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 expressionresult. 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 notdetermined 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
postit 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 viceversa 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 “postit”
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?