Abstract The purpose of this study was to develop mathematical games to assist children in learning addition and subtraction computation. The investigator designed games and experimented through four extra lessons, then studied children¡¦s mathematical learning after the lessons. Participants of this study were second grade children from two elementary schools in Kaohsiung. The experimentation was divided into three phases: pilot phase, first edition phase, and, second edition phase. Data collection and administration included pre-test and post-test; questionnaire of children¡¦s attitude towards mathematics; questionnaire of children¡¦s attitude towards games; record sheets for loan of games; children¡¦s worksheets; learning diaries; researcher¡¦s observation notes; and, researcher¡¦s reflections on design of games. The researcher referred to the above data source, later, evaluated according to the results of experimentation, and completed final revisions. Data analyses and results were three: Design, implementation, and revision. Design. When designing a game, the materials followed that of children¡¦s real life experiences. The characteristics of the four games were: Cooperation, challenge, education, and, chance and luck. Implementation. Children changed from ¡§individual responsibility¡¨ to ¡§co-ordination and cooperation in group¡¨, from ¡§assurance¡¨ to ¡§lack of confidence¡¨, from ¡§marginal participation¡¨ to ¡§eager to play with laughter¡¨, and ¡§no control of shouting¡¨. From the data analyses, children performed well in cognition, attitude, and skills. Revision. The principles of modification were to evaluate games¡¦ strengths and weaknesses, and adjusting games design. The respective revisions were tutoring play, children as key persons, formulating regulations, and, enriching contents. The findings of this study suggested that children benefit from mathematical games and constitute mathematical learning in two-digit number computation. If instructors try hard to design games, be keen in administration, and adjusted games, children will learn through games and result more effective mathematics learning. Finally, it is concluded that the goal of enhancing mathematics learning through games-based instruction as feasible. Key word¡Gmathematics games, addition and subtraction computation

This diploma thesis deals with third-graders from one of Prague's elementary schools and their understanding of numbers. In addition to the Prodos textbooks, some didactical ideas were taken from the Fraus textbooks, which are aimed at the development of children's understanding mathematics. The experiments were carried out in a classroom and lasted one and a half year. Seven pupils were chosen to take part. The experiments were aimed at gauging their semantic and structural understanding of numbers. The pupils worked individually and were individually interviewed after the experiments. The assumption that pupils use strategies that are not taught in the classroom was confirmed, as well as the fact that the operation of subtraction is much more complicated than addition in mental arithmetic and also in written counting. However, the assumption that the pupils perform better when faced with financial matters as opposed to other situations was not confirmed. It came to light that when creating a verbal task, it is necessary to consider the pupil's understanding of the vocabulary used. Key Words: addition, subtraction, decomposition of number, computation, mathematics, calculative strategies

The aim of this study is to investigate the changes in problem-solving abilities and learning attitudes of two elementary second graders after integrating picture books into problem-posing instruction and administering tests on addition and subtraction word problems. The case study approach was adopted; where two second graders, having difficulty in comprehending addition and subtraction word problems, were selected from the researcher¡¦s class. Both students were given addition and subtraction word problems problem-solving pretests and were interviewed afterwards to further analyze their problem-solving abilities and attitudes so as to refer to remedial instruction. After instructional activities came to an end, addition and subtraction word problems problem-solving posttests were then given. Students were interviewed to explore the changes in problem-solving abilities and learning attitudes. Data were collected by observations, interviews and collections of related documents (including two cases¡¦ addition and subtraction word problems pretest and posttest, problem-solving worksheets, problem-posing worksheets, learning diaries, researcher¡¦s teaching introspection record table, audio-taping, video-taping, etc.). The changes in cases¡¦ problem-solving abilities and learning attitudes were analyzed and the findings indicated were four. First, picture books create problem-solving context that can raise cases¡¦ learning motives. Second, with problem-posing process of all types of addition and subtraction word problems and practice, problem-solving abilities of the cases were improved in four dimensions (Being able to locate unknown in problem targets and the known conditions from word problems; being able to select appropriate schematic drawing and to identify its relationship with the problem; being able to set up equations; and, being able to examine whether answers were reasonable.). Third, the remedial instructions designed were able to raise cases¡¦ level of confidence in problem-solving. Fourth, implementing remedial instruction by integrating picture books into problem-posing instruction enables not only transforming one case¡¦s passive learning attitudes to positive but also maintaining the other case¡¦s original positive attitudes. The above four results indicated that implementing addition and subtraction word problems remedial instruction to two second graders by integrating picture books into problem-posing instruction helped to improve children¡¦s problem-solving abilities and learning attitudes. Implications on research and practice were also given.

O presente trabalho tem como objetivo verificar o papel do ensino na aprendizagem da relação inversa entre adição e subtração e se o cálculo relacional pode ajudar no entendimento dessa relação inversa. Os objetivos específicos são: verificar diferenças na compreensão da relação inversa entre adição e subtração antes e após intervenção; e identificar, após a intervenção, as influências do entendimento do cálculo relacional na compreensão da relação inversa. A proposta caracterizou-se por uma pesquisa-intervenção com abordagem qualiquantitativa, realizada com alunos do PROEJA FIC do Instituto Federal Farroupilha, Campus São Borja/RS. O trabalho foi desenvolvido em quatro sessões, com oficinas de problemas matemáticos e aplicação de testes: um pré-teste, aplicado antes da primeira sessão; um pósteste, aplicado após a última sessão; e um pós-teste tardio, aplicado três meses após a última sessão. Para a análise quantitativa utilizamos o método de Análise de Variância (ANOVA) e realizamos uma análise qualitativa das observações, resolução dos problemas e das estratégias utilizadas nos problemas. No pré-teste houve uma pequena diferença de acertos em relação ao pós-teste, já na comparação com o pós-teste tardio os alunos retornaram ao ponto inicial, com o mesmo número de acertos do pré-teste. Ao verificarmos os tipos de problemas, percebemos que no bloco de problemas diretos os alunos tiveram o maior número de acertos. O bloco de problemas indiretos de início desconhecido foi os que os alunos mais erraram, porém no bloco de problemas indiretos de adendo desconhecido houve um aumento do número de acertos do pré-teste para o pós-teste tardio. Na análise dos dados, destacamos que os alunos do PROEJA FIC não compreendem a relação inversa entre adição e subtração, pois ao representar os problemas erraram o resultado por não entender a relação apontada no enunciado. Isso indica desconhecimento do cálculo relacional, porque ao escolherem incorretamente o cálculo para resolver um problema é evidente que as estratégias mentais utilizadas não estão adequadas. Diante disso, constatamos que os alunos do PROEJA FIC estudados não compreendem a relação inversa entre adição e subtração, devido ao fato de ainda não entenderem as relações e os conceitos que envolvem a estrutura aditiva. Assim, as quatro sessões de intervenção foram insuficientes com relação às necessidades apresentadas pelos alunos, sendo que para obter um resultado mais eficaz, com adultos, sugerimos um maior número de intervenções. / This study aims to determine the role of teaching in learning the inverse relationship between addition and subtraction and the relational calculus can help in understanding this inverse relationship. The specific objectives are: to determine differences in understanding the inverse relationship between addition and subtraction before and after intervention; and to identify, after the intervention, the influences of knowing relational calculus in order to understand the inverse relationship. The proposal was characterized by an intervention-research with qualiquantitative approach, conducted with PROEJA FIC students from the Instituto Federal Farroupilha, São Borja Câmpus/RS. The study was carried out in four sessions with mathematical problems workshops and tests application: a pre-test, applied before the first session; a post-test, applied after the last session; and a delayed post-test, applied three months after the last session. For the quantitative analysis we used the method of analysis of variance (ANOVA) and performed a qualitative analysis of the observations, problems solving and strategies used in the problems. During the pre-test there was a small difference in hits in relation to the post-test. When compared to the delayed post-test, the students returned to their starting point, with the same number of hits from the pre-test. When we observed the types of problems involved, we noticed that the students had the highest number of hits on the block of direct problems. The block of indirect problems with unknown beginning was the one the students missed the most, but in the block of indirect problems with unknown addendum there was an increased number of hits from the pre-test to the delayed post-test. During the data analysis, we highlight that the PROEJA FIC students did not understand the inverse relationship between addition and subtraction, since when representing the problems they failed by not understanding the relationship indicated in the statement. This demonstrates a lack of knowledge on relational calculus, because by choosing incorrectly the calculus to solve a problem it is clear that the mental strategies used are not appropriate. Therefore, we observed that the PROEJA FIC students do not understand the inverse relationship between addition and subtraction, due to the fact they are still not able to understand the relationships and concepts involving the additive structure. Thus, the four intervention sessions were insufficient regarding the needs presented by the students. In order to obtain a more effective result with adults, we suggest a higher number of interventions.

O presente trabalho tem como objetivo verificar o papel do ensino na aprendizagem da relação inversa entre adição e subtração e se o cálculo relacional pode ajudar no entendimento dessa relação inversa. Os objetivos específicos são: verificar diferenças na compreensão da relação inversa entre adição e subtração antes e após intervenção; e identificar, após a intervenção, as influências do entendimento do cálculo relacional na compreensão da relação inversa. A proposta caracterizou-se por uma pesquisa-intervenção com abordagem qualiquantitativa, realizada com alunos do PROEJA FIC do Instituto Federal Farroupilha, Campus São Borja/RS. O trabalho foi desenvolvido em quatro sessões, com oficinas de problemas matemáticos e aplicação de testes: um pré-teste, aplicado antes da primeira sessão; um pósteste, aplicado após a última sessão; e um pós-teste tardio, aplicado três meses após a última sessão. Para a análise quantitativa utilizamos o método de Análise de Variância (ANOVA) e realizamos uma análise qualitativa das observações, resolução dos problemas e das estratégias utilizadas nos problemas. No pré-teste houve uma pequena diferença de acertos em relação ao pós-teste, já na comparação com o pós-teste tardio os alunos retornaram ao ponto inicial, com o mesmo número de acertos do pré-teste. Ao verificarmos os tipos de problemas, percebemos que no bloco de problemas diretos os alunos tiveram o maior número de acertos. O bloco de problemas indiretos de início desconhecido foi os que os alunos mais erraram, porém no bloco de problemas indiretos de adendo desconhecido houve um aumento do número de acertos do pré-teste para o pós-teste tardio. Na análise dos dados, destacamos que os alunos do PROEJA FIC não compreendem a relação inversa entre adição e subtração, pois ao representar os problemas erraram o resultado por não entender a relação apontada no enunciado. Isso indica desconhecimento do cálculo relacional, porque ao escolherem incorretamente o cálculo para resolver um problema é evidente que as estratégias mentais utilizadas não estão adequadas. Diante disso, constatamos que os alunos do PROEJA FIC estudados não compreendem a relação inversa entre adição e subtração, devido ao fato de ainda não entenderem as relações e os conceitos que envolvem a estrutura aditiva. Assim, as quatro sessões de intervenção foram insuficientes com relação às necessidades apresentadas pelos alunos, sendo que para obter um resultado mais eficaz, com adultos, sugerimos um maior número de intervenções. / This study aims to determine the role of teaching in learning the inverse relationship between addition and subtraction and the relational calculus can help in understanding this inverse relationship. The specific objectives are: to determine differences in understanding the inverse relationship between addition and subtraction before and after intervention; and to identify, after the intervention, the influences of knowing relational calculus in order to understand the inverse relationship. The proposal was characterized by an intervention-research with qualiquantitative approach, conducted with PROEJA FIC students from the Instituto Federal Farroupilha, São Borja Câmpus/RS. The study was carried out in four sessions with mathematical problems workshops and tests application: a pre-test, applied before the first session; a post-test, applied after the last session; and a delayed post-test, applied three months after the last session. For the quantitative analysis we used the method of analysis of variance (ANOVA) and performed a qualitative analysis of the observations, problems solving and strategies used in the problems. During the pre-test there was a small difference in hits in relation to the post-test. When compared to the delayed post-test, the students returned to their starting point, with the same number of hits from the pre-test. When we observed the types of problems involved, we noticed that the students had the highest number of hits on the block of direct problems. The block of indirect problems with unknown beginning was the one the students missed the most, but in the block of indirect problems with unknown addendum there was an increased number of hits from the pre-test to the delayed post-test. During the data analysis, we highlight that the PROEJA FIC students did not understand the inverse relationship between addition and subtraction, since when representing the problems they failed by not understanding the relationship indicated in the statement. This demonstrates a lack of knowledge on relational calculus, because by choosing incorrectly the calculus to solve a problem it is clear that the mental strategies used are not appropriate. Therefore, we observed that the PROEJA FIC students do not understand the inverse relationship between addition and subtraction, due to the fact they are still not able to understand the relationships and concepts involving the additive structure. Thus, the four intervention sessions were insufficient regarding the needs presented by the students. In order to obtain a more effective result with adults, we suggest a higher number of interventions.

O presente trabalho tem como objetivo verificar o papel do ensino na aprendizagem da relação inversa entre adição e subtração e se o cálculo relacional pode ajudar no entendimento dessa relação inversa. Os objetivos específicos são: verificar diferenças na compreensão da relação inversa entre adição e subtração antes e após intervenção; e identificar, após a intervenção, as influências do entendimento do cálculo relacional na compreensão da relação inversa. A proposta caracterizou-se por uma pesquisa-intervenção com abordagem qualiquantitativa, realizada com alunos do PROEJA FIC do Instituto Federal Farroupilha, Campus São Borja/RS. O trabalho foi desenvolvido em quatro sessões, com oficinas de problemas matemáticos e aplicação de testes: um pré-teste, aplicado antes da primeira sessão; um pósteste, aplicado após a última sessão; e um pós-teste tardio, aplicado três meses após a última sessão. Para a análise quantitativa utilizamos o método de Análise de Variância (ANOVA) e realizamos uma análise qualitativa das observações, resolução dos problemas e das estratégias utilizadas nos problemas. No pré-teste houve uma pequena diferença de acertos em relação ao pós-teste, já na comparação com o pós-teste tardio os alunos retornaram ao ponto inicial, com o mesmo número de acertos do pré-teste. Ao verificarmos os tipos de problemas, percebemos que no bloco de problemas diretos os alunos tiveram o maior número de acertos. O bloco de problemas indiretos de início desconhecido foi os que os alunos mais erraram, porém no bloco de problemas indiretos de adendo desconhecido houve um aumento do número de acertos do pré-teste para o pós-teste tardio. Na análise dos dados, destacamos que os alunos do PROEJA FIC não compreendem a relação inversa entre adição e subtração, pois ao representar os problemas erraram o resultado por não entender a relação apontada no enunciado. Isso indica desconhecimento do cálculo relacional, porque ao escolherem incorretamente o cálculo para resolver um problema é evidente que as estratégias mentais utilizadas não estão adequadas. Diante disso, constatamos que os alunos do PROEJA FIC estudados não compreendem a relação inversa entre adição e subtração, devido ao fato de ainda não entenderem as relações e os conceitos que envolvem a estrutura aditiva. Assim, as quatro sessões de intervenção foram insuficientes com relação às necessidades apresentadas pelos alunos, sendo que para obter um resultado mais eficaz, com adultos, sugerimos um maior número de intervenções. / This study aims to determine the role of teaching in learning the inverse relationship between addition and subtraction and the relational calculus can help in understanding this inverse relationship. The specific objectives are: to determine differences in understanding the inverse relationship between addition and subtraction before and after intervention; and to identify, after the intervention, the influences of knowing relational calculus in order to understand the inverse relationship. The proposal was characterized by an intervention-research with qualiquantitative approach, conducted with PROEJA FIC students from the Instituto Federal Farroupilha, São Borja Câmpus/RS. The study was carried out in four sessions with mathematical problems workshops and tests application: a pre-test, applied before the first session; a post-test, applied after the last session; and a delayed post-test, applied three months after the last session. For the quantitative analysis we used the method of analysis of variance (ANOVA) and performed a qualitative analysis of the observations, problems solving and strategies used in the problems. During the pre-test there was a small difference in hits in relation to the post-test. When compared to the delayed post-test, the students returned to their starting point, with the same number of hits from the pre-test. When we observed the types of problems involved, we noticed that the students had the highest number of hits on the block of direct problems. The block of indirect problems with unknown beginning was the one the students missed the most, but in the block of indirect problems with unknown addendum there was an increased number of hits from the pre-test to the delayed post-test. During the data analysis, we highlight that the PROEJA FIC students did not understand the inverse relationship between addition and subtraction, since when representing the problems they failed by not understanding the relationship indicated in the statement. This demonstrates a lack of knowledge on relational calculus, because by choosing incorrectly the calculus to solve a problem it is clear that the mental strategies used are not appropriate. Therefore, we observed that the PROEJA FIC students do not understand the inverse relationship between addition and subtraction, due to the fact they are still not able to understand the relationships and concepts involving the additive structure. Thus, the four intervention sessions were insufficient regarding the needs presented by the students. In order to obtain a more effective result with adults, we suggest a higher number of interventions.

Läromedel har en styrande roll i många matematikklassrum vilket innebär att läromedel påverkar vilka uppgifter som elever erbjuds i undervisningen. Men vad är det som erbjuds i läromedel? Syftet med studien var att undersöka hur uppgifter i svenska läromedel för årskurs 1 behandlar tals additiva del-helhetsrelationer. Teorin om handlingserbjudanden och begrepp från variationsteorin har använts i syfte att utifrån ett funktionellt perspektiv undersöka vad som erbjuds i läromedel. Studien genomfördes genom en kvantitativ innehållsanalys och en kvalitativ dataanalys av fem läromedelsserier. Den kvantitativa analysen visar förekomsten av uppgifter som behandlar tals additiva del-helhetsrelationer. Den visar också att elever erbjuds olika representationsformer samt att växla mellan dessa. Språk, symbol och bild är de som erbjuds i störst utsträckning medan konkret modell endast erbjuds i ett fåtal uppgifter. Vidare visar den kvalitativa analysen skillnader i hur uppgifter är utformade för att synliggöra tals additiva del-helhetsrelationer. Uppgifternas form erbjuder elever skillnader i struktur, systematik samt i vilket utsträckning de olika sätt på vilket ett tal kan delas upp synliggörs. För att undervisningen ska främja elevers förståelse för tal och relationen mellan tal behöver läraren vara medveten om de möjligheter som läromedel erbjuder samt eventuella begränsningar. Detta för att kunna rikta elevers uppmärksamhet mot det som avses att läras. / Mathematics textbooks have a governing role in many mathematics classrooms which means that textbooks affect the tasks that learners are afforded in teaching. But what is afforded in these textbooks? The aim of the study was to investigate how Swedish textbooks for first grade treats additive part-whole relations of numbers. The theory of affordances and concepts from the variation theory has been used for the purpose, that based on a functional perspective to investigate what is afforded in textbooks. The study was conducted through a quantitative content analysis and a qualitative data analysis of five series of textbooks. The quantitative analysis shows the existence of tasks that treat additive part-whole relations of numbers. It also shows that learners are afforded different forms of representations and to switch between them. Language, symbol and image are afforded to the greatest extent while concrete model is only afforded in a few tasks. Furthermore, the qualitative analysis shows differences in how tasks are designed to make the additive part-whole relations of numbers visible. In the design of tasks, learners are afforded differences in structure, systematics and the extent to which the different ways in which a number can be divided are made visible. To promote learners´ understanding of additive part-whole relations of numbers, the teacher needs to be aware of the opportunities that textbooks can afford, as well as limitations. This is to be able to draw the learners´ attention to what is intended to be learned.

In his keynote address to the National Council of Teachers of Mathematics research pre-session, Sloane (2006b) challenged mathematics education researchers to ‘quantify qualitative insights’. This quasi-experimental study used blended methods to investigate the development of two-digit addition and subtraction strategies. Concurrent classroom teaching experiments were conducted in two intact first grade classrooms (n = 41) in a mid-Atlantic American public school. From a pragmatic emergent perspective, design research (Gravemeijer & Cobb, 2006) was used to develop local instructional theory. An amplified theoretical framework for early base-ten strategies is explicated. Multilevel modelling for repeated measures was used to evaluate the differences in strategy usage between classes across occasions and the association of particular pedagogical practices with the emergence of incrementing and decrementing by ten (N10) or decomposition (1010) strategies (Beishuizen, Felix, & Beishuizen, 1990).The two matched classes were not different in terms of gender, poverty, race, pre-assessment performance, and special education services. After the first unit of instruction with differentiated pedagogical tools, the collection class was significantly (p = .001) more likely to use 1010 than the linear class. No difference was demonstrated during the post-assessment. Students in both classes were more likely to use N10 during the last structured interview than in the first (p < .0001). Furthermore, there was no difference between the two classes in using any advanced strategy; however, students in both classes were more likely to use an advanced strategy at the conclusion of the study than they were initially (p = .033). The order of emergence of 1010 and N10 was not associated with the ability to develop both strategies, but there was an association (p < .001) between use of an advanced strategy and success on a district-mandated written assessment of two-digit addition and subtraction.Two original instructional sequences of contextually-based investigations are presented. Protocols transcribed from videotaped lessons and dynamic assessment interviews are presented to illuminate specific constructs detected and to illustrate the pedagogical techniques. An amplified framework for early place value constructs is proposed. Recommendations for future studies, curricular changes, and the need of early intervention are discussed.

In his keynote address to the National Council of Teachers of Mathematics research pre-session, Sloane (2006b) challenged mathematics education researchers to ‘quantify qualitative insights’. This quasi-experimental study used blended methods to investigate the development of two-digit addition and subtraction strategies. Concurrent classroom teaching experiments were conducted in two intact first grade classrooms (n = 41) in a mid-Atlantic American public school. From a pragmatic emergent perspective, design research (Gravemeijer & Cobb, 2006) was used to develop local instructional theory. An amplified theoretical framework for early base-ten strategies is explicated. Multilevel modelling for repeated measures was used to evaluate the differences in strategy usage between classes across occasions and the association of particular pedagogical practices with the emergence of incrementing and decrementing by ten (N10) or decomposition (1010) strategies (Beishuizen, Felix, & Beishuizen, 1990).The two matched classes were not different in terms of gender, poverty, race, pre-assessment performance, and special education services. After the first unit of instruction with differentiated pedagogical tools, the collection class was significantly (p = .001) more likely to use 1010 than the linear class. No difference was demonstrated during the post-assessment. Students in both classes were more likely to use N10 during the last structured interview than in the first (p < .0001). Furthermore, there was no difference between the two classes in using any advanced strategy; however, students in both classes were more likely to use an advanced strategy at the conclusion of the study than they were initially (p = .033). The order of emergence of 1010 and N10 was not associated with the ability to develop both strategies, but there was an association (p < .001) between use of an advanced strategy and success on a district-mandated written assessment of two-digit addition and subtraction.Two original instructional sequences of contextually-based investigations are presented. Protocols transcribed from videotaped lessons and dynamic assessment interviews are presented to illuminate specific constructs detected and to illustrate the pedagogical techniques. An amplified framework for early place value constructs is proposed. Recommendations for future studies, curricular changes, and the need of early intervention are discussed.

In his keynote address to the National Council of Teachers of Mathematics research pre-session, Sloane (2006b) challenged mathematics education researchers to ‘quantify qualitative insights’. This quasi-experimental study used blended methods to investigate the development of two-digit addition and subtraction strategies. Concurrent classroom teaching experiments were conducted in two intact first grade classrooms (n = 41) in a mid-Atlantic American public school. From a pragmatic emergent perspective, design research (Gravemeijer & Cobb, 2006) was used to develop local instructional theory. An amplified theoretical framework for early base-ten strategies is explicated. Multilevel modelling for repeated measures was used to evaluate the differences in strategy usage between classes across occasions and the association of particular pedagogical practices with the emergence of incrementing and decrementing by ten (N10) or decomposition (1010) strategies (Beishuizen, Felix, & Beishuizen, 1990).The two matched classes were not different in terms of gender, poverty, race, pre-assessment performance, and special education services. After the first unit of instruction with differentiated pedagogical tools, the collection class was significantly (p = .001) more likely to use 1010 than the linear class. No difference was demonstrated during the post-assessment. Students in both classes were more likely to use N10 during the last structured interview than in the first (p < .0001). Furthermore, there was no difference between the two classes in using any advanced strategy; however, students in both classes were more likely to use an advanced strategy at the conclusion of the study than they were initially (p = .033). The order of emergence of 1010 and N10 was not associated with the ability to develop both strategies, but there was an association (p < .001) between use of an advanced strategy and success on a district-mandated written assessment of two-digit addition and subtraction.Two original instructional sequences of contextually-based investigations are presented. Protocols transcribed from videotaped lessons and dynamic assessment interviews are presented to illuminate specific constructs detected and to illustrate the pedagogical techniques. An amplified framework for early place value constructs is proposed. Recommendations for future studies, curricular changes, and the need of early intervention are discussed.