Problem Posing in Mathematics

In the 21st century, creativity has been emphasized as one of the crucial skills needed to tackle challenges stemming from technology advancement and rapid changes in society and the labor market. Therefore, various countries include the development of creativity in their education policies and curricula. Like many other countries, the Mongolian national curriculum aims to develop creativity across the subjects. Specifically, the math curriculum emphasizes mathematical creativity as a learning outcome to be measured. The intended policies and the curricula emphasize creativity; at the same time, it has been a challenging task for educators to boost and measure creativity. The present study attempts to examine mathematical creativity among Mongolian students. Research participants were 278 ninth-grade students from urban, suburban, and rural schools in Mongolia. The students completed a mathematical problem-posing test. The study found Mongolian ninth-grade students have lower mathematical creativity. The findings suggest that students' ability to think in a variety of and novel ways should be supported through various effective teaching strategies.

Our presentation considers the problems associated with assessment of problem-solving group work. In the experience carried out in our own classrooms with students aged 15-17 the assessment of open-ended group work is possible by means of dividing activities into phases and classifying the expected results. This building up of research routes involves a delicate process of didactical planning but it allows us to evaluate reliably through suitable grids. The production of these grids is possible if we analyse the cognitive categories implied in the development of problems set for the students. Cette communication présente les problèmes qui sont associés à l'èvaluation du travail de groupe "problem-solving". Dans les expériences réelement faites dans nos classes avec des élèves de 15-17 ans on a vu que l'evaluation du travail de groupe est possible si l'on partage les activitées en phases et on classifie les résultats que l'on s'attend. Cette construction de parcours de recherche est un moment délicat qui doit etre planifié du point de vue didactique et cela nous conduit à une évaluation plus objective par des grilles convenables. La production de ces grilles est possibles si on analyse les catégories cognitives qui sont impliquées dans le développement des problèmes prèsentes aux élèves.

The aim of this study is to examine the views of pre-service mathematics teacher on mathematical reasoning and problem-solving skills. The study will contribute to the determination of how, where, when and for what purpose pre-service mathematics teachers can use their mathematical reasoning skills in teaching activities and the reasons why learners can make these associations during the problem solving process. The data on which the study is based are thought to contain important signs of how pre-service mathematics teachers can gain mathematical reasoning and problem solving processes to students and how to use them successfully. The study was designed in phenomenological model of qualitative research methods. The study was carried out on 107 pre-service teachers who were studying in the department of primary mathematics education and selected for purposive sampling method. In order to collect data, the Semi-Structured Interview Form which was developed by the researchers was used to determine the pre-service teachers' views on mathematical reasoning and problem solving. The data obtained through this form were analyzed by the content analysis method. When the research results are examined; It is determined that pre-service teachers do not have enough knowledge about reasoning, they need such education and they have positive attitudes towards reasoning. On the other hand, pre-service teachers stated the problem as a pending action and problem. In addition, it is thought that deeper investigation is still needed on the phenomenon of reasoning in terms of learner groups.

As an introduction to the special issue on problem posing, the paper presents a brief overview of the research done on this topic in mathematics education. Starting from this overview, the authors acknowledge important issues that need to be taken into account in the developing field of problem posing and identify new directions of research, some of which are addressed by the collection of the papers included in this volume.

Esercizi e problemi con soluzioni e svolgimenti
Seconda edizione aggiornata 2013

Con il moltiplicarsi dei corsi di laurea nei quali è presente l’esame propedeutico di Matematica Generale si è assistito a una proliferazione di pubblicazioni che illustrano le principali nozioni di Algebra, Analisi e Geometria. Questi testi, se da un lato rappresentano un efficace strumento di supporto alle lezioni teoriche, dall'altro non forniscono tutti gli strumenti necessari per superare effettivamente l'esame finale.
Il presente volume nasce con il tentativo di ovviare a tale inconveniente fornendo abbondante materiale su tutti i principali argomenti di un esame di Matematica Generale: insiemi, funzioni, successioni, limiti e integrali, sia nella loro veste classica che in quella applicativa. Infine, a tutti coloro che vivono la matematica come un'interessante sfida intellettuale, con la quale confrontarsi e divertirsi, è dedicata l'ultima sezione che vi lasciamo piacevolmente scoprire!

Indice:

Introduzione

1. Se volete un pretesto...
2. Insiemi e relazioni
3. Matematica... ma con discrezione
4. Un primo sguardo alle funzioni
5. Successioni
6. Serie numeriche
7. Limiti di funzioni
8. Derivate e dintorni
9. Disegnare una funzione
10. Interpretare un grafico
11. Ancora sulle funzioni
12. Integrali
13. Algebra lineare
14. Funzioni a più variabili
15. Quando il gioco si fa duro

Soluzioni
Appendice

Nota sugli Autori:

Claudio Marini
Nato a Orvieto nel 1978, si laurea in Matematica nell’Ateneo senese nel 2001. Nella stessa Università consegue un Master e un Dottorato in Logica Matematica e Informatica Teorica. Parallelamente a un variegato lavoro di ricerca (si occupa di logiche fuzzy, giochi a informazione incerta, linguaggi formali, ecc.) porta avanti da anni, con scrupolo ed entusiasmo, un’intensa attività didattica che spazia su più fronti: dall’insegnamento nel carcere ai corsi serali per adulti, passando per l’università e la scuola secondaria.

Giuseppe Scianna
Nato a Livorno nel 1965 ma cresciuto scientifi camente a Pisa, si laurea in Matematica nel 1988, perfezionandosi successivamente a Trento con un Dottorato di Ricerca. Dal 1992 è di ruolo presso l’Ateneo di Siena e tiene corsi di base e corsi avanzati di Matematica nella Facoltà di Economia e in quella di Scienze Matematiche Fisiche e Naturali. Sino ad ora ha avuto prevalenti interessi di ricerca in calcolo delle variazioni, teoria geometrica della misura ed equazioni differenziali alle derivate parziali.

Los Programas de Estudio de Matemática en Costa Rica, proponen la Resolución de Problemas en contextos reales como estrategia metodológica principal y el Planteamiento de Problemas como uno de los cinco procesos matemáticos. Así, este estudio analiza algunos elementos que intervienen en el proceso de enseñanza y aprendizaje de contenidos matemáticos empleando dicha estrategia y el papel del planteamiento de problemas como actividad complementaria en dicho proceso. Los resultados muestran la importancia del trabajo del profesor como organizador y guía de la clase y del estudiante como responsable de resolver el problema; así como del gran valor educativo que tiene el planteamiento de problemas en el proceso de Resolución de Problemas

This international review offers current findings on the art and science of problem posing and its multiple contributions to mathematics teaching, learning, training, and curriculum
design. Diverse perspectives on problem posing frame the concept as a springboard for scientific inquiry and the process as a means to promote mathematical understanding across the primary and secondary grades. Problem posing is demonstrated as enhancing students' problem-solving skills, bolstering knowledge retention, and improving attitudes toward mathematics, and the book provides evidence-based strategies for its integration into both classroom work and teacher education. This information is particularly critical as
mathematics-based knowledge continues to dominate technology and the sciences.

The Riemann zeta function is one of the most Leonhard  Euler important and fascinating functions in mathematics. Analyzing the matter of conjecture of Riemann divide our analysis in the zeta function and in the proof of conjecture, which has consequences on the distribution of prime numbers.

Matematika merupakan mata pelajaran yang diajarkan di setiap tingkatan sekolah, mulai dari sekolah dasar sampai perguruan tinggi. Matematika merupakan ilmu yang mempunyai peran penting dalam memajukan daya pikir manusia yang mendasari perkembangan teknologi modern dan digunakan sebagai alat penting di berbagai bidang, termasuk ilmu alam, teknik, kedokteran/medis, dan ilmu sosial. Oleh karena itu, pembelajaran matematika sangat perlu diberikan di sekolah dasar sesuai dengan Standar Isi yang tertera pada Permendiknas No 22 Tahun 2006, yang menyatakan bahwa:

Here are two related conjectures, each called the twin prime conjecture. The first version states that there are an infinite number of pairs of twin primes (Guy 1994, p. 19). It is not known if there are an infinite number of such primes (Wells 1986, p. 41; Shanks 1993, p. 30), but it seems almost certain to be true. Arenstorf (2004) published a purported proof of the conjecture (Weisstein 2004). Unfortunately, a serious error was found in the proof. As a result, the paper was retracted and the twin prime conjecture remains fully open. According to theorem 5, which is based on the previous ones, when we go up to the order of magnitude of numbers, the number of twin primes becomes infinite.

In this paper, we describe the process of students’ self-assessment of their creativity and its development in the context of posing mathematical problems, presuming that such a process would support the development of their creativity. Examination of two case studies reveals that self-assessment of creativity may support its development provided that one possesses specific personal recourses; however, this process might suppress the creativity of those lacking the needed resources. Therefore, we suggest that self-assessment of creativity cannot stand on its own, and should be supplemented by teachers’ feedback or other environmental 'scaffolding'.

For any intuitionistic multi-fuzzy set A = { < x , µ A (x) , ν A (x) > : x∈X} of an universe set X, we study the set [A] (α, β) called the (α, β)–lower cut of A. It is the crisp multi-set { x∈X : µ i (x) ≤ α i , ν i (x) ≥ β i , ∀i } of X. In this paper, an attempt has been made to study some algebraic structure of intuitionistic multi-anti fuzzy subgroups and their properties with the help of their (α, β)–lower cut sets. Keywords Intuitionistic fuzzy set (IFS), Intuitionistic multi-fuzzy set (IMFS), Intuitionistic multi-anti fuzzy subgroup (IMAFSG), Intuitionistic multi-anti fuzzy normal subgroup (IMAFNSG), (,)–lower cut, Homomorphism.

La invención de problemas es una actividad con valor reconocido dentro de la experiencia matemática educativa, por otra parte, desde el punto de vista de la investigación la caracterización de problemas matemáticos planteados por estudiantes permite ahondar en sus conocimientos matemáticos, en este caso, sobre el concepto de multiplicación. Específicamente se estudian tipos de contextos, tipos de problemas multiplicativos y complejidad matemática de las situaciones multiplicativas creadas por 109 estudiantes chilenos de sexto año básico. De un total de 327 producciones, se identifican 177 problemas matemáticos multiplicativos en los cuales prevalecen contextos personales, problemas multiplicativos del tipo proporcionalidad simple y complejidad matemática de reproducción.

The links between the mathematical and cognitive models that interact during problem solving are explored with the purpose of developing a reference framework for designing problem-posing tasks. When the process of solving is a successful one, a solver successively changes his/her cognitive stances related to the problem via transformations that allow different levels of description of the initial wording. Within these transformations, the passage between successive phases of the problem-solving process determines four operational categories: decoding (transposing the text into more explicit relations among the data and the operating schemes, induced by the constraints of the problem), representing (transposing the problem via a generated mental model), processing (identifying an associated mathematical model based on the mental configurations suggested by the problem and own mathematical competence), and implementing (applying identified mathematical techniques to the particular situation of the problem, with the purpose of drafting a conventional solution). The study of this framework in action offers insights for more effective teaching and can be used in problem posing and problem analysis in order to devise questions more relevant for deep learning.

This article provides an illustration of the explanatory and discovery functions of proof with an original geometric conjecture made by Clough, a Grade 11 student in relation to Viviani's theorem. After logically explaining (proving) the result geometrically and algebraically, the result is generalised to other polygons by further reflection on the proof(s). Different proofs are given, each giving different insights that lead to further generalisations. The underlying heuristic reasoning is carefully described in order to provide an exemplar for designing learning trajectories to engage students with these functions of proof. Some interactive Java sketches regarding the variation and generalizations are available at: http://frink.machighway.com/~dynamicm/clough.html

Learning models that can improve critical thinking, skills collaborate, communicate, and creative thinking are needed in the 21st-century education era. Critical and creative thinking are the two essential competencies of the four skills required in the 21st century. However, both are still difficult to achieve well by students due to a lack of thinking skills during mathematics learning. This study was conducted to determine the model of learning that is appropriate to develop students' critical and creative thinking skills. The study used three-class samples from eighth grade. The first class is given the problem-posing lesson; the second class is given contextual learning and third class as a control class. The results of the study indicate that improving students' critical and creative thinking skills are included in the moderate category for types using contextual learning and problem-posing. Also, it is found that contextual learning is more effective for improving critical thinking skills when compared with learning problem posing and expository learning. Meanwhile, learning problem posing is more useful to enhance creative thinking skills compared with contextual and expository learning.

This paper is the author's personal side project and is of no affiliation to any institution or courses. The paper is a rookie attempt to deriving mathematical expressions of the well-known puzzle, the bridge and torch problem. The problem is about finding the shortest time needed for n people to cross a bridge at night. The catch is the bridge can only support at most two persons at a time, and the two persons will cross the bridge at the speed of the slower one. Moreover, a torch is needed for each and every crossing, and the torch has a finite time of burning. Thus, the puzzle is a problem to determine the possible shortest time from the choice of the two persons per crossing against the finite burning time of the torch. The result of this paper gives into fruition of a set of three equations--the vector form describing the mechanics of the logic solution, even-valued n function and odd-valued n function.

The aim of this study was to investigate the effects of modelling, collabora-tive and game-based learning on geometry success in third-grade students. These approaches were applied to geometry instruction in nature on the success of students in geometry. The students' views about geometry activities were also examined. Explanatory design, one of the mixed methods in which qualitative and quantitative methods are both used, was used in the study. The study used a randomized pretest-posttest control group design for the quantitative data; phenomenological study, one of the qualitative research designs, provided the study's qualitative dimension. The study group consisted of 101 third-grade students attending three different public primary schools. There were 65 participants in the experimental group and 36 in the control group. The quantitative data were collected by a geometry success test and the qualitative data were collected through a structured interview form. The quantitative findings obtained at the end of the study revealed that the students' success in geometry was greatest in the modelling group. Qualitative findings showed that geometry activities in nature were more effective than in-class activities.

Problem solving and problem posing are leading mathematical activities that stimulate mathematical thinking. From the theoretical point of view, these activities are very complex, partly due to the various issues that describe/define problem solving and problem posing and their role in the process of teaching and learning mathematics. Problem solving and problem posing are interrelated activities; we could say that they are in an interdependent relationship: we solve the problems we pose, we pose the problems in a way that we can solve them. However, the two processes are not equally present in every situation. Research into problem solving focuses mainly on the following areas: the basic characteristics of a mathematical problem; the nature (conceptual, procedural) and role of representation (interplay between internal and external) of a mathematical problem; mental schemas for problem solving; heuristics as principles, methods and (cognitive) tools for solving problems; types of generalisations and reasoning (abductive, narrative, naïve, arithmetic, algebraic); problem solving as a challenging activity for mathematically gifted students; and the role of the teacher in guiding problem solving as a way of implementing student problem solving in the classroom. Regarding problem posing, there are also some critical questions: How can the existing definitions of problem posing be categorised? How is problem posing conceived by the research community in relation to other mathematical constructs? What are the possible ways of implementing problem posing in research and teaching settings? Regarding problem solving, problem posing is formulated/used in research findings for generating (formulating, finding, creating) new problems; reformulating existing problems; creating and/or reformulating problems; raising questions and viewing old questions from a new angle; and an act of modelling. Research has demonstrated and frequently confirmed that (mathematical) problem posing and solving possess great potential for learners, but the reality in terms of teaching practice, external examinations, teaching material, and mathematics curricula seems out of alignment with the research findings. In this focus issue, we have considered two aspects of problem posing and problem solving: conceptualisation and implementation in the mathematics classroom. This issue contains five articles that address the issues of problem posing and problem-solving. The authors come from different backgrounds (Greece, Croatia, Hungary, Germany), which means that diverse perspectives and research

El talento matemático ha sido estudiado generalmente empleando tareas de resolución de problemas y pocas investigaciones abordan el tema mediante actividades de invención de problemas. Por tanto, en este documento se presenta un estudio teórico bibliográfico que evidencia la viabilidad de estudiar el talento matemático mediante actividades de invención de problemas.

This paper concerns the role of mathematical problems in the epistemology of Jean Cavaillès. Most occurrences of the term “problem” in his texts refer to mathematical problems, in the sense in which mathematicians themselves use the term: for an unsolved question which they hope to solve. Mathematical problems appear as breaking points in the succession of mathematical theories, both giving a continuity to the history of mathematics and illuminating the way in which the history of mathematics breaks up into successive theories with different kinds of operations and, in a sense, different kinds of a prioris. We briefly compare mathematical becoming to the succession of episteme in Foucault’s Les Mots et les choses. We then come back to the necessity that Cavaillès attributes to mathematical becoming, and which the position of mathematical problems illustrates, in order to discuss its various consequences in Cavaillès’ later works but also in Canguilhem’s discussion of Cavaillès’ role in the Resistance. Finally, we study other types of problems, in Cavaillès’ writings: philosophical problems and what we will call “questions” rather than “problems,” and which contrary to mathematical problems, as Cavaillès uses the term, cannot be solved but pervade the whole of the history of mathematics. We will put these “questions” in relation to Lautman’s Ideas.

PADA KALI INI SAYA SEBAGAI PENERIMA AMANAH, TELAH DIBERIKAN TUHAN "PISAU YANG DAPAT MEMBEDAH ALAM METAFISIKA YAITU PISAU MATEMATIKA METAFISIKA ATAU PISAU METAMATIKA LAODE (L) YANG MEMBOR DAN MENCINCANG METAFISIKA CIPTAAN TUHAN". DAN PISAU INI SAYA AKAN BERIKAN KEPADA SELURUH UMMAT MANUSIA UNTUK DAPAT MEMBEDAH SENDIRI "HAKEKAT CIPTAAN TUHAN, DAN AGAR SUPAYA SELURUH MANUSIA DAPAT YAKIN DAN PERCAYA ADANYA TUHAN PENCIPTA SEGALA SESUATU" • KEPADA SAUDARA SAUDARAKU YANG "ATHEIS" DAN "POLITHEIS/MUSYRIK" MAUPUN JUGA YANG "MONOTHEIS" MARILAH KITA GUNAKAN PISAU INI "SUPAYA MENGETAHUI RAHASIA CIPTAAN TUHAN DAN AKHIRNYA AKAN MENGAKUI BAHWA TUHAN ADA DAN TUHAN ITU HANYA SATU (ESA), TUHAN PENCIPTA SEGALA SESUATU, TUHAN PEMILIK SELURUH ALAM SEMESTA ". • PADA KESEMPATAN PERTAMA INI BEBERAPA SERI METAMATIKA YANG DITERBITKAN SECARA BERSERI, SAYA AKAN MEMBEDAH MENGENAI TURUNNYA ADAM DAN HAWA SEBAGAI KAKEK MOYANG SELURUH MANUSIA BUMI, BERPINDAH DARI SURGA LANSUNG KEBUMI INDONESIA. DAN AKAN DILANJUTKAN DENGAN TOPIK TOPIK LAINNYA YANG MASIH MERUPAKAN MISTERI METAFISIKA YANG MASIH GELAP DALAM DUNIA SAINS. • SEMOGALAH SEMUA INI MEMBERIKAN MANFAAT KEPADA SELURUH UMMAT MANUSIA DIBUMI. JAKARTA, DESEMBER 2008

This paper presents the results of an experiment in which 4 th to 6 th graders with above average mathematical abilities modified a given problem. The experiment found evidence of links between problem posing and cognitive flexibility. Emerging from organizational theory, cognitive flexibility is conceptualized through three primary constructs: cognitive variety, cognitive novelty, and changes in cognitive framing. Among these components, changes in cognitive framing could be effectively detected in problem posing situations, giving a relevant indication for students' creative potential. The student's capacity to generate coherent and consistent problems in the context of problem modification may indicate the existence of a strategy of functional type for generalizations, which seems to be specific to mathematical creativity.

The purpose of this research are to reviewing: degree of student activity, degree of student problem solving, and degree of creativity student problem solving, on learning use STAD with problem posing model aid two dimentional figure puzzle. This research use descriptive with quantitative approximation. Object of this research is STAD with problem posing model aid two dimentional figure puzzle on rectangular and triangle. Obtained data rources from result of observation student activity and result individual tes for problem solving and creativity student problem solving. The result showing that learning of STAD with problem posing model aid two dimentional figure puzzle is : student activity included in the high category with percentage 71,1 %, the percentage of student problem solving is 43,56% or included in middle category , and creativity of student problem solving included in the middle category with percentage 40,3 %,.

This paper reviews the conclusions of a few studies focused on cognitive flexibility of 4 to 6 graders in relation to their problem posing abilities. It seems that, in problem posing situations, high achievers in mathematics develop cognitive frames that make them cautious in changing the parameters of their new problems, even when they make interesting generalizations. These students display a kind of cognitive variety that is mathematically specialized, of a functional type. They show awareness of a need for mathematical consistency, which make them persevere in carefully controlling the changes.

Too few students consider mathematics to be interesting, not to mention enjoyable or challenging. Integrating strategy games into mathematics classes can make them more exiting, enabling students to learn or practice various topics in a different manner while developing their mathematical thinking and improving their knowledge. 
One example of such a game is NIM, which is a two-player game, and according to Gardner (1983), one of the oldest mathematical games in the world. In this paper, I present a different perspective for the analysis of the game, using more common school knowledge – arithmetic series.

The aim of this paper is to study the class of β-normal spaces. The relationships among s-normal spaces, p-normal spaces and β-normal spaces are investigated. Moreover, we study the forms of generalized β-closed functions. We obtain characterizations of β-normal spaces, properties of the forms of generalized β-closed functions and preservation theorems.

The Wiles proof on the Frey curve is true, but Frey hypothesis inconsistent; the Fermat’s proof is reduced to few pages.
  Appendix: The error of the curve of Frey, and Fermat's demonstration Notebook on the Wiles deception.
  The demonstration of Wiles for the Fermat’s theorem is certainly wrong: he bases oneself on the hypothesis for which if and only if the integer equation zª = xª+yª has solutions (= theorem of Fermat false), the Frey elliptic function Y² = X(X-yª)(X+xª) exist and vice versa. In particular, since elliptic, the cubic curve must be also modular (by the Taniyama-Shimura conjecture) what if it is not such, the Fermat theorem may be true.

A look into students’ misconceptions help explain the very low geometric thinking and may assist teachers in correcting errors to aid students in reaching a higher van Hiele geometric thinking level. In this study, students’ geometric thinking was described using the van Hiele levels and misconceptions on triangles. Participants (N=30) were Grade 9 students in the Philippines. More than half of the participants were in the van Hiele’s visualization level. Most students had imprecise use of terminologies. A few had misconceptions on class inclusion, especially when considering isosceles right triangles and obtuse triangles. Very few students correctly recognized the famous Pythagorean Theorem. Implications for more effective geometry teaching are considered.

Mihaela Berindeanu, Gyuszi Szep

Lucrarea cuprinde exercitii si probleme structurate in functie de itemii programei de olimpiada, clasa a VIII-a. La fiecare capitol exista o parte de teorie urmata de exercitii rezolvate si apoi exercitii propuse spre rezolvare.
Modul de abordare a informatiei, selectarea din multiple surse a problemelor propuse, transforma aceasta lucrare intr-un ghid de pregatire la matematica in vederea participarii cu succes la concursuri si olimpiada. Dorim succes tuturor elevilor pasionati de matematica!

Cuprins:
Programa Olimpiadei de matematica pentru clasa a VIII-a
Probleme recapitulative din clasa a VII-a
Multimea numerelor reale
Calcul algebric
Functii
Ecuatii. Inecuatii. Sisteme de ecuatii
Inegalitati
Bibliografie

Keywords The purpose of this study, to determine problem-posing skills and algebraic thinking levels and whether there is a relationship between problem-posing skills and algebraic thinking levels. In this research, correlation was used from quantitative research methods. In the fall semester of the 2016-2017 academic year, 308 students (151 girls, 157 boys) attending 7th and 8th grade in three secondary schools in the province of Konya were participated in the research. In the research, to determine the algebraic thinking levels of students, developed by Hart et al. (1998) and "Algebraic Thinking Level Test" adapted to Turkish by Altun (2005) was used. In addition, to measure the problem-posing skills of the students, Problem Posing Test was used as a data collection tool. The Spearman's Rank Correlation coefficient technique was used to determine the relationship between problem-posing skills and algebraic thinking levels. As a result of the research, while there is an accumulated at level 0 and level 1 among 7th grade students, it is accumulated at level 2 and level 4 among 8th grade students. In addition, it was determined that there is a strong correlation between students' algebraic thinking level and problem-posing skills in the positive direction. Math education Problem posing Algebraic thinking level

This paper explores a 1949 geometry exam question, generalizing it by 'looking back' at the solution in Polya style, as well as considering some interesting variations of it, including an optimization problem that is solved purely geometrically, without recourse to calculus.

In the present study we explore changes in perceptions of our class of prospective mathematics teachers (PTs) regarding their mathematical knowledge. The PTs engaged in problem
posing activities in geometry, using the “What If Not?” (WIN) strategy, as part of their work on computerized inquiry-based activities. Data received from the PTs’ portfolios reveals that
they believe that engaging in the inquiry-based activity enhanced both their mathematical and meta-mathematical knowledge. As to the mathematical knowledge, they deepened their knowledge regarding the geometrical concepts and shapes involved, and during the process of creating the problem and checking its validity and its solution, they deepened their understanding of the interconnections among the concepts and shapes involved. As to
meta-mathematical knowledge, the PTs refer to aspects such as the meaning of the givens and their relations, validity of an argument, the importance and usefulness of the definitions
of concepts and objects, and the importance of providing a formal proof.

We present a simple, fully probabilistic, Bayesian solution to -sample omnibus tests for comparison, with the Behrens-Fisher problem as a special case, which is free from the many defects found in the standard, classical, frequentist, likelihoodist and Bayesian approaches to those problems. We solve the main measure-theoretic difficulty for degenerate problems with continuous parameters of interest and Lebesgue-negligible point null hypothesis by approximating the corresponding continuous random variables by sequences of discrete ones defined on partitions of the parameter spaces and by taking the limit of the prior-to-posterior ratios of the probability of the null hypothesis for the corresponding discrete problems. Those limits are well defined under proper technicalities thanks to the Henstock-Kurzweil integral that is as powerful as the Lebesgue integral but still relies on Riemann sums, which are essential in the present approach. The solutions to the relative continuous proble...