<?xml version="1.0" encoding="ISO-8859-1"?><article xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">
<front>
<journal-meta>
<journal-id>2448-8089</journal-id>
<journal-title><![CDATA[Educación matemática]]></journal-title>
<abbrev-journal-title><![CDATA[Educ. mat.]]></abbrev-journal-title>
<issn>2448-8089</issn>
<publisher>
<publisher-name><![CDATA[Sociedad Mexicana de Investigación y Divulgación de la Educación Matemática A.C.; Universidad de Guadalajara]]></publisher-name>
</publisher>
</journal-meta>
<article-meta>
<article-id>S2448-80892023000300082</article-id>
<article-id pub-id-type="doi">10.24844/em3503.03</article-id>
<title-group>
<article-title xml:lang="es"><![CDATA[De la expresión de la razón con dos números naturales, a su expresión con una fracción. Dos experiencias de ingeniería didáctica en el nivel básico]]></article-title>
<article-title xml:lang="en"><![CDATA[From the expression of the ratio with two natural numbers, to its expression with a fraction. Two didactic experiences at primary school and high school]]></article-title>
</title-group>
<contrib-group>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Block Sevilla]]></surname>
<given-names><![CDATA[David]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Ramos Banda]]></surname>
<given-names><![CDATA[Daniela]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Sosa]]></surname>
<given-names><![CDATA[Juan José]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
</contrib-group>
<aff id="Af1">
<institution><![CDATA[,Instituto Politécnico Nacional Centro de Investigación y de Estudios Avanzados Departamento de Investigaciones Educativas]]></institution>
<addr-line><![CDATA[ ]]></addr-line>
<country>Mexico</country>
</aff>
<aff id="Af2">
<institution><![CDATA[,danielabanda7@gmail.com  ]]></institution>
<addr-line><![CDATA[ ]]></addr-line>
<country>Mexico</country>
</aff>
<aff id="Af3">
<institution><![CDATA[,Universidad Nacional del Nordeste  ]]></institution>
<addr-line><![CDATA[ ]]></addr-line>
<country>Argentina</country>
</aff>
<pub-date pub-type="pub">
<day>00</day>
<month>00</month>
<year>2023</year>
</pub-date>
<pub-date pub-type="epub">
<day>00</day>
<month>00</month>
<year>2023</year>
</pub-date>
<volume>35</volume>
<numero>3</numero>
<fpage>82</fpage>
<lpage>111</lpage>
<copyright-statement/>
<copyright-year/>
<self-uri xlink:href="http://www.scielo.org.mx/scielo.php?script=sci_arttext&amp;pid=S2448-80892023000300082&amp;lng=en&amp;nrm=iso"></self-uri><self-uri xlink:href="http://www.scielo.org.mx/scielo.php?script=sci_abstract&amp;pid=S2448-80892023000300082&amp;lng=en&amp;nrm=iso"></self-uri><self-uri xlink:href="http://www.scielo.org.mx/scielo.php?script=sci_pdf&amp;pid=S2448-80892023000300082&amp;lng=en&amp;nrm=iso"></self-uri><abstract abstract-type="short" xml:lang="es"><p><![CDATA[Resumen: Se presentan los resultados de un estudio didáctico sobre el tránsito de la expresión de una razón con dos números naturales, a su expresión con una fracción. En este tránsito las fracciones adquieren el significado de razones. El estudio consta de dos experiencias cortas de ingeniería didáctica, con alumnos de entre 11 y 13 años, una en sexto grado de primaria y otra en primer grado de secundaria (séptimo grado). Los resultados dan cuenta de cierto potencial de las situaciones utilizadas para favorecer la consideración de razones y para dar lugar a su expresión con fracciones. Así mismo, dejan ver puntos débiles, y sugieren vías para seguir explorando.]]></p></abstract>
<abstract abstract-type="short" xml:lang="en"><p><![CDATA[Abstract:The results of a didactic study about the transition from the expression of a ratio with two natural numbers, to its expression with a fraction and with a percentage, are presented. In this transition, the fractions acquire the meaning of ratios, that is, of relationships between quantities. The study consists of two short experiences of didactic engineering, with students between 11 and 13 years old, one in the sixth grade of primary school and the other in the first grade of high school (seventh grade). The results are encouraging in terms of the potential of the situations used to favor the consideration of ratios and to give rise to their expression with fractions. Likewise, they reveal weak points, and suggest ways to continue exploring.]]></p></abstract>
<kwd-group>
<kwd lng="es"><![CDATA[Enseñanza de las matemáticas]]></kwd>
<kwd lng="es"><![CDATA[Aritmética]]></kwd>
<kwd lng="es"><![CDATA[Razones]]></kwd>
<kwd lng="es"><![CDATA[Fracciones]]></kwd>
<kwd lng="en"><![CDATA[Mathematics education]]></kwd>
<kwd lng="en"><![CDATA[Arithmetic]]></kwd>
<kwd lng="en"><![CDATA[Ratios]]></kwd>
<kwd lng="en"><![CDATA[Fractions]]></kwd>
</kwd-group>
</article-meta>
</front><back>
<ref-list>
<ref id="B1">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Adjiage]]></surname>
<given-names><![CDATA[R.]]></given-names>
</name>
<name>
<surname><![CDATA[Pluvinage]]></surname>
<given-names><![CDATA[F.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[An experiment in teaching ratio and proportion]]></article-title>
<source><![CDATA[Educational Studies in Mathematics]]></source>
<year>2007</year>
<volume>65</volume>
<numero>2</numero>
<issue>2</issue>
<page-range>149-75</page-range></nlm-citation>
</ref>
<ref id="B2">
<nlm-citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Artigue]]></surname>
<given-names><![CDATA[M.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Ingeniería didáctica]]></article-title>
<person-group person-group-type="editor">
<name>
<surname><![CDATA[Gómez]]></surname>
<given-names><![CDATA[P.]]></given-names>
</name>
</person-group>
<source><![CDATA[Ingeniería didáctica en educación matemática. Un esquema para la investigación y la innovación en la enseñanza y el aprendizaje de las matemáticas]]></source>
<year>1995</year>
<page-range>33-59</page-range><publisher-name><![CDATA[Grupo Editorial Iberoamérica]]></publisher-name>
</nlm-citation>
</ref>
<ref id="B3">
<nlm-citation citation-type="">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Block]]></surname>
<given-names><![CDATA[D.]]></given-names>
</name>
</person-group>
<source><![CDATA[La noción de razón en las matemáticas de la escuela primaria. Un estudio didáctico]]></source>
<year>2006</year>
</nlm-citation>
</ref>
<ref id="B4">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Block]]></surname>
<given-names><![CDATA[D.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Se cambian fichas por estampas. Un estudio didáctico sobre la noción de razón &#8220;múltiplo&#8221; y su vinculación con la multiplicación de números naturales]]></article-title>
<source><![CDATA[Educación Matemática]]></source>
<year>2006</year>
<volume>18</volume>
<numero>2</numero>
<issue>2</issue>
<page-range>5-36</page-range></nlm-citation>
</ref>
<ref id="B5">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Block]]></surname>
<given-names><![CDATA[D.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[&#8220;Los saltos de las ranas&#8221;. Estudio de una secuencia didáctica de proporcionalidad, con problemas de comparación de razones, en quinto grado de primaria.]]></article-title>
<source><![CDATA[Educación Matemática]]></source>
<year>2021</year>
<volume>33</volume>
<numero>2</numero>
<issue>2</issue>
<page-range>115-46</page-range></nlm-citation>
</ref>
<ref id="B6">
<nlm-citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Bosch]]></surname>
<given-names><![CDATA[M.]]></given-names>
</name>
</person-group>
<source><![CDATA[La dimensión ostensiva en la actividad matemática. El caso de la proporcionalidad]]></source>
<year>1995</year>
<publisher-name><![CDATA[Departent de Matemàtiques. Facultat de Ciències, Univeristat Autònoma de Barcelona]]></publisher-name>
</nlm-citation>
</ref>
<ref id="B7">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Brousseau]]></surname>
<given-names><![CDATA[G.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Problèmes de didactique des décimaux]]></article-title>
<source><![CDATA[Recherches en Didactique des Mathématiques]]></source>
<year>1981</year>
<volume>2</volume>
<numero>1</numero>
<issue>1</issue>
<page-range>37-127</page-range></nlm-citation>
</ref>
<ref id="B8">
<nlm-citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Brousseau]]></surname>
<given-names><![CDATA[G.]]></given-names>
</name>
</person-group>
<source><![CDATA[Théorie des situations didactiques. Didactique des mathématiques 1970-1990]]></source>
<year>1998</year>
<publisher-name><![CDATA[La Pensée Sauvage, Éditions]]></publisher-name>
</nlm-citation>
</ref>
<ref id="B9">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Brousseau]]></surname>
<given-names><![CDATA[G.]]></given-names>
</name>
<name>
<surname><![CDATA[Brousseau]]></surname>
<given-names><![CDATA[N.]]></given-names>
</name>
<name>
<surname><![CDATA[Warfield]]></surname>
<given-names><![CDATA[V.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Rationals and decimals as required in the school curriculum. Part 1: Rationals as measurement]]></article-title>
<source><![CDATA[The Journal of Mathematical Behavior]]></source>
<year>2004</year>
<volume>23</volume>
<numero>1</numero>
<issue>1</issue>
<page-range>1-20</page-range></nlm-citation>
</ref>
<ref id="B10">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Clark]]></surname>
<given-names><![CDATA[M. R.]]></given-names>
</name>
<name>
<surname><![CDATA[Berenson]]></surname>
<given-names><![CDATA[S. B.]]></given-names>
</name>
<name>
<surname><![CDATA[Cavey]]></surname>
<given-names><![CDATA[L. O.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[A comparison of ratios and fractions and their roles as tools in proportional reasoning]]></article-title>
<source><![CDATA[Journal of Mathematical Behavior]]></source>
<year>2003</year>
<volume>22</volume>
<numero>3</numero>
<issue>3</issue>
<page-range>297-317</page-range></nlm-citation>
</ref>
<ref id="B11">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Davis]]></surname>
<given-names><![CDATA[G. E.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[From parts and wholes to proportional reasoning]]></article-title>
<source><![CDATA[Journal of Mathematical Behavior]]></source>
<year>2003</year>
<volume>22</volume>
<numero>3</numero>
<issue>3</issue>
<page-range>213-2016</page-range></nlm-citation>
</ref>
<ref id="B12">
<nlm-citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Freudenthal]]></surname>
<given-names><![CDATA[H.]]></given-names>
</name>
</person-group>
<source><![CDATA[Didactical Phenomenology of Mathematical Structures]]></source>
<year>1983</year>
<publisher-name><![CDATA[Kluwer Academic Publishers]]></publisher-name>
</nlm-citation>
</ref>
<ref id="B13">
<nlm-citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Hernández]]></surname>
<given-names><![CDATA[S.]]></given-names>
</name>
</person-group>
<source><![CDATA[Aritmética y Nociones de Geometría]]></source>
<year>1954</year>
<edition>8ª</edition>
<publisher-name><![CDATA[Herrero Hnos. Sucs., S. A.]]></publisher-name>
</nlm-citation>
</ref>
<ref id="B14">
<nlm-citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Kieren]]></surname>
<given-names><![CDATA[T.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Personal knowledge of rational numbers: Its intuitive and formal development]]></article-title>
<person-group person-group-type="editor">
<name>
<surname><![CDATA[Hiebert]]></surname>
<given-names><![CDATA[J.]]></given-names>
</name>
<name>
<surname><![CDATA[Behr]]></surname>
<given-names><![CDATA[M.]]></given-names>
</name>
</person-group>
<source><![CDATA[Number concepts and operations in the middle grades]]></source>
<year>1988</year>
<volume>2</volume>
<publisher-name><![CDATA[Lawrence Erlbaum Associates]]></publisher-name>
</nlm-citation>
</ref>
<ref id="B15">
<nlm-citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Lamon]]></surname>
<given-names><![CDATA[S. J.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Rational numbers and proportional reasoning: Toward a theoretical framework for Research]]></article-title>
<person-group person-group-type="editor">
<name>
<surname><![CDATA[Lester]]></surname>
<given-names><![CDATA[F. K.]]></given-names>
</name>
</person-group>
<source><![CDATA[Second handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics]]></source>
<year>2007</year>
<page-range>629-68</page-range><publisher-name><![CDATA[Information Age Publishing]]></publisher-name>
</nlm-citation>
</ref>
<ref id="B16">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Parker]]></surname>
<given-names><![CDATA[M.]]></given-names>
</name>
<name>
<surname><![CDATA[Leinhardt]]></surname>
<given-names><![CDATA[G.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Percent: A Privileged Proportion]]></article-title>
<source><![CDATA[Review of Educational Research]]></source>
<year>1995</year>
<volume>65</volume>
<numero>4</numero>
<issue>4</issue>
<page-range>421-81</page-range></nlm-citation>
</ref>
<ref id="B17">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Pedersen]]></surname>
<given-names><![CDATA[P. L.]]></given-names>
</name>
<name>
<surname><![CDATA[Bjerre]]></surname>
<given-names><![CDATA[M.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Two conceptions of fraction equivalence]]></article-title>
<source><![CDATA[Educational Studies in Mathematics]]></source>
<year>2021</year>
<volume>107</volume>
<page-range>135-57</page-range></nlm-citation>
</ref>
<ref id="B18">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Noelting]]></surname>
<given-names><![CDATA[G.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[The development of proportional reasoning and the ratio concept. Part I. Differentiation of stages]]></article-title>
<source><![CDATA[Educational Studies in Mathematics]]></source>
<year>1980</year>
<volume>11</volume>
<numero>2</numero>
<issue>2</issue>
<page-range>217-53</page-range></nlm-citation>
</ref>
<ref id="B19">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Noelting]]></surname>
<given-names><![CDATA[G.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[The development of proportional reasoning and the ratio concept. Part II. Problem structure at successive stages. Problem solving strategies and the mechanism of adaptive restructuring]]></article-title>
<source><![CDATA[Educational Studies in Mathematics]]></source>
<year>1980</year>
<volume>11</volume>
<numero>3</numero>
<issue>3</issue>
<page-range>331-63</page-range></nlm-citation>
</ref>
<ref id="B20">
<nlm-citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Ramos]]></surname>
<given-names><![CDATA[D.]]></given-names>
</name>
</person-group>
<source><![CDATA[La fracción como vía de expresión de una razón y de un cociente. Análisis de una experiencia didáctica]]></source>
<year>2014</year>
<publisher-name><![CDATA[Departamento de Investigaciones Educativas, Centro de Investigación y de Estudios Avanzados]]></publisher-name>
</nlm-citation>
</ref>
<ref id="B21">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Ramos]]></surname>
<given-names><![CDATA[D.]]></given-names>
</name>
<name>
<surname><![CDATA[Block]]></surname>
<given-names><![CDATA[D.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[&#8220;Por cada tres naranjas que recojas, te doy dos&#8221;. Una propuesta didáctica para trabajar con razones y expresarlas con fracciones]]></article-title>
<source><![CDATA[Revista para maestr@s de educación básica. Digital &#8220;Entre Maestr@s&#8221;]]></source>
<year>2016</year>
<volume>16</volume>
<numero>57</numero>
<issue>57</issue>
<page-range>54-63</page-range></nlm-citation>
</ref>
<ref id="B22">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Schadl]]></surname>
<given-names><![CDATA[C.]]></given-names>
</name>
<name>
<surname><![CDATA[Ufer]]></surname>
<given-names><![CDATA[S.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Beyond linearity: Using IRT-scaled level models to describe the relation between prior proportional reasoning skills and fraction learning outcomes]]></article-title>
<source><![CDATA[Child Development, (vista temprana)]]></source>
<year>2023</year>
<page-range>1-17</page-range></nlm-citation>
</ref>
<ref id="B23">
<nlm-citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Schwartz]]></surname>
<given-names><![CDATA[J. L.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Intensive quantity and referent transforming arithmetic operations]]></article-title>
<person-group person-group-type="editor">
<name>
<surname><![CDATA[Hiebert]]></surname>
<given-names><![CDATA[J.]]></given-names>
</name>
<name>
<surname><![CDATA[Behr]]></surname>
<given-names><![CDATA[M.]]></given-names>
</name>
</person-group>
<source><![CDATA[Number concepts and operations in the middle grades]]></source>
<year>1988</year>
<volume>2</volume>
<page-range>41-52</page-range><publisher-name><![CDATA[Lawrence Erlbaum Associates]]></publisher-name>
</nlm-citation>
</ref>
<ref id="B24">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Simon]]></surname>
<given-names><![CDATA[M. A.]]></given-names>
</name>
<name>
<surname><![CDATA[Placa]]></surname>
<given-names><![CDATA[N.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Reasoning about intensive quantities in whole-number multiplication? A possible basis for ratio understanding]]></article-title>
<source><![CDATA[For the Learning of Mathematics]]></source>
<year>2012</year>
<volume>32</volume>
<numero>2</numero>
<issue>2</issue>
<page-range>35-41</page-range></nlm-citation>
</ref>
<ref id="B25">
<nlm-citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Sosa]]></surname>
<given-names><![CDATA[J.]]></given-names>
</name>
</person-group>
<source><![CDATA[La probabilidad como lugar de encuentro de razones, fracciones y decimales. Un estudio didáctico en primer grado de nivel secundaria]]></source>
<year>2018</year>
<publisher-name><![CDATA[Departamento de Investigaciones Educativas, Centro de Investigación y de Estudios Avanzados]]></publisher-name>
</nlm-citation>
</ref>
<ref id="B26">
<nlm-citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Vergnaud]]></surname>
<given-names><![CDATA[G.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Multiplicative Structures]]></article-title>
<person-group person-group-type="editor">
<name>
<surname><![CDATA[Hiebert]]></surname>
<given-names><![CDATA[J.]]></given-names>
</name>
<name>
<surname><![CDATA[Behr]]></surname>
<given-names><![CDATA[M.]]></given-names>
</name>
</person-group>
<source><![CDATA[Number concepts and operations in the middle grades]]></source>
<year>1988</year>
<volume>2</volume>
<page-range>141-61</page-range><publisher-name><![CDATA[Lawrence Erlbaum Associates]]></publisher-name>
</nlm-citation>
</ref>
<ref id="B27">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Wonziak]]></surname>
<given-names><![CDATA[F.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Fondements du travail épistémologique du professeur]]></article-title>
<source><![CDATA[Recherches en Didactique des Mathématiques]]></source>
<year>2019</year>
<volume>39</volume>
<numero>1</numero>
<issue>1</issue>
<page-range>15-50</page-range></nlm-citation>
</ref>
</ref-list>
</back>
</article>
