Related papers: Making mathematical claims
"Math is not a spectator sport." "Lecturing is educational malpractice." Slogans like these rally some mathematicians to teach classes that feature "active learning", where lecturing is eschewed for student participation. Yet as much as I…
While proof is a central component of postsecondary mathematical study, proof construction has historically posed significant difficulties for students who intend to earn mathematics degrees at the undergraduate level. This work is…
Real-life conjectures do not come with instructions saying whether they they should be proven or, instead, refuted. Yet, as we now know, in either case the final argument produced had better be not just convincing but actually verifiable in…
Even if students can make the blend, interpret physics correctly in mathematical symbology and graphs, they still need to be able to apply that knowledge in productive and coherent ways. As instructors, we can show our solutions to complex…
Many mathematicians find mathematics aesthetically beautiful and even comparable to art forms such as music or painting. On the other hand, every year a great number of school students leave mathematics with total disillusionment and…
A close look at students' written work on examinations offers a wealth of information about their performance, their knowledge of the subject, their strengths, weaknesses and misconceptions, and their overall level of mathematical skills…
Practicing mathematicians often assume that mathematical claims, when they are true, have good reasons to be true. Such a state of affairs is "unreasonable", in Wigner's sense, because basic results in computational complexity suggest that…
In this note we report on an implementation of discovery-oriented problems in courses on Real Analysis and Differential Equations. We explain a type of task-design that gives students the opportunity to conjecture, refute and prove. What is…
Consider the following story: A teacher announces to her students a test for the following week, such that the test will be ``surprising''. The students use this as the basis for a ``logical derivation'' and reach a contradiction, which…
This note presents reflections drawn from my recent experiences in teaching a course on mathematics and sustainability, with a particular emphasis on raising awareness of the topic and its broader implications. The lectures were structured…
"Mathematicians, like physicists, are pushed by a strong fascination. Research in mathematics is hard, it is intellectually painful even if it is rewarding, and you would not do it without some strong urge." [D. Ruelle]. We shall give some…
The paper examines the construction of a course in mathematical analysis at a pedagogical university, aimed at developing the ability of future mathematics teachers to detect and solve problems related to finding proofs. Key words: teaching…
It is nowadays common to consider that proof must be part of the learning of mathematics from Kindergarten to University1. As it is easy to observe, looking back to the history of mathematical curricula, this has not always been the case…
One challenge (or opportunity!) that many instructors face is how varied the backgrounds, abilities, and interests of students are. In order to simultaneously instill confidence in those with weaker preparations and still challenge those…
Constructivists (and intuitionists in general) asked what kind of mental construction is needed to convince ourselves (and others) that some mathematical statement is true. This question has a much more practical (and even cynical)…
In this article we describe special type of mathematical problems that may help develop teaching methods that motivate students to explore patterns, formulate conjectures and find solutions without only memorizing formulas and procedures.…
In this paper we propose a very specific educational challenge that teachers can use to motivate ambitious and enthusiastic mathematics students who have mastered basic trigonometry and trig functions. The objective is to lead students to a…
A significant amount of research has considered mathematical proofs, the students who learn them, and the instructors that teach them, from a variety of perspectives. This paper considers this topic from four main perspectives: students'…
The learning of mathematics starts early but remains far from any theoretical considerations: pupils' mathematical knowledge is first rooted in pragmatic evidence or conforms to procedures taught. However, learners develop a knowledge which…
Mathematicians judge proofs to possess, or lack, a variety of different qualities, including, for example, explanatory power, depth, purity, beauty and fit. Philosophers of mathematical practice have begun to investigate the nature of such…