| 1. |
- Bergqvist, Tomas, 1962-, et al.
(author)
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Creative and algorithmic reasoning – the role of strategy choices in practice and test
- 2022
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In: Nordisk matematikkdidaktikk, NOMAD. - Göteborg : Nationellt centrum för matematikutbildning (NCM). - 1104-2176. ; 27:1, s. 5-25
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Journal article (peer-reviewed)abstract
- This study is based on a framework of algorithmic and creative mathematical rea- soning and focuses on students’ strategy choices in both practice and test. Previous research indicates that students that practice mathematics with tasks with given solution methods are outperformed in later test by students that have to construct solution methods during practice. Video recordings, students’ written solutions, and student interviews from ten university students provides data on strategy choices. The analysis was carried out to capture students’ strategy choices and reasons for these choices. The results showed that there was no real difference in how the stu- dents solved the tasks in the test. Regardless of practice condition, more or less the same solution strategies were used in the test situation.
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| 2. |
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| 3. |
- Jäder, Jonas, 1971-
(author)
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Med uppgift att lära : om matematikuppgifter som en resurs för lärande
- 2019
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Doctoral thesis (other academic/artistic)abstract
- Elevers möjligheter att utveckla sin kunskap i matematik påverkas av de uppgifter de arbetar med. Det är möjligt att göra en distinktion mellan rutinuppgifter och matematiska problem. En rutinuppgift är en uppgift som en elev kan lösa genom att använda en välbekant metod, eller genom att imitera en förlaga. För att lösa ett matematiskt problem behöver däremot eleven konstruera en för henne ny lösningsmetod. För att utveckla sin matematiska kunskap behöver elever möta såväl rutinuppgifter som matematiska problem. Problemlösning kan skapa förutsättningar för en elev att utveckla såväl en kreativ problemlösningsförmåga, som en konceptuell, matematisk förståelse.Avhandlingen består av fem studier med ett fokus på matematikuppgifter, där studie 1-3 syftade till att undersöka vilka möjligheter att arbeta med matematisk problemlösning som elever i gymnasieskolan erbjuds. Detta undersöktes genom läroboksanalyser, studier av elevers arbete med uppgifter och av elevers uppfattningar om matematik. Uppgifter i läroböcker från 12 länder analyserades (studie 1) och ungefär 10 procent av dessa var matematiska problem. Eleverna arbetade (studie 2) nästan uteslutande med de uppgifter som av läroboksförfattarna kategoriserats som enkla och utan att arbeta problemlösande. Bland dessa uppgifter var andelen matematiska problem 4 procent. Inte heller bland uppgifter som kategoriserats som till exempel ’problemlösning’ eller ’utforska’ var matematiska problem i övervikt. Resultaten var relativt lika för de tolv ländernas läroböcker. Elevers uppfattningar om att rutinarbete är säkrare och något som är rimligt att förvänta sig i matematik (studie 3) kan ha en ytterligare påverkan på deras möjligheter att arbeta problemlösande. Med tanke på de positiva effekter som påvisats för elever som arbetar med problemlösning verkar elevers möjligheter att arbeta med problemlösning begränsade. Det finns potential i att såväl utveckla innehållet i läroböckerna för att öka andelen matematiska problem, som i ett medvetet uppgiftsurval från dessa läroböcker.Syftet med studie 4 och 5 var att fördjupa förståelsen för problemlösning. Ett analytiskt ramverk har utvecklats för att identifiera kreativa, konceptuella och andra utmaningar i elevers problemlösning. Respektive utmaning karaktäriserades för att ytterligare fördjupa förståelsen för dessa och för problemlösning. Elevers arbete med matematiska problem (studie 4) och lärares förväntningar på de utmaningar elever möter vid problemlösning (studie 5) studerades. Konceptuella och kreativa utmaningar visade sig vara de mest centrala vid elevers problemlösning. Genom den karaktäristik som knöts till respektive utmaning kan svårigheter med att identifiera, framför allt kreativa utmaningar, och relationen mellan uppgift och utmaning diskuteras.
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| 4. |
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| 5. |
- Norqvist, Mathias, 1971-
(author)
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Explanations do not improve algorithmic reasoning tasks : Volume 1
- 2016
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In: Proceedings of the 40th Conference of the International Group for the Psychology of Mathematics Education. - Szeged, Hungary : International Group for the Psychology of Mathematics Education. ; , s. 110-110
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Conference paper (peer-reviewed)
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| 6. |
- Norqvist, Mathias, 1971-
(author)
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On Mathematical Reasoning : being told or finding out
- 2016
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Doctoral thesis (other academic/artistic)abstract
- School-mathematics has been shown to mainly comprise rote-learning of procedures where the considerations of intrinsic mathematical properties are scarce. At the same time theories and syllabi emphasize competencies like problem solving and reasoning. This thesis will therefore concern how task design can influence the reasoning that students apply when solving tasks, and how the reasoning during practice is associated to students’ results, cognitive capacity, and brain activity. In studies 1-3, we examine the efficiency of different types of reasoning (i.e., algorithmic reasoning (AR) or creative mathematically founded reasoning (CMR)) in between-groups designs. We use mathematics grade, gender, and cognitive capacity as matching variables to get similar groups. We let the groups practice 14 different solution methods with tasks designed to promote either AR or CMR, and after one week the students are tested on the practiced solution methods. In study 3 the students did the test in and fMRI-scanner to study if the differing practice would yield any lasting differences in brain activation. Study 4 had a different approach and focused details in students’ reasoning when working on teacher constructed tasks in an ordinary classroom environment. Here we utilized audio-recordings of students’ solving tasks, together with interviews with teachers and students to unravel the reasoning sequences that students embark on. The turning points where the students switch subtask and the reasoning between these points were characterized and visualized. The behavioral results suggest that CMR is more efficient than AR, and also less dependent on cognitive capacity during the test. The latter is confirmed by fMRI, which showed that AR had higher activation than CMR in areas connected to memory retrieval and working memory. The behavioral result also suggested that CMR is more beneficial for cognitively less proficient students than for the high achievers. Also, task design is essential for both students’ choice of reasoning and task progression. The findings suggest that: 1) since CMR is more efficient than AR, students need to encounter more CMR, both during task solving and in teacher presentation, 2) cognitive capacity is important but depending on task design, cognitive strain will be more or less high during test situations, 3) although AR-tasks does not prohibit the use of CMR they make it less likely to occur. Since CMR-tasks can emphasize important mathematical properties, are more efficient than AR- tasks, and more beneficial for less cognitively proficient students, promoting CMR can be essential if we want students to become mathematically literate.
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| 7. |
- Norqvist, Mathias, 1971-, et al.
(author)
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Shifts in student attention on algorithmic and creative practice tasks
- 2023
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In: Educational Studies in Mathematics. - : Springer. - 0013-1954 .- 1573-0816.
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Journal article (peer-reviewed)abstract
- In mathematics classrooms, it is common practice to work through a series of comparable tasks provided in a textbook. A central question in mathematics education is if tasks should be accompanied with solution methods, or if students should construct the solutions themselves. To explore the impact of these two task designs on student behavior during repetitive practice, an eye-tracking study was conducted with 50 upper secondary and university students. Their eye movements were analyzed to study how the two groups shifted their gaze both within and across 10 task sets. The results show that when a solution method was present, the students reread this every time they solved the task, while only giving minute attention to the illustration that carried information supporting mathematical understanding. Students who practiced with tasks without a solution method seemed to construct a solution method by observing the illustration, which later could be retrieved from memory, making this method more efficient in the long run. We discuss the implications for teaching and how tasks without solution methods can increase student focus on important mathematical properties.
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| 9. |
- Norqvist, Mathias, 1971-
(author)
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The effect of explanations on mathematical reasoning tasks
- 2018
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In: International journal of mathematical education in science and technology. - Abingdon : Taylor & Francis. - 0020-739X .- 1464-5211. ; 49:1, s. 15-30
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Journal article (peer-reviewed)abstract
- Studies in mathematics education often point to the necessity for students to engage in more cognitively demanding activities than just solving tasks by applying given solution methods. Previous studies have shown that students that engage in creative mathematically founded reasoning to construct a solution method, perform significantly better in follow up tests than students that are given a solution method and engage in algorithmic reasoning. However, teachers and textbooks, at least occasionally, provide explanations together with an algorithmic method, and this could possibly be more efficient than creative reasoning. In this study, three matched groups practiced with either creative, algorithmic, or explained algorithmic tasks. The main finding was that students that practiced with creative tasks did, outperform the students that practiced with explained algorithmic tasks in a post-test, despite a much lower practice score. The two groups that got a solution method presented, performed similarly in both practice and post-test, even though one group got an explanation to the given solution method. Additionally, there were some differences between the groups in which variables predicted the post-test score.
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| 10. |
- Wikström Hultdin, Ulrika, 1977-
(author)
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Between symbols and words : structural connections in mathematics texts and their effect on reading
- 2024
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Doctoral thesis (other academic/artistic)abstract
- While students progress through their school years, they are expected to develop reading skills in all academic subjects, including mathematics. Mathematics texts, being multisemiotic, require readers to make meaning not only from written language but also from mathematical symbols and visualizations. Integrating content presented through different sign systems is essential for creating coherence. Thus, the organizational structure of these texts becomes critically important when designing texts for learning. The purpose of this thesis is to build knowledge of the organization of mathematical symbols and written language, and to achieve better understanding of how this organization influences the reading of mathematics texts. First, the structural connections between mathematical symbols and written language in mathematics texts designed for students are characterized. Five distinct categories of such connections—Interwoven, Chunked, Marked, Adjoined, and Referenced—are identified, ranging from connections in which mathematical symbols are integrated into sentences (Interwoven), to those based solely on the proximity between two text sequences (Adjoined). The prevalence of these connection categories in textbooks from different school levels is also investigated. The results indicate a progression in the use of structural connections, with a shift from reliance on proximity in early school years towards a preference for symbols interwoven in sentences between years 2 and 5, suggesting that all students eventually need to navigate texts with interwoven symbols. Additionally, changes can be seen in how symbols are being connected to more detailed meanings. Second, the reading of mathematics texts employing two distinct text designs inspired by the new framework is compared: one design features only sentences with interwoven symbols, whereas the other uses a graphic to highlight key connections between symbols and words. The reading processes and experiences of students are investigated by analyzing gaze measurements and interviews. The results indicate that the two designs have different advantages depending on the situation. While the graphic design can facilitate reading and interpretation by drawing attention to the connections between symbols and words, enabling quicker content matching, the symbols interwoven in sentences might provide better access to details or allow more efficient reading in other contexts. Moreover, individual differences in processing and experiences were noted: while some readers benefitted from the graphic design, others did not. Yet, as reading becomes more complex, the graphic is increasingly appreciated. It is concluded that while readers generally prefer text designs that enhance readability, the optimal design varies based on the reader and the context. The discussion includes what text design benefits whom and under what circumstances.
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