Assessig prospective primary school teachers didacticmathematical kowledge o elemetary algebraic reasoig 1 Evaluació de coocimietos didáctico-matemáticos sobre razoamieto algebraico elemetal de futuros maestros DOI: 10.4438/1988-592X-RE-2015-370-303 Jua D. Godio Miguel R. Wilhelmi Teresa Neto Teresa F. Blaco Ágel Cotreras Carme Díaz-Bataero Atoio Estepa Aitzol Lasa Uiversidad de Graada Abstract Differet studies suggest the eed to itroduce elemetary algebraic reasoig at a early age, which i curret curricular proposals is oly taught i Secodary Educatio. I this paper, the resposes to a questioaire evaluatig prospective primary school teachers didactic-mathematical kowledge regardig elemetary algebraic reasoig are aalysed. The aim is to provide a diagosis of prospective primary school teachers kowledge regardig elemetary algebra ad its teachig. The results will be used to develop a traiig programme for teachers, (1) The research reported i this article was carried out as part of the followig projects: EDU2012-31869 ad EDU2013-41141-P, Miistry of Ecoomy ad Competitiveess (MINECO, Spai). It has bee also sposored by FCT/MEC through atioal fuds (PIDDAC) ad co-sposored by FEDER through COMPETE - Programa Operacioal Fatores de Competitividade withi the PEst-C/CED/UI0194/2013 project. 188 Revista de Educació, 370. October-December 2015, pp. 188-215
which ultimately esures a effective study process i primary educatio. The sample cosisted of 597 studets from the Uiversities of Graada, Jaé, Public Uiversity of Navarra, Satiago de Compostela i Spai, ad the Uiversity of Aveiro i Portugal. The questioaire comprises 25 items that assess algebraic ad pedagogical cotet kowledge i primary educatio. Takig ito accout that i these uiversities the groups are ad extra homogeeous ad ad itro heterogeeous, etire groups were itetioally selected at each uiversity. The quatitative aalysis of the results eabled the estimatio of the istrumet psychometric properties (difficulty ad discrimiatio idexes, reliability ad validity). The compariso of the traiig programmes i mathematics ad the way i which the subject is taught i the differet uiversities reveals that curret curriculum emphasizes o the psycho-pedagogical area. It also reveals a usatisfactory discipliary traiig that does ot iclude algebraic reasoig. Results show sigificat differeces amog uiversities i the participats level of kowledge i the various didactic - mathematical kowledge compoets. We coclude that it is ecessary to improve the teachers educatio programmes so they ca progressively foster their pupils algebraic reasoig. To achieve this goal these programmes should iclude specific traiig activities o elemetary algebraic reasoig. Keywords: teacher traiig, evaluatio, algebraic kowledge, didactical kowledge, comparative study Resume Diferetes estudios sugiere la ecesidad de icorporar desde edades tempraas el razoamieto algebraico elemetal, que e las propuestas curriculares clásicas se iicia e la Educació Secudaria Obligatoria. E este trabajo se aaliza los resultados de aplicar u cuestioario de evaluació de coocimietos didáctico - matemáticos sobre razoamieto algebraico elemetal a ua muestra de estudiates del Grado e Maestro e Educació Primaria. El objetivo es la elaboració de u diagóstico sobre la competecia algebraica elemetal y su didáctica de los futuros maestros, que permita emarcar u programa formativo para estos, que garatice fialmete procesos de estudio efectivos e la educació primaria. La muestra está compuesta por 597 estudiates de las Uiversidades de Graada, Jaé, Pública de Navarra, Satiago de Compostela e España y de Aveiro e Portugal. E estas uiversidades, los grupos se costituye homogéeos ad extra y heterogéeos ad itro. Se propoe pues la selecció itecioal de grupos completos e cada uiversidad. El cuestioario costa de 25 ítems que evalúa tato coocimietos algebraicos como coocimietos sobre la eseñaza y apredizaje del álgebra e Educació Primaria. El aálisis cuatitativo de los resultados ha permitido explorar las características psicométricas del istrumeto (ídices de dificultad, discrimiació, fiabilidad y validez). La comparació de los programas de formació e matemáticas y su didáctica etre las distitas uiversidades participates revela el éfasis psicopedagógico del Pla de estudios vigete y Revista de Educació, 370. October-December 2015, pp. 188-215 189
muestra ua formació discipliar deficiete, que, e particular, o icluye el bloque de razoamieto algebraico. Los resultados muestra u bajo ivel de coocimietos geeralizado e las distitas compoetes del coocimieto didáctico - matemático, co diferecias sigificativas etre las uiversidades. Se cocluye que es ecesario revisar los programas de formació y plaificar el diseño de accioes formativas específicas sobre los coteidos algebraicos elemetales, a fi de capacitar a los futuros maestros para que pueda promover e los alumos de primaria el progresivo desarrollo del pesamieto algebraico. Palabras clave: formació de maestros, evaluació, coocimieto algebraico, coocimieto didáctico, estudio comparado. Itroductio Teacher traiig i mathematics ad didactics is essetial for the improvemet of both teachig ad learig. Therefore, it is a field of research ad developmet that has bee icluded i mathematics educatio. I Spai, this traiig occurs i the Faculty of Educatio ad it is affected by a chagig regulatory eviromet, as well as by the lack of huma, techological ad time resources. However, the implemetatio of a Teacher s Degree i Primary Educatio (TDPE) i compliace with the framework of the Europea Higher Educatio Area (EHEA) ad the developmet of research i the departmets of specific didactics are opeig ew possibilities for improvemet ad developmet of teacher traiig programmes. This paper presets the results of a research project o Evaluatio ad developmet of prospective primary school teachers didacticmathematical kowledge o elemetary algebraic reasoig. The objective is to cotribute with ew kowledge, teachig resources ad methodological proposals for the improvemet of iitial traiig i mathematics, ad its teachig at the level of Primary Educatio, takig ito accout the Spaish educatioal cotext as well as the ew curricula. This paper is closely related to the TEDS-M 2 study (Teacher Educatio ad Developmet Study i Mathematics) (Tatto et al., 2008; INEE, 2012; (2) TEDS-M was the first large-scale iteratioal comparative study i higher educatio, focusig o iitial traiig of primary ad early secodary educatio mathematics teachers. The TEDS-M data were collected i 2008, so the situatio described relates to the curriculum ad coditios at that time. Sice the Spaish uiversities have udergoe sigificat chages i the curriculum for teacher traiig. 190 Revista de Educació, 370. October-December 2015, pp. 188-215
Saz & Martí, 2014). Although, it is limited to a specific aspect of the didactic-mathematical traiig of prospective teachers, «elemetary algebraic reasoig», usig a sample of five istitutios specialised i teacher traiig: four of them Spaish uiversities (Graada, Jaé, Public Uiversity of Navarra, Satiago) ad a Portuguese uiversity (Aveiro). Algebraic reasoig, perceived as mathematical modellig tool, is applicable to differet cotet blocks (arithmetic, geometry, measuremet, data aalysis ad probability). Additioally, geeralisatio ad symbolisatio processes, as well as the cocepts of relatio, variable, equatio ad fuctio are idispesable tools i mathematics, both at a professioal ad at a school level. The Priciples ad Stadards for School Mathematics of the Natioal Coucil of Mathematics Teachers (NCTM, 2000) recommed the itroductio of algebra i Preschool as oe of five cotet blocks: Numbers ad Operatios, Geometry, Measuremet, Data Aalysis, ad Probability. The ew Spaish curricular guidelies (MECD, 2014) ecourage the developmet of algebraic thikig i Primary Educatio; amely with the itet to achieve that all studets, after completig primary educatio, are able to describe ad aalyse situatios of chage, fid patters ad mathematical laws i umerical, geometric ad fuctioal cotexts, evaluatig their usefuless i predictio-makig»(p. 19387). Thus, itroductio to algebra is ow part of the cotiuig uclei, i.e., «the cetral aspects of mathematics teachig ad learig which presumably remai uchaged over time» (Wilhelmi, 2014, 33). Keepig this i mid, if algebra is to be icluded i mathematics at school as a ew trasversal cotet, teachers should embrace this ew broadeed visio ad acquire basic skills i elemetary algebraic reasoig. I previous papers, (Godio, Castro, Aké & Wilhelmi, 2012; Godio, Aké, Gozato & Wilhelmi, 2014) we have preseted prelimiary results regardig this part of the project which focuses o the study of didactical problems i algebra ad its didactics at the level of traiig i the TDPE. I Godio et al. (2015) we preset the desig of a questioaire used to evaluate relevat aspects of prospective teachers didactic-mathematical kowledge (DMK) regardig Elemetary Algebraic Reasoig (EAR). The use of the DMK-EAR questioaire with a sample of prospective teachers provides a isight ito the status of their didactic-mathematical kowledge i elemetary algebraic reasoig, as well as criteria for the developmet of tailored traiig programmes. Revista de Educació, 370. October-December 2015, pp. 188-215 191
This paper describes the mathematical ad didactical kowledge of TDPE studets about EAR ad its relatio with the traiig programmes they are attedig; i.e. there is a approach to the issue of diagosig prospective teachers didactic-mathematical kowledge o EAR based o the existig traiig programmes. Therefore, followig this geeral itroductio, there is a descriptio of the theoretical framework supportig the developmet of the questioaire, as well as the variables take ito accout. Subsequetly, there is a characterisatio of the cotext ad studet sample aswerig the questioaire. The, there is a presetatio of the global ad partial results of the quatitative study, idicatig some psychometric properties of the istrumet ad distiguishig algebraic ad didactical kowledge. Fially, the coclusios ad a discussio of the implicatios of the study are preseted. Theoretical Framework This research is based o two theoretical framework compoets: 1) Nature of algebraic reasoig i primary educatio (Carraher & Schliema, 2007; Cai & Kuth, 2011); ad 2) didactic-mathematical kowledge of the Maths Teacher (Godio, 2009). These compoets support the evaluatio istrumet used i this research. Godio, Ake, Gozato & Wilhelmi (2014) preset a model of EAR structured ito four algebraisatio levels. This model takes ito accout the objects ad processes ivolved i mathematical activity. At level 0 there are o algebraic traits i mathematical activity, whereas level 3 is clearly algebraic. The itermediate levels 1 ad 2, cosidered as progressive algebraisatio levels, showcase some objects ad processes which are algebraic i ature. Regardig the modellig tool adopted for the teacher s didacticmathematical kowledge 3, two types of variable were take ito accout: algebraic cotet ad didactical cotet. I terms of the algebraic cotet three values or categories with differet subcategories were take ito cosideratio: (2) TEDS-M was the first large-scale iteratioal comparative study i higher educatio, focusig o iitial traiig of primary ad early secodary educatio mathematics teachers. The TEDS-M data were collected i 2008, so the situatio described relates to the curriculum ad coditios at that time. Sice the Spaish uiversities have udergoe sigificat chages i the curriculum for teacher traiig. 192 Revista de Educació, 370. October-December 2015, pp. 188-215
Structures (equivalece relatio; properties of operatios, equatios,...) Fuctios (arithmetic patters, geometric patters; liear, affie, quadratic fuctios,...) Modellig tool (cotext problems solved via equatios or fuctio relatioships) Regardig the variable didactical cotet (based o a algebraic cotet, associated with primary school level or higher) the followig categories were cosidered: Epistemic: recogitio of objects ad algebraic processes (represetatios, cocepts, procedures, properties, geeralisatio, modellig); recogitio of algebraisatio levels. Cogitive: persoal meaigs iferred by the studets (kowledge, uderstadig ad competece i elemetary algebraic cotets); learig coflicts at the level of algebraic objects ad processes. Istructioal: Resources for teachig algebra i primary school (situatio-problem, techical resources), as well as its adaptatio to the school curriculum. Method Istrumet The istrumet used for data collectio was the DMK-EAR questioaire (see Appedix) composed of a set of 10 tasks, each of which cosists of items that evaluate differet aspects of algebraic ad didactic algebraic cotet. The descriptio of the questioaire developmet process ad the discussio of its validity ad reliability are available i Godio et al. (2015). Table I shows a classificatio of the differet items of the questioaire accordig to the categories of didactic-mathematical kowledge o EAR. This istrumet will ot evaluate all categories of didactic-mathematical kowledge proposed by the DMK-EAR model, as it is desiged to obtai the iformatio provided by the studets i writig withi a limited period of time (2h approximately). Revista de Educació, 370. October-December 2015, pp. 188-215 193
Oe should take ito accout that the DMK-EAR categories are ot disjoit or exclusive; thus, the same item may appear i more tha oe category. For example, item 10a), «Name a variable of the problem that eables the study of fuctios», ivolves the cotet «fuctios» but also «modellig». Additioally, the studet is asked to state oe problem variat with a specific didactic purpose, leadig prospective teachers to apply kowledge about a «istructioal resource.» Moreover, the item also ivolves specialised kowledge of the mathematical cotet itself, i.e. the epistemic facet i its situatioal ad regulatory compoets (cocept of fuctio ad its represetatios). For this reaso, i table I, some items are icluded i more tha oe cell. TABLE I. Cotets evaluated by each item of the questioaire source: Godio et al. (2015) Depedet variables The first objective is to defie the quatitative variable «degree of accuracy of the aswers give to the 25 items of the questioaire». Partially correct aswers were assessed positively, so the score assiged to each item was: 0 poits, if the aswer is icorrect. 1 poit, if it is partially correct, 2 poits, if it is correct. 194 Revista de Educació, 370. October-December 2015, pp. 188-215
I additio to the variable «total score», i.e. sum of scores obtaied i all 25 items (0-50 poits), quatitative variables are defied regardig «commo ad advaced mathematical kowledge» ad «didactical cotet», represetig the sum by rows or colums of the variables provided i Table I. a) Cocerig commo ad advaced mathematical kowledge A1_ALG: Assess kowledge of «algebra» characteristic i Primary school (commo kowledge) or Secodary school (advaced kowledge). This scale icludes items 3a, 4a, 5a, 6a, 7a, 8a, 9a, ad10b. A2_EST: Evaluates kowledge related to properties of the algebraic structures used i equatio solvig. It icludes items 1a, 1b, 2a, 2b, 3a, 3b, 4a, 4b, 4c, 7a, 7b, 7c, ad 8b. A3_FUN: Icludes kowledge related to geometric patters ad fuctios. It icludes items 5a, 5b, 5c, 6a, 6c, 8a, 8b, 9a, 10a, ad 10b. A4_MOD: Evaluates kowledge related to algebraic modellig (usig equatios or fuctios). It icludes item, 8b, 9a, 9b, 9c, ad 10a. b) Cocerig didactical cotet CD1_EPI: Icorporate kowledge o the epistemic facet of the DMK EAR ad icludes items 2b, 4b, 5c, 6b, 7b, 7c, 9b, ad 10b. CD2_COG: Cotai kowledge o the cogitive facet ad icludes items 1a, 1b, 2a, 3b, 4c, ad 6c. CD3_INS: Assess kowledge o the istructioal facet ad icludes items 5b, 8b, 9c, ad10a. The differet depedet variables comprise a umber of differet items scorig 0, 1 or 2, so its variatio rage is distict. To ease the compariso ad iterpretatio of the scores they were covert to the iterval [0, 10]. To esure reliability of the studet respose codig process, a criteria protocol was developed with illustrative examples of the type of respose. Additioally, ambiguous iterpretatio was discussed withi the research group. Revista de Educació, 370. October-December 2015, pp. 188-215 195
Explaatory variables To iterpret the results of the evaluatio of studet kowledge, the followig explaatory variables were take ito accout: Admissio rate to the Teacher s Degree i Elemetary Educatio (ratio betwee the umber of places available ad the umber of applicatios for admissio) Cut-off score i the admissio exam. Period of time at which the test was doe: 1. it was applied at the begiig of the traiig process; 2. it was applied at the ed of the traiig process. Cotext ad sample descriptio The target populatio of this research are Spaish studets of the Teacher s Degree i Primary Educatio (TDPE) (MECD, 2007). The questioaire has bee used with several groups of studets from the Uiversities of Graada, Jaé, Public Uiversity of Navarra, Satiago de Compostela, attedig the TDPE uder the ew curriculum. The questioaire has also bee used with a sample of studets from the Uiversity of Aveiro, which allows for a comparative study. Both Spaish ad Portuguese curricula have bee reviewed withi the framework of the Europea Higher Educatio Area (EHEA). The umber of credits assiged to Mathematics Educatio i the Spaish curriculum guidelies is 18 (Pla 2010) for the Uiversities of Jaé, Public of Navarra ad Satiago; i Graada the umber is 22. Although the teachig course has goe from 3 to 4 years ad that, therefore, the total umber of credits has icreased from 180 to 240, these studies cotiue to have a strog psychological ad pedagogical compoet. The percetage of credits allocated to traiig i mathematics ad its didactics is 7.5%, which is similar to the 1991-2010 Pla (6.7% accordig to the study Rico, Gomez & Cañadas, 2014). I this study, it was ot possible to iclude a detailed aalysis of the cotets of mathematics ad its didactics offered i the five uiversities, as was doe by Rico et al. (2014). However, we foud that algebra does ot appear explicitly i the syllabus of the correspodig subjects. Emphasis is o the study of umbers, measuremet, geometry ad data 196 Revista de Educació, 370. October-December 2015, pp. 188-215
aalysis. O the other had, at the Uiversity of Aveiro the course, which is divided ito 3 years, icludes several subjects that seek to itesify the mathematical traiig of prospective teachers. However, algebraic kowledge is, oly briefly discussed i the subject Didactics of Mathematics ad Educatioal Techology. I the Uiversities of Graada ad Aveiro, the questioaire was give to last-year studets. I the remaiig uiversities the questioaire was give to studets i the begiig of the degree. Nevertheless, give that throughout the remaider of these courses there is o explicit traiig i EAR, results show the studets professioal kowledge about EAR, i.e. that which they will be usig throughout their career. The questioaire was used with etire classes ot oly to gather iformatio eablig the evaluatio of studet kowledge, but also to use the task solvig activity as a device for further traiig. The preset study refers oly to the evaluatio results. Table II shows the sample size i each uiversity ad the values of three possible variables explaiig the differet scores obtaied i each uiversity. TABLE II. sample distributio accordig to uiversity ad explaatory variables source: Authors. The data relatig to the cut-off ad access quota were take from the followig sites: Aveiro (Direcção Geral de Esio superior, DGEs): http://www.acessoesiosuperior.pt/detcursopi.asp?codc=9853&code=0300 Graada Jaé (Juta de Adalucía): http://www.jutadeadalucia.es/ecoomiaiovacioyciecia/sguit U. public of Navarra (Uiversidad pública de Navarra): http://www.uavarra.es/estudios/acceso-y-matricula/grados/otas-de-corte satiago (Comisió Iteruiversitaria de Galicia): http://ciug.cesga.es/otascorte.html Revista de Educació, 370. October-December 2015, pp. 188-215 197
Data aalysis For the aalysis of the items, descriptive statistics techiques (mea, dispersio, ad box plots) were used. After checkig the accurateess of the total scores ad subscales, parametric iferetial tests were computed to compare groups, icludig Aalysis of Variace. Fially, a factor aalysis was performed to explore the structure of the participats resposes to the items. Results of the quatitative study I this sectio some psychometric characteristics of the DMK EAR questioaire are discussed. This study, which adds to the study of Godio et al. (2015), is crucial because the sample size is sigificatly larger (597 studets) ad it allows for a better adjustmet of some idicators. Next, there is a aalysis of the results cocerig the level of kowledge o algebra ad didactics of algebra, relatig such kowledge to the existig learig opportuities offered by the several uiversities. Some psychometric properties of the istrumet Item Aalysis Table III presets the difficulty idices of the items icluded i the questioaire calculated o the etire sample. This idex does ot correspod to the percetage of correct resposes as scores of 0, 1 ad 2 were ascribed based o the degree of correctio. The average of scores for each item was calculated ad, to make the aalysis simpler, they were coverted to iterval [0-100]. As idicated i Table III, the mea score was 32.5 (stadard error 1.43) (o a scale of 0 to 100), idicatig that the level of kowledge is isufficiet. I fact, oly 5 items have a difficulty idex above 50, whereas the difficulty idex of 21 items is below this level; i additio, 11 items have a difficulty idex below 30 (at least 7 out of 10 aswers is icorrect). 198 Revista de Educació, 370. October-December 2015, pp. 188-215
TABLE III. Difficulty idex of the items i the DMk EAR questioaire (=597) source: Authors I the secod colum, the discrimiatio idex of the items is icluded, amely the differece i meas betwee the low performace group (percetile 33) ad the high performace group (percetile 66) resultig i the proper discrimiatio of all items. (p <. 001). Reliability ad validity The reliability coefficiet obtaied for the scale (Crobach s alpha), applied to the sample of 597 studets was.783 (> 0.7), cosidered acceptable (Hair, Aderso, Tatham & Black, 2010; George & Mallery, 2003). Revista de Educació, 370. October-December 2015, pp. 188-215 199
I order to obtai proof of the costruct validity of the istrumet, a factor aalysis of mai compoets ad varimax rotatio (Kaiser, 1958) were performed, after verificatio of the adequacy of the data to this type of aalysis. Eight factors were obtaied i result. Table 4 shows the rotated compoet matrix, i which all factor loadigs below.33 were elimiated to eable a better aalysis of the results. Each factor is liked to the items icluded i each task, thus reiforcig the idea that there is a close relatioship betwee mathematical kowledge ad specific didactical kowledge ivolved i the correspodig mathematical activity. There is a high correlatio betwee factor ad task i the 8 factors, except 4 ad 5. Factor 4 basically matches task 4 (hexagoal patter) except for item 2b, which is also icluded i this factor; item 2b requests idetificatio of properties ivolved i the equality used as equivalece. Compoet 5 assembled items from three tasks: Task 8 (iterpretatio of algebraic expressios ad problem statemet), Task 3 (sum of three cosecutive umbers, types of explaatio which may be give by studets), item 2a (explaatio of the use of equality as equivalece). A possible explaatio for this groupig is that i the three items (2a, 3 ad 8) the studet is expected to aalyse a task, support their aswer or iterpret its meaig, i.e. meta-mathematical tasks ivolvig ot oly a high degree of abstractio but also a wide rage of didactical resources for the aalysis of the correspodig mathematical activity. The remaiig factors ca be described as follows: Factor 1: Didactic mathematical kowledge related to the mathematical activity of recogisig geometric patters. Factor 2: Didactic mathematical kowledge related to the modellig of problems with arithmetic algebraic structure. Factor 3: Didactic mathematical kowledge related to the iterpretatio of liear fuctios. Factor 6: Didactic mathematical kowledge related to liear fuctioal modellig. Factor 7: Didactic mathematical kowledge related to the use of equality as equivalece. Factor 8: Didactic mathematical kowledge related to the kowledge of arithmetic algebraic properties of the decimal system. 200 Revista de Educació, 370. October-December 2015, pp. 188-215
These results support the idea that learig i didactics of mathematics should be separated from other mathematical cotets through a teachig i actio (Wilhelmi, 2005) that is based o the eeds arisig from prospective teachers mathematical ad didactical progress. TABLEA IV. Rotated compoet matrix Method: Aalysis of mai compoets. Rotatio method: varimax with kaiser ormalisatio. source: Authors Revista de Educació, 370. October-December 2015, pp. 188-215 201
Comparative aalysis of studets kowledge Table V presets the meas ad stadard deviatios of the variable «Total Score» ad the sample size i the differet uiversities. To facilitate compariso ad iterpretatio of the total ad partial scores, the scale was chaged to the iterval [0, 10]. I figure I there is a compariso of the frequecy distributios of the total score obtaied i the five uiversities usig box plots. TABLE V. Average scores ad stadard deviatios of the total score per uiversity source: Authors I all cases, the average score is below 5. However, there are sigificat differeces i the mea scores obtaied i the differet uiversities (F = 37.05, p =.000) amely the Public Uiversity of Navarra (Figure I). Moreover, ulike other uiversities, i Public of Navarra there is o register of outliers, therefore, if they were suppressed i the other uiversities differeces i meas would ehace. 202 Revista de Educació, 370. October-December 2015, pp. 188-215
FIGURE 1. Compariso of the total score usig box plots source: Authors The correlatio coefficiet betwee the average scores of the uiversities ad the cut-off grade to eter the correspodig Faculty is.27, whereas correlatio with the admissio rate is -.07. I both cases, the relatioship is ot statistically sigificat at a 95% cofidece level. Additioally, there were o statistically sigificat differeces i average scores betwee the group of subjects who aswered the questioaire i their last year of traiig (M = 3.20; SD = 1.45; Aveiro ad Graada) ad those who aswered it i the begiig (M = 3.26, SD = 1.39; Jaé, Public Uiversity of Navarra ad Satiago). The result of the t test comparig idepedet samples was ot sigificat (t = 0.46; p =.644). A plausible explaatio for this result is that the algebraic cotets are ot explicitly studied i ay of the uiversities. Revista de Educació, 370. October-December 2015, pp. 188-215 203
Aalysis of partial variables: algebraic ad didactical kowledge The aalysis of variace used to study the differeces i meas i the total score variables ad the partial variables betwee differet uiversities (all trasformed to the iterval [0, 10]), suggests that there are sigificat differeces i all variables, as the p-value is <.001 i all cases (Table VI). TABLE VI. Descriptive statistics ad compariso of meas betwee uiversities source: Authors Scores below 5 highlight the difficulties faced by studets aswerig questios of epistemic ature (CD1_EPI), i.e. those that require specialised kowledge of algebraic cotet. Figure II shows the frequecy distributio of the total score ad the 7 partial variables usig box plots. Figure II shows there are outliers for all variables, except for A1_ALG. If they were suppressed from the calculatios the mea values would be eve lower tha those preseted i Table IV. This fidig supports the thesis idicatig deficiet didacticmathematical kowledge ad the eed to take cocrete measures to address the existig weakesses. Figure III eables the aalysis of the differeces i studet kowledge, accordig to costructed partial variables. The same graph presets the meas of scores obtaied for the 7 variables accordig to the five 204 Revista de Educació, 370. October-December 2015, pp. 188-215
uiversities. There is a existig tred i each uiversity, i.e. if a uiversity s score icreases whe movig from oe variable to aother the remaiig uiversities scores also icrease. Similarly, if a uiversity s score decreases whe movig from oe variable to aother the others also decrease, give that the polygoal lie of the lie graph for each uiversity rarely itersects, a fact that cofers coherece ad rigour to the study. FIGURE II. Compariso of the distributio of the differet variables usig box plots source: Authors Whe aalysed, these variables show a differetiated behaviour at the Public Uiversity of Navarra, which presets higher score i all variables, i additio to o itersectio beig registered with the lie graph of the remaiig uiversities. This fact is cosistet with the compariso of the Revista de Educació, 370. October-December 2015, pp. 188-215 205
total score displayed i the box plots, cotributig to the iteral validity of the results. Aalysis of algebraic kowledge The aalysis of the level of algebraic kowledge was doe accordig to the followig variables: A1_ALG: Commo ad advaced kowledge i algebra. A2_EST: Kowledge of equatios ad relatios. A3_FUN: Kowledge of fuctios A4_MOD: Kowledge of modellig. The mea score obtaied i the metioed variables was isufficiet. The best result was obtaied i A1_ALG (4.14 poits) probably because its items are of a procedural ature, i.e. the studet must carry out a series of calculatios ad come up with a result. Followig is A4_MOD (3.71 poits), which shows the highest degree of dispersio (highest stadard deviatio), although it is a eve dispersio (few outliers). The items i this variable refer to simple models studied i secodary school. The follows the score i A2_EST (3.32 poits). This variable icludes the structures, relatioships ad properties of mathematical framework, which are ot mastered by all studets. Fially, the lowest score was obtaied i A3_FUN (2.18 poits), which has may high outliers, justifyig the coclusio that some studets master the subject whereas the majority has clear difficulties. 206 Revista de Educació, 370. October-December 2015, pp. 188-215
FIGURE III. Meas of scores obtaied i the partial variables source: Authors Aalysis of didactical kowledge The aalysis of the level of didactical kowledge was carried out accordig to the followig variables: CD1_EPI: Kowledge of epistemic aspects CD2_COG: Kowledge of cogitive aspects. CD3_INS: Kowledge of istructioal aspects. The mea scores obtaied i these variables are also isufficiet. The best score was obtaied i CD2_COG (4.63 poits), with the secod best score if we cosider all 7 variables. It is evely distributed, showig few outliers: 50% of studets are withi a score (4-6) ad the remaiig 50% are evely distributed i the two whiskers (box plot). Revista de Educació, 370. October-December 2015, pp. 188-215 207
The other two variables obtaied the lowest score, eve cosiderig the seve variables. The lowest score was obtaied i CD1_EPI (1.49 poits). Keepig i mid the box plot, 75% of studets have scored less tha 2 ad 25% have scored betwee 2 ad 8, which results i may outliers. Oe possible explaatio is that the questios of the items icluded are ot commo i pre-uiversity educatio, so studets are ot used to aswerig such questios. Lastly, CD3_INS (2.38 poits) is secod i the lower scores, idicatig our studets lack of istructioal practice. 75% of studets have scored less tha 4, the remaiig 25% were rated betwee 6 ad 10, with few outliers. There are o differeces at the level of the period i which the questioaire was admiistered i the uiversities, suggestig that the existig difficulties are ot resolved throughout their traiig path. Summary ad implicatios for teacher traiig This paper provides meaigful iformatio regardig the level of kowledge of a sigificat sample of prospective primary educatio teachers o elemetary algebraic reasoig (EAR). Therefore, it supports previous studies o the ature of EAR (Godio et al., 2014) ad the developmet of measuremet istrumets (Godio et al., 2015). Overall, we uveiled major weakesses i the kowledge of prospective teachers who are curretly studyig i Spai uder the Curriculum established i 2010 (MEC, 2007), i accordace with the guidelies of the Europea Higher Educatio Area (Bologa pla). Although the Teacher s Degree i Primary Educatio has icluded a fourth year (which meat goig from 180 to 240 ECTS), it has ot substatially altered the geeral ad psycho-educatioal profile of teacher traiig. There is o itetio to questio the curret orietatios for the teachertutor, who is maily resposible for teachig Laguage, Mathematics ad Natural ad Social scieces, but to highlight the urget eed to esure traiig i the various subjects thus eablig the teacher to successfully face the existig epistemological challeges. The low scores obtaied by the Portuguese studets studyig uder a pla of studies which is quite differet from the Spaish oe, supports the belief that the developmet of didactic mathematical kowledge o EAR requires special attetio. 208 Revista de Educació, 370. October-December 2015, pp. 188-215
Rico et al. (2014, p 58) state that the results of the TEDS-M study poit out the eed to place greater emphasis o the topics taught i didactics of mathematics ad broade the topics taught i school mathematics: this should be take ito accout i the future at both istitutioal ad traiig levels. I this paper there is experimetal data supportig these studies. As a result, the preset situatio is clearly usatisfactory, at least regardig algebraic cotet, despite the implemetatio of ew programmes based o the guidelies of the Europea Higher Educatio Area. The subsequet aalysis of the TED-M results by Gutierrez-Gutierrez Gomez & Rico (2014), regardig didactical kowledge o umbers of prospective Spaish primary educatio teachers traied accordig to the 1991 pla, highlights weakesses at the level of such kowledge, which «should be corrected i forthcomig programmes» (p. 296). The results of our study, although it is cetred o didactic-mathematical kowledge regardig algebra usig our ow istrumet, prevet us from beig optimistic about the improvemet of teacher traiig with the ew programmes. I this paper we have tried to explai the differeces i mea scores obtaied i the five participatig uiversities by meas of its potetial relatioship with the cut-off required for admissio ad the admissio rate. Although the correlatios differ from zero (positive with the cut-off poit ad egative with the admissio rate), they were ot sigificat. I future developmets of this study, iformatio should be gathered regardig other potetial explaatory variables, amely the studets admissio grade, or the measuremet of academic achievemet i mathematics i previous courses. O the other had, from a methodological poit of view, it is worth metioig that the samples were chose purposefully. This does ot decrease the represetativeess of the groups i the participatig uiversities, as studets are distributed radomly, so it is possible to assume their represetativeess with respect to the uiversities atteded. However, a future developmet of this study, cocerig the evaluatio of prospective primary teachers level of kowledge o EAR, should iclude a wider rage of uiversities explicitly cotrollig the radom ad represetative distributio of selected samples. Lastly, the variable CD3_INS, based o prospective teachers kowledge of istructioal aspects, shows very low scores. This hits Revista de Educació, 370. October-December 2015, pp. 188-215 209
towards the eed to iclude didactical ad pedagogical strategies i mathematics to eable TDPE studets to predict their pupils behaviour ad come up with itervetios for the maagemet ad cotrol of the study process. I may cases, these activities deped o the umber of studets per group, which prevets the use of more participatory ad collaborative methods, cosistet with the basic assumptios of the ew Europea Higher Educatio Area. As future lies of work i the evaluatio problem of didactic mathematical kowledge o algebra ad its didactics, ad uder the DMK-EAR model, we ca metio the desig of items relatig to the facets: Affective, that is, kowledge of future teachers o attitudes, motivatio, emotios, beliefs ad their ifluece o the learig of elemetary algebra. Ecological, curricular aspects ad liks to other cotets. Mediacioal, usig techological resources i teachig algebra. This last aspect is closely related to the TPACK 4 models that idetify the types of kowledge that teachers eed to master to itegrate techology i a effective way i their teachig. Refereces Cai, J. & Kuth, E. (2011). Early algebraizatio. A global dialogue from multiple perspectives. Berli: Spriger-Verlag. Carraher, D. W. & Schliema, A. L. (2007). Early algebra ad algebraic reasoig. E, F. Lester (Ed.), Secod Hadbook of Research o Mathematics Teachig ad Learig (Vol. 2, 669-705). Charlotte, N.C: Iformatio Age Publishig, Ic. y NCTM. (4) Techological Pedagogical Cotet Kowledge. The TPACK approach exteds the theoretical otio itroduced by Shulma (1986) of «pedagogical cotet kowledge» (PCK). A brief descriptio of TPACK is available o the website (http://www.tpack.org/), where there is access to specialized literature. 210 Revista de Educació, 370. October-December 2015, pp. 188-215
George, D. & Mallery, P. (2003). SPSS for Widows step by step: A Simple Guide ad Referece. 11.0 Update, 4.ª ed. Bosto: Ally & Baco. Godio J. D. (2009). Categorías de aálisis de los coocimietos del profesor de matemáticas. UNIÓN: Revista Iberoamericaa de Educació Matemática, 20, 13-31. Godio, J. D. Aké, L., Gozato, M. & Wilhelmi, M. R. (2014). Niveles de algebrizació de la actividad matemática escolar. Implicacioes para la formació de maestros. Eseñaza de las Ciecias, 32.1, 199-219. Godio, J. D., Castro, W., Aké, L. & Wilhelmi, M. D. (2012). Naturaleza del razoamieto algebraico elemetal. Boletim de Educação Matemática- BOLEMA, 26 (42B), 483-511. Godio, J. D., Aké, L., Cotreras, A., Díaz, C., Estepa, A. Lacasta, E., Lasa, A., Neto, T., Oliveras, M. L. & Wilhelmi, M. D. (2015). Diseño de u cuestioario para evaluar coocimietos didáctico-matemáticos sobre razoamieto algebraico elemetal. Eseñaza de las Ciecias, 33.1, 127-150. Gutiérrez-Gutiérrez, A., Gómez, P. & Rico, L (2014) Spaish preservice primary school teachers pedagogical kowledge of umbers: results from TEDS-M / Coocimieto didáctico de los estudiates españoles de magisterio sobre úmeros: resultados e TEDS-M. Cultura y Educació: Culture ad Educatio, 26 (2), 265-297. Hair, J.F., Aderso, W.C., Tatham, R.L., & Black, W.C. (2010). Multivariate data aalysis : a global perspective, 7th ed. Upper Saddle River (NJ) : Pearso Pretice Hall. INEE. (2012). TEDS-M. Estudio iteracioal sobre la formació iicial e matemáticas de los maestros. Iforme español. Madrid: Autor. Kaiser, H. F. (1958). The varimax criterio for aalytic rotatio i factor aalysis. Psychometrika, 23(3), 187-200. MECD (2014). Real Decreto 126/2014, de 28 de febrero, por el que se establece el currículo básico de la Educació Primaria (BOE 01/03/2014). MECD (2007). Orde ECI/3857/2007, de 27 de diciembre, por la que se establece los requisitos para la verificació de los títulos uiversitarios oficiales que habilite para el ejercicio de la profesió de Maestro e Educació Primaria. Boletí Oficial del Estado (BOE) 312, de 29 de diciembre, 53747-53750. Natioal Coucil of Teachers of Mathematics (2000). Priciples ad stadards for school mathematics. Resto, VA: Autor. Revista de Educació, 370. October-December 2015, pp. 188-215 211
Rico, L., Gómez, P. & Cañadas, M. C. (2014). Formació iicial e educació matemática de los maestros de Primaria e España, 1991-2010. Revista de Educació, 363, 35-59. Saz, I., & Martí, R. (2014). El estudio TEDS-M de la IEA e el marco del Istituto Nacioal de Evaluació Educativa (INEE). I M. T. Gozález, M. Codes, D. Arau y T. Ortega (Eds.), Ivestigació e Educació Matemática XVIII (pp. 67-81). Salamaca: SEIEM. Shulma, L. S. (1986). Those who uderstad: kowledge growth i teachig. Educatioal Researcher, 15(2), 4-14. Tatto, M. T., Schwille, J., Sek, S., Igvarso, L., Peck, R., & Rowley, G. (2008). Teacher Educatio ad Developmet Study i Mathematics (TEDS-M): Policy, practice, ad readiess to teach primary ad secodary mathematics. Coceptual framework. East Lasig, MI: Teacher Educatio ad Developmet Iteratioal Study Ceter, College of Educatio, Michiga State Uiversity. Wilhelmi, M. R. (2005). Papel de la didáctica de las matemáticas e la formació de profesores de secudaria. Gaceta de la Real Sociedad Matemática Española 8(1), 159-179. Wilhelmi, M. R. (2014). Matemáticas y su didáctica e Magisterio. UNO, Revista de Didáctica de las Matemáticas 66, 28-35. Appedix: DMK-EAR questioaire 1. Cosider the followig questio posed to a pupil of the first cycle of primary educatio: What umber should be placed i the box so that equality is true? 8 + 4 = + 5 A pupil aswers that the umber is 12, a) Explai the possible reasoig that led the pupil to give that aswer. b) Which iterpretatio of the sig = is beig doe by the pupil? 2. A pupil was asked to idicate whether the expressio «13 + 11 = 12 + 12» is true or false. The pupil aswers the followig: 212 Revista de Educació, 370. October-December 2015, pp. 188-215
It is true because we subtract oe to twelve ad add it to the other twelve, the result is what is there (o the left). a) Explai the reasoig that could lead the pupil to come up with this respose. b) Which properties of additio led the pupil to justify their respose? 3. A pupil made the followig hypothesis: «I add three cosecutive atural umbers. If I divide the result by three I always get the secod umber a) Is the statemet valid for all atural umbers? Why? b) I your opiio, what kid of justificatio could a primary school pupil give to this hypothesis? 4. Carefully aalyse the followig sum, ad determie the umber represetig each letter. Cosider that each letter has a differet value. A B C A B C + A B C 2 A C C a) What are the umerical values of A, B ad C? How do you kow they are correct? Explai your reasoig. b) Ca you solve the task usig a algebraic procedure? What would that resolutio be ad which algebraic cocepts would be used? c) What kid of respose ad justificatio do you thik a elemetary school studet could give to this problem? 5. Cosider the followig sequece. a) Represet the ext two terms of the sequece ad idicate the umber of segmets eeded to build each oe of them. Explai how you did it. Revista de Educació, 370. October-December 2015, pp. 188-215 213
b) How would you chage the statemet of the task to hit a resolutio procedure which ivolved algebraic kowledge? c) What would be the algebraic kowledge ivolved? 6. Cosider the followig sequece of three shapes defied by dots: a) Determie the umber of dots that the shape placed o the twetyfifth (25th) positio of this sequece will have, assumig it cotiues with the same rule of formatio. Support your aswer. b) Idicate techiques or differet ways to solve the problem. c) Do you cosider that this task could be proposed to studets i the 3 rd cycle of primary school? How could they reach a solutio? 7. A pupil received a certai amout of moey to eat for 40 days from his parets. However, he foud places where he could save 4 euro a day o food. Thus, the iitial budget lasted 60 days. a) How much Moey did he receive? b) Ca you solve the problem usig oly arithmetic procedures? How? c) Ca you solve the problem usig algebraic kowledge? How? 8. Aalyse the followig expressios ad aswers: 1. 4x + 5 = 25 2. y = 2x +1 3. P = 2c + 2l a) Describe your iterpretatio of each of the above expressios. b) Come up with three problems that may be proposed to elemetary studets ad whose solutio leads to these expressios. 9. To fill a cotaier with a maximum capacity of 90 litres with water a faucet whose flow is costat ad equal to 18 litres per miute is used. a) Idicate which of the three graphical represetatios correspods to the situatio described above, beig that the X axis represets time i miutes ad the Y axis the volume of water i litres. 214 Revista de Educació, 370. October-December 2015, pp. 188-215
Aswer: ; Justificatio: b) What mathematical kowledge or other type of kowledge is used to solve this task? c) Do you cosider that this task could be give to primary school childre? If so, what cycle? Support your aswers. 10. A teacher presets the followig problem to his pupils: At a store they sell each kilo of pears for 2 ad charge 10 cets per bag. How much would a bag with 4 kg of pears cost? a) Come up with a variat of the problem which could be used to itroduce liear fuctios. Suppose each bag carries 4 kg. b) Solve the problem you stated ad ame the algebraic skills used. Cotact address: Jua D. Godio. Uiversidad de Graada, Facultad de Ciecias de la Educació, Departameto de Didáctica de la Matemática. Facultad de Educació. Campus de Cartuja. 18071 Graada. España. E-mail: jgodio@ugr.es Revista de Educació, 370. October-December 2015, pp. 188-215 215