Submitted by Jorge Silva (jorgelmsilva@ufrrj.br) on 2017-10-17T17:35:44Z No. of bitstreams: 1 2016 - Ernane Tarcisio Martins Gomes.pdf: 1573845 bytes, checksum: 78069c6459bd2b6aba951ba371ae74fd (MD5) / Made available in DSpace on 2017-10-17T17:35:44Z (GMT). No. of bitstreams: 1 2016 - Ernane Tarcisio Martins Gomes.pdf: 1573845 bytes, checksum: 78069c6459bd2b6aba951ba371ae74fd (MD5) Previous issue date: 2016-08-17 / CNPq / Soil organic matter can be derived from the decomposition of animal or vegetable. Most commercial crops show photosynthetic mechanisms C3 and C4, which set different carbon isotopes C12 or C13. Through isotopic analysis can determine the origin and prevalence of type of carbon that make up the soil organic matter (SOM) in a given area. Depending on the chemical fractionation MOS can also be divided into fractions FAF, FAH and FHU, these fractions have specific characteristics depending on their origin training and ambient conditions. The objective of this study was to evaluate the influence of plant origin C3 or C4 in the formation of the chemical and molecular structure of humic substances extracted from four areas with different vegetation cover, but submitted to agroecological management. To investigate this influence were extracted humic substances (HS) of these soils, being held chemical fractionation and chemical characterization by NMR analysis, FTIR, quantification and ?13C e ?15N isotopes in the AH fraction, 13 C-CP / MAS NMR and elemental determination. From the results obtained from these analyzes it was concluded that the plant does not influence qualitative differences, but in quantitative between the chemical components of humic substances (HS), the type of photosynthetic cycle does not seem to be a determining factor in the formation and incorporation of the types of structures in HA, but rather the quantities of these structures and their humic organizational distribution. The chemometric theoretical analyzes indicate that the development and preservation of humic these soils may be possible in intensive management systems such as rotation upon application of exogenous sources of stabilized SH, as well as favoring the humidification process in less managed systems as pasture and SAF. In this study it was demonstrated by characterization and chemometrics applied to AH of an agro-ecological system, the current state of MOS in terms of stability and structural enrichment and being proposed possible avenues for recovery and preservation of SH in mineral soils that may contribute to the increase in conditions fertility and thus agricultural production. / A mat?ria org?nica do solo pode ser oriunda da decomposi??o de animais ou vegetais. A maioria das culturas comerciais apresentam mecanismos fotossint?ticos C3 ou C4, que fixam diferentes is?topos de carbono C12 ou C13. Atrav?s de an?lises isot?picas ? poss?vel determinar a origem e predomin?ncia do tipo de carbono que comp?em a mat?ria org?nica do solo (MOS) em determinada ?rea. Em fun??o do fracionamento qu?mico a MOS ainda pode ser dividida nas fra??es FAF, FAH e FHU, estas fra??es apresentam caracter?sticas especificas em fun??o da sua origem de forma??o e condi??es ambientes. O objetivo deste trabalho foi avaliar a influ?ncia da origem vegetal C3 ou C4 na forma??o da estrutura qu?mica e molecular das substancias h?micas extra?das de quatro ?reas com cobertura diferentes vegetais, por?m submetidas ao manejo agroecol?gico. Para averiguar esta influ?ncia foram extra?das as substancias h?micas (SH) destes solos, sendo realizado o fracionamento qu?mico e caracteriza??o qu?mica atrav?s de an?lises de RMN, FTIR, quantifica??o de is?topos ?13C e?15N na fra??o de AH, 13C-CP/MAS NMR e a determina??o elementar. A partir dos resultados obtidos a partir dessas an?lises foi poss?vel concluir que a origem vegetal n?o influencia em diferen?as qualitativas e sim em quantitativas entre os componentes qu?micos das substancias h?micas (SH), o tipo de ciclo fotossint?tico n?o parece ser um fator determinante na forma??o e incorpora??o dos tipos de estruturas nos AH, mas sim nas quantidades destas estruturas e na sua distribui??o organizacional h?mica. As an?lises te?ricas quimiom?tricas indicam que a incorpora??o e preserva??o de MO humificada nestes solos pode ser poss?vel em sistemas de manejo intensivo como a rota??o, mediante aplica??es de fontes ex?genas de SH estabilizada, assim como tamb?m o favorecimento dos processos de humifica??o em sistemas menos manejados como pastagem e SAF. Neste estudo foi comprovado mediante carateriza??o e quimiometria aplicadas aos AH de um sistema agroecol?gico, o estado atual da MOS em termos de estabilidade e enriquecimento estrutural e sendo propostas poss?veis vias para a recupera??o e preserva??o das SH em solos minerais que possam contribuir ao aumento das condi??es de fertilidade e consequentemente da produ??o agr?cola.

Thesis (M.Sc. (Biochemistry)) --University of Limpopo, 2011 / Dicerocaryum senecioides is a crawling herb that is found growing mostly in sandy areas of southern and south-eastern Africa and its small, hairy leaves have been used over the years as food, shampoo, and for treatment of various ailments. In this study, the dichloromethane (D2) fraction was prepared from a crude methanol extract of D. senecoides leaves, and its effect on the proliferation of RAW 264.7 murine macrophages was investigated. Treatment of the macrophages with the extract resulted in a dose- and time-dependent decrease in cell viability as determined by the MTT assay and real time cell analysis. Cytotoxicity of the D2 fraction on the macrophages was demonstrated to be due to apoptosis by staining the cells with DAPI nucleic acid stain. Anti-inflammatory activity of D2 fraction on RAW cells was determined by evaluating intracellular ROS production by the DCFH-DA fluorescent assay. Cells treated with the D2 fraction and stimulated with PMA were found to have a lower fluorescence intensity compared to untreated, stimulated cells; thus mimicking the response observed in the resting cells. The percentage fluorescence in untreated, stimulated cells doubled, while no significant change was observed in the D2-treated cells. The effect of the D2 fraction on iNOS activity was also assessed. The fraction reduced the NO synthesised by iNOS in cells treated with the D2 fraction and stimulated with LPS dose-dependently. The D2 fraction was further fractionated by semi-preparative HPLC; and thin layer chromatography was used to analyse phytocompounds of the 96 HPLC sub-fractions as well as to screen these sub-fractions for anti-oxidative activity. Sub-fractions 1-7 and 33-39 showed an intensely pronounced DPPH-scavenging compound and this scavenging ability was confirmed by a quantitative DPPH assay that provided parallel results. The reducing potential of the sub-fractions was assessed by evaluating their Fe3+-reducing ability through the FRAP assay. Sub-fractions 1-7 and 33-39 displayed remarkable reducing potential. Taken together with the DPPH-scavenging activity, these findings suggest that HPLC sub-fractions 1-7 and 33-39 possess a compound(s) with impressive antioxidant activity. These findings merit the D2 fraction as an extract that can be used to control chronic inflammation as it does not only inhibit free radical production, but also scavenges excessive ROS and has the ability to induce apoptosis in the macrophages responsible for dysregulated production of the free radicals. The extract also has commendable chemoprotective and chemotherapeutic potential as it demonstrated pro-apoptotic activity along with prevention of excess free-radical production. / National Research Foundation and the University of Limpopo Research Office

In recent years, combinatorial properties of finite and infinite words have become increasingly important in fields of physics, biology, mathematics, and computer science. In particular, the fascinating family of Sturmian words has become an extremely active subject of research. These infinite binary sequences have numerous applications in various fields of mathematics, such as symbolic dynamics, the study of continued fraction expansion, and also in some domains of physics ( quasicrystal modelling ) and computer science ( pattern recognition, digital straightness ). There has also been a recent surge of interest in a natural generalization of Sturmian words to more than two letters - the so - called episturmian words, which include the well - known Arnoux - Rauzy sequences. This thesis represents a significant contribution to the study of Sturmian and episturmian words, and related objects such as generalized Thue - Morse words and substitutions on a finite alphabet. Specifically, we prove some new properties of certain palindromic factors of the infinite Fibonacci word; establish generalized ' singular ' decompositions of suffixes of certain morphic Sturmian words; completely describe where palindromes occur in characteristic Sturmian words; explicitly determine all integer powers occurring in a certain class of k-strict episturmian words ( including the k-bonacci word ) ; and prove that certain episturmian and generalized Thue - Morse continued fractions are transcendental. Lastly, we begin working towards a proof of a characterization of invertible substitutions on a finite alphabet, which generalizes the fact that invertible substitutions on two letters are exactly the Sturmian morphisms. / Thesis (Ph.D.)--School of Mathematical Sciences, 2006.

Cette thèse présente une contribution à l'amélioration de certains résultats concernant les algorithmes en Algèbre linéaire et plus particulièrement les algorithmes sur les matrices structurées. Nous présentons un nouvel algorithme de diagonalisation par blocs des matrices de Hankel, particulièrement efficace. Dans le cas où la matrice de Hankel correspond à une suite récurrente linéaire, nous retrouvons ainsi l'algorithme de Berlekamp-Massey, mais dans une version simplifiée (plus facile à expliquer et à programmer) et accélérée par des troncatures. En outre notre version permet une gestion dynamique des données. Notre diagonalisation par blocs, qui s'applique sur un corps arbitraire, nous permet de donner une démonstration purement algébrique et simple d'un délicat théorème de Frobenius pour la signature d'une forme de Hankel réelle. Nous donnons également une étude approfondie de l'algorithme d'Euclide signé et de ses versions matricielles pour les matrices de Hankel et de Bezout associées à un couple de polynômes. Nous expliquons les rapports existants entre différents algorithmes connus dans la littérature.

Ziel dieser Arbeit ist es, ein sequentielles Extraktionsverfahren zur Erfassung unterschiedlich stabiler Anteile der orgnischen Bodensubstanz (OBS) zu entwickeln und zu klären, ob ein Zusammenhang zwischen Löslichkeit und Stabilität besteht. Darüber hinaus sollen der Einfluss von Bewirtschaftungsmaßnahmen auf Menge und Zusammensetzung dieser OBS-Anteile und Zusammenhänge zwischen Fourier Transform Infrarot (FT-IR)-Spektroskopiedaten und der Kationenaustauschkapazität (KAK) der OBS analysiert werden. <br><br> Für die Untersuchungen wurden Böden der Langzeitfeldexperimente (LFE) in Halle, Bad Lauchstädt und Rotthalmünster beprobt. Zur Erfassung unterschiedlicher OBS-Fraktionen wurden im ersten Schritt die wasserlöslichen OBS-Anteile (OBS(W)-Fraktion) aus den Böden isoliert. Im zweiten Schritt wurden aus den Extraktionsrückständen der Wasserextraktion OBS-Anteile mit einer Natrium (Na)-Pyrophosphatlösung extrahiert (OBS(PY)-Fraktion). Die Stabilität der OBS-Fraktionen wurde anhand von &#948;13C-Bestimmungen und 14C-Messungen untersucht. Die Charakterisierung der Zusammensetzung der OBS-Fraktionen erfolgte mittels FT-IR Spektroskopie. <br><br> Generell wird mit der OBS(PY)-Fraktion ein größerer Anteil am organischen Kohlenstoffgehalt der Böden erfasst als mit der OBS(W)-Fraktion. Die &#948;13C- und 14C-Daten zeigen, dass die OBS(W)-Fraktion höhere Anteile jungen organischen Materials als die OBS(PY)-Fraktion enthält. Das entwickelte sequentielle Extraktionsverfahren ist also prinzipiell geeignet unterschiedlich stabile OBS-Anteile anhand ihrer Löslichkeit zu isolieren. <br><br> Mittels FT-IR spektroskopischer Untersuchungen wird festgestellt, dass Bewirtschaftungsmaßnahmen, wie die Düngung, sowie Standorteigenschaften die Zusammensetzung der OBS-Fraktionen beeinflussen. Für die OBS(PY)-Fraktion ist dies stärker ausgeprägt als für die OBS(W)-Fraktion. Die KAK der OBS(PY)-Fraktion aus den Böden der LFE in Halle und Bad Lauchstädt ist positiv mit der Absorptionsintensität der C=O-Bande in den FT-IR Spektren dieser OBS-Fraktion korreliert. / The aim of this work is the development of a sequential extraction procedure for the isolation of different stable parts of soil organic matter (SOM) according to their solubility and to prove the relationship between solubility and stability. Additionally, the effects of management on amount and composition of different SOM fractions as well as relationships between data sets from Fourier Transform Infrared (FT-IR) spectroscopy and cation exchange capacitity (CEC) of SOM should be analysed. <br><br> Soils from long term field experiments at Halle, Bad Lauchstädt and Rotthalmünster were sampled from the soil surface up to 30 cm depth. To obtain different SOM fractions water soluble SOM (SOM(W) fraction) were extracted at first before Na-pyrophosphat soluble SOM (SOM (PY) fraction) were isolated from the residue of water extraction. Stability of different SOM fractions were analyzed using 14C- and &#948;13C-measurements. Functional composition was studied with FT-IR spectroscopy. <br><br> Generally, SOM(PY) represents a larger part of soil organic carbon (SOC) than SOM(W) in relation to the total SOC-content of the investigated soils. &#948;13C- and 14C-measurements show that SOM(W) contains more younger material than SOM(PY). The findings suggest that different stable SOM fractions can be obtained according to their solubility by the sequential extraction procedure. <br><br> FTIR spectra show that SOM composition is influenced by management options like fertilization and field conditions. Such effects are more pronounced for SOM(PY) than for SOM(W). Furthermore data from FT-IR spectra and CEC measurements of SOMPY) show that the CEC is closely related to the C=O absorption intensities.

Canola fibre-protein and can-sugar are the two by-products arising from a process for separating high quality protein fractions from canola meal. In the first trial chemical characteristics of fibre-protein and can-sugar were examined in comparison with commercial canola and soy meal. In the second trial in situ rumen degradability and kinetics of test feed was studied. Based on the findings of those two trials, available energy values were estimated based on NRC (2001) while protein contents potentially absorbable at small intestine were predicted using both NRC (2001) and DVE/OEB models. Subsequently a mixture of fibre-protein and can-sugar was used as an additive to dehydrated alfalfa pellet and two dairy cow trials were conducted to determine the palatability and examine effect on lactation performances of blended alfalfa pellet feeding in comparison with standard alfalfa pellet. Palatability difference was evaluated by Paterson -two choice alternating access method through a 7 day experimental period using 6 lactating Holstein cows. In the lactating performance trial, 6 cows were randomly assigned into two groups and two treatments were allocated over three experimental periods in a switchback design. <p>Can-sugar consisted of water soluble components (CP 15.6 %DM; SCP 96.2 %CP; NFC 99.9 %CHO) with non-protein nitrogen as the main CP fraction (NPN 96.2 %CP). Fibre-protein was a highly fibrous material (NDF: 55.6%; ADF: 46.3%; ADL: 24.1%) comparing to canola meal (NDF: 25.4%, ADF: 21.2%, ADL: 9.0%) due to presence of higher level of seed hulls in fibre-protein. Comparing to canola meal, fibre-protein contained 9% less CP and 1/4 of that consisted of undegradable ADIP. Rumen degradability of can-sugar was assumed as immediate and total as it was water soluble. Most of the ruminally undegradable nutrient components present in canola meal appeared to be concentrated into fibre protein during the manufacturing process and as a result fibre-protein has shown a consistently lower effective degradability of DM, OM, CP NDF and ADF comparing to both canola and soy meal. Available energy content in can-sugar was marginally higher than that of canola meal while fibre-protein contained only 2/3 that of canola meal. The predicted absorbable protein content at small intestine was about 1/2 that of canola meal. These results indicate that fibre-protein can be considered as a secondary source of protein in ruminant feed and a mixture of fibre-protein and can-sugar would nutritionally complement each other to formulate into a cheaper ingredient in ruminant ration. In the palatability study, there was no significant difference (P>0.05) in intake preference or finish time between the blended and standard alfalfa pellets. The results from the lactation study showed that there was no significant difference (P>0.05) in milk yield, dairy efficiency or milk composition between the blended and standard alfalfa pellets. The results from the two studies indicated that fibre-protein and can-sugar fractions could be used as an additive to alfalfa dehydrated pellet at 15% inclusion rate without compromising its palatability or the performance of dairy cows. <p>For future studies it is proposed to conduct feeding trials with varying levels of inclusions to alfalfa pellet to know the nutritional effect of fibre-protein and can-sugar while ascertain optimum inclusion rate.

Canola fibre-protein and can-sugar are the two by-products arising from a process for separating high quality protein fractions from canola meal. In the first trial chemical characteristics of fibre-protein and can-sugar were examined in comparison with commercial canola and soy meal. In the second trial in situ rumen degradability and kinetics of test feed was studied. Based on the findings of those two trials, available energy values were estimated based on NRC (2001) while protein contents potentially absorbable at small intestine were predicted using both NRC (2001) and DVE/OEB models. Subsequently a mixture of fibre-protein and can-sugar was used as an additive to dehydrated alfalfa pellet and two dairy cow trials were conducted to determine the palatability and examine effect on lactation performances of blended alfalfa pellet feeding in comparison with standard alfalfa pellet. Palatability difference was evaluated by Paterson -two choice alternating access method through a 7 day experimental period using 6 lactating Holstein cows. In the lactating performance trial, 6 cows were randomly assigned into two groups and two treatments were allocated over three experimental periods in a switchback design. <p>Can-sugar consisted of water soluble components (CP 15.6 %DM; SCP 96.2 %CP; NFC 99.9 %CHO) with non-protein nitrogen as the main CP fraction (NPN 96.2 %CP). Fibre-protein was a highly fibrous material (NDF: 55.6%; ADF: 46.3%; ADL: 24.1%) comparing to canola meal (NDF: 25.4%, ADF: 21.2%, ADL: 9.0%) due to presence of higher level of seed hulls in fibre-protein. Comparing to canola meal, fibre-protein contained 9% less CP and 1/4 of that consisted of undegradable ADIP. Rumen degradability of can-sugar was assumed as immediate and total as it was water soluble. Most of the ruminally undegradable nutrient components present in canola meal appeared to be concentrated into fibre protein during the manufacturing process and as a result fibre-protein has shown a consistently lower effective degradability of DM, OM, CP NDF and ADF comparing to both canola and soy meal. Available energy content in can-sugar was marginally higher than that of canola meal while fibre-protein contained only 2/3 that of canola meal. The predicted absorbable protein content at small intestine was about 1/2 that of canola meal. These results indicate that fibre-protein can be considered as a secondary source of protein in ruminant feed and a mixture of fibre-protein and can-sugar would nutritionally complement each other to formulate into a cheaper ingredient in ruminant ration. In the palatability study, there was no significant difference (P>0.05) in intake preference or finish time between the blended and standard alfalfa pellets. The results from the lactation study showed that there was no significant difference (P>0.05) in milk yield, dairy efficiency or milk composition between the blended and standard alfalfa pellets. The results from the two studies indicated that fibre-protein and can-sugar fractions could be used as an additive to alfalfa dehydrated pellet at 15% inclusion rate without compromising its palatability or the performance of dairy cows. <p>For future studies it is proposed to conduct feeding trials with varying levels of inclusions to alfalfa pellet to know the nutritional effect of fibre-protein and can-sugar while ascertain optimum inclusion rate.

The aim of this study has been to investigate students’ understanding of fractions when they occur in the practical subject home economics. In order to investigate this, the learning study method was used. Learning study can be understood as a method to gain deeper understanding of what is learnt by students during a lesson to increase learning outcome. The variation theory framework has been utilized to analyze the results of this study. The object of learning was chosen to be addition of fractions bigger than one half when they appear in home economics. The results indicate that in order for the learners to experience the object of learning it requires that they discern some critical aspects; to distinguish between quarters and four parts, the relationship between the part and the whole in fractions, varied presentations of the same fraction along with an understanding of the meaning of the concept to double.   Moreover the students have carried out a math test which tested the students’ theoretical understanding of fractions to see whether they can transfer the knowledge from one learning context to another. The result evinces that most of the learners show a good understanding of fractions in a theoretical context as well as a practical. The learning study has increased the students learning according to the object of learning

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.   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.   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. / 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.   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.   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.

The purpose of this study was to examine pre-service mathematics teachers&amp / #8217 / subject matter knowledge, pedagogical content knowledge, and the relationships between the two on multiplication and division of fractions. For this purpose, pre-service teachers&amp / #8217 / understanding of key facts, concepts, principles, and proofs, their knowledge on common (mis)conceptions and difficulties held by the elementary students, their strategies of teaching these concepts, and representations they use to reason their understanding on multiplication and division of fractions were examined. Qualitative case study was performed where / data was collected from the pre-service teachers at the end of the spring semester of 2004-2005. Pre-service teachers were senior students enrolled in a teacher education program at a public university. Results revealed that pre-service teachers could easily symbolize and solve the basic questions on multiplication and division of fractions. However, in terms of pre-service teachers&amp / #8217 / interpretation and reasoning of key facts and principles on multiplication and division of fractions, their subject matter knowledge could not be regarded as conceptually deep. Furthermore, although pre-service teachers have strong belief that they should teach multiplication and division of fractions conceptually, where the logical background of the operations is explained, they do not have sufficient knowledge to represent and explain these topics and relationships conceptually.