The roles and nature of proof and justification across Mathematics Education from kindergarten through graduate study
From Kaput Center Wiki
Proof in mathematics is a convincing demonstration of the propositional truth or falsity of some mathematical statement. Proof is considered informal if it is persuasively expressed in natural language. A formal proof, which is the focus of this Wiki, primarily uses a formal language consisting of mathematical symbols. The semantical and syntactical (terms which will be elaborated on below) rules of manipulation allow for the construction of proofs that are not only unambiguous but also timeless. The validity of the geometric theorems that Euclid and Pythagoras constructed 2500 years ago are as acceptable today as they were when they were first proved.
A mathematical statement that has been proved is called a theorem. Once a rigorous mathematical proof has been constructed, it is universally accepted as valid and can then be used to prove other theorems. A theorem is sometimes called a lemma, a minor theorem, when it is used as an intermediate argument in the proof of another (major) theorem.
The one feature that sets mathematics apart from other sciences is the notion of mathematical proof. No other scientific discipline, except those that are purely mathematically based like mathematical physics, is as dependent on proof as is mathematics. Proof is essential in mathematics. A proof in mathematics establishes the timeless validity of a mathematical argument and also arguments that are dependent on them. The certainty and confidence in proved theorems are dependent only upon the terminology, definitions and axioms upon which they are formulated.
On the other hand, the certainty of proofs and theorems in the physical world is the concern of empirical scientists who can never be absolutely certain about empirical results as the discovery of new facts oftentimes nullifies prior results. This observation is summarized in a famous quote of Albert Einstein regarding the laws of mathematics and certainty. He said that as "far as the laws of mathematics refer to reality they are not certain, and as far as they are certain they do not refer to reality."
[edit] Relevant geometry and mathematics of the 21st Century
In the 19th century the world was introduced to non-Euclidean geometries which are geometries that are independent of Euclid's fifth postulate, otherwise called the parallel postulate. These geometries are called hyperbolic (Lobachevskian) and elliptical (Riemannian). Elementary and high school curricula cover parabolic (Euclidean) and analytic (coordinate) geometries but the axioms and proofs in these geometries can only be applied in flat space. Theorems in both hyperbolic and elliptic geometry, however, are applicable to curved spaces.
Hyperbolic geometry was proven to be logically consistent in 1868. Einstein used it in his theory of General Relativity (GR) in 1915. Physical theories, like GR, are dependent on well-formulated mathematical theorems. The mathematics gives physicists the confidence that their theories are 'mathematically' sound and allows for the unification of physical themes. Galileo unified the concepts of rest and motion using the mathematics of mechanics, Newton unified celestial and terrestrial mechanics using mathematics of motion (the calculus), Maxwell unified electricity and magnetism using the vector calculus.
Mathematicians have also established unified concepts in their discipline. Felix Klein showed that projective geometry was a unifying framework for affine, metric, and Euclidean geometries. By the early 20th century the axiomatization of geometry spread to other branches of mathematics with an improvement in the standards of formal proof. Proof from the mathematical perspective has become very rigorous indeed. David Hilbert proposed a set of 20 axioms in 1899 as the foundation for a modern treatment of Euclidean geometry. Euclid's parallel postulate is known to be equivalent to the equidistance postulate, Playfair's axiom, Proclus' axiom, the triangle postulate, and the Pythagorean theorem. Some theorems have become axioms and no longer need to be proved. In the world of computer science higher order pattern unification of algorithmic code has rendered some proof irrelevant by selectively hiding portions of a proof making the resulting code much more efficient to execute.
There is an obvious concern that students are not understanding or learning proof in the traditional classroom or the electronic learning environment. The real issue may turn out to be that students are not being taught proof methods that are required for working with 21st century mathematical structures and geometries. Vector algebra, elliptical geometry, and higher order unification concepts used in electromagnetism, general relativity, and the lambda calculus may be relevant for teaching proof in the 21st century.
[edit] What method of proof do we use synthetic, analytic, or vector?
By the time they reach college students have been introduced, at least conceptually, to several types of geometries: plane (synthetic) and coordinate (analytic), affine and projective, euclidean and non-euclidean. Proof in each of these geometries requires, speaking euphemistically, an appropriate frame of reference.
Consider the proof that the medians of a triangle are concurrent. We could prove it synthetically, analytically, or using vector methods. The synthetic proof requires more steps than the analytic proof and many more steps than vector proof. It goes without saying that proof principles from one method inform, from the pedagogical point of view off view, alternative proof techniques but primarily from the perspective of the teacher. The student can be confused trying to use arguments, or heuristics, from multiple perspectives. Recall the difficulty that students routinely have when professors introduce geometrical arguments to complete a proof in the calculus.
Abstractions have obvious advantages in proof. The algebra and topology, symbolic and predicate logic eliminate the need for numerical measurements as well as geometrical figures allowing teachers to focus on the formal (essential) properties involved in proof. Symbols (syntactical instruments = the bricks) can be manipulated but the conceptual proof schema (semantic instruments = the mortar) is used to cement the bricks together. This analogy from masonry also reveals another relationship between bricks (syntax) and mortar (semantics) in the construction of buildings. The mortar is normally hidden in order to highlight the aesthetic of the bricks!
[edit] Mathematical cognition, semiotic representations, and cognitive paradoxes
In February 2007 Stephen Hegedus, Director of the James J. Kaput Center for Research and Innovation in Mathematics Education, presented a paper on TECHNOLOGY THAT MEDIATES AND PARTICIPATES IN MATHEMATICAL COGNITION[1].
Data and analyses of students working with dynamic geometry software and software that linked "multiple representations of function" was presented at the conference. The focus was on the role of both software and integrated hardware, called dynamic mathematics, that opens up "new forms of expressivity and new forms of understanding for students."
Student investigations, explorations, and cognition of mathematical concepts are built upon the capacity of the virtual environment to react to actions, a dynamic coaction of student and digital platform environment. A "recursive exploration space, action–reaction loop, forces cognition to be distributed in the space defined by the agent (the student) and the environment. The emergent distributed intelligence is made tangible by a responsive environment (increasing in agency) and the digital tools in front of the agent-role of the student."
The presence and intelligent use of digital technologies in mathematics education has revived an interest in understanding ways of conceiving mathematics and, in particular, understanding mathematical cognition. Researchers, like Duval and Hegedus, regard mathematical cognition as an emergence and transformation of "successive and evolving representational systems."
Dynamic software, like Cabri and Sketchpad, and SimCalc, software that links multiple representations of function in interactive ways across networks, are referred to as Dynamic Technological Environments (DTE). DTEs have three distinguishing attributes:
Attribute 1: Recursive Exploration Space
Attribute 2: Sharing of the mathematical objects that students create (networking).
Attribute 3: Executability of the semiotic representations embodied in the environment.
Semiotic representations are a central issue in discussing proof in mathematics. At the CERME5 conference Dr. Hegedus referred to a historical deep question concerning mathematical objects and their semiotic representations.
The question, posed by Duval (2001, 2006), was set in the context of the cognitive paradox of comprehension in mathematics in which there is a confusion of an object and its (semiotic) representation. Duval (p.107) emphasizes that in mathematics one can "confuse an object and its representation ... in contrast to other domains of scientific knowledge" and it is precisely this "that poses a major problem. Because mathematical objects, in contrast to phenomena of astronomy, physics, chemistry, biology, etc., are never accessible by perception or by instruments (microscopes, telescopes, measurement apparatus,…). Access to mathematical objects must necessarily pass by way of semiotic representations. This explains moreover why the development of mathematical knowledge led to the development and diversification of registers of representations. One can therefore formulate the paradox of comprehension in mathematics in the following way: : how can one not confuse an object and its representation if one has no access to this object apart from its representation?"
[edit] Teaching & learning proof: K-16 curriculum
In March 2009 Routledge Education co-published Teaching and Learning Proof Across the Grades: A K-16 Perspective.[2] for the National Council of Teachers of Mathematics (NCTM). The publication highlights the main ideas that have recently emerged on proof research and, more importantly, defines an agenda for future study. The chapters in the book discuss (1) how forms of proof evolve chronologically and cognitively, and (2) how curricula and instruction can support the development of students’ understanding of proof.
Almost six years ago, in the June 2003 issue of the MAA Online Research Sampler column of The Mathematical Association of America, Keith Weber, who teaches in the Graduate School of Education at Rutgers University, reminded us that proofs in mathematics are a "notoriously difficult mathematical concept for students." He referenced the empirical studies that showed "that many students emerge from proof-oriented courses such as high school geometry, introduction to proof, real analysis, and abstract algebra unable to construct anything beyond very trivial proofs. Furthermore, most university students do not know what constitutes a proof and cannot determine whether a purported proof is valid."[3]
[edit] Thinking/Understanding, Semantic/Syntactic, and other Dualities
[edit] References
Hegedus, S., Dalton, S., & Moreno-Armella, L. (2007). Technology that mediates and participates in mathematical cognition. Paper presented at 5th Congress of ERME, the European Society for Research in Mathematics Education, CERME5, Larnaka, Cyprus.
Duval R., (2001). The cognitive analysis of problems of comprehension in the learning of mathematics. Paper presented at the Semiotics Discussion Group of the 25th Conference of the International Group for the Psychology of Mathematics Education, Utrecht, Freudenthal Institute, The Netherlands.
Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61: pp. 103-131.
Hilbert, D. (1971). Foundations of Geometry. 10th English edition, translation of the second German edition by L. Unger. Chicago: Open Court.
