Planning and evaluation of a learning model for deepining studentsʹ understanding of percentages

Mula, Melinda (author), Hodnik, Tatjana (mentor)

Odstotek je eden najbolj razširjenih matematičnih pojmov, ki se na različne načine uporablja v šolah in v resničnem življenju. Učenci se zavedajo, da je računanje z odstotki in poznavanje njihovega pomena zelo koristna in uporabna spretnost, a raziskave kažejo, da ima veliko učencev težave pri reševanju problemov, povezanih z odstotki. Kritike za omenjene težave so usmerjene predvsem v poučevanje, ki v središče postavi razvijanje učenčevih spretnosti pri računanju z odstotki, v veliko manjši meri pa nameni pozornost učenčevemu poglobljenemu razumevanju teh vsebin in njenih pomenov v različnih kontekstih. V pričujoči študiji prikazujemo nov model za poučevanje odstotkov v sedmem razredu osnovne šole, izhajajoč iz predpostavke, da so učenci pri tej starosti (13 do 14 let), glede na kognitivni razvoj, zmožni konceptualnega razumevanja odstotkov in povezovanja teh vsebin z znanjem o ulomkih in decimalnih številih. V izhodišču oblikovanja modela smo analizirali pomen različnih reprezentacij za razumevanje določenega vidika pojma odstotek oz. njihovo medsebojno dopolnjevanje s ciljem razviti model, s katerim bi spodbujali učenčevo poglobljeno razumevanje odstotkov. Model je bil zasnovan na temelju koncepta 'metoda oblikovanja modela', katerega izhodišče je procesno načrtovanje in interveniranje pri poučevanju, končna evalvacija modela pa je temeljila na pedagoškem eksperimentu. Metoda oblikovanja modela je omogočala induktivni pristop: oblikovana sta bila dva pilotna modela (PM 1 in PM 2), ki smo jih preverili na izbranem vzorcu učencev. Najprej smo preverili PM 1, ga ovrednotili, na osnovi dognanj oblikovali PM 2, ki smo ga ponovno preverili na drugem vzorcu sedmošolcev. Izpopolnjena pilotna modela smo razvili v PGBE model za poučevanje in učenje odstotkov. Kratica PGBE (P-poster method, G-grids, B-bar model, E-extended bar model) zaobjema prepletenost reprezentacij odstotkov in njihov pomen za razvijanje posamezne taksonomske ravni znanja: reprezentiranje v obliki posterja za intuitivno znanje, mreža za konceptualno znanje, prikazovanje odstotkov s stolpci za proceduralno znanje in razširjena reprezentacija s stolpci za razvijanje problemskega znanja. Učinkovitost PGBE modela za poučevanje in učenje odstotkov je bila preverjena na vzorcu 473 sedmošolcev iz desetih šol treh regij na Kosovem. Pri ugotavljanju učinkovitosti modela smo uporabili naslednje instrumente: preizkus znanja za ugotavljanje predznanja, preizkus za ugotavljanje znanja po izvedenem pristopu, opazovalni list za spremljanje poteka učnega procesa, učiteljevi dnevniški zapisi, učenčeve izhodne kartice, pol-strukturirani intervju za učitelje. Rezultati pilotske raziskave, v katerih smo zaporedno preverili dva pilotna modela (PM 1 in PM 2) so pokazali učinkovitost načina povezovanja reprezentacij in vlogo interakcije tako med učenci in učitelji kot tudi med učenci. Temeljita analiza izvedb pilotnih modelov je rezultirala v model PGBE. Četudi je bilo pilotiranje izvedeno dvakrat, smo model preverili na izbranem vzorcu učencev z eksperimentalno metodo pedagoškega raziskovanja in izvedbo ustreznih statističnih postopkov za vrednotenje rezultatov učencev na preizkusih znanja (ugotavljanje predznanja, znanja po izvedbi pristopa in na pot-testu, dva meseca po zaključku eksperimenta). Učenčevi rezultati na preizkusih znanja so izpolnjevali pogoje za parametrični test, zato smo uporabili t-test za analizo rezultatov preizkusa znanja o odstotkih, za statistično analizo rezultatov preizkusa znanja glede na raven znanja (intuitivno, konceptualno, proceduralno in problemsko) pa smo uporabili Mann-Whitney U test. Statistična analiza učenčevih rezultatov na preizkusih znanja pokaže, da so učenci eksperimentalne skupine dosegli boljše rezultate pri 26-ih ocenjevalnih enotah od skupaj 28 enot na preizkusu znanja po izvedenem pristopu in pri 21-ih ocenjevalnih enotah od 28 enot na post-testu, ki je bil z učenci obeh skupin izveden dva meseca po izvedbi pristopa. To z drugimi besedami pomeni, da so bili dosežki učencev eksperimentalne skupine boljši pri 92,9 % vseh ocenjevalnih enot na končnem preizkusu znanja in pri 75 % vseh enot post-testa. Glede na taksonomske ravni znanja lahko zaključimo, da so učenci eksperimentalne skupine na post-testu dosegli boljše rezultate od učencev kontrolne skupine pri 16-ih ocenjevalnih enotah od skupaj 22 enot, ki so preverjale višje taksonomske ravni znanja o odstotkih. Omenjeni rezultati so jasni indikatorji, da je implementacija poučevanja odstotkov po PGBE modelu vplivala na poučevanje in učenje odstotkov, natančneje, vplivala je na poglobljeno razumevanje teh vsebin in na trajnost osvojenega znanja. Na osnovi rezultatov te študije lahko upravičeno priporočamo izvedbo učnega pristopa po PGBE modelu za poučevanje in učenje odstotkov. PGBE model podpira učence pri učenju odstotkov z razumevanjem, pri usvajanju proceduralnih znanj in jih spodbuja k prefinjenemu matematičnemu razmišljanju in sklepanju o odstotkih ter drugih, podobnih konceptih, na primer ulomkih in decimalnih številih.

• Language: Slovenian
• Keywords: odstotek
• Work type: Dissertation
• Typology: 2.08 - Doctoral Dissertation
• Organization: PEF - Faculty of Education
• Year: 2020
• PID: 20.500.12556/RUL-118096
• COBISS.SI-ID: 25514499
• Publication date in RUL: 25.08.2020

Secondary language

• Language: English

Abstract

Although percentage is one of the most ubiquitous mathematical concepts that applies widely in school and real-life situations and students are aware about benefits in learning it, there are evidences showing that a considerable number of students struggle to solve different percentage problems. The criticism is mainly addressed to teachers for being focused on mastering students’ skills for calculating percentages and not so much on building students’ principled-conceptual knowledge about percentages, as an effective and sustainable investment for solving percentage problems from various contexts. This study addressed the necessity to develop a new learning model for teaching and learning percentages in Grade 7, when students’ cognitive development ensures the conceptual understanding of percentages as proportional statements and offers the possibility for more effective matching of them with students’ prior knowledge about multiplicative structures, such as fractions and decimal numbers. This study explores various instructional models that are acknowledged for maximizing the development of a certain type of students’ mathematical knowledge about percentages and their effective interrelation into a learning model for facilitating students’ in-depth understanding of percentages. As the intention behind designing the new learning model was interventionist and process oriented for improving the introduction of percentages, the design research method based on teaching experiment was used for designing it. This method offered the possibility to use the inductive approach in designing two Hypothetical Learning Trajectories (HLTs), pilot them consecutively as teaching experiments in two different classrooms and refine them into a new learning model (the PGBE model) for teaching and learning percentages. The abbreviation PGBE presents the interrelation of main methods and instructional models for developing four types of students’ mathematical knowledge related to percentages, such as: P for the poster method for recalling students’ intuitive knowledge about them, G for different patterns of grids for building concrete type of knowledge about percentages, B for bar model for developing students’ procedural knowledge for calculating percentages and E for extended bar model that fosters students’ principled-conceptual understanding of percentages. The effectiveness of the implementation of the PGBE model in students’ learning of percentages was assessed through two cycles of piloting and the empirical study conducted with 473 students of Grade 7 classes of ten different schools in three municipalities, analysis of the knowledge and retention tests results, classroom observation protocols, teachers’ reflective diaries, students’ exit cards, and semi-structured interviews with teachers. Piloting offered the possibility to assess the effectiveness of the interrelation of different research-based instructional models into the development path of the HLT 1 and HLT 2 and the social interaction between teacher and students, and also between students themselves during their implementation. The feedback from consecutive piloting of both HLTs was used to finalize the PGBE model. Even though the piloting was done twice, the effectiveness of the implementation of PGBE model was assessed by organizing the experimental method and conducting statistical analysis of students’ results in the knowledge test and retention test on percentages. Students’ results in the knowledge test and retention test fulfilled the assumptions of the parametric tests, so the independent t-test was used to perform statistical analysis of students’ results in both of them, whereas for performing statistical analyses of students’ results in the knowledge test on percentages related to four types of mathematical knowledge a Mann-Whitney U test was used. The statistical analysis of students’ results shows that students of the experimental group outperformed students of the control group in 26 out of 28 items of the knowledge test, and in 21 out of 28 items of the retention test. This indicator proves that students of the experimental group outperformed their counterparts in 92.9 % of all items of the knowledge test and in 75 % of all items of the retention test conducted two months later. In particular, students’ results of the experimental group are higher in 16 out of 22 items of the retention test that are related to the highest levels of students’ mathematical knowledge about percentages. All these indicators regarding the knowledge test and retention test prove that the implementation of the PGBE for teaching and learning percentages had impacted the learning of students of Grade 7, since it influenced their in-depth learning and a long-lasting knowledge about percentages. As a result of this research study, the implementation of the PGBE model for deepening students’ understanding of percentages is recommended. This learning model supports students in gaining a deeper understanding about percentages, in mastering their computational skills, and sophisticating their mathematical thinking and reasoning about percentages and other similar mathematical concepts, such as fractions and decimal numbers.

• Keywords: Percentage