Since teachers have to be clear about the mathematical and technological competencies they want to evaluate, in order to assess their students’ learning, I summarise these two kinds of competencies in the table below for each of the assessment tasks.
Tasks Mathematical competencies Technological competencies The
Counter- example
Correct use of examples in proving or refuting conjectures; generalizing theorems and finding exceptions; applying mathematical reasoning; constructing mathematical representations.
Making a draw with the tools; appropriate use of the instruments; selecting and using technological tools to construct a diagram.
The Dog Exploring; problem solving; applying mathematical reasoning; pattern- seeking; ability of converting a diagram in algebraic operations; connecting mathematical ideas to everyday experiences.
Exploring the situation with the tools; appropriate use of the instruments; ability to use effectively digital technologies in mathematics to solve unfamiliar problems and make rational conjectures; developing visualization
skills to assist in processing information.
The Right Triangle
Reading a diagram; explaining the mathematical reasoning with own words; using a property in a particular example; stating concepts; generalizing; proving; ‘noticing’.
Dragging objects to find invariance; dragging to test the ‘stability’ of a property as along they are working on a diagram; developing visualization skills to assist in processing information.
The Ball Understanding what is going on (interpreting the situation); exploring; predicting; finding invariance and properties; problem solving; making assumptions; using a property in a particular context; developing and apply new mathematical knowledge through problem solving; explaining the mathematical reasoning with own words; generalizing; proving; guessing; pattern-seeking; making connections; ‘noticing’; connecting mathematical ideas to everyday experiences, and to other disciplines.
Exploring the situation; ‘make experiments’ with the tools; trial-and- error techniques; selecting and using appropriately technological tools; interpreting the ‘behaviour’ of a construction to make inferences and deductions about it; using the tools to test the validity of conjectures; ability to use effectively digital technologies in mathematics to solve unfamiliar problems and make rational conjectures; developing visualization skills to assist in processing information.
Table 3: Mathematical and Technological Competencies
It is important to notice that the same question could be an exploration task for one person, and an exercise or an application of a well known property for another, depending on students’ previous knowledge. The competencies that a task aims to evaluate could be different for different classes or students, only a good knowledge of the students and their ‘history of learning’ can allow a teacher to design a task that evaluates the competencies that he/she wants to test.
In the tasks illustrated above I described the ways in which they follow Sinclair’s suggestion in the design of the sketch depending on the aim of the task. Is there something that makes a task better than another one? A good task is composed of a good question and a good design.
I contend that the task on the counter-example is a good question because it asks students to think about a property of the circle they are supposed to know, to see
if they are able to find an exception, or if the property is always correct. The sketch aims to make students reflect through the tools, but there are not pre-constructed diagrams, students should draw the diagram that represents the property or the diagram that represents the exception to the property. In this context the technology is not very useful, because students could draw a diagram also with paper and pencil, without using the tools of the DGE.
The task with the dog makes students reflect on an image of a dog roaming around a house. In order to find the area of the surface where the dog can go without breaking the rope, students need firstly to figure out where the dog is allowed to go, and then to represent that surface with geometrical shapes with known area. Since all the sketches are related to circle geometry, and this task in particular offers three circles as tools to find the area, then it is easy for students to recognize that they need to use circles to represent the area. However, it is not immediately clear where to place the three circles; also, finding the exact number of the area is not a simple request, because students need to add and subtract portions of the circles. Here we have a good questions and a good design, and the aim of the design is helping students exploring the situation with a dynamic representation of the problem.
The task on the triangle asks students to think about a particular case of a general property. The question is quite simple, but is formulated in a way that does not imply the direct connection to the property of the central angles and the inscribed angles subtended by the same arc. The DGE has a fundamental role, since the questions prompt students to drag the points on the circle. The task could be seen also as an exploration task for students who do not remember the property - they could drag the vertex on the circle and find the invariance of the right inscribed angle when the other two points are the end points of a diameter.
The task on the ball is definitely a good question, and the DGE is the core of the problem: the movement and the tools are the key elements of the sketch. There may be some doubt as to whether it is an assessment question or not. It is an exploration task where students are asked to investigate the situation and explain a certain behaviour of the sketch with a property of the circle. This task evaluates the ‘noticing’, and the problem solving competence, since students need to observe and to make connections. It also evaluates the technological competence of exploring with the tools, and using them to draw the dynamic diagram that represents the hidden construction of the situation.