Bridging Mathematics with Word Problems

The aim of this thesis was to explore several important aspects of word problems: the nature of word problems used in school mathematics textbooks and the difficulty level of different types of word problems. The specific goals were to investigate students’ performance when solving various types of word problems and to determine whether students’ word-problem skills and their beliefs about word problem-solving can be improved by enriching word problems used in mathematics teaching. To achieve the goals, this thesis reports on five original studies, as follows. Study I showed a comparison between the characteristics of word problems presented in Thai and Finnish school mathematics textbooks. The analyses included 1,565 word problems from a series of second- to fourth-grade Thai and Finnish mathematics textbooks. The overall results show that the nature of word problems used in Finnish textbooks vary from Thai textbooks in many ways. Finnish textbooks contain more multistep word problems, while in Thai textbooks, one-step word problems appear more frequently. Thai textbooks have a smaller percentage of repetitive sections (ones that include only the same type of problems) than Finnish textbooks. In both countries, the percentage of word problems requiring the use of realistic considerations is extremely low, less than five percent of the total. Studies II and III presented the impacts of a Word Problem Enrichment (WPE) programme, developed to encourage teachers to use innovative self-created word problems to improve student mathematical modelling and problem-solving skills. Participants comprised 10 classroom teachers and their 170 students from fourth and sixth grades, from elementary schools in southwest Finland. In Study II, the intervention effectiveness on student problem-solving performance was investigated. The results suggested that enriching word problems used in mathematics teaching is a promising method for improving student problem-solving skills when solving non-routine and application word problems. However, it is not known if WPE has an effect on student beliefs about word problem-solving, and how the programme works for students with different initial motivation in learning mathematics. Study III examined the effectiveness of WPE on student beliefs about word problem-solving by using latent profile analysis (LPA) and structural equation modelling (SEM) to analyse relationships among the different cognitive, motivation, and belief factors. Results indicated that the impacts of WPE are various depending upon the initial motivation level of students. The effects of WPE on student beliefs appeared only in students with a low initial motivation level, while its impacts on student problem-solving performance were found only in students with a high initial motivation level. Studies IV and V were conducted to examine hypotheses regarding (1) the dimensionality of students’ performance on word problems and (2) difficulty level of three types of word problems: routine, non-routine and application word problems by utilizing item response theory (IRT) modelling. The data used in Study IV was collectedas part of the Word Problem project (Studies II and III). Participants comprised 170 fourth- and sixth-grade students. Students’ problem-solving performance was assessed with a word problem-solving test, including five word problems: one routine, three non-routine, and one application. The results of Study IV show that students’ performance on word problems can be seen as a unidimensional construct that denies the original assumption. The results of the IRT model indicate that the theoretically demanding application word problem has a higher difficulty level than non-routine and routine word problems. Nevertheless, the results are obscure if this application word problem (used in Study IV) is harder because of its demand for realistic considerations or other possibly relevant factors (e.g. decimal numbers included, division, more problem-solving steps required). Moreover, the sample size of Study IV could be considered relatively small for this kind of complicated IRT model. Therefore, Study V uses a larger sample size and a bigger set of word problems with more variety in application and non-routine word problems. The data used in Study V was collected as part of the Quest for Meaning project. Participants comprised 891 fourth-grade students (446 boys and 445 girls) from different elementary schools situated in cities, small towns, and rural communities in southern Finland. On the same lines as Study IV, the results of Study V indicated that students’ performance on word problems can be seen as a unidimensional construct. Concerning item difficulty level, the results of the IRT model do not show a clear distinction among word-problem types and reject the hypothesis that application word problems have a higher difficulty level than non-routine word problems. Some non-routine word problems appear to be more difficult than the application word problem, even though other characteristics of these two types of word problems were very similar (e.g., they required the same type of operation and the same number of problem-solving steps). The results of the five studies reveal that even though the mathematics textbooks were highly regarded in Thailand and Finland, most given word problems frequently include a simple goal without demanding any realistic considerations. These results strongly suggest that more innovative application word problems are definitely needed in classroom mathematics. In our study, we developed the WPE to encourage teachers to develop their own meaningful non-routine and applications word problems, and to use these self-created word problems to improve mathematical modelling and students’ word problem-solving performance. The results show that WPE is a promising approach to improve not only student problem-solving skills but also student beliefs about word problem-solving. The impacts of WPE are different depending upon students’ initial motivation level. The impacts of WPE on student beliefs were found only in students with a low initial motivation level, while its impacts on student problem-solving performance were found only in students with a high initial motivation level. These results suggest that in classroom practice, it is important that teachers provide enough support for students to be more confident and feel less overwhelmed when facing non-routine and application word problems. Teachers should be aware of differences of word-problem types and utilise this information in planning how to scaffold students’ word problem-solving by giving word problems based on their difficulty level.Väitöskirjatyö kohdistuu matematiikan sanallisten tehtävien tärkeisiin ominaisuuksiin: koulumatematiikassa hyödynnettävien sanallisten tehtävien luonteen sekä erityyppisten sanallisten tehtävien vaikeustason tarkasteluun. Keskeisinä tavoitteina oli tarkastella oppilaiden suoriutumista heidän ratkaistessaan erityyppisiä sanallisia tehtäviä ja selvittää, voidaanko oppilaiden sanallisten tehtävien ratkaisutaitoja ja heidän uskomuksiaan sanallisten tehtävien ratkaisuun liittyen parantaa rikastamalla matematiikan opetuksessa käytettäviä sanallisia tehtäviä. Näiden tavoitteiden saavuttamiseksi tässä väitöstutkimuksessa toteutettiin viisi osatutkimusta. Osatutkimuksessa I vertailtiin suomalaisissa ja thaimaalaisissa matematiikan oppikirjoissa käytettävien sanallisten tehtävien ominaisuuksia. Tutkimuksessa analysoitiin 1565 sanallista tehtävää suomalaisista ja thaimaalaisista eri oppikirjasarjojen toisen–neljännen luokan matematiikan oppikirjoista. Tulokset osoittivat, että suomalaisissa oppikirjoissa esiintyvät sanalliset tehtävät eroavat monin tavoin Thaimaassa käytössä olevien oppikirjojen tehtävistä. Suomalaisissa oppikirjoissa on enemmän useita välivaiheita sisältäviä sanallisia tehtäviä, kun taas thaimaalaisissa oppikirjoissa esiintyy enemmän yksivaiheisia sanallisia tehtäviä. Thaimaalaisissa oppikirjoissa on prosentuaalisesti vähemmän toistavia osioita (sisältävät ainoastaan tietyn tyyppisiä tehtäviä) kuin suomalaisissa oppikirjoissa. Molempien vertailtavien maiden oppikirjoissa sellaisten tehtävien osuus, joiden ratkaiseminen vaatii todellisten arkielämän näkökohtien huomioimista, on todella vähäinen, vain noin viisi prosenttia kaikista sanallisista tehtävistä. Osatutkimukset II ja III esittelivät niin sanotun Sanallisten Tehtävien Rikastaminen (STR) –ohjelman vaikutuksia, joka kehitettiin tarkoituksena rohkaista opettajia hyödyntämään opetuksessaan innovatiivisia, itse kehittelemiään sanallisia ongelmia parantamaan oppilaiden matemaattisen mallintamisen ja ongelmanratkaisun taitoja. Tutkittavina oli 10 luokanopettajaa ja heidän 170 oppilastaan neljänneltä ja kuudennelta luokalta varsinaissuomalaisista kouluista. Osatutkimuksessa II selvitettiin intervention vaikuttavuutta suhteessa oppilaiden ongelmanratkaisutaitoihin. Tulokset osoittivat, että matematiikan opetuksessa sanallisten tehtävien rikastaminen on lupaava menetelmä oppilaiden ongelmanratkaisutaitojen parantamiseksi, kun ratkaistaan ei-rutiininomaisia ja soveltamista vaativia sanallisia ongelmia. Tässä osatutkimuksessa jäi kuitenkin vielä epäselväksi, onko STR:llä vaikutusta oppilaiden uskomuksiin sanallisten ongelmanratkaisutehtävien ratkaisua kohtaan ja kuinka ohjelma vaikuttaa erilaisen motivaation matematiikan opiskelua kohtaan omaavien oppilaiden oppimiseen. Osatutkimuksessa III selvitettiin STR-ohjelman vaikuttavuutta oppilaiden uskomuksiin sanallisiin ongelmanratkaisutehtäviin liittyen hyödyntäen latenttia profiilianalyysia (LPA) ja rakenneyhtälömallinnusta (structural equation modelling, SEM), joiden avulla analysoitiin erilaisten kognitiivisten, motivationaalisten ja uskomuksiin liittyvien tekijöiden välisiä suhteita. Tulokset indikoivat, että STR-ohjelman vaikutukset ovat erilaisia riippuen oppilaiden motivaatiotasosta matematiikan opiskelua kohtaan. STR:n vaikutukset uskomuksiin näkyivät ainoastaan niiden oppilaiden kohdalla, joilla oli alhainen motivaatio, kun taas ohjelmalla oli vaikutuksia ongelmanratkaisutaitojen tasoon vain sellaisten oppilaiden osalta, joiden motivaatio oli korkea. Osatutkimuksissa IV ja V selvitettiin (1) sijoittuvatko oppilaiden suoritukset sanallisissa tehtävissä yhdelle vaikeusdimensiolle vai onko sanallisten tehtävien vaikeudessa eri dimensioita ja (2) kolmen tyyppisten sanallisten tehtävien (rutiininomaisetn, ei-rutiininomaiset ja soveltamista vaativat tehtävät) vaikeustasoa hyödyntämällä modernia testiosioiden mallinnusmenetelmääa (item response theory modelling, IRT). Tutkimuksen IV aineisto kerättiin osana sanallisten tehtävien interventioprojektia (vrt. Osatutkimukset II ja III). Tutkittavina oli 170 neljännen ja kuudennen luokan oppilasta. Oppilaiden suoriutumista sanallisista tehtävistä arvioitiin ongelmanratkaisutestillä, joka piti sisällään viisi sanallista tehtävää: yhden rutiininomaisen tehtävän, kolme ei-rutiininomaista tehtävää ja yhden soveltamista vaativan tehtävän. Osatutkimuksen IV tulokset osoittavat, että oppilaiden suoriutuminen sanallisista tehtävistä voidaan odotusten vastaisesti nähdä yksiulotteisena rakenteena. IRT-mallin tulokset antavat viitteitä, että teoreettisesti vaativampi soveltamista vaativa sanallinen tehtävä on vaikeustasoltaan haastavampi kuin ei-rutiininomaiset ja rutiininomaiset tehtävät. Tulosten avulla ei kuitenkaan voitu vielä selittää, johtuiko soveltamista vaativan tehtävän (vrt. Osatutkimus IV) vaikeus siitä, että sen ratkaiseminen edellytti realististen näkökohtien huomioimista vai mahdollisesti jotkin muut relevantit tekijät (esim. desimaalilukujen tai jakolaskujen sisältyminen, monivaiheisempi ongelmanratkaisuprosessi). Tämän lisäksi otoskoko Osatutkimuksessa IV oli suhteellisen pieni monimutkaisen testiosioiden mallinnusmenetelmän hyödyntämiseen. Tästä syystä Osatutkimuksessa V hyödynnettiin suurempaa otoskokoa ja laajempaa sanallisten tehtävien joukkoa, joka sisälsi monipuolisempia rutiininomaisia ja ei-rutiininomaisia tehtäviä. Osatutkimuksen V aineistona oli aiemmassa Merkitystä etsimässä –projektisa koottu laaja aineisto. Tutkittavina oli 891 neljännen luokan oppilasta (446 poikaa ja 445 tyttöä) suurehkoissa kaupungeissa, pikkukaupungeissa ja maaseudulla sijaitsevista alakouluista eripuolilta eteläistä Suomea. Linjassa Osatutkimuksen IV tulosten kanssa, myös Osatutkimuksen V tulokset antoivat viitteitä, että oppilaiden suoriutuminen sanallisissa tehtävissä voidaan selittää yksiulotteisella rakenteella. IRT-mallin tulokset eivät osoita selkeää eroa sanallisten tehtävien eri vaikeustasotyyppien välillä ja hylkäävät hypoteesin siitä, että soveltamista vaativien sanallisten tehtävien vaikeustaso olisi korkeampi kuin ei-rutiininomaisten tehtävien. Jotkut ei-rutiininomaiset sanalliset tehtävät näyttivät olevan vaikeampia kuin soveltamista vaativat tehtävät, vaikka muut ominaisuudet näiden kahden erityyppisten sanallisten tehtävien välillä olivat hyvin samankaltaiset (esim. vaativat samanlaisia laskutoimintoja ja yhtä monta välivaihetta). Viiden osatutkimuksen tulokset paljastavat, että vaikka matematiikan oppikirjoja pidetään yleisesti korkeatasoisina Thaimaassa ja Suomessa, suurin osa niissä olevista sanallisista tehtävistä sisältävät yksinkertaisen tavoitteen ilman että ne edellyttäisivät todellisen elämän tilanteiden huomioon ottamista Nämä tulokset osoittavat selkeästi, että kouluissa tarvitaan innovatiivisempia, soveltamista vaativia matematiikan sanallisia tehtäviä. Tutkimuksissamme kehitimme STR-ohjelman rohkaisemaan opettajia kehittämään itse omia ei-rutiininomaisia ja soveltamista vaativia tehtäviä ja hyödyntämään näitä itse kehitettyjä sanallisia tehtäviä parantaakseen matemaattista mallintamista ja oppilaiden sanallisissa ongelmanratkaisutehtävissä suoriutumista. Tulokset osoittavat, että STR tarjoaa lupaavan lähestymistavan parantaa oppilaiden ongelmanratkaisutaitojen lisäksi myös oppilaiden uskomuksia sanallisten tehtävien ratkaisemiseen liittyen. STR:n vaikutukset olivat erilaisia riippuen oppilaiden motivaatiotasosta. STR vaikutti vain sellaisten oppilaiden uskomuksiin, joilla oli alhainen motivaatio, kun taas ohjelman vaikutukset ongelmanratkaisutehtävissä suoriutumiseen oli nähtävissä ainoastaan niiden oppilaiden keskuudessa, joilla oli korkea motivaatio. Näiden tulosten mukaisesti on tärkeää, että opettajat tarjoavat riittävästi tukea oppilaille, jotta oppilaiden itsevarmuus parantuisi ja he tuntisivat itsensä vähemmän lannistuneiksi kohdatessaan ei-rutiininomaisia ja soveltamista vaativia sanallisia tehtäviä. Opettajien tulisi olla tietoisia erityyppisistä sanallisista tehtävistä ja hyödyntää tätä tietoa suunnitellessaan, kuinka tukea oppilaiden sanallisten tehtävien ongelmanratkaisua tarjoamalla vaikeustasoltaan erilaisia sanallisia tehtäviä.Siirretty Doriast

The impact of digital pedagogy training on in-service teachers' attitudes towards digital technologies

To prepare schools for the demands of the twenty-first century, teachers have been challenged to expand their use of digital technologies in their teaching. This study is a part of OpenDigi, which aimed to create teachers' communities for enhancing digital pedagogy skills and the use of digital assessment tools. The aim of this study is to investigate the impact of digital pedagogy training on in-service teachers' attitudes towards digital technologies. Self-report questionnaires were collected from 98 elementary and lower secondary school in-service teachers in southern Finland. A total of 22 in-service teachers completed both pre- and post-test. The results showed that the impact of the digital pedagogy training depended on teachers' ICT confidence level. Teachers who had low confidence in ICT use showed an increased ICT confidence level after the programme, while teachers who already had high confidence in ICT use showed no significant changes in their confidence level. Moreover, the results indicated that the need for ICT support was lower after the training for the teachers in high confidence group, while there were no significant changes in the need for ICT support for the teachers in the low confidence group. These results suggested that the training could fulfil teachers' needs for ICT support, particularly for the teachers in the high confidence group. The challenges teachers face in expanding the use of digital technologies in practice and their commitment to the digital culture are discussed

Primary School Student Teachers’ Perceived and Actual Knowledge in Biology

Individuals’ perceptions of their knowledge can have an important role in shaping their cognition and influencing their behaviour. However, there has been a scarcity of studies in biology on how perceived knowledge relates to actual knowledge. The focus of this article is on quantitative results analysing and interpreting student teachers’ perceived knowledge of biological content in relation to their actual animal and species name knowledge linked to the ecosystem in which they live. K-means cluster analysis and ANOVA were used. The results show a high- and low-level perceived knowledge cluster group among the participants. They further indicate that the difference in actual animal and species name knowledge between these cluster groups remained the same during the five years of the study. The student teachers with a higher level of perceived knowledge tended to have better actual animal and species name knowledge than those in the low-level group. The actual animal name knowledge in these cluster groups was similar with regard to the local Finnish ecosystems but differed concerning the exotic species by year. The year that the participants enrolled in the study programme had an impact on their actual animal and species name knowledge. Strategies for coping with work-related demands and maintaining engagement in one’s career would be important additions to the teacher education curriculum. </p

What makes mathematical word problem solving challenging? Exploring the roles of word problem characteristics, text comprehension, and arithmetic skills

In this study we investigated word-problem (WP) item characteristics, individual differences in text comprehension and arithmetic skills, and their relations to mathematical WP-solving. The participants were 891 fourth-grade students from elementary schools in Finland. Analyses were conducted in two phases. In the first phase, WP characteristics concerning linguistic and numerical factors and their difficulty level were investigated. In contrast to our expectations, the results did not show a clear connection between WP difficulty level and their other characteristics regarding linguistic and numerical factors. In the second phase, text comprehension and arithmetic skills were used to classify participants into four groups: skilful in text comprehension but poor in arithmetic; poor in text comprehension but skilful in arithmetic; very poor in both skills; very skilful in both skills. The results indicated that WP-solving performance on both easy and difficult items was strongly related to text comprehension and arithmetic skills. In easy items, the students who were poor in text comprehension but skilful in arithmetic performed better than those who were skilful in text comprehension but poor in arithmetic. However, there were no differences between these two groups in WP-solving performance on difficult items, showing that more challenging WPs require both skills from students.</p

The Role of Beliefs and Motivational Variables in Enhancing Word Problem Solving

A Word Problem Enrichment programme (WPE) has been found to increase student word problem solving performance when facing non-routine and application problems. However, it is unknown if the WPE has an impact on student beliefs about word problem solving, and how the WPE works for students with different motivation in learning mathematics. This study investigated the impact of the WPE on student beliefs about word problem solving by using advanced statistical methods (LPA and SEM) to analyse relations among the different cognitive, motivation, and belief factors. A total of 170 fourth- and sixth-grade students from elementary schools participated. Results showed that the effects of WPE are various depending on students’ initial motivation level. The impacts of the WPE on student beliefs were found only in students with a low initial motivation level, while its impacts on student problem-solving performance were found only in students with a high initial motivation level.</p

Adaptive number knowledge and its relation to arithmetic and pre-algebra knowledge

Traditionally measured skills with arithmetic are not related to later algebra success at levels that would be expected given the close conceptual relation between arithmetic and algebra. However, adaptivity with arithmetic may be one aspect of arithmetic competences that can account for additional variation in algebra attainment. With this in mind, the present study aims to present evidence for the existence and relevance of a newly acknowledged component of adaptivity with arithmetic, namely, adaptive number knowledge. In particular, we aim to examine whether there are substantial individual differences in adaptive number knowledge and to what extent these differences are related to arithmetic and pre-algebra skills and knowledge. Adaptive number knowledge is defined as the well-connected knowledge of numerical characteristics and relations. A large sample of 1065 Finnish late primary school students completed measures of adaptive number knowledge, arithmetic conceptual knowledge, and arithmetic fluency. Three months later they completed a measure of pre-algebra skills. Substantial individual differences in adaptive number knowledge were identified using latent profile analysis. The identified profiles were related to concurrent arithmetic skills and knowledge. As well, adaptive number knowledge was found to predict later pre-algebra skills, even after taking into account arithmetic conceptual knowledge and arithmetic fluency. These results suggest that adaptive number knowledge is a relevant component of mathematical development, and may help account for disparities in algebra development.</p

Effects of a mathematics game-based learning environment on primary school students' adaptive number knowledge

Developing adaptive expertise with arithmetic problem solving is a much desired aim of primary school mathematics education. However, there are very few practical tools for teachers that would aid reaching this complex mathematical learning goal. The aim of the present study was to test the effects of a game-based learning environment in supporting primary school students' adaptive number knowledge and related arithmetic skills. Participants were 1168 students in grades four, five, and six. Classes were randomized in two conditions: in the experimental group regular mathematics teaching was enriched with gameplay using the Number Navigation Game (NNG), and in the control group students continued according to their regular math curriculum. An experimental design with pre- and post-test was used to measure students' adaptive number knowledge, arithmetic fluency, and pre-algebra knowledge. Overall, results showed that the experimental group outperformed the control group on adaptive number knowledge and math fluency. Results showed varying effects of the training in different grade levels, with more pronounced improvement of students' adaptive number knowledge in grade five. Game performance was related to the experimental group's post-test scores even after controlling for pre-test scores and grade. Results suggest that the NNG is effective in enhancing different types of arithmetic skills and knowledge in different grades of primary school education and can provide teachers with a practical and flexible tool to extend their regular classroom practice

Predstava bodocih uciteljev razrednega pouka o njihovem znanju biologije in njihovo dejansko znanje biologije

Individuals’ perceptions of their knowledge can have an important role in shaping their cognition and influencing their behaviour. However, there has been a scarcity of studies in biology on how perceived knowledge relates to actual knowledge. The focus of this article is on quantitative results analysing and interpreting student teachers’ perceived knowledge of biological content in relation to their actual animal and species name knowledge linked to the ecosystem in which they live. K-means cluster analysis and ANOVA were used. The results show a high- and low-level perceived knowledge cluster group among the participants. They further indicate that the difference in actual animal and species name knowledge between these cluster groups remained the same during the five years of the study. The student teachers with a higher level of perceived knowledge tended to have better actual animal and species name knowledge than those in the low-level group. The actual animal name knowledge in these cluster groups was similar with regard to the local Finnish ecosystems but differed concerning the exotic species by year. The year that the participants enrolled in the study programme had an impact on their actual animal and species name knowledge. Strategies for coping with work-related demands and maintaining engagement in one’s career would be important additions to the teacher education (DIPF/Orig.

Primary School Student Teachers’ Perceived and Actual Knowledge in Biology

Individuals’ perceptions of their knowledge can have an important role in shaping their cognition and influencing their behaviour. However, there has been a scarcity of studies in biology on how perceived knowledge relates to actual knowledge. The focus of this article is on quantitative results analysing and interpreting student teachers’ perceived knowledge of biological content in relation to their actual animal and species name knowledge linked to the ecosystem in which they live. K-means cluster analysis and ANOVA were used. The results show a high- and low-level perceived knowledge cluster group among the participants. They further indicate that the difference in actual animal and species name knowledge between these cluster groups remained the same during the five years of the study. The student teachers with a higher level of perceived knowledge tended to have better actual animal and species name knowledge than those in the low-level group. The actual animal name knowledge in these cluster groups was similar with regard to the local Finnish ecosystems but differed concerning the exotic species by year. The year that the participants enrolled in the study programme had an impact on their actual animal and species name knowledge. Strategies for coping with work-related demands and maintaining engagement in one’s career would be important additions to the teacher education curriculum