Abstract The fundamental purpose of this research is to discuss the solution strategies and error types when seventh-grade students in middle school solved simple equations with one unknown and application problems; we hope results can provide reference for teachers in remedial teaching and in improving instructional. The items in the examination paper used in this research were taken from 12 past papers of basic competency test in Taiwan (2001 to 2006, twice a year). The subjects were students from Kaohsiung City Y Middle School (City region), Kaohsiung County T Middle School (Ordinary town) and Pingtung County Z Middle School (Near seacoast) forming a convenience sample. The investigators selected 2 classes from each school by random, with a total of 188 students. The main results of this research are as follows: 1. There are ten questions and the average proportion of correct answers is 49.10%; it shows that, at least 51% students have difficulties in solving symbolic operation and linear equations with one unknown. 2. The solution strategy student used most frequently is Removal of Term. Equal Axiom and Enumeration Evaluation Method were used occasionally. No student used Substitution method, Reduction method and Hidden method. 3. The main error types are: missing knowledge of using symbol; do not understand meanings of symbols; combine dissimilar terms freely; do not understand the meaning of bracket; cannot do calculation involving bracket; and, cannot simplify equation etc. 4. The main reasons for failures are: do not understand the meaning of equation; do not understand combination rules of like terms, position of symbol, unknown variables; whether containing fractions in reducing equation will also cause an error. Keywords: linear equations with one unknown, solution strategies, error types.

<p>In this qualitative study of students’ methods of calculating subtractions, I have used interviews, subtraction exercises and analysis of teaching material. The purpose of my study was to explore which methods students in grade three uses when calculating subtractions. I also wanted to highlight which strategies the students use and their comprehension of the concept of subtraction. In the study, I also highlight the different pedagogical ideas on which the teaching material is based on and the students’ choice of methods.  The study shows that the students choose to use the methods”deduct” and “kind of number” independently. Further it also shows that the students choose to switch from the method “deduct” to “kind of number” when the numbers in the exercises are further up on the number axis. When asked, the students answered that subtraction means “minus” which they in turn explained as removing something, an explanation confirmed by the Swedish Academy dictionary. The students’ choice of methods and teaching material is based on different fundamental pedagogical view.</p><p> </p>

In this qualitative study of students’ methods of calculating subtractions, I have used interviews, subtraction exercises and analysis of teaching material. The purpose of my study was to explore which methods students in grade three uses when calculating subtractions. I also wanted to highlight which strategies the students use and their comprehension of the concept of subtraction. In the study, I also highlight the different pedagogical ideas on which the teaching material is based on and the students’ choice of methods.  The study shows that the students choose to use the methods”deduct” and “kind of number” independently. Further it also shows that the students choose to switch from the method “deduct” to “kind of number” when the numbers in the exercises are further up on the number axis. When asked, the students answered that subtraction means “minus” which they in turn explained as removing something, an explanation confirmed by the Swedish Academy dictionary. The students’ choice of methods and teaching material is based on different fundamental pedagogical view.

Syftet med denna studie är att undersöka vilka svårigheter som finns för lärande inom multiplikation i årskurs 3. För att besvara studiens syfte tillämpades en diagnos samt semistrukturerade intervjuer som metod. Diagnosen genomfördes i två klasser i årskurs 3 och två pedagoger intervjuades för vidare undersökning. För att tydliggöra hur olika räkneuppgifter samt svårigheter hänger samman med olika fenomen tillämpades variationsteorin. Med utgångspunkt i variationsteorin samt begreppen kontrast, separation, generalisering och fusion kategoriseras det insamlade materialet utifrån analysverktyget meningskoncentrering. Resultatet visar att eleverna har svårigheter med följande: upptäcka likheter kring ett lärandeobjekt, skapa förståelse för de enskilda kritiska aspekterna av lärandeobjektet, se skillnad mellan två värden samt att välja en lämplig strategi. Vad gäller pedagogernas uppfattningar och beskrivningar av elevernas lösningsstrategier lyfts följande aspekter fram: stress och slarvfel, bristande förmåga att hitta ett samband mellan multiplikation och addition, bristfällig läs- och begreppsförståelse samt svårigheter att tillämpa en fungerande strategi. För att främja elevernas utveckling inom multiplikation påvisar resultatet vikten av att pedagoger i sin framtida undervisning arbetar med konkret material, didaktiska verktyg såsom areaformen och tallinje, visualisering av multiplikation genom addition, begreppsförståelse och samtal i grupp för att utöka matematikspråket. / The purpose of this study is to investigate the difficulties that exist for learning in multiplication in year 3. To answer the purpose of the study, a diagnosis and semi-structured interviews were used as a method. The diagnosis was performed in two classes in year 3 and two educators were interviewed for further investigation. In order to clarify how different arithmetic tasks and difficulties are related to different phenomena, the theory of variation was applied. Based on the theory of variation and the concepts of contrast, separation, generalization and fusion, the collected material is categorized on the basis of the analysis tool meaning concentration. The results show that the students have difficulties with the following: discover similarities around a learning object, create an understanding of the individual critical aspects of the learning object, see the difference between two values and choose an appropriate strategy. Regarding the educators 'perceptions and descriptions of the students' solution strategies, the following aspects are highlighted: stress and carelessness, lack of ability to find a connection between multiplication and addition, inadequate reading and comprehension and difficulties in applying a working strategy. To promote students' development in multiplication, the results show the importance of educators in their future teaching working with concrete material, didactic tools such as area form and number line, visualization of multiplication through addition,concept understanding and group discussions to expand the language of mathematics.

Darbo tyrimų objektas – neįgalių moksleivių, konfliktų sprendimo strategijos, sportinės veiklos motyvacija ir sav��s vertinimas. Tyrimo mokslinė problema - atskleisti skirtingą negalę turinčių asmenų konfliktų sprendimo strategijų, sportinės veiklos motyvų pasirinkimą ir savęs vertinimo lygį. Tyrimo tikslas – šiame darbe tiriame skirtingą negalią turinčių jaunuolių (silpnaregių, aklųjų, fiziškai neįgaliųjų ir su cerebriniu paralyžiumi) gebėjimo spręsti konfliktus, motyvaciją dalyvauti sportinėje veikloje ir jų savęs vertinimo ypatumus. Mūsų darbo uždaviniai: 1. Nustatyti skirtingą negalę turinčių jaunuolių, konflikto sprendimo strategijų, sportinės veiklos motyvų ir savęs vertinimo lygio raiškos ypatumus. 2. Ištirti skirtingo fizinio aktyvumo jaunuolių, konflikto sprendimo strategijų, sportinės veiklos motyvų ir savęs vertinimo lygio raiškos ypatumus. 3. Nustatyti sąsajas tarp konflikto sprendimo strategijų, sportinės veiklos motyvų ir savęs vertinimo lygio. Hipotezės: 1. Keliame prielaidą, kad negalės rūšis turi ryšį su neįgalių 15 – 20 metų jaunuolių konfliktų sprendimu, sportinės veiklos motyvų pasirinkimu ir savęs vertinimu. 2. Keliame prielaidą, kad fizinis aktyvumas turi ryšį su neįgalių 15 – 20 metų jaunuolių konfliktų sprendimu, sportinės veiklos motyvų pasirinkimu ir savęs vertinimu. Išvados: 1. Tiriant skirtingą negalią turinčių neįgalių jaunuolių, konflikto sprendimo strategijų, sportinės veiklos motyvų ir savęs vertinimo lygio... [toliau žr. visą tekstą] / Object of the research – conflicts solution strategies, sport activity motivation and the self-confidence of disabled leaners. Scientific problem of research - work scientific problem researches that reveal the self-confidence level of people having different disabilities, the choice of sport activity reasons and ability to solve conflicts. Research goal – young people‘s, having different disabilities (the weak-eyed, the blind, phisically disabled and having cerebral palsy), motivation to take part in sport activities and their peculiarities of self-confidence, and capability to solve conflicts were researched in this work. Objectives of the research: 1. To establish the peculiarities of young people having different disabilities, conflict solution strategies, sport activity motives and the level of self-confidence expression. 2. To research the peculiarities of young people having different phisical activity, conflict sollution strategies, sport activity motives and self-confidence level expression. 3. To establish whether there is a statistically reliable connection among conflict solution strategies, sport activity reasons and self-confidence. Hypotheses: 1. We are raising an assumption that the kind of disability influences motives‘ choice of self-confidence level, conflict sollution strategies and sport activity of 15-20 years old disabled young people. 2. We are raising an assumption that thysical activity influences motives‘ choice of self-confidence level... [to full text]

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abstract: The purpose of this study is to impact the teaching and learning of math of 2nd through 4th grade math students at Porfirio H. Gonzales Elementary School. The Cognitively Guided Instruction (CGI) model serves as the independent variable for this study. Its intent is to promote math instruction that emphasizes problem-solving to a greater degree and facilitates higher level questioning of teachers during their instructional dialogue with students. A mixed methods approach is being employed to see how the use of the CGI model of instruction impacts the math achievement of 2nd through 4th grade students on quarterly benchmark assessments administered at this school, to see how students problem-solving abilities progress over the duration of the study, and to see how teacher practices in questioning progress. Quantitative methods are used to answer the first of these research questions using archival time series (Amrein & Berliner, 2002) to view trends in achievement before and after the implementation of the CGI model. Qualitative methods are being used to answer questions around students' progression in their problem-solving abilities and teacher questioning to get richer descriptions of how these constructs evolve over the course of the study. / Dissertation/Thesis / Ed.D. Educational Leadership and Policy Studies 2012

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This study used Polya’s problem-solving model to map the level of development of the algebraic problem solving skills of Grade 12 learners from the Oshana Region in Northern Namibia. Deficiencies in problem solving skills among students in Namibian tertiary institutions have highlighted a possible knowledge gap between the Grade 12 and tertiary mathematics curricula (Fatokun, Hugo & Ajibola, 2009; Miranda, 2010). It is against this background that this study investigated the problem solving skills of Grade 12 learners in an attempt to understand the difficulties encountered by the Grade 12 learners in the problem solving process. Although there has been a great deal of effort made to improve student problem solving throughout the educational system, there is no standard way of evaluating written problem solving that is valid, reliable and easy to use (Docktor & Heller, 2009). The study designed and employed a computer aided algebraic problem solving assessment (CAAPSA) tool to map the algebraic problem solving skills of a sample of 210 Grade 12 learners during the 2010 academic year. The assessment framework of the learners’ problem solving skills was based on the Trends in International Mathematics and Science Study (TIMSS), Schoenfeld’s (1992) theory of metacognition and Polya’s (1957) problem solving model. The study followed a mixed methods triangulation design, in which both quantitative and qualitative data were collected and analysed simultaneously. The data collection instruments involved a knowledge base diagnostic test, an algebraic problem solving achievement test, an item analysis matrix for evaluating alignment of examination content to curriculum assessment objectives, a purposively selected sample of learners’ solution snippets, learner questionnaire and task-based learner interviews. The study found that 83.8% of the learners were at or below TIMSS level 2 (low) of algebraic problem solving skills. There was a moderate correlation between the achievement in the knowledge base and algebraic problem solving test (Pearson r = 0.5). There was however a high correlation between the learners’ achievement in the algebraic problem solving test and achievement in the final Namibia Senior Secondary Certificate (NSSC) examination of 2010 (Pearson r = 0.7). Most learners encountered difficulties in Polya’s first step, which focuses on the reading and understanding of the problem. The algebraic strategy was the most successfully employed solution strategy. / Mathematics Education / M. Sc. (Mathematics, Science and Technology Education (Mathematics Education))

The purpose of this study was to explore solution strategies that can enhance the achievement of low-performing Grade 12 learners in the following mathematical aspects: finding the general term of a quadratic sequence, factorising third degree polynomials, determining the centre and radius of a circle, and calculating the angle between two lines. A convenience sample of twenty-five low-performing Grade 12 learners from a secondary school in Capricorn District of Limpopo Province participated in the study which adopted a repeated-measures research design. Learners were exposed to multiple solution strategies and data were collected using achievement tests. Findings indicated significant differences in learners‟ average scores due to the solution strategies used. In determining the general term of a quadratic sequence, learners‟ scores were significantly higher when they used formula and the table method than with the method of residues and solving simultaneous equations. Synthetic division made learners to achieve better scores than long division and equating coefficients in factorising third degree polynomials. The use of formulae to find the centre and radius of a circle made learners to have better achievement scores than completing the square. In calculating the angle between two lines learners‟ scores were better using formula and the cosine rule than using theorems. It was concluded that exposing low-performing Grade 12 learners to multiple solution strategies would enhance their achievement in the mathematical aspects explored in the study. Some of the solution strategies that made learners to achieve better results were not in the prescribed mathematics textbooks. The study therefore recommends that mathematics teaching should not be textbook-driven and that low-performing Grade 12 learners should not be regarded as beyond redemption. / Mathematics Education / M.Sc. (Mathematics, Science and Technology Education)