The purpose of this study was to investigate the prospective teachers&rsquo / knowledge of mistakes held by elementary students in fractions and their proposed strategies to overcome those mistakes. The data were collected from 149 prospective elementary mathematics teachers enrolled in the elementary mathematics education programs from a public university in Central Anatolian Region. Fraction Knowledge Questionnaire was used to accomplish the purpose of the study. The data collection tool included nine open ended questions, and each question had two sub-tasks. In this study, the items in the &ldquo / Fraction Knowledge Questionnaire&rdquo / were analyzed in-depth in order to reach a detailed description of prospective teachers&rsquo / knowledge about students&rsquo / mistakes on fractions. The results of this study revealed that prospective elementary mathematics teachers mostly could identify the students&rsquo / mistakes. However, although prospective teachers could notice the students&rsquo / mistakes, they could give superficial reasons for these mistakes. Furthermore, verbal explanations, using area representation, using real life model, reviewing prior knowledge, teaching standard algorithm, asking guided questions, using simple examples, using counter examples, using drill and practice, making students aware of their mistakes, and increasing students&rsquo / motivation were the suggested strategies by prospective teachers in order to overcome students&rsquo / mistakes in fractions.

<p>Syftet med den här studien är att undersöka undervisning och lärande av lärandeobjektet, bråk med olika nämnare, i årskurs 4, årskurs 6 och årskurs 9. Då specifikt jämförelsen av bråk med olika nämnare och addition med hjälp av gemensam nämnare. Studien avser att söka vad som är kritiskt för att kunna lära sig detta.</p><p> </p><p>Vi genomförde en learning study som är en typ av aktionsforskning där studien genomförs i klassrumsmiljö och fokus ligger på vad eleverna ges möjlighet att lära sig. Vanligtvis genomförs en learning study i tre cykler där undervisningen gradvis förbättras mellan cyklerna. Vår första cykel genomfördes i årskurs fyra, den andra cykeln i årskurs sex och den sista och avslutande cykeln i årskurs nio. Lektionen i årskurs fyra var en introduktion till bråk där fokus låg på nämnarens funktion. Lektionerna i årskurs sex och nio fokuserade till stor del på addition av bråk med olika nämnare. Processen i varje cykel började med ett förtest som sedan följdes upp med en planerad lektion utifrån utfallet, lektionen i sin tur videofilmades utifrån det iscensatta lärandeobjektet, och som sista del tog vi reda på det erfarna lärandeobjektet med hjälp av eftertest. Materialet (förtest, lektion och eftertest) från de tre lektionerna analyserades därefter och jämfördes för att få fram kritiska aspekter vid inlärning av bråk med olika nämnare.</p><p> </p><p>Resultatet indikerar att de kritiska aspekterna för elever i årskurs fyra är innebörden av täljare och nämnare samt relationen mellan dem. Denna kritiska aspekt lyckades vi komma åt under lektionen. I de senare cyklerna visar resultatet att den viktigaste kritiska aspekten är relationen mellan täljare och nämnare och insikten att de står i proportionalitet till varandra, dessa aspekter kom vi endast till viss del åt under lektionerna i årskurs sex och årskurs nio. I vissa fall försämrades deras förståelse efter lektionerna. Eleverna i årskurs sex och årskurs nio uppvisade samma svårigheter och vi kan inte påvisa någon förbättring mellan årskurserna; eleverna i årskurs sex och nio låg på samma nivå kunskapsmässigt då det gäller bråk med olika nämnare.</p> / <p>The aim of this study is to investigate the teaching and learning of fractions with different denominators as an object of learning in grade 4, 6 and 9. We concentrate more specifically on the comparison of fractions with different denominators as well as performing additions through transformation to a common denominator. Moreover, the study aims at finding out what is critical in order to learn this.</p><p> </p><p>We carried out a learning study, which is a kind of action research where the study is carried out in the classroom environment and where focus is on what the pupils is given the opportunity to learn. Moreover, a learning study is normally conducted in three cycles where the aim is to improve teaching during and between the different cycles. Our first cycle was conducted in grade four, the second cycle in grade six and the final and concluding one in grade nine. The lesson in grade four served as an introduction to fractions since the pupils were unfamiliar with the concept of fractions, yet the focus was on the function of the denominator, whereas the focus of the lessons in grade six and grade nine was on performing addition of fractions with different denominators. The process in each of the cycles in the learning study began with a pretest. The lesson was then planned according to the assumed critical aspects shown in the pretest before carried out. Each lesson was videotaped, and followed by a posttest about a week after the lesson. The material (pre-tests, lessons, and post-tests) from the three cycles were analyzed and thereafter also compared in order to find and compare critical aspects when it comes to the learning of fractions with different denominators.</p><p> </p><p>The result indicates that the critical aspects for pupils in grade four were the meaning of the denominator and the numerator and their relationship. These aspects were targeted in the lesson and the pupils’ understanding of this was improved after the lesson. Moreover, the result implies that the most important critical aspect in grade six, as well as in grade nine, appears to be to understand the proportional relationship between the denominator and the numerator. This aspect was targeted at the respective lessons, yet the pupils only showed little or even no better understanding after the lessons, and in some cases their understanding even became impaired. It appears as if the pupils in grade six and grade nine have the same difficulties when dealing with fractions with unequal denominators, and moreover they were on the same level when dealing with the fractions.</p>

Cette thèse présente l'analyse des pratiques de cinq enseignants de CM2 sur le chapitre " fractions ", ces professeurs utilisant le même manuel écrit par des didacticiens (Euromaths). Dans la première partie de ce travail nous nous intéressons aux savoirs à enseigner présentés dans le manuel choisi. Ils sont caractérisés à partir d'une analyse s'appuyant sur la théorie la dialectique outil-objet et de la comparaison d'Euromaths avec cinq autres ouvrages. La seconde partie de la thèse présente les résultats de l'analyse des pratiques de ces cinq professeurs des écoles qui se sont servis du manuel pour construire leur enseignement. En adoptant le point de vue théorique de la " double approche " didactique et ergonomique des pratiques des enseignants, selon lequel leur analyse ne peut pas se réduire à l'analyse des apprentissages potentiels des élèves qu'on peut déduire de leurs pratiques, un cadrage théorique issue de la théorie de l'activité a été mis en place pour approcher les utilisations du manuel. Ce cadrage semble convenir aux nécessaires aller-retour entre le local (ce qui se passe effectivement en classe) et le global (l'inscription de quelques séances dans le processus d'enseignement des rationnels, compte tenu du manuel) et a permis d'obtenir un spectre des utilisations possibles du manuel par les cinq enseignants dont les séances ont été observées. De fait, les pratiques enseignantes analysées s'avèrent différentes les unes des autres et révèlent, pour certains enseignants, des difficultés de mise en œuvre du manuel. En particulier, il semblerait qu'aux connaissances mathématiques et didactiques nécessaires à l'appréciation du processus d'enseignement et à la mise en œuvre des situations d'apprentissage en classe, s'ajoute la nécessité d'utiliser de gestes professionnels très précis pour leur réalisation. Au vu des résultats se pose la question d'un juste milieu concernant la part d'ouverture, mais aussi d'implicite, des manuels proposés aux professeurs des écoles et plus particulièrement celle de la part d'explicitation des tâches prescrites aux enseignants (telles qu'elles apparaissent dans le manuel et le livre du professeur) pour que l'ouverture des tâches prévues pour les élèves subsiste.

Milk is provided to new borns as their first food source and it contains essential nutrients, vitamins and other beneficial components, such as enzymes and antibodies that are required for rapid growth and development of the new born and for sustained growth over time. Milk contains two main types of proteins; casein proteins and whey proteins. Although casein proteins account for up to 80% of the proteins found in bovine milk, it is the whey protein that has become of high interest because of its bioactive content. Whey, a very watery mixture of lactose, proteins, minerals and trace amounts of fat, is formed from milk when the milk is coagulated and/or the casein proteins are removed from the milk. Bovine whey protein, including both the acidic and basic fractions (low and high isoelectric point, respectively), has previously been studied in vitro (cell based) and in vivo (using rats) for its impact on bone to determine if it can help improve bone mineral density and help reduce the risk of developing bone diseases, such as osteoporosis. Bone is constantly undergoing a remodelling process of being dissolved and reformed and the two main cell types responsible for this bone remodelling process are mature osteoclasts, which dissolve (resorb) bone, and osteoblasts, which reform the bone. Prior work has shown that acidic protein fractions derived from different sources of whey protein concentrate (WPC) have both in vivo and in vitro activity on bone, particularly anti-resorptive properties. However, the component(s) which confer activity have not yet been identified. In this thesis, work was undertaken to better understand the analytical composition of three types of WPC (cheese, mineral acid and lactic acid) and their associated acidic protein fractions and relate this to bone activity in the hope of identifying where the activity lies. Bone activity was assessed using in vitro screening with osteoblast cells (MC3T3-E1) and osteoclast cells (RAW 264.7). Comparison of the cell-based bone activity of the parent WPCs and corresponding acidic fractions indicated that the acidic fractions derived from both mineral acid and lactic WPC were superior in their ability to inhibit osteoclast development. Although compositional data was complex and definitive correlations with both bone bioactivities could not be made, it appeared that elements common to both the acidic fractions were a higher proportion of GLYCAM-1 and bone sialoprotein-1 (osteopontin). Further studies to more closely investigate the bone bioactivity of the acidic fractions are warranted.

Milk is provided to new borns as their first food source and it contains essential nutrients, vitamins and other beneficial components, such as enzymes and antibodies that are required for rapid growth and development of the new born and for sustained growth over time. Milk contains two main types of proteins; casein proteins and whey proteins. Although casein proteins account for up to 80% of the proteins found in bovine milk, it is the whey protein that has become of high interest because of its bioactive content. Whey, a very watery mixture of lactose, proteins, minerals and trace amounts of fat, is formed from milk when the milk is coagulated and/or the casein proteins are removed from the milk. Bovine whey protein, including both the acidic and basic fractions (low and high isoelectric point, respectively), has previously been studied in vitro (cell based) and in vivo (using rats) for its impact on bone to determine if it can help improve bone mineral density and help reduce the risk of developing bone diseases, such as osteoporosis. Bone is constantly undergoing a remodelling process of being dissolved and reformed and the two main cell types responsible for this bone remodelling process are mature osteoclasts, which dissolve (resorb) bone, and osteoblasts, which reform the bone. Prior work has shown that acidic protein fractions derived from different sources of whey protein concentrate (WPC) have both in vivo and in vitro activity on bone, particularly anti-resorptive properties. However, the component(s) which confer activity have not yet been identified. In this thesis, work was undertaken to better understand the analytical composition of three types of WPC (cheese, mineral acid and lactic acid) and their associated acidic protein fractions and relate this to bone activity in the hope of identifying where the activity lies. Bone activity was assessed using in vitro screening with osteoblast cells (MC3T3-E1) and osteoclast cells (RAW 264.7). Comparison of the cell-based bone activity of the parent WPCs and corresponding acidic fractions indicated that the acidic fractions derived from both mineral acid and lactic WPC were superior in their ability to inhibit osteoclast development. Although compositional data was complex and definitive correlations with both bone bioactivities could not be made, it appeared that elements common to both the acidic fractions were a higher proportion of GLYCAM-1 and bone sialoprotein-1 (osteopontin). Further studies to more closely investigate the bone bioactivity of the acidic fractions are warranted.

Milk is provided to new borns as their first food source and it contains essential nutrients, vitamins and other beneficial components, such as enzymes and antibodies that are required for rapid growth and development of the new born and for sustained growth over time. Milk contains two main types of proteins; casein proteins and whey proteins. Although casein proteins account for up to 80% of the proteins found in bovine milk, it is the whey protein that has become of high interest because of its bioactive content. Whey, a very watery mixture of lactose, proteins, minerals and trace amounts of fat, is formed from milk when the milk is coagulated and/or the casein proteins are removed from the milk. Bovine whey protein, including both the acidic and basic fractions (low and high isoelectric point, respectively), has previously been studied in vitro (cell based) and in vivo (using rats) for its impact on bone to determine if it can help improve bone mineral density and help reduce the risk of developing bone diseases, such as osteoporosis. Bone is constantly undergoing a remodelling process of being dissolved and reformed and the two main cell types responsible for this bone remodelling process are mature osteoclasts, which dissolve (resorb) bone, and osteoblasts, which reform the bone. Prior work has shown that acidic protein fractions derived from different sources of whey protein concentrate (WPC) have both in vivo and in vitro activity on bone, particularly anti-resorptive properties. However, the component(s) which confer activity have not yet been identified. In this thesis, work was undertaken to better understand the analytical composition of three types of WPC (cheese, mineral acid and lactic acid) and their associated acidic protein fractions and relate this to bone activity in the hope of identifying where the activity lies. Bone activity was assessed using in vitro screening with osteoblast cells (MC3T3-E1) and osteoclast cells (RAW 264.7). Comparison of the cell-based bone activity of the parent WPCs and corresponding acidic fractions indicated that the acidic fractions derived from both mineral acid and lactic WPC were superior in their ability to inhibit osteoclast development. Although compositional data was complex and definitive correlations with both bone bioactivities could not be made, it appeared that elements common to both the acidic fractions were a higher proportion of GLYCAM-1 and bone sialoprotein-1 (osteopontin). Further studies to more closely investigate the bone bioactivity of the acidic fractions are warranted.

Milk is provided to new borns as their first food source and it contains essential nutrients, vitamins and other beneficial components, such as enzymes and antibodies that are required for rapid growth and development of the new born and for sustained growth over time. Milk contains two main types of proteins; casein proteins and whey proteins. Although casein proteins account for up to 80% of the proteins found in bovine milk, it is the whey protein that has become of high interest because of its bioactive content. Whey, a very watery mixture of lactose, proteins, minerals and trace amounts of fat, is formed from milk when the milk is coagulated and/or the casein proteins are removed from the milk. Bovine whey protein, including both the acidic and basic fractions (low and high isoelectric point, respectively), has previously been studied in vitro (cell based) and in vivo (using rats) for its impact on bone to determine if it can help improve bone mineral density and help reduce the risk of developing bone diseases, such as osteoporosis. Bone is constantly undergoing a remodelling process of being dissolved and reformed and the two main cell types responsible for this bone remodelling process are mature osteoclasts, which dissolve (resorb) bone, and osteoblasts, which reform the bone. Prior work has shown that acidic protein fractions derived from different sources of whey protein concentrate (WPC) have both in vivo and in vitro activity on bone, particularly anti-resorptive properties. However, the component(s) which confer activity have not yet been identified. In this thesis, work was undertaken to better understand the analytical composition of three types of WPC (cheese, mineral acid and lactic acid) and their associated acidic protein fractions and relate this to bone activity in the hope of identifying where the activity lies. Bone activity was assessed using in vitro screening with osteoblast cells (MC3T3-E1) and osteoclast cells (RAW 264.7). Comparison of the cell-based bone activity of the parent WPCs and corresponding acidic fractions indicated that the acidic fractions derived from both mineral acid and lactic WPC were superior in their ability to inhibit osteoclast development. Although compositional data was complex and definitive correlations with both bone bioactivities could not be made, it appeared that elements common to both the acidic fractions were a higher proportion of GLYCAM-1 and bone sialoprotein-1 (osteopontin). Further studies to more closely investigate the bone bioactivity of the acidic fractions are warranted.

Milk is provided to new borns as their first food source and it contains essential nutrients, vitamins and other beneficial components, such as enzymes and antibodies that are required for rapid growth and development of the new born and for sustained growth over time. Milk contains two main types of proteins; casein proteins and whey proteins. Although casein proteins account for up to 80% of the proteins found in bovine milk, it is the whey protein that has become of high interest because of its bioactive content. Whey, a very watery mixture of lactose, proteins, minerals and trace amounts of fat, is formed from milk when the milk is coagulated and/or the casein proteins are removed from the milk. Bovine whey protein, including both the acidic and basic fractions (low and high isoelectric point, respectively), has previously been studied in vitro (cell based) and in vivo (using rats) for its impact on bone to determine if it can help improve bone mineral density and help reduce the risk of developing bone diseases, such as osteoporosis. Bone is constantly undergoing a remodelling process of being dissolved and reformed and the two main cell types responsible for this bone remodelling process are mature osteoclasts, which dissolve (resorb) bone, and osteoblasts, which reform the bone. Prior work has shown that acidic protein fractions derived from different sources of whey protein concentrate (WPC) have both in vivo and in vitro activity on bone, particularly anti-resorptive properties. However, the component(s) which confer activity have not yet been identified. In this thesis, work was undertaken to better understand the analytical composition of three types of WPC (cheese, mineral acid and lactic acid) and their associated acidic protein fractions and relate this to bone activity in the hope of identifying where the activity lies. Bone activity was assessed using in vitro screening with osteoblast cells (MC3T3-E1) and osteoclast cells (RAW 264.7). Comparison of the cell-based bone activity of the parent WPCs and corresponding acidic fractions indicated that the acidic fractions derived from both mineral acid and lactic WPC were superior in their ability to inhibit osteoclast development. Although compositional data was complex and definitive correlations with both bone bioactivities could not be made, it appeared that elements common to both the acidic fractions were a higher proportion of GLYCAM-1 and bone sialoprotein-1 (osteopontin). Further studies to more closely investigate the bone bioactivity of the acidic fractions are warranted.

Milk is provided to new borns as their first food source and it contains essential nutrients, vitamins and other beneficial components, such as enzymes and antibodies that are required for rapid growth and development of the new born and for sustained growth over time. Milk contains two main types of proteins; casein proteins and whey proteins. Although casein proteins account for up to 80% of the proteins found in bovine milk, it is the whey protein that has become of high interest because of its bioactive content. Whey, a very watery mixture of lactose, proteins, minerals and trace amounts of fat, is formed from milk when the milk is coagulated and/or the casein proteins are removed from the milk. Bovine whey protein, including both the acidic and basic fractions (low and high isoelectric point, respectively), has previously been studied in vitro (cell based) and in vivo (using rats) for its impact on bone to determine if it can help improve bone mineral density and help reduce the risk of developing bone diseases, such as osteoporosis. Bone is constantly undergoing a remodelling process of being dissolved and reformed and the two main cell types responsible for this bone remodelling process are mature osteoclasts, which dissolve (resorb) bone, and osteoblasts, which reform the bone. Prior work has shown that acidic protein fractions derived from different sources of whey protein concentrate (WPC) have both in vivo and in vitro activity on bone, particularly anti-resorptive properties. However, the component(s) which confer activity have not yet been identified. In this thesis, work was undertaken to better understand the analytical composition of three types of WPC (cheese, mineral acid and lactic acid) and their associated acidic protein fractions and relate this to bone activity in the hope of identifying where the activity lies. Bone activity was assessed using in vitro screening with osteoblast cells (MC3T3-E1) and osteoclast cells (RAW 264.7). Comparison of the cell-based bone activity of the parent WPCs and corresponding acidic fractions indicated that the acidic fractions derived from both mineral acid and lactic WPC were superior in their ability to inhibit osteoclast development. Although compositional data was complex and definitive correlations with both bone bioactivities could not be made, it appeared that elements common to both the acidic fractions were a higher proportion of GLYCAM-1 and bone sialoprotein-1 (osteopontin). Further studies to more closely investigate the bone bioactivity of the acidic fractions are warranted.

Milk is provided to new borns as their first food source and it contains essential nutrients, vitamins and other beneficial components, such as enzymes and antibodies that are required for rapid growth and development of the new born and for sustained growth over time. Milk contains two main types of proteins; casein proteins and whey proteins. Although casein proteins account for up to 80% of the proteins found in bovine milk, it is the whey protein that has become of high interest because of its bioactive content. Whey, a very watery mixture of lactose, proteins, minerals and trace amounts of fat, is formed from milk when the milk is coagulated and/or the casein proteins are removed from the milk. Bovine whey protein, including both the acidic and basic fractions (low and high isoelectric point, respectively), has previously been studied in vitro (cell based) and in vivo (using rats) for its impact on bone to determine if it can help improve bone mineral density and help reduce the risk of developing bone diseases, such as osteoporosis. Bone is constantly undergoing a remodelling process of being dissolved and reformed and the two main cell types responsible for this bone remodelling process are mature osteoclasts, which dissolve (resorb) bone, and osteoblasts, which reform the bone. Prior work has shown that acidic protein fractions derived from different sources of whey protein concentrate (WPC) have both in vivo and in vitro activity on bone, particularly anti-resorptive properties. However, the component(s) which confer activity have not yet been identified. In this thesis, work was undertaken to better understand the analytical composition of three types of WPC (cheese, mineral acid and lactic acid) and their associated acidic protein fractions and relate this to bone activity in the hope of identifying where the activity lies. Bone activity was assessed using in vitro screening with osteoblast cells (MC3T3-E1) and osteoclast cells (RAW 264.7). Comparison of the cell-based bone activity of the parent WPCs and corresponding acidic fractions indicated that the acidic fractions derived from both mineral acid and lactic WPC were superior in their ability to inhibit osteoclast development. Although compositional data was complex and definitive correlations with both bone bioactivities could not be made, it appeared that elements common to both the acidic fractions were a higher proportion of GLYCAM-1 and bone sialoprotein-1 (osteopontin). Further studies to more closely investigate the bone bioactivity of the acidic fractions are warranted.