Understanding of inverse proportional reasoning in pre-service teachers

From an early age, understanding proportional reasoning is a fundamental pillar in mathematics education, and therefore, teachers should have a thorough knowledge of it. Despite its significance, there are few studies that analyse the difficulties that student teachers have in understanding proporti...

Full description

Autores:
Cabero, Ismael
Santágueda-Villanueva, María
Villalobos-Antúnez, Jose Vicente
Roig, Ana Isabel
Tipo de recurso:
Article of journal
Fecha de publicación:
2020
Institución:
Corporación Universidad de la Costa
Repositorio:
REDICUC - Repositorio CUC
Idioma:
eng
OAI Identifier:
oai:repositorio.cuc.edu.co:11323/7226
Acceso en línea:
https://hdl.handle.net/11323/7226
https://repositorio.cuc.edu.co/
Palabra clave:
Inverse proportionality
Problem-solving strategies
Intensive quantity
Extensive quantity
Proportional reasoning
Tabular and graphic representation
Pre-service teachers
Rights
openAccess
License
CC0 1.0 Universal
id RCUC2_36fa7432e51d57450e5805bd4302b295
oai_identifier_str oai:repositorio.cuc.edu.co:11323/7226
network_acronym_str RCUC2
network_name_str REDICUC - Repositorio CUC
repository_id_str
dc.title.spa.fl_str_mv Understanding of inverse proportional reasoning in pre-service teachers
title Understanding of inverse proportional reasoning in pre-service teachers
spellingShingle Understanding of inverse proportional reasoning in pre-service teachers
Inverse proportionality
Problem-solving strategies
Intensive quantity
Extensive quantity
Proportional reasoning
Tabular and graphic representation
Pre-service teachers
title_short Understanding of inverse proportional reasoning in pre-service teachers
title_full Understanding of inverse proportional reasoning in pre-service teachers
title_fullStr Understanding of inverse proportional reasoning in pre-service teachers
title_full_unstemmed Understanding of inverse proportional reasoning in pre-service teachers
title_sort Understanding of inverse proportional reasoning in pre-service teachers
dc.creator.fl_str_mv Cabero, Ismael
Santágueda-Villanueva, María
Villalobos-Antúnez, Jose Vicente
Roig, Ana Isabel
dc.contributor.author.spa.fl_str_mv Cabero, Ismael
Santágueda-Villanueva, María
Villalobos-Antúnez, Jose Vicente
Roig, Ana Isabel
dc.subject.spa.fl_str_mv Inverse proportionality
Problem-solving strategies
Intensive quantity
Extensive quantity
Proportional reasoning
Tabular and graphic representation
Pre-service teachers
topic Inverse proportionality
Problem-solving strategies
Intensive quantity
Extensive quantity
Proportional reasoning
Tabular and graphic representation
Pre-service teachers
description From an early age, understanding proportional reasoning is a fundamental pillar in mathematics education, and therefore, teachers should have a thorough knowledge of it. Despite its significance, there are few studies that analyse the difficulties that student teachers have in understanding proportionality, and even less so inverse proportionality. We emphasised inverse missing-value problems by analysing them according to the type of unknown and the representation used. We checked which strategies they use to solve them and related them to other generic problems of proportional reasoning. For such purposes, we used a combined quantitative and qualitative empirical study applied to how pre-service teachers solve fifteen problems. The results show that the representations used in the statements aid their understanding and help solve the problems. Similarly, it is shown here that certain problem-solving strategies complicate proportional reasoning in pre-service teachers.
publishDate 2020
dc.date.accessioned.none.fl_str_mv 2020-11-09T19:28:28Z
dc.date.available.none.fl_str_mv 2020-11-09T19:28:28Z
dc.date.issued.none.fl_str_mv 2020-09-29
dc.type.spa.fl_str_mv Artículo de revista
dc.type.coar.fl_str_mv http://purl.org/coar/resource_type/c_2df8fbb1
dc.type.coar.spa.fl_str_mv http://purl.org/coar/resource_type/c_6501
dc.type.content.spa.fl_str_mv Text
dc.type.driver.spa.fl_str_mv info:eu-repo/semantics/article
dc.type.redcol.spa.fl_str_mv http://purl.org/redcol/resource_type/ART
dc.type.version.spa.fl_str_mv info:eu-repo/semantics/acceptedVersion
format http://purl.org/coar/resource_type/c_6501
status_str acceptedVersion
dc.identifier.issn.spa.fl_str_mv 2227-7102
dc.identifier.uri.spa.fl_str_mv https://hdl.handle.net/11323/7226
dc.identifier.doi.spa.fl_str_mv doi:10.3390/educsci10110308
dc.identifier.instname.spa.fl_str_mv Corporación Universidad de la Costa
dc.identifier.reponame.spa.fl_str_mv REDICUC - Repositorio CUC
dc.identifier.repourl.spa.fl_str_mv https://repositorio.cuc.edu.co/
identifier_str_mv 2227-7102
doi:10.3390/educsci10110308
Corporación Universidad de la Costa
REDICUC - Repositorio CUC
url https://hdl.handle.net/11323/7226
https://repositorio.cuc.edu.co/
dc.language.iso.none.fl_str_mv eng
language eng
dc.relation.references.spa.fl_str_mv 1. Lesh, R.; Post, T.R.; Behr, M. Proportional reasoning. In Number Concepts and Operations in the Middle Grades; National Council of Teachers of Mathematics, Lawrence Erlbaum Associates: Reston, VA, USA, 1988; pp. 93–118.
2. Ben-Chaim, D.; Keret, Y.; Ilany, B.S. Ratio and Proportion; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2012.
3. National Council of Teachers of Mathematics; Commission on Standards for School Mathematics. Curriculum and Evaluation Standards for School Mathematics; National Council of Teachers of Mathematics: Reston, VA, USA, 1989.
4. Riley, K. Teachers understanding of proportional reasoning. In Proceedings of the 32nd Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, , Columbus, OH, USA, 28–31 October 2010, Volume 6, pp. 1055–1061.
5. Arican, M. Preservice middle and high school mathematics teachers’ strategies when solving proportion problems. Int. J. Sci. Math. Educ. 2018, 16, 315–335. [CrossRef]
6. Lamon, S.J. Rational numbers and proportional reasoning: Toward a theoretical framework for research. Second. Handb. Res. Math. Teach. Learn. 2007, 1, 629–667.
7. Izsák, A.; Jacobson, E. Understanding Teachers’ Inferences of Proportionality Between Quantities that Form a Constant Difference or Constant Product; National Council of Teachers of Mathematics Research Presession: Denver, CO, USA, 2013.
8. Akar, G. Different levels of reasoning in within state ratio conception and the conceptualization of rate: A possible example. In Proceedings of the 32nd Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, , Columbus, OH, USA, 28–31 October 2010, Volume 4, pp. 711–719.
9. Simon, M.A.; Blume, G.W. Mathematical modeling as a component of understanding ratio-as-measure: A study of prospective elementary teachers. J. Math. Behav. 1994, 13, 183–197. [CrossRef]
10. Harel, G.; Behr, M. Teachers’ Solutions for Multiplicative Problems. Hiroshima J. Math. Educ. 1995, 3, 31–51.
11. Orrill, C.H.; Izsák, A.; Cohen, A.; Templin, J.; Lobato, J. Preliminary Observations of Teachers’ Multiplicative Reasoning: Insights from Does It Work and Diagnosing Teachers’ Multiplicative Reasoning Projects; Kaput Center for Research and Innovation in STEM Education, University of Massachusetts: Dartmouth, MA, USA, 2010.
12. Post, T.R.; Harel, G.; Behr, M.; Lesh, R. Intermediate Teachers’ Knowledge of Rational Number Concepts. In Integrating Research on Teaching and Learning Mathematics; State University of NY Press: New York, NY, USA, 1991; pp. 177–198.
13. Cramer, K.A.; Post, T.; Currier, S. Learning and teaching ratio and proportion: Research implications: Middle grades mathematics. In Research Ideas for the Classroom: Middle Grades Mathematics; Macmillan Publishing Company: New York, NY, USA, 1993; pp. 159–178.
14. Lim, K.H. Burning the candle at just one end. Math. Teach. Middle Sch. 2009, 14, 492–500.
15. Arican, M. Exploring Preservice Middle and High School Mathematics Teachers’ Understanding of Directly and Inversely Proportional Relationships. Ph.D. Thesis, University of Georgia, Athens, GA, USA, 2015.
16. Fisher, L.C. Strategies used by secondary mathematics teachers to solve proportion problems. J. Res. Math. Educ. 1988, 19 , 157–168. [CrossRef]
17. Lobato, J.; Ellis, A.; Zbiek, R.M. Developing Essential Understanding of Ratios, Proportions, and Proportional Reasoning for Teaching Mathematics: Grades 6–8; ERIC: Washington, DC, USA, 2010.
18. Sowder, J.; Armstrong, B.; Lamon, S.; Simon, M.; Sowder, L.; Thompson, A. Educating teachers to teach multiplicative structures in the middle grades. J. Math. Teach. Educ. 1998, 1, 127–155. [CrossRef]
19. Stemn, B.S. Building middle school students’ understanding of proportional reasoning through mathematical investigation. Education 3–13 2008, 36, 383–392. [CrossRef]
20. Becerra, M.V.; Pancorbo, L.; Martínez, R.; Rodríguez, R. Matemáticas 2◦ ESO; McGraw-Hill: Madrid , Spain 1997.
21. Sallán, J.M.G.; Vizcarra, R.E. Proporcionalidad aritmética: Buscando alternativas a la enseñanza tradicional. Suma Rev. Sobre Ense NAnza Aprendiz. Las MatemÁTicas 2009, 62, 35–48.
22. Initiative, C.C.S.S. Common Core State Standards for Mathematics. 2010. Available online: http://www. corestandards.org/assets/CCSSI_Math%20Standards.pdf (accessed on 25 July 2020).
23. Beckmann, S. Mathematics for Elementary Teachers, 3rd ed.; Pearson Addison-Wesley: Boston, MA, USA, 2011.
24. NCTM. Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics; NCTM: Reston, VA, USA, 2006.
25. Lamon, S.J. Teaching Fractions and Ratios for Understanding: Essential Content Knowledge and Instructional Strategies for Teachers; Routledge: London, UK, 2020.
26. Vergnaud, G. Multiplicative Structures; Lesh, R., Landau, M., Eds.; Academic Press: Cambridge, MA, USA, 1983; pp. 127–174.
27. Vergnaud, G. Multiplicative Structures; Hiebert, J., Behr, M., Eds.; National Council of Teachers of Mathematics: Reston, VA, USA, 1988; pp. 141–161.
28. Beckmann, S.; Izsák, A. Two perspectives on proportional relationships: Extending complementary origins of multiplication in terms of quantities. J. Res. Math. Educ. 2015, 46, 17–38. [CrossRef]
29. Arican, M. Preservice mathematics teachers’ understanding of and abilities to differentiate proportional relationships from nonproportional relationships. Int. J. Sci. Math. Educ. 2019, 17, 1423–1443. [CrossRef]
30. Cabero, I.; Epifanio, I. Finding Archetypal Patterns for Binary Questionnaires; SORT: Barcelona, Spain 2020.
31. Sinn, R.; Spence, D.; Poitevint, M. Rates Teaching Research Project; San Diego, CA, USA. 2010. Available online: http://faculty.ung.edu/djspence/Presentations/PBRateProblems/PatternBlockRatesWorksheet.pdf (accessed on 29 October 2020).
32. Canada, D.; Gilbert, M.; Adolphson, K. Conceptions and misconceptions of elementary preservice teachers in proportional reasoning. In Proceedings of the 32th conference of the International Group for the Psychology of Mathematics Education, Columbus, OH, USA, 28–31 October 2010; Figueras, O., Cortina, J., Alatorre, S., Sepulveda, T.R.A., Eds.; Volume 2, pp. 249–256.
33. Cox, D.C. Similarity in middle school mathematics: At the crossroads of geometry and number. Math. Think. Learn. 2013, 15, 3–23. [CrossRef]
34. Tourniaire, F.; Pulos, S. Proportional reasoning: A review of the literature. Educ. Stud. Math. 1985, 16, 181–204. [CrossRef]
35. Karplus, R. Proportional reasoning of early adolescents. In Acquisition of Mathematics Concepts and Processes; Academic Press: London, UK, 1983; pp. 45–90.
36. Lamon, S.J. Ratio and proportion: Connecting content and children’s thinking. J. Res. Math. Educ. 1993, 24, 41–61. [CrossRef]
37. Schwartz, J.L. Intensive quantity and referent transforming arithmetic operations. Res. Agenda Math. Educ. Number Concepts Oper. Middle Grades 1988, 2, 41–52.
38. Kaput, J.; West, M.M. Missing-Value Proportional Problems: Factors Affecting Informal Reasoning Patterns. In The Development of Multiplicative Reasoning in the Learning of Mathematics; SUNY Press: Albany, NY, USA, 1994; pp. 235–287
39. Thompson, P. A Theoretical Model of Quantity-Based Reasoning in Arithmetic and Algebra; Center for Research in Mathematics & Science Education, San Diego State University: San Diego, CA, USA, 1990.
40. Weiland, T.; Orrill, C.H.; Nagar, G.G.; Brown, R.E.; Burke, J. Framing a robust understanding of proportional reasoning for teachers. J. Math. Teach. Educ. 2020, 1–24. [CrossRef]
41. Lamon, S. Second handbook of research on mathematics teaching and learning. Inf. Age Publ. Ration. Proportional Reason. Towar. Theor. Framew. Res. 2007, 1, 629–668.
42. Burgos, M.; Beltrán-Pellicer, P.; Giacomone, B.; Godino, J.D. Conocimientos y competencia de futuros profesores de matemáticas en tareas de proporcionalidad. Educação e Pesqui. 2018, 44. [CrossRef]
43. Martínez Juste, S.; Muñoz Escolano, J.M.; Oller Marcén, A.M.; Ortega del Rincón, T. Análisis de problemas de proporcionalidad compuesta en libros de texto de 2o de ESO. Rev. Latinoam. Investig. MatemÁTica Educ. 2017, 20, 95–122. [CrossRef]
44. López, M.J.G.; Guzmán, P.G. Magnitudes y medida: Medidas directas. In Matemáticas Para Maestros de Educación Primaria; Pirámide, Madrid, Spain, 2011; pp. 351–374.
45. Valverde, G. Competencias Matemáticas Promovidas Desde la Razón y la Proporcionalidad en la Formación Inicial de Maestros de Educación Primaria; Universidad de Granada: Granada, Spain, 2013.
46. Kelle, U.; Buchholtz, N. The combination of qualitative and quantitative research methods in mathematics education: A “mixed methods” study on the development of the professional knowledge of teachers. In Approaches to Qualitative Research in Mathematics Education; Springer: Berlin/Heidelberg, Germany, 2015; pp. 321–361.
47. R Core Team. R: A Language and Environment for Statistical Computing; R Foundation for Statistical Computing: Vienna, Austria, 2020.
48. Clement, J. Algebra word problem solutions: Thought processes underlying a common misconception. J. Res. Math. Educ. 1982, 13, 16–30. [CrossRef]
49. Karplus, R.; Adi, H.; Lawson, A.E. Intellectual development beyond elementary school VIII: proportional, probabilistic, and correlational reasoning. Sch. Sci. Math. 1980, 80, 673–83. [CrossRef]
50. Rupley, W.H. The effects of numerical characteristics on the difficulty of proportion problems. Diss. Abstr. Int. 1982, 254–263.
51. Buforn, Á.; Llinares, S.; Fernández, C. Características del conocimiento de los estudiantes para maestro españoles en relación con la fracción, razón y proporción. Rev. Mex. Investig. Educ. 2018, 23, 229–251.
52. Burgos, M.; Godino, J.D. Prospective primary school teachers’ competence for analysing the difficulties in solving proportionality problem. Math. Educ. Res. J. 2020, 1–23. [CrossRef]
53. Son, J.W. How preservice teachers interpret and respond to student errors: Ratio and proportion in similar rectangles. Educ. Stud. Math. 2013, 84, 49–70. [CrossRef]
54. Sztajn, P.; Confrey, J.; Wilson, P.H.; Edgington, C. Learning trajectory based instruction: Toward a theory of teaching. Educ. Res. 2012, 41, 147–156. [CrossRef]
55. Weiland, T.; Orrill, C.H.; Brown, R.E.; Nagar, G.G. Mathematics teachers’ ability to identify situations appropriate for proportional reasoning. Res. Math. Educ. 2019, 1–18. [CrossRef]
dc.rights.spa.fl_str_mv CC0 1.0 Universal
dc.rights.uri.spa.fl_str_mv http://creativecommons.org/publicdomain/zero/1.0/
dc.rights.accessrights.spa.fl_str_mv info:eu-repo/semantics/openAccess
dc.rights.coar.spa.fl_str_mv http://purl.org/coar/access_right/c_abf2
rights_invalid_str_mv CC0 1.0 Universal
http://creativecommons.org/publicdomain/zero/1.0/
http://purl.org/coar/access_right/c_abf2
eu_rights_str_mv openAccess
dc.format.mimetype.spa.fl_str_mv application/pdf
dc.publisher.spa.fl_str_mv Corporación Universidad de la Costa
dc.source.spa.fl_str_mv Education Sciences
institution Corporación Universidad de la Costa
dc.source.url.spa.fl_str_mv https://www.mdpi.com/2227-7102/10/11/308/xml
bitstream.url.fl_str_mv https://repositorio.cuc.edu.co/bitstreams/4889c582-e7e7-49ff-946c-ffe751d7c782/download
https://repositorio.cuc.edu.co/bitstreams/0b55420d-612d-48ec-9ef4-ea73228f0c37/download
https://repositorio.cuc.edu.co/bitstreams/845dc473-817a-40f1-9ba9-e75f291941e6/download
https://repositorio.cuc.edu.co/bitstreams/3bbe473a-606a-4959-926f-34019b25f57c/download
https://repositorio.cuc.edu.co/bitstreams/4d0b1b27-f55b-4349-a841-486f3adee2fc/download
bitstream.checksum.fl_str_mv a0f1167261d6cd126c4081106d04200b
42fd4ad1e89814f5e4a476b409eb708c
e30e9215131d99561d40d6b0abbe9bad
1382797296831fe6f7a547654fbb420f
f5c53bd97dabb0001f8fb4088d431c07
bitstream.checksumAlgorithm.fl_str_mv MD5
MD5
MD5
MD5
MD5
repository.name.fl_str_mv Repositorio de la Universidad de la Costa CUC
repository.mail.fl_str_mv repdigital@cuc.edu.co
_version_ 1811760786496815104
spelling Cabero, IsmaelSantágueda-Villanueva, MaríaVillalobos-Antúnez, Jose VicenteRoig, Ana Isabel2020-11-09T19:28:28Z2020-11-09T19:28:28Z2020-09-292227-7102https://hdl.handle.net/11323/7226doi:10.3390/educsci10110308Corporación Universidad de la CostaREDICUC - Repositorio CUChttps://repositorio.cuc.edu.co/From an early age, understanding proportional reasoning is a fundamental pillar in mathematics education, and therefore, teachers should have a thorough knowledge of it. Despite its significance, there are few studies that analyse the difficulties that student teachers have in understanding proportionality, and even less so inverse proportionality. We emphasised inverse missing-value problems by analysing them according to the type of unknown and the representation used. We checked which strategies they use to solve them and related them to other generic problems of proportional reasoning. For such purposes, we used a combined quantitative and qualitative empirical study applied to how pre-service teachers solve fifteen problems. The results show that the representations used in the statements aid their understanding and help solve the problems. Similarly, it is shown here that certain problem-solving strategies complicate proportional reasoning in pre-service teachers.Cabero, Ismael-will be generated-orcid-0000-0003-1839-7205-600Santágueda-Villanueva, María-will be generated-orcid-0000-0002-5472-7972-600Villalobos-Antúnez, Jose VicenteRoig, Ana Isabel-will be generated-orcid-0000-0003-4522-8327-600application/pdfengCorporación Universidad de la CostaCC0 1.0 Universalhttp://creativecommons.org/publicdomain/zero/1.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2Education Scienceshttps://www.mdpi.com/2227-7102/10/11/308/xmlInverse proportionalityProblem-solving strategiesIntensive quantityExtensive quantityProportional reasoningTabular and graphic representationPre-service teachersUnderstanding of inverse proportional reasoning in pre-service teachersArtículo de revistahttp://purl.org/coar/resource_type/c_6501http://purl.org/coar/resource_type/c_2df8fbb1Textinfo:eu-repo/semantics/articlehttp://purl.org/redcol/resource_type/ARTinfo:eu-repo/semantics/acceptedVersion1. Lesh, R.; Post, T.R.; Behr, M. Proportional reasoning. In Number Concepts and Operations in the Middle Grades; National Council of Teachers of Mathematics, Lawrence Erlbaum Associates: Reston, VA, USA, 1988; pp. 93–118.2. Ben-Chaim, D.; Keret, Y.; Ilany, B.S. Ratio and Proportion; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2012.3. National Council of Teachers of Mathematics; Commission on Standards for School Mathematics. Curriculum and Evaluation Standards for School Mathematics; National Council of Teachers of Mathematics: Reston, VA, USA, 1989.4. Riley, K. Teachers understanding of proportional reasoning. In Proceedings of the 32nd Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, , Columbus, OH, USA, 28–31 October 2010, Volume 6, pp. 1055–1061.5. Arican, M. Preservice middle and high school mathematics teachers’ strategies when solving proportion problems. Int. J. Sci. Math. Educ. 2018, 16, 315–335. [CrossRef]6. Lamon, S.J. Rational numbers and proportional reasoning: Toward a theoretical framework for research. Second. Handb. Res. Math. Teach. Learn. 2007, 1, 629–667.7. Izsák, A.; Jacobson, E. Understanding Teachers’ Inferences of Proportionality Between Quantities that Form a Constant Difference or Constant Product; National Council of Teachers of Mathematics Research Presession: Denver, CO, USA, 2013.8. Akar, G. Different levels of reasoning in within state ratio conception and the conceptualization of rate: A possible example. In Proceedings of the 32nd Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, , Columbus, OH, USA, 28–31 October 2010, Volume 4, pp. 711–719.9. Simon, M.A.; Blume, G.W. Mathematical modeling as a component of understanding ratio-as-measure: A study of prospective elementary teachers. J. Math. Behav. 1994, 13, 183–197. [CrossRef]10. Harel, G.; Behr, M. Teachers’ Solutions for Multiplicative Problems. Hiroshima J. Math. Educ. 1995, 3, 31–51.11. Orrill, C.H.; Izsák, A.; Cohen, A.; Templin, J.; Lobato, J. Preliminary Observations of Teachers’ Multiplicative Reasoning: Insights from Does It Work and Diagnosing Teachers’ Multiplicative Reasoning Projects; Kaput Center for Research and Innovation in STEM Education, University of Massachusetts: Dartmouth, MA, USA, 2010.12. Post, T.R.; Harel, G.; Behr, M.; Lesh, R. Intermediate Teachers’ Knowledge of Rational Number Concepts. In Integrating Research on Teaching and Learning Mathematics; State University of NY Press: New York, NY, USA, 1991; pp. 177–198.13. Cramer, K.A.; Post, T.; Currier, S. Learning and teaching ratio and proportion: Research implications: Middle grades mathematics. In Research Ideas for the Classroom: Middle Grades Mathematics; Macmillan Publishing Company: New York, NY, USA, 1993; pp. 159–178.14. Lim, K.H. Burning the candle at just one end. Math. Teach. Middle Sch. 2009, 14, 492–500.15. Arican, M. Exploring Preservice Middle and High School Mathematics Teachers’ Understanding of Directly and Inversely Proportional Relationships. Ph.D. Thesis, University of Georgia, Athens, GA, USA, 2015.16. Fisher, L.C. Strategies used by secondary mathematics teachers to solve proportion problems. J. Res. Math. Educ. 1988, 19 , 157–168. [CrossRef]17. Lobato, J.; Ellis, A.; Zbiek, R.M. Developing Essential Understanding of Ratios, Proportions, and Proportional Reasoning for Teaching Mathematics: Grades 6–8; ERIC: Washington, DC, USA, 2010.18. Sowder, J.; Armstrong, B.; Lamon, S.; Simon, M.; Sowder, L.; Thompson, A. Educating teachers to teach multiplicative structures in the middle grades. J. Math. Teach. Educ. 1998, 1, 127–155. [CrossRef]19. Stemn, B.S. Building middle school students’ understanding of proportional reasoning through mathematical investigation. Education 3–13 2008, 36, 383–392. [CrossRef]20. Becerra, M.V.; Pancorbo, L.; Martínez, R.; Rodríguez, R. Matemáticas 2◦ ESO; McGraw-Hill: Madrid , Spain 1997.21. Sallán, J.M.G.; Vizcarra, R.E. Proporcionalidad aritmética: Buscando alternativas a la enseñanza tradicional. Suma Rev. Sobre Ense NAnza Aprendiz. Las MatemÁTicas 2009, 62, 35–48.22. Initiative, C.C.S.S. Common Core State Standards for Mathematics. 2010. Available online: http://www. corestandards.org/assets/CCSSI_Math%20Standards.pdf (accessed on 25 July 2020).23. Beckmann, S. Mathematics for Elementary Teachers, 3rd ed.; Pearson Addison-Wesley: Boston, MA, USA, 2011.24. NCTM. Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics; NCTM: Reston, VA, USA, 2006.25. Lamon, S.J. Teaching Fractions and Ratios for Understanding: Essential Content Knowledge and Instructional Strategies for Teachers; Routledge: London, UK, 2020.26. Vergnaud, G. Multiplicative Structures; Lesh, R., Landau, M., Eds.; Academic Press: Cambridge, MA, USA, 1983; pp. 127–174.27. Vergnaud, G. Multiplicative Structures; Hiebert, J., Behr, M., Eds.; National Council of Teachers of Mathematics: Reston, VA, USA, 1988; pp. 141–161.28. Beckmann, S.; Izsák, A. Two perspectives on proportional relationships: Extending complementary origins of multiplication in terms of quantities. J. Res. Math. Educ. 2015, 46, 17–38. [CrossRef]29. Arican, M. Preservice mathematics teachers’ understanding of and abilities to differentiate proportional relationships from nonproportional relationships. Int. J. Sci. Math. Educ. 2019, 17, 1423–1443. [CrossRef]30. Cabero, I.; Epifanio, I. Finding Archetypal Patterns for Binary Questionnaires; SORT: Barcelona, Spain 2020.31. Sinn, R.; Spence, D.; Poitevint, M. Rates Teaching Research Project; San Diego, CA, USA. 2010. Available online: http://faculty.ung.edu/djspence/Presentations/PBRateProblems/PatternBlockRatesWorksheet.pdf (accessed on 29 October 2020).32. Canada, D.; Gilbert, M.; Adolphson, K. Conceptions and misconceptions of elementary preservice teachers in proportional reasoning. In Proceedings of the 32th conference of the International Group for the Psychology of Mathematics Education, Columbus, OH, USA, 28–31 October 2010; Figueras, O., Cortina, J., Alatorre, S., Sepulveda, T.R.A., Eds.; Volume 2, pp. 249–256.33. Cox, D.C. Similarity in middle school mathematics: At the crossroads of geometry and number. Math. Think. Learn. 2013, 15, 3–23. [CrossRef]34. Tourniaire, F.; Pulos, S. Proportional reasoning: A review of the literature. Educ. Stud. Math. 1985, 16, 181–204. [CrossRef]35. Karplus, R. Proportional reasoning of early adolescents. In Acquisition of Mathematics Concepts and Processes; Academic Press: London, UK, 1983; pp. 45–90.36. Lamon, S.J. Ratio and proportion: Connecting content and children’s thinking. J. Res. Math. Educ. 1993, 24, 41–61. [CrossRef]37. Schwartz, J.L. Intensive quantity and referent transforming arithmetic operations. Res. Agenda Math. Educ. Number Concepts Oper. Middle Grades 1988, 2, 41–52.38. Kaput, J.; West, M.M. Missing-Value Proportional Problems: Factors Affecting Informal Reasoning Patterns. In The Development of Multiplicative Reasoning in the Learning of Mathematics; SUNY Press: Albany, NY, USA, 1994; pp. 235–28739. Thompson, P. A Theoretical Model of Quantity-Based Reasoning in Arithmetic and Algebra; Center for Research in Mathematics & Science Education, San Diego State University: San Diego, CA, USA, 1990.40. Weiland, T.; Orrill, C.H.; Nagar, G.G.; Brown, R.E.; Burke, J. Framing a robust understanding of proportional reasoning for teachers. J. Math. Teach. Educ. 2020, 1–24. [CrossRef]41. Lamon, S. Second handbook of research on mathematics teaching and learning. Inf. Age Publ. Ration. Proportional Reason. Towar. Theor. Framew. Res. 2007, 1, 629–668.42. Burgos, M.; Beltrán-Pellicer, P.; Giacomone, B.; Godino, J.D. Conocimientos y competencia de futuros profesores de matemáticas en tareas de proporcionalidad. Educação e Pesqui. 2018, 44. [CrossRef]43. Martínez Juste, S.; Muñoz Escolano, J.M.; Oller Marcén, A.M.; Ortega del Rincón, T. Análisis de problemas de proporcionalidad compuesta en libros de texto de 2o de ESO. Rev. Latinoam. Investig. MatemÁTica Educ. 2017, 20, 95–122. [CrossRef]44. López, M.J.G.; Guzmán, P.G. Magnitudes y medida: Medidas directas. In Matemáticas Para Maestros de Educación Primaria; Pirámide, Madrid, Spain, 2011; pp. 351–374.45. Valverde, G. Competencias Matemáticas Promovidas Desde la Razón y la Proporcionalidad en la Formación Inicial de Maestros de Educación Primaria; Universidad de Granada: Granada, Spain, 2013.46. Kelle, U.; Buchholtz, N. The combination of qualitative and quantitative research methods in mathematics education: A “mixed methods” study on the development of the professional knowledge of teachers. In Approaches to Qualitative Research in Mathematics Education; Springer: Berlin/Heidelberg, Germany, 2015; pp. 321–361.47. R Core Team. R: A Language and Environment for Statistical Computing; R Foundation for Statistical Computing: Vienna, Austria, 2020.48. Clement, J. Algebra word problem solutions: Thought processes underlying a common misconception. J. Res. Math. Educ. 1982, 13, 16–30. [CrossRef]49. Karplus, R.; Adi, H.; Lawson, A.E. Intellectual development beyond elementary school VIII: proportional, probabilistic, and correlational reasoning. Sch. Sci. Math. 1980, 80, 673–83. [CrossRef]50. Rupley, W.H. The effects of numerical characteristics on the difficulty of proportion problems. Diss. Abstr. Int. 1982, 254–263.51. Buforn, Á.; Llinares, S.; Fernández, C. Características del conocimiento de los estudiantes para maestro españoles en relación con la fracción, razón y proporción. Rev. Mex. Investig. Educ. 2018, 23, 229–251.52. Burgos, M.; Godino, J.D. Prospective primary school teachers’ competence for analysing the difficulties in solving proportionality problem. Math. Educ. Res. J. 2020, 1–23. [CrossRef]53. Son, J.W. How preservice teachers interpret and respond to student errors: Ratio and proportion in similar rectangles. Educ. Stud. Math. 2013, 84, 49–70. [CrossRef]54. Sztajn, P.; Confrey, J.; Wilson, P.H.; Edgington, C. Learning trajectory based instruction: Toward a theory of teaching. Educ. Res. 2012, 41, 147–156. [CrossRef]55. Weiland, T.; Orrill, C.H.; Brown, R.E.; Nagar, G.G. Mathematics teachers’ ability to identify situations appropriate for proportional reasoning. Res. Math. Educ. 2019, 1–18. [CrossRef]PublicationORIGINALUnderstanding of Inverse Proportional Reasoning in Pre-Service Teachers..pdfUnderstanding of Inverse Proportional Reasoning in Pre-Service Teachers..pdfapplication/pdf285452https://repositorio.cuc.edu.co/bitstreams/4889c582-e7e7-49ff-946c-ffe751d7c782/downloada0f1167261d6cd126c4081106d04200bMD51CC-LICENSElicense_rdflicense_rdfapplication/rdf+xml; charset=utf-8701https://repositorio.cuc.edu.co/bitstreams/0b55420d-612d-48ec-9ef4-ea73228f0c37/download42fd4ad1e89814f5e4a476b409eb708cMD52LICENSElicense.txtlicense.txttext/plain; charset=utf-83196https://repositorio.cuc.edu.co/bitstreams/845dc473-817a-40f1-9ba9-e75f291941e6/downloade30e9215131d99561d40d6b0abbe9badMD53THUMBNAILUnderstanding of Inverse Proportional Reasoning in Pre-Service Teachers..pdf.jpgUnderstanding of Inverse Proportional Reasoning in Pre-Service Teachers..pdf.jpgimage/jpeg67350https://repositorio.cuc.edu.co/bitstreams/3bbe473a-606a-4959-926f-34019b25f57c/download1382797296831fe6f7a547654fbb420fMD54TEXTUnderstanding of Inverse Proportional Reasoning in Pre-Service Teachers..pdf.txtUnderstanding of Inverse Proportional Reasoning in Pre-Service Teachers..pdf.txttext/plain70563https://repositorio.cuc.edu.co/bitstreams/4d0b1b27-f55b-4349-a841-486f3adee2fc/downloadf5c53bd97dabb0001f8fb4088d431c07MD5511323/7226oai:repositorio.cuc.edu.co:11323/72262024-09-17 11:09:08.168http://creativecommons.org/publicdomain/zero/1.0/CC0 1.0 Universalopen.accesshttps://repositorio.cuc.edu.coRepositorio de la Universidad de la Costa CUCrepdigital@cuc.edu.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

Publicaciones similares