Automated analysis of the difficulty of secondary school geometry theorems

Recent curriculum reforms underline mathematical activity related to proof validation , but few studies have explicitly addressed proof validation at the secondary school level. This chapter reports on our study of this issue. We suggest a specific kind of task for introducing proof validation in secondary school geometry and define the meanings of proof validation and proof modification in terms of Lakatos’s notion of the local counterexample . We briefly report on a classroom-based intervention implemented using such tasks in a lower secondary school in Japan. We then analyze the results of a task-based questionnaire conducted after the intervention to investigate how well the students did in proof validation and modification. The analysis shows that student failure in proof validation arose mainly from their difficulty with producing diagrams that satisfied the condition of the proof problem.

This study was undertaken to determine if there is a relationship between final grades received in secondary school technically-oriented courses and the final grade received in a college private pilot ground school course. Ninety-six first-year students enrolled in the ground school course comprised the sample, with data for 51 aviation pre-major students and 45 non-aviation pre-major students examined both as two separate samples and as a single sample. Final grades earned by these students in secondary school elementary algebra, geometry and advanced algebra, the mean of the three mathematics final grades, the mean of the secondary science final grades and the mean of the secondary English final grades were used as independent variables. The final grades earned in Technology 101, Private Pilot Ground School, were used as the dependent variable. From the analysis, it may be concluded that, when grades earned by aviation pre-major students and non-aviation pre-major students were analyzed as one sample, there was a relationship for all students between the private pilot ground school course and all of the secondary school technically-related courses. When the grades earned by aviation pre-majors were analyzed alone, there were relationships between only the ground school course and geometry and the mean of the final grades earned in secondary mathematics courses. The analysis of the grades earned by non-aviation pre-majors revealed relationships between the ground school course and all of the secondary school technically-related courses. There was no relationship between the ground school course and the mean of the final grades earned in secondary English courses for any of the three samples studied. This study suggests that secondary school mathematics plays a role in preparing all students for a college private pilot ground school course. Secondary school geometry appears to playa particularly Significant role in preparing pre-aviation majors who are taking a private pilot ground school course as an initial professional course.

Secondary school geometry is perhaps most succinctly described as the study of shape. Many aspects of shape are studied, such as properties of and relationships among shapes, location of shapes, transformations of shapes, and reasoning about shape. Consider an important counterpoint to this shape story or perhaps chapter zero in the story—the study of vertex-edge graphs, which are geometric objects for which shape is not an essential characteristic.

The purpose of this study was to identify the secondary school geometry portions in mathematics text book in terms of the van Hiele level and to investigate the van Hiele level of understanding geometry of third year mathematics students in Addis Ababa University who were completed geometry courses with respect to sex and program. The 25 multiple-choice item test was developed by the Cognitive Development and Achievement in Secondary School Geometry Project (Usiskin, 1982) based on the van Hiele Theory of Geometric Thinking. The data were analyzed based on the ‘4 of 5 criterion’ and the results showed that the presentation of grade 9 and grade 10 geometry topics correspond with the expected level stated by the National Council of Teachers of Mathematics Principles and Standards for School Mathematics. Most of the regular and extension students were below Level 3 that is they were not adequate to instruct at secondary school level. The result also indicates as there is no statistically significant difference in geometric reasoning levels between the male and female regular mathematics students but there is statistically significant difference as in geometric reasoning levels between the male and female extension mathematics students. That is extension male mathematics students are numerically higher van Hiele level than that of extension female mathematics students. There is also a statistically significant difference as in geometric reasoning levels between the regular mathematics students and extension mathematics students. That is extension mathematics students are numerically higher van Hiele level than that of regular mathematics students.

Proof validation and modification in secondary school geometry

In contrast to existing research that has typically addressed the process from example generation to proof construction, this study aims at enhancing empirical examination after proof construction leading to revision of statements and proofs in secondary school geometry. The term “empirical examination” refers to the use of examples or diagrams to investigate whether a statement is true or a proof is valid. Although empirical examination after proof construction is significant in school mathematics in terms of cultivating students’ critical thinking and achieving authentic mathematical practice, how this activity can be fostered remains unclear. This paper shows the strength of a particular kind of mathematical task, proof problems with diagrams, and teachers’ roles in implementing the tasks, by analysing two classroom-based interventions with students in the eighth and ninth grades. In the interventions, the tasks and the teachers’ actions successfully prompted the students to discover a case rejecting a proof and a case refuting a statement, modify the proof, properly restrict the domain of the statement by disclosing its hidden condition, and invent a more general statement that was true even for the refutation of the original statement.

With the transition to the digital age, changes have emerged in the skills expected from the individuals of the 21st century, and accordingly, the preparation of curricula to develop these skills has become the main goal of all countries in the world. In our country, studies have been carried out to develop curricula in this direction, and with this research, it is aimed to examine the secondary education mathematics (2010, 2011, 2013 and 2017) and geometry (2011) teaching programs in the context of 21st century skills. The research is a survey study aimed at examining the secondary school mathematics and geometry course curriculum in terms of 21st century skills. As the data source of the research, secondary school mathematics course and secondary school geometry course curricula shared on the official website of the Ministry of National Education were taken. Document analysis method was used in the collection and analysis of data in the research in which these teaching programs were accepted as documents. Curriculums specified within the scope of document analysis were analyzed with descriptive analysis method based on 21st century skills within the scope of Partnership for 21st Century Learning [P21]. The skills included in the curricula were supported by direct quotations from the curricula. According to the results of this research, it has been determined that the curriculum is not qualified to cover all 21st century skills. The fact that media literacy, leadership and responsibility skills are not included in the curriculum, and that the evaluation elements of the programs are insufficient in the context of 21st century skills are among the remarkable results. The findings obtained at the end of the research were discussed with the support of the literature and suggestions were made for future research.

This paper explores the application of GeoGebra software to assist teachers and students in lessons on fixed points of families of lines and circles, a relatively complex topic in secondary school geometry that requires logical reasoning and advanced proof skills. GeoGebra, as a visual modeling tool, enables the discovery and proof of fixed points in problems involving families of lines and circles. The selected illustrations are derived from problems featured in competitions for high-achieving students and specialized 10th-grade entrance exams in recent years. To evaluate the effectiveness of this teaching method, experiments were conducted with 59 Grade 9 students at Newton Secondary and High School (Bac Tu Liem, Ha Noi). A t-test on students’ test scores after being taught fixed-point problems showed statistically significant differences between the GeoGebra method group (M = 7.07, SD = 1.64) and the traditional method group (M = 5.6, SD = 1.3) [t(59) = 1.987934, p = 0.001134 < 0.05]. Additionally, students in the experimental group demonstrated greater interest in the presented problems. The findings from this study highlight the potential of using GeoGebra to teach advanced geometrical problems within the Vietnamese educational context.

At one time Latin was considered such an essential part of the school curriculum that everyone was required to study it. If justification were needed for the requirement, the argument would likely be given that the study of Latin “trained the mind”. Is secondary school geometry in the same category that Latin formerly was, or are there legitimate reasons for requiring that geometry be studied by most secondary school students? Perhaps one reason for studying geometry is that it improves problem-solving ability in general. The specific proposition that geometry improves problem-solving ability would seem to be subject to experimental verification, if indeed the study of geometry does improve problem-solving ability. The evidence herein seems to indicate that the kind of reasoning used in studying geometry improves the ability to solve not only geometric problems but other types of problems as well.

The van Hiele Levels at Czech Secondary Schools

The study explored gender and school location-related differences with respect to difficulties in geometry among students. The study is an analytic survey research design, because, it attempted to compare the statuses of two groups of subjects in a given tribute. The effect or observation investigated in this study was students’ areas of difficulties in geometry. Three research questions and two hypotheses guided the study. The population of the study was 9,200 senior secondary school three students from Obollo and Nsukka Education zones of Enugu state, Nigeria. The sample of the study, using cluster proportionate random sampling technique was 1,000 students made up of 492 boys and 508 girls, clustered as 515 urban and 485 rural students from the two education zones. The instrument for data collection was Test on Secondary School Geometry (TOSSG) developed by the researchers, using Test Blue Print to ensure content validity. The 30-multiple choice test instrument was validated by two experts from the department of Science Education, University of Nigeria, Nsukka and trial tested on 20 students from a co-educational school in Enugu education zone of the state. The reliability coefficient of the instrument was 0.91, Using Kuder-Richardson 20 (KR-20). Data were analyzed using mean and standard deviation to answer research questions, while the null hypotheses were tested at 0.05 level of significance, using the Z-test statistic. The result of the study indicated differences in achievement with respect to gender and school location. Boys experienced less difficulty than girls, while urban students experienced less difficulty than their rural counterparts. It was recommended that teachers should adopt a variety of pedagogical strategies that are gender and culture-sensitive, capable of securing girl-child education and addressing different learning styles within instructional environments.

Secondary school students possess un- tapped talents for making conjectures and doing proofs. Opportunities for guessing, and arriving at, tentative conclusions are essential to their mathematical develop- ment. Through a careful sequence of activities, students can gain experience in conjecturing and in writing proofs that mirror their logical thinking. The purpose of this article is to share two orientation problems and six sample worksheets that stimulate students to use a complete conjecturing-proving sequence.

Higher Course Geometry (being Parts IV and V of A School Geometry). By H. G. Forder. Pp. x, 264. 6s. In separate parts: Part IV, 2s. 6d.; Part V, 4s. 1931. (Cambridge) - A School Geometry. By C. O. Tuckey and P. W C. Hollowell. Pp. xvi, 340. 4s. 6d. 1931. (Christophers) - A Geometry for Advanced Division, Central and Secondary Schools. Part I. By J. W. M. Gunn. Pp. viii, 124. 2s. 1931. (Rivington) - A Companion to Elementary Geometry. By G. H. Hamilton. Pp. vi, 87. 2s. 6d. 1931. (Blackie) - Volume 16 Issue 219

Higher Course Geometry (being Parts IV and V of A School Geometry). By H. G. Forder. Pp. x, 264. 6s. In separate parts: Part IV, 2s. 6d.; Part V, 4s. 1931. (Cambridge) - A School Geometry. By C. O. Tuckey and P. W C. Hollowell. Pp. xvi, 340. 4s. 6d. 1931. (Christophers) - A Geometry for Advanced Division, Central and Secondary Schools. Part I. By J. W. M. Gunn. Pp. viii, 124. 2s. 1931. (Rivington) - A Companion to Elementary Geometry. By G. H. Hamilton. Pp. vi, 87. 2s. 6d. 1931. (Blackie) - Volume 16 Issue 219

Ponte Academic JournalJun 2019, Volume 75, Issue 6 THE EFFECT OF GEOBOARD USE ON LEARNERS' MOTIVATION FOR LEARNING OF GEOMETRY THEOREMS Author(s): Mandlenkosi Richard SibiyaJ. Ponte - Jun 2019 - Volume 75 - Issue 6 doi: 10.21506/j.ponte.2019.6.14 Abstract:This study investigated the effects of using a Geoboard on learners" motivation for learning geometry theorems. The study was a qualitative research project, which included twenty (n = 20) participants from two secondary schools in King Cetshwayo District in KwaZulu-Natal Province (selected by means of convenience sampling). Eight (n = 8) of the top learners, four (n = 4) of the middle learners, and eight (n = 8) of the bottom learners were randomly selected from the mark list for the test on geometry theorems to form four focus groups. Each focus group comprised five (n = 5) participants. All twenty (n = 20) participants were taught geometry theorems using a Geoboard for two weeks. The data collected from the four focus groups were analysed using the Attention, Relevance, Confidence, Satisfaction (ARCS) theory of motivation, following an interview schedule. The results revealed that learners became motivated to learn geometry theorems after being taught these theorems using a Geoboard. The study further revealed that only the Attention, Relevance, and Confidence aspects of the ARCS theory increased learners" motivation, and not the Satisfaction aspect. Further findings of the study were that participants became more confident and showed interest in learning geometry theorems, especially in writing geometry theorems reasons correctly as corresponding with ARCS theory. It is recommended that external awards such as incentives must be used, along with Geoboards to teach geometry theorems which can subsequently increase extrinsic motivation and learners" satisfaction. Download full text:Check if you have access through your login credentials or your institution Username Password