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Herbst, Patricio

doi:10.1023/a:1020264906740

Cited by 105 publications

(28 citation statements)

References 17 publications

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“…For mathematics education, the didactical object in line with this approach is the wellknown two columns proof which facilitates the evaluation of a proof by both the student and the teacher (see Herbst 1999Herbst , 2002a; also González and Herbst, 2006). As it is very often the case, this process of didactification ''killed'' he initial intention of which Fawcett was a characteristic example.…”

Section: Introduction To An Epistemological Discussionmentioning

confidence: 99%

The role of the researcher’s epistemology in mathematics education: an essay on the case of proof

Balacheff

2008

ZDM Mathematics Education

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Is there a shared meaning of ''mathematical proof'' among researchers in mathematics education? Almost all researchers may agree on a formal definition of mathematical proof. But beyond this minimal agreement, what is the state of our field? After three decades of activity in this area, being familiar with the most influential pieces of work, I realize that the sharing of keywords hides important differences in the understanding. These differences could be obstacles to scientific progress in this area, if they are not made explicit and addressed as such. In this essay I take a sample of research projects which have impacted the teaching and learning of mathematical proof, in order to describe where the gaps are. Then I suggest a possible scientific programme which aspires to strengthen the research practice in this domain. Eventually, I make the additional claim that this programme could hold for other areas of research in mathematics education.

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Section: Introduction To An Epistemological Discussionmentioning

confidence: 99%

The role of the researcher’s epistemology in mathematics education: an essay on the case of proof

Balacheff

2008

ZDM Mathematics Education

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“…Whereas, Chapter 4, "Congruent triangles", has seven sections: Congruent Figures, Triangles congruence by SSS and SAS, Triangle congruence by ASA and AAS, Using congruent triangles: CPCTC, Isosceles and equilateral triangles, Congruence in right triangles, Using corresponding parts of congruence triangle. The three chapters were selected because the teachers noted these were the chapters students were generally taught about proof and the research literature suggests such there exist a likelihood that students can be exposed to proof for these mathematical topics (Donoghue, 2003;Herbst, 2002).…”

Section: Textbookmentioning

confidence: 99%

The Implications of a Pacing Guide on the Development of Students Ability to Prove in Geometry

Sears

2018

Int Elect J Math Ed

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This study examined the influence of a departmental decision to use the same pacing guide on the planning and enactment of proof tasks of the district-adopted textbook (Prentice Hall Geometry). Quantitative data were collected from a textbook analysis and the tasks students were assigned, and the qualitative data were collected from classroom observations, teachers' artifacts and interviews. The results indicate that teachers adhere to their departmental pacing guide by assigning the same tasks, however, there existed variation in the enacted lessons. Additionally, the results suggest the proof tasks assigned in the pacing guide generally required little cognitive rigor. This study has implication on the development of students' proof skills.

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“…One of the main stakes of geometry instruction has been, since the establishment of such a course in the US high school curriculum, the expectation that students learn geometric proofs. Various arguments have been made over time to justify the value of such a curricular stake; these arguments virtually associate it to each of the notions of authenticity noted above (González and Herbst 2006;Herbst 2002a). Thus, for example, early in the nineteenth century students were expected to learn the proofs of theorems because those were understood as part and parcel of the mathematical content they were supposed to acquire, a text that embodied the mathematical values of the discipline (in the sense of AM D ).…”

mentioning

confidence: 99%

“…We hypothesize that time has shaped curriculum and practice in such a way as to avoid conflict and rather allocate to each notion of authentic mathematics jurisdiction over different domains of instructional practice. 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).…”

mentioning

confidence: 99%

“…First, mathematicians generally write proofs in order to establish the truth of a proposition, and the propositions that are proved are usually instrumental in solving other problems, or in producing subsequent proofs of other propositions. In contrast, scholars have noted that the usual proof exercises done by students in the geometry classroom involve claims that are already known to be true, are inconsequential for later work, and separate proving from other opportunities for problem solving (Herbst 2002a;Schoenfeld 1988).…”

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confidence: 99%

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Teachers’ perspectives on “authentic mathematics” and the two-column proof form

2008

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We investigate experienced high school geometry teachers' perspectives on "authentic mathematics" and the much-criticized two-column proof form. A videotaped episode was shown to 26 teachers gathered in five focus groups. In the episode, a teacher allows a student doing a proof to assume a statement is true without immediately justifying it, provided he return to complete the argument later. Prompted by this episode, the teachers in our focus groups revealed two apparently contradictory dispositions regarding the use of the two-column proof form in the classroom. For some, the two-column form is understood to prohibit a move like that shown in the video. But for others, the form is seen as a resource enabling such a move. These contradictory responses are warranted in competing but complementary notions, grounded on the corpus of teacher responses, that teachers hold about the nature of authentic mathematical activity when proving. This paper explores geometry teachers' perspectives on how the two-column proof form can be both a resource and a constraint in engaging students in proving. The two-column form has been used in the USA to teach students how to do proofs since early in the twentieth century (Herbst 2002a). But the two-column form has been criticized for perpetrating a formalistic image of proving, and calls for reform in the 1980s (NCTM 1989) had recommended that teachers decrease attention to two-column proving. In this article we examine a counterproposal that emerges from listening to teachers: namely, that the twocolumn form can actually be productive when engaging students in proving, by acting not only a constraint, but also a resource enabling complex and non-linear reasoning.

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Cited by 105 publications

References 17 publications

The role of the researcher’s epistemology in mathematics education: an essay on the case of proof

The role of the researcher’s epistemology in mathematics education: an essay on the case of proof

The Implications of a Pacing Guide on the Development of Students Ability to Prove in Geometry

Teachers’ perspectives on “authentic mathematics” and the two-column proof form

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