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Agiria diferente? Se pergunte: Como vou me recuperar? Comece a agir. Escreva sua lista. Como o seu. Que mudou a sua vida! Sullivan, M. Pre al ulus. Ninth Edition. Pearson Edu ation. Weinberg, A, and E. The out omes of this proje t are a set of instruments for data olle tion for a larger study that will also investigate orrelations of resour e use and student learning. Context The UTMOST proje t fo uses on the development and use of open sour e omputational resour es in the tea hing and learning of mathemati s at undergraduate level.
The proje t ontinues work in the development of four inter onne ted suites of te hnologi al resour es: Notably, SMC tra ks a student's time spent at various tasks su h as viewing a textbook, using the intera tive features of a textbook, working on exer ises, editing LATEX do uments, dis ussing the ourse with a fellow student, or dis ussing the ourse with the instru tor.
UTMOST improves these resour es and has a resear h omponent asso iated with their use in real settings.
The proje t investigates the ways in whi h textbooks in the open sour e platform are used by tea hers and students in Linear Algebra and Abstra t Algebra over three semesters. There are three overar hing aims of the resear h omponent of the proje t: To identify the tea hers' and students' use of textbooks that are in the open sour e platform, and the tea hers' and students' use of identi al PDF versions of the same textbooks 2.
To propose measures of student learning that would potentially identify the impa t of these resour es. We hara terize the resour es used by instru tors and students depending on availability of multi-media e.
Methods In the pilot phase we olle t in depth data from four lassrooms, three using the dynami version of the textbooks via observations, interviews, and self-reports of use, and ompare those data with automati ally generated data that an be aptured via the online platform. Other forms of data in lude tea her surveys and logs of work, student surveys and logs of work, video-re ordings of tea hers' planning sessions and lesson ena tment, interviews with tea hers, fo us groups with students and tests for students' performan e.
By the time we present in May, we will be able to draw on our pilot data 4 tea hers at three universities in two states. SMC makes these apabilities possible.
Referen es Brown, Matthew W. Theorizing the design and use of ur- ri ulum materials. Gueudet, Ghislaine, and Lu Trou he. An example from tertiary mathemati s instru tors. Rabardel, Pierre, and Yvonne Waern. Rezat, Sebastian. Mathemati s urri ulum materials and tea her development, edited by Ghislaine Gueudet, Birgit Pepin and Lu Trou he, Dordre th: Students' utilization s hemes of mathemati s text- books related to self-regulated pra ti ing.
Yin, Robert K. Case study resear h design and methods. Thousand Oaks, CA: The limitations and the potential of the presentational media were always part of the mathemati s ulture. Des ribing the ommuni ation of Greek mathemati ians, Netz suggests that the limita- tions of the media available wetted sand, dusted surfa e or wax tablets were essentially similar to those of modern books: The limitations of the media available suggest rather, the preparation of the diagram prior to the ommuni ative a t- a onsequen e of the inability to erase.
An intera tive diagram ID , is a relatively small unit of an intera tive text in e-textbook or another material. An ID presents information and expli itly requires the viewer to take a tion and hange the text within given limitations. What is it that textbooks provide pedagogi ally and epistemologi ally, besides a reminder of the weight of the past?
How might they hange in the future, and how ould su h hanges serve the interests of publishers, authors, students, and edu ators? This issue requires s holars to develop lens for analyzing pedagogi al design and tea hing-learning pro esses with intera tive texts. There are three fun tions of ID in the framework: The presentational fun tion fo uses on what and how is being illustrated by the diagram. Three types of examples are widely used: They serve as a dynami illustration that helps analyze the situation without being able to hange the information.
The tone in whi h the text addresses the learner is subje t to design de isions having to do with the orientational fun tion. An a urate ID has ri hness of detail, but ompleteness of detail in sket h means that the user has to work in order to see through the whole, to make onta t with and examine details.
The IDs' design made it possible to address the given graphs as a sket h, but at the same time the sket h an be intera tively unfolded into a detailed a urate diagram, whi h auses students to hange their fo us from data testing to hoosing the ne essary data. Illustrating, Elaborating, Guiding. Illustrating IDs are simply-operated, unsophisti ated representations.
The important omponents in the design of the Elaborating IDs are ri h tools and linked representations that enable various dire tions in the sear h for a solution. We use the term Guiding IDs in relation to guided inquiry. This kind of ID provides the means for students to explore new ideas. In addition to providing resour es that promote inquiry, they also set the boundaries and provide a framework for the pro ess of working with the task.
The goals of the resear h were: What are the hara teristi s of a tivity onsisted of reading and solving tasks whi h are presented as IDs, as a way for students to explore texts of the task?
How do the hara terizations of pro esses vary in a ordan e to the three designed organizational fun tions of IDs: Ea h series in ludes a preliminary task and three omparable tasks; ea h was designed upon a semioti framework.
We found that even the minimal intera tion designed in the illustrating ID an be helpful in onsolidating relevant knowledge that is not adequately stru tured yet.
Students who worked with the ID looked for ways to bypass the designed onstrains: Regarding guiding IDs we found that it an be a form of instru tion toward development of new mathemati al ideas. The guiding IDs' design limits the student's a tion and at the same time provides an open spa e for student's ideas.
Referen es Friesen, Norm.
Naftaliev, Elena, and Yerushalmy Mi hal. The Role Played by Resour es and Constraints. Springer International Publishing.
Netz, Reviel. The Shaping of Dedu tion in Greek Mathemati s: A Study in Cognitive History. Cambridge University Press, UK. Taking into onsideration the fa t that textbooks have some representation of their ontent, e. In most ases methodology used is omparative analysis of ontent analysis or ase studies of long observation on a single or few tea hers, and the analysis of textbooks is done a ording to the resear h question. This study in ludes development and design based resear h of the browser extension.
The initial oding ategories and rational were designed, among other aspe ts, to address te hnology related hara teristi s. These ategories serve as an initial state when attempting to listen to and learn from tea hers while still providing them with a literature based staring point. Referen es Olsher, S. Changes suggested by tea hers to the math textbook they use in lass".
Jones, C. Bokhove, G. Southampton, GB: University of Southampton. Pepin, B. Remillard, J. Charlotte, NC: Information age publishing. This learning environment aims at assisting students' work with fra tions and understanding their problem-solving pra ti es. Moreover, it supports tea hers in diagnosing students' learning pro esses. Based on the work of 6th-graders who used this e-book during a four-week lass on fra tions we provide pro ess data allowing the analysis of students' a tions in some detail.
In parti ular, the data show students' answers as well as their ways to these answers. A ordingly, it is possible to get insights in their understanding or misunderstanding of fra tions. In this presentation, we will introdu e typi al strategies of students' when a quiring the on ept of fra tion. Hen e it is of major importan e that the textbooks support that tenden y. Do textbooks in lude ontexts, ontents, results that meet the standard of proof?
If yes, are the arguments appropriate for students at the proposed level? Are they orre t? Are they oherent to the previous ontents of the textbook? We also invite submissions of reports on experien es fo using on mathemati al arguing in the lassroom, based or not on textbooks, using or not symboli language.
The minutes of this symposium will be used for short presentations followed by dis ussions aiming to get an overview of the state of the art on erning reasoning in textbooks. This study seeks to map the mathemati al statements that are presented to the students and how the authors prove or demonstrate the validity of those statements.
The results obtained were analyzed in terms of the mathemati al stru ture of the set Z. A esso: PNLD If we ompute with algebrai expressions without making use of the properties of the arithmeti operations, we loose an unique opportunity of using mathemati al thinking with the students, mainly the one related to abstra tion.
In fa t, algebra in Elementary S hool is losely related to the mathemati al way of arguing. Using symbols as representatives of numbers, we an establish arithmeti assertions and justify them using generi thinking, that is, arguing making use of those symbols and not of parti ular instan es.
Algebra enables us to write shortly and oherently general relations It is an essential tool for proving properties, des ribing patterns and solving problems. For that, abstra tion is ne essary to free us from ontexts and parti ular ases. Martinez , p. In this work, we analyse books from other ountries with respe t to the introdu tion of algebra, in luding textbooks from Portugal, Chile, Fran e, and Cape Verde and also books from Fran e and Chile written for Elementary S hool tea hers.
Referen es Carvalho, Sandro A. Masters diss. O tober 21, A essed: Usiskin, Zalman. In Algebrai Thinking, Grades K Barbara Moses Reston, Va.: National Coun il of Tea hers of Mathemati s , SM Chile S. In fa t, in addition to students not being exposed to a tivities related to argumentation, students often also do not have the opportunity to be involved in the tea hing of geometry, whi h boils down to the tea hing of measures, however "it is a fa t that geometri questions tend to arouse the interest of adoles ents and young people in a natural and spontaneous way.
Brasil, This order is natural: With regard to the dedu tion of the formula for the area of the re tangle, for example, whi h we believe to be one of the most important dedu tions to be made, sin e it is from it that we an dedu e all the other formulas, none of the analyzed authors makes use of generi thinking: In the example given, the author shows a re tangle with 5 m of base and 3 m of height, and only informs that to ount the squares we an multiply 5 by 3, without making any allusion to the meaning of re tangular arrangement asso iated with multipli ation not even is there a suggestion ommented in tea her's book: We believe that this argumentation would be within the rea h of students, as already shown in Meinerz We also believe that this would be a good starting point to later extend the formulas to rational and real measures, making use of the on ept of these numbers.
Referen es a o Bian hini, E. Para que ensinar e aprender Geometria no Ensino Fundamental? Iezzi, Gelson. Dol e, Osvaldo.
Ma hado, Antonio. Ensino Funda- mental. Editora Atual. Obra Coletiva, Editora Moderna, 3. Meinerz, Fran iele. Undergraduate diss. November 14, While with addition, subtra tion and multipli ation we have two initial values and obtain a third value, whi h is the result of the operation, the division of natural numbers gives us two resulting values: Hen e, the student is led to think that division also has only one out ome, the quotient.
Is it natural to start the dis ussion on division of natural numbers only with dividends that are multiples of the divisors? Are the divisions with zero remainder a tually the most frequent ases of division in the students' real life? Therefore, in those textbooks, the hypotheses for this operation, whi h are a ru ial part of mathemati al reasoning, are not emphasized. Figure 2 shows a situation whi h is familiar to the students, and whi h demonstrates that this assertion is false.
The Eu lidean division is the only division that makes sense in the universe N, and very often the remainder plays an essential role in the problem we are trying to solve.
Thus, this mis on eption should be avoided in elementary edu ation. Referen es a Isolani, C. Ripoll, C. Rio de Janeiro: Soppelsa, J. Su h representations involve equations and oordinates as a means to give the student a distin t way to deal with geometry. Besides, the national urri ulum also re ommends that: The tea hing of analyti geometry should be presented in an arti ulate manner with algebra, enlarging the ability of visualization.
Brasil, , p. It is interesting to stress the operational importan e of the textbook in the Brazilian edu ation system and keep in mind that su h an analysis should be a onstant task.
Choppin expresses the need of this task when he states that: A textbook is not a book that we read, but an instrument that we use. It is the seizing of the ons ien e of the dynami dimension of the textbook it only really exists through the uses we give it! Choppin, , p.
In ea h of these topi s, we initially verify how the ontent is presented: We lose ea h topi he king whether the proposed a tivities instigate the mathemati al thinking of the student.
Figura 1: Our analysis, however, shows a big gap in the textbooks with respe t to the presentation of the dedu tion of formulas involved in analyti geometry. Referen es Brasil. MEC, A esso em: Base Na ional Comum Curri ular: Choppin, Alain. O historiador e o livro es olar. Moderna, , v. It has the obje tive of evaluating, downloading and free distributing the textbooks that are going to be used in publi s hools.
First, a ommittee hosen by this program evaluate the books and de ide whi h ones is quality produ ed to possibly be used at s hools. Se ond, the program reates a guide that presents the most important ontent and approa h of the approved textbooks, so the tea hers an hoose whi h one they want to work with. Finally, the government sends the textbooks to all publi s hools. Referen es Fan, Lianghuo. ZDM Mathemati s Edu ation, Reid, David A.
Proof in Mathemati s Edu ation: Sense Publishers. We believe that the development of dedu tive reasoning depends more on tea hers' a tions than on dida ti al materials. The questions had been applied in three publi s hools, to students of 8th and 9th grades of basi edu ation, from 12 to 17 years of age. The on lusion was that the level of argumentation of this group is still ingenuous and informal.
In this form, the parti ipants of the inquiry evaluated the answers given by the pupils, justifying the marks given. In the analyses arried through, we verify that tea hers have great in lination for arguments approa hing to the formal proof. More in ipient and ingenuous proposals of argumentation had not been so well valued, and pragmati answers BALCHEFF, , were onsidered una eptable as proofs, under the point of view of the mathemati al rigor.
Data analysis, whi h also took into a ount 10 undergraduate students who parti ipated in ompleting the form, indi ates that tea hers in this group, in general, are not in lined to foster the development of a tivities in the lassroom, in order to onstru t skills and abilities to argue and prove in mathemati s.
Textbooks ould help in this issue, sin e they a t as guides for the tea hing pra ti e. Martins and Mandarino analyzed the Guide of Textbooks of , fo using on the hapters on erning Geom- etry. Bolema, Rio Claro SP , v. PIMM, D. London, GB, p. Base Na ional Comum Curri ular. The Mathemati s Tea her, v. For the Learning of Mathemati s, 17 1 , p. Rio de Janeiro RJ. A esso em Representing studies of tea her' use of resour es from China, Fran e, South Afri a, Sweden, and the United States, the symposium explores the relationship between individual and olle tive tea her apa ity and the design of resour es as fa tors that shape the ena ted urri ulum.
The following questions will guide the session: How might we understand the pro esses by whi h tea hers engage with urri ulum resour es to design instru tion? How do resour e features ontribute to tea hers' urri ulum designs? How do tea hers' urri ulum designs ontribute to the ena ted urri ulum? Resting on so io- ultural analyses of the agent-tool relationship Vygot- sky, , this perspe tive on eptualises resour es as ultural tools that mediate tea hers' urri ulum design work and are produ ts of this work.
Tea hers' intelle tual and ultural resour es also mediate this pro ess. Finally, tea hers' urri ulum design work o urs in a ontext, often with other tea hers, and unfolds over time, leading to new designs, new apa ities, and new urri ulum ena tments. In , the types of resour es used by tea hers are diverse and in lude print, digital, and online tools. Symposium Parti ipants and Organization The symposium is intended to highlight the work of early- areer resear hers.
Four resear hers have agreed to submit abstra ts by O t. Up to 5 additional papers an be a epted. Referen es Gueudet, Ghislaine, and Trou he, Lu , Theorizing the design and use of urri ulum materials. Remillard, et al. Conne ting urri ulum materials and lassroom instru tion pp. Remillard, Janine T. Vygotsky, Lev S. Mind in So iety. Cambridge, MA: Harvard University Press. They have being fundamental to tea hers de ision on whi h ontents must be taught as well as the instru tional approa h to be developed in lass Lajolo, It is a national program responsible for the pro ess of sele tion, evaluation and hoi e of the textbook.
Ea h review, in the ase of mathemati s, omprises analysis of: These do uments are also used to give tea her and their s hool ommunity the support for dis ussion and development of edu ational proje ts and also referen es for analyzing and sele ting dida ti materials and te hnologi al resour es. In that sense, Remillard on eptualized modes and forms of engagement for what the tea hers do in their transa tions with a urri ulum resour e: Nonetheless, in order to prepare a math lass, for example, how are the pro esses by whi h tea hers engage with urri ulum and resour es: Interviews were undertaken with these tea hers and a report was demanded about the initial intentions when building their worksheets.
The data analysis had some steps: Nevertheless, the pro ess by whi h ea h one engaged with these urri ular resour es has its parti ularity. They revealed personal urri ulum resour es and forms of engagement.
Gueudet, Ghislaine, and Trou he, Lu. Lajolo, Marisa. Kieran, Tanguay, and Armando Solares. Shaping Both the Impli it and the Expli it". Remillard, Janine. The instru - tional sequen e is based on a onje tured learning traje tory on fra tions as measures, and was developed through a series of lassroom design experiments, following Cobb and olleagues' methodologi al guidelines Gravemeijer and Cobb Third, the resour e guided Irene in supporting students' sense making of important fra tion notions during lassroom intera tions.
We explain how the support the instru tional sequen e provided at ea h of the three layers ontributed to its viability as a resour e for Irene's tea hing. During o-planning of the instru tional sessions, the sequen e be ame a resour e for Irene's a assessment of students' prior parti ipation, b anti ipation of student reasoning in the oming session, and planning responses to students' ontributions and emerging ideas.
Finally, during the instru tional sessions, the sequen e be ame a resour e for making in-the-moment de isions about guiding lassroom intera tions, making them more manageable. In this ontext, the ase we analyze sheds light into the role that instru tional sequen es that are a produ t of areful design and experimentation in lassrooms, an play in supporting tea hers who work with low performing students.
While our ase highlights the resour e use by a single tea her, the des ribed layers of support are onsistent with our earlier analysis of the role that similar instru tional sequen e played in supporting the learning of a group of 12 middle s hool tea hers in the USA over 5 years Visnovska, Cobb, and Dean We dis uss impli ations of our analysis for on eptualization and design of edu ative urri ulum materials Davis and Kraj ik Visnovska, and C.
Supporting students' reasoning about the inverse order relationship.
Davis, E. Kraj ik. Gravemeijer, K. The design, development and evaluation of programs, pro esses and produ ts, edited by J. Gravemeijer, S. M Kenney and N. Nieveen, Organisation for E onomi Co-operation and Development.
PISA results: What students know and an do - student performan e in mathemati s, reading and s ien e Vol. Visnovska, J. Cobb, and C. What does it take? Mathemati s urri ulum materials and tea her development, edited by G. Gueudet, B. Pepin and L. Trou he, Dordre ht: The study situated in so io- ultural theory and notion of mediating artefa ts, regards tea hing as mediation between the mathemati s taught and the learners.
The analyti al approa h adopted utilises a lesson as a unit of analysis instead of a tea her to allow patterns of use a ross all twenty lessons regardless of the tea her. Further ategorization of the omissions into produ tive and riti al omissions; and inje tions into robust and distra tive, illuminate two key issues pertaining to tea hers' mobilisation of the textbook.
Referen es Brown, Matthew. Theorizing the Design and Use of Curri ulum Materials. Remillard, Beth A. Herbel-Eisenmann and Gwendolyn M.
Routledge, Erlbaum, Leshota, Moneoang. Leshota, Moneoang, and J Adler. Polanyi, M. The Ta it Dimension. An hor Books, Vygotsky, L. Harvard University Press, Werts h, James V.
Mind as A tion. Oxford University Press, Freudenthal, , but resear h suggests that tea hers may la k a nuan ed understanding of how CPs an support learning of new mathemati s Lee, This ase study hara terizes the role of ontextualized problems in one ena ted algebra unit to provide insight into how the tea her's apa ities and design pra ti es medi- ated students' experien es with the modelling-based urri ulum.
Theoreti al Framework I frame my study using Stein, Remillard and Smith's des ription of the temporal phases of urri ulum use: Then, the tea her and students intera t with ea h other and the written urri ulum during the lesson to reate the ena ted urri ulum.
Using this framework, I tra e a modeling-based algebra unit from the written urri ulum phase through to the ena ted urri ulum. Methods The dataset in ludes the tea her interviews, the written urri ulum, the tea hers' plans, and trans ripts of lassroom dis ourse.
Then, I ode ea h portion of the tea hers' plans to determine the extent to whi h her plans follow the sequen es in the written urri ulum. I also oded ea h utteran e in the trans ript of the lassroom to identify whether lassroom dis ourse around one type of task in luded referen es to other types. At other points in the unit, the tea her resequen ed or omitted parti ular portions of the urri ulum so that the ontextualized problems were no longer used as introdu tions.
Be ause the units were sequen ed with the intent that students would leverage ontextualized problem solving experien es to help them solve non- ontextualized problems, students were left to work the non- ontextualized problem without a key resour e intended by the developers. Dis ussion The tea her's design apa ity has learly been developed through her repeated use of the urri u- lum.
These previous ena tments developed two key omponents of her pedagogi al design apa ity: Her reordering of the urri ulum, though, represents the presen e of omponent 1 and the absen e of omponent 2. Routledge Freudenthal, Hans. Revisiting mathemati s edu ation. China Le tures Springer. Lee, Ji-Eun. Lester Ed. Information Age Publishing. Our work examines a domain of the work of tea hing not widely explored in the literature-using urri ulum resour es to design mathemati s instru tion-and proposes a framework for identifying the mathemati al demands of the work and the knowledge tea hers draw on to navigate them.
We use the term knowledge of urri ulum embedded mathemati s KCEM to refer to the knowledge a tivated by tea hers when reading and interpreting mathemati al tasks, instru tional designs, and representations in mathemati s urri ulum materials. Among them, there is general agreement that tea hing is a omplex a tivity requiring knowledge, deliberation, and skilled a tion Shulman, We examined omponents of the tea her's guides and interviewed tea hers when planning lessons with urri ulum resour es to onsider two related questions: How are mathemati al ideas embedded in mathemati s tea her's guides?
What do tea hers need to reason about in order to surfa e and make produ tive use of these ideas when planning with the guides? Foundational mathemati al ideas refers to understanding of the foundational mathemati al ideas that underlie instru tional formats. Re ognizing mathemati al learning pathways in a urri ulum in ludes understanding how a parti ular goal is situated within a set of ideas that develop over time.
Further, the four dimensions point to routine features of urri ulum resour es around whi h knowledge development might o ur. When made expli it, these dimensions an be examined by tea hers to support their understanding of the mathemati al omponents of these tools. Content knowledge for tea hing: What makes it spe ial? Journal of Tea her Edu ation, 59 5 , Beyond mere knowledge of mathemati s: The importan e of knowing-to a t in the moment.
Edu ational Studies in Mathemati s, 38, Rowland, Tim. The knowledge quartet: The genesis and appli ation of a framework for analyzing mathemati s tea hing and deepening tea hers' mathemati s knowledge.
Journal of Edu- ation, 1 3 , Shulman, L. Those who understand: Knowledge growth in tea hing. Edu ational Resear her, 15 2 , Knowledge and tea hing: Foundations of the new reform.
Harvard Edu a- tional Review, 57 1 , Currently, it analyzes and restru tures of way of deep mathemati al knowledge of the urri ulum. We proposed various ourses that promote hanges in relation with knowledge, through the problematization of s hool mathemati s psm.
The psm promotes for tea hers a distin t look at tea hing resour es, su h as developing a on rete idea about the design of learning situations, favoring the transversality of mathemati s, a epting the diversity of students' answers to the same question.
This training aims to impa t tea hers' a tivity in their own lassrooms with their students. PIDPDM program a ompanies the tea hing pro ess in order to support urri ulum hanges, towards implementation in lassrooms. Introdu tion The Mexi an s hool system, at the high-s hool level, is divided into te hnologi al subsystems, general and preparation for work that is grouped in several departments: The PIDPDM is a program that takes ourses, workshops and seminars for professional tea her development in mathemati s and is dire ted to all the subsystems.
For that, introdu ing the pms as an indispensable resour e for instru tional mathemati s in the lassroom. This resour e would be fun tional for all tea hers, who subsequently developed designs of intervention for the mathemati s lassroom where a tivities will be ontextualized in the ulture of the students.
The problematization of s hool mathemati s psm The psm is a me hanism for ollaboration and profound, professional questioning and understanding of s hool mathemati s that traditionally is worked in their lassrooms. The Ante edent: