Levels of Proof in Lower Secondary School Mathematics

Computability is one of the fundamental notions of mathematics and computer science, trying to capture the effective content of mathematics and its applications. Computability Theory explores the frontiers and limits of effectiveness and algorithmic methods. It has its origins in Godel's Incompleteness Theorems and the formalization of computability by Turing and others, which later led to the emergence of computer science as we know it today. Computability Theory is strongly connected to other areas of mathematics and theoretical computer science. The core of this theory is the analysis of relative computability and the induced degrees of unsolvability; its applications are mainly to Kolmogorov complexity and randomness as well as mathematical logic, analysis and algebra. Current research in computability theory stresses these applications and focuses on algorithmic randomness, computable analysis, computable model theory, and reverse mathematics (proof theory). Recent advances in these research directions have revealed some deep interactions not only among these areas but also with the core parts of computability theory. The goal of this Dagstuhl Seminar is to bring together researchers from all parts of computability theory and related areas in order to discuss advances in the individual areas and the interactions among those.

A study of school mathematics curriculum enacted by competent teachers in Singapore secondary schools is a programmatic research project at the National Institute of Education (NIE) funded by the Ministry of Education (MOE) in Singapore through the Office of Education Research (OER) at NIE. The main goal of the project is to collect a set of data that would be used by two studies to research the enacted secondary school mathematics curriculum. The project aims to examine how competent experienced secondary school teachers implement the designated curriculum prescribed by the MOE in the 2013 revision of curriculum. It does this firstly by examining the video recordings of the classroom instruction and interactions between secondary school mathematics teachers and their students, as it is these interactions that fundamentally determine the nature of the actual mathematics learning and teaching that take place in the classroom. It also examines content through the instructional materials used—their preparation, use in classroom and as homework. The project comprises a video segment and a survey segment. Approximately 630 secondary mathematics teachers and 600 students are participating in the project. The data collection for the video segment of the project is guided by the renowned complementary accounts methodology while the survey segment adopts a self-report questionnaire approach. The findings of the project will serve several purposes. They will provide timely feedback to mathematics specialists in the MOE, inform pre-service and professional development programmes for mathematics teachers at the NIE and contribute towards articulation of “Mathematics pedagogy in Singapore secondary schools” that is evidence based.

School Science and MathematicsVolume 18, Issue 2 p. 112-123 VALID AIMS AND PURPOSES FOR THE STUDY OF MATHEMATICS IN SECONDARY SCHOOLS Alfred Davis, Alfred Davis Francis W. Parker School, SSO Webster Ave., ChicagoSearch for more papers by this author Alfred Davis, Alfred Davis Francis W. Parker School, SSO Webster Ave., ChicagoSearch for more papers by this author First published: February 1918 https://doi.org/10.1111/j.1949-8594.1918.tb02098.xAboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Volume18, Issue2February 1918Pages 112-123 RelatedInformation

This JiM project set out to research the current state of understanding of mathematical justification by both Singapore secondary school students and Mathematics teachers across the different content strands in the Singapore secondary Mathematics curriculum. This timely and relevant project was started to complement the current goal of the Ministry of Education (MOE) on fostering reasoning in the learning of Mathematics in schools. While MOE focuses on the reasoning process, our study examined the outcomes of this reasoning process. For many years, mathematical reasoning and communication have been two key process skills in the Singapore Mathematics Framework under the Processes component. Thus, mathematical reasoning and justification are essential components of any mathematical activity that students should be familiar with. Yet from our personal communications with secondary school mathematics teachers between 2013 and 2016 and evidence from GCE O and N Level examiners' reports, we obtained the impression that many students, as well as mathematics teachers, struggled to navigate mathematical justification successfully. Mathematics teachers seemed to point students' difficulties in justification to several student-related factors. Other than those factors, we, however, speculated that there might be other potential obstacles, which contribute to students' poor performance in justification tasks. Therefore, this JiM project was undertaken to explore the quality of written justifications constructed by secondary school students and their mathematics teachers, with the aim of proposing a professional development (PD) programme that would allow the teachers to better engage their students in mathematical reasoning and justification more effectively, thereby improving the quality of written justifications provided by their students.

In the Mathematical Community there is a real interest in the mathematics that our children arc being taught and all of us have a sincere desire to help in planning a sound mathematical curriculum. Many of you, I am sure, are aware of the several groups which unite the talents of professional mathematicians and expert teachers in an effort to make a substantial contribution to elementary and secondary school mathematics. I refer particularly to the School Mathematics Study Group under the direction of Professor E. G. Beglc of Yale University, to the Commission on Mathematics of the College Entrance Examination Board, and to the University of Illinois chool of Mathematics under the direction of Professor Max Beberman. I have been privileged to work with the School Mathematics Swdy Group and I am extremely enthusiastic and optimistic about the series of textbooks which are being tried out this year in 40 different centers. These textbooks range from the 7th to the 12th grades.

Learning mathematics in the secondary school – that is, learning the content of the secondary school mathematics curriculum – has, until recently, been rarely offered to learners with Down syndrome. This chapter aims to change that, showing how mathematics might be made accessible to learners with Down syndrome, and why this is important to do. Teachers who have done this have described this as some of the best teaching they have done! In this chapter Is it possible? Even if it is, should we do this? Examples of adjustments to secondary mathematics topics Using the tools of mathematics (e.g. calculators, spreadsheets, graphing software) to access the curriculum Problem-solving, reasoning and justifying – the processes of mathematics Assessing learning

Mathematical Proof and Genre Theory

The purpose of this study was to determine the levels of mathematics and mathematics education students’ views about mathematical proof and find out whether there was a significant difference among students’ views according to their departments or not. The study was conducted with senior students (n=275) studying in final year at the department of Primary Mathematics Teacher Training (n=98), Secondary Mathematics Teacher Training (n=87) and Mathematics (n=90) at a state university at the beginning of the spring semester of 2014-2015 academic terms in East Anatolia Region of Turkey. Quantitative research approach was adopted in this study. The model of the study was comparative research which was one of the model in the nonexperimental research designs. The data were collected with Conception Scale for Mathematical Proof developed by Doruk and Güler (2014). Results of the study showed that secondary mathematics teacher training students had more negative overall views, more negative views regarding benefits obtained from mathematical proof in mathematical sense, and less self-confidence, and more negative views about meaning assigned to proof than other students. Moreover, it was revealed that mathematics students’ views regarding necessity of proof in mathematics and mathematics education were more positive than other students. ÖzetBu çalışmanın amacı matematik ve matematik eğitimi öğrencilerinin matematiksel ispata yönelik görüşlerinin düzeylerini belirlemek ve öğrenci görüşlerinin öğrenim görülen bölümlere göre farklılık gösterip göstermediğini ortaya çıkarmaktır. Çalışma 2014-2015 öğretim yılının bahar yarıyılının başında, Doğu Anadolu Bölgesi’ndeki bir devlet üniversitesinin son sınıflarında öğrenim gören toplam 275 ilköğretim matematik öğretmenliği (n=98), ortaöğretim matematik öğretmenliği (n=87) ve matematik bölümü öğrencileri (n=90) ile yürütülmüştür. Çalışmada nicel araştırma yaklaşımı benimsenmiştir. Çalışmanın modeli deneysel olmayan desenlerden biri olan karşılaştırmalı araştırma modelidir. Çalışmanın verileri, Doruk ve Güler (2014) tarafından geliştirilen Matematiksel İspata Yönelik Görüş Ölçeği ile toplanmıştır. Çalışma sonucunda, ortaöğretim matematik öğretmenliği bölümü öğrencilerinin ispata yönelik genel görüşlerinin, ispatların kendilerine matematiksel anlamda sağladığı faydalara yönelik görüşlerinin, ispata karşı özgüvenlerinin ve ispata yükledikleri anlamın diğer öğrencilere göre daha olumsuz olduğu tespit edilmiştir. Matematik bölümü öğrencilerinin ispatın matematik ve matematik eğitimindeki gerekliliğine yönelik görüşlerinin ise diğer öğrencilere nazaran daha olumlu olduğu ortaya çıkmıştır.

The national secondary school mathematics curriculum of Turkey, promulgated in 2013, encourages teachers to integrate dynamic geometry software into their classroom practices. One of the reasons for this request is that this kind of software offer teachers the opportunity to design learning environments in which students can experience the processes that constitutes mathematical exploration and proof. However, in order to exploit the possible advantages of integrating such software, teachers need the necessary technical knowledge about the software to be used and concrete examples of how it can be utilized as well. This study aims to contribute to the latter need by analyzing the solution steps of an open-ended geometry problem and exemplifying the role of the software in the solution process.

هدفت الدراسة إلى قياس مدى اكتساب طلبة الصف الثامن الأساسي لمهارات التفكير الرياضي، والكشف عما إذا كان هناك اختلاف في مستوى أداء الطلبة على مهارات التفكير الرياضي باختلاف مستوى التحصيل والجنس. ولتحقيق اهداف الدراسة تم استخدام المنهج الوصفي التحليلي عن طريق اختبار تحصيلي مكون من (20) فقرة موزعة على مهارات التفكير الرياضي السبعة وهي (التعبير بالرموز، التعميم، الاستقراء، الاستنتاج، التفكير المنطقي، النمذجة، البرهان الرياضي) على عينة تكونت من (620) طالبا وطالبة من طلبة الصف الثامن الأساسي. وأظهرت نتائج الدراسة وجود تدني في مستوى أداء طلبة الصف الثامن الأساسي على مهارات التفكير الرياضي، وعدم وجود فروق ذات دلالة إحصائية تعزى إلى الجنس على مهارات التفكير الرياضي، ووجود أثر يعزى لمستوى التحصيل على مهارات التفكير الرياضي. وفي ضوء هذه النتائج اوصت الدراسة بضرورة تبني برامج تدريبية تستند إلى توجهات الدراسة الدولية في الرياضيات في اثناء تنفيذ منهاج الرياضيات والعلوم TIMSS.

This volume is dedicated to Prof. Dag Prawitz and his outstanding contributions to philosophical and mathematical logic. Prawitz's eminent contributions to structural proof theory, or general proof theory, as he calls it, and inference-based meaning theories have been extremely influential in the development of modern proof theory and anti-realistic semantics. In particular, Prawitz is the main author on natural deduction in addition to Gerhard Gentzen, who defined natural deduction in his PhD thesis published in 1934. The book opens with an introductory paper that surveys Prawitz's numerous contributions to proof theory and proof-theoretic semantics and puts his work into a somewhat broader perspective, both historically and systematically. Chapters include either in-depth studies of certain aspects of Dag Prawitz's work or address open research problems that are concerned with core issues in structural proof theory and range from philosophical essays to papers of a mathematical nature. Investigations into the necessity of thought and the theory of grounds and computational justifications as well as an examination of Prawitz's conception of the validity of inferences in the light of three dogmas of proof-theoretic semantics are included. More formal papers deal with the constructive behaviour of fragments of classical logic and fragments of the modal logic S4 among other topics. In addition, there are chapters about inversion principles, normalization of proofs, and the notion of proof-theoretic harmony and other areas of a more mathematical persuasion. Dag Prawitz also writes a chapter in which he explains his current views on the epistemic dimension of proofs and addresses the question why some inferences succeed in conferring evidence on their conclusions when applied to premises for which one already possesses evidence.

As part of a PhD researching teacher’s positive emotions in secondary mathematics education, I am exploring the role of play as both instigator and product of emotions. Play has recognised social, creative, cognitive and especially emotional benefits for young children. However, as students mature, do they still respond to play? Teaching mathematics in secondary school is not normally associated with play, but as illustrated below, can appear in many forms. I will argue that, in the hands of a skilled practitioner, playfulness can benefit teacher, students, and their mathematics learning. This paper explores the unique nature of play in everyday classrooms using a sample of observations and interviews with experienced UK teachers. I conclude that play appears in many subtle forms, playing various roles in a mathematics classroom.

Teachers' knowledge for teaching mathematics in secondary school is incomplete without the inclusion of knowledge of technology integration in teaching. Developing technology pedagogical content knowledge (TPACK) of pre-service teachers is essential in teacher preparation programs because PSTs gain knowledge of how to use technology in mathematics teaching before beginning their practice. This study investigates how preservice mathematics teachers' TPACK emerges during the secondary mathematics methods and field-based courses. The instructors supported the growth of PSTs' knowledge for teaching secondary mathematics and integrating technology in the method courses. The evidence of TPACK knowledge included data from lesson plans and lesson reflections, technology presentations, and semi-structured exit interviews through content analysis. Recommendations for policy and practice are elaborated.

MODERN MATHEMATICS IN SECONDARY SCHOOLS

Remedying secondary school students’ illusion of linearity: a teaching experiment aiming at conceptual change