ACTA UNIVERSITATIS PALACKIANAE OLOMUCENSIS FACULTAS PAEDAGOGICA MATHEMATICA VII MATEMATIKA 4. Matematické vzdělávání v kontextu proměn primární školy

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1 ACTA UNIVERSITATIS PALACKIANAE OLOMUCENSIS FACULTAS PAEDAGOGICA MATHEMATICA VII MATEMATIKA 4 Matematické vzdělávání v kontextu proměn primární školy Sborník příspěvků z konference s mezinárodní účastí Mathematical Education in a Context of Changes in Primary School The Conference Proceedings OLOMOUC 2010

2 Anotace Sborník obsahuje p ísp vky ú astník v decké konference s mezinárodní ú astí Elementary Mathematics Education 2010, která se pod názvem "Matematické vzd lávání v kontextu prom n primární školy konala ve dnech na Pedagogické fakult UP v Olomouci. Výsledky v deckovýzkumné, odborné a pedagogické innosti ú astník konference jsou zam eny na aktuální problémy matematické p ípravy u itel primárních škol i školské praxe. Cíl a zam ení konference Prezentace p vodních výsledk v oblasti elementární matematiky a didaktiky matematiky, zam ené k využití v teorii i praxi primární školy a v pregraduální p íprav u itel. Teoretické reflexe i výstupy výzkum na pracovištích v R a zahrani í v souvislosti se zm nami v kurikulu i metodách vyu ování matematice. Hlavní témata konference Inovativní p ístupy k profesní p íprav u itele matematiky primární školy. Reflexe obsahových i metodických prom n matematické komponenty primárního vzd lávání. Abstract The proceedings contain contributions from all the participants of the scientific conference Elementary Mathematics Education 2010 with international participation that took place at the Faculty of Education of Palacký University in Olomouc. The results of scientific research, professional work, and pedagogical activities of conference participants are focused on current problems in the mathematical preparation of primary school teachers and school practice. Aims of the conference Presentation of previous results in elementary mathematics and mathematical didactics focused on its usage in primary school theory and practice and in the mathematical preparation of primary school teachers. Theoretical reflection and research outputs at workplaces in the Czech Republic and abroad in connection with changes in curriculum and teaching mathematics methods. Programový výbor/programme committee Georg Malaty (Joensuu, Finsko), Helena Siwek (Krakow, Polsko), Jolanta Seidel (Bydgoszcz, Polsko), Adam Plocki (Krakow, Polsko), Ondrej Šedivý (Nitra, Slovensko), Pavol Hanzel (Banská Bystrica, Slovensko), Jana Coufalová (Plze, eská republika), Alena Hošpesová ( eské Bud jovice, eská republika), Bohumil Novák (Olomouc, eská republika) Organiza ní výbor/organizing committee Anna Stopenová, Jitka Laitochová, David Nocar, Dita Navrátilová, Kate ina Šupíková, Eva Hotová, Jitka Hoda ová, Tomáš Zdráhal Za p vodnost a správnost jednotlivých p ísp vk odpovídají jejich auto i. P ísp vky neprošly redak ní ani jazykovou úpravou. Editor Martina Uhlí ová, 2010 ISSN ISBN

3 OBSAH Úvodem... 7 Plenární přednášky Eveleigh Frances The Role of Assessment in Effective Pedagogy in Primary Mathematics. 10 Malaty George Developments in Using Visualizations, Functional Materials and Actions in Teaching Mathematics: understanding of relations, making generalizations and solving problems. 18 Swoboda Ewa Ile różnych klocków domina można ułożyć z kafelków wisienki Vondráková Eva Otevřená mysl jako podmínka efektivního fungování péče o nadané Příspěvky Andrzejewska Jolanta Edukacja matematyczna dziecka w przedszkolu analiza porównawcza treści programowych Beránek Jaroslav Využití historických témat ve vyučování matematice na 1. stupni základní školy Bilewicz-Kuźnia Barbara, Parczewska Teresa Kształtowanie pojęć matematycznych na etapie edukacji elementarnej Blažková Růžena, Vaňurová Milena Charakteristika nadaného žáka s poruchou učení z hlediska matematických úloh Brinckova Jaroslava Edukačné koncepcie rozvoja matematickej gramotnosti v príprave učiteliek Predškolskej pedagogiky Cachová Jana Kalkulačka a její místo v aritmetice primární školy Círus Lukáš Jak britské standardy pro oblast ICT pomáhají matematice 72 Coufalová Jana, Chmelová Miroslava Individuální předpoklady žáků prvního stupně pro další rozvoj jejich matematického myšlení Coufalová Jana, Chmelová Miroslava, Hrabětová Regina Žáci, učitelé a algoritmy 84 Dušková Jana, Pěchoučková Šárka Práce s textem slovní úlohy Gerová Ľubica Podmienky pre rozvoj matematickej gramotnosti študentov pred vstupom na vysokú školu.. 94

4 Heřmánková Pavla Matematika a současnost Híc Pavel, Pokorný Milan Skúsenosti s kombinovanou formou výučby predmetu logika, množiny, relácie 105 Hodaňová Jitka Modely a modelování v matematice v primárním vzdělávání Holubová Drahomíra Odraz duševní pohody tělesné zdraví Hošpesová Alena, Matějů Bohuna, Fantová Šárka Mozaiky jako podnětná výuková prostředí Hotová Eva Identifikace matematicky nadaných žáků Jakubowicz-Bryx Anna Miejsce edukacji matematycznej w nowych programach kształcenia zintegrowanego Jančařík Antonín Rozvoj početních dovedností u mladších školních dětí Janků Magdaléna Didaktické hry v hodinách matematiky na 1. stupni ZŠ 137 Kaslová Michaela Rezervy inovací vyučování matematice Klenovčan Pavel Prestavba vyučovania matematiky Kováčik Štefan Rovnice v matematike a ich význam pre učivo fyziky Kováčik Štefan, Sebínová Katrína Možnosti využitia Excelu v matematike Krejčová Eva O některých zkušenostech ze soutěživých metod a forem práce v hodinách matematiky na 1. stupni ZŠ 162 Laitochová Jitka Matematika anglicky. 167 Lucká Mária Interaktívne testovanie ako nástroj motivácie vo vyučovaní matematiky v 4. ročníku ZŠ Lucká Mária, Malčík Martin, Pokorný Milan, Švecová Libuše Diagnostika stavu vedomostí žiakov 3. a 5. ročníka ZŠ z matematiky Mokriš Marek Priestor a tvar pohľad na matematickú gramotnosť študentov odboru predškolská a elementárna pedagogika 182 Nawolska Barbara O dobrych wynikach i błędnych rozwiązywaniach zadań 187 Nocar David Inovovaný koncept matematické složky profesní přípravy učitelů primární školy na Pedagogické fakultě Univerzity Palackého v Olomouci. 192 Novák Bohumil Nestandardní aplikační úloha, její reflexe a interpretace budoucími učiteli primární školy

5 Nováková Eva Pět let se Cvrčkem. 205 Nowak Jolanta Działaj - odkrywaj - konstruuj czyli kształtowanie pojęcia liczby naturalnej w umyśle dziecka Nowak Zbigniew Burza pytań jako metodyka pracy z tekstem matematycznym Partová Edita Očakávania a realita efektivity interaktívnej digitálnej tabule Partová Edita, Žilková Katarína Rozvíjanie pojmu relácia v predškolskom veku prostriedkami IKT Gabriela Pavlovičová Výroková logika vo vyučovaní elementárnej matematiky 230 Perný Jaroslav Rozvíjení geometrické představivosti žáků Prídavková Alena, Šimčíková Edita Návrh Stimulačného programu z matematiky pre deti zo sociálne znevýhodňujúceho prostredia Příhonská Jana Shodné a rozdílné aspekty matematické edukace v primární škole na Slovensku a v České republice Příhonská Jana, Vilimovská Lucie, Čálková Tereza Aktivizující metody ve výuce s využitím IKT technologií na základní škole. 250 Roubíček Filip Geometrická představivost jako složka matematické gramotnosti Seidel Jolanta, Sobieszczyk Maria Zajęcia komputerowe na etapie wczesnoszkolnej edukacji - poziom umiejętności i wspomaganie ich rozwoju Shromáždilová Miluše Využití interaktivní tabule v matematice primární školy Scholtzová Iveta Niektoré aspekty matematickej gramotnosti študentov odboru Predškolská a elementárna pedagogika Sotáková Kristína Model zlomku v otvorenom softvérovom prostredí Stopenová Anna Geometrie v učivu primární školy Šedivý Ondrej Symetria v prírode a symetria v školskej matematike Šťastná Alena Řízené objevování v matematice na 1. stupni ZŠ Tichá Marie Podnětná výuková prostředí pro přirozenou diferenciaci Trnková Veronika Matematika z pohledu začínajících učitelů Uhlířová Martina Model of the teacher development with re-spect to the level of ICT implementation

6 Vršanská Lenka Didaktické pomůcky v matematice primární školy Zdráhal Tomáš Geometrická interpretace největšího společného dělitele Zeľová Veronika, Gombár Miroslav Zmena, vzťahy, závislosť pohľad na matematickú gramotnosť študentov odboru Predškolská a elementárna pedagogika Postery Olšáková Věra, Molnár Josef Práce s talenty Janků Magdalena, Šťastná Alena, Šupíková Kateřina 1. ročník studnetské grantové soutěže na UP prezentace schválených projektů na katedře matematiky Recenze

7 ÚVODEM Po roční přestávce se v Olomouci opět setkávají učitelé vysokých i základních škol, jejichž vědecká, odborná i pedagogická práce je věnovaná aktuálním problémům matematiky v primárním vzdělávání a vysokoškolské přípravě učitelů matematiky u nás i v zahraničí. Tématem letošní olomoucké konference Elementary Mathematics Education 2010 se stalo Matematické vzdělávání v kontextu proměn primární školy. Proměna školy znamená nejen změnu kurikulárního rámce vzdělávání, ale přináší zřetelně se zvyšující nároky na osobnostní a profesní kvality učitelů. Řešení složitých situací ve vzdělávacím procesu spojených se změnou přístupu k dítěti, vytváření prostředí poskytující dostatek podnětů pro jeho osobnostní rozvoj, diferenciace a individualizace výuky, integrace dětí se speciálními vzdělávacími potřebami, posílení mezipředmětové integrace, ale i změny v sociálních poměrech dětí, to vše přináší nové požadavky na profesní přípravu učitelů. Rovněž vliv evropské dimenze vzdělávání, zavádění nových technologií do škol a další faktory vyžadují, aby kvalifikační požadavky na učitele primární školy byly reflektovány v koncepci přípravného a dalšího vzdělávání. Je skutečností školskou praxí neustále potvrzovanou, že učitele nestačí vybavit nějakými (např. matematickými) poznatky, nějakými recepty, jednou provždy. Orientace studenta učitelství v prostředí akademického poznání, jeho způsobilost vybírat, aktualizovat, aplikovat, způsobilost adekvátním způsobem komunikovat se širokým spektrem partnerů (žáky, kolegy, rodiči, veřejností) se ukazuje jako nezbytná pro jeho praxi. Uvedené myšlenky byly reflektovány a na příkladech z konkrétní edukační reality konkretizovány v různé souvislosti v převážné většině příspěvků konference. Záštitu nad konferencí Matematické vzdělávání v kontextu proměn primární školy přijala děkanka Pedagogické fakulty UP prof. PaedDr. Libuše Ludíková, CSc., jíž patří dík za podporu organizátorů z Katedry matematiky PdF UP v Olomouci i všech účastníků. Spolupořadatelem je Společnost učitelů matematiky JČMF, která organizací pověřila pracovní skupinu pro matematiku na 1. stupni ZŠ. Organizátoři zpracovali pro účastníky konference a další zájemce recenzovaný sborník plných textů příspěvků v jazyce prezentace jako jeden z titulů řady Acta Universitatis Palackianae Olomucensis a sborník abstrakt příspěvků v anglickém jazyce. Oba sborníky jsou vydány Vydavatelstvím Univerzity Palackého. Programový i organizační výbor konference věří, že konference úspěšně naváže na předchozí ročníky a přispěje k dalšímu rozvoji didaktiky matematiky. V Olomouci dne Bohumil Novák 7

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9 PLENÁRNÍ PŘEDNÁŠKY

10 ACTA UNIVERSITATIS PALACKIANAE OLOMUCENSIS FACULTAS PAEDAGOGICA 2010 MATHEMATICA VII THE ROLE OF ASSESSMENT IN EFFECTIVE PEDAGOGY IN PRIMARY MATHEMATICS Frances EVELEIGH Abstract This paper discusses the role that large scale systemic assessment programs can play in informing classroom practice and enhancing student learning outcomes. Although large scale assessment provides only a brief snapshot of a student s ability it plays a vital role in informing teachers about mathematical learning and pedagogy. The experiences of international programs such as the Trends in Mathematics and Science Study (TIMSS), as well as the Australian National Assessment Program Literacy and Numeracy (NAPLAN) provide examples of how large scale assessment programs can provide diagnostic information to school administrators and teachers to improve teaching and learning. Introduction Over the past three decades the star of large scale testing has continued to rise. As governments and educational bodies have become more accountable to key stakeholders in schooling systems worldwide, the role of system-wide assessment has gained strength and importance. In an attempt to gaze through the lens of primary school mathematics teachers and extract that which is appropriate, useful and even critical from these assessment programs, the data provided by international and national assessments such as the Trends in Mathematics and Science Study (TIMSS) and the Australian National Assessment Program Literacy and Numeracy (NAPLAN) are examined with the intent of provoking thought toward the opportunities to improve current practices and teaching opportunities within the classroom. The role of large-scale assessment in international and national contexts is complex. In the case for national assessments, Davey (1992, 1) states Any proposed national examination must be part of a broader plan, a plan that also integrates objectives, standards, teaching, assessment, and accountability for results. There is a tension between those who advocate that the purpose of large scale assessment is for learning and those who deem that the purpose of these assessments is accountability; (the idea of holding schools, principals and jurisdictions responsible for results.) As much funding, planning and energy is expended on any large scale assessment, stakeholders will understandably attempt to gain as much information as possible from their exercise. For many countries a national test may attempt to serve both purposes; learning and accountability, providing direction for governments and policy makers and simultaneously reporting diagnostically at class and student level. The attempted union of 10

11 these two purposes may bring dissention as it did in the United Kingdom. The Cambridge Report (Alexander 2009), a comprehensive enquiry into primary education, recommends that the two aims should be uncoupled. There are inherent common strengths and weaknesses in large scale assessments. Opponents of national testing justifiably argue that too much importance is often placed upon a point-in-time assessment that has limited scope. Students are judged and reported on a single piece test in which they are expected to perform at their peak. In terms of the testing instrument it is reasonable to assume that there may be limitations on curriculum coverage. The rich and varied mathematics curricula that is hopefully offered in the classroom cannot be covered in a national test designed to cater for a whole population. There are limitations in the item types presented in test forms and some concepts, in particular spatial ideas, can be difficult to test in a pen and paper medium. There are also implications for validity and reliability which begs the question: are the tests well-targeted and actually testing what they purport to test? Adams (2003) believes that whilst point-in-time tests can provide accurate information about systems (and possibly schools), standardised tests do not necessarily provide reliable information about individual students once measurement errors are taken into account. There is also the view that system-imposed testing may adversely affect students and teachers, adding load and reducing the time that should be concentrated on what might be considered as more valuable learning time. On the other hand, however, national testing has the potential to provide useful information on many levels to a wide variety of stakeholders. For government departments, the information provided may guide policy development, allocation of funding and resources, the implementation of professional development programs and decision-making for the future. For principals and teachers, diagnostic information at school and classroom level can provide information to guide data-driven school-level interventions and initiatives. The information provided by large scale assessment programs can provide insight into the necessary emphasis of future school programs. In conjunction with other information, the effectiveness of implemented strategies can be measured over time, highlighting school-specific needs that require further targeting to improve the quality of education. Parents and students can be provided with an individual snapshot of a student s performance highlighting areas of excellence and those of greater need. Large scale testing can also allow for comparability of results between countries, jurisdictions and provide reliable information at these and school levels. Consistency can be maintained and comparisons can be made over time. The aim is to combine traditional top-down models of monitoring with bottom-up approaches to pupil assessment at school level, so that the quality of education is gauged more effectively and thus improved (Eurydice 2009, 20). The Context of National Testing in Europe and Australia Eurydice (2009) contends that throughout Europe the greater incidence of school autonomy has coincided with the need to regulate the monitoring of education systems and hence the widespread implementation of national tests. In some countries this is still being developed whilst others embraced the idea at the end of the last century. National testing manifests itself in different forms in various European countries where differences involve the age at and frequency with which students are tested. With different 11

12 educational priorities, each country approaches assessment with a combination of formative and summative forms of assessment, corresponding with varying levels of mandatory education. Table 1 shows both the types of tests and the frequency with which they were administered in primary and lower secondary in 2008/2009, as published in National Testing of Pupils in Europe: Objectives, Organisation and Use of Results (Eurydice 2009). Table 1 Number and type of national tests and school years in which they are administered, in primary and lower secondary 2008/2009. Table 1 shows that whilst the majority of European countries administer large scale cohort testing, the countries of Belgium (German speaking), Czech Republic, Greece, Wales and Liechtenstein have not yet embraced this assessment form, with no national tests in the primary or lower secondary years or equivalent ages. According to the Eurydice report (2009), this investigation into the national testing scene was requested by the Czech Presidency of the Council of the European Union during the first half of The Czech Republic s interest has heightened due to debate at 12

13 a national policy level regarding the introduction of national testing as an instrument for improving the quality of education, with national tests being viewed as a strong possibility for the future. Research projects have been undertaken at the primary and lower secondary school years with results and data being examined. Whilst its neighbours Slovakia, Germany and Austria have embarked on national testing programs, the Czech Republic has held onto the principle of constant pupil assessment with its main aim being to determine learning and attainment levels ascertained internally at school level. Introduced in Australia in 2008, the National Assessment Program Literacy and Numeracy (NAPLAN) replaced state-based large cohort assessments administered by the individual states and territories. Each student in Years 3, 5 7 and 9 completes a suite of tests in Literacy (Reading, Writing and Language) and Mathematics (Numeracy). In Mathematics (Numeracy), systemic differences needed to be addressed to take account of differing state-based curricula. In some states for example, the Year 7 cohort comprises the last year of the primary system, whilst in other jurisdictions, it is the first year of secondary schooling; with implications for content in the areas of Algebra and Statistics. The national test in Australia continues to evolve. The various state departments have slowly relinquished their past programs and accepted the permanency of NAPLAN. Energies have subsequently turned to programs aimed at supporting teachers and assisting schools in accessing, understanding, manipulating and using the rich data generated by this program. With the launch of an Australian curriculum programmed for early 2011 and several years scheduled for its complete implementation, the national mathematics test can theoretically grow to be more closely aligned with learning in the classroom. Implications for primary mathematics teachers In the classroom setting, it is the primary teacher who is best placed to determine the needs of students and capture evidence of learning, employing many tools and forms of assessment as learning growth of students is monitored. Greater focus on methods and training for enhancing teacher assessment practices should be paramount. These should include tools for engaging with the data produced by large scale testing programs. As teachers who become practiced in applying research findings from programs such as TIMSS, move through their careers and eventually take leadership roles within schools, acceptance and application of large scale data will be commonplace and a key component of a student s profile. This information should enhance the picture of each student that has already evolved out of daily formative assessment processes such as class tests, informal interaction and observation. Despite these variations in approaches to pupil assessment, the process of assessing learning outcomes forms part of the overall structure of education systems. In all countries, pupil assessment forms an integral part of teaching and learning and thus, ultimately, an instrumental factor in improving the quality of education (DePascale 2003, 11). Rather than fearing research outcomes from large scale testing, it is instead an opportunity for learning and selectively taking the components transferable and applicable to the classroom. It is a widespread practice among countries in Europe to provide information enabling schools to measure themselves against the national average results achieved by pupils in national tests, and to make improvements on the basis of that comparison. This is true of 13

14 most national tests designed to monitor schools or the education system as a whole (Eurydice 2009). As primary teachers of mathematics become further exposed to national testing, the gains are numerous. The testing instruments for mathematics are written and designed by expert test developers, undergoing much rigour in their development that would be difficult to replicate in school or classroom contexts. In Australia, the items are repeatedly panelled and reviewed by interest groups and representatives of key stakeholders, including teachers. All items are trialled and carefully analysed with only a small percentage of the overall trial pool selected for inclusion in the final tests. The matrix of the test is complex and a fine balance of all the constraints and specifications needs to be maintained. A spread of difficulty within the test is necessary to discriminate between the lower ability students and those who are more able. This needs to occur in sub-strands, with testing of the mathematical areas of number, space (geometry), measurement and chance and data (statistics) covered at both ends of the difficulty scale. Test items developed under such scrutiny provide exemplars for teachers to follow. Classroom professionals can adapt the principles of sound test constructions and apply them in the smaller context. Assessment, whether formal or informal, at the classroom level is a crucial aspect of monitoring both effective classroom practice and the quality of learning. A great advantage of national mathematics tests is that they have the potential to allow primary teachers to compare the effectiveness of their teaching across time. This feature is difficult using Traditional Test Theory models. The measurement principles that underpin large scale assessments such as TIMSS and NAPLAN allow for the construction of a Numeracy scale which remains constant over time. By using internationally recognised analysis programs and techniques grounded in the Rasch measurement model, a calibrated interval scale, independent of the test items and student sample, can be constructed to measure student performance against a stable ruler. Future tests can be linked to the scale and student performance measured against the same ruler to allow valid and defensible comparisons. As the programs of learning and even individual lessons delivered by teachers are evaluated and refined, the objective feedback provided in the form of student results in international and national assessments can assist teachers in self monitoring, leading to greater efficacy. Schools are able to monitor the success of initiatives or specific areas that may have been targeted and help determine allocation of resources for future programs. Cohorts of students are also able to be scrutinised. In NAPLAN, students in Year 3 can be compared to current students in Year 5. Alternatively over time, the results of students in Year 5 can be compared to the results that these same students achieved two years ago when in Year 3. This means that as a student does the NAPLAN tests every two years, comparison of the results from subsequent tests can be made and progress on the scale observed. In mathematics, this can provide great insight into a school and classroom. Particular year groups may demonstrate weaknesses in mathematics or in specific areas of the domain. The reporting format for NAPLAN allows teachers to view class results at substrand level allowing quick recognition of areas for concern even to the less experienced teacher. To use an example, some students may demonstrate fewer correct items in the items that are specifically testing spatial concepts. Further and often frequently, whole class or year groups may demonstrate a weakness in a specific sub-strand or content area, in this 14

15 case, Space. It may then be apparent that Year 3 and Year 5 demonstrate less ability in this sub-strand when compared to a national group. In turn this directs focus for schools and classrooms with new strategies for improved learning in this target area or for a target group. A specific program put into action can then be monitored over the next few courses of national testing and the effectiveness of implemented strategies evaluated. This promotes sound teacher pedagogical practices of identifying and addressing need, implementing strategies, monitoring, evaluating and re-evaluating. For classroom teachers, specific items in the mathematics tests can be studied and in the case of multiple choice items, distracters (incorrect options) can be analysed. Wellcrafted multiple choice options provide rich feedback as the distracters can operate on the different levels of student learning. Quality diagnostic information that is easily accessible to teachers can provide further pointers to student, class, year or school weaknesses and strengths. Sound and practical reporting is a crucial aspect of a large-scale testing program. The information must be readily accessible, easily interpretable and suit the needs of the key stakeholders. Schools and primary teachers require reports in a format that is easy to access and provides a greater level of detail. In Australia, this information is made available to schools and teachers either online or on CD. By interacting with the data, teachers can closely examine trends at the different school levels, comparing their school and class results with national and state means, and clusters of like schools. Perhaps the most prominent international mathematics large-scale assessment involving primary school students is the Trends in International Mathematics and Science Study (TIMSS), undertaken every four years with Year 4 and Year 8 students, providing data about national and international trends in mathematics and science achievement (Thomson 2007). The curriculum focus of TIMSS is threefold. The intended curriculum, the implemented curriculum and the attained curriculum are all examined, with the implemented curriculum being that which is interpreted and delivered by classroom teachers and considers the opportunities that are provided for students to learn mathematics, the varied instructional practices among countries and the factors which influence these variations (Thomson 2007). The released TIMSS items, along with the scoring guide for each item and item almanacs, are currently available. These include the statistics for each item shown for each participating country; the number of students attempting each item, the percent who chose the correct response, percentage of students who chose each incorrect option in multiple choice items and the percentage who responded to various marking codes for constructed response items; extending to the percentage of boys and girls in each country who selected the correct answer. This type of information allows teachers to engage with the data, gaining insight into how items in international assessments are constructed and scored, then transferring what is applicable to their own testing. Challenges for Large Scale Systemic Assessments A danger for teachers overly concentrating on the results of large-scale assessment is the potential to inadvertently narrow the curriculum being taught, choosing to focus instead only on the content which is covered by such tests. Another temptation for primary teachers of mathematics may be to rush from one topic to another, in essence to tick off one content area, to ensure all topics have been reached by the appointed date of the national 15

16 test. The specific needs of each class in terms of the programming, sequencing and delivering of learning experiences should not be driven by impending test dates. Primary teachers are well-positioned to determine the depth and coverage of content areas for the students in their care. Well-supported teachers provided with professional development opportunities can develop greater capacity in challenging all students within the basic topics including advanced students. Students, teachers and researchers can all be considered to be learners; partners in an international collaboration to develop new knowledge and to understand and improve the practices and outcomes of our classrooms (Clarke and Novotna 2008, 89). In order for teachers to implement effective early mathematics education, they need to be supported by better teacher preparation and ongoing professional development opportunities. The teacher is the key to effective, high-quality mathematics education in early childhood classrooms (Lee and Ginsburg 2009). Concluding comments There are challenges specific to primary school teachers in the area of mathematics. Mathematics is a subject in which it is socially acceptable to profess inability or failure. There is a desperate need to not only raise standards in mathematics but to also raise the profile of mathematics (Hoyles 2010). A fundamental mathematical competency is the capacity to think and reason mathematically. This involves asking probing and exploratory questions about what is possible, what could happen under certain conditions, how one might go about investigating a certain situation, and analysing logically the connections among problem elements (PISA 2003, 32). Surely this characteristic is to be valued in mature independent functioning citizens in society and should be encouraged. The primary years are vital for fostering and maintaining positive attitudes towards mathematics. The importance of ability in early mathematics cannot be overstated with success in mathematics a predictor of academic success in later school years in both mathematics and English (Duncan et al 2007). Hoyles (2010) also believes that those who are denied access to opportunities to succeed in mathematics have fewer opportunities in later life. Standardised tests are shaped by and evolve in accordance with national policy agendas and educational structures. They have emerged as an important instrument in education policy and are used for measuring and monitoring the performance of individual pupils, schools and education systems (Eurydice 2009, 63). Despite its detractors, the presence of large scale assessment appears fixed with few signs of abating. Whilst these instruments are administered to a large cohort, the assessment materials and the results generated in the hands of teacher professionals can be dissected to have a positive effect at student level. By encouraging closer inspection of mathematics assessment instruments and the data provided by programs such as TIMSS and NAPLAN, teachers knowledge can be further expanded with a sharpened view of individual student capabilities. The challenge that faces all mathematics teachers is to learn the skills that enable them to engage with the data and information that come from large scale programs and use these to improve teaching, learning, pedagogy and student outcomes. 16

17 References 1. ADAMS, R. J. (2003) Cohort Testing: Are you expecting too much? Keynote address: 2003 Joint Roundtable Conference of the Australasian Accountability Network and the PMRT National Measurement Advisory Group. July 14 16, Darwin, Northern Territory. 2. BLACK, Paul; William, Dylan (1999) Assessment for Learning: Beyond the Black Box. Assessment Reform Group. University of Cambridge. 3. CLARKE, D.J. & Novotna, J. (2008) Classroom Research in Mathematics Education as a Collaborative Enterprise for the International Research Community: The Leaner's Perspective Study In O. Figueras, J.L. Cortina, S. Alatorre, T. Rojano, & A. Sepulveda (Eds.), Proceedings of the Joint Meeting of PME32 and PME-NA XXX, Volume 1, DAVEY, Lynn (1992) The case for a national testing system Practical Assessment, Research & Evaluation, 3(1). Retrieved February 11, 2010 from This paper has been viewed 21,333 times since 11/13/ DUNCAN, G. J., Dowsett, C. J., Claessens, A., Magnuson, K., Huston, A. C., Klebanov, P., et al. (2007) School readiness and later achievement Developmental Psychology, 43(6), EACEA; Eurydice (2009) National Testing of Pupils in Europe: Objectives, Organisation and Use of Results. Brussels: Eurydice Network. 7. HOUSE OF COMMONS (2007), Testing and assessment: Third report of Session , Children,Schools and Families Committee, London 8. LEE, J.S; Ginsburg, H.P (2009) Early childhood teachers misconceptions about mathematics education for young children in the United States Australasian Journal of Early Childhood 34(4) December MONS, Nathalie (2009) Theoretical and Real Effects of Standardised Assessment, literature review. Eurydice Network. 10. OECD (2005) Formative Assessment Improving Learning in Secondary Classrooms. Paris: OECD, THOMSON, S; Buckley, S (2009) Informing mathematics pedagogy : TIMSS 2007 Australia and the world. Australian Council for Educational Research, Melbourne. Contact address Ms Frances Eveleigh; MEd UNSW, BEd UNSW, Dip Teach Syd CAE Research Fellow, Australian Council for Educational Research (ACER) 1/140 Bourke Road, Alexandria NSW 2015 Phone: eveleigh@acer.edu.au 17

18 ACTA UNIVERSITATIS PALACKIANAE OLOMUCENSIS FACULTAS PAEDAGOGICA 2010 MATHEMATICA VII DEVELOPMENTS IN USING VISUALIZATION, FUNCTIONAL MATERIALS AND ACTIONS IN TEACHING MATHEMATICS: UNDERSTANDING OF RELATIONS, MAKING GENERALIZATIONS AND SOLVING PROBLEMS (PART I) George MALATY Including figures in textbooks had been a difficult task in the 19 th century. Therefore, in teaching mathematics, Geometric textbooks were the only ones having figures. The development of print technology in the 20 th century has encouraged publishers and authors to include figures in mathematics textbooks, up to using figures unrelated to the content discussed and even unrelated to mathematics. In Finland, our work with teachers through in-service and pre-service education, and as well our work in mathematical clubs, started from Joensuu, have made figures use of special meaning in visualizing mathematical ideas. Visualization developed, by us, has become of variety of roles. From one hand it assists children to construct and understand mathematical concepts and relations, and from the other hand it is a tool to demonstrate a mathematical problem to children and as well a tool for children to solve mathematical problems and pose mathematical problems. We do use a type of dynamic visualization, where the main property it has is being isomorphic to the visualized mathematical entity. For more accuracy, our approaches to achieve understanding and leading to discovery are not depending on only visualization, but among others functional materials and actions. Indeed, visualization, functional materials and actions could be alternatives, but in different cases two or the three are needed. In using more than one of these facilitators, one could have a major role, but also the used facilitator could be of equal value. These facilitators can be used effectively for not only Primary School, but as well, in some cases, in Secondary School and Higher Education. About the terms used by us, we do use functional materials instead of the commonly used term manipulatives. This is related to our type of using these materials, where most of them have to be made by teachers and students for use in specific cases. About actions, we mean the physical participation of students in a performance designed by the teacher. Finally, we have to remember that there is no facilitator, which can bring alone understanding and leading to discovery of concepts and relations, the main facilitator is the teacher, who is the architect of the learning process. Visualization, geometry teaching and geometry development The oldest Finnish mathematical textbook, known to me, is a Geometry textbook written in Swedish by Henrik Heikel. The Second Edition of 1844 is the one available to me (Heikel 1844). This textbook includes 92 pages without any single geometric figure. After these pages, the textbook has 7 pages of 121 geometric figures. To make the textbook readable, figures are numerated, while in the margins of text pages the needed figures are mentioned 18

19 (Fig. 1). In addition, the size of each of figures page is of doubled size of text page. The left half of figures page is empty, and only the right half has figures and folded over the left one. This manual printing technique makes the reading of the book possible by opening the needed folded figures page to bring the figure or figures needed close to the corresponding text (Fig. 2). For instance Fig.81 is mentioned in page 66 at the left margin (Fig. 1), while in the sixth page of figures pages, Fig. 81 is presented (Fig. 2). Three years later the problem of including figures with text in the same page was solved in the Finnish translation of the same textbook (Heikel 1847, 77). Here I cannot avoid the attraction of making comment on the elegancy of the problems provided in this textbook. For instance, in Fig. 88 among others, it is easy to prove that there are four congruent quadrilaterals, and the intersection of two equivalent hexagons in a right triangle leads to an elegant proof of the Pythagorean theorem. Fig.1. Heikel 1844 textbook, p.66 Fig.2. Sixth page of the 7 figures pages of Heikel s 1844 textbook 19

20 Having figures in printed textbook in the 19 th Century wasn t easy task, but this was inevitable task for Geometry textbooks. This can be seen from the manual work in printing the first Geometry textbook in Finland (Fig. 2). Visualization had been the way through which not only geometry has been developed. But Geometry, with its visual images, had led the mankind to find a new way of thinking. This is because deductive thinking, which is much wider than syllogism, had been emerged through working in geometry. Since Euclid (fl. 300 BC), and even before him, visualization in geometry has been of essential role in providing images of abstracts, starting from visualizing a point, that which has no part by the first definition of Euclid s Elements (Fitzpatrick 2008, 6). Visualization had given a relevant environment for thinking, to move from intuition to deduction. Euclid s definitions, starting from that of a point wouldn t be exceptions. In addition, visualization and its use in searching for proofs had assisted the progress in building a deductive system to discover the need for axioms. In the way of writing the axioms, Euclid didn t use figures in presenting his axioms, but the statements he wrote reflects imagination of corresponding figures. This was also the case for Hilbert, 2300 years later, in writing, the regarded, more rigorous form of Euclid s axioms. But, in addition, Hilbert has used figures in presenting some of his axioms. The first two axioms of Group II, Axioms of Order, are good examples. The First one states that If A, B, C are points of a straight line and B lies between A and C, then B lies also between C and A (Hilbert 1902, 3). This statement would be enough to get imaginary visualization of the corresponding figure, but Hilbert had provided a corresponding figure as in Fig.3 (Hilbert 1902, 4). Fig.3. The figure used by Hilbert in his book: The Foundations of geometry (English translation, 1902, p.4) Going back to Euclid s definition of a point, despite the fact that the Elements of Euclid starts with the definition of a point, no doubt that this was a synthesis of earlier findings in geometry, started to be seen in Greece three hundreds before Euclid s elements. This means that Euclid s Elements, among others his definition of a point was effected by a work based on interaction between visual thinking and abstract thinking. Visualization in teaching arithmetic Despite the progress achieved in the 19 th century, in using figures in Geometry textbooks, till the end of that century no figures has appeared in neither arithmetic, nor Algebra textbooks, at least in Finland. Nevertheless, in 1888 arithmetic textbook of Ernst Bonsdorff we can see a way of visualizing arithmetical ideas by using mathematical text. This we can see among others in using letters S, K, Y to mean Hundreds, Tens and Ones to justify the performance of addition in a column, which the textbook presented at the bottom of the same page, as seen in Fig. 4 (Bonsdorff 1888, 6). 20

21 Fig.4. Bonsdorff 1888 textbook, p.6 Fig.5. Ceder 1912 textbook, p.32 In the first two decades of the 20 th century figures started to be a tool to present arithmetic ideas, but the progress of printing gave a chance to include other types of figures in mathematics textbooks. One of our best textbooks authors of the 20 th century, Kaarlo Rober Ceder, had used a cube to represent the relation between volume unites: cube meter, cube decimeter, cube centimeter and cube millimeter in his arithmetic textbook of 1912 (Ceder 1912, 32). An example of the other type of use, we can see from a textbook of Kaarlo Aukusti Merikoski (Merikoski 1963, 30), where in giving simple word problems, related to animals, a figure of a field and cows were presented. The solution of each word problem was in need of making a subtraction, and the animals mentioned Fig.6. Merikoski 1963 textbook, p. 30 in word problems were not always cows. These two types of visualization have been used in the first half of the 20 th century and both have continued to exist up to the date. In some more details: the 1970s textbooks, of the New Math era were of less figures of the type of Fig. 6, but in the 1980s, at the time of Back-to-Basics era, such figures, and even others of no relation neither to the text nor to the discussed topic, had increased. In the 1970s, for Primary School textbooks, the focus was on presenting figures of sets, where this presentation was not always correct. For instance, for the addition of two numbers (Fig. 7), in an artificial way two disjoint sets 21

22 were presented and then encircled to get the so-called union of the two sets (Rikala, Suni, Virtanen 1970, 15). For Secondary Schools, sets presentations were used to form typical Venn Diagrams, and other types alike were used, among them arrow diagrams in studying the topic of Relations and Functions (Wolff, Holmström & Paltakari 1974, 109). Here to notice that, mathematics textbooks since the time of the New Math are written mainly by teachers and not mathematicians as before. The main reason was the opposition of mathematicians, to the New math movement. A good example of this opposition can be seen from the paper of Rolf Nevanlinna (Nevanlinna 1966). Fig Rikala, Suni & Virtanen 1970 textbook, p. 15 ( tai means or ) Fig.8. Wolff, Holmström & Paltakari textbook 1974, p.109 In the 1980s, where Sets based curriculum was already abandoned, the development of color print technology encouraged textbooks authors and publishers, to present colorful figures in textbooks, related or not related to the discussed topic. But, where the problems were simple, the exaggeration of using colors made the perception of the figures difficult for both students and teachers. Functional materials use in teaching arithmetic In our work, instead of the commonly used term manipulatives, we are using the term Functional Materials. The reason is that the materials we use haven t to manipulate children, but assist them to understand and discover. Till the beginning of the second half of the 1980s, the use of Functional Materials wasn t common in Finnish schools. Nevertheless, here to mention that one of our best authors of Primary School textbooks of the 1950s and 1960s, Toivo Vahervuo, has used a type of visualization in his textbooks, which he turned into a type of functional materials named Satatalo, i.e. Hundred House. Figures 9, 10 and 11 are some of Vahervuo s textbooks. They are good enough to present the idea of constructing Satatalo and its possible uses (Vahervuo 1965, 95, 106, 101). Fig.9. Vahervuo 1965 textbook, p.95 Fig.10. Vahervuo 1965 textbook, 22

23 Fig.11. Vhervuo 1965 textbook, p.101 Figure 10 is closed to the form of Satatalo and in Figure 11 the letter K stands for Kymmenen, which means Ten. In Satatalo, plastic rectangular material replaced each rectangle in Fig. 10. On the back of each rectangular material, the letter K was written. The ten red circles of each rectangle in Fig.10 were replaced in Satatalo by 10 circular plastic materials. Each plastic rectangular material of Satatalo has 10 relevant pockets, to allow the fixing of the needed number of circular materials. Having two faces for both of the rectangular and circular materials explains one of the advantages of functional materials, which do not easy to achieve in using figures. In Satatalo one face of the circular material was black and the other face was white, and this gives the chance to make a needed discrimination. Satatalo was easy to fold. For individual use, smaller form of Satatalo was available. A use of other two Functional Materials had started in the 1970s. These two are the older version of Dienes Multibase Arithmetic Blocks (MAB) and Unifix blocks. To some extent the visits of Tamas Varga s ( ) to Finland in the 1970s had been of affect in bringing these Functional Materials to Finland. Varga was deeply impressed by the work his fellow Hungarian Zoltan P. Dienes had made outside Hungary, in particular Dienes s uses of Functional Materials (Servais, W. & Varga, T., 1971). The older version of MAB is the wooden one of the late 1950s. Where the use of MAB was not common in any time in Finnish Schools, with the exception of the 10-base blocks set, which has gained popularity in the last 10 years, and are named Decimal System tools. Unifix materials had been of more use in the 1970s, and it is still used. One of the advantages emphasized of using Unifix blocks was the significantly less noise it brings to classroom, in contrast to the case of using wooden materials. The other emphasized advantage was the easy way of stringing the blocks. These two properties, of Unifix materials, were the ones motivated Charles Tracey to invent them in England at the beginning of the 1950s. About the using of functional materials in Finland, and worldwide, we may need to give some brief background. At least, three names cannot be forgotten, one is Friedrich Froebel ( ), the second is Maria Montessori ( ) and the Third is Lev Vygotsky ( ). They are not only pioneers in this field, but in addition they have effected on others work later, in particular the affect of Froebel on Cuisenaire s and Montessori on Dienes s. At the time of the New Math, some of those involved in curriculum reform were interested in finding a way to make the abstract structure of the new curriculum achievable for children, and accepted by their parents. The funds gave for curriculum reform, at that time, gave the chance to the spread of materials developed earlier than the New Math movement. Besides Unifix Blocks, we can mention to the spread of Cuisenaire Rods, Geoboard and Vygotsky s Logical Blocks. As the case of Unifix Blocks, Cuisenaire Rods and Geoboard were 23

24 developed at the beginning of the 1950s. The Belgium primary school teacher Georges Cuisenaire ( ) had developed his rods with the help of the Egyptian-French mathematician Caleb Gattegno ( ), who is the inventor of the Geoboard (Servais & Varga 1971, Malaty 1988). Regards Finland, Cuisenaire Rods, Geoboard, Vygotsky s Logic Blocks and others were known to those involved in New Math reform, but these materials hadn t got a wide spread in schools at the time of the New Math, in particular Cuisenaire Rods were found complicated. Actions role in teaching mathematics and practical-theoretical reasons Besides visualization and Functional Materials, I am speaking about actions use in teaching mathematics in the title of my paper. The reason for this addition is that achieving understanding is not always possible by using visualization, functional materials or both. By actions we mean the physical participation of students in tasks designed by the teacher to achieve specific goal. This may sound relevant to young children. This in general true, but in our experience we have found that even for adult students, actions could be of special role in leading to discovery. As an example, in teaching university students, with no background in permutations, I have found that, making a row of three chairs in front of students in auditorium and asking three students to investigate the possible ways of their sitting on the chairs have been of special affect in understanding the permutation concepts. After this first investigation, I asked the students to return to their places, and then I asked the entire group to think about the number of possible permutations of sitting three students on three chairs. When some of the students failed in find the correct number, I asked again the three students to come to the front for assistance. This time, I asked the whole group about the alternative cases of sitting of one of the three students on the so-called first chair. This was an easy question to them, but the simple actions of students had been a key issue to find the formula n P r = n ( n 1) ( n 2 )...( n r+ 1), for the number of permutations of r objects taken from n distinct objects. This doesn t mean that using of actions lead by itself to discovery. Teacher s heuristic questions and the way of writing students answers are of a vital role. The teacher has to think and write in details about his plans, where this plan should be of a manuscript type. Here we have to say that in our work we do underline teacher s role as the main facilitator. In using visualization, functional materials or actions, teacher s work is the work of the architect of learning, and learning for us is that of understanding and discovering. Developing the use of visualization, functional materials and actions Since the middle of the 1980s a large part of my work has become related to Primary School Mathematics. This, among others, has included pre-service and inservice education of teachers, and the establishment of mathematical clubs, started by the one established by me in 1988 at the Normal School, i.e. the Practice School of the University of Joensuu. Through this work, developing the use of visualization, functional materials and actions in Primary School was one of my main interests, and part of this use I am going to present in my paper. This explains, why my previous discussion was related to the time before the middle of the 1980s. Some of my work was related to developing the use of Satatalo and Unifix materials, but some of that 24