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O Rio de Janeiro em O Mestre do Amor. Paulo Que pode ser aquilo. Ernest Hemingway: Is it natural to start the dis ussion on division of natural numbers only with dividends that are multiples of the divisors?

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: We arried out a work of problematization of s hool mathemati s that treats the diverse themes with the hara teristi s that were fundamental for tea her empowerment Reyes-Gasperini, Proportionality, representational fun tion, trigonometri al modelation, Thinking and Variational Language before linear and nonlinear.

Among them were 1 used as resour es spe ialized bibliographies to onfront that whi h the textual books given as a fa t, as for example refutes phrases su h as this: The urrent proje ts of professional tea her development sin e the PIDPDM During the year we performed an intervention in three major proje ts with a total of six-thousand parti ipating tea hers. The se ond, approa hed the urri ular materials that realized the problematization of thoughts: When the tea hers return to fa e their evaluation, they will have new tools whi h they an defend, to give their answers.

The third, analyzed learning situations designed under a so ioepeistemologi al foundation, in order that the tea hers later design and support their own situations to implement them in the lassroom.

Referen es Cantoral, R. Supporting maths tea hers in the so ial onstru tion of knowledge. In In lusive edu ation in a tion: Empowering tea hers: Disponible en http: In Fran e, these new means are extensively used. This proje t a ts as in ubator of on epts Trou he Spe ially, it leads to on eptual developments for the do umentational approa h to dida ti s Gueudet and Trou he Gueudet and Trou he propose the term resour e in a omprehensive a eptation, as everything nourishing tea hers' work and the term do ument as a hybrid entity, resulting of the tea her do umentation work, asso iating a resour e developed for a given dida ti al goal and a s heme piloting the pro ess of design and usage of this resour e.

This knowledge takes into a ount stru turing features of lassroom pra ti e: We propose to ombine this approa h with the do umentational approa h to dida ti s, introdu ing the notion of raft do umentational knowledge as the professional knowledge that arises from and in tea hers' do umentation work.

This raft do umentational knowledge develops throughout the development of s hemes. Tea hers' do umentation work develops in so ial ontext, in a way depending on their own history with tea hing resour es. In this sense, to understand how tea hers build their raft do umentational knowledge, the analysis of their do umentational traje tory through a olle tive lens takes an important pla e. Then, we dis uss here: This work takes pla e in a riti al moment in Fran e, a new urri ulum for grades 1 to 9, being implemented in September , proposing deep hanges: We follow Anna and Viviane's preparation of the progression related to this new topi algorithmi and programming in their ordinary environment.

We asked them to map their do umentational traje tory. We ross the data from their preparation of a new lesson with the mapping of their do umentational traje tory. Referen es Fle k, Ludwik. Chi ago: University of Chi ago Press original edition, Do umentational Geneses and Professional Geneses. Springer Netherlands, Ro ha, Katiane D. Germany, Hamburg, July , Ruthven, Kenneth.

The Example of Dynami Geometry. English and David Kirshner, Human development, 52 2: Trou he, Lu. A essed O tober 14, As urri ulum programs are widely used in Sweden Boesen et al. Brown et al. The aim is twofold. Des ription of modules and envisioned tea her support The two modules onsist of four lessons and fo us on one mathemati al topi grade one: The third and fourth lessons relate to an open response lesson 3 and reengagement lesson lesson 4.

We did so be ause we anti ipate that tea hers have to modify the written lessons and be ause Choppin underlined the importan e of tea hers' related understanding for produ tive urri ulum adaptations.

Support in relation to the instru tional sequen e over the lessons in lude an introdu tion to the module onsisting of a des ription of the lesson goals, how the module onne ts to previous and future ontent, and how the lessons in the module relate to ea h other. These two types of support have been des ribed earlier in a study about redesigning an existing s ien e urri ulum program Davis et al.

Referen es Atanga, N. Developing mathemati al ompeten e: From the intended to the ena ted urri ulum.

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The Journal of Mathemati al Behavior, 33, Brown, S. Re on eiving Fidelity of implementa- tion: Journal for Resear h in Mathemati s Edu ation, 40 4 , Choppin, J. Learned adaptations: Tea hers' understanding and use of urri ulum re- sour es. Journal of Mathemati s Tea her Edu ation, 14 5 , Designing Edu ative Curri ulum Materials: A Theoreti ally and Empiri ally Driven Pro ess.

Harvard Edu ational Review, 84 1 , Hemmi, K. Journal of Mathemati s Tea her Edu ation, 18 6 , A ross- ultural analysis of the voi e of six tea her's guides from three ultural ontexts. Stein, M. Greenwi h, CT: Van Steenbrugge, H. This is likely to hange tea hers' working modes, and at the same time provides hallenges for dealing with these resour es. The result, developed along this genesis, is named a do ument omposed of resour es ombined to rea h a tea hing obje tive and of a s heme Vergnaud, onne ting this obje tive, the organization of a tion, and professional knowledge.

The resour es a tea her works with onstitute her resour e system. Through do umentation work, tea hers are linked not only to resour es, but also to other tea hers. We propose the notion of do umentation expertise DE to represent the expertise in tea her do - umentation work, the s hemes for better resour e work whi h onsist by skills organization of a tion and related knowledge. Brown , p. Fur- ther study of Remillard detailed the pro edures of per eiving the urri ular resour es the same way for non- urri ular resour es as identifying the mathemati al purpose or point underlying mathemati al- instru tional obje ts: PDC is the ability to per eive the underlying stru tures of the designed resour es with respe t to their instru tional intensions.

Compared with PDC, DE involves a wider range of re- sour es, ompassing a omplete pro ess of do umentation genesis: How do mathemati s tea hers show and develop their DE in their olle tive do umentation work?

Within a olle tive working ulture des ribed by Yang , a Tea hing Resear h Group TRG represents the s hool-based tea her organization and provides institutional supports for tea hers' olle tive work. It has been a knowledged as a ru ial fa tor for explaining the top performan e of Chinese students in international evaluations Wang, This is also eviden ed by one of our previous studies with three Chinese mathemati s tea hers Pepin, Xu, Trou he, Wang, China; and Fran e.

This is being explored in two dimensions: The Fren h ase is situated in the national urri ulum reform, and we pay attention on the sele ted tea hers' olle tive resour e work for a new tea hing topi , algorithmi. From February to April , another data olle tion will take pla e in a Chinese middle s hool. In terms of methodology, we will validate the data olle tion instruments e. Referen es Adler, J. Con eptualising resour es as a theme for tea her edu ation, Journal of Mathe- mati s Tea her Edu ation, 3, Brown, M.

The tea her-tool relationship: Remillard, B. Lloyd, Eds. Learning by expanding. An a tivity-theoreti al approa h to developmental resear h. Gueudet, G. From text to 'lived' resour es: Mathemati s urri ulum materials and tea her development. Mathemati s tea hers as designers: Developing a deeper understanding of math- emati s tea hing expertise: Chinese mathemati s tea hers' resour e systems as windows into their work and expertise.

Edu ational Studies in Mathemati s, http: Understanding tea her-resour e intera tions: Wang, J. Mathemati s edu ation in China: Jiangsu Edu ation Publishing House.

Wang, C. Analyzing tea hers' expertise, resour es and olle tive work throughout Chinese and Fren h windows. Yang, X. Con eption and hara teristi s of expert mathemati s tea hers in China. We proposed dimensions of reading geometri diagrams gleaned from analyzing the geometri tasks in multiple elementary and se ondary textbooks, in luding traditional and reform urri ula from multiple ountries.

These dimensions represent distin t aspe ts of geometri diagrams that students are expe ted to pay attention to and interpret as they negotiate the meanings of mathemati al tasks. In this workshop, we will share what we have learned from our re ent study of the expe tations for how students read geometri diagrams.

We will introdu e a framework des ribing the expe tations for students while reading geometri diagrams and will explain several examples from U.

We will end this session with a dis ussion on the impli ations these hanging expe tations have for the tea hing and learning of geometry. With eviden e that students are expe ted to develop sophisti ated ways to negotiate meaning from diagrams, we argue that edu ators need to raft opportunities for students to develop strategies for reading geometri diagrams.

In the study that will be dis ussed, we ompared the geometri diagrams found in a Grade 1 U. To ode the reading expe tations of the geometri diagrams, we developed six odes to des ribe the geometri reading of diagrams, identi- fying: Using these odes, we report how the expe tations of reading geometri ally hange as students progress through s hool.

Referen es Dietiker, Leslie, and Aaron Brakonie ki. University of Southampton, UK. Pimm, David. New York, NY: This English-language, resear h-fo used workshop will be omplemented by a parallel workshop in Portuguese aimed at Brazilian tea hers who are involved as partners in the Um Livro Aberto proje t as well as a several related resear h paper presentations. Referen es Even, Ruhama, and Shai Olsher. The Integrated Mathemati s Wiki-book Proje t. Learning in the Te hnologi al Era, edited by Y.

Eshet-Alkalai, A. Caspi, S. Eden, N. Geri, Y. Yair, and Y. Kalman, The Open University of Israel. Freudenthal, Hans. Mathemati s as an edu ational task. Gravemeijer, Koeno, and Dolly van Eerde.

Livro Aberto de Matemati a. Sour ed from: Joseph Jenifer J. Whether the pro esses are utilized in isolation or as a onne ted omponent, the pro ess of writing an be demonstrated throughout these strands. For example, in order to problem solve one an write an explanation or des ription of the problem solving pro ess by reasoning and proving one's mathe- mati al thinking. Students an also write to des ribe the pro ess of onne ting the mathemati s ontent in addition to providing an explanation of a parti ular mathemati al representation.

The textbook publishing industry, as well as urri ulum proje ts funded by the National S ien e Foun- dation NSF , moved qui kly to develop urri ulum materials i.

Thus, the mathemati s textbook is typi ally resear hed as the dominant tool in lassroom instru tion Tarr et al. However, the language in the types of prompts has not been investigated. In order to onstru t a response in mathemati s, the student must be able to omprehend the prompt while produ ing pre ise language to respond to the prompt.

O'Connell et al. The omprehension of mathemati s en ompasses not only vo abulary terms, but also the understanding of symbols Thompson et al. These types of vo abulary have the potential to make the omprehension of mathemati s a omplex pro ess. The PSSM NCTM, pla es an emphasis on vo abulary under the pro ess strand of ommu- ni ation by re ommending that students use mathemati al vo abulary to express mathemati al ideas in a pre ise manner.

The Haggarty and Pepin study examined and ompared the layout of the mathemati al textbooks used in Fran e, Germany and England. Additionally, the study investigated the opportunities students had to perform mathemati al pro esses through the use of the vo abulary and language in the dire tions. More importantly the resear her examined the linguisti hoi es i. The areas explored were developed from resear h in writing and textbook use.

The development of a tool in the form of a matrix assisted with the oding of prompts in these parti ular areas. Ea h one of these workshop omponents has had national re ognition in onferen e presentations and manus ripts for publi ation. Retrieved from http: Haggarty, Linda. An investigation of mathemati s textbooks and their use in English, Fren h and German lassrooms: Who gets an opportunity to learn what? British Edu ational Resear h Journal, 28 4 , Herbel-Eisenmann, Beth.

From intended urri ulum to written urri ulum: Journal for Resear h in Mathemati s Edu ation, 38 4 , National Coun il of Tea hers of Mathemati s, ed. Prin iples and standards for s hool math- emati s. National Coun il of Tea hers of Mathemati s. O'Connell, Susan R. Beyea, Susan S. Denvir, Leila A. Dowdall, Nikki G. Friedland, and Joelle D.

Lessons Learned about Writing in Mathemati s. Senk, Sharon L. Re ommenda- tions and issues. What are they: Tarr, James E. Reys, Barbara J. Weiss, I. Pasley, P. Smith, E. Banilower, and D. Horizon Resear h.

Math tea hing went from New Math to the ontemporary tea hing pra ti e. Is there less mathemati s now than there used to be during the German New Math? In parti ular, the normative riteria for the tea hing of mathemati s are not a matter that mathemati ians an larify among themselves.

Ernest lo. In his analysis, Ernest , p. In the workshop, we onsider tasks in several textbook series for German pre-university edu ation in the light of Ernest's interest groups perspe tive. Here in Figure 1 we give an example. Figure 1: The underground parking is only a essible via a steep doorway. The passage should be as smooth as possible and the maximal slope should not be too big, su h that it is traversable by an ordinary passenger ar.

Make a planning by using an appropriate fun tion graph. Compare your solution with those of others. We ould think of several a tivities: In this way, we get an impression of the intended mathemati al awareness f. In the spe trum of aspe ts that appear by this analysis, we try to re ognize the fa ets of the interest groups as Ernest had des ribed them. Referen es Ernest, Paul. The Philosophy of Mathemati s Edu ation. Falmer Press. Ernest, Paul. Why tea h mathemati s? In Why Learn Maths?

Bedford Way Papers. Mathemati al awareness by linguisti analysis of variable substitution. Kaenders, Rainer, and Kvasz, Ladislav. Mathematis hes Bewusstsein. Genette Nossa es olha por este tipo de material deve-se a dois fatores: Maio 26 - A essado em 19 de outubro, Paratextos Editoriais. Cotia, SP: Thompson, John B. Ideologia e Cultura Moderna: Editora Vozes. Chevallard, Yves. L'appro he anthropologique. A es- sado no site http: Dom Quixote, p.

Fra tions in Realisti Mathemati s Edu ation: A Paradigm of Developmental Resear h. Kluwer A ademi Publishers, Conforme Valente , p. O uso de ferramentas interativas, segundo Gravina e Basso , p. Porto Alegre: Evangraf, Cap. Valente org. O Computador na So iedade do Conhe imento. Henrique N. Manual da dis iplina: Agrade imento: Universidade de Lisboa. Universidade do Porto. Tese de Doutoramento. V dir. Ripoll, Cydara C. A metodologia empregada parte dos estudos de John B.

Belo Horizonte: A olonialidade do saber: Buenos Aires: Ideologia e ultura moderna: Laguna, A. Menezes, E. Midiamix, D, and Paulo F. Carvalho, J. Ro ha, Katiane, and Lu Trou he.

Re ife: Universidade Federal de Pernambu o 6, What Makes It Spe ial? A essed November 09, Pinto, Nayara K. Shulman, Lee S. Mathemati s textbooks for tea hing: An analysis of ontent knowledge and pedagogi al ontent knowledge on erning algebra in mathemati s textbooks in Swedish upper se ondary edu ation.

A essed November 13, In this relation the sign is asso iated with the obje t via index, symbol or i on. Thus, a sign is not only a stru ture, but is also a pro ess. In this ontext, a ording to Otte , the symbol is the pro ess of the ir ular intera tion between indexi al and i oni elements, realized as an endless pro ess of interpretation.

In the edu ational ontext, the on eptualization of the mathemati al obje t, in general, is an hored in its presentation in the textbook. In this regard, the analysis of the book may indi ate how the mathemati al obje t approa h ontemplates that sign relationship.

That is, how the signs used by the authors an a t as an i on, index or a symbol aiming at understanding the mathemati al obje t, parti ularly, the derivative of a real fun tion of one independent variable.

In fa t, there are indi ations in the book that the derivative symbol is the result of a pro ess whi h arti ulates indexes and i ons. Referen es Peir e, C. Es ritos Coligidos. Nova Cultural. Peir e, C. Perspe tiva. Otte, M. Semioti s and Philosophy in Charles Sanders Peir e. Fabbri hesi, R. Cambridge S holars Publishing. The only viable a ess in su h a ase is the analysis of the 1 This resear h is related to a proje t funded by the CNPq.

The ase in question is the number line, whi h appears today as a natural, evident and unproblemati notion. A loser analysis shows, however, that this notion is intimately onne ted to the on ept of negative numbers; pra ti ing the number line reveals therefore ne essarily the ompli ated pro ess of development and establishment of the on ept of negative numbers.

An a ess to this pro ess is provided by the analysis of mathemati s textbooks published during the eighteenth entury and the early nineteenth entury, both at the se ondary level and the level of higher edu ation in various ountries. We fo us on original sour es of Fren h and German mathemati s, whi h stand out in this period: Fis her, Ernst G.

Lehrbu h der Elementar-Mathematik. Vierter Theil: Prony, Gaspard Ri he de. The Guide presents reviews of the olle tions, the result of an evaluation pro ess arried out by tea hers in edu ational institutions all over Brazil.

Its obje tive is to guide tea hers in hoosing textbooks. Furthermore, it presents the riteria used in the evaluation and the guiding ompeten ies. This author further proposes a model for urri ulum interpretation organized in six developmental levels. It expresses a on ern that the data be real, onne ted to physi al or so ial reality, with a resear h perspe tive, involving olle tion, sele tion, organization, representation, and inferential analysis based on qualitative information or numeri al data.

In the PNLD Guide, we observed a se ond hange, whi h establishes resear h as the stru turing axis of statisti s edu ation, highlighting the importan e of experien ing resear h and all its stages. We observed that these obje tives are presented in the tea hers' manuals of all the olle tions analyzed sin e However, analyses of the a tivities presented in the same olle tions are still being arried out to see if, in fa t, the stated obje tives are borne out in the a tivities proposed.

Porto Alegre, RS: We argue that along with the inves- tigation of the textbooks for lassroom use and their manuals to tea hers, the advan es of mathemati s edu ation resear h on the knowledge of tea hers to tea h imply the ne essity of textbooks for tea her edu ation ourses.

The methodology of designing the ontent of the textbook follows losely the Lesson Study prin iples, and based in Mathe- mati al Thinking skills in the lassrooms Isoda, Katagiri, We onsider that the interest of tea her edu ators that are re ognizing the book as potential tea hing material in tea her edu ation ourses is an indi ation of the importan e to produ e textbooks for tea hers to learn how to tea h.

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Ball, Deborah and Bass, Hyman. Davis and E. Simmt Eds Pro eedings of the annual meeting of the Canadian Math. Mathemati al Thinking: How to Develop it in the Classroom. Silver, Ed. Even and D. Shulman, Lee.

Knowledge Growth in Tea hing". Edu ational Resear her, Vol 5, n. A key design prin iple is to rigorously spe ify the stru ture of a book and then leanly separate the ontent from the presentation.

As an extreme example, onversion to Nemeth Braille has been su essful in an early prototype. Not only do these in lude standard elements like audio and video, but they an also allow a reader to explore mathemati al ideas with tools like GeoGebra, JSXGraph, and Sage ells, in addition to WeBWorK automated homework exer ises.

These books an all be read with a desktop or laptop omputer, but also work well on small tablets and on smartphones that are smaller still. We will demonstrate the nature of the online output with several textbooks made available with open li enses. Primarily these are undergraduate mathemati s textbooks, spanning the urri ulum from developmental mathemati s, through al ulus, and up to more advan ed ourses su h as number theory and abstra t algebra.

However, other dis iplines are now being represented, in luding omputer s ien e, poetry, and musi theory. More about the proje t an be found at mathbook. It is omplex be ause it is hybrid and requires the arti ulation of physi al as well as digital elements.

In this sense, we propose to approa h in this presentation the engineering pro ess that allowed, through the integration of the theoreti al and methodologi al prin iples mentioned above, to develop an instrumentation of the do umentation pro ess for the tea hing of mathemati s. We will present in parti ular two notions, and their omputational modilizations in the engineering pro ess: Con eptualising resour es as a theme for tea her edu ation.

Journal of Mathe- mati s Tea her Edu ation, 3, Do Trabalho do umental dos professores: Ro ha, Ea h three years, Brazilian Ministry of Edu ation evaluates the textbook olle tions proposed by the publishers and elaborates a guide with reviews of the approved olle tions on the basis of the evaluation pro ess.

For ea h area of s hool knowledge, tea hers' team of ea h state s hool sele ts a olle tion of textbooks among those approved and listed in the guide, to use a ordingly their pedagogi al needs. Until last year, the guide was provided in a digital and printed ri h text format with a volume per subje t area. Sin e the PNLD , an intera tive guide version with all areas was developed. In this presentation, we present how we built the digital guide of the PNLD Brazil, , onsidering all the involved parameters, to providing an intera tive guide as useful as possible to support tea her's hoi es of textbooks, onsidering their requirements to use in their pra ti e.

The onsidered parameters for the elaboration of the digital guide an be organized in the four following topi s related to the various dimensions of the engineering pro ess of the digital guide elaboration: We will present the engineering pro ess and its theoreti al and te hni al ba kground.

A ording to the resear h aim, one theoreti al tool may be more pertinent than another may. The aim of this arti le is to present a methodologi al path omposed of the following phases: A essed at http: Up to s, textbooks began tea hing arithmeti from s rat h, adhering to the tradition of libri d'abba o, des ribed by van Egmond During ss, the arithmeti textbook tradition ontinued in two phases, intended for primary and lower se ondary edu ation.

In this study we shall fo us on both ends of this sequen e. The matter is however, more ompli ated as Arithmeti a Islandi a may have more models, or it may be seen as a free and adapted translation. Furthermore, exer ises, ontained in Arithmeti a Islandi a are also found in other younger arithmeti manus ripts. We shall in parti ular study the history of the tea hing of ratio and proportions in the form of the Rule of Three and ompare its tea hing in Arithmeti a Islandi a and other books through the three enturies to that of ratio and proportions in Skali 3B by Tofteberg et al.

Referen es Frommius, Geo. Arithmeti a Islandi a Tofteberg, Grete N. In print. Skali 3B. Van Egmond, Warren Pra ti al mathemati s in the Italian Renaissan e: Istituto e Museo di Storia della S ienza. The Brazilian National Textbook Programme PNLD , sin e the late s, has been on erned with distin t aspe ts to be onsidered in textbook evaluations and Carvalho and Lima observed advan es in textbook quality.

An important aspe t of assessment on erns how symboli representations are dealt with in text- books as means to think about mathemati al on epts. To not onfuse a mathemati al on ept with its representation, Duval emphasizes the importan e of semioti representation transformations: This study analyses three textbook olle tions: The analysis on ern whi h ombinatorial situations are presented and what symboli representations onversions are required.

Early s hooling textbooks tend to use natural language with drawings, tree diagrams or tables to aid the understanding of ombinatorial situations. Positive aspe ts were observed in ombinatorial situations proposed, su h as onversions of varied symboli representations, but a better distribution of problem types and more symboli representation onversions still need to be present in textbooks.

Duval, Raymond. Peter Lang. More than just a substitute vehi le for a ess to traditional paper textbooks ontents, literature has pointed to how the use of e-textbooks through this type of te hnology an potentially transform tea hing and learning of mathemati s Pepin et al.

It is worth mentioning that the Brazilian government's largest program of free distribution of Basi S hool textbooks the PNLD program requires ompeting publishers to deliver a digital version of the textbook with intera tive resour es. In this ontext, digital manipulatives are a key element! How to make su h digital manipulatives for mathemati s and statisti s e-textbooks? In our presentation, we will make an overview of the nowadays major user-friendly systems, that is, systems that use little or no dire t programming.

We hope that our analysis will be useful for professionals and publishers interested in the produ tion of math e-textbooks. Referen es Pepin, Birgit et al. English, and David Kirshner, Yerushalmy, Mi hal. Springer, First, it is being onstru ted in permanent onta t with dozens of tea hers of regular s hools in Brazil, who are testing the material in their lassrooms.

Referen es David F. Robitaille and Kenneth J. Douglas A. Grouws New York: Ma millan, , Haggarty, Linda and Pepin, Birgit. Marshall S. Smith and Catherine M. Nilson J. Despite ample eviden e that images help guide our intuition and help design a logi al hain of arguments needed in the proving pro ess, there is a la k of resear h on how to expli itly formulate the hara teristi s of an image to foster this ognitive pro ess.

What kind of illustrations are urrently present in mathemati s textbooks? How an we help students interpret visual information and gain a ess to the information that is oded in every image? These are the questions we investigate and for whi h we provide viable answers. Potentially this may lead to additional hallenges imposed on students to orre tly on eptu- alize a given mathemati al obje t, due to the restri tive nature of the image itself.

We indi ate onfusing hara teristi s stemming from typi al images and provide suggestions for over oming these obsta les. We list possible venues for further resear h on evaluating images in mathemati s textbooks. Norris and John S. Ma nab. Visualization in Mathemati s, Reading and S ien e Edu ation. Presmeg, Norma. In Hand- book of resear h on the psy hology of mathemati s edu ation, edited by Angel Gutierrez and Paolo Boero, Sen e. Vinner, Shlomo, and Rina Hershkowitz. In Pro eedings of the fourth international onferen e for the psy hology of mathemati s edu ation pp.

In this sense, this paper aims to present and dis uss a very parti ular type of use of dida ti material in the pre-servi e mathemati s tea hers online distan e edu ation. The result of it, IDDM, will be presented in the next paragraph, and is based on the notion of humans-with-media Borba The resear h was qualitative, entered on the strand alled Grounded Theory Strauss and Corbin Details an be found in Chiari Agatha Christie - A morte da Sra.

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