Olof Steinthorsdottir | University of Northern Iowa (original) (raw)

Papers by Olof Steinthorsdottir

Middle School Journal, 2016

Mathematics Education Research Journal, Feb 1, 2009

Understanding of ratio and proportion is critical to the development of higher level mathematical... more Understanding of ratio and proportion is critical to the development of higher level mathematical skills. Following Carpenter, Gomez, et al.'s (1999) proposal of a four-level trajectory in the development of proportional reasoning, a 12-week investigation was undertaken of the developmental trajectory of proportional reasoning of girls in two 5th-grade classes in Iceland. Students in these classes were accustomed to instructional practices that encouraged them to devise and explain their own solutions to mathematical problems. Results of the study confirm the learning trajectory with the addition of a further distinct level of development between Level 2 and Level 3. Results showed that girls moved easily, with minimum scaffolding, from Level 1 to 2 and from Level 2 to 3. The transition to Level 4, which involves explicit awareness of 'within' and 'between' multiplicative relationships, took greater time and effort. Teacher awareness of the four-level learning strategy, with the new emerging Level 3, assists in the design of appropriate problems, class structure, and teaching strategies. Building on Lamon's notions of unitizing and norming as by Carpenter et al.'s (1999) developmental model, this study contributes to our comprehension of students' understanding of proportionality and how it develops.

This study focused on 26 girls' development of proportional reasoning in two fifthgrade classroom... more This study focused on 26 girls' development of proportional reasoning in two fifthgrade classrooms in Iceland. The students were used to instructional practices that encouraged them to devise their own solutions to mathematical problems. The results supported four levels of proportional reasoning. Level 1, girls showed limited ratio knowledge. Level 2, they perceived the given ratio as an indivisible unit. Level 3, students conceived of the given ratio as a reducible unit. And at Level 4 students no longer thought of ratios exclusively as unit quantities, but understood the proportion in terms of multiplicative relations. The results suggest that students can reach level 3 reasoning with less struggle than it takes to achieve level 4, which suggests that the knowledge needed to operate on level 3 was within their reach. OBJECTIVES This study investigates the developmental of proportional reasoning of girls in two fifth-grade classes in Iceland. The purposes of this study was to further investigate four levels of proportional reasoning identified in a pilot study that the author conducted in collaboration prior to the study reported here (Carpenter et al. 1999). In particular, do the four levels describe the pathway of a population of Icelandic girls before, during, and after they have engaged in a unit focused on proportional reasoning? Secondly, what evidence is there for the existence of Level 2, Level 3, and Level 4 ways of reasoning in students' verbal protocols? And finally how does instruction that is focused on students' reasoning help students make the transition from level to level? BACKGROUND AND THEORETICAL ORIENTATION Proportional reasoning represents a cornerstone in the development of children's mathematical thinking (Inhelder & Piaget, 1958; Resnick & Singer, 1993). Ratio and proportion are critical ideas for students to understand; however, although young children demonstrate foundations for proportional reasoning, students are slow to attain mastery of these concepts. Many studies on children's proportional reasoning provide evidence of various influences on students' thinking about proportion. Among these influential factors are the problem numerical structures 1. The number structure refers to the multiplicative relationship within and between ratios in a proportional setting. A "within"

Icelandic 5th grade girls developmental trajectories in proportional reasoning

Understanding of ratio and proportion is critical to the development of higher level mathematical... more Understanding of ratio and proportion is critical to the development of higher level mathematical skills. Following Carpenter, Gomez, et al.’s (1999) proposal of a four-level trajectory in the development of proportional reasoning, a 12-week investigation was undertaken of the developmental trajectory of proportional reasoning of girls in two 5th-grade classes in Iceland. Students in these classes were accustomed to instructional practices that encouraged them to devise and explain their own solutions to mathematical problems. Results of the study confirm the learning trajectory with the addition of a further distinct level of development between Level 2 and Level 3. Results showed that girls moved easily, with minimum scaffolding, from Level 1 to 2 and from Level 2 to 3. The transition to Level 4, which involves explicit awareness of ‘within ’ and ‘between’ multiplicative relationships, took greater time and effort. Teacher awareness of the four-level learning strategy, with the ne...

Investigations in Mathematics Learning, 2017

This study explores the effects of number structure characteristics on student thinking in solvin... more This study explores the effects of number structure characteristics on student thinking in solving missing-value proportion problems. Prior research has documented that the presence of an integer ratio is beneficial, particularly if the integer relationship is within the same measure space. Less information on student performance is available, however, on shrink problems, in which the value to be found is smaller than the given value. In the current work, we systematically investigate student success rates and strategy use on problems that vary in the presence/absence of integer ratios in either measure space and in both enlarge and shrink problems. This study provides evidence and direction to educators seeking to advance their students' proportional reasoning via judiciously chosen tasks. We start from a distribution analysis of student strategies, which allows us to propose a hierarchy of problem difficulty. We then discuss apparent differences in student thinking based on these number structure characteristics and identify problem types for which more sophisticated proportional reasoning is needed. Directions for further research and implications for teaching are discussed.

Mathematics Teaching in the Middle School, 2014

While students are solving a proportion problem, their work in a measure space will enable teache... more While students are solving a proportion problem, their work in a measure space will enable teachers to take the measure of their thinking.

The Proceedings of the 12th International Congress on Mathematical Education, 2015

While mathematics are universal, it appears that delicate process in the classroom, but not only ... more While mathematics are universal, it appears that delicate process in the classroom, but not only there, lead boys and girls to perceive things differently. And from this perception at school depends the future of the jobs. If the teacher, male or female, is conscious of this, what can he/she do to provide to each pupil or student, boy or girl, the opportunity of understanding, participating and finally appreciating mathematics at best? The subject is not new: it merges explicitly at ICME3 in Karlsruhe (Germany) in 1976. «[…] Moreover, it is recommended that the theme 'Women ans Mathematics' be an explicit theme of ICME 1980.»: this ends the third and last resolution of the Congress. This recommandation became realised at ICME4 in Berkeley in 1980 and goes on since. From the proposals received for ICME12 from all over the world, the reflection at Topic Study Group «Gender and Education» was organised along four themes: gender issues in research and learning environmental; student's achievement, assessment and classroom activities; self-efficacy and attitudes; gendered views of mathematics.

Encyclopedia of Mathematics Education, 2014

This study focused on 26 girls' development of proportional reasoning in two fifthgrade classroom... more This study focused on 26 girls' development of proportional reasoning in two fifthgrade classrooms in Iceland. The students were used to instructional practices that encouraged them to devise their own solutions to mathematical problems. The results supported four levels of proportional reasoning. Level 1, girls showed limited ratio knowledge. Level 2, they perceived the given ratio as an indivisible unit. Level 3, students conceived of the given ratio as a reducible unit. And at Level 4 students no longer thought of ratios exclusively as unit quantities, but understood the proportion in terms of multiplicative relations. The results suggest that students can reach level 3 reasoning with less struggle than it takes to achieve level 4, which suggests that the knowledge needed to operate on level 3 was within their reach. OBJECTIVES This study investigates the developmental of proportional reasoning of girls in two fifth-grade classes in Iceland. The purposes of this study was to further investigate four levels of proportional reasoning identified in a pilot study that the author conducted in collaboration prior to the study reported here (Carpenter et al. 1999). In particular, do the four levels describe the pathway of a population of Icelandic girls before, during, and after they have engaged in a unit focused on proportional reasoning? Secondly, what evidence is there for the existence of Level 2, Level 3, and Level 4 ways of reasoning in students' verbal protocols? And finally how does instruction that is focused on students' reasoning help students make the transition from level to level? BACKGROUND AND THEORETICAL ORIENTATION Proportional reasoning represents a cornerstone in the development of children's mathematical thinking (Inhelder & Piaget, 1958; Resnick & Singer, 1993). Ratio and proportion are critical ideas for students to understand; however, although young children demonstrate foundations for proportional reasoning, students are slow to attain mastery of these concepts. Many studies on children's proportional reasoning provide evidence of various influences on students' thinking about proportion. Among these influential factors are the problem numerical structures 1. The number structure refers to the multiplicative relationship within and between ratios in a proportional setting. A "within"

Gender and proportional strategy use in Iceland 2 ABSTRACT This study was conducted to investigat... more Gender and proportional strategy use in Iceland 2 ABSTRACT This study was conducted to investigate the influence of semantic type and number structure on individuals' use of strategies in solving missing value proportion problems, and to examine gender differences in strategy use. Fifty-three eighth graders in one school in Reykjavik, Iceland, participated in this study. Twenty-seven females and twenty-six males, were individually interviewed as they solved sixteen missing value proportion problems. The problems represented four semantic structures: Well chunked (W-C), part-part whole (P-P-W), associated sets (AS), and symbolic (S-P). For each semantic structure there were four problems, each representing a distinct number structures: Integer-integer with an integer answer findings in this study indicate that number structure influenced strategy use and success to a greater extent than semantic type. Girls were more successful than boys in associated sets and symbolic problems, ...

Zdm, 2008

Students’ mathematical achievement in Iceland, as reported in PISA 2003, showed significant and (... more Students’ mathematical achievement in Iceland, as reported in PISA 2003, showed significant and (by comparison) unusual gender differences in mathematics: Iceland was the only country in which the mathematics gender gap favored girls. When data were broken down and analyzed, the Icelandic gender gap appeared statistically significant only in the rural areas of Iceland, suggesting a question about differences in rural and urban educational communities. In the 2007 qualitative research study reported in this paper, the authors interviewed 19 students from rural and urban Iceland who participated in PISA 2003 in order to investigate these differences and to identify factors that contributed to gender differences in mathematics learning. Students were asked to talk about their mathematical experiences, their thoughts about the PISA results, and their ideas about the reasons behind the PISA 2003 results. The data were transcribed, coded, and analyzed using techniques from analytic induction in order to build themes and to present both male and female student perspectives on the Icelandic anomaly. Strikingly, youth in the interviews focused on social and societal factors concerning education in general rather then on their mathematics education.

Interchange, 2007

... thinking skills, communication and leadership skills to guarantee higher capital and manageri... more ... thinking skills, communication and leadership skills to guarantee higher capital and managerialmobility in the ... The work of Gutstein sets a brilliant and necessary example for a pedagogy of ... in the US) to increase numbers of female students in graduate programs, necessitates ...

This study was conducted to investigate the influence of contextual structure and number structur... more This study was conducted to investigate the influence of contextual structure and number structure on individuals' use of strategies and success rate in solving missing value proportion problems. Fifty-three eighth graders in one school in Reykjavik, Iceland, participated in this study. Twenty-seven females and twenty-six males were individually interviewed as they solved sixteen missing value proportion problems. The problems number structure was carefully manipulated within planned parameters of complexity. The number complexity formed a parallel hierarchy among the contextual structure. The findings in this study indicate that number structure influenced strategy use and success to a greater extent than contextual structure.

This study was conducted to investigate the influence of contextual and number structures on indi... more This study was conducted to investigate the influence of contextual and number structures on individuals' use of strategies in solving missing value proportion problems, and to examine gender differences in strategy use. Fifty-three eighth graders in one school in Reykjavik, Iceland, participated in this study. Twenty-seven females and twenty-six males were individually interviewed as they solved sixteen missing value proportion problems. The problems represented four contextual structures: wellchunked (W-C), part-part-whole (P-P-W), associated sets (A-S), and symbolic problems (S-P). For each contextual structure there were four problems, each representing a distinct number structures: integer-integer with an integer answer (I-I-I), integernoninteger with an integer answer (I-N-I or N-I-I), noninteger-noninteger with an integer answer (N-N-I), and noninteger-noninteger with a non-integer answer (N-N-N). The findings in this study indicate that the number structure influenced strategy use and success to a greater extent than did the contextual structure. The easiest problems for students to solve were I-I-I tasks and the most difficult problems were the N-N-N tasks. Most students used multiplicative strategies on the I-I-I problems and I-N-I problems, but resorted to less sophisticated reasoning on the N-N-I and N-N-N problems. No gender differences were identified in the overall success rate. However, girls were more successful than boys in handling associated sets and symbolic problems, and boys were more successful than girls in part-part-whole problems. Moreover, the data suggest that the contextual structures influence females' choice of strategy more than that of males.

Middle School Journal, 2016

Mathematics Education Research Journal, Feb 1, 2009

Understanding of ratio and proportion is critical to the development of higher level mathematical... more Understanding of ratio and proportion is critical to the development of higher level mathematical skills. Following Carpenter, Gomez, et al.'s (1999) proposal of a four-level trajectory in the development of proportional reasoning, a 12-week investigation was undertaken of the developmental trajectory of proportional reasoning of girls in two 5th-grade classes in Iceland. Students in these classes were accustomed to instructional practices that encouraged them to devise and explain their own solutions to mathematical problems. Results of the study confirm the learning trajectory with the addition of a further distinct level of development between Level 2 and Level 3. Results showed that girls moved easily, with minimum scaffolding, from Level 1 to 2 and from Level 2 to 3. The transition to Level 4, which involves explicit awareness of 'within' and 'between' multiplicative relationships, took greater time and effort. Teacher awareness of the four-level learning strategy, with the new emerging Level 3, assists in the design of appropriate problems, class structure, and teaching strategies. Building on Lamon's notions of unitizing and norming as by Carpenter et al.'s (1999) developmental model, this study contributes to our comprehension of students' understanding of proportionality and how it develops.

This study focused on 26 girls' development of proportional reasoning in two fifthgrade classroom... more This study focused on 26 girls' development of proportional reasoning in two fifthgrade classrooms in Iceland. The students were used to instructional practices that encouraged them to devise their own solutions to mathematical problems. The results supported four levels of proportional reasoning. Level 1, girls showed limited ratio knowledge. Level 2, they perceived the given ratio as an indivisible unit. Level 3, students conceived of the given ratio as a reducible unit. And at Level 4 students no longer thought of ratios exclusively as unit quantities, but understood the proportion in terms of multiplicative relations. The results suggest that students can reach level 3 reasoning with less struggle than it takes to achieve level 4, which suggests that the knowledge needed to operate on level 3 was within their reach. OBJECTIVES This study investigates the developmental of proportional reasoning of girls in two fifth-grade classes in Iceland. The purposes of this study was to further investigate four levels of proportional reasoning identified in a pilot study that the author conducted in collaboration prior to the study reported here (Carpenter et al. 1999). In particular, do the four levels describe the pathway of a population of Icelandic girls before, during, and after they have engaged in a unit focused on proportional reasoning? Secondly, what evidence is there for the existence of Level 2, Level 3, and Level 4 ways of reasoning in students' verbal protocols? And finally how does instruction that is focused on students' reasoning help students make the transition from level to level? BACKGROUND AND THEORETICAL ORIENTATION Proportional reasoning represents a cornerstone in the development of children's mathematical thinking (Inhelder & Piaget, 1958; Resnick & Singer, 1993). Ratio and proportion are critical ideas for students to understand; however, although young children demonstrate foundations for proportional reasoning, students are slow to attain mastery of these concepts. Many studies on children's proportional reasoning provide evidence of various influences on students' thinking about proportion. Among these influential factors are the problem numerical structures 1. The number structure refers to the multiplicative relationship within and between ratios in a proportional setting. A "within"

Icelandic 5th grade girls developmental trajectories in proportional reasoning

Understanding of ratio and proportion is critical to the development of higher level mathematical... more Understanding of ratio and proportion is critical to the development of higher level mathematical skills. Following Carpenter, Gomez, et al.’s (1999) proposal of a four-level trajectory in the development of proportional reasoning, a 12-week investigation was undertaken of the developmental trajectory of proportional reasoning of girls in two 5th-grade classes in Iceland. Students in these classes were accustomed to instructional practices that encouraged them to devise and explain their own solutions to mathematical problems. Results of the study confirm the learning trajectory with the addition of a further distinct level of development between Level 2 and Level 3. Results showed that girls moved easily, with minimum scaffolding, from Level 1 to 2 and from Level 2 to 3. The transition to Level 4, which involves explicit awareness of ‘within ’ and ‘between’ multiplicative relationships, took greater time and effort. Teacher awareness of the four-level learning strategy, with the ne...

Investigations in Mathematics Learning, 2017

This study explores the effects of number structure characteristics on student thinking in solvin... more This study explores the effects of number structure characteristics on student thinking in solving missing-value proportion problems. Prior research has documented that the presence of an integer ratio is beneficial, particularly if the integer relationship is within the same measure space. Less information on student performance is available, however, on shrink problems, in which the value to be found is smaller than the given value. In the current work, we systematically investigate student success rates and strategy use on problems that vary in the presence/absence of integer ratios in either measure space and in both enlarge and shrink problems. This study provides evidence and direction to educators seeking to advance their students' proportional reasoning via judiciously chosen tasks. We start from a distribution analysis of student strategies, which allows us to propose a hierarchy of problem difficulty. We then discuss apparent differences in student thinking based on these number structure characteristics and identify problem types for which more sophisticated proportional reasoning is needed. Directions for further research and implications for teaching are discussed.

Mathematics Teaching in the Middle School, 2014

While students are solving a proportion problem, their work in a measure space will enable teache... more While students are solving a proportion problem, their work in a measure space will enable teachers to take the measure of their thinking.

The Proceedings of the 12th International Congress on Mathematical Education, 2015

While mathematics are universal, it appears that delicate process in the classroom, but not only ... more While mathematics are universal, it appears that delicate process in the classroom, but not only there, lead boys and girls to perceive things differently. And from this perception at school depends the future of the jobs. If the teacher, male or female, is conscious of this, what can he/she do to provide to each pupil or student, boy or girl, the opportunity of understanding, participating and finally appreciating mathematics at best? The subject is not new: it merges explicitly at ICME3 in Karlsruhe (Germany) in 1976. «[…] Moreover, it is recommended that the theme 'Women ans Mathematics' be an explicit theme of ICME 1980.»: this ends the third and last resolution of the Congress. This recommandation became realised at ICME4 in Berkeley in 1980 and goes on since. From the proposals received for ICME12 from all over the world, the reflection at Topic Study Group «Gender and Education» was organised along four themes: gender issues in research and learning environmental; student's achievement, assessment and classroom activities; self-efficacy and attitudes; gendered views of mathematics.

Encyclopedia of Mathematics Education, 2014

This study focused on 26 girls' development of proportional reasoning in two fifthgrade classroom... more This study focused on 26 girls' development of proportional reasoning in two fifthgrade classrooms in Iceland. The students were used to instructional practices that encouraged them to devise their own solutions to mathematical problems. The results supported four levels of proportional reasoning. Level 1, girls showed limited ratio knowledge. Level 2, they perceived the given ratio as an indivisible unit. Level 3, students conceived of the given ratio as a reducible unit. And at Level 4 students no longer thought of ratios exclusively as unit quantities, but understood the proportion in terms of multiplicative relations. The results suggest that students can reach level 3 reasoning with less struggle than it takes to achieve level 4, which suggests that the knowledge needed to operate on level 3 was within their reach. OBJECTIVES This study investigates the developmental of proportional reasoning of girls in two fifth-grade classes in Iceland. The purposes of this study was to further investigate four levels of proportional reasoning identified in a pilot study that the author conducted in collaboration prior to the study reported here (Carpenter et al. 1999). In particular, do the four levels describe the pathway of a population of Icelandic girls before, during, and after they have engaged in a unit focused on proportional reasoning? Secondly, what evidence is there for the existence of Level 2, Level 3, and Level 4 ways of reasoning in students' verbal protocols? And finally how does instruction that is focused on students' reasoning help students make the transition from level to level? BACKGROUND AND THEORETICAL ORIENTATION Proportional reasoning represents a cornerstone in the development of children's mathematical thinking (Inhelder & Piaget, 1958; Resnick & Singer, 1993). Ratio and proportion are critical ideas for students to understand; however, although young children demonstrate foundations for proportional reasoning, students are slow to attain mastery of these concepts. Many studies on children's proportional reasoning provide evidence of various influences on students' thinking about proportion. Among these influential factors are the problem numerical structures 1. The number structure refers to the multiplicative relationship within and between ratios in a proportional setting. A "within"

Gender and proportional strategy use in Iceland 2 ABSTRACT This study was conducted to investigat... more Gender and proportional strategy use in Iceland 2 ABSTRACT This study was conducted to investigate the influence of semantic type and number structure on individuals' use of strategies in solving missing value proportion problems, and to examine gender differences in strategy use. Fifty-three eighth graders in one school in Reykjavik, Iceland, participated in this study. Twenty-seven females and twenty-six males, were individually interviewed as they solved sixteen missing value proportion problems. The problems represented four semantic structures: Well chunked (W-C), part-part whole (P-P-W), associated sets (AS), and symbolic (S-P). For each semantic structure there were four problems, each representing a distinct number structures: Integer-integer with an integer answer findings in this study indicate that number structure influenced strategy use and success to a greater extent than semantic type. Girls were more successful than boys in associated sets and symbolic problems, ...

Zdm, 2008

Students’ mathematical achievement in Iceland, as reported in PISA 2003, showed significant and (... more Students’ mathematical achievement in Iceland, as reported in PISA 2003, showed significant and (by comparison) unusual gender differences in mathematics: Iceland was the only country in which the mathematics gender gap favored girls. When data were broken down and analyzed, the Icelandic gender gap appeared statistically significant only in the rural areas of Iceland, suggesting a question about differences in rural and urban educational communities. In the 2007 qualitative research study reported in this paper, the authors interviewed 19 students from rural and urban Iceland who participated in PISA 2003 in order to investigate these differences and to identify factors that contributed to gender differences in mathematics learning. Students were asked to talk about their mathematical experiences, their thoughts about the PISA results, and their ideas about the reasons behind the PISA 2003 results. The data were transcribed, coded, and analyzed using techniques from analytic induction in order to build themes and to present both male and female student perspectives on the Icelandic anomaly. Strikingly, youth in the interviews focused on social and societal factors concerning education in general rather then on their mathematics education.

Interchange, 2007

... thinking skills, communication and leadership skills to guarantee higher capital and manageri... more ... thinking skills, communication and leadership skills to guarantee higher capital and managerialmobility in the ... The work of Gutstein sets a brilliant and necessary example for a pedagogy of ... in the US) to increase numbers of female students in graduate programs, necessitates ...

This study was conducted to investigate the influence of contextual structure and number structur... more This study was conducted to investigate the influence of contextual structure and number structure on individuals' use of strategies and success rate in solving missing value proportion problems. Fifty-three eighth graders in one school in Reykjavik, Iceland, participated in this study. Twenty-seven females and twenty-six males were individually interviewed as they solved sixteen missing value proportion problems. The problems number structure was carefully manipulated within planned parameters of complexity. The number complexity formed a parallel hierarchy among the contextual structure. The findings in this study indicate that number structure influenced strategy use and success to a greater extent than contextual structure.

This study was conducted to investigate the influence of contextual and number structures on indi... more This study was conducted to investigate the influence of contextual and number structures on individuals' use of strategies in solving missing value proportion problems, and to examine gender differences in strategy use. Fifty-three eighth graders in one school in Reykjavik, Iceland, participated in this study. Twenty-seven females and twenty-six males were individually interviewed as they solved sixteen missing value proportion problems. The problems represented four contextual structures: wellchunked (W-C), part-part-whole (P-P-W), associated sets (A-S), and symbolic problems (S-P). For each contextual structure there were four problems, each representing a distinct number structures: integer-integer with an integer answer (I-I-I), integernoninteger with an integer answer (I-N-I or N-I-I), noninteger-noninteger with an integer answer (N-N-I), and noninteger-noninteger with a non-integer answer (N-N-N). The findings in this study indicate that the number structure influenced strategy use and success to a greater extent than did the contextual structure. The easiest problems for students to solve were I-I-I tasks and the most difficult problems were the N-N-N tasks. Most students used multiplicative strategies on the I-I-I problems and I-N-I problems, but resorted to less sophisticated reasoning on the N-N-I and N-N-N problems. No gender differences were identified in the overall success rate. However, girls were more successful than boys in handling associated sets and symbolic problems, and boys were more successful than girls in part-part-whole problems. Moreover, the data suggest that the contextual structures influence females' choice of strategy more than that of males.