´Now I Get It!´ : Developing a Real-World Design Solution for Understanding Equation-Solving Concepts

Strong conceptual understanding contributes to mathematics learning. Manipulatives (i.e., hands-on learning tools that allow for mathematical concept exploration through different senses) can facilitate students' understanding of mathematical concepts when used meaningfully. However, a body of research has demonstrated that although teachers have considered manipulatives pedagogically beneficial, when it comes to everyday classroom practice, they often prefer traditional teacher-centred and paper-and-pencil instruction. This doctoral research aims to develop a manipulative and its appropriate use to promote not only students' understanding of mathematical concepts, but also classroom adoption. Solving one-variable linear equations in primary school classrooms was used as a case study. An educational design research (EDR) approach was used throughout three phases of a 6-year enquiry: initial research, concept development, and design development. Phase 1 (initial research) was undertaken to gain a theoretical and contextual understanding and investigate existing manipulatives. In Phase 2 (concept development), four manipulative concepts were generated based on the Phase 1 findings. Each concept was then evaluated in terms of its pedagogical benefits and compatibility with school and classroom practice. During Phase 3 (design development), informed by the Phase 2 findings, a design solution (i.e., a tangible manipulative allowing physical input and providing digital output, student worksheets, teacher guides, and class activities) was developed. The developed design solution was then implemented and evaluated in classrooms. Empirical research was conducted in Finnish comprehensive schools. Altogether, 18 teachers, 98 primary school students, and 65 lower secondary school students took part in different phases of the research. The data were collected using mixed methods, including class interventions, paper-based tests, thinking aloud, questionnaires, and interviews. Qualitative and quantitative data collected from various methods and data sources were simultaneously analysed and then compared and combined to holistically understand the research results. Together, multiple iterations (of investigation, design, and assessment) resulted in practical and theoretical outcomes. The research-based design solution, which promotes students' understanding of equation-solving concepts and classroom practice, is the practical outcome of this research to directly improve educational practice. Additionally, the research contributes to three types of theories: domain theories, design frameworks, and design methodologies. The first theoretical outcome is a domain theory yielding two types of knowledge, that is, context and outcomes knowledge. The context knowledge describes the challenges and opportunities of using manipulatives in mathematics classrooms, as well as strengths and limitations of existing manipulatives. The outcomes knowledge describes outcomes of implementing the design solution: the developed tangible manipulative accompanied by the instructional materials enhanced students' understanding of equation-solving concepts through discovery learning, social interaction, and multimodal expression of mathematical thinking; the manipulative is likely to be adopted in the classroom because of its pedagogical benefits and compatibility with school and classroom practice. The second theoretical outcome is a design Jramework for real-world educational technologies. Content, pedagogy, practice, and technology should be taken into consideration when designing real- world educational technologies to ensure their educational benefits, utilisation, adoption, and feasibility. The third theoretical outcome is a design methodo/ogy built on firsthand experience from undertaking this EDR. The guidelines for conducting EDR guides how to embrace opportunities and overcome challenges that may emerge. This research contributes to a link between research and practice in mathematics education. It provides researchers with knowledge of how multimodal interaction with manipulatives enhances mathematics learning and guidelines for conducting EDR. It guides educational designers to take various aspects into consideration when designing educational technologies to improve real-world practice. Moreover, this research also has practical implications. First, it encourages teacher educators to prepare pre- and in-service teachers for successful incorporation of manipulatives in their mathematics classrooms. Second, it guides practitioners on how to support their students to benefit from manipulatives. Third, it urges schools to support the acquisition and utilisation of manipulatives. Finally, it calls on school curricula to encourage the use of manipulatives in the mathematics classroom to promote students' conceptual understanding

Luokanopettajaopiskelijoiden arviointeja ei-rutiinimaisista matematiikan tehtävistä

Matematiikan harjoitustehtävät ovat pysyneet oppimateriaaleissa rakenteiltaan samankaltaisinaniiden historian ajan. Tehtävätyypit ovat lähinnä suljettuja tehtäviä, jotkauseimmiten kehittävät vain jo aikaisemmin opittua ja tukevat siis proseduraalista sujuvuutta.Tehtävät voisivat kehittää entistä enemmän myös käsitteellistä ymmärrystä jastrategista sekä metakognitiivisia taitoja. Tässä tutkimuksessa on kehitetty matematiikanperinteisistä tehtävistä poikkeavia tehtävätyyppejä. Keväällä 2016 tekijöiden yliopiston1. vuosikurssin luokanopettajaopiskelijat (N=82) kokeilivat neljä uutta tehtävätyyppiäkurssin harjoituksissa ja arvioivat niiden matemaattisen sekä monilukutaidon osaamisensuhteen. Artikkelissamme kuvaamme neljää tehtäväympäristöä, opiskelijoiden arviointejaniistä sekä yhden tehtävän ratkaisuja matemaattisten käsitteiden käytön näkökulmasta

Constructing a design framework and design methodology from educational design research on real-world educational technology development

Educational design research (EDR) seeks to contribute to both practice and theory by developing solutions that improve educational practice and generating usable and generalisable knowledge. Most EDR researchers tend to focus on reporting their research contributions to educational practice. Therefore, there is a need for disseminating research that pays more attention to the theoretical contributions of EDR so that those outside a particular EDR project can benefit. This paper focuses on the theoretical contributions, particularly the design framework and design methodological knowledge, of a 6-year EDR enquiry that aimed to develop educational technologies that promote primary school mathematics learning and classroom practice. Informed by the literature and direct experiences of working in collaboration with teachers and various disciplines during this iterative study, a design framework for developing real-world educational technologies and guidelines for conducting EDR are proposed. The design framework highlights four essential aspects—content, pedagogy, practice, and technology—that should be considered when developing educational technologies to ensure their educational benefits, feasibility, and successful real-world utilisation and adoption. The proposed guidelines for conducting EDR, such as exploring design alternatives and employing appropriate design construction and evaluation methods, can assist other researchers, including a single doctoral student, in embracing opportunities and overcoming the challenges that may emerge.publishedVersionPeer reviewe

Learning equation solving using a technological manipulative and languaging

Kirjallisuuden ja aikaisempien tutkimusten mukaan perinteisesti koulumatematiikan opetuksessa painotettu proseduraalinen osaaminen ei yksinään riitä matematiikan oppimisen onnistumiseen. Vuoden 2014 perusopetuksen opetussuunnitelman perusteet painottavat oppilaiden matemaattisten käsitteiden ymmärryksen merkitystä. Tässä artikkelissa esitellään Lehtosen väitöstutkimus esimerkkinä, kuinka oppimisteknologian käyttö voi tukea alakoululaisten matemaattisten käsitteiden ymmärrystä. Väitöstutkimuksen aikana kehitettiin moniesitysmuotoinen oppimisväline alakoululaisten yhtälönratkaisun oppimista varten. Kehitetty oppimisväline yhdistää fyysisten ja digitaalisten välineiden vahvuudet: fyysisten osien liikkuminen saa oppilaat ajattelemaan omaa toimintaa, kun taas digitaaliset osat motivoivat oppilaita oppimaan sekä mahdollistavat reaaliaikaisen ohjauksen ja palautteen saamisen. Luokkakokeilussa osoitettiin, että kehitetyn oppimisvälineen käyttö yhdessä matemaattisen ajattelun kielentämisen mallin kanssa tuki neljäsluokkalaisten yhtälönratkaisun käsitteiden oppimista. Kehitetty toimintamateriaali toimi oppilaiden oppimisen, kommunikoinnin ja vuorovaikutuksen välineenä. Oppilaat pitivät kehitettyä välinettä oman oppimisen kannalta hyödyllisenä, helppokäyttöisenä ja miellyttävänä käyttää. Lisäksi he kokivat tekemällä oppimisen ja yhteisöllisen kielentämisen mielekkääksi työtavaksi tulevaisuudessa matematiikan oppimiseen.Procedural competence traditionally emphasised in school mathematics teaching alone is not sufficient to succeed in learning mathematics. The Finnish National Core Curriculum for Basic Education 2014 has underlined the importance of stu-dents’ understanding of mathematical concepts. This article uses Lehtonen’s disser-tation to demonstrate how educational technology utilisation can promote stu-dents’ mathematical concepts understanding. A multimodal manipulative was de-veloped for primary school students to learn how to solve equations with under-standing. The manipulative combines the strengths of physical and digital tools: physical component manipulation helps students focus on their activities; digital components motivate students to learn and provide them with real-time guidance and feedback. Class intervention results reveal that the developed manipulative to-gether with languaging of mathematical thinking helped fourth graders learn key concepts in equation solving. The manipulative functioned as students’ learning, communication, and collaborative tool. Students perceived it as beneficial, easy, and pleasant to use. In the future, they would like to learn mathematics by doing and languaging with peers.publishedVersionNon peer reviewe

Luokanopettajaopiskelijoiden arviointeja ei-rutiinimaisista matematiikan tehtävistä

Matematiikan harjoitustehtävät ovat pysyneet oppimateriaaleissa rakenteiltaan samankaltaisinaniiden historian ajan. Tehtävätyypit ovat lähinnä suljettuja tehtäviä, jotkauseimmiten kehittävät vain jo aikaisemmin opittua ja tukevat siis proseduraalista sujuvuutta.Tehtävät voisivat kehittää entistä enemmän myös käsitteellistä ymmärrystä jastrategista sekä metakognitiivisia taitoja. Tässä tutkimuksessa on kehitetty matematiikanperinteisistä tehtävistä poikkeavia tehtävätyyppejä. Keväällä 2016 tekijöiden yliopiston1. vuosikurssin luokanopettajaopiskelijat (N=82) kokeilivat neljä uutta tehtävätyyppiäkurssin harjoituksissa ja arvioivat niiden matemaattisen sekä monilukutaidon osaamisensuhteen. Artikkelissamme kuvaamme neljää tehtäväympäristöä, opiskelijoiden arviointejaniistä sekä yhden tehtävän ratkaisuja matemaattisten käsitteiden käytön näkökulmasta

Multimodal Communication and Peer Interaction during Equation-Solving Sessions with and without Tangible Technologies

Despite the increasing use of technologies in the classroom, there are concerns that technology-enhanced learning environments may hinder students’ communication and interaction. In this study, we investigated how tangible technologies can enhance students’ multimodal communication and interaction during equation-solving pair work compared to working without such technologies. A tangible app for learning equation solving was developed and tested in fourth- and fifth-grade classrooms with two class teachers and 24 students. Video data of the interventions were analysed using deductive and inductive content analysis. Coded data were also quantified for quantitative analysis. Additionally, teacher interview data were used to compare and contrast the findings. The findings showed that the tangible app better promoted students’ multimodal communication and peer interaction than working only with paper and pencil. When working in pairs, tangible-app students interacted with one another much more often and in more ways than their paper-and-pencil peers. The implications of this study are discussed in terms of its contributions to research on tangible technologies for learning, educational technology development, and the use of tangibles in classrooms to support students’ multimodal communication and peer interaction.peerReviewe