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We build upon a concept of understanding in the hermeneutic tradition of philosophy. To discuss why the frame concept is helpful for the discussion of mathematical knowledge, we must first discuss the concept of understanding (see Sect. In terms of frames this can be explained by saying that the student has not recognized which high-level structural frames were involved in the proof. In such a case we would normally not ascribe to that student understanding of that proof. In terms of the frame model, these goals are achieved when certain frames required for the understanding of proof texts – as well as the ability to apply these frames in practice – have been acquired.Īs an illustration of how the frame concept can be fruitfully applied to shed light on the nature of proof understanding, consider the case when a mathematics student has laboriously checked all details in a complex proof but does not see the big picture of how all these proof steps work together. Note that among the goals of our mathematical education are learning about various kinds of mathematical structures as well as learning various proof techniques in the sense of being able to adapt them to other contexts or problems. These expectations can be modeled through different kinds of frames, namely structural frames that specify expectations about how proofs that use various proof methods are usually structured, and ontological frames that specify mathematical structures and objects and expectations about how they are usually referred to. ![]() We apply this concept to the context of mathematical proofs, which readers interpret by complementing the explicitly given information with their expectations about how proofs are usually structured depending on the applied proof method and how mathematical structures and objects are usually referred to in the text. 2) is used in linguistics, cognitive science and artificial intelligence to model how explicitly given information is combined with expectations deriving from background knowledge. The frame concept (further explained in Sect. What does it mean to say that a mathematician has understood a certain mathematical proof? In this paper we approach the topic of mathematical understanding by combining analytical tools from various academic disciplines, most notably the frame concept.
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