Establishing a custom of proving in american school geometry: evolution of the two-column proof in the early twentieth century

By Patricio G. Herbst. 2002

163 citations recorded by the source corpus.

Subfields: Education, History and Philosophy of Science, Theoretical Computer Science

View Paper: https://www.semanticscholar.org/paper/66400264c495441ad81fff1ad8619f0fa8875c7f

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Citations in the corpus

Competing Arguments for the Geometry Course: Why Were American High School Students Supposed to Study Geometry in the Twentieth Century? (2006, background): “His hope that students would engage in crafting original proofs seems to be a reminiscence of expectations in the report of the Committee of Ten and the practice of many, simple proof-exercises that became standard shortly thereafter (Herbst, 2002).”
Interactions with Diagrams and the Making of Reasoned Conjectures in Geometry (2004, background): “Elsewhere (Herbst 2002b) I have described this as the logical ordering of a list of descriptive assertions about a figure, noting in particular how some geometric assertions—those involving incidence and separation—are often invisible; and hence very rarely will students be held accountable for…”
Enacting proof-related tasks in middle school mathematics: Challenges and opportunities (2010, background): “228), proof in the high school geometry curriculum has evolved to emphasize verification (Herbst, 2002b).”
The Need for Proof and Proving: Mathematical and Pedagogical Perspectives (2011, background): “Learners face a problematic situation when they confront a problem that cannot be routinely solved by their current O. Zaslavsky (*) Department of Teaching and Learning, New York University , New York , NY , USA Department of Education in Technology and Science Technion – Israel Institute of Technol”
What Do We Mean by Mathematical Proof (2011, background): “Students are traditionally introduced to the term “mathematical proof” in high school geometry classes, where they are taught that proofs are highly organized, usually using two columns, in which each line must be deduced from the lines above using some “rule of geometry” [30].”
The Mathematical Nature of Reasoning-and-Proving Opportunities in Geometry Textbooks (2014, background): “For geometry, however, which typically represents the high point of reasoning-and-proving opportunities in the school curriculum (Hanna & de Bruyn, 1999; Herbst, 2002), these quantities of reasoning-and-proving opportunities may be viewed as limited in light of the calls for reasoning-and-proving to”
Proof in School Mathematics: Insights from Psychological Research into Students' Ability for Deductive Reasoning (2008, background): “…play a role in facilitating students’ use of deductive reasoning in the context of proof (e.g., Balacheff, 1991; Ball & Bass, 2003; Boero et al., 1996; Herbst, 2002a; Jones, 2000; Lampert, 1992; Mariotti, Bussi, Boero, Ferri, & Garutti, 1997; Stylianides, 2007b, 2007c; Weber, 2001; Yackel & Cobb,…”
Proving and Doing Proofs in High School Geometry Classes: What Is It That Is Going On for Students? (2006, background): “A second group of relevant studies focused on the history and epistemology of proof, including accounts of the role that proof plays in the discipline of mathematics (Hanna, 1989, 1995; Lakatos, 1976; Otte, 1994; Rav, 1999) and of the role of proof in the school curriculum (Hanna, 1983; Herbst, 2002”
Teachers’ perspectives on “authentic mathematics” and the two-column proof form (2009, methodology): “The existing custom of proving in high school geometry, embodied in the twocolumn form, exemplifies how practice has evolved historically to avoid that tension ( Herbst 2002a ).”
Teaching Geometry With Problems: Negotiating Instructional Situations and Mathematical Tasks (2006, methodology): “The first consisted of an analysis a priori (Comiti, Grenier, & Margolinas, 1995; Herbst, 2002a, 2003) of the task that could develop around the problem chosen.”
How Mathematicians Obtain Conviction: Implications for Mathematics Instruction and Research on Epistemic Cognition (2014, background): “In these courses, proofs are typically written in a two-column format (see Herbst, 2002), where every statement in a proof is something known to be true (e.g., an axiom, a definition, an assumption from the theorem being proven) or is a direct deduction from earlier assertions in the proof using a…”
Opportunities to Learn Reasoning and Proof in High School Mathematics Textbooks (2012, unknown): “In the early 20th century, deductive reasoning was almost exclusively studied in geometry courses, and students were asked to memorize proofs similar to those in Euclid’s Elements (Herbst, 2002; Sinclair, 2008).”
When, how, and why prove theorems? A methodology for studying the perspective of geometry teachers (2008, background): “old knowledge, particularly to apply theorems, definitions, and postulates, into short chains of deductive reasoning that proved rather useless claims about configurations ( Herbst, 2002b ).”
Engaging Students in Proving: A Double Bind on the Teacher (2002, background): “At the same time that it provided a way to show similarities between the work of the teacher who was proving theorems and the work of the student who was doing proof-exercises, it permitted their work to be differentiated according to instructional needs ( Herbst, in press ).”
Approaching Proof in a Community of Mathematical Practice (2008, background): “Hersh (1993) describes some variations in proof standards in applied mathematics and pure mathematics and finds great differences in the rigour between them, but even in pure mathematics itself.”
The Semiotic Structure of Geometry Diagrams: How Textbook Diagrams Convey Meaning (2015, background): “385), it has been relatively common in mathematics education to analyze textbooks to document historical changes in the curriculum (e.g., Baker et al., 2010; Fujita & Jones, 2003; Herbst, 2002; Van Sickle, 2011).”