Terms and expressions of research on teaching-learning proof

Work in progress

This page is created in support of the work of the Research gate group 

"Proof in Mathematics Education: Reflection and Institutionalization"

Argument

Base argument

"In a classroom situation where students engage in a proving activity, the teacher who manages the situation or a researcher who studies the situation can use the three major components of an argument to identify and compare two arguments in that situation. The first argument is what I call the base argument, and it represents the prevailing student argument at the initial stages of the proving activity." (Stylianides 2007 p.299)

Classroom community

"... the term "classroom community" requires some clarification. I consider the classroom community to consist primarily of the students. The teacher has a special membership status in this community as the representative of the discipline of mathematics (Yackel & Cobb, 1996) and as the person who has a special role to play in trying to connect students with broader mathematical knowledge (Ball, 1993; Lampert, 1992)." (Stylianides 2007 p.292)

Empirical argument 

"... by empirical arguments I mean arguments based on the use of examples that offer confirming, yet incomplete, evidence that a mathematical claim is true." (Stylianides 2007 p.298)

Ensuing argument

"The first argument is the base argument, which marks the starting point of an instructional intervention. The second argument, which I call the ensuing argument, is the principal argument that seems to result from, or mark the endpoint of, the instructional intervention." (Stylianides 2007 p.314)

Formal Proof

"The term rigorous proof or formal proof ... is understood here to mean a proof in mathematics or logic which satisfies two conditions of explicitness. First, every definition, assumption, and rule of inference appealed to in the proof has been, or could be, explicitly stated; in other words, the proof is carried out within the frame of reference of a specific known axiomatic system. Second, every step in the chain of deductions which constitutes the proof is set out explicitly." (Hanna 1983 p. 3)

Level of mathematical rigor

"By level of mathematical rigor of an argument I mean the extent to which the three major components of an argument -- the set of accepted statements, the modes of argumentation, and the modes of argument representation -- meet the disciplinary standards for these components in the development of proofs." (Stylianides 2007 p.394) 

Proof

Stylianides definition meets "(1) the intellectual-honesty principle, which states that the notion of proof in school mathematics should be conceptualized so that it is, an once, honest to mathematics as a discipline and honoring of students as mathematical learners; and (2) the continuum principle, which states that there should be continuity in how the notion of proof is conceptualized in different grade levels so that students' experiences with proof in school have coherence." (Stylianides 2007) : 

"Proof is a mathematical argument, a connected sequence of assertions for or against a mathematical claim, with the following characteristics:

1. It uses statements accepted by the classroom community (set of accepted statements) that are true and available without further justification;
2. It employs forms of reasoning (modes of argumentation) that are valid and known to, or within the conceptual reach of, the classroom community; and
3. It is communicated with forms of expression (modes of argument representation) that are appropriate and known to, or within the conceptual reach of, the classroom community." (Stylianides 2007 p.291)

"I claim that the definition of proof is appropriate for use in school mathematics primarily because it has the following four characteristics: (1) it considers both mathematics as a discipline and students as mathematical learners; (2) it promotes a consistent meaning of proof throughout the grades; (3) it prevents empirical arguments from being considered as proofs; and (4) it supports analysis of classroom instruction related to proof and study of the role of teachers in managing their students' proving activity." (Stylianides 2007 p.293-294) 

"The definition excludes the possibility of empirical arguments being considered as proofs at any level of schooling, where by empirical arguments I mean arguments based on the use of examples that offer confirming, yet incomplete, evidence that a mathematical claim is true." (Stylianides 2007 p.298)

Proof threshold

"[...] how the base and ensuing arguments are situated with respect to the critical value of mathematical rigor of an argument that I call proof threshold. If an argument meets the definition of proof on all three components of an argument, the level of rigor of the argument exceeds the proof threshold and so the argument is said to meet the standard of proof." (Stylianides 2007 p.314)

Proving task 

Rigorous proof

See "Formal proof"