Starting from the above literature review, theoretical background and the use cases, we are refining and designing the Interactive Semi-open Question (ISQ) as a digital interactive application that consists in a like open-ended question posed to the student and in some virtual objects, being language tiles. The student is expected to build the answer by choosing, dragging and juxtaposing some of the available language tiles. These latter ones allow to construct various sentences, some of them can be acceptable as correct answers to the posed question, some other maybe only partially acceptable, and finally some others not acceptable as they are incorrect answers. The ISQ is not a very open-ended question, but it can be used as a scaffolding to convert the student’s thinking into a communicable sentence, under some pre-fixed rules that can be both socio-mathematical and linguistic. In case of argumentation, the ISQ can foster the passage from reasoning as process, that is production of arguments, to expression of arguments. Moreover, being the student in front of a question in style open-ended, she is brought naturally to think about the answer in a free manner. Then, if language tiles to be made available are appropriately chosen, the student will be able to reformulate her own thinking in a sentence which can be built with some of the given tiles.

The application is born in an educational context whose goal was to foster the argumentative competence scaffolding the moving towards the expression of arguments in a literate register (Ferrari, 2004). In such context, framed in an e-learning platform, there is the need of recognizing automatically the answer given by the students, as we were interested in the supporting the construction of answers including causal proposition which makes explicit the reason of the main statement (Albano, Dello Iacono, Mariotti, 2017). This is why open-ended questions are not manageable and, on the other hand, close-ended questions show many educational limits, as they do not allows to focus on the causal nor deductive structure of the sentences at stake. It is worthwhile to note that the ISQ, we implemented by using GeoGebra, and embedded into Moodle platform, can be integrated into any other web environment, supplying it with a new kind of question (Albano & Dello Iacono, 2018b).

In the near future we plan to pilot testing the tool and analyze the results from an educational perspective to the case studies cited in the previous section. In addition, we want to take further interesting cases known in literature, of which we report below a brief preliminary theoretical analysis.

Let us see the problem (Boero, 2017, p. 3): “Consider all the products of three consecutive natural numbers. What is their GCD? Prove that it is their GCD”.

Boero (2017, p. 4) reported the following transcript of a student:

*1·2·3 = 6 2·3·4 = 24 3·4·5 = 60 10·11·12 = 1320 it is evident that 6 is the GCD of the first three products, because it is the greatest divisor of the first product and a divisor of the other products. Is it a divisor of 1320? … Yes, 1320 is an even number divisible by 3 because the sum of its digits is a multiple of 3. Then 6 might be the divisor of all the other products too. But why? Probably, by looking at these four products, all the products are even… But why? OK, one factor is always even! Even numbers go two by two, thus among three numbers one number … one number at least is even, and they may be two, like in the case of 2·3·4. Look at, three is there! And a multiple of three is in the last product! Why? In the case of 2, multiples go two by two … In the case of 3, numbers go three by three. That is the reason! Now I try to write down the general reasoning: The greatest common divisor is 6 because every product is divisible by 6 because every three consecutive numbers contain one even number (multiple of 2) and one multiple of 3, because multiples of 2 go two by two, and multiples of 3 go three by three (The teacher writes the following question: Why greatest?) (after a while) Because the first product is divisible by 6, and no greater divisor is there.*

Starting from the above transcript, some tiles can be foreseen for an ISQ shown in Fig. 5:

Note that the language tiles can be grouped together according some educational criteria. For instance, in Fig. 5 the first line contains numerical examples, the fourth and fifth lines refer to the concept of divisibility, the last line contains causal conjunctions. Showing language tiles according some criteria allows to pose the learner’s attention on some key points and guide somehow thinking, such as focus on the need of justifying (causal conjunctions) which is not natural, even in front of explicit demand “Justify your answer”, or on some mathematical concept involved or needed for proof, such as divisibility criteria. Some of the tiles, such as causal conjunctions or articles, have been made more copies as they can be needed more times, so when the student drags a copy, a new one appears in the same place.

The language tiles in Fig. 5, besides the sentence reported in Boero, allow to build further various correct sentences. For instance, the row 1 and 2 of the Fig. 6 show two alternative possible arguments which can be built. Both of them can continue with the sentence shown in row 3. We note that the tiles available do not contain distractors, but the student should use all the tiles, acting and arranging them as in a puzzle.

Let us now consider the following problem (Fig. 7) from Haj-Yahya, Hershkowitz, and Dreyfus (2014). The *Ramie’s proof* is given to the students (p. 219): *“ABCD is a parallelogram, therefore AD and BC are parallel. BG is part of BC and AF is part of AD, hence AF and BG are parallel (parts of parallel sides). We found a pair of opposite parallel sides, therefore ABGF is a trapezium”*, posing the question: “*Did Ramie give a correct and complete proof?”*

The outcomes reported in the cited paper show that some students are in trouble with the definition of a parallelogram as limit case of trapezium: “Ramie’s proof is correct, he found and proved that there is a pair of parallel sides” (student a13, p. 219), “not correct because it is incomplete, he should prove that the other pair are not parallel, AB is not parallel to FG because they intersect in point H” (student b43, p. 220). In this case the ISQ can be used both for constructing a proof (e.g. Ramie’s one) and for detecting such kind of difficulties, planning suitably the tiles, as done in Fig. 8, as they allow to build both the previous students’ sentences.

The ISQ toolkit presented in this paper allows to implement a scaffolding methodology for fostering the argumentative competence, with reference both to re-arrange arguments into a deductive chain and to communicate arguments in a verbal-semantic expression. The *language tiles*, made available by the tool, can be freely dragged and juxtaposed for building sentences, assuming the language as a manipulative artefact. This manipulative feature of the toolkit allows the students to activate thinking and communication processes. In this respect, the use of the toolkit can favour the improvement of mathematical communication, as for what concerns the passage from colloquial registers to literate ones (Ferrari, 2004), and the agreement to socio-mathematics norms which regulate communication among mathematicians (Mariotti, 2006). Thus it supports the students in a shift from own reasoning to expressing themselves in and about mathematics at a more theoretical and technical precision by means of the use of literate registers. Such improvement in using literate registers impacts on advanced mathematical thinking, in the frame of discursive approach to mathematical learning (Sfard, 2001). We also assume that improving communication means also improving the production of mathematical arguments, as the two things are strictly linked, in a back and forth moving. It is worthwhile to note that some language tiles of the toolkit can recall mathematical knowledge (concepts, properties, theorems). So when the student chooses some tiles, she activates not only a cognitive process concerning the construction of sentence she has in mind but, looking at the “mathematical” tiles, she needs to recover her knowledge and to activate connections. In other words, she is somehow forced to deepen mathematical contents.

The ISQ toolkit also permits to preserve the added value of posing an open-ended question instead of other close-ended questions, so having the possibility of working on argumentative competency and assessing argumentative texts. In fact, the student is left free of answering by means of sentences built by language tiles offered by the tool. The causal structure of the sentence, as well as the conditional structure or whatever else, can be highlighted and fostered by appropriate tiles dedicated to various conjunctions. We observe that, starting from the same set of tiles, the student can build more than one sentences showing correct argumentations or proofs. Different sentences can allow the teacher to be informed on what the student knows and not only on what she lacks, and use this information to steer personalized learning trajectories. From the argumentative point of view, the choice of the tiles to be made available is a crucial for educational effectiveness. The ISQ can be implemented in various typologies, for instance making available: all the tiles which are needed to construct the argumentative sentence, such as a puzzle (see the above first problem); also tiles which can bring to build wrong sentences, such as distractors (see subsection
*Using ISQ to foster expression of arguments*
); appropriate tiles to produce a formal proof and also to reason on limit cases concerning mathematical concepts (see the above second problem). Moreover, the ISQ can be used for automatic assessment in computer-based environment. In fact, it is possible to label the tiles in order to automatically re-construct the sentence (see subsection
*Use of ISQ in automatic assessment*
) and the assess its correctness.

From the educational point of view, the toolkit can be used at every level of school degrees, from primary school to university, choosing appropriate language tiles. The toolkit can be framed into an inquiry-based approach, where students are steered to explore and to conjecture and then they are supported in proving or refuting their findings, also constructing counter-examples.

Anyway, some alerts in using the ISQ methodology can be depicted. The teacher should be careful in choosing the tiles to allow the student to effectively put in words her thinking without restricting her own style of expressing. In fact, increasing the number of tiles greatly increases the number of sentences that can be constructed. As shown in the previous section, the available tiles allow to construct further sentences expressing the same thinking. As drawback, in constructing their sentences, the students can be guided by the choice of words which sound attractive from mathematical point of view, without reflecting on their correctness in the case at stake. This is the case of “direct proportionality”, shown in the subsection
*Using ISQ to foster expression of arguments*
. Thus, it is important that the teacher makes an a priori analysis of all the sentences that can be constructed by the available tiles, so to be able to manage the educational gaps that can be emerged.