with their rules of use) and their referents; it orients the structural/relational interpretation of the technique.
Exploiting the possibilities of visualization, computation, dynamicity and interaction made available by digital technology, it is possible to instrument a mathematical technique. The instrumentation of a mathematical technique concerns a new way to use signs in the construction and interpretation of the technique. This fact is due to the new operative and representative order mediated by the technology. In order to analyse the differences between an instrumented and a standard technique, it is necessary to study how the role of the signs changes in relation to the new operative and representative order mediated by the technology. To this aim, it is useful to refer to Peirce’s distinction between three kinds of signs, i.e. symbol, icon and index. Peirce’s distinction is defined on the basis of the relation that each of these kinds of signs established with its object of reference (Peirce, 2003). Let us consider symbols, for example. Peirce says that these are signs that can be related with their objects of reference on the basis of specific conventional rules. By constrast, an icon is a sign whose shape and structure reflects the structure and the properties of the object of reference. It is important to stress that for Peirce symbols and icons are strictly linked. He says that the rules characterizing mathematical symbols include an iconic link with a property of the represented object. In addition to symbols and icons, Peirce also considers the indexes, namely signs whose link to the objects of reference is due to the fact that the sign is partially determined by the object itself (for instance, the  windsock is an index of wind direction, in the sense that its direction is determined by its object of reference). These functions of signs are at the basis on algebraic thinking.
For example, Peirce says that an algebraic expression can be considered an icon or an index sinceit can reveal new important aspects about the object of reference. In the natural integer domain, an algebraic expression such as 2x+1 is index of the the generic odd number because the form of this expression is determined by the property of this number “to be successors of even numbers”. As a matter of fact, the structure of the expression reflects iconically such property. By applying associative and distributive rules it is possible to transform the given expression into the following one: x+(x+1). Also in this form the espression is index of the generic odd number. Nevertheless, in this case the structure of this new expression iconically reflects another property of odd numbers: "to be addition of two consecutive numbers". It is not easy to understand the iconical connection established between algebraic rules and conventions and abstraict properties of the object of reference; it is not easy to understand the iconical relationship between a representative event and the object that determine it. 
Learning algebra means learning manipulation of algebraic symbols on the basis of rules and conventions as well as recognising the objects they are index of and the properties of these objects whose signs are icons.
Thus, in learning algebra the development of semiotic processes to recognise iconic and indexical aspects that characterize algebraic signs, is crucial. We think that technology can be exploit to make emerge the iconical and indexical aspects of a mathematical object. In particular, our hypothesys is that technology can make available new representative and operative possibilities to structure a new phenomenological space for algebraic activity, thus allowing teachers to reconfigure algebraic knowledge into an object of investigation for students and supplying students with new mediating tools to improve their learning.

The use of the Activity theory frame

In algebra didactical practices the operative and representative changes due to the introduction of technology allow to shape in other way the mathematical knowledge. Thus, in these new didactical practices, mathematical knowledge became an object of investigation. The operative and representative changes in the use of a technique established a new social order in its use too. With social order related to the use of a technique we mean the activities in which participants communicate, sharing and discuss ideas and meanings about mathematical phenomena.

In order to analyse the changes in the social order related to the use of a technique, we consider the activity in which this technique is used as unity of analysis and we use the Activity theory frame to study the changes which appear in this activity due to the instrumentation of the technique.

Activity Theory is a philosophical and cross-disciplinary theory adopted for studying different forms of human practice, such as teaching/learning, that are seen as development processes mediated by artefacts, where individual and social levels are simultaneously interlinked (Kuutti, 1996).
In Activity Theory, an activity is a form of acting directed towards an object, and it is the object that distinguishes one activity from another. Transforming the object into an outcome is what motivates the existence of an activity. According to the theory, activities consist of actions or chains of actions, which in turn consist of operations. If we consider Activity Theory applied to the educational field, the object of an activity is the learning of a given knowledge or the development of a given ability; the outcome of this activity, the motive for which the activity is developed, is students' acquisition of that knowledge or that ability (Bellamy, 1996).
Cole and Engeström (1991) have devised a systemic model expressing the complex relationships between elements in an activity (see figure below) which is particularly useful for studying the relationships that take place in teaching/learning activity (see also Engeström, 1987; 1991). Their model highlights three mutual relationships involved in every activity, namely the relationship between subject and object, that between subject and community, and that between community and object. Each of these relationships is mediated by a third entity. The relationship between subject and the object is mediated by artefacts that both enable and constrain the subject’s action. The relationship between subject and community is mediated by rules (explicit or implicit norms, conventions and social interactions), while that between community and object is mediated by the division of labour (different roles characterizing labour organization). 
The model depicted in the figure also shows that each entity mediates all the relationships described in the model. Artefacts used in the activity mediate not only the relationship between the subject and the object but also that between subject and community and that between community and object. Moreover, mediating entities are not mutually independent but rather exert influence over one another. For example, the introduction of a new artefact in an activity influences both the norms regulating participant interaction in the activity and the roles that the participants can assume.

This model allows us to analyze the mediation role played by the artifact during the classroom activities described in this pedagogical plan.
The artefact adopted in the plan is ALNUSET, and in particular two specific components are used: the algebraic line component and the manipulator component. These components and their integration mediate the student’s actions associated to the activity object. The details of this mediation role are described elsewhere in relation to the specific artefact-based learning activities: here we shall broadly outline the mediation role in general terms.

For a brief summary of Activity Theory, click here.

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Population show tooltip helpexplode

School level show tooltip help

Second year of upper secondary school

Age range show tooltip help

15-16 years old

Population description show tooltip help

This pedagogical plan has been designed for students who have just developed syntactic competencies in the algebraic domain and who, according to our hypothesis, are still weak in terms of semantic competencies.

Nevertheless, this pedagogical plan could also be tackled, with appropriate modifications, by students engaged in the first approach to algebraic language. In this case the pre-test should not be proposed to students but could be transformed into a post-test.

Student prerequisites show tooltip help

  • Knowledge of the operations with numbers (integers and rational numbers)
  • Basic skills in literal calculus 
  • Basic skills in the solution of first degree equations 
  • 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 ALNUSET 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 first and second year of upper secondary school. Link to national maths curriculum document.

Socio-cultural context show tooltip help

Goals show tooltip helpexplode

Curricular goals show tooltip help

Learn how to:

  • Perform calculations with integer and rationals numbers;
  • Treat monomials and operations;
  • Factorise polynomials;
  • Solve first- and second-degree integer equations and inequations

These are goals considered by the actual italian curricula. We want to pursue these goals, not only on the operative plan but also on the semantic plan.

Content-epistemological goals show tooltip help

  • Learn that a letter in algebra indicates the elements of the numerical set where it is defined.
  • Learn that a literal algebraic expression indicates the resul of the operations performed in sequence.
  • Learn that two equivalent expressions indicates the same value.
  • Learn that algebraic transformation preserve the equivalence among the forms of expression. 
  • Learn to recognize and to construct an opposite or reciprocal expression to an assigned one. 
  • Learn that the sum of opposite expressions is 0 while the product of reciprocal expressions is 1
  • Learn to justify rules of transformation involving opposite or reciprocal expressions.
  • Learn that a polynomial is an algebraic expression consituted by constants and variables combined through three operations: addition, subtraction and multiplication. 
  • Learn that polynomial roots are defined as the set of values that, if substituted for the variables, give the value 0 to the polynomial.
  • Learn to use the root of a polynomial to factorize it
  • Learn that solving an equation means finding the truth set that makes true the equality between two expressions.
  • Learn that equivalent equations are characterized by the same truth set
  • Learn that the truth set of equation whose members are two equivalent expressions is an identity
  • Learn that the truth set of an equation whose members are  never equal  is the empty set 

Cognitive goals show tooltip help

  • Learn how to generalize and to test a hypothesis

Social-affective goals show tooltip help

Develop the willingness and capacity to:

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

Instrumental goals show tooltip help

  • Learn how to use the algebraic line component of ALNUSET
  • Learn how to use the symbolic manipulator component of ALNUSET
  • Learn how to use both components in a coordinated fashion.

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

Theoretical framework show tooltip help

The new operative and representative order mediated by Alnuset

In this session we analyze the role of Alnuset, and in particular the role of the Algebraic line component, in defining a new operative and representative order of the algebraic techniques.
To  this aim we use the Peirce semiotic frame previously described.

In the Algebraic Line, the drag of the mobile point is index of what a variable denotes in a chosen numerical set. The link between the variable and its object of reference carried out by the drag, can be didactically exploited to construct a more concrete idea of the algebraic variable notion.

In this component, three geometrical models are available to construct additions, multiplications and powers (and respectively their opposite operations) between numbers and expressions already constructed in the algebraic line. By means of these models the operation between two expressions associated to two points on the algebraic line, is reflected in iconic way in the geometrical construction of the point associated to the expression-result. This iconic link can be exploited to attach meaning to the algebraic writing (for example, to attach meaning to the writings such as 3*x; x+3; x^3 understanding their differences). 

If an algebraic expression constructed and represented on the algebraic line not contains a variable, then the associated point is index of what the expression denotes in a determined way. On the contrary, if an algebraic expression constructed and represented on the algebraic line contains a variable, the associated point is index of what the expression denotes in not-determined way. We note that particular visual functions allow to keep control by visual perception on the relation among the variable, the expression containing that variable and the mathematical event of which the expression is index. This occurs even if the mobile point associated to the variable and the mobile point associated to the expression are spatially very distant on the algebraic line.

In algebra learning an important algebraic phenomena is making zero the polynomial value. This phenomena is strictly linked to the development of techniques which allow to find the polynomial roots. The algebraic line component of Alnuset provides a command which allows students to find the polynomial roots. This command allows student to keeps control of the technique on the base of her/his spatial, perceptive and motor experience. This command allows student to exploit the visualization, dynamism and interactivity properties of the system to find an approximate value of the root which makes null the polynomial. An algorithm of the system calculates the exact root on the base of this approximate value. Moreover, this algorithm provides a dynamic, graphical and numeric feedback of the computation process which it performs. The result found by this computational process is the polynomial root.
We note that this representative event mediated by the technology, can be considered an index of the algebraic phenomena  to make zero the polynomial value.

In this component the equivalence between two algebraic expressions is characterized by two representative events: the two expressions are associated to the same point and they are contained in the same post-it for each value associated to their variable moving the corresponding point on the line. These representative events are index of equivalence between the two expressions.

The equality between two expressions can define a proposition that can be an equation or an identity. We can observe that the equivalence phenomena is characterized as identity if:

  • it concerns two expressions in which the variable is not contained
  • it stay stable for any drags of the point corresponding to the variable contained in the expressions. In this last case, the two expressions can be manipulated and transformed one in the other. 

We can observe that the equivalence phenomena is characterized as conditioned equality between two expressions if the representative event appears only for some of the values corresponding to the variable point. In this case, the search of the conditions which allow the representative event to appear, is the solution of the equation defined as the equality between the two expressions.

In the AL, an algebraic proposition constructed by the editor, appears in a specific space and it is associated to a green or red ball. The ball color depends on the specific value of the unknown which defines the truth value of the proposition (red/false, green/truth). Moreover, by means of an graphical approach, a specific function allows student to define the truth set of the proposition on the algebraic line. The system translates the set in formal language and it attaches a colored ball. The color of the ball depends on the specific value of the unknown. This value makes conditional on the belonging of the unknown value to the set defined by the student (red/not belonging, green/belonging). The concordance of color between the two balls when the point associate to the unknown is dragged on any values of the algebraic line, is index of the fact that the edited set is the truth set of the proposition.

Algebraic Manipulator

As the Algebraic Line, the Algebraic Manipulator properties are useful to establish an iconic and index relation between a representative event and an algebraic phenomena.
In the following, we consider some of these representative events.

One of the most important representative event in Algebraic Manipulator is the exploration of the structure characterizing the algebraic expression or proposition which one want to manipulate. We observe that the position of the mouse pointer on a sign (number/letter, operator, brackets) of the expression sets up the framing of the part of the expression that that sign characterizes, on the hierarchical point of view, in the structure of the expression.

So, the representative dynamic event that is realized on the expression signs through the drag of the mouse is index of the possible decomposition of the expression in its constituent parts. These parts are characterized, on the hierarchical plan, by the element which is pointed by the mouse from time to time.

After selecting by mouse a part of expression, only commands that can be applied on it are highlighted to be available. This representative event is index of a structural analogy, that is an iconic link between the rule associated to the highlighted command and the selected part of expression. The application of the command produces the transformation according to the rule associated to it.

Some specific functions allow an integrated use between the two components. For example, there are two functions allowing to transfer expressions respectively from the algebraic manipulator to the algebraic line and vice-versa form the algebraic line to the manipulator.  Moreover, some commands of algebraic manipulations such as “Insert from AL” exploit results found in the algebraic line component to perform a transformation (for example to factorize a polynomial it is necessary to find its roots in the algebraic line).

These representative events related to the integrated use of the two components can be index of important algebraic phenomena. For example, the representation of an expression and its transformed expression (by the manipulator commands) on the algebraic line, allows to make emerge the representative event “post-it” which contains two algebraic equivalent expressions, as previously described.


New Pedagogical strategies mediated by Alnuset

In this pedagogical plan, ALNUSET is used according to two different pedagogical strategies:

  • Task solution based on the use of  pen & paper vs. task solution based on use of the tool
  • Explorative use of ALNUSET to find a solution to a task, followed by the question: What conclusion can you make?

Both pedagogical strategies can be a source of contradictions, and overcoming these can be the motor of learning.
We use Activity Theory to explain and to model the learning process mediated by these two pedagogical strategies.

Strategy 1: Task solution based on the use of  pen & paper vs. task solution based on use of the tool

Strategy 2: Explorative use of ALNUSET to find a solution to a task, followed by the question: What conclusion can you make?

Alnuset

ALNUSET (ALgebra on the NUmerical SETs) is a new educational digital artefact devoted to students of lower and upper secondary school (ages from 12-13 to 16-17). It includes three components: the Algebraic Line component, the Algebraic Manipulator componentand the Cartesian Plan component. The former is oriented to the development of an algebra of quantities, the second component deals with an algebra of operations and the third component is oriented to develop an algebra of functions. The educational significance of the system can emerge through the integrated use of these components in school practice.

The Algebraic line is an operative and representative environment for the construction of mathematical expressions involving numbers and letters defined on a specific numerical set (natural integers, relative integers, rational numbers, rational numbers extended to rational powers) and for the representation of lettering expressions as mobile points on the line. The Symbolic Manipulator component is a structured symbolic calculation and multiple representation environment for the manipulation of symbolic algebraic expressions, and for the solution of equations and inequations.

In this pedagogical plan the third component of Alnuset, the Cartesian Plan, will not be considered

Tool access show tooltip help

NIL

The Algebraic Line component [Component]

The Algebraic line component provides the opportunity to use letters as names of mobile points on the line. It provides geometrical models for graphically constructing sums and differences, products and ratios, powers and roots of the numbers, letters and expressions already represented on the line.

Every new expression constructed in this way is associated to a point on the Algebraic line that indicates the result of operations performed in sequence. The algebraic line is based on a representation built by mathematicians over previous centuries and is distinct from the number line, which is of a profoundly different nature. Indeed, the very nature of the Algebraic Line is algebraic, while the very nature of a number line is arithmetic. This algebraic nature is given by operational and representational features, i.e. by the fact that in ALNUSET this artefact is able to:

  • manage mobile points corresponding to algebraic variables;
  • manage the construction of literal expressions involving variables and their representation as points on the line;
  • manage the dragging of mobile points corresponding to algebraic variables and the movement of the points of the expressions involving those variables.

Since these characteristics are possible only through digital technology, we believe that the Algebraic Line can exist only as a digital artefact.

The Symbolic Manipulator component [Component]

The Algebraic Manipulator component is a structured symbolic calculation and multiple representation environment for the manipulation of symbolic algebraic expressions, and for the solution of equations and inequations.

In general, structured symbolic calculation environments are designed for educational activity. They allow the user to act on expressions and equations with step-by-step transformations. These characteristics make such environments quite different from CAS (Computer Algebra Systems). CAS are designed for professional mathematical activities; they allow the user to handle algebraic transformation with black-box algorithms, producing solutions that in many cases are hard for students to understand.

The Algebraic Manipulator of ALNUSET is designed to allow students to approach algebraic manipulation by constructing meanings that are well founded in algebraic activity and have a clear theoretical basis.

This component manages algebraic manipulations of a single expression and the solution of equations and inequations of a degree less than or equal to 4. The interface provides a set of fundamental commands for symbolic manipulation that is specific to the numerical domain chosen by the user (natural integers, relative integers, rational numbers, full range domain). The basic commands for algebraic transformation have been designed to allow the user to control each step in the algebraic transformation solution process. The commands are related to a system of operational properties and calculation rules; if they are applied to one part of an expression that can be suitably manipulated with those commands, then the entire expression is transformed in accordance with the embedded transformation rules. The set of algebraic manipulation commands available to the user is open ended: new commands can be added to those already available and used in subsequent manipulations.
The commands structure comprises:

  • a command for editing the text of the symbolic expression to be transformed or the equation/inequation to be solved
  • basic commands incorporating symbolic rewriting rules related to the properties of operations and of specific computation rules
  • a command for turning the manipulation that the user has just performed into a new symbolic rewriting rule
  • algebraic manipulation commands created by the user
  • commands for solving equations/inequations.

The symbolic rewriting rule associated to each command generates an expression that is equivalent to the original one.

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

20 hours

Actors' roles show tooltip help

STUDENTS

  • 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

  • Algebraic and polynomial expressions, comprising 4 activities:
  1. Exploring what an expression denotes through an algebra of quantities
  2. Exploring equivalent expressions by integrating an algebra of operations with an algebra of quantities
  3. Exploring opposite and reciprocal expressions
  4. Exploring roots of polynomials 

MODULE 3

  • Equations, comprising 2 activities:
  1. Exploring equations as conditioned equality between two expressions
  2. Exploring particular kinds of equations (involving two equivalent expressions or two never equal expresions or two opposite or two reciprocal expressions as members)

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

G. P. Chiappini, B. Pedemonte, E. Robotti, F. Vannucci

Subject domains show tooltip help

  • Algebra

Topics show tooltip help

  • Algebraic expressions
  • Polynomials
  • Equations

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

  • Expressions
  • Equations
  • Equivalence
  • Equality

Description show tooltip help

This module comprises a test to evaluate students' pre-knowledge of expressions and equations, particularly equality and equivalence notions in treating algebraic expressions and equations. This test should be proposed to students who have previously dealt with expressions and equations topics in school.

Rationale show tooltip helpexplode

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Population show tooltip helpexplode

School level show tooltip help

Second year of upper secondary school

Age range show tooltip help

15-16 years old

Population description show tooltip help

This pedagogical plan has been designed for students who have just developed syntactic competencies in the algebraic domain and who, according to our hypothesis, are still weak in terms of semantic competencies.

Nevertheless, this pedagogical plan could also be tackled, with appropriate modifications, by students engaged in the first approach to algebraic language. In this case the pre-test should not be proposed to students but could be transformed into a post-test.

Student prerequisites show tooltip help

  • Knowledge of operations with numbers (integers and rational numbers)
  • Familiarity with expression resolution.

Teacher prerequisites show tooltip help

No specialised mathematics knowledge is necessary.

Context show tooltip helpexplode

Physical context show tooltip help

 The test can be done in the classroom.

Goals show tooltip helpexplode

Content-epistemological goals show tooltip help

Many meanings that students build can easily be lost in traditional teaching. The aim of the test is not to highlight students' competencies about expressions, polynomials and equations, but to check the meanings students have  acquired about these topics.

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

Resources show tooltip helpexplode

test [Resource for students]

Work plan show tooltip helpexplode

Setting show tooltip help

The test is proposed to the class by the teacher. The students have to work individually. 

Time show tooltip help

2 hours

Actors' roles show tooltip help

Students solve the tasks set in the test.

The teacher does not intervene.

The researcher observes.

What to do and how show tooltip help

Each student in the class is given a test to complete without copying from schoolmates. No suggestions are to be given by the teacher. At the end of the work, the teacher collects the test papers.
The test papers are not to be corrrected. Over the course of the pedagogical plan, the tasks contained in the test will be proposed and corrected

Algebraic and polynomial expressions

Identity show tooltip helpexplode

Authors show tooltip help

G. P. Chiappini, B. Pedemonte, E. Robotti, F. Vannucci

Subject domains show tooltip help

  • Algebra

Topics show tooltip help

  • Algebraic expressions
  • Equivalent, opposite and reciprocal expressions
  • Polynomial expressions and polynomial roots

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

  • Algebraic line
  • Algebraic Manipulator
  • Activity theory
  • Algebra of quantities
  • Algebra of operations
  • Algebraic expressions
  • Polynomials

Description show tooltip help

This module focuses on algebraic and polynomial expressions with the aim to develop:

  • conceptualisation of what an expression denotes
  • comprehension of what characterizes two equivalent expressions, two opposite expressions or two reciprocal expressions
  • comprehension of the notion of polynomial roots and the capability to use them to factorize polynomial expression or to solve problems.

These objectives are pursued with the mediation of the two components of Alnuset so as to favour integration between an Algebra of Quantities and an Algebra of Operations. An Algebra of Quantities focuses on the denotative aspects of algebraic expressions while an Algebra of operations focuses on the operations which preserve formal properties of expressions within a numeric domain.

Rationale show tooltip helpexplode

Many learning difficulties in algebra are rooted in the passage from arithmetic to algebra, and in the way in which mathematical expressions are used in these two domains. A cognitive gap emerges when students pass from operating with numerical expressions composed only of numbers that are typical of arithmetic, to literal expressions composed also of letters that characterize the Algebra domain. Generally speaking, a mathematical expression is a writing composed of numbers and/or letters connected by the symbols of mathematical operations (addition, subtraction, multiplication, division, power of a number, extraction of root); this expression indicates the result of the operations performed in sequence.
We observe that a numerical expression indicates, in a determinate way, the number that is the result of the operations performed in sequence and it denotes this number (e.g. “3+2*5” and “11+2” denote the same number), while a literal expression indicates, in a indeterminate way, the result of the operations performed in sequence and it denotes the function that associates the result of the operations performed to each value of the letters (e.g. “2a+1”denotes the function f={ x --> 2x+1}).
Operating with numerical expressions, it is always possible to verify that two different expressions indicating two different sequences of operations can be equivalent by observing that they produce the same result, which they denote. This verification cannot be accomplished with literal expressions because the result of the sequence of operations is indeterminate. Hence their equivalence can only be demonstrated by transforming one of the two expressions into the other using rules of transformation that preserve their formal properties. This is one of the main aspects that characterize algebra as a science of formal operations and differentiates it from arithmetic as a science of quantities.
If the passage from numerical expressions to literal expressions is not mediated in an appropriate way on the didactical plane, difficulties and obstacles can emerge in the development of genuine algebraic thought, as witnessed by several researchers.
Our hypothesys is that the passage from arithmetic of quantities to algebra of formal operations should be mediated by algebra of quantities. Algebra of quantities is an algebra that focuses the attention on numerical quantities indicated in an indeterminate way by a literal symbol or by a literal expression. This need for algebra of quantities has a justification on the historical plane as well as on the didactical one. As a matter of fact, to give algebra a status of autonomous discipline, around 1830 Peacock distinguished between arithmetical and symbolic algebra - in the former attention is focused on quantities denoted by literal symbols and expressions while in the latter there is a separation from what the literal symbols and expressions denote, and an exclusive attention is paid to operations. This analysis has been at the basis of our research work that has oriented the design and the accomplishment of a new system named ALNUSET to support the teaching and learning processes in the algebraic domain. This PP shows how Alnuset can be exploit on the didactical plan to  take a deep innovation in the approach to algebraic and polynomial expressions. 

Theoretical framework show tooltip help

This pedagogical plan can be defined and justified by referring to two theoretical frameworks. The first is Peirce’s semantic frame.It is used to frame the ways mathematical objects and their interaction are represented in Alnuset and the ways representations of Alnuset can be acted on.

The second theoretical framework of reference is the Activity Theory. It is used to frame the pedagogical strategy used in the PP, to analyze the contradictions that can emerge in  the development of the didactical activities and the mediating role of Alnuset representations to overcome them. 

The use of the Peirce's semiotic frame

This pedagogical plan is based on the assumption that mathematical knowledge is the result of a social construction. This social construction is realized by a continuous dialectic between the development on the operational/procedural level of mathematical techniques, and their conceptualisation on a structural/relational level.
A mathematical technique is, first of all, a way to solve a problem. Moreover, every technique is composed of mechanic and automatic reasoning and operations (Chevallard, 1992; Artigue, 2002). Mathematical techniques are realized by the use of signs; they are characterized by operative and representative orders which are shared socially.
The operative order concerns the rules of use of signs; it orients the development of the technique on the procedural level.
The representative order concerns the relationship among signs (with their rules of use) and their referents; it orients the structural/relational interpretation of the technique.
Exploiting the possibilities of visualization, computation, dynamicity and interaction made available by digital technology, it is possible to instrument a mathematical technique. The instrumentation of a mathematical technique concerns a new way to use signs in the construction and interpretation of the technique. This fact is due to the new operative and representative order mediated by the technology. In order to analyse the differences between an instrumented and a standard technique, it is necessary to study how the role of the signs changes in relation to the new operative and representative order mediated by the technology. To this aim, it is useful to refer to Peirce’s distinction between three kinds of signs, i.e. symbol, icon and index. Peirce’s distinction is defined on the basis of the relation that each of these kinds of signs established with its object of reference (Peirce, 2003). Let us consider symbols, for example. Peirce says that these are signs that can be related with their objects of reference on the basis of specific conventional rules. By constrast, an icon is a sign whose shape and structure reflects the structure and the properties of the object of reference. It is important to stress that for Peirce symbols and icons are strictly linked. He says that the rules characterizing mathematical symbols include an iconic link with a property of the represented object. In addition to symbols and icons, Peirce also considers the indexes, namely signs whose link to the objects of reference is due to the fact that the sign is partially determined by the object itself (for instance, the  windsock is an index of wind direction, in the sense that its direction is determined by its object of reference). These functions of signs are at the basis on algebraic thinking.
For example, Peirce says that an algebraic expression can be considered an icon or an index sinceit can reveal new important aspects about the object of reference. In the natural integer domain, an algebraic expression such as 2x+1 is index of the the generic odd number because the form of this expression is determined by the property of this number “to be successors of even numbers”. As a matter of fact, the structure of the expression reflects iconically such property. By applying associative and distributive rules it is possible to transform the given expression into the following one: x+(x+1). Also in this form the espression is index of the generic odd number. Nevertheless, in this case the structure of this new expression iconically reflects another property of odd numbers: "to be addition of two consecutive numbers". It is not easy to understand the iconical connection established between algebraic rules and conventions and abstraict properties of the object of reference; it is not easy to understand the iconical relationship between a representative event and the object that determine it. 
Learning algebra means learning manipulation of algebraic symbols on the basis of rules and conventions as well as recognising the objects they are index of and the properties of these objects whose signs are icons.
Thus, in learning algebra the development of semiotic processes to recognise iconic and indexical aspects that characterize algebraic signs, is crucial. We think that technology can be exploit to make emerge the iconical and indexical aspects of a mathematical object. In particular, our hypothesys is that technology can make available new representative and operative possibilities to structure a new phenomenological space for algebraic activity, thus allowing teachers to reconfigure algebraic knowledge into an object of investigation for students and supplying students with new mediating tools to improve their learning.

The use of the Activity theory frame

In algebra didactical practices the operative and representative changes due to the introduction of technology allow to shape in other way the mathematical knowledge. Thus, in these new didactical practices, mathematical knowledge became an object of investigation. The operative and representative changes in the use of a technique established a new social order in its use too. With social order related to the use of a technique we mean the activities in which participants communicate, sharing and discuss ideas and meanings about mathematical phenomena.

In order to analyse the changes in the social order related to the use of a technique, we consider the activity in which this technique is used as unity of analysis and we use the Activity theory frame to study the changes which appear in this activity due to the instrumentation of the technique.

Activity Theory is a philosophical and cross-disciplinary theory adopted for studying different forms of human practice, such as teaching/learning, that are seen as development processes mediated by artefacts, where individual and social levels are simultaneously interlinked (Kuutti, 1996).
In Activity Theory, an activity is a form of acting directed towards an object, and it is the object that distinguishes one activity from another. Transforming the object into an outcome is what motivates the existence of an activity. According to the theory, activities consist of actions or chains of actions, which in turn consist of operations. If we consider Activity Theory applied to the educational field, the object of an activity is the learning of a given knowledge or the development of a given ability; the outcome of this activity, the motive for which the activity is developed, is students' acquisition of that knowledge or that ability (Bellamy, 1996).
Cole and Engeström (1991) have devised a systemic model expressing the complex relationships between elements in an activity (see figure below) which is particularly useful for studying the relationships that take place in teaching/learning activity (see also Engeström, 1987; 1991). Their model highlights three mutual relationships involved in every activity, namely the relationship between subject and object, that between subject and community, and that between community and object. Each of these relationships is mediated by a third entity. The relationship between subject and the object is mediated by artefacts that both enable and constrain the subject’s action. The relationship between subject and community is mediated by rules (explicit or implicit norms, conventions and social interactions), while that between community and object is mediated by the division of labour (different roles characterizing labour organization). 
The model depicted in the figure also shows that each entity mediates all the relationships described in the model. Artefacts used in the activity mediate not only the relationship between the subject and the object but also that between subject and community and that between community and object. Moreover, mediating entities are not mutually independent but rather exert influence over one another. For example, the introduction of a new artefact in an activity influences both the norms regulating participant interaction in the activity and the roles that the participants can assume.

This model allows us to analyze the mediation role played by the artifact during the classroom activities described in this pedagogical plan.
The artefact adopted in the plan is ALNUSET, and in particular two specific components are used: the algebraic line component and the manipulator component. These components and their integration mediate the student’s actions associated to the activity object. The details of this mediation role are described elsewhere in relation to the specific artefact-based learning activities: here we shall broadly outline the mediation role in general terms.

For a brief summary of Activity Theory, click here.

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Population show tooltip helpexplode

School level show tooltip help

Second year of upper secondary school

Age range show tooltip help

15-16 years old

Population description show tooltip help

This module has been designed for students who have just developed syntactic competencies in the algebraic domain and, according to our hypothesis, have semantic competencies that are still weak. Nevertheless, this module could also be tackled, with appropriate modifications, to students engaged in the first approach to algebraic language. 

Student prerequisites show tooltip help

  • Knowledge of operations with numbers (integers and rational numbers)
  • Basic skills in literal calculus 
  • 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 ALNUSET 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 first and second year of upper secondary school. Link to national maths curriculum document.

Socio-cultural context show tooltip help

Goals show tooltip helpexplode

Curricular goals show tooltip help

Learn how to:

  • Perform calculations with integer and rationals numbers;
  • Treat monomials and operations;
  • Factorise polynomials;

These are goals considered by the actual italian curricula. We want to pursue these goals, not only on the operative plan but also on the semantic plan.

Content-epistemological goals show tooltip help

  • Learn that a letter in algebra indicates the elements of the numerical set where it is defined.
  • Learn that a literal algebraic expression indicates the resul of the operations performed in sequence.
  • Learn that two equivalent expressions indicates the same value.
  • Learn that algebraic transformation preserve the equivalence among the forms of expression. 
  • Learn to recognize and to construct an opposite or reciprocal expression to an assigned one. 
  • Learn that the sum of opposite expressions is 0 while the product of reciprocal expressions is 1
  • Learn to justify rules of transformation involving opposite or reciprocal expressions.
  • Learn that a polynomial is an algebraic expression consituted by constants and variables combined through three operations: addition, subtraction and multiplication. 
  • Learn that polynomial roots are defined as the set of values that, if substituted for the variables, give the value 0 to the polynomial.
  • Learn to use the root of a polynomial to factorize it

Cognitive goals show tooltip help

  • Learn how to generalize and to test a hypothesis

Social-affective goals show tooltip help

Develop the willingness and capacity to:

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

Instrumental goals show tooltip help

 Learn how to use the following functions of the algebraic line component of ALNUSET:

  • the insertion of mobile points
  • the selection of a numerical set
  • the post-it
  • the polynomial root

Learn how to use the the following functions of the Algebraic Manipulator component of ALNUSET:

  • Selection of a part of an expression
  • Application of a rule to a selected part of an expression
  • Construction and insertion of a "User rule"
  • the factorise command.

Learn how to use the "send to line function" and the "send to manipulator function" to integrate the two components of Alnuset.

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

 

Theoretical framework show tooltip help

The new operative and representative order mediated by Alnuset

In this session we analyze the role of Alnuset, and in particular the role of the Algebraic line component, in defining a new operative and representative order of the algebraic techniques.
To  this aim we use the Peirce semiotic frame previously described.

In the Algebraic Line, the drag of the mobile point is index of what a variable denotes in a chosen numerical set. The link between the variable and its object of reference carried out by the drag, can be didactically exploited to construct a more concrete idea of the algebraic variable notion.

In this component, three geometrical models are available to construct additions, multiplications and powers (and respectively their opposite operations) between numbers and expressions already constructed in the algebraic line. By means of these models the operation between two expressions associated to two points on the algebraic line, is reflected in iconic way in the geometrical construction of the point associated to the expression-result. This iconic link can be exploited to attach meaning to the algebraic writing (for example, to attach meaning to the writings such as 3*x; x+3; x^3 understanding their differences). 

If an algebraic expression constructed and represented on the algebraic line not contains a variable, then the associated point is index of what the expression denotes in a determined way. On the contrary, if an algebraic expression constructed and represented on the algebraic line contains a variable, the associated point is index of what the expression denotes in not-determined way. We note that particular visual functions allow to keep control by visual perception on the relation among the variable, the expression containing that variable and the mathematical event of which the expression is index. This occurs even if the mobile point associated to the variable and the mobile point associated to the expression are spatially very distant on the algebraic line.

In algebra learning an important algebraic phenomena is making zero the polynomial value. This phenomena is strictly linked to the development of techniques which allow to find the polynomial roots. The algebraic line component of Alnuset provides a command which allows students to find the polynomial roots. This command allows student to keeps control of the technique on the base of her/his spatial, perceptive and motor experience. This command allows student to exploit the visualization, dynamism and interactivity properties of the system to find an approximate value of the root which makes null the polynomial. An algorithm of the system calculates the exact root on the base of this approximate value. Moreover, this algorithm provides a dynamic, graphical and numeric feedback of the computation process which it performs. The result found by this computational process is the polynomial root.
We note that this representative event mediated by the technology, can be considered an index of the algebraic phenomena  to make zero the polynomial value.

In this component the equivalence between two algebraic expressions is characterized by two representative events: the two expressions are associated to the same point and they are contained in the same post-it for each value associated to their variable moving the corresponding point on the line. These representative events are index of equivalence between the two expressions.

The equality between two expressions can define a proposition that can be an equation or an identity. We can observe that the equivalence phenomena is characterized as identity if:

  • it concerns two expressions in which the variable is not contained
  • it stay stable for any drags of the point corresponding to the variable contained in the expressions. In this last case, the two expressions can be manipulated and transformed one in the other. 

We can observe that the equivalence phenomena is characterized as conditioned equality between two expressions if the representative event appears only for some of the values corresponding to the variable point. In this case, the search of the conditions which allow the representative event to appear, is the solution of the equation defined as the equality between the two expressions.

In the AL, an algebraic proposition constructed by the editor, appears in a specific space and it is associated to a green or red ball. The ball color depends on the specific value of the unknown which defines the truth value of the proposition (red/false, green/truth). Moreover, by means of an graphical approach, a specific function allows student to define the truth set of the proposition on the algebraic line. The system translates the set in formal language and it attaches a colored ball. The color of the ball depends on the specific value of the unknown. This value makes conditional on the belonging of the unknown value to the set defined by the student (red/not belonging, green/belonging). The concordance of color between the two balls when the point associate to the unknown is dragged on any values of the algebraic line, is index of the fact that the edited set is the truth set of the proposition.

Algebraic Manipulator

As the Algebraic Line, the Algebraic Manipulator properties are useful to establish an iconic and index relation between a representative event and an algebraic phenomena.
In the following, we consider some of these representative events.

One of the most important representative event in Algebraic Manipulator is the exploration of the structure characterizing the algebraic expression or proposition which one want to manipulate. We observe that the position of the mouse pointer on a sign (number/letter, operator, brackets) of the expression sets up the framing of the part of the expression that that sign characterizes, on the hierarchical point of view, in the structure of the expression.

So, the representative dynamic event that is realized on the expression signs through the drag of the mouse is index of the possible decomposition of the expression in its constituent parts. These parts are characterized, on the hierarchical plan, by the element which is pointed by the mouse from time to time.

After selecting by mouse a part of expression, only commands that can be applied on it are highlighted to be available. This representative event is index of a structural analogy, that is an iconic link between the rule associated to the highlighted command and the selected part of expression. The application of the command produces the transformation according to the rule associated to it.

Some specific functions allow an integrated use between the two components. For example, there are two functions allowing to transfer expressions respectively from the algebraic manipulator to the algebraic line and vice-versa form the algebraic line to the manipulator.  Moreover, some commands of algebraic manipulations such as “Insert from AL” exploit results found in the algebraic line component to perform a transformation (for example to factorize a polynomial it is necessary to find its roots in the algebraic line).

These representative events related to the integrated use of the two components can be index of important algebraic phenomena. For example, the representation of an expression and its transformed expression (by the manipulator commands) on the algebraic line, allows to make emerge the representative event “post-it” which contains two algebraic equivalent expressions, as previously described.


New Pedagogical strategies mediated by Alnuset

In this pedagogical plan, ALNUSET is used according to two different pedagogical strategies:

  • Task solution based on the use of  pen & paper vs. task solution based on use of the tool
  • Explorative use of ALNUSET to find a solution to a task, followed by the question: What conclusion can you make?

Both pedagogical strategies can be a source of contradictions, and overcoming these can be the motor of learning.
We use Activity Theory to explain and to model the learning process mediated by these two pedagogical strategies.

Strategy 1: Task solution based on the use of  pen & paper vs. task solution based on use of the tool

Strategy 2: Explorative use of ALNUSET to find a solution to a task, followed by the question: What conclusion can you make?

ALNUSET

ALNUSET (ALgebra on the NUmerical SETs) is a new educational digital artefact devoted to students of lower and upper secondary school (ages from 12-13 to 16-17). It includes three components: the Algebraic Line component, the Algebraic Manipulator componentand the Cartesian Plan component. The former is oriented to the development of an algebra of quantities, the second component deals with an algebra of operations and the third component is oriented to develop an algebra of functions. The educational significance of the system can emerge through the integrated use of these components in school practice.

The Algebraic line is an operative and representative environment for the construction of mathematical expressions involving numbers and letters defined on a specific numerical set (natural integers, relative integers, rational numbers, rational numbers extended to rational powers) and for the representation of lettering expressions as mobile points on the line. The Symbolic Manipulator component is a structured symbolic calculation and multiple representation environment for the manipulation of symbolic algebraic expressions, and for the solution of equations and inequations.

In this pedagogical plan the third component of Alnuset, the Cartesian Plan, will not be considered.

Tool access show tooltip help

NIL

The Algebraic Line component [Component]

The Algebraic line component provides the opportunity to use letters as names of mobile points on the line. It provides geometrical models for graphically constructing sums and differences, products and ratios, powers and roots of the numbers, letters and expressions already represented on the line.

Every new expression constructed in this way is associated to a point on the Algebraic line that indicates the result of operations performed in sequence. The algebraic line is based on a representation built by mathematicians over previous centuries and is distinct from the number line, which is of a profoundly different nature. Indeed, the very nature of the Algebraic Line is algebraic, while the very nature of a number line is arithmetic. This algebraic nature is given by operational and representational features, i.e. by the fact that in ALNUSET this artefact is able to:

  • manage mobile points corresponding to algebraic variables;
  • manage the construction of literal expressions involving variables and their representation as points on the line;
  • manage the dragging of mobile points corresponding to algebraic variables and the movement of the points of the expressions involving those variables.

Since these characteristics are possible only through digital technology, we believe that the Algebraic Line can exist only as a digital artefact.

The Symbolic Manipulator component [Component]

The Algebraic Manipulator component is a structured symbolic calculation and multiple representation environment for the manipulation of symbolic algebraic expressions, and for the solution of equations and inequations.

In general, structured symbolic calculation environments are designed for educational activity. They allow the user to act on expressions and equations with step-by-step transformations. These characteristics make such environments quite different from CAS (Computer Algebra Systems). CAS are designed for professional mathematical activities; they allow the user to handle algebraic transformation with black-box algorithms, producing solutions that in many cases are hard for students to understand.

The Algebraic Manipulator of ALNUSET is designed to allow students to approach algebraic manipulation by constructing meanings that are well founded in algebraic activity and have a clear theoretical basis.

This component manages algebraic manipulations of a single expression and the solution of equations and inequations of a degree less than or equal to 4. The interface provides a set of fundamental commands for symbolic manipulation that is specific to the numerical domain chosen by the user (natural integers, relative integers, rational numbers, full range domain). The basic commands for algebraic transformation have been designed to allow the user to control each step in the algebraic transformation solution process. The commands are related to a system of operational properties and calculation rules; if they are applied to one part of an expression that can be suitably manipulated with those commands, then the entire expression is transformed in accordance with the embedded transformation rules. The set of algebraic manipulation commands available to the user is open ended: new commands can be added to those already available and used in subsequent manipulations.

The commands structure comprises:

  • a command for editing the text of the symbolic expression to be transformed or the equation/inequation to be solved
  • basic commands incorporating symbolic rewriting rules related to the properties of operations and of specific computation rules
  • a command for turning the manipulation that the user has just performed into a new symbolic rewriting rule
  • algebraic manipulation commands created by the user
  • commands for solving equations/inequations.

The symbolic rewriting rule associated to each command generates an expression that is equivalent to the original one.

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

13 hours

Actors' roles show tooltip help

STUDENTS

  • 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 module contains 4 activities:

  1. Exploring what an expression denotes through an algebra of quantities
  2. Exploring equivalent expressions by integrating an algebra of operations with an algebra of quantities
  3. Exploring opposite and reciprocal expressions
  4. Exploring roots of polynomials 

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

Exploring what an expression denotes through an algebra of quantities

Identity show tooltip helpexplode

Authors show tooltip help

G. P. Chiappini, B. Pedemonte, E. Robotti, F. Vannucci

Subject domains show tooltip help

  • Algebra

Topics show tooltip help

  • Algebraic expressions

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

  • Algebraic expression
  • Algebra of quantities
  • Algebraic line
  • Numeric set

Description show tooltip help

This PP focuses on algebraic expressions with the aim to conceptualise what an expression denotes from a quantitative point of view.
In this activity the Algebraic line of Alnuset is used as a mediation tool of an Algebra of Quantities. 

Rationale show tooltip helpexplode

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Population show tooltip helpexplode

School level show tooltip help

Second year of upper secondary school

Age range show tooltip help

15-16 years old

Population description show tooltip help

This module has been designed for students who have just developed syntactic competencies in the algebraic domain and, according to our hypothesis, have semantic competencies that are still weak. Nevertheless, this module could also be tackled, with appropriate modifications, to students engaged in the first approach to algebraic language. 

Student prerequisites show tooltip help

  • Knowledge of operations with numbers (integers and rational numbers)
  • Basic skills in literal calculus 
  • 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 ALNUSET 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 first and second year of upper secondary school. Link to national maths curriculum document.

Goals show tooltip helpexplode

Curricular goals show tooltip help

Learn how to:

  • Perform calculations with integer and rationals numbers;
  • Treat monomials and operations;

These are goals considered by the actual italian curricula. We want to pursue these goals, not only on the operative plan but also on the semantic plan.

Content-epistemological goals show tooltip help

  • Learn that a letter in algebra indicates the elements of the numerical set where it is defined.
  • Learn that a literal algebraic expression indicates the resul of the operations performed in sequence. 

Cognitive goals show tooltip help

  • Learn how to generalize and to test a hypothesis

Social-affective goals show tooltip help

Develop the willingness and capacity to:

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

Instrumental goals show tooltip help

Learn how to use the following functions of the algebraic line component of ALNUSET:

  • selection of the numerical set 
  • insertion of mobile points

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

Theoretical framework show tooltip help

The activities of the card 1 can be used to develop learning through contradictions that can emerge during the tasks solution. These contradictions can emerge in two following ways:

  • comparing students’ task solutions made first with pen and paper and then on the algebraic line of ALNUSET
  • comparing students’ task solutions made by different students exploring activities in ALNUSET.

Learning model based on Activity Theory: task solution with paper and pen vs use of Alnuset

Learning model based on Activity Theory: task solution based on explorative use of ALNUSET

Alnuset

ALNUSET (ALgebra on the NUmerical SETs) is a new educational digital artefact devoted to students of lower and upper secondary school (ages from 12-13 to 16-17). It includes three components: the Algebraic Line component, the Algebraic Manipulator componentand the Cartesian Plan component. The former is oriented to the development of an algebra of quantities, the second component deals with an algebra of operations and the third component is oriented to develop an algebra of functions. The educational significance of the system can emerge through the integrated use of these components in school practice.

The Algebraic line is an operative and representative environment for the construction of mathematical expressions involving numbers and letters defined on a specific numerical set (natural integers, relative integers, rational numbers, rational numbers extended to rational powers) and for the representation of lettering expressions as mobile points on the line. The Symbolic Manipulator component is a structured symbolic calculation and multiple representation environment for the manipulation of symbolic algebraic expressions, and for the solution of equations and inequations.

In this pedagogical plan the third component of Alnuset, the Cartesian Plan, will not be considered

Tool access show tooltip help

NIL

The Algebraic line component [Component]

The Algebraic line component provides the opportunity to use letters as names of mobile points on the line. It provides geometrical models for graphically constructing sums and differences, products and ratios, powers and roots of the numbers, letters and expressions already represented on the line.

Every new expression constructed in this way is associated to a point on the Algebraic line that indicates the result of operations performed in sequence. The algebraic line is based on a representation built by mathematicians over previous centuries and is distinct from the number line, which is of a profoundly different nature. Indeed, the very nature of the Algebraic Line is algebraic, while the very nature of a number line is arithmetic. This algebraic nature is given by operational and representational features, i.e. by the fact that in ALNUSET this artefact is able to:

  • manage mobile points corresponding to algebraic variables;
  • manage the construction of literal expressions involving variables and their representation as points on the line;
  • manage the dragging of mobile points corresponding to algebraic variables and the movement of the points of the expressions involving those variables.

Since these characteristics are possible only through digital technology, we believe that the Algebraic Line can exist only as a digital artefact.

Selection of the numerical set [Component Feature]

In the Algebraic line component it is possible to select a specific numerical set. This function is available via a command on the Domain popup menu. It allows the user to choose the numerical set (N, Z, Q, Q extended to rational power ) in which he/she wants to operate. The choice of numerical set modifies how data are displayed on the Algebraic Line: for example, if the domain of natural integers is selected, only positive numbers are displayed. The choice also poses limits on operations that are not closed in the chosen domain and on the movement of any variable point constructed on the line.

Insertion of mobile points [Component Feature]

In the Algebraic line component it is possible to insert a mobile point on the lines as a variable point, and to assign it a name in the form of a letter. Variable points can be dragged along the lines with the mouse in accordance with the restraints imposed by the chosen numerical domain: for example, if the domain of natural numbers has been selected, the variable point can only be dragged over natural numbers.

Addition/subtraction operations [Component Feature]

A specific function of the Algebraic line component allows the user to perform additions and subtractions on expressions represented on the Algebraic Line and to represent the result as a new expression associated to a point on the same line. The user can perform these operations by exploiting the geometrical model embedded in the function. This model constitutes the dynamic and interactive interface device through which two expressions can be instantiated for constructing a new expression that indicates the result of the operation performed.

The two images below show how the geometrical model can be used  for selecting the operation to be performed (addition or subtraction) and for instantiating the expression to be added or subtracted.

Selection of the operation to be performed (addition or subtraction) depends on the choice of the two points of the model used to instantiate the expressions. Instantiation of the expressions is performed by dragging the point of the geometrical model onto the point of the Algebraic line corresponding to the expression one wants to add or subtract.

The choice of numerical domain poses limits on subtraction. In the set of natural integers, the point “a-b” can only constructed for values of b whereby b<=a.

Multiplication/division operations [Component Feature]

A specific function of the Algebraic line component allows the user to perform multiplications and divisions on expressions represented on the Algebraic Line and to represent the result as a new expression associated to a point on the same line.

The user can perform these operations by exploiting the geometrical model embedded in the function. This model constitutes the dynamic and interactive interface device through which two expressions can be instantiated for constructing a new expression that indicates the result of the operation performed.

The two images below show how the geometrical model for multiplication or division embedded in this function can be used as an interface device for selecting the operation to be performed (multiplication or division) and for instantiating the expressions to be multiplied or divided.

Selection of the operation to be performed (multiplication or division) depends on the choice of the two points of the model used to instantiate the expressions. Instantiation of the expressions is performed by dragging the point of the geometrical model onto the point of the Algebraic line corresponding to the expression one wants to multiply or divide.

The choice of numerical domain poses limits on division. In the set of natural and relative integers, the point “a:b” can only constructed for values of a and b whereby the result of a:b belongs to the chosen domain.

Integer Power/Rational Power operations [Component Feature]

A specific function of the Algebraic line component allows the user to perform the integer power or rational power of an expression represented on the Algebraic Line and to represent the result as a new expression associated to a point on the same line.

The user can perform these operations by exploiting the geometrical model embedded in the function. This model constitutes the dynamic and interactive interface device through which the expressions can be instantiated for constructing a new expression that indicates the result of the operation performed.

The two images below show how the geometrical model for integer power or rational power can be used as an interface device for selecting the operation to be performed (integer power or rational power), for defining its grade and for instantiating the expression on which to perform the selected operation. The grade of the operation can be dynamically increased or decreased by using two specific buttons. Selection of the operation to be performed (integer power and rational power) depends on the choice of the point of the model used to instantiate the expression.

Instantiation of the expression is performed by dragging the point of the geometrical model onto the point of the Algebraic line corresponding to the expression of which one wants to determine the integer power or the rational power.


The choice of numerical domain poses limits on the rational power. In the set of natural and relative integers and of rational numbers, the point corresponding to a rational power can only be constructed for numerical values whereby the result of the rational power belongs to the chosen domain.

Resources show tooltip helpexplode

General description show tooltip help

The main aim of this card is to highlight what a letter and/or an expression denotes in a specific numeric set.

This aim will be achieved by 7 tasks.

Card 1 [Resource for students]

The aim of this card is to develop knowledge about what expression denotes in a particular numerical set.

There are 7 tasks in this card: some of them highlight the different denotations of an expression if considered in different numeric sets (tasks: a), b)), some other tasks highlight denotation of an expression stressing the quantitative point of view (tasks c), d), e), f)), some tasks highlight denotation of expressions observing their writings (task e)) .

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

3 hours

Actors' roles show tooltip help

STUDENTS

  • 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 arranged in pairs at each computer where Alnuset is running. The teacher delivers each pair the card containing the tasks, which are to be solved working with Alnuset. The answers to the task are to be written in the students' notebooks or in the card.

At the beginning the teacher gives some information about the Algebraic line of Alnuset. In particular he/she explains how to insert a mobile point on the line and to change numerical domain.

Students then solve the tasks following indications given in the card. The teacher can propose a discussion about the results found.

Exploring equivalent expressions integrating an algebra of operations with an algebra of quantities

Identity show tooltip helpexplode

Authors show tooltip help

G. P. Chiappini, B. Pedemonte, E. Robotti, F. Vannucci

Subject domains show tooltip help

  • Algebra

Topics show tooltip help

  • Algebraic expressions
  • Equivalent expressions

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

  • Algebra of quantities
  • Algebra of operations
  • Equivalent expressions
  • Algebraic line
  • Algebraic Manipulator

Description show tooltip help

This pedagogical plan focuses on algebraic expressions with the aim of developing comprehension of the aspects characterizing two equivalent expressions: this is achieved through integration of an algebra of quantities and an algebra of operations.
An Algebra of Quantities focuses on the denotative aspects of algebraic expressions while an Algebra of operations focuses on the operations which preserves equivalence of expressions within a numeric domain.To favour integration of these, the two components of Alnuset are used: the Algebraic Line component and the Symbolic Manipulator component.

Rationale show tooltip helpexplode

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Population show tooltip helpexplode

School level show tooltip help

Second year of upper secondary school

Age range show tooltip help

15-16 years old

Population description show tooltip help

This module has been designed for students who have just developed syntactic competencies in the algebraic domain and, according to our hypothesis, have semantic competencies that are still weak. Nevertheless, this module could also be tackled, with appropriate modifications, to students engaged in the first approach to algebraic language. 

Student prerequisites show tooltip help

  • Knowledge of operations with numbers (integers and rational numbers)
  • Basic skills in literal calculus 
  • 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 ALNUSET 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 first and second year of upper secondary school. Link to national maths curriculum document.

Goals show tooltip helpexplode

Curricular goals show tooltip help

Learn how to:

  • Perform calculations with integer and rationals numbers;
  • Treat monomials and operations;

These are goals considered by the actual italian curricula. We want to pursue these goals, not only on the operative plan but also on the semantic plan.

Content-epistemological goals show tooltip help

  • Learn that two equivalent expressions indicates the same value.
  • Learn that algebraic transformation preserve the equivalence among the forms of expression.  

Cognitive goals show tooltip help

  • Learn how to generalize and to test a hypothesis

Social-affective goals show tooltip help

Develop the willingness and capacity to:

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

Instrumental goals show tooltip help

Learn how to use the following functions of the algebraic line component of ALNUSET:

  • the insertion of mobile points
  • the post-it

Learn how to use the the following functions of the Algebraic Manipulator component of ALNUSET:

  • Selection of a part of expression
  • Application of a rule to a selected part of an expression
  • construction and insertion of a "User rule" 

Learn how to use the "send to line function"

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

This activity focuses on an algebra of operations that preserve formal properties within a numeric domain.
The aim of the activity is comprehension of the equivalence of expressionsby means of a symbolic transformation approach integrated with a quantitative approach. In this context, two expressions are  deemed equivalent by virtue of the fact that they denote the same numerical value and they can be symbolically transformed into each other. The transformation can be performed by operations which preserve formal properties.

Theoretical framework show tooltip help

The tasks of this card mediate learning of the equivalence between expressions. Two expressions are equivalent if one can be transformed into the other by means of symbolic transformation. The symbolic transformation is performed in the algebraic manipulator component of ALNUSET: in this case the equivalence is verified within an algebra of operations. Some tasks of this card require students to verify the equivalence of these expressions in the algabraic line. In this case, equivalence is verified within an algebra of quantities.

Description of these tasks according to Cole and Engestrom‘s model is as follows

Alnuset

ALNUSET (ALgebra on the NUmerical SETs) is a new educational digital artefact devoted to students of lower and upper secondary school (ages from 12-13 to 16-17). It includes three components: the Algebraic Line component, the Algebraic Manipulator componentand the Cartesian Plan component. The former is oriented to the development of an algebra of quantities, the second component deals with an algebra of operations and the third component is oriented to develop an algebra of functions. The educational significance of the system can emerge through the integrated use of these components in school practice.

The Algebraic line is an operative and representative environment for the construction of mathematical expressions involving numbers and letters defined on a specific numerical set (natural integers, relative integers, rational numbers, rational numbers extended to rational powers) and for the representation of lettering expressions as mobile points on the line. The Symbolic Manipulator component is a structured symbolic calculation and multiple representation environment for the manipulation of symbolic algebraic expressions, and for the solution of equations and inequations.

In this pedagogical plan the third component of Alnuset, the Cartesian Plan, will not be considered

Tool access show tooltip help

NIL

The Algebraic Line component [Component]

The Algebraic line component provides the opportunity to use letters as names of mobile points on the line. It provides geometrical models for graphically constructing sums and differences, products and ratios, powers and roots of the numbers, letters and expressions already represented on the line.

Every new expression constructed in this way is associated to a point on the Algebraic line that indicates the result of operations performed in sequence. The algebraic line is based on a representation built by mathematicians over previous centuries and is distinct from the number line, which is of a profoundly different nature. Indeed, the very nature of the Algebraic Line is algebraic, while the very nature of a number line is arithmetic. This algebraic nature is given by operational and representational features, i.e. by the fact that in ALNUSET this artefact is able to:

  • manage mobile points corresponding to algebraic variables;
  • manage the construction of literal expressions involving variables and their representation as points on the line;
  • manage the dragging of mobile points corresponding to algebraic variables and the movement of the points of the expressions involving those variables.

Since these characteristics are possible only through digital technology, we believe that the Algebraic Line can exist only as a digital artefact.

Post-it [Component Feature]

The constructed expressions on the Algebraic Line are displayed in specific spaces named post-it. In general, a post-it contains all the equivalent expressions  constructed by the user and associated to a specific point.

The figure below show a post-it containing equivalent expressions corresponding to the same point.

The constructed expressions are also visualised in a specific space "Values" at the bottom of the Algebraic Line.

The Symbolic Manipulator component [Component]

The Algebraic Manipulator component is a structured symbolic calculation and multiple representation environment for the manipulation of symbolic algebraic expressions, and for the solution of equations and inequations.

In general, structured symbolic calculation environments are designed for educational activity. They allow the user to act on expressions and equations with step-by-step transformations. These characteristics make such environments quite different from CAS (Computer Algebra Systems). CAS are designed for professional mathematical activities; they allow the user to handle algebraic transformation with black-box algorithms, producing solutions that in many cases are hard for students to understand.

The Algebraic Manipulator of ALNUSET is designed to allow students to approach algebraic manipulation by constructing meanings that are well founded in algebraic activity and have a clear theoretical basis.

This component manages algebraic manipulations of a single expression and the solution of equations and inequations of a degree less than or equal to 4. The interface provides a set of fundamental commands for symbolic manipulation that is specific to the numerical domain chosen by the user (natural integers, relative integers, rational numbers, full range domain). The basic commands for algebraic transformation have been designed to allow the user to control each step in the algebraic transformation solution process. The commands are related to a system of operational properties and calculation rules; if they are applied to one part of an expression that can be suitably manipulated with those commands, then the entire expression is transformed in accordance with the embedded transformation rules. The set of algebraic manipulation commands available to the user is open ended: new commands can be added to those already available and used in subsequent manipulations.

The commands structure comprises:

  • a command for editing the text of the symbolic expression to be transformed or the equation/inequation to be solved
  • basic commands incorporating symbolic rewriting rules related to the properties of operations and of specific computation rules
  • a command for turning the manipulation that the user has just performed into a new symbolic rewriting rule
  • algebraic manipulation commands created by the user
  • commands for solving equations/inequations.

The symbolic rewriting rule associated to each command generates an expression that is equivalent to the original one.

Selection of a part of expression [Component Feature]

Operators, brackets and literal or numerical elements define the parts of the expression and their hierarchical organization. When the mouse pointer is positioned over any part of the expression, the system dynamically outlines the chunk of the expression at the hierarchical level determined by the activated element, ranging from an individual elements through to the entire expression. This facilitates the user in the subsequent step of dynamically highlighting a given part of the expression at a given hierarchical level by clicking and dragging the cursor. The selected part of the expression is thus highlighted in yellow, as is any applicable rewriting rule of the interface that can be applied to it.

Application of a rule to a selected part of an expression [Component Feature]

When a part of the symbolic representation is selected for manipulation, it comes under the system’s control and only the rewriting rules that can be applied to that part are available. This is illustrated in the figure below: the entire symbolic representation has been selected and only some rules are made available for manipulation (highlighted in yellow).

 
When a command incorporating a symbolic rewriting rule is applied to the selected part of the symbolic representation, the rule that has been applied is displayed beside the transformed expression.

Create a user rule [Component Feature]

The function “Create User Rule” creates a new symbolic manipulation command associated to the rule E <=> F, once the algebraic manipulation proving that expression E is equivalent to expression F has been completed.

The system automatically associates the last transformation of the expression to F. The new command is immediately added to the list of user-created commands in the User Rules space.

Send to line [Component Feature]

This function makes it possible to send a selected expression or a selected part of it to the Algebraic Line. The expression automatically appears in the Values space of Algebraic Line component and associated to a point on the line as a label .

Resources show tooltip helpexplode

Card 2 [Resource for students]

The aim of this card is to develop knowledge about equivalence between expressions.
This aim will be achieved by 6 tasks which are useful for thinking over specific aspects characterizing the concept of equivalent expressions.

Some tasks of this card concern the equivalence between expressions mainly within a quantitative point of view (tasks a), b)) and some other tasks concern this topic within an algebra of operations (tasks c), d), e), f)). 

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

STUDENTS

  • 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 organised in pairs working at each computer where Alnuset is running. The teacher delivers the card containing the tasks to each pair.Tasks are to be solved working with Alnuset. The answer to each is to be written in the students' notebooks or on the card.

At the beginning the teacher gives some information about the Algebraic Manipulator component of Alnuset. In particular he/she explains how to:

  • change numerical domain;
  • insert an expression; 
  • apply a specific rule to the expression by selecting a part of it;
  • create a new rule
  • how to send an expression from the Algebraic Line component to the Algebraic Manipulator component. 

Then, students solve tasks proposed in the card following the given indications. At the end of this work the teacher could propose a discussion concerning equivalnt expressions.

Exploring opposite and reciprocal expressions

Identity show tooltip helpexplode

Authors show tooltip help

G. P. Chiappini, B. Pedemonte, E. Robotti, F. Vannucci

Subject domains show tooltip help

  • Algebra

Topics show tooltip help

  • Algebraic expressions
  • Opposite expressions
  • Reciprocal expressions

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

Description show tooltip help

This pedagogical plan focuses on algebraic expressions with the aim to develop the comprehension of what characterize two opposite expressions and two reciprocal expressions through an integration between an algebra of quantities and an algebra of operations.

These objectives are pursued with the mediation of the two components of Alnuset so as to favour integration between an Algebra of Quantities and an Algebra of Operations. An Algebra of Quantities focuses on the denotative aspects of algebraic expressions while an Algebra of operations focuses on the operations which preserve formal properties of expressions within a numeric domain.

Rationale show tooltip helpexplode

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Population show tooltip helpexplode

School level show tooltip help

Second year of upper secondary school

Age range show tooltip help

15-16 years old

Population description show tooltip help

This module has been designed for students who have just developed syntactic competencies in the algebraic domain and, according to our hypothesis, have semantic competencies that are still weak. Nevertheless, this module could also be tackled, with appropriate modifications, to students engaged in the first approach to algebraic language. 

Student prerequisites show tooltip help

  • Knowledge of operations with numbers (integers and rational numbers)
  • Basic skills in literal calculus 
  • 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 ALNUSET 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 first and second year of upper secondary school. Link to national maths curriculum document.

Goals show tooltip helpexplode

Curricular goals show tooltip help

Learn how to:

  • Perform calculations with integer and rationals numbers;
  • Treat monomials and operations;

These are goals considered by the actual italian curricula. We want to pursue these goals, not only on the operative plan but also on the semantic plan.

Content-epistemological goals show tooltip help

  • Learn to recognize and to construct an opposite or reciprocal expression to an assigned one. 
  • Learn that the sum of opposite expressions is 0 while the product of reciprocal expressions is 1
  • Learn to justify rules of transformation involving opposite or reciprocal expressions. 

Cognitive goals show tooltip help

  • Learn how to generalize and to test a hypothesis

Social-affective goals show tooltip help

Develop the willingness and capacity to:

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

Instrumental goals show tooltip help

 Learn how to use the following functions of the algebraic line component of ALNUSET:

  • the insertion of mobile points
  • the selection of a numerical set
  • the post-it

Learn how to use the the following functions of the Algebraic Manipulator component of ALNUSET:

  • Selection of a part of an expression
  • Application of a rule to a selected part of an expression
  • construction and insertion of a "User rule"

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

The main aim of this activity is developing the comprehension of what characterize two opposite expressions, and two reciprocal expressions.

This aim will be achieved through an integration between an algebra of quantities and an algebra of operations by means of three cards: card 3, card 4 and card 5.

Theoretical framework show tooltip help

The aim of the card 3 is mediating the learning of opposite expressions. The aim of the card 4 is mediating the learning of reciprocal expressions and the aim of the card 5 is mediating the learning of the two specific rules A+-A=0 and A*1/A=1 and their connection respectively with opposite expressions and reciprocal expressions.

These cards ask students to represent these mathematical concepts on the Algebraic line component and to demontrate them in the Algebraic Manipulator component. 

The description of these cards by  the Cole and Engestrom‘s model will be presented in the following.

ALNUSET

ALNUSET (ALgebra on the NUmerical SETs) is a new educational digital artefact devoted to students of lower and upper secondary school (ages from 12-13 to 16-17). It includes three components: the Algebraic Line component, the Algebraic Manipulator componentand the Cartesian Plan component. The former is oriented to the development of an algebra of quantities, the second component deals with an algebra of operations and the third component is oriented to develop an algebra of functions. The educational significance of the system can emerge through the integrated use of these components in school practice.

The Algebraic line is an operative and representative environment for the construction of mathematical expressions involving numbers and letters defined on a specific numerical set (natural integers, relative integers, rational numbers, rational numbers extended to rational powers) and for the representation of lettering expressions as mobile points on the line. The Symbolic Manipulator component is a structured symbolic calculation and multiple representation environment for the manipulation of symbolic algebraic expressions, and for the solution of equations and inequations.

In this pedagogical plan the third component of Alnuset, the Cartesian Plan, will not be considered.

Tool access show tooltip help

NIL

The Algebraic Line component [Component]

The Algebraic line component provides the opportunity to use letters as names of mobile points on the line. It provides geometrical models for graphically constructing sums and differences, products and ratios, powers and roots of the numbers, letters and expressions already represented on the line.

Every new expression constructed in this way is associated to a point on the Algebraic line that indicates the result of operations performed in sequence. The algebraic line is based on a representation built by mathematicians over previous centuries and is distinct from the number line, which is of a profoundly different nature. Indeed, the very nature of the Algebraic Line is algebraic, while the very nature of a number line is arithmetic. This algebraic nature is given by operational and representational features, i.e. by the fact that in ALNUSET this artefact is able to:

  • manage mobile points corresponding to algebraic variables;
  • manage the construction of literal expressions involving variables and their representation as points on the line;
  • manage the dragging of mobile points corresponding to algebraic variables and the movement of the points of the expressions involving those variables.

Since these characteristics are possible only through digital technology, we believe that the Algebraic Line can exist only as a digital artefact.

Insertion of mobile points [Component Feature]

In the Algebraic line component it is possible to insert a mobile point on the lines as a variable point, and to assign it a name in the form of a letter.

Variable points can be dragged along the lines with the mouse in accordance with the restraints imposed by the chosen numerical domain. The possibility to drag a variable point is useful for seeing the relationship between an expression and the opposite expression. As the figure below shows, the user can understand the relationship between the expression x and the expression -x by experiencing them through actions.

Post-it [Component Feature]

The constructed expressions on the Algebraic Line are displayed in specific spaces named post-it. In general, a post-it contains all the equivalent expressions  constructed by the user and associated to a specific point.

The figure below show a post-it containing equivalent expressions corresponding to the same point.

The constructed expressions are also visualised in a specific space "Values" at the bottom of the Algebraic Line.

Selection of the numerical set [Component Feature]

In the Algebraic line component it is possible to select a specific numerical set.
This function is available via a command in the Domain popup menu. It allows the user to choose the numerical set (N, Z, Q, Q extended to rational power ) in which he/she wants to operate. The choice of numerical set modifies how data are displayed on the Algebraic Line: for example, if the domain of relative integers is selected, rational numbers are not visualised on the line, as shown in the figure below. The choice also poses limits on operations that are not closed in the chosen domain and on the movement of any variable point constructed on the line.

 

The Algebraic Manipulator component [Component]

The Algebraic Manipulator component is a structured symbolic calculation and multiple representation environment for the manipulation of symbolic algebraic expressions, and for the solution of equations and inequations.

In general, structured symbolic calculation environments are designed for educational activity. They allow the user to act on expressions and equations with step-by-step transformations. These characteristics make such environments quite different from CAS (Computer Algebra Systems). CAS are designed for professional mathematical activities; they allow the user to handle algebraic transformation with black-box algorithms, producing solutions that in many cases are hard for students to understand.

The Algebraic Manipulator of ALNUSET is designed to allow students to approach algebraic manipulation by constructing meanings that are well founded in algebraic activity and have a clear theoretical basis.

This component manages algebraic manipulations of a single expression and the solution of equations and inequations of a degree less than or equal to 4. The interface provides a set of fundamental commands for symbolic manipulation that is specific to the numerical domain chosen by the user (natural integers, relative integers, rational numbers, full range domain). The basic commands for algebraic transformation have been designed to allow the user to control each step in the algebraic transformation solution process. The commands are related to a system of operational properties and calculation rules; if they are applied to one part of an expression that can be suitably manipulated with those commands, then the entire expression is transformed in accordance with the embedded transformation rules. The set of algebraic manipulation commands available to the user is open ended: new commands can be added to those already available and used in subsequent manipulations.

The commands structure comprises:

  • a command for editing the text of the symbolic expression to be transformed or the equation/inequation to be solved
  • basic commands incorporating symbolic rewriting rules related to the properties of operations and of specific computation rules
  • a command for turning the manipulation that the user has just performed into a new symbolic rewriting rule
  • algebraic manipulation commands created by the user
  • commands for solving equations/inequations.

The symbolic rewriting rule associated to each command generates an expression that is equivalent to the original one.

Application of a rule to a selected part of an expression [Component Feature]

A part of an expression has to be selected in order to apply a specific rule to it. When a part of the symbolic representation is selected for manipulation, only those commands that can be applied to that part are available. The figure below illustrates the manipulation of the expression 5-7.

When a command incorporating a symbolic rewriting rule is applied to the selected part of the symbolic representation, the rule that has been applied is displayed beside the transformed expression.

Create a user rule [Component Feature]

The function “Create User Rule” creates a new symbolic manipulation command associated to the rule E <=> F, once the algebraic manipulation proving that expression E is equivalent to expression F has been completed. The system automatically associates the last transformation of the expression to F. The new command is immediately added to the list of user-created commands in the User Rules space. The figure below shows the transformation of the expression (a+b)*(a-b) into expression a2-b2: the rule has been added in the user rules space.

Resources show tooltip helpexplode

General description show tooltip help

The main aim of this activity is to propose an innovative approach to opposite expressions and to reciprocal expressions based on the integrated use of algebra of quantities and algebra of operations.

Card 3 [Resource for students]

The aim of this card is to develop the concept of opposite expressions

This aim will be achieved by 6 tasks useful for thinking over specific aspects characterizing the concept of opposite expressions. Some tasks concern the exploration of some specific caracteristics of the opposite expressions on the AL (task b)). Some tasks (task a), task c)) concern the exploration of properties that caracterise opposite expressions: a-b=a+-b; a+-a=0; -a=-1*a. This exploration is based on the integrated use between algebra of quantities and algebra of operation. Some tasks require to recognise opposite expressions by their form (task d)) and to justify theyr answer from a theoretical point of view (task e), task f)). 

Card 4 [Resource for students]

The aim of this card is to develop the concept of reciprocal expressions.

This aim will be achieved by 8 tasks useful for thinking over specific aspects characterizing the concept of reciprocal expressions. Some tasks concern the exploration of some specific caracteristics of the reciprocal expressions on the AL (task b), task e)). Some tasks (task a), task c)) concern the exploration of properties that caracterise reciprocal expressions: a*1/a=1; a^(-1)=1/a. This exploration is based on the integrated use between algebra of quantities and algebra of operation. Some tasks require to recognise opposite expressions by their form (task d), task h)) and to justify theyr answer from a theoretical point of view (task f), task g)). 

Card 5 [Resource for students]

The aim of this card is to highlight the two properties a+-a=0 and a*1/a =1 that respectively caracterise opposite and reciprocal expressions.

This aim will be achieved by 7 tasks useful for thinking over specific aspects characterizing the concept of reciprocal expressions with some restrictions. Some tasks introduce these rules instantiated on specific examples (task a), task b)), some tasks allow students to reflect about the direct and indirect use of these rules (task c), taskd), task e)). Finally there are two tasks to strenght the introduced concepts in this PP (task f), task g)).

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

6 hours: approximately 2 hours for each card

Actors' roles show tooltip help

STUDENTS

  • 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

It could be better to deliver the three cards in three different sections. These cards concern important topics, usually not easy for students to understand.

Students are arranged in pairs at each computer where Alnuset is running. The teacher delivers card 3 containing the tasks to each pair. The tasks are to be solved working with Alnuset. The answer to the task is to be written in the students' notebooks or on the card.

Card 4 is about reciprocal expressions. We can suppose that at this point students are probably familiar with the tool and with the kinds of activity proposed in the card. So this card can probably be filled in by students and a discussion about reciprocal expressions can be planned when students have finished the tasks.

Card 5 allow to strenght the previously developed concepts highlighting on specific rules. This card could be used by the teacher to verify whether students have acquired some important aspects of the topics addressed.

Exploring roots of polynomials

Identity show tooltip helpexplode

Authors show tooltip help

G. P. Chiappini, B. Pedemonte, E. Robotti, F. Vannucci

Subject domains show tooltip help

  • Algebra

Topics show tooltip help

  • Polynomial expressions
  • Polynomial roots

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

Description show tooltip help

This pedagogical plan focuses on algebraic and polynomial expressions with the aim of developing comprehension of the notion of polynomial roots and the capability to use them to factorize polynomial expression or to solve problems.
These objectives are pursued with the mediation of the two components of Alnuset so as to favour integration between an Algebra of Quantities and an Algebra of Operations. An Algebra of Quantities focuses on the denotative aspects of algebraic expressions while an Algebra of operations focuses on the operations which preserve formal properties of expressions within a numeric domain.

Rationale show tooltip helpexplode

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Population show tooltip helpexplode

School level show tooltip help

Second year of upper secondary school

Age range show tooltip help

15-16 years old

Population description show tooltip help

This module has been designed for students who have just developed syntactic competencies in the algebraic domain and, according to our hypothesis, have semantic competencies that are still weak. Nevertheless, this module could also be tackled, with appropriate modifications, to students engaged in the first approach to algebraic language. 

Student prerequisites show tooltip help

  • Knowledge of operations with numbers (integers and rational numbers)
  • Basic skills in literal calculus 
  • 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 ALNUSET 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 first and second year of upper secondary school. Link to national maths curriculum document.

Goals show tooltip helpexplode

Curricular goals show tooltip help

Learn how to factorise polynomials.

This is a goal considered by the actual italian curricula. We want to pursue these goals, not only on the operative plan but also on the semantic plan.

Content-epistemological goals show tooltip help

  • Learn that a polynomial is an algebraic expression consituted by constants and variables combined through three operations: addition, subtraction and multiplication. 
  • Learn that polynomial roots are defined as the set of values that, if substituted for the variables, give the value 0 to the polynomial.
  • Learn to use the root of a polynomial to factorize it

Cognitive goals show tooltip help

  • Learn how to generalize and to test a hypothesis

Social-affective goals show tooltip help

Develop the willingness and capacity to:

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

Instrumental goals show tooltip help

Learn how to use the "polynomial root function" of the algebraic line component of ALNUSET.
Learn how to use the "factorise command" of the Algebraic Manipulator component of ALNUSET.
Learn how to use the "send to line function" and the "send to manipulator function" to integrate the two components of ALNUSET

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

This PP is about polynomial expressions. In particular, the focus of this activity to highlight the polynomial roots concept and the factorisation concept for polynomial expression.

Theoretical framework show tooltip help

Taking into account a specific numeric set, the roots of a polynomial expression are the values of x in which the polynomial expression becomes null. Thus, by means an algebra of quantities, these values of x are found by means an activity of exploration in AL and by means the Roots function in AL component. Through algebra of operations,  the factorisation concept of an polynomial expression is mediated by the factorisation function in AM. This function is available only if the polynomial roots are already found in AL.

In this activity ALNUSET  is used to explore the possible solutions of the tasks.

The description of this card by  the Cole and Engestrom‘s model will be presented in the following.

Alnuset

ALNUSET (ALgebra on the NUmerical SETs) is a new educational digital artefact devoted to students of lower and upper secondary school (ages from 12-13 to 16-17). It includes three components: the Algebraic Line component, the Algebraic Manipulator componentand the Cartesian Plan component. The former is oriented to the development of an algebra of quantities, the second component deals with an algebra of operations and the third component is oriented to develop an algebra of functions. The educational significance of the system can emerge through the integrated use of these components in school practice.

The Algebraic line is an operative and representative environment for the construction of mathematical expressions involving numbers and letters defined on a specific numerical set (natural integers, relative integers, rational numbers, rational numbers extended to rational powers) and for the representation of lettering expressions as mobile points on the line. The Symbolic Manipulator component is a structured symbolic calculation and multiple representation environment for the manipulation of symbolic algebraic expressions, and for the solution of equations and inequations.

In this pedagogical plan the third component of Alnuset, the Cartesian Plan, will not be considered.

Tool access show tooltip help

NIL

The Algebraic Line component [Component]

The Algebraic line component provides the opportunity to use letters as names of mobile points on the line. It provides geometrical models for graphically constructing sums and differences, products and ratios, powers and roots of the numbers, letters and expressions already represented on the line.

Every new expression constructed in this way is associated to a point on the Algebraic line that indicates the result of operations performed in sequence. The algebraic line is based on a representation built by mathematicians over previous centuries and is distinct from the number line, which is of a profoundly different nature. Indeed, the very nature of the Algebraic Line is algebraic, while the very nature of a number line is arithmetic. This algebraic nature is given by operational and representational features, i.e. by the fact that in ALNUSET this artefact is able to:

  • manage mobile points corresponding to algebraic variables;
  • manage the construction of literal expressions involving variables and their representation as points on the line;
  • manage the dragging of mobile points corresponding to algebraic variables and the movement of the points of the expressions involving those variables.

Since these characteristics are possible only through digital technology, we believe that the Algebraic Line can exist only as a digital artefact.

Polynomial root [Component Feature]

This function allows the user to find real roots of a polynomial starting from an approximate value the user has found by dragging the polynomial variable to a point on the Algebraic line where the polynomial is null or tends to be null. The figure below illustrates the computation process performed by the system to find a root of the polynomial x2-2.

A graphical animation (red triangle vs green triangle) indicates the automatic computation process performed by the system to determine the numerical value of the root. The root of the polynomial is presented as a label associated to the x point on the blue line. To accept the result, the user has to click on the label.

Once the results have been accepted, the polynomial roots are displayed underneath the corresponding polynomial in the Roots window under the Algebraic line (see the figure below).

The results obtained from this function can be used by the “factorize” function in the Algebraic Manipulator to build the factorized polynomial.

Send to manipulator [Component Feature]

This function allows the user to send an expression constructed in the Algebraic line component to the Manipulator component. This can done by clicking the right mouse button on the expression on the line or on the expression in the "Value" window. An example is shown in the figure below.

 

We observe that if the user has already found the root of the polynomial P(x) in the Algebraic Line component and wishes to factorise it, s/he can send the expression to the Manipulator component so as to apply the factorize command to that expression.

The Symbolic Manipulator Component [Component]

The Algebraic Manipulator component is a structured symbolic calculation and multiple representation environment for the manipulation of symbolic algebraic expressions, and for the solution of equations and inequations.

In general, structured symbolic calculation environments are designed for educational activity. They allow the user to act on expressions and equations with step-by-step transformations. These characteristics make such environments quite different from CAS (Computer Algebra Systems). CAS are designed for professional mathematical activities; they allow the user to handle algebraic transformation with black-box algorithms, producing solutions that in many cases are hard for students to understand.

The Algebraic Manipulator of ALNUSET is designed to allow students to approach algebraic manipulation by constructing meanings that are well founded in algebraic activity and have a clear theoretical basis.

This component manages algebraic manipulations of a single expression and the solution of equations and inequations of a degree less than or equal to 4. The interface provides a set of fundamental commands for symbolic manipulation that is specific to the numerical domain chosen by the user (natural integers, relative integers, rational numbers, full range domain). The basic commands for algebraic transformation have been designed to allow the user to control each step in the algebraic transformation solution process. The commands are related to a system of operational properties and calculation rules; if they are applied to one part of an expression that can be suitably manipulated with those commands, then the entire expression is transformed in accordance with the embedded transformation rules. The set of algebraic manipulation commands available to the user is open ended: new commands can be added to those already available and used in subsequent manipulations.

The commands structure comprises:

  • a command for editing the text of the symbolic expression to be transformed or the equation/inequation to be solved
  • basic commands incorporating symbolic rewriting rules related to the properties of operations and of specific computation rules
  • a command for turning the manipulation that the user has just performed into a new symbolic rewriting rule
  • algebraic manipulation commands created by the user
  • commands for solving equations/inequations.

The symbolic rewriting rule associated to each command generates an expression that is equivalent to the original one.

Factorise command [Component Feature]

The "Factor roots" command makes it possible to factorize polynomials of the 2nd, 3rd or 4th degree in a variable and with integer coefficients on the basis of its roots found in the Algebraic component. We note that the factorize command is available in all Computer Algebra Systems. These systems calculate polynomial roots using a black-box command, in other words a complex series of internal procedures that are not transparent to the user. By contrast, in the ALNUSET symbolic manipulator, the factorize command draws on the results obtained with the “polynomial root” command in the Algebraic Line component for performing polynomial factorization. We recall that this command allows the student to determine the exact real root of a polynomial using the approximate root value s/he has already found by dragging the polynomial variable to a point on the line where the polynomial becomes, or tends to become, null.
Let’s examine how the factorize command works in ALNUSET. If the student has already found the root of the polynomial P(x) in the Algebraic Line component and wishes to factorise it, s/he applies the factorize command to it in the Algebraic Manipulator and obtains the factorisation based on the original root. As shown in the figure below, this command is available (yellow colour) only if at least one root of the polynomial expression has been found in the Algebraic line component.

Resources show tooltip helpexplode

Card 6 [Resource for students]

The main aim of task a) is to highlight the factorisation concept for polynomial expressions by means of an exploration of polynomial roots. The task requires the student to find roots by the computational function present in the system. Task b) require to find roots of a polynomial in the AL in order to factorise it in the Algebraic Manipulator component  Task c) is another similar activity. The aim of  task c) is to introduce the concept of conditioned equality between two expressions.
The two last tasks of the card introduce the concept of equation which will be developed in the next PP.

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

STUDENTS

  • 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 arranged in pairs at each computer where Alnuset is running. The teacher delivers the card containing the tasks to each pair. Tasks are to be solved working with Alnuset. The answer to the tasks are to be written in the students' notebooks or on the card.

At the beginning the teacher explains the way to find the roots of a poynomial on the Algebraic line and the way to factorise the polynomial in the Algebraic Manipulator component. She also shows the command which allows the user to send an expression from the Algebraic line component to the Symbolic Manipulator component.

After that, students solve alone the proposed tasks a).  At the end of this work the teacher could propose a discussion in order to explicate the emerged results and to underline some important concepts concerning roots of polynomials.

Equations

Identity show tooltip helpexplode

Authors show tooltip help

G. P. Chiappini, B. Pedemonte, E. Robotti, F. Vannucci

Subject domains show tooltip help

  • Algebra

Topics show tooltip help

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

  • Algebraic Line
  • Algebraic Manipulator
  • Activity theory
  • Algebra of quantities
  • Algebra of operations
  • Algebraic expressions
  • Polynomials
  • Equations
  • Alnuset

Description show tooltip help

This pedagogical plan focuses on equations with the aim to develop:

  • the conceptualisation of the  equation in term of contitioned equality between two expressions
  • the conceptualisation  that an equation denote a truth value 
  • the capability to use a quantitative and a formal approach in finding the truth set of an equation
  • the capability to face equations involving as their members respectively two equivalent expressions or two never equal expressions, or two opposite expressions or two reciprocal expressions 

This pedagogical plan is constitued by two sub PP, each of them is characterized by a working card for students. 

Rationale show tooltip helpexplode

An equation (or algebraic proposition) is a piece of writing that expresses a relationship between two algebraic expressions by means of the signs of comparison (=, >,<). It denotes a truth value (true/false) that is univocally determined if the two expressions of the proposition are numerical (i.e the proposition 2*3+2=10-2 denotes “true”; the proposition 2*3=5 denote “false”) while it can be conditioned by the value assumed by the letter when the proposition is literal (i.e. the proposition “2x-5=x-1” denotes “true” for x=4 while it denotes “false” for all the other numerical values of x). Research works have highlighted great difficulties when students pass from the use of numerical propositions to the use of lettering propositions. Difficulties concern the control of what is the truth value of a proposition according to the value assumed by the letters that characterize it.
How can the equal sign between two algebraic expressions be interpreted? Can an algebraic expression be equal to another one without being equivalent to it? How can the transformation of an algebraic expression and the solution of an equation be interpreted? Answering these questions might not be easy for students, as well as justifying appropriately their replies or fulfilling tasks that require a good mastery of the notions of equivalence and equality involved in these questions. In the current teaching practice, students mainly learn to use algebraic rules properly, but they often show difficulties both in justifying them and assigning meanings to the previously mentioned notions. In order to develop these competencies changes and new levels of mediation in teaching are necessary. A first essential change might consist in focusing the students’ attention either on the numerical quantities indicated by a literal symbol or expression or on the numerical quantities that can condition the truth value of an equality. Our hypothesys is that the current approach to algebra teaching based on formal operations, should be deeply modified and integrated with algebra of quantities. Algebra of quantities is an algebra that focuses the attention on numerical quantities indicated in an indeterminate way by a literal symbol or by a literal expressionor by a literal expression, or on the numerical quantities (truth set) that condition the equality or inequality of a proposition. This need for algebra of quantities has a justification on the historical plane as well as on the didactical one. As a matter of fact, to give algebra a status of autonomous discipline, around 1830 Peacock distinguished between arithmetical and symbolic algebra - in the former attention is focused on quantities denoted by literal symbols and expressions while in the latter there is a separation from what the literal symbols and expressions denote, and an exclusive attention is paid to operations. This analysis has been at the basis of our research work that has oriented the design and the accomplishment of a new system named ALNUSET to support the teaching and learning processes in the algebraic domain. This PP shows how Alnuset can be exploit on the didactical plan to take a deep innovation in the approach to equations. 

Theoretical framework show tooltip help

This pedagogical plan can be defined and justified by referring to two theoretical frameworks. The first is Peirce’s semantic frame.It is used to frame the ways mathematical objects and their interaction are represented in Alnuset and the ways representations of Alnuset can be acted on.

The second theoretical framework of reference is the Activity Theory. It is used to frame the pedagogical strategy used in the PP, to analyze the contradictions that can emerge in  the development of the didactical activities and the mediating role of Alnuset representations to overcome them. 

The use of the Peirce's semiotic frame

This pedagogical plan is based on the assumption that mathematical knowledge is the result of a social construction. This social construction is realized by a continuous dialectic between the development on the operational/procedural level of mathematical techniques, and their conceptualisation on a structural/relational level.
A mathematical technique is, first of all, a way to solve a problem. Moreover, every technique is composed of mechanic and automatic reasoning and operations (Chevallard, 1992; Artigue, 2002). Mathematical techniques are realized by the use of signs; they are characterized by operative and representative orders which are shared socially.
The operative order concerns the rules of use of signs; it orients the development of the technique on the procedural level.
The representative order concerns the relationship among signs (with their rules of use) and their referents; it orients the structural/relational interpretation of the technique.
Exploiting the possibilities of visualization, computation, dynamicity and interaction made available by digital technology, it is possible to instrument a mathematical technique. The instrumentation of a mathematical technique concerns a new way to use signs in the construction and interpretation of the technique. This fact is due to the new operative and representative order mediated by the technology. In order to analyse the differences between an instrumented and a standard technique, it is necessary to study how the role of the signs changes in relation to the new operative and representative order mediated by the technology. To this aim, it is useful to refer to Peirce’s distinction between three kinds of signs, i.e. symbol, icon and index. Peirce’s distinction is defined on the basis of the relation that each of these kinds of signs established with its object of reference (Peirce, 2003). Let us consider symbols, for example. Peirce says that these are signs that can be related with their objects of reference on the basis of specific conventional rules. By constrast, an icon is a sign whose shape and structure reflects the structure and the properties of the object of reference. It is important to stress that for Peirce symbols and icons are strictly linked. He says that the rules characterizing mathematical symbols include an iconic link with a property of the represented object. In addition to symbols and icons, Peirce also considers the indexes, namely signs whose link to the objects of reference is due to the fact that the sign is partially determined by the object itself (for instance, the  windsock is an index of wind direction, in the sense that its direction is determined by its object of reference). These functions of signs are at the basis on algebraic thinking.
For example, Peirce says that an algebraic expression can be considered an icon or an index sinceit can reveal new important aspects about the object of reference. In the natural integer domain, an algebraic expression such as 2x+1 is index of the the generic odd number because the form of this expression is determined by the property of this number “to be successors of even numbers”. As a matter of fact, the structure of the expression reflects iconically such property. By applying associative and distributive rules it is possible to transform the given expression into the following one: x+(x+1). Also in this form the espression is index of the generic odd number. Nevertheless, in this case the structure of this new expression iconically reflects another property of odd numbers: "to be addition of two consecutive numbers". It is not easy to understand the iconical connection established between algebraic rules and conventions and abstraict properties of the object of reference; it is not easy to understand the iconical relationship between a representative event and the object that determine it. 
Learning algebra means learning manipulation of algebraic symbols on the basis of rules and conventions as well as recognising the objects they are index of and the properties of these objects whose signs are icons.
Thus, in learning algebra the development of semiotic processes to recognise iconic and indexical aspects that characterize algebraic signs, is crucial. We think that technology can be exploit to make emerge the iconical and indexical aspects of a mathematical object. In particular, our hypothesys is that technology can make available new representative and operative possibilities to structure a new phenomenological space for algebraic activity, thus allowing teachers to reconfigure algebraic knowledge into an object of investigation for students and supplying students with new mediating tools to improve their learning.

The use of the Activity theory frame

In algebra didactical practices the operative and representative changes due to the introduction of technology allow to shape in other way the mathematical knowledge. Thus, in these new didactical practices, mathematical knowledge became an object of investigation. The operative and representative changes in the use of a technique established a new social order in its use too. With social order related to the use of a technique we mean the activities in which participants communicate, sharing and discuss ideas and meanings about mathematical phenomena.

In order to analyse the changes in the social order related to the use of a technique, we consider the activity in which this technique is used as unity of analysis and we use the Activity theory frame to study the changes which appear in this activity due to the instrumentation of the technique.

Activity Theory is a philosophical and cross-disciplinary theory adopted for studying different forms of human practice, such as teaching/learning, that are seen as development processes mediated by artefacts, where individual and social levels are simultaneously interlinked (Kuutti, 1996).
In Activity Theory, an activity is a form of acting directed towards an object, and it is the object that distinguishes one activity from another. Transforming the object into an outcome is what motivates the existence of an activity. According to the theory, activities consist of actions or chains of actions, which in turn consist of operations. If we consider Activity Theory applied to the educational field, the object of an activity is the learning of a given knowledge or the development of a given ability; the outcome of this activity, the motive for which the activity is developed, is students' acquisition of that knowledge or that ability (Bellamy, 1996).
Cole and Engeström (1991) have devised a systemic model expressing the complex relationships between elements in an activity (see figure below) which is particularly useful for studying the relationships that take place in teaching/learning activity (see also Engeström, 1987; 1991). Their model highlights three mutual relationships involved in every activity, namely the relationship between subject and object, that between subject and community, and that between community and object. Each of these relationships is mediated by a third entity. The relationship between subject and the object is mediated by artefacts that both enable and constrain the subject’s action. The relationship between subject and community is mediated by rules (explicit or implicit norms, conventions and social interactions), while that between community and object is mediated by the division of labour (different roles characterizing labour organization). 
The model depicted in the figure also shows that each entity mediates all the relationships described in the model. Artefacts used in the activity mediate not only the relationship between the subject and the object but also that between subject and community and that between community and object. Moreover, mediating entities are not mutually independent but rather exert influence over one another. For example, the introduction of a new artefact in an activity influences both the norms regulating participant interaction in the activity and the roles that the participants can assume.

This model allows us to analyze the mediation role played by the artifact during the classroom activities described in this pedagogical plan.
The artefact adopted in the plan is ALNUSET, and in particular two specific components are used: the algebraic line component and the manipulator component. These components and their integration mediate the student’s actions associated to the activity object. The details of this mediation role are described elsewhere in relation to the specific artefact-based learning activities: here we shall broadly outline the mediation role in general terms.

For a brief summary of Activity Theory, click here.

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Population show tooltip helpexplode

School level show tooltip help

Second year of upper secondary school

Age range show tooltip help

15-16 years old

Population description show tooltip help

This module has been designed for students who have just developed syntactic competencies in the algebraic domain. Our hypothesis is that semantic competencies are weak in this field for these students. Nevertheless, this module could also be proposed, with appropriate modifications, to students in the first approach to algebraic language. 

Student prerequisites show tooltip help

  • Knowledge of the operations with numbers (integers and rational numbers)
  • Basic skills in literal calculus 
  • Basic skills in the solution of first degree equations 
  • 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 ALNUSET 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 first and second year of upper secondary school. Link to national maths curriculum document.

Goals show tooltip helpexplode

Curricular goals show tooltip help

Learn how to solve first- and second-degree integer equations and inequations

This is a goal considered by the actual italian curricula. We want to pursue these goals, not only on the operative plan but also on the semantic plan.

Content-epistemological goals show tooltip help

  • Learn that solving an equation means finding the truth set that makes true the equality between two expressions.
  • Learn that equivalent equations are characterized by the same truth set
  • Learn that the truth set of equation whose members are two equivalent expressions is an identity
  • Learn that the truth set of an equation whose members are  never equal  is the empty set 

Cognitive goals show tooltip help

  • Learn how to generalize and to test a hypothesis

Social-affective goals show tooltip help

Develop the willingness and capacity to:

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

Instrumental goals show tooltip help

Learn how to use the following functions of the algebraic line component of ALNUSET:

  • the truth set function
  • the post-it
  • the polynomial root

Learn how to use the rules of the Algebraic Manipulator component and in particular the equations and inequations rules. Learn how to use the "send to line function" and the "send to manipulator function" to integrate the two components of Alnuset.

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

Theoretical framework show tooltip help

 The learning activities of the card 10 focus on contradictions which can emerge from hypothesis performed in paper and pen and their validation /confutation performed in AL or in AM. Overcoming these contradictions learning can be realized.

In the following we will present a learning model based on Activity Theory which describe how learning can be realized

The new operative and representative order mediated by Alnuset

In this session we analyze the role of Alnuset, and in particular the role of the Algebraic line component, in defining a new operative and representative order of the algebraic techniques.
To  this aim we use the Peirce semiotic frame previously described.

In the Algebraic Line, the drag of the mobile point is index of what a variable denotes in a chosen numerical set. The link between the variable and its object of reference carried out by the drag, can be didactically exploited to construct a more concrete idea of the algebraic variable notion.

In this component, three geometrical models are available to construct additions, multiplications and powers (and respectively their opposite operations) between numbers and expressions already constructed in the algebraic line. By means of these models the operation between two expressions associated to two points on the algebraic line, is reflected in iconic way in the geometrical construction of the point associated to the expression-result. This iconic link can be exploited to attach meaning to the algebraic writing (for example, to attach meaning to the writings such as 3*x; x+3; x^3 understanding their differences). 

If an algebraic expression constructed and represented on the algebraic line not contains a variable, then the associated point is index of what the expression denotes in a determined way. On the contrary, if an algebraic expression constructed and represented on the algebraic line contains a variable, the associated point is index of what the expression denotes in not-determined way. We note that particular visual functions allow to keep control by visual perception on the relation among the variable, the expression containing that variable and the mathematical event of which the expression is index. This occurs even if the mobile point associated to the variable and the mobile point associated to the expression are spatially very distant on the algebraic line.

In algebra learning an important algebraic phenomena is making zero the polynomial value. This phenomena is strictly linked to the development of techniques which allow to find the polynomial roots. The algebraic line component of Alnuset provides a command which allows students to find the polynomial roots. This command allows student to keeps control of the technique on the base of her/his spatial, perceptive and motor experience. This command allows student to exploit the visualization, dynamism and interactivity properties of the system to find an approximate value of the root which makes null the polynomial. An algorithm of the system calculates the exact root on the base of this approximate value. Moreover, this algorithm provides a dynamic, graphical and numeric feedback of the computation process which it performs. The result found by this computational process is the polynomial root.
We note that this representative event mediated by the technology, can be considered an index of the algebraic phenomena  to make zero the polynomial value.

In this component the equivalence between two algebraic expressions is characterized by two representative events: the two expressions are associated to the same point and they are contained in the same post-it for each value associated to their variable moving the corresponding point on the line. These representative events are index of equivalence between the two expressions.

The equality between two expressions can define a proposition that can be an equation or an identity. We can observe that the equivalence phenomena is characterized as identity if:

  • it concerns two expressions in which the variable is not contained
  • it stay stable for any drags of the point corresponding to the variable contained in the expressions. In this last case, the two expressions can be manipulated and transformed one in the other. 

We can observe that the equivalence phenomena is characterized as conditioned equality between two expressions if the representative event appears only for some of the values corresponding to the variable point. In this case, the search of the conditions which allow the representative event to appear, is the solution of the equation defined as the equality between the two expressions.

In the AL, an algebraic proposition constructed by the editor, appears in a specific space and it is associated to a green or red ball. The ball color depends on the specific value of the unknown which defines the truth value of the proposition (red/false, green/truth). Moreover, by means of an graphical approach, a specific function allows student to define the truth set of the proposition on the algebraic line. The system translates the set in formal language and it attaches a colored ball. The color of the ball depends on the specific value of the unknown. This value makes conditional on the belonging of the unknown value to the set defined by the student (red/not belonging, green/belonging). The concordance of color between the two balls when the point associate to the unknown is dragged on any values of the algebraic line, is index of the fact that the edited set is the truth set of the proposition.

Algebraic Manipulator

As the Algebraic Line, the Algebraic Manipulator properties are useful to establish an iconic and index relation between a representative event and an algebraic phenomena.
In the following, we consider some of these representative events.

One of the most important representative event in Algebraic Manipulator is the exploration of the structure characterizing the algebraic expression or proposition which one want to manipulate. We observe that the position of the mouse pointer on a sign (number/letter, operator, brackets) of the expression sets up the framing of the part of the expression that that sign characterizes, on the hierarchical point of view, in the structure of the expression.

So, the representative dynamic event that is realized on the expression signs through the drag of the mouse is index of the possible decomposition of the expression in its constituent parts. These parts are characterized, on the hierarchical plan, by the element which is pointed by the mouse from time to time.

After selecting by mouse a part of expression, only commands that can be applied on it are highlighted to be available. This representative event is index of a structural analogy, that is an iconic link between the rule associated to the highlighted command and the selected part of expression. The application of the command produces the transformation according to the rule associated to it.

Some specific functions allow an integrated use between the two components. For example, there are two functions allowing to transfer expressions respectively from the algebraic manipulator to the algebraic line and vice-versa form the algebraic line to the manipulator.  Moreover, some commands of algebraic manipulations such as “Insert from AL” exploit results found in the algebraic line component to perform a transformation (for example to factorize a polynomial it is necessary to find its roots in the algebraic line).

These representative events related to the integrated use of the two components can be index of important algebraic phenomena. For example, the representation of an expression and its transformed expression (by the manipulator commands) on the algebraic line, allows to make emerge the representative event “post-it” which contains two algebraic equivalent expressions, as previously described.


New Pedagogical strategies mediated by Alnuset

In this pedagogical plan, ALNUSET is used according to two different pedagogical strategies:

  • Task solution based on the use of  pen & paper vs. task solution based on use of the tool
  • Explorative use of ALNUSET to find a solution to a task, followed by the question: What conclusion can you make?

Both pedagogical strategies can be a source of contradictions, and overcoming these can be the motor of learning.
We use Activity Theory to explain and to model the learning process mediated by these two pedagogical strategies.

Strategy 1: Task solution based on the use of  pen & paper vs. task solution based on use of the tool

Strategy 2: Explorative use of ALNUSET to find a solution to a task, followed by the question: What conclusion can you make?

Alnuset

ALNUSET (ALgebra on the NUmerical SETs) is a new educational digital artefact devoted to students of lower and upper secondary school (ages from 12-13 to 16-17). It includes three components: the Algebraic Line component, the Algebraic Manipulator componentand the Cartesian Plan component. The former is oriented to the development of an algebra of quantities, the second component deals with an algebra of operations and the third component is oriented to develop an algebra of functions. The educational significance of the system can emerge through the integrated use of these components in school practice.

The Algebraic line is an operative and representative environment for the construction of mathematical expressions involving numbers and letters defined on a specific numerical set (natural integers, relative integers, rational numbers, rational numbers extended to rational powers) and for the representation of lettering expressions as mobile points on the line. The Symbolic Manipulator component is a structured symbolic calculation and multiple representation environment for the manipulation of symbolic algebraic expressions, and for the solution of equations and inequations.

In this pedagogical plan the third component of Alnuset, the Cartesian Plan, will not be considered.

Tool access show tooltip help

NIL

The Algebraic line component [Component]

The Algebraic line component provides the opportunity to use letters as names of mobile points on the line. It provides geometrical models for graphically constructing sums and differences, products and ratios, powers and roots of the numbers, letters and expressions already represented on the line.

Every new expression constructed in this way is associated to a point on the Algebraic line that indicates the result of operations performed in sequence. The algebraic line is based on a representation built by mathematicians over previous centuries and is distinct from the number line, which is of a profoundly different nature. Indeed, the very nature of the Algebraic Line is algebraic, while the very nature of a number line is arithmetic. This algebraic nature is given by operational and representational features, i.e. by the fact that in ALNUSET this artefact is able to:

  • manage mobile points corresponding to algebraic variables;
  • manage the construction of literal expressions involving variables and their representation as points on the line;
  • manage the dragging of mobile points corresponding to algebraic variables and the movement of the points of the expressions involving those variables.

Since these characteristics are possible only through digital technology, we believe that the Algebraic Line can exist only as a digital artefact.

The Symbolic Manipulator component [Component]

The Algebraic Manipulator component is a structured symbolic calculation and multiple representation environment for the manipulation of symbolic algebraic expressions, and for the solution of equations and inequations.

In general, structured symbolic calculation environments are designed for educational activity. They allow the user to act on expressions and equations with step-by-step transformations. These characteristics make such environments quite different from CAS (Computer Algebra Systems). CAS are designed for professional mathematical activities; they allow the user to handle algebraic transformation with black-box algorithms, producing solutions that in many cases are hard for students to understand.

The Algebraic Manipulator of ALNUSET is designed to allow students to approach algebraic manipulation by constructing meanings that are well founded in algebraic activity and have a clear theoretical basis.

This component manages algebraic manipulations of a single expression and the solution of equations and inequations of a degree less than or equal to 4. The interface provides a set of fundamental commands for symbolic manipulation that is specific to the numerical domain chosen by the user (natural integers, relative integers, rational numbers, full range domain). The basic commands for algebraic transformation have been designed to allow the user to control each step in the algebraic transformation solution process. The commands are related to a system of operational properties and calculation rules; if they are applied to one part of an expression that can be suitably manipulated with those commands, then the entire expression is transformed in accordance with the embedded transformation rules. The set of algebraic manipulation commands available to the user is open ended: new commands can be added to those already available and used in subsequent manipulations.

The commands structure comprises:

  • a command for editing the text of the symbolic expression to be transformed or the equation/inequation to be solved
  • basic commands incorporating symbolic rewriting rules related to the properties of operations and of specific computation rules
  • a command for turning the manipulation that the user has just performed into a new symbolic rewriting rule
  • algebraic manipulation commands created by the user
  • commands for solving equations/inequations.

The symbolic rewriting rule associated to each command generates an expression that is equivalent to the original one.

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

7 hours

Actors' roles show tooltip help

STUDENTS

  • 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 module comprises 2 activity phases:

  1. Exploring equations as conditioned equality between two expressions
  2. Exploring particular kinds of equations (involving two equivalent expressions or two never equal expresions or two opposite or two reciprocal expressions as members) 

Teaching activities and the way to propose them in class are described elsewhere.

Exploring equations as conditioned equality between two expressions

Identity show tooltip helpexplode

Authors show tooltip help

G. P. Chiappini, B. Pedemonte, E. Robotti, F. Vannucci

Subject domains show tooltip help

  • Algebra

Topics show tooltip help

  • Equations

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

Description show tooltip help

This pedagogical plan focuses on equations with the aim to develop:

  • the conceptualisation of the equation in term of contitioned equality between two expressions
  • the conceptualisation that an equation denotes a truth value 
  • the capability to use a quantitative and a formal approach in finding the truth set of an equation

Rationale show tooltip helpexplode

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Population show tooltip helpexplode

School level show tooltip help

Second year of upper secondary school

Age range show tooltip help

15-16 years old

Population description show tooltip help

This module has been designed for students who have just developed syntactic competencies in the algebraic domain. Our hypothesis is that semantic competencies are weak in this field for these students. Nevertheless, this module could also be proposed, with appropriate modifications, to students in the first approach to algebraic language. 

Student prerequisites show tooltip help

  • Knowledge of the operations with numbers (integers and rational numbers)
  • Basic skills in literal calculus 
  • Basic skills in the solution of first degree equations 
  • 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 ALNUSET 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 first and second year of upper secondary school. Link to national maths curriculum document.

Goals show tooltip helpexplode

Curricular goals show tooltip help

Learn how to solve first- and second-degree integer equations and inequations

This is a goal considered by the actual italian curricula. We want to pursue these goals, not only on the operative plan but also on the semantic plan.

Content-epistemological goals show tooltip help

  • Learn that solving an equation means finding the truth set that makes true the equality between two expressions.

Cognitive goals show tooltip help

  • Learn how to generalize and to test a hypothesis

Social-affective goals show tooltip help

Develop the willingness and capacity to:

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

Instrumental goals show tooltip help

Learn how to use the following functions of the algebraic line component of ALNUSET:

  • the truth set function
  • the polynomial root

Learn how to use the equations and inequations rules of the Algebraic Manipulator component. Learn how to use the "send to line function" and the "send to manipulator function" to integrate the two components of Alnuset.

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

The main aim of this PP is to conceptualise equations in term of conditioned equality between two expressions and  that an equation denotes atruth set of values

Theoretical framework show tooltip help

The action of inserting the equal sign between two algebraic expressions leads to structuring an algebraic equality. On the morphological plan an algebraic equality is a writing composed of two algebraic expressions connected through the comparison sign “=”. On the semantic plan, the algebraic equality is a propositional function that indicates a truth value (true/false) in a particular numerical domain. We observe that the truth value of an equality can be conditioned by numerical values assigned to the variables it contains. The truth value concerns the equality relationship between two properties which are iconically reflected in the form of the two expressions making up the equality.

The learning activities of the card 7 focus on contradictions which emerge from pedagogical strategies. These pedagogical strategies are performed in solution of the tasks based on the use of paper and pen vs. the use of AL of ALNUSET and on the explorative use of ALNUSET. Overcoming these contradictions learning can be realized.

In the following we will present learning models based on Activity Theory which mediate these two pedagogical strategies.

Learning model based on Activity Theory which represents  the task solution  based on the use of  paper and pen vs. the use of Alnuset.

Learning model based on Activity Theory which represents  the task solution  based on the explorative use of ALNUSET.

Peirce
According to Peirce the icon sign serves "to help direct attention to other properties of the object than those the sign-vehicle is conventionally interpreted to represent" (79). To show this, Peirce suggests that two algebraic equations which consist of similar characters--"especially when we put resembling letters for corresponding coefficients"--resemble one another as an icon. 

Alnuset

ALNUSET (ALgebra on the NUmerical SETs) is a new educational digital artefact devoted to students of lower and upper secondary school (ages from 12-13 to 16-17). It includes three components: the Algebraic Line component, the Algebraic Manipulator componentand the Cartesian Plan component. The former is oriented to the development of an algebra of quantities, the second component deals with an algebra of operations and the third component is oriented to develop an algebra of functions. The educational significance of the system can emerge through the integrated use of these components in school practice.

The Algebraic line is an operative and representative environment for the construction of mathematical expressions involving numbers and letters defined on a specific numerical set (natural integers, relative integers, rational numbers, rational numbers extended to rational powers) and for the representation of lettering expressions as mobile points on the line. The Symbolic Manipulator component is a structured symbolic calculation and multiple representation environment for the manipulation of symbolic algebraic expressions, and for the solution of equations and inequations.

In this pedagogical plan the third component of Alnuset, the Cartesian Plan, will not be considered.

Tool access show tooltip help

NIL

The Algebraic line component [Component]

The Algebraic line component provides the opportunity to use letters as names of mobile points on the line. It provides geometrical models for graphically constructing sums and differences, products and ratios, powers and roots of the numbers, letters and expressions already represented on the line.

Every new expression constructed in this way is associated to a point on the Algebraic line that indicates the result of operations performed in sequence. The algebraic line is based on a representation built by mathematicians over previous centuries and is distinct from the number line, which is of a profoundly different nature. Indeed, the very nature of the Algebraic Line is algebraic, while the very nature of a number line is arithmetic. This algebraic nature is given by operational and representational features, i.e. by the fact that in ALNUSET this artefact is able to:

  • manage mobile points corresponding to algebraic variables;
  • manage the construction of literal expressions involving variables and their representation as points on the line;
  • manage the dragging of mobile points corresponding to algebraic variables and the movement of the points of the expressions involving those variables.

Since these characteristics are possible only through digital technology, we believe that the Algebraic Line can exist only as a digital artefact.

Comparison function [Component Feature]

To explore and define the truth set of algebraic propositions, namely the truth set of equations and inequations, the user has to define the proposition s/he wants to solve by means of the Comparison function.

When "Comparison" is clicked, an icon appears in the window and the values can be istantiated.

The user can instantiate the two expressions of the proposition by dragging the two green points represented on the line on their corresponding points.

The proposition defined by the user is automatically reported in the Sets window underneath the Algebraic line.

Truth Set function [Component Feature]

This function allows the user to explore and define the truth set of algebraic propositions, namely the truth set of equations and inequations. Once the user has selected this command, s/he has to define the proposition s/he wants to solve. To this aim the "Comparison" function is made available for defining the two expressions of the proposition and the type of relation between them. The proposition defined in this way by the user is automatically reported in a window at the bottom of the screen. Clicking the right mouse button on the proposition and selecting “Edit Set” displays a new line on the screen between the two lines already shown.

The user can then drag on this new line the points already represented on the other lines that s/he considers important for defining the truth set of the proposition. 


In this way, the user can define on this line the interval where the defined proposition is “true”. This definition is completely mouse based. The action performed by the user on the line (dragging of points, definition of intervals) are automatically expressed in formal language by the system and reported in the window containing the proposition. A "green light" shown beside the expression in this window indicates that the solution is correct and complete (see below).

Send to Manipulator [Component Feature]

This function allows the user to send an expression constructed in the Algebraic line component to the Manipulator component. Clicking the right mouse button on the expression on the line or on the expression in the "Sets" window sends that expression to the Manipulator component. An example is shown in the figure below.

Polynomial root [Component Feature]

This function allows the user to find real roots of a polynomial starting from an approximate value the user has found by dragging the polynomial variable to a point on the Algebraic line where the polynomial is null or tends to be null. In the figure below you can see the computation process performed by the system to find a root of the polynomial x^2-2x-1.

A graphical animation (red triangle vs green triangle) indicates the automatic computation process performed by the system to determine the numerical value of the root. The root of the polynomial is presented as a label associated to the x point on the blue line. To accept the result, the user has to click on the label.

Once the results have been accepted the polynomial roots are displayed in the Roots window underneath the corresponding polynomial below the Algebraic line (see the figure below).

The Symbolic Manipulator component [Component]

The Algebraic Manipulator component is a structured symbolic calculation and multiple representation environment for the manipulation of symbolic algebraic expressions, and for the solution of equations and inequations.

In general, structured symbolic calculation environments are designed for educational activity. They allow the user to act on expressions and equations with step-by-step transformations. These characteristics make such environments quite different from CAS (Computer Algebra Systems). CAS are designed for professional mathematical activities; they allow the user to handle algebraic transformation with black-box algorithms, producing solutions that in many cases are hard for students to understand.

The Algebraic Manipulator of ALNUSET is designed to allow students to approach algebraic manipulation by constructing meanings that are well founded in algebraic activity and have a clear theoretical basis.

This component manages algebraic manipulations of a single expression and the solution of equations and inequations of a degree less than or equal to 4. The interface provides a set of fundamental commands for symbolic manipulation that is specific to the numerical domain chosen by the user (natural integers, relative integers, rational numbers, full range domain). The basic commands for algebraic transformation have been designed to allow the user to control each step in the algebraic transformation solution process. The commands are related to a system of operational properties and calculation rules; if they are applied to one part of an expression that can be suitably manipulated with those commands, then the entire expression is transformed in accordance with the embedded transformation rules. The set of algebraic manipulation commands available to the user is open ended: new commands can be added to those already available and used in subsequent manipulations.

The commands structure comprises:

  • a command for editing the text of the symbolic expression to be transformed or the equation/inequation to be solved
  • basic commands incorporating symbolic rewriting rules related to the properties of operations and of specific computation rules
  • a command for turning the manipulation that the user has just performed into a new symbolic rewriting rule
  • algebraic manipulation commands created by the user
  • commands for solving equations/inequations.

The symbolic rewriting rule associated to each command generates an expression that is equivalent to the original one.

Application of a rule to a selected part of expression [Component Feature]

When a part of the symbolic representation is selected for manipulation, it comes under the system’s control and only the commands that can be applied to that part are available. The figure below shows a selected equation (right) and the commands that can be applied to the equation highlighted in yellow (left).

The figure below illustrates the manipulation of an equation. It can be noted that, in accordance with the current selection, only some commands are made available.

Send to line [Component Feature]

This function makes it possible to send a selected equation or a selected part of it to the Algebraic Line. The expression automatically appears in the Sets space of the Algebraic Line component and as a label associated to a point on the line.

Resources show tooltip helpexplode

Card 7 [Resource for students]

The aim of this card is to develop the concept of equation as conditioned equality between two expressions.

This aim will be achieved by 7 tasks useful for thinking over specific aspects characterizing this concept. The first tasks (task a), task b), task c)) concern the exploration of equation from a quantitative point of view highlighting that the action of inserting the equal sign between two algebraic expressions leads to structuring an algebraic equality. The second part of the tasks concern the truth set of equation highlighting in particular, that an equation and its transformed have the same truth set. 

 

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

3 hours

Actors' roles show tooltip help

STUDENTS

  • 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 arranged in pairs at each computer where Alnuset is running. The teacher delivers the card containing the tasks to each pair. Tasks have to be solved working with Alnuset. The answer to the task is to be written in the students' notebooks or on the card.

At this point of the experiment students are probably able to solve the whole card. Nevertheless,  at the end of the card, the teacher could propose a discussion about the results found. This discussion could be useful to make explicit some important concepts that emerge in solving the card.

Exploring particular kinds of equations

Identity show tooltip helpexplode

Authors show tooltip help

G. P. Chiappini, B. Pedemonte, E. Robotti, F. Vannucci

Subject domains show tooltip help

  • Algebra

Topics show tooltip help

  • Equations
  • Equivalent expressions as members of equation
  • Never equal expressions as members of equation
  • Opposite expressions as members of equation
  • Reciprocal expressions as members of equation

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

Description show tooltip help

This pedagogical plan focuses on equations with the aim to develop:

  • the conceptualisation of the equation in term of contitioned equality between two expressions
  • the conceptualisation that an equation denote a truth value 
  • the capability to use a quantitative and a formal approach in finding the truth set of an equation
  • the capability to face equations involving as their members respectively two equivalent expressions or two never equal expressions 
  • the capability to face equations involving as their members respectively two opposite expressions or two reciprocal expressions

Rationale show tooltip helpexplode

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Population show tooltip helpexplode

School level show tooltip help

Second year of upper secondary school

Age range show tooltip help

15-16 years old

Population description show tooltip help

This module has been designed for students who have just developed syntactic competencies in the algebraic domain. Our hypothesis is that semantic competencies are weak in this field for these students. Nevertheless, this module could also be proposed, with appropriate modifications, to students in the first approach to algebraic language. 

Student prerequisites show tooltip help

  • Knowledge of the operations with numbers (integers and rational numbers)
  • Basic skills in literal calculus 
  • Basic skills in the solution of first degree equations 
  • 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 ALNUSET 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 first and second year of upper secondary school. Link to national maths curriculum document.

Goals show tooltip helpexplode

Curricular goals show tooltip help

Learn how to solve first- and second-degree integer equations and inequations

This is a goal considered by the actual italian curricula. We want to pursue these goals, not only on the operative plan but also on the semantic plan.

Content-epistemological goals show tooltip help

  • Learn that equivalent equations are characterized by the same truth set
  • Learn that the truth set of equation whose members are two equivalent expressions is an identity
  • Learn that the truth set of an equation whose members are  never equal  is the empty set 

Cognitive goals show tooltip help

  • Learn how to generalize and to test a hypothesis

Social-affective goals show tooltip help

Develop the willingness and capacity to:

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

Instrumental goals show tooltip help

Learn how to use the following functions of the algebraic line component of ALNUSET:

  • the post-it
  • the polynomial root
  • the truth set function

Learn how to use the rules of the Algebraic Manipulator component and in particular the equations and inequations rules.

Learn how to use the "send to line function" and the "send to manipulator function" to integrate the two component of Alnuset.

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

The main aim of this pedagogical plan is:

  • developing the capability to use a quantitative and a formal approach in finding the truth set of an equation
  • developing the capability to face equations involving as their members respectively two specific finds of expressions (equivalent expressions; never equal expressions; opposite expressions; reciprocal expressions)

Theoretical framework show tooltip help

The learning activities of the card 8 focus on contradictions which can emerge from solutions of the tasks performed in paper and pen and solutions of the same tasks performed in AL, that is to say in operative way, and in AM, that is to say in formal way. Overcoming these contradictions learning can be realized.

In the following we will present a learning model based on Activity Theory which describe how learning can be realized

In particularm the learning model based on Activity Theory  describes the interpaly between the tasks solution based on the use of  paper and pen and the tasks solution based on the  Alnuse's use.

Learning model based on Activity Theory which describes the evolution of the educational activity to achieve learning using AL and AM of Alnuset.

Alnuset

ALNUSET (ALgebra on the NUmerical SETs) is a new educational digital artefact devoted to students of lower and upper secondary school (ages from 12-13 to 16-17). It includes three components: the Algebraic Line component, the Algebraic Manipulator componentand the Cartesian Plan component. The former is oriented to the development of an algebra of quantities, the second component deals with an algebra of operations and the third component is oriented to develop an algebra of functions. The educational significance of the system can emerge through the integrated use of these components in school practice.

The Algebraic line is an operative and representative environment for the construction of mathematical expressions involving numbers and letters defined on a specific numerical set (natural integers, relative integers, rational numbers, rational numbers extended to rational powers) and for the representation of lettering expressions as mobile points on the line. The Symbolic Manipulator component is a structured symbolic calculation and multiple representation environment for the manipulation of symbolic algebraic expressions, and for the solution of equations and inequations.

In this pedagogical plan the third component of Alnuset, the Cartesian Plan, will not be considered.

Tool access show tooltip help

NIL

The Algebraic line component [Component]

The Algebraic line component provides the opportunity to use letters as names of mobile points on the line. It provides geometrical models for graphically constructing sums and differences, products and ratios, powers and roots of the numbers, letters and expressions already represented on the line.

Every new expression constructed in this way is associated to a point on the Algebraic line that indicates the result of operations performed in sequence. The algebraic line is based on a representation built by mathematicians over previous centuries and is distinct from the number line, which is of a profoundly different nature. Indeed, the very nature of the Algebraic Line is algebraic, while the very nature of a number line is arithmetic. This algebraic nature is given by operational and representational features, i.e. by the fact that in ALNUSET this artefact is able to:

  • manage mobile points corresponding to algebraic variables;
  • manage the construction of literal expressions involving variables and their representation as points on the line;
  • manage the dragging of mobile points corresponding to algebraic variables and the movement of the points of the expressions involving those variables.

Since these characteristics are possible only through digital technology, we believe that the Algebraic Line can exist only as a digital artefact.

Polynomial root [Component Feature]

This function allows the user to find real roots of a polynomial starting from an approximate value already found by dragging the polynomial variable to a point on the Algebraic line where the polynomial is null or tends to be null. The figure shows the computation process performed by the system to find a root of the polynomial 5x-4.

 

A graphical animation (red triangle vs green triangle) indicates the automatic computation process performed by the system to determine the numerical value of the root. The root of the polynomial is presented as a label associated to the x point on the blue line. To accept the result, the user has to click on the label.

Once the results have been accepted, the polynomial roots are displayed under the corresponding polynomial in the Roots window below the Algebraic line (see the figure below).

Post-it [Component Feature]

The expressions constructed on the Algebraic Line are displayed in specific spaces named post-its. In general, a post-it contains all the equivalent expressions constructed by the user that have been associated to a specific point.
The figure below shows a post-it containing equivalent expressions corresponding to the same point.

The constructed expressions are also displayed in the "Values" window underneath the Algebraic Line.

The truth set function [Component Feature]

This function allows the user to explore and define the truth set of algebraic propositions, namely the truth set of equations and inequations. Once the user has selected this command, s/he has to define the proposition s/he wants to solve (in this case 3x-2=-3x+2). To this aim the interface makes available the "Comparison" function for defining the two expressions of the proposition and the type of relation between them. The proposition that the user has defined in this way is automatically reported in a specific space at the bottom of the screen. Clicking the right mouse button on the proposition and selecting “Edit Set” displays a new line on the screen between the two existing lines.

The user can drag onto this new line the points already represented on the other lines that s/he considers important points for defining the truth set of the proposition. In this way, the user can define on this line the interval where the defined proposition is “true”. This definition is completely mouse based. The actions performed by the user on the line (dragging of points, definition of intervals) are automatically expressed in formal language by the system and reported in the specific space containing the proposition. Feedback is given to let the user know whether the solution is correct and complete.

Send to Manipulator [Component Feature]

This function allows the user to send an expression constructed in the Algebraic line component to the Manipoulator component. This is done by clicking the right mouse button on the expression on the line or on the expression on the "Sets" window. An example is shown in the figure below.

The Symbolic Manipulator component [Component]

The Algebraic Manipulator component is a structured symbolic calculation and multiple representation environment for the manipulation of symbolic algebraic expressions, and for the solution of equations and inequations.

In general, structured symbolic calculation environments are designed for educational activity. They allow the user to act on expressions and equations with step-by-step transformations. These characteristics make such environments quite different from CAS (Computer Algebra Systems). CAS are designed for professional mathematical activities; they allow the user to handle algebraic transformation with black-box algorithms, producing solutions that in many cases are hard for students to understand.

The Algebraic Manipulator of ALNUSET is designed to allow students to approach algebraic manipulation by constructing meanings that are well founded in algebraic activity and have a clear theoretical basis.

This component manages algebraic manipulations of a single expression and the solution of equations and inequations of a degree less than or equal to 4. The interface provides a set of fundamental commands for symbolic manipulation that is specific to the numerical domain chosen by the user (natural integers, relative integers, rational numbers, full range domain). The basic commands for algebraic transformation have been designed to allow the user to control each step in the algebraic transformation solution process. The commands are related to a system of operational properties and calculation rules; if they are applied to one part of an expression that can be suitably manipulated with those commands, then the entire expression is transformed in accordance with the embedded transformation rules. The set of algebraic manipulation commands available to the user is open ended: new commands can be added to those already available and used in subsequent manipulations.

The commands structure comprises:

  • a command for editing the text of the symbolic expression to be transformed or the equation/inequation to be solved
  • basic commands incorporating symbolic rewriting rules related to the properties of operations and of specific computation rules
  • a command for turning the manipulation that the user has just performed into a new symbolic rewriting rule
  • algebraic manipulation commands created by the user
  • commands for solving equations/inequations.

The symbolic rewriting rule associated to each command generates an expression that is equivalent to the original one.

Application of a rule to a selected part of expression [Component Feature]

When a part of the symbolic representation is selected for manipulation, it comes under the system’s control and only the commands that can be applied to that part are available. The figure below illustrates the manipulation of an equation: in accordance with the current selection, only some commands are available.

The figure below shows the transformation of an equation (3x-2=-3x+2) involving two opposite expressions as members. This transformation is the proof that the equation 3x-2=-3x+2 is equivalent to the equation 2(3x-2)=0.

Send to line [Component Feature]

This function makes it possible to send a selected equation or a selected part of it to the Algebraic Line.

The equation automatically appears in the Sets window in the Algebraic Line component and is associated as a label to a point on the line.

Using the truth set function it is possible to show that the two equations have  the same truth set as their solution.

Resources show tooltip helpexplode

Card 8 [Resource for students]

This card is the last card of the PP. Its aim is to deep the concept of equation considering some specific kinds of equation and observing the different truth sets that correspond to each case. The task a) takes into account an equation constituted by equivalent expressions as members. The task b) considers an equation constituted by two never equal expressions as members. The task c) and the task d) considers equations respectively constituted by opposite and reciprocal expressions as members.

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

STUDENTS

  • 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 arranged in pairs at each computer where Alnuset is running. The teacher delivers the card containing the tasks to each pair. Tasks are to be solved working with Alnuset. The answers to the task are to be written in the students' notebooks or on the card.

The teacher sets the tasks on the card. These are explorative tasks. At the end of this work, the teacher could propose a discussion to make explicit some concepts about an equation constituted by two specific kinds of expressions as members (equivalent expressions, never equal expressions, opposite expressions, reciprocal expressions). The aim of this discussion is to istituzionalise emerged concepts about this topic.