 are mere tautologies or analytical truths without substantial content and hence incapable of sustaining any inferences leading to new and even surprising discoveries, as the detections of sleuths like Sherlock Holmes or Nero Wolfe were supposed to lead. (p. 25) From the educational side, we also find two possible approaches. Some studies illustrate a substantial didactical discontinuity between argumentations and proofs, highlighting a jump between the two (which we have called the basic gap). Some scholars highlight a form of cognitive discontinuity between them; others highlight an epistemic discontinuity between the respective statuses of knowledge; often the two forms of discontinuity are both stressed in the same research. In any case, all of them say that it is very difficult to find an effective way of teaching and learning of proofs in the classroom. As examples, see the research studies touched upon in their chapter in the Compendium by Stylianides et al. (2017), and others, like those of Balacheff (1987), Duval (1991), and Thompson et al. (2012). However, in the panorama of the didactic research on proof, we also find opposite positions. The studies by Boero et al. (1996), Garuti et al. (1996Garuti et al. (, 1998, Mariotti (2006), Pedemonte (2007), Baccaglini-Frank and Mariotti (2010) and others (see for example the discussion and the related bibliography in the papers by Boero et al. 2010;Guala and Boero 2017) highlight the possibility of forms of cognitive continuity between the construction of a conjecture and the construction of the proof, which they call cognitive unity: During a problem-solving process, an argumentation activity is usually developed to produce a conjecture. The hypothesis of cognitive unity is that in some cases this argumentation can be used by the student in the construction of proof by organising in a logical chain some of the previously produced arguments. (Pedemonte 2007, p. 24) The studies mentioned above show that proof is more 'accessible' to students if an argumentation activity is developed for the construction of a conjecture and, conversely, the construction of proof is more difficult if such a cognitive unity is not achieved.
adjective 'epistemic' to indicate the programs of scientific investigation, and the related theories, pursued and implemented by different schools and authors. In other words, 'epistemological' refers to the subject who studies epistemic matters, whereas 'epistemic' merely refers to knowledge, justification and belief. The distinction between 'ontics' and 'ontology' is proper to the Continental philosophy (contrasted with the Analytical one); the difference was discussed by Heidegger (1927): ontic is what makes something what it is; while ontological refers to one's own first-person, subjective, phenomenological experience of being.
Other research shows that the Dynamic Geometry Environments, DGE, (e.g. Cabri Géomètre, Geogebra, Sketchpad, etc.) can support forms of cognitive continuity, at least in the case of elementary geometry (Arzarello et al. 2012: see the discussion about abductions in the final section).
In any case, the pedagogical problem of how to teach/learn proof at school is still open, provided that it is a teaching goal in the school (e.g. it is so in the Italian Licei 3 ). It requires the taking of a position with respect to the previous dilemmas (be they epistemic, cognitive, or both) on the value of the logical deductions with respect to argumentations.

The Theoretical Basis of Our Proposal
In the chapter we continually use the terms argumentation and proof. Clarifying their meaning and mutual relationships for learning purposes is among the objectives of this contribution. To orient the reader, we give now two definitions and will discuss the reasons why we have chosen this formulation.
Following Toulmin (, 1974, Toulmin et al. (1984), we use the term argumentation (or reasoning) to refer to a text made of one or more concatenated argumentative steps. An argumentative step is identifiable through the presence of a Fact, a Claim and a Warrant that justifies the validity (possibly with a certain degree of probability) of the Claim, because of the Fact (Fig. 10.1).
The Warrant can explicitly or implicitly refer to a set of knowledge, principles, etc. possibly organized in the system (Backing) on which the Warrant is based. Sometimes in the argumentation there are exceptions (Rebuttal) according to which the Claim does not follow from the data under the Warrant. Figure 10.2 exemplifies this situation.
For the notion of proof, we instead refer to Rav (1999): "Proofs are the mathematician's way to display the mathematical machinery for solving problems and to justify that a proposed solution to a problem is indeed a solution" (p. 13).
It should be noted that often in the literature and in the practice of teaching some people tend to identify proofs with derivations (purely syntactic objects); but a proof is never reducible to this specific aspect only (see Rav 1999, p. 12 and the comments on the so called DTP model, definition-theorem-proof, by Thurston 1994).
To deepen the analysis on the theme continuity-discontinuity between arguments and proofs we refer to two theoretical models. First of all, we consider Toulmin's model, sketched above, since it is widely used in mathematics education research (for example, by Boero et al. 1996Boero et al., 2010Boero 2011, andby Knipping andReid 2015). Its critical analysis from an epistemic point of view led us to consider another model: the Logic of Inquiry by Hintikka (logician). For space reasons, we cannot introduce a third important model, which is due to an adaptation (Boero and Planas 2014;Guala and Boero 2017) of the construct of rationality by Habermas (we invite readers with an interest in the discursive practices related to proofs to consult these references).

The Model of Stephen E. Toulmin
We have already introduced the Toulmin model for our definition of argumentation. He calls his model "substantial" and "practical" in opposition to the "analytical" and "theoretical" character of the syllogism, which for him fully represents logical deduction.
For Toulmin, an argumentation makes sense only if it is contextualized: the English scholar typically thinks of it as a discourse that takes place between two people, in disagreement about the proposed statement; in this way, one of them disputes the Claim, and the other, who made the initial affirmation, tries to justify it by giving increasing Warrants with their Backings. The model is inspired by legal practice and this is the reason why Toulmin calls it jurisprudential  pp. 41-43): in fact, the five components of the arguments assume their full meaning in a debate that takes place in a courtroom during the discussion of a specific legal case. The goal is to convince the adversary, not the search/explanation of the truth, which is instead the goal of scientific argument, in particular in mathematics. The full meaning of the Toulmin model is therefore constituted by the intertwining of the two components: (a) the structure (Claim-Fact-Warrant-Backing-Rebuttal) (b) the context in which the argument takes place.
The structure of part (a) makes sense only if modeled by the dynamic process of part (b) and its elements are seen according to this perspective (Zarębski 2009).
Unfortunately, most of the literature that uses Toulmin's model in mathematical education somewhat neglects the analysis of the relationships between the two components (a) and (b). This is not by chance. A first reason is that the jurisprudential model is quite different from that of scientific investigation, mathematics in particular: the distinction between Warrant and Backing is important in the jurisprudential model but it does not appear to fit so well with the mathematical field. Toulmin (1975) illustrated "how the sort of backing called for by our arguments varies from one field of argument to another" (p. 97). For example, the three different warrants 'A whale will be a mammal', 'A Bermudan will be a Briton', 'A Saudi Arab will be a Muslim' rely on different backings: the first warrant is supported by referring it to a taxonomic classification system, the second by referring to the rules governing the nationality of British colonies (at the time when Toulmin wrote), and the third to the statistics recording the distribution of religious faiths among people of different nationalities. The relative backings therefore refer to texts of Zoology, to the English laws in effect in a certain year, to statistical data collected following certain protocols.
A second reason is that the model in all the exemplifications refers exclusively to the forms of syllogistic reasoning as an example of analytical reasoning: this is a very serious limitation of its use in the analysis of mathematical reasoning, which, as it is known, cannot be reduced to syllogisms alone (Börger et al. 1997). This has important consequences for the structure of mathematical and scientific arguments in general, as we now illustrate with an example.
It is well known that in mathematics we have many sentences like "for all x, there exists a y such that…". (89-form). A typical example is the continuity of a function at a point. Let's consider the function f x ð Þ ¼ x 2 and suppose we want to show that it is continuous in x 0 ¼ 1. To show