Proof and Proving

Proof, Proving, and Teacher-Student Interaction: Theories and Contexts

“…They have found that, as a group, teachers have resources to justify positive appraisals of certain elements of the work of proving: the use of an unproven conjecture as a premise in proving a target conclusion; the identification of new mathematical concepts and their properties from objects introduced and observations made in justifying a construction; the deductive derivation of a conditional statement connecting two concomitant facts about a diagram; the prediction of an empirical fact by operating algebraically with symbols representing the quantities to be measured; the breaking up of a complicated proof problem into smaller problems (lemmas); the application of a specific proving technique (e.g., reduction to a previously proven case); and the establishment of equivalence relationships among a set of concomitantly true statements. Herbst, Miyakawa and Chazan (2010) have proposed that teachers might use the various functions of mathematical proof documented in the literature (e.g., verification, explanation, discovery, communication, systematization, development of an empirical theory, and container of techniques) (de Villiers, 1990;Hanna & Barbeau, 2008;Hanna & Jahnke, 1996) to attach contractual value to actions like those listed above. There remain two questions; whether classroom exchanges are possible (manageable) between these actions and the elements of currency; and whether the exchanges can be contained within instances of the 'doing proofs' situation or otherwise whether they require more explicit negotiations of the didactical contract.…”

Informal prerequisites for informal proofs

“…There is a wide range of investigations starting from empirical investigations in the students' proof ability (e.g., Senk 1985;Healy & Hoyles, 1998;Reiss, Klieme & Heinze, 2001), to the development of theoretical models of the proving process (Balacheff, 1988;Boero, 1999), philosophical background about proving in mathematics and mathematics classroom (Hanna, 1983;Hanna & Jahnke, 1996;Hanna, 1997) and the identification of individual proof schemes (Harel & Sowder, 1998). Furthermore, in the Third International Mathematics and Science Study (TIMSS) (Beaton et.…”

Razonamiento Configural Y Procedimientos De Verificación en Contexto Geométrico

“…Tras las reformas educativas de los años cincuenta y sesenta del pasado siglo, la prueba fue prácticamente relegada a la heurística (Hanna & Jahnke, 1996). Sin embargo, hoy en día las orientaciones curriculares de distintos países, entre ellos España, y las recomendaciones de organizaciones como el National Council of Teachers of Mathematics en Estados Unidos destacan como un objetivo fundamental de la educación matemática el desarrollo de la capacidad de razonamiento matemático y, más en particular, de la capacidad de efectuar demostraciones matemáticas (NCTM, 2000).…”

“…Sin embargo, hoy en día las orientaciones curriculares de distintos países, entre ellos España, y las recomendaciones de organizaciones como el National Council of Teachers of Mathematics en Estados Unidos destacan como un objetivo fundamental de la educación matemática el desarrollo de la capacidad de razonamiento matemático y, más en particular, de la capacidad de efectuar demostraciones matemáticas (NCTM, 2000). Lograr este objetivo es un asunto complejo que se enfrenta con dificultades; hay investigaciones que han documentado que los alumnos no sienten la necesidad de la demostración deductiva (Hanna & Jahnke, 1996) y no distinguen entre diferentes formas de razonamiento matemático (explicación, argumentación, verificación y demostración) (Dreyfus, 1999).…”