What Does “Without Loss of Generality” Mean,and How Do We Detect It

When one goes from a geometrical statement to an algebraic statement, the immediate translation is to replace every point by a pair of coordinates, if in the plane (or more as required). A statement with N points is then a statement with 2N (or 3N or more) variables, and the complexity of tools like cylindrical algebraic decomposition is doubly exponential in the number of variables. Hence one says “without loss of generality, A is at (0,0)” and so on. How might one automate this, in particular the detection that a “without loss of generality” argument is possible, or turn it into a procedure (and possibly even a formal proof)?     

Is this a coincidence? The role of examples in fostering a need for proof

It is widely known that students often treat examples that satisfy a certain universal statement as sufficient for showing that the statement is true without recognizing the conventional need for a general proof. Our study focuses on special cases in which examples satisfy certain universal statements, either true or false in a special type of mathematical task, which we term “Is this a coincidence?”. In each task of this type, a geometrical example was chosen carefully in a way that appears to illustrate a more general and potentially surprising phenomenon, which can be seen as a conjecture. In this paper, we articulate some design principles underlying the choice of examples for this type of task, and examine how such tasks may trigger a need for proof. Our findings point to two different kinds of ways of dealing with the task. One is characterized by a doubtful disposition regarding the generality of the observed phenomenon. The other kind of response was overconfidence in the conjecture even when it was false. In both cases, a need for “proof” was evoked; however, this need did not necessarily lead to a valid proof. We used this type of task with two different groups: capable high school students and experienced secondary mathematics teachers. The findings were similar in both groups.     

Proof validation in real analysis: Inferring and checking warrants

In the study reported here, we investigate the skills needed to validate a proof in real analysis, i.e., to determine whether a proof is valid. We first argue that when one is validating a proof, it is not sufficient to make certain that each statement in the argument is true. One must also check that there is good reason to believe that each statement follows from the preceding statements or from other accepted knowledge, i.e., that there is a valid warrant for making that statement in the context of this argument. We then report an exploratory study in which we investigated the behavior of 13 undergraduates when they were asked to determine whether or not a particular flawed mathematical argument is a valid mathematical proof. The last line of this purported proof was true, but did not follow legitimately from the earlier assertions in the proof. Our findings were that only six of these undergraduates recognized that this argument was invalid and only two did so for legitimate mathematical reasons. On a more positive note, when asked to consider whether the last line of the proof followed from previous assertions, a total of 10 students concluded that the statement did not and rejected the proof as invalid.     

On null geodesically complete spacetimes under NEC and NGC: Is the Gao-Wald “time dilation” a topological effect?

We review a theorem of Gao-Wald on a kind of a gravitational “time delay” effect in null geodesically complete spacetimes under NEC and NGC, and we observe that it is not valid anymore throughout its statement, as well as a conclusion that there is a class of cosmological models where particle horizons are absent, if one substituted the manifold topology with a finer (spacetime) topology. Since topologies of the Zeeman-Göbel class incorporate the causal, differential, and conformal structure of a spacetime and there are serious mathematical arguments in favour of such topologies and against the manifold topology, there is a strong evidence that “time dilation” theorems of this kind are topological in nature rather than having a particular physical meaning.     

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

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.     

Are You Convinced? Middle‐Grade Students' Evaluations of Mathematical Arguments

Students learn norms of proving by observing teachers generating proofs, engaging in proving, and generalizing features of proofs deemed convincing by an authority, such as a textbook. Students at all grade levels have difficulties generating valid proof; however, little research exists on students' understandings about what makes a mathematical argument convincing prior to more formal instruction in methods of proof. This study investigated middle‐school students' (ages 12–14) evaluations of arguments for a statement in number theory. Students evaluated both an empirical and a general argument in an interview setting. The results show that students tend to prefer empirical arguments because examples enhance an argument's power to show that the statement is true. However, interview responses also reveal that a significant number of students find arguments to be most convincing when examples are supported with an explanation that “tells why” the statement is true. The analysis also examined the alignment of students' reasons for choosing arguments as more convincing along with the strategies they employ to make arguments more convincing. Overall, the findings show middle‐school students' conceptions about what makes arguments convincing are more sophisticated than their performance in generating arguments suggests.     

Problem solving heuristics,affect, and discrete mathematics

It has been suggested that activities in discrete mathematics allow a kind of new beginning for students and teachers. Students who have been “turned off” by traditional school mathematics, and teachers who have long ago routinized their instruction, can find in the domain of discrete mathematics opportunities for mathematical discovery and interesting, nonroutine problem solving. Sometimes formerly low-achieving students demonstrate mathematical abilities their teachers did not know they had. To take maximum advantage of these possibilities, it is important to know what kinds of thinking during problem solving can be naturally evoked by discrete mathematical situations—so that in developing a curriculum, the objectives can include pathways to desired mathematical reasoning processes. This article discusses some of these ways of thinking, with special attention to the idea of “modeling the general on the particular.” Some comments are also offered about students' possible affective pathways and structures.     

Proofs as bearers of mathematical knowledge

Yehuda Rav’s inspiring paper “Why do we prove theorems?” published in Philosophia Mathematica (1999, 7, pp. 5–41) has interesting implications for mathematics education. We examine Rav’s central ideas on proof—that proofs convey important elements of mathematics such as strategies and methods, that it is “proofs rather than theorems that are the bearers of mathematical knowledge”and thus that proofs should be the primary focus of mathematical interest—and then discuss their significance for mathematics education in general and for the teaching of proof in particular.     

Prototype Proofs in Type Theory

The proofs of universally quantified statements, in mathematics, are given as “schemata” or as “prototypes” which may be applied to each specific instance of the quantified variable. Type Theory allows to turn into a rigorous notion this informal intuition described by many, including Herbrand. In this constructive approach where propositions are types, proofs are viewed as terms of λ‐calculus and act as “proof‐schemata”, as for universally quantified types. We examine here the critical case of Impredicative Type Theory, i. e. Girard's system F, where type‐quantification ranges over all types. Coherence and decidability properties are proved for prototype proofs in this impredicative context.     

Possible Student Justification of Proofs

The National Council of Teachers of Mathematics calls for an increased emphasis on proof and reasoning in school mathematics curricula. Given such an emphasis, mathematics teachers must be prepared to structure curricular experiences so that students develop an appreciation for both the value of proof and for those strategies that will assist them in developing proving skills. Such an outcome is more likely when the teacher feels secure in his/her own understanding of the concept of “mathematical proof” and understands the ways in which students progress as they take on increasingly more complex mathematical justifications. In this article, a model of mathematical proof, based on Balacheff's Taxonomy of Mathematical Proof, outlining the levels through which students might progress as they develop proving skills is discussed. Specifically, examples of the various ways in which students operating at different levels of skill sophistication could approach three different mathematical proof tasks are presented. By considering proofs under this model, teachers are apt to gain a better understanding of each student's entry skill level and so effectively guide him/her toward successively more sophisticated skill development.     

Mathematikdidaktik: Die Erforschung theoretischen Wissens in sozialen Kontexten des Lernens und Lehrens

The problem of “defining” mathematics education as a proper scientific discipline has been discussed controversely for more than 20 years now. The paper tries to clarify some important aspects especially for answering the question of what makes mathematics education a specific scientific discipline and a field of research. With this aim in mind the following two dimensions are investigated: On the one hand, one has to be aware that mathematics is not “per se” the object of research in mathematics education, but that mathematical knowledge always has to be regarded as being “situated” within a social context. This means that mathematical knowledge only gains its specific epistemological meaning within a social context and that the development and understanding of mathematical knowledge is strongly influenced by the social context. On the other hand the specificity of the theory-practice-problem poses an essential demand on the scientific work in mathematics education.     

The interplay of empirical and deductive reasoning in proving “if” and “only if” statements in a Dynamic Geometry environment

This paper is situated within the ongoing enterprise to understand the interplay of students’ empirical and deductive reasoning while using Dynamic Geometry (DG) software. Our focus is on the relationships between students’ reasoning and their ways of constructing DG drawings in connection to directionality (i.e., “if” and “only if” directions) of geometry statements. We present a case study of a middle-school student engaged in discovering and justifying “if” and “only if” statements in the context of quadrilaterals. The activity took place in an online asynchronous forum supported by GeoGebra. We found that student's reasoning was associated with the logical structure of the statement. Particularly, the student deductively proved the “if” claims, but stayed on empirical grounds when exploring the “only if” claims. We explain, in terms of a hierarchy of dependencies and DG invariants, how the construction of DG drawings supported the exploration and deductive proof of the “if” claims but not of the “only if” claims.     

Extending Independent Sets to Bases and the Axiom of Choice

We show that the both assertions “in every vector space B over a finite element field every subspace V ? B has a complementary subspace S” and “for every family ?? of disjoint odd sized sets there exists a subfamily ?={F_j:j ?ω} with a choice function” together imply the axiom of choice AC. We also show that AC is equivalent to the statement “in every vector space over ? every generating set includes a basis”.     

Distinguishing Features of Mechanical and Human Problem-Solving

Over the years, research in mathematical problem-solving has examined expert/novice problem-solving performance on various types of problems and subjects. In particular, DeFranco examined two groups of Ph.D. mathematicians as they solved four mathematics problems and found that although all were content experts, only one group were problem-solving experts. Based on this study, this article posits the notion that one distinguishing feature between experts and novices is that experts tend to look for special features of a problem and use algorithms only as a “fail-safe” system while novices act like “machines” relying on algorithms to solve the problems. Why? The article explores the idea that novice problem solvers learned to solve problems the way they learned proof, that is, in a formal, abstract and mechanizable way. Beliefs about proof and the culture in which it is practiced help frame a mathematician's view of the discipline and ultimately impacts classroom practice. The authors believe that current classroom instruction tends to create a culture that fosters algorithmic proficiency and a “machine-like” approach to the learning of mathematics and problem-solving. Further, they argue that mathematicians need to be aware of the distinction between knowing a proof is true and explaining why it is true. When these distinctions are appreciated and practiced during classroom instruction, then and only then will students begin to acquire the mathematical knowledge to become better problem solvers.     

Young students’ subjective interpretations of mathematical diagrams: elements of the theoretical construct “frame-based interpreting competence”

During the last few decades several studies have showed that mathematical visual aids are not at all self-explanatory. Nevertheless, students do make sense of those representations spontaneously and—as a matter of course—cannot avoid their own sense-making. Further, the function of visual aids as “re-presentation” of a given structure is complemented through an epistemological function to explore mathematical structures and generate new meaning. But in which way do socially learned interpreting schemes (frames) influence children’s subjective interpretations of mathematical diagrams? The CORA project investigates which frames can be reconstructed in young pupils’ interpretations of visual diagrams. This paper presents central ideas, theoretical background and—by means of short sequences from pre- and post-interviews—first aspects of “frame-based interpreting competence”. We describe children’s subjective frames in a range between “object-oriented” (focus on the diagram’s visible elements) and “system-oriented” (focus on relation between those elements).     

Is the definition of mathematics as used in the PISA assessment Framework applicable to the HarmoS Project?

The project known as the “Harmonisation of the Obligatory School”, or in its shortened form as “HarmoS”, has a high priority for Switzerland's educational policy in the coming years. Its purpose is to determine levels of competency, valid throughout Switzerland, for specific areas of study and including the subject of mathematics. The general theoretical basis of the overall HarmoS Project is constituted by the expertise written under the direction of Eckhard Klieme and entitled “Zur Entwicklung nationaler Bildungsstandards” (Klieme 2003) [i.e. “On the Development of National Education Standards”]. The proposal announcing the HarmoS partial project devoted to Mathematics includes references to the results and subsequent analysis of PISA 2003. It thus seems appropriate for us to begin our work on HarmoS with a critical consideration of the definition of mathematics and mathematical literacy as they are used in the PISA Study. In a first part, we want to describe the core ideas of HarmoS. In a second part, we will address the meaning of general educational goals for the development of competency models and education standards to the extent that it is necessary to properly locate our problem. In a third part we will analyse the concept of mathematics which is at the basis of the PISA Study (OECD 2004) and more precisely defined in the publication “Assessment Framework” (OECD 2003) In the fourth and last part, we will try to provide a differentiated answer to the question posed in the title of this paper.     

Advertising a new product in a segmented market

We bring some market segmentation concepts into the statement of the “new product introduction” problem with Nerlove-Arrow’s linear goodwill dynamics. In fact, only a few papers on dynamic quantitative advertising models deal with market segmentation, although this is a fundamental topic of marketing theory and practice. In this way we obtain some new deterministic optimal control problems solutions and show how such marketing concepts as “targeting” and “segmenting” may find a mathematical representation. We consider two kinds of situations. In the first one, we assume that the advertising process can reach selectively each target group. In the second one, we assume that one advertising channel is available and that it has an effectiveness segment-spectrum, which is distributed over a non-trivial set of segments. We obtain the explicit optimal solutions of the relevant problems.     

Syllogisms and 5-Square of Opposition with Intermediate Quantifiers in Fuzzy Natural Logic

In this paper, we provide an overview of some of the results obtained in the mathematical theory of intermediate quantifiers that is part of fuzzy natural logic (FNL). We briefly introduce the mathematical formal system used, the general definition of intermediate quantifiers and define three specific ones, namely, “Almost all”, “Most” and “Many”. Using tools developed in FNL, we present a list of valid intermediate syllogisms and analyze a generalized 5-square of opposition.     

Traveling down the road: from cognitive neuroscience to mathematics education … and back

In this commentary paper to the special issue on “Cognitive Neuroscience and Mathematics Education”, we reflect on the connection between cognitive neuroscience and mathematics education from an educational research point of view. The current issue highlights that cognitive neuroscience offers a series of tools, methodologies and theories to investigate cognitive processes that take place during mathematical thinking and learning. This might complement and extend our knowledge that has been obtained on the basis of behavioral data only, the common approach in educational research. At the same time, we note that the existing neuroscientific studies have investigated mathematical performance in relative isolation from the educational context. The characteristics of this context have, however, a large influence on mathematical performance and its correlated brain activity, an issue that should be addressed in future research. We contend that traveling back and forth from cognitive neuroscience to mathematics education might yield a better understanding of how mathematical learning takes place and how it can be influenced.     

Cardinalities of Residue Fields of Noetherian Integral Domains

We determine the relationship between the cardinality of a Noetherian integral domain and the cardinality of a residue field. One consequence of the main result is that it is provable in Zermelo–Fraenkel Set Theory with Choice (ZFC) that there is a Noetherian domain of cardinality ?₁ with a finite residue field, but the statement “There is a Noetherian domain of cardinality ?₂ with a finite residue field” is equivalent to the negation of the Continuum Hypothesis.