Teaching mathematics and programming : new approaches with empirical evaluation

Programming and mathematics are core areas of computer science (CS) and consequently also important parts of CS education. Introductory instruction in these two topics is, however, not without problems. Studies show that CS students find programming difficult to learn and that teaching mathematical topics to CS novices is challenging. One reason for the latter is the disconnection between mathematics and programming found in many CS curricula, which results in students not seeing the relevance of the subject for their studies. In addition, reports indicate that students' mathematical capability and maturity levels are dropping. The challenges faced when teaching mathematics and programming at CS departments can also be traced back to gaps in students' prior education. In Finland the high school curriculum does not include CS as a subject; instead, focus is on learning to use the computer and its applications as tools. Similarly, many of the mathematics courses emphasize application of formulas, while logic, formalisms and proofs, which are important in CS, are avoided. Consequently, high school graduates are not well prepared for studies in CS. Motivated by these challenges, the goal of the present work is to describe new approaches to teaching mathematics and programming aimed at addressing these issues: Structured derivations is a logic-based approach to teaching mathematics, where formalisms and justifications are made explicit. The aim is to help students become better at communicating their reasoning using mathematical language and logical notation at the same time as they become more confident with formalisms. The Python programming language was originally designed with education in mind, and has a simple syntax compared to many other popular languages. The aim of using it in instruction is to address algorithms and their implementation in a way that allows focus to be put on learning algorithmic thinking and programming instead of on learning a complex syntax. Invariant based programming is a diagrammatic approach to developing programs that are correct by construction. The approach is based on elementary propositional and predicate logic, and makes explicit the underlying mathematical foundations of programming. The aim is also to show how mathematics in general, and logic in particular, can be used to create better programs.

Enabling constructive alignment in programming instruction

In this thesis, simple methods have been sought to lower the teacher’s threshold to start to apply constructive alignment in instruction. From the phases of the instructional process, aspects that can be improved with little effort by the teacher have been identified. Teachers have been interviewed in order to find out what students actually learn in computer science courses. A quantitative analysis of the structured interviews showed that in addition to subject specific skills and knowledge, students learn many other skills that should be mentioned in the learning outcomes of the course. The students’ background, such as their prior knowledge, learning style and culture, affects how they learn in a course. A survey was conducted to map the learning styles of computer science students and to see if their cultural background affected their learning style. A statistical analysis of the data indicated that computer science students are different learners than engineering students in general and that there is a connection between the student’s culture and learning style. In this thesis, a simple self-assessment scale that is based on Bloom’s revised taxonomy has been developed. A statistical analysis of the test results indicates that in general the scale is quite reliable, but single students still slightly overestimate or under-estimate their knowledge levels. For students, being able to follow their own progress is motivating, and for a teacher, self-assessment results give information about how the class is proceeding and what the level of the students’ knowledge is.

Conceptions of mathematics teacher education : thoughts among teacher educators in Tanzania

This study addresses the question of teacher educators’ conceptions of mathematics teacher education (MTE) in teacher colleges in Tanzania, and their thoughts on how to further develop it. The tension between exponents of content as opposed to pedagogy has continued to cause challenging conceptual differences, which also influences what teacher educators conceive as desirable in the development of this domain. This tension is connected to the dissatisfaction of parents and teachers with the failure of school mathematics. From this point of view, the overall aim was to identify and describe teacher educators’ various conceptions of MTE. Inspired by the debate among teacher educators about what the balance should be between subject matter and pedagogical knowledge, it was important to look at the theoretical faces of MTE. The theoretical background involved the review of what is visible in MTE, what is yet to be known and the challenges within the practice. This task revealed meanings, perspectives in MTE, professional development and assessment. To do this, two questions were asked, to which no clear solutions satisfactorily existed. The questions to guide the investigation were, firstly, what are teacher educators’ conceptions of MTE, and secondly, what are teacher educators’ thoughts on the development of MTE? The two questions led to the choice of phenomenography as the methodological approach. Against the guiding questions, 27 mathematics teacher educators were interviewed in relation to the first question, while 32 responded to an open-ended questionnaire regarding question two. The interview statements as well as the questionnaire responses were coded and analysed (classified). The process of classification generated patterns of qualitatively different ways of seeing MTE. The results indicate that MTE is conceived as a process of learning through investigation, fostering inspiration, an approach to learning with an emphasis on problem solving, and a focus on pedagogical knowledge and skills in the process of teaching and learning. In addition, the teaching and learning of mathematics is seen as subject didactics with a focus on subject matter and as an organized integration of subject matter, pedagogical knowledge and some school practice; and also as academic content knowledge in which assessment is inherent. The respondents also saw the need to build learner-educator relationships. Finally, they emphasized taking advantage of teacher educators’ neighbourhood learning groups, networking and collaboration as sustainable knowledge and skills sharing strategies in professional development. Regarding desirable development, teacher educators’ thoughts emphasised enhancing pedagogical knowledge and subject matter, and to be determined by them as opposed to conventional top-down seminars and workshops. This study has revealed various conceptions and thoughts about MTE based on teacher educators´ diverse history of professional development in mathematics. It has been reasonably substantiated that some teacher educators teach school mathematics in the name of MTE, hardly distinguishing between the role and purpose of the two in developing a mathematics teacher. What teacher educators conceive as MTE and what they do regarding the education of teachers of mathematics revealed variations in terms of seeing the phenomenon of interest. Within limits, desirable thoughts shed light on solutions to phobias, and in the same way low self-esteem and stigmatization call for the building of teacher educator-student teacher relationships.

Mathematics inspired by Darwin. Adaptive dynamics of dispersal and cooperation

In 1859, Charles Darwin published his theory of evolution by natural selection, the process occurring based on fitness benefits and fitness costs at the individual level. Traditionally, evolution has been investigated by biologists, but it has induced mathematical approaches, too. For example, adaptive dynamics has proven to be a very applicable framework to the purpose. Its core concept is the invasion fitness, the sign of which tells whether a mutant phenotype can invade the prevalent phenotype. In this thesis, four real-world applications on evolutionary questions are provided. Inspiration for the first two studies arose from a cold-adapted species, American pika. First, it is studied how the global climate change may affect the evolution of dispersal and viability of pika metapopulations. Based on the results gained here, it is shown that the evolution of dispersal can result in extinction and indeed, evolution of dispersalshould be incorporated into the viability analysis of species living in fragmented habitats. The second study is focused on the evolution of densitydependent dispersal in metapopulations with small habitat patches. It resulted a very surprising unintuitive evolutionary phenomenon, how a non-monotone density-dependent dispersal may evolve. Cooperation is surprisingly common in many levels of life, despite of its obvious vulnerability to selfish cheating. This motivated two applications. First, it is shown that density-dependent cooperative investment can evolve to have a qualitatively different, monotone or non-monotone, form depending on modelling details. The last study investigates the evolution of investing into two public-goods resources. The results suggest one general path by which labour division can arise via evolutionary branching. In addition to applications, two novel methodological derivations of fitness measures in structured metapopulations are given.

Assessment of Young Learners’ English Proficiency in Bilingual Content Instruction CLIL

This applied linguistic study in the field of second language acquisition investigated the assessment practices of class teachers as well as the challenges and visions of language assessment in bilingual content instruction (CLIL) at primary level in Finnish basic education. Furthermore, pupils’ and their parents’ perceptions of language assessment and LangPerform computer simulations as an alternative, modern assessment method in CLIL contexts were examined. The study was conducted for descriptive and developmental purposes in three phases: 1) a CLIL assessment survey; 2) simulation 1; and 3) simulation 2. All phases had a varying number of participants. The population of this mixed methods study were CLIL class teachers, their pupils and the pupils’ parents. The sampling was multi-staged and based on probability and random sampling. The data were triangulated. Altogether 42 CLIL class teachers nationwide, 109 pupils from the 3rd, 4th and 5th grade as well as 99 parents from two research schools in South-Western Finland participated in the CLIL assessment survey followed by an audio-recorded theme interview of volunteers (10 teachers, 20 pupils and 7 parents). The simulation experimentations 1 and 2 produced 146 pupil and 39 parental questionnaires as well as video interviews of volunteered pupils. The data were analysed both quantitatively using percentages and numerical frequencies and qualitatively employing thematic content analysis. Based on the data, language assessment in primary CLIL is not an established practice. It largely appears to be infrequent, incidental, implicit and based on impressions rather than evidence or the curriculum. The most used assessment methods were teacher observation, bilingual tests and dialogic interaction, and the least used were portfolios, simulations and peer assessment. Although language assessment was generally perceived as important by teachers, a fifth of them did not gather assessment information systematically, and 38% scarcely gave linguistic feedback to pupils. Both pupils and parents wished to receive more information on CLIL language issues; 91% of pupils claimed to receive feedback rarely or occasionally, and 63% of them wished to get more information on their linguistic coping in CLIL subjects. Of the parents, 76% wished to receive more information on the English proficiency of their children and their linguistic development. This may be a response to indirect feedback practices identified in this study. There are several challenges related to assessment; the most notable is the lack of a CLIL curriculum, language objectives and common ground principles of assessment. Three diverse approaches to language in CLIL that appear to affect teachers’ views on language assessment were identified: instrumental (language as a tool), dual (language as a tool and object of learning) and eclectic (miscellaneous views, e.g. affective factors prioritised). LangPerform computer simulations seem to be perceived as an appropriate alternative assessment method in CLIL. It is strongly recommended that the fundamentals for assessment (curricula and language objectives) and a mutual assessment scheme should be determined and stakeholders’ knowledge base of CLIL strengthened. The principles of adequate assessment in primary CLIL are identified as well as several appropriate assessment methods suggested.

Portfolio as an Indicator of Young Learners’ English Proficiency in Mainstream Language Instruction (EFL) and Bilingual Content Instruction (CLIL)

Tämä soveltavan kielitieteen ja kielitaidon arvioinnin toimintatutkimus tarkasteli kieliportfolion ominaisuuksia ja mahdollisuuksia nuorten oppijoiden englannin kielen arvioinnissa kahdessa eri oppimiskontekstissa: englanti oppiaineena (EFL) ja kaksikielinen sisällönopetus (CLIL). Tutkielman itsenäiset, kahteen eri englannin kielen rekisteriin (arkikieli ja akateeminen kieli) kohdistuneet portfoliokokeilut olivat erillisiä tapaustutkimuksia. Molemmat portfoliot perustuivat väljästi Eurooppalaiseen kielisalkkumalliin, ja ne olivat osa tutkielmantekijän luokkaopetusta ja -toimintaa. EFL -portfoliokokeilu 9-10-vuotiaille kolmasluokkalaisille toteutettiin marraskuun 2011 ja toukokuun 2012 välisenä aikana, kun CLIL -portfoliokokeilu n. 7-9-vuotiallle ensimmäisen ja toisen luokan oppilaille kesti kaksi lukuvuotta 2012–2014. Molemmissa kokeiluissa myös oppilaiden vanhemmat kuuluivat tutkimusjoukkoon, samoin CLIL -portfolion toteutuksessa avustaneet ja opettajanäkökulmaa edustaneet opettajaopiskelijat. Portfoliokokeilun aloitti myös kaksi muuta CLIL -opettajaa, mutta kumpikin kokeilu päättyi alkuvaiheeseensa. Tarkemman tarkastelun kohteina olivat tutkimuksen osallistujien kokemukset ja mielipiteet portfoliokokeiluista. Erityisesti tavoitteena oli selvittää, miten informatiivisena englannin kielitaidon indikaattorina kieliportfoliota pidettiin. Myös kehitysehdotuksia kerättiin. Trianguloitu aineisto koottiin sekä puolistrukturoiduin kyselyin että vapaaehtoisin teemahaastatteluin, jotka äänitettiin. EFL -aineisto koostui 18 oppilaskyselystä, 17 huoltajakyselystä ja 7 oppilashaastattelusta. CLIL -aineistoon sisältyi 19 oppilaskyselyä, 18 huoltajakyselyä, 7 oppilashaastattelua ja yksi opettajaopiskelijoiden (N=3) ryhmähaastattelu. Aineisto analysoitiin pääosin kvalitatiivisin menetelmin temaattisen sisältöanalyysin keinoin, mutta myös laskien frekvenssejä ja prosenttisosuuksia. Osallistujien mielipiteet ja kokemukset olivat hyvin samankaltaiset ja positiiviset kummassakin portfoliokokeilussa. Merkittävä enemmistö sekä oppilaista että huoltajista koki, että portfolion avulla on mahdollista osoittaa englannin kielitaitoa ja sen kehittymistä. Oppilaat kuvailivat portfoliotyötä hauskaksi ja kivaksi, ja heidän mielestään portfoliotehtävien pitäisi olla tarpeeksi haastavia, sisältää taiteellisia ja luovia elementtejä sekä kohdistua tuttuihin, mielenkiintoisiin aiheisiin. He totesivat, että portfolion avulla voi oppia lisää kieltä. Vanhempien mielestä portfolio kertoo koulun vieraiksi jääneistä oppisisällöistä, auttaa ymmärtämään lapsen ajatusmaailmaa ja motivaatiotasoa sekä paljastaa heidän kielitaidostaan uusia ulottuvuuksia. Opettajaopiskelijat havaitsivat, että portfolion avulla voi tutustua oppilaiden kieli- ja kulttuuritaustoihin sekä kartoittaa heidän kielellisiä tarpeitaan. Tämän tutkielman teoreettisen tarkastelun ja tulosten mukaan kieliportfolio tukee erinomaisesti uuden Perusopetuksen Opetussuunnitelman (NCC 2014) tavoitteita ja arvioinnin uudistuspyrkimyksiä sekä lainsäädännön arvioinnille asettamia edellytyksiä. Portfolio on erittäin suositeltava nuorten oppijoiden kielitaidon arviointimenetelmä perinteisten rinnalle.

Utilising student profiles in mathematics course arrangements

This thesis develops a method for identifying students struggling in their mathematical studies at an early stage. It helps in directing support to students needing and benefiting from it the most. Thus, frustration felt by weaker students may decrease and therefore, hopefully, also drop outs of potential engineering students. The research concentrates on a combination of personality and intelligence aspects. Personality aspects gave information on conation and motivation for learning. This part was studied from the perspective of motivation and self-regulation. Intelligence aspects gave information on declarative and procedural knowledge: what had been taught and what was actually mastered. Students answered surveys on motivation and self-regulation in 2010 and 2011. Based on their answers, background information, results in the proficiency test, and grades in the first mathematics course, profiles describing the students were formed. In the following years, the profiles were updated with new information obtained each year. The profiles used to identify struggling students combine personality (motivation, selfregulation, and self-efficacy) and intelligence (declarative and procedural knowledge) aspects at the beginning of their studies. Identifying students in need of extra support is a good start, but methods for providing support must be found. This thesis also studies how this support could be taken into account in course arrangements. The methods used include, for example, languaging and scaffolding, and continuous feedback. The analysis revealed that allocating resources based on the predicted progress does not increase costs or lower the results of better students. Instead, it will help weaker students obtain passing grades.