Husserl's approach to transcendental intersubjectivity

It is our intention in the course of the development of this thesis to give an account of how intersubjectivity is "eidetically" constituted by means of the application of the phenomenological reduction to our experience in the context of the thought of Edmund Husserl; contrasted with various representative thinkers in what H. Spiegelberg refers to as "the wider scene" of phenomenology. That is to say, we intend to show those structures of both consciousness and the relation which man has to the world which present themselves as the generic conditions for the possibility of overcoming our "radical sol itude" in order that we may gain access to the mental 1 ife of an Other as other human subject. It is clear that in order for us to give expression to these accounts in a coherent manner, along with their relative merits, it will be necessary to develop the common features of any phenomenological theory of consdousness whatever. Therefore, our preliminary inquiry, subordinate to the larger theme, shall be into some of the epistemological results of the application of the phenomenological method used to develop a transcendental theory of consciousness. Inherent in this will be the deliniation of the exigency for making this an lIintentional ll theory. We will then be able to see how itis possible to overcome transcendentally the Other as an object merely given among other merely given objects, and further, how this other is constituted specifically as other ego. The problem of transcendental intersubjectivity and its constitution in experience can be viewed as one of the most compelling, if not the most polemical of issues in phenomenology. To be sure, right from the beginning we are forced to ask a number of questions regarding Husserl's responses to the problem within the context of the methodological genesis of the Cartesian Meditations, and The Crisis of European Sciences and Transcendental Phenomenology. This we do in order to set the stage for amplification. First, we ask, has Husserl lived up to his goal, in this connexion, of an apodictic result? We recall that in his Logos article of 1911 he adminished that previous philosophy does not have at its disposal a merely incomplete and, in particular instances, imperfect doctrinal system; it simply has none whatever. Each and every question is herein controverted, each position is a matter of individual conviction, of the interpretation given byaschool, of a "point of view". 1. Moreover in the same article he writes that his goal is a philosophical system of doctrine that, after the gigantic preparatory work. of generations, really be- . gins from the ground up with a foundation free from doubt and rises up like any skilful construction, wherein stone is set upon store, each as solid as the other, in accord with directive insights. 2. Reflecting upon the fact that he foresaw "preparatory work of generations", we perhaps should not expect that he would claim that his was the last word on the matter of intersubjectivity. Indeed, with 2. 'Edmund Husserl, lIPhilosophy as a Rigorous Science" in Phenomenology and theCrisis6fPhilosophy, trans". with an introduction by Quentin Lauer (New York.: Harper & Row, 1965) pp. 74 .. 5. 2Ibid . pp. 75 .. 6. 3. the relatively small amount of published material by Husserl on the subject we can assume that he himself was not entirely satisfied with his solution. The second question we have is that if the transcendental reduction is to yield the generic and apodictic structures of the relationship of consciousness to its various possible objects, how far can we extend this particular constitutive synthetic function to intersubjectivity where the objects must of necessity always remain delitescent? To be sure, the type of 'object' here to be considered is unlike any other which might appear in the perceptual field. What kind of indubitable evidence will convince us that the characteristic which we label "alter-ego" and which we attribute to an object which appears to resemble another body which we have never, and can never see the whole of (namely, our own bodies), is nothing more than a cleverly contrived automaton? What;s the nature of this peculiar intentional function which enables us to say "you think just as I do"? If phenomenology is to take such great pains to reduce the takenfor- granted, lived, everyday world to an immanent world of pure presentation, we must ask the mode of presentation for transcendent sub .. jectivities. And in the end, we must ask if Husserl's argument is not reducible to a case (however special) of reasoning by analogy, and if so, tf this type of reasoning is not so removed from that from whtch the analogy is made that it would render all transcendental intersubjective understandtng impos'sible? 2. HistoticalandEidetic Priority: The Necessity of Abstraction 4. The problem is not a simple one. What is being sought are the conditions for the poss ibili:ty of experi encing other subjects. More precisely, the question of the possibility of intersubjectivity is the question of the essence of intersubjectivity. What we are seeking is the absolute route from one solitude to another. Inherent in this programme is the ultimate discovery of the meaning of community. That this route needs be lIabstract" requires some explanation. It requires little explanation that we agree with Husserl in the aim of fixing the goal of philosophy on apodictic, unquestionable results. This means that we seek a philosophical approach which is, though, not necessarily free from assumptions, one which examines and makes explicit all assumptions in a thorough manner. It would be helpful at this point to distinguish between lIeidetic ll priority, and JlhistoricallJpriority in order to shed some light on the value, in this context, of an abstraction.3 It is true that intersubjectivity is mundanely an accomplished fact, there havi.ng been so many mi.llions of years for humans to beIt eve in the exi s tence of one another I s abili ty to think as they do. But what we seek is not to study how this proceeded historically, but 3Cf• Maurice Natanson;·TheJburne in 'Self, a Stud in Philoso h and Social Role (Santa Cruz, U. of California Press, 1970 . rather the logical, nay, "psychological" conditions under which this is possible at all. It is therefore irrelevant to the exigesis of this monograph whether or not anyone should shrug his shoulders and mumble IIwhy worry about it, it is always already engaged". By way of an explanation of the value of logical priority, we can find an analogy in the case of language. Certainly the language 5. in a spoken or written form predates the formulation of the appropriate grammar. However, this grammar has a logical priority insofar as it lays out the conditions from which that language exhibits coherence. The act of formulating the grammar is a case of abstraction. The abstraction towards the discovery of the conditions for the poss; bi 1 ity of any experiencing whatever, for which intersubjective experience is a definite case, manifests itself as a sort of "grammar". This "grammar" is like the basic grammar of a language in the sense that these "rulesil are the ~ priori conditions for the possibility of that experience. There is, we shall say, an "eidetic priority", or a generic condition which is the logical antecedent to the taken-forgranted object of experience. In the case of intersubjectivity we readily grant that one may mundanely be aware of fellow-men as fellowmen, but in order to discover how that awareness is possible it is necessary to abstract from the mundane, believed-in experience. This process of abstraction is the paramount issue; the first step, in the search for an apodictic basis for social relations. How then is this abstraction to be accomplished? What is the nature of an abstraction which would permit us an Archimedean point, absolutely grounded, from which we may proceed? The answer can be discovered in an examination of Descartes in the light of Husserl's criticism. 3. The Impulse for Scientific Philosophy. The Method to which it Gives Rise. 6. Foremost in our inquiry is the discovery of a method appropriate to the discovery of our grounding point. For the purposes of our investigations, i.e., that of attempting to give a phenomenological view of the problem of intersubjectivity, it would appear to be of cardinal importance to trace the attempt of philosophy predating Husserl, particularly in the philosophy of Descartes, at founding a truly IIscientific ll philosophy. Paramount in this connexion would be the impulse in the Modern period, as the result of more or less recent discoveries in the natural sciences, to found philosophy upon scientific and mathematical principles. This impulse was intended to culminate in an all-encompassing knowledge which might extend to every realm of possible thought, viz., the universal science ot IIMathexis Universalis ll •4 This was a central issue for Descartes, whose conception of a universal science would include all the possible sciences of man. This inclination towards a science upon which all other sciences might be based waS not to be belittled by Husserl, who would appropriate 4This term, according to Jacab Klein, was first used by Barocius, the translator of Proclus into Latin, to designate the highest mathematical discipline. . 7. it himself in hopes of establishing, for the very first time, philosophy as a "rigorous science". It bears emphasizing that this in fact was the drive for the hardening of the foundations of philosophy, the link between the philosophical projects of Husserl and those of the philosophers of the modern period. Indeed, Husserl owes Descartes quite a debt for indicating the starting place from which to attempt a radical, presupositionless, and therefore scientific philosophy, in order not to begin philosophy anew, but rather for the first time.5 The aim of philosophy for Husserl is the search for apodictic, radical certitude. However while he attempted to locate in experience the type of necessity which is found in mathematics, he wished this necessity to be a function of our life in the world, as opposed to the definition and postulation of an axiomatic method as might be found in the unexpurgated attempts to found philosophy in Descartes. Beyond the necessity which is involved in experiencing the world, Husserl was searching for the certainty of roots, of the conditi'ons which underl ie experience and render it pOssible. Descartes believed that hi~ MeditatiOns had uncovered an absolute ground for knowledge, one founded upon the ineluctable givenness of thinking which is present even when one doubts thinking. Husserl, in acknowledging this procedure is certainly Cartesian, but moves, despite this debt to Descartes, far beyond Cartesian philosophy i.n his phenomenology (and in many respects, closer to home). 5Cf. Husserl, Philosophy as a Rigorous Science, pp. 74ff. 8 But wherein lies this Cartesian jumping off point by which we may vivify our theme? Descartes, through inner reflection, saw that all of his convictions and beliefs about the world were coloured in one way or another by prejudice: ... at the end I feel constrained to reply that there is nothing in a all that I formerly believed to be true, of which I cannot in some measure doubt, and that not merely through want of thought or through levity, but for reasons which are very powerful and maturely considered; so that henceforth I ought not the less carefully to refrain from giving credence to these opinions than to that which is manifestly false, if I desire to arrive at any certainty (in the sciences). 6 Doubts arise regardless of the nature of belief - one can never completely believe what one believes. Therefore, in order to establish absolutely grounded knowledge, which may serve as the basis fora "universal Science", one must use a method by which one may purge oneself of all doubts and thereby gain some radically indubitable insight into knowledge. Such a method, gescartes found, was that, as indicated above by hi,s own words, of II radical doubt" which "forbids in advance any judgemental use of (previous convictions and) which forbids taking any position with regard to their val idi'ty. ,,7 This is the method of the "sceptical epoche ll , the method of doubting all which had heretofor 6Descartes,Meditations on First Philosophy, first Med., (Libera 1 Arts Press, New York, 1954) trans. by L. LaFl eur. pp. 10. 7Husserl ,CrisiS of Eliroeari SCiences and Trariscendental Phenomenology, (Northwestern U. Press, Evanston, 1 7 ,p. 76. 9. been considered as belonging to the world, including the world itself. What then is left over? Via the process of a thorough and all-inclusive doubting, Descartes discovers that the ego which performs the epoche, or "reduction", is excluded from these things which can be doubted, and, in principle provides something which is beyond doubt. Consequently this ego provides an absolute and apodictic starting point for founding scientific philosophy. By way of this abstention. of bel ief, Desca'rtes managed to reduce the worl d of everyday 1 ife as bel ieved in, to mere 'phenomena', components of the rescogitans:. Thus:, having discovered his Archimedean point, the existence of the ego without question, he proceeds to deduce the 'rest' of the world with the aid of innate ideas and the veracity of God. In both Husserl and Descartes the compelling problem is that of establ ishing a scientific, apodictic phi'losophy based upon presuppos itionless groundwork .. Husserl, in thi.s regard, levels the charge at Descartes that the engagement of his method was not complete, such that hi.S: starting place was not indeed presupositionless, and that the validity of both causality and deductive methods were not called into question i.'n the performance of theepoche. In this way it is easy for an absolute evidence to make sure of the ego as: a first, "absolute, indubitablyexisting tag~end of the worldll , and it is then only a matter of inferring the absolute subs.tance and the other substances which belon.g to the world, along with my own mental substance, using a logically val i d deductive procedure. 8 8Husserl, E.;' Cartesian 'Meditation;, trans. Dorion Cairns (Martinus Nijhoff, The Hague, 1970), p. 24 ff.

Introduction to engineering : a case study of an interdisciplinary course in Mathematics, Science and Technology

This thesis research was a qualitative case study of a single class of Interdisciplinary Studies: Introduction to Engineering taught in a secondary school. The study endeavoured to explore students' experiences in and perceptions of the course, and to investigate the viability of engineering as an interdisciplinary theme at the secondary school level. Data were collected in the form of student questionnaires, the researcher's observations and reflections, and artefacts representative of students' work. Data analysis was performed by coding textual data and classifying text segments into common themes. The themes that emerged from the data were aligned with facets of interdisciplinary study, including making connections, project-based learning, and student engagement and affective outcomes. The findings of the study showed that students were positive about their experiences in the course, and enjoyed its project-driven nature. Content from mathematics, physics, and technological design was easily integrated under the umbrella of engineering. Students felt that the opportunity to develop problem solving and teamwork skills were two of the most important aspects of the course and could be relevant not only for engineering, but for other disciplines or their day-to-day lives after secondary school. The study concluded that engineering education in secondary school can be a worthwhile experience for a variety of students and not just those intending postsecondary study in engineering. This has implications for the inclusion of engineering in the secondary school curriculum and can inform the practice of curriculum planners at the school, school board, and provincial levels. Suggested directions for further research include classroom-based action research in the areas of technological education, engineering education in secondary school, and interdisciplinary education.

Youth Involvement in Organizational Decision Making: The Connection to Youth Initiative and Organizational Functioning

Studying positive adolescent development requires an examination of the mutually beneficial associations between youth and their environment. These youthcontext relations include both the contributions that youth make to others and society and the youth-context interactions that might predict positive youth outcomes. Community and youth-serving organizations, where youth may be involved in decision-making roles such as service delivery, advocacy, or on boards of directors, can provide one important context for youth contributions and for positive adolescent development. Research on the outcomes of youth involvement in organizational decision-making, however, is limited, and largely consists of exploratory qualitative studies. This dissertation is formatted as an integrated article dissertation. It begins with a review of the literature on contexts of structured youth activities and positive youth development. This review is intended to describe theory on development-context relations, in which development is considered an interactive process that occurs between individuals and their contexts, as it pertains the positive development of youth who are involved in various structured activities (e.g., volunteering). This description follows with a review of current research, and conclusions and rationale for the current studies. Following this theoretical and research background, the dissertation includes reports of two studies that were designed to address gaps in the research on youth involvement in organizational decision-making. The first was a qualitative research synthesis to elucidate and summarize the extant qualitative research on the outcomes of youth involvement in organizational decision making on adults and organizations. Results of this study suggested a number of outcomes for service provision, staff, and broader organizational functioning, including both benefits to organizations as well as some costs. The second study was a quantitative analysis of the associations among youth involvement, organizations' learning culture, and youth initiative, and relied on survey data gathered from adults and youth in community-based organizations with youth involvement. As expected, greater youth involvement in organizational decision making was associated with higher learning culture within the organization. Two dimensions of youth involvement, greater program engagement and relationships with adults, were related to greater youth initiative. A third dimension, sense of ownership, was not- .-.- associated with youth's level of initiative. Moreover, the association between relationships with adults and youth initiative was only significant in organizations with relatively low learning culture. Despite some limitations, these studies contribute to the research literature by providing some indication of the potential benefits and costs of youth involvement and by making an important contribution toward the early stages of context-level analyses of youth development. Findings have important implications for practitioners, funders, future research, and lifespan development theory.

Mathematical modelling in mathematics education and instruction

This paper aims at giving a concise survey of the present state-of-the-art of mathematical modelling in mathematics education and instruction. It will consist of four parts. In part 1, some basic concepts relevant to the topic will be clarified and, in particular, mathematical modelling will be defined in a broad, comprehensive sense. Part 2 will review arguments for the inclusion of modelling in mathematics teaching at schools and universities, and identify certain schools of thought within mathematics education. Part 3 will describe the role of modelling in present mathematics curricula and in everyday teaching practice. Some obstacles for mathematical modelling in the classroom will be analysed, as well as the opportunities and risks of computer usage. In part 4, selected materials and resources for teaching mathematical modelling, developed in the last few years in America, Australia and Europe, will be presented. The examples will demonstrate many promising directions of development.

Formulating generalised 'goal games' against nature: An illustration from decision-making under uncertainty in agriculture

The games-against-nature approach to the analysis of uncertainty in decision-making relies on the assumption that the behaviour of a decision-maker can be explained by concepts such as maximin, minimax regret, or a similarly defined criterion. In reality, however, these criteria represent a spectrum and, the actual behaviour of a decision-maker is most likely to embody a mixture of such idealisations. This paper proposes that in game-theoretic approach to decision-making under uncertainty, a more realistic representation of a decision-maker's behaviour can be achieved by synthesising games-against-nature with goal programming into a single framework. The proposed formulation is illustrated by using a well-known example from the literature on mathematical programming models for agricultural-decision-making. (c) 2005 Elsevier Inc. All rights reserved.

Healing the wounds of war: memories of violence and the making of history in Zimbabwe's most recent past.

How does a society less than two decades after a liberation war which involved large sections of the population come to terms with the memories of violence and war — a war in which there was no clear distinction between insurgent and counter‐insurgent, liberator and oppressor and in which the majority of the casualties can be found among the rural civilian population? This was a predicament not exclusive to Zimbabwe, but one which also applies to Mozambique, South Africa and, more recently, to Rwanda. Since its independence Zimbabwe has been a prime example of successful reconciliation. Ranger has argued that spiritual healing has contributed importantly to coming to terms with the trauma of war through turning violence into history. Here it will be argued that an analysis of the intersections between memories of violence, healing, and history reveals a twofold process. Social healing is made possible by a shift from conviction and compensation to revealing without convicting. At the same time healing provides an arena for communities in which competing and contesting memories of violence are renegotiated. Through these processes sense is being made of the past; history is being made.

An investigation into feature construction to assist word sense disambiguation

Identifying the correct sense of a word in context is crucial for many tasks in natural language processing (machine translation is an example). State-of-the art methods for Word Sense Disambiguation (WSD) build models using hand-crafted features that usually capturing shallow linguistic information. Complex background knowledge, such as semantic relationships, are typically either not used, or used in specialised manner, due to the limitations of the feature-based modelling techniques used. On the other hand, empirical results from the use of Inductive Logic Programming (ILP) systems have repeatedly shown that they can use diverse sources of background knowledge when constructing models. In this paper, we investigate whether this ability of ILP systems could be used to improve the predictive accuracy of models for WSD. Specifically, we examine the use of a general-purpose ILP system as a method to construct a set of features using semantic, syntactic and lexical information. This feature-set is then used by a common modelling technique in the field (a support vector machine) to construct a classifier for predicting the sense of a word. In our investigation we examine one-shot and incremental approaches to feature-set construction applied to monolingual and bilingual WSD tasks. The monolingual tasks use 32 verbs and 85 verbs and nouns (in English) from the SENSEVAL-3 and SemEval-2007 benchmarks; while the bilingual WSD task consists of 7 highly ambiguous verbs in translating from English to Portuguese. The results are encouraging: the ILP-assisted models show substantial improvements over those that simply use shallow features. In addition, incremental feature-set construction appears to identify smaller and better sets of features. Taken together, the results suggest that the use of ILP with diverse sources of background knowledge provide a way for making substantial progress in the field of WSD.

Vernacular meaning making : Examples of narrative impact in fiction film questioning the 'banal' notion in mediatization of religion theory

The outcome of an audience study supports theories stating that stories are a primary means by which we make sense of our experiences over time. Empirical examples of narrative impact are presented in which specific fiction film scenes condense spectators' lives, identities and beliefs. One conclusion is that spectators test the emotional realism of the narative for greater significance, connecting diegetic fiction experiences with their extra-diegetic world in their quest for meaning, self and identity. The 'banal' notion of the mediatization of religion theory is questioned as unsatisfactory in the theoretical context of individualized meaning-making processes. As a semantically negatively charged concept, it is problematic when analyzing empirical examples of spectators' use of fictional narratives, especially when trying to characterize the idiosyncratic and complex interplay between spectators' fiction emotions and their testing of mediated narratives in an exercise to find moral significance in extra-filmic life. Instead vernacular meaning-making is proposed.

Integrating a sense of coherence into the neonatal environment

Background: Family centred care (FCC) is currently a valued philosophy within neonatal care; an approach that places the parents at the heart of all decision-making and engagement in the care of their infant. However, to date, there is a lack of clarity regarding the definition of FCC and limited evidence of FCCs effectiveness in relation to parental, infant or staff outcomes. Discussion: In this paper we present a new perspective to neonatal care based on Aaron Antonovksy's Sense of Coherence (SOC) theory of well-being and positive health. Whilst the SOC was originally conceptualised as a psychological-based construct, the SOCs three underpinning concepts of comprehensibility, manageability and meaningfulness provide a theoretical lens through which to consider and reflect upon meaningful care provision in this particular care environment. By drawing on available FCC research, we consider how the SOC concepts considered from both a parental and professional perspective need to be addressed. The debate offered in this paper is not presented to reduce the importance or significance of FCC within neonatal care, but, rather, how consideration of the SOC offers the basis through which meaningful and effective FCC may be delivered. Practice based implications contextualised within the SOC constructs are also detailed. Summary: Consideration of the SOC constructs from both a parental and professional perspective need to be addressed in FCC provision. Service delivery and care practices need to be comprehensible, meaningful and manageable in order to achieve and promote positive well-being and health for all concerned.

Ensino de matemática, história da matemática e artefatos: Possibilidade de interligar saberes em cursos de formação de professores da Educação Infantil e anos iniciais do Ensino Fundamental.

This work is located at the shield of research that defends the use of Mathematics History, based on the utilization of historical artifacts at teaching activities, at Mathematics classrooms, and at graduation courses for teachers of Elementary School and of the first grades of High School. The general objective is to examine the possibility of the use of historical artifacts, at teaching activities, at graduation courses for teachers of Elementary School and of the first grades of High School. Artifact, at this work, is comprehended as objects, documents, monuments, images and other kinds of materials that make sense to the Human actions at the past and that represent what have been said and done at the Human history. At the construction of the theoretical-methodological way of the research we have based ourselves upon the ideas of the authors that are engaged at the teachers formation; at researchers adherents to the use of Mathematics History (MH) as a methodological resource, and at studies accomplished that elucidate the role of the artifacts at the history and as a mediatory element of learning. We defend the thesis that the utilization of historical artifacts at teaching activities enables the increasing of the knowledge, the development of competencies and essential abilities to the teacher acting, as well as interact at different areas of the knowledge, that provides a conception of formation where the teacher improves his learning, learning-doing and learning-being. We have adopted a qualitative research approach with a theoretical and pratic study disposition about the elements that contribute to the teachers works at the classroom, emphasizing the role of the Mathematics history at the teacher s formation and as a pedagogical resource at the mathematics classroom; the knowledge, the competencies and abilities of the historical artifacts as an integrative link between the different areas of the knowledge. As result, we emphasize that the proposition of using the MH, through learning activities, at the course of teacher graduation is relevant, because it allows the investigation of ideas that originate the knowledge generated at every social context, considering the contribution of the social and cultural, political and economical aspects at this construction, making easy the dialog among the areas and inside of each one The historical artifact represents a research source that can be deciphered, comprehended, questioned, extracting from it information about knowledge of the past, trace and vestiges of the culture when it was created, consisting of a testimony of a period. These aspects grant to it consideration to be explored as a mediatory element of the learning. The artifacts incorporated at teaching activities of the graduation courses for teachers promote changes on the view about the Mathematics teaching, in view of to privilege the active participation of the student at the construction of his knowledge, at the reflection about the action that has been accomplished, promoting stimulus so the teachers can create their own artifacts, and offer, either, traces linking the Mathematics with others knowledge areas.

Vídeo-aula de história da matemática: uma possibilidade didática para o ensino de matemática

In this work we are disagreeing with the possibility of production and of the use of video-class for the disciplines of history of mathematics by the teachers of elementary e middle school as a way to contribute to the development of their classes. Our goal is to provide to the mathematics teachers the option of connecting social, scientific, cognitive, and didactic aspects of topics in math thoughts to their students. That shall be based in the presence of mathematics in the history of humanity. Thus, we consider possible that teachers and their students can link and relate mathematics to other sciences, education culture, and reflect about the many ways of represent them, as well as the patters of organization of nature and of culture. In this way, they shall be able to observe and interpret situations that involve mathematical questions associated to the various means of historic-epistemological studies already done by other researchers and scholars in the field of history of mathematics who works in creating video-classes. In addition to that, we can use all the available media in order to give edifying dynamics to the mathematical formulations established throughout history. In this sense, we are based and focused on the objectives, which are sustained by educational computer technology, techniques for video making, as well as mathematical teaching proposals and the historical inquiring made by Mendes (2001, 2009a, 2009b). The validating experimentation allowed us to conclude that the techniques we used in the production of the history of mathematics video-classes proved they to be valid ones. They are able to be executed with the minimum of technological resources. In addition, they have the same efficacy as far as classroom use

Um estudo de práticas matemáticas históricas e sugestões de uso na matemática escolar

This dissertation aims to contribute on teaching of mathematics for enabling learning connected to the relationship among science, society, culture and cognition. To this end, we propose the involvement of our students with social practices found in history, since. Our intention is to create opportunities for school practices that these mathematical arising from professional practice historical, provide strategies for mathematical thinking and reasoning in the search for solutions to problematizations found today. We believe that the propose of producing Basic Problematization Units, or simply UBPs, in math teacher formation, points to an alternative that allows better utilization of the teaching and learning process of mathematics. The proposal has the aim of primary education to be, really forming the citizen, making it critical and society transformative agent. In this sense, we present some recommendations for exploration and use of these units for teachers to use the material investigated by us, in order to complement their teaching work in mathematics lessons. Our teaching recommendations materialized as a product of exploration on the book, Instrumentos nuevos de geometria muy necessários para medir distancia y alturas sem que interuengan numeros como se demuestra em la practica , written by Andrés de Cespedes, published in Madrid, Spain, in 1606. From these problematizations and the mathematics involved in their solutions, some guidelines for didactic use of the book are presented, so that the teacher can rework such problematizations supported on current issues, and thus use them in the classroom

Student Problem Solving

The purpose of this study is to determine if students solve math problems using addition, subtraction, multiplication, and division consistently and whether students transfer these skills to other mathematical situations and solutions. In this action research study, a classroom of 6th grade mathematics students was used to investigate how students solve word problems and how they determine which mathematical approach to use to solve a problem. It was discovered that many of the students read and re-read a question before they try to find an answer. Most students will check their answer to determine if it is correct and makes sense. Most students agree that mastering basic math facts is very important for problem solving and prefer mathematics that does not focus on problem solving. As a result of this research, it will be emphasized to the building principal and staff the need for a unified and focused curriculum with a scope and sequence for delivery that is consistently followed. The importance of managing basic math skills and making sure each student is challenged to be a mathematical thinker will be stressed.