Mathematics I: General Mathematics

semester2Mathematics I:General MathematicsCOURSE GUIDEAssociate Degree in Education/B.Ed. (Hons) Elementary2012

This product has been made possible by the support of the American People through the United States Agency forInternational Development (USAID). The contents of this report are the sole responsibility of the authors, and donot necessarily reflect the views of USAID or the United States Government.Technical Support: Education Development Center (EDC); Teachers College, Columbia University

ForewordTeacher education in Pakistan is leaping into the future. This updated Scheme of Studies is the latestmilestone in a journey that began in earnest in 2006 with the development of a National Curriculum,which was later augmented by the 2008 National Professional Standards for Teachers in Pakistan andthe 2010 Curriculum of Education Scheme of Studies. With these foundations in place, the HigherEducation Commission (HEC) and the USAID Teacher Education Project engaged faculty across thenation to develop detailed syllabi and course guides for the four-year B.Ed. (Hons) Elementary and thetwo-year Associate Degree in Education (ADE).The syllabi and course guides have been reviewed by the National Curriculum Review Committee(NCRC ) and the syllabi are approved as the updated Scheme of Studies for the ADE and B.Ed. (Hons)Elementary programmes.As an educator, I am especially inspired by the creativity and engagement of this updated Scheme ofStudies. It offers the potential for a seismic change in how we educate our teachers and ultimately ourcountry’s youngsters. Colleges and universities that use programmes like these provide their studentswith the universally valuable tools of critical thinking, hands-on learning, and collaborative study.I am grateful to all who have contributed to this exciting process; in particular the faculty and stafffrom universities, colleges, and provincial institutions who gave freely of their time and expertise forthe purpose of preparing teachers with the knowledge, skills, and dispositions required for nurturingstudents in elementary grades. Their contributions to improving the quality of basic education inPakistan are incalculable. I would also like to thank the distinguished NCRC members, who helpedfurther enrich the curricula by their recommendations. The generous support received from theUnited States Agency for International Development (USAID) enabled HEC to draw on technicalassistance and subject-matter expertise of the scholars at Education Development Center, Inc., andTeachers College, Columbia University. Together, this partnership has produced a vitally importantresource for Pakistan.PROF. DR SOHAIL NAQVIExecutive DirectorHigher Education CommissionIslamabad

How this course guide was developedAs part of nation-wide reforms to improve the quality of teacher education, theHigher Education Commission (HEC) with technical assistance from the USAIDTeacher Education Project engaged faculty across the nation to develop detailedsyllabi and course guides for the four-year B.Ed. (Hons) Elementary and two-yearAssociate Degree in Education (ADE).The process of designing the syllabi and course guides began with a curriculumdesign workshop (one workshop for each subject) with faculty from universitiesand colleges and officials from provincial teacher education apex institutions.With guidance from national and international subject experts, they reviewed theHEC Scheme of Studies, organized course content across the semester, developeddetailed unit descriptions and prepared the course syllabi. Although the coursesyllabi are designed primarily for Student Teachers, they are useful resources forteacher educators too.In addition, participants in the workshops developed elements of a courseguide. The course guide is designed for faculty teaching the B.Ed. (Hons)Elementary and the ADE. It provides suggestions for how to teach the contentof each course and identifies potential resource materials. In designing both thesyllabi and the course guides, faculty and subject experts were guided by theNational Professional Standards for Teachers in Pakistan 2009 and the NationalCurriculum 2006. The subject experts for each course completed the initial draftsof syllabi and course guides.Faculty and Student Teachers started using drafts of syllabi and course guides andthey provided their feedback and suggestions for improvement. Final drafts werereviewed and approved by the National Curriculum Review Committee (NCRC).The following faculty were involved in designing this course guide: ShabanaSaeed, GCET (F) Rawalakot; Saima Khan, University of Education, Lahore;Khalid Pervez, GCET Kasur; Dr Shahid Farooq, IER University of the Punjab,Lahore; Muhammad Zaman, BoC Sindh, Jamshoro; Muhammad Rauf, IERUniversity of Peshawar; Noor Alam, GCET (M) Lalamusa; Shereen Taj,University of Balochistan, Quetta; Zakia Ishaq, GCEE (F) Pishin; M. Nadeem,RITE (M) DI Khan; Zohra Khatoon, University of Sindh; Shoukat Usmani,GCET (M) Muzaffarabad; Ijaz-Ur-Rauf, GCET Shahpur Sadar, Muhammad Asim,University of Karachi; Rashid Ahmed Noor, RITE (M) Peshawar; MuhammadRafique, GCET Mirpur; Farjana Memon, GECE (W) Hyderabad; Abdul Khaliq,BoC, Quetta; Muhammad Wasim Uddin, RITE (M) Haripur; Muhammad Afzal,University of Education, Lahore; Gul Muhammad, GCEE Quetta; ShabanaiiiHOW THIS COURSE GUIDE WAS DEVELOPED

Hyder, GECE (W) Hussainabad, Karachi; Dr Iqbal Majoka, Hazara University;Ibad Ur Rehman, GCET (M) Jamrud; Ghulam Abbass, University of Education,Lahore; Safia Khatoon, GCET (F) Jamrud; and Maria Akhtar, Fatima JinnahWomen University, Rawalpindi.Subject expert guiding course design: Loretta Heuer, Senior Research andDevelopment Associate, Education Development Center (EDC).Date of NCRC review: 3 March 2012NCRC Reviewers: Dr Imran Yousuf, Arid Agriculture University, Rawalpindi; andDr Tayyab, Foundation University, Islamabad.ASSOCIATE DEGREE IN EDUCATION/B.ED. (HONS) ELEMENTARYiv

Table of contentsAbout the course guide07Syllabus08Planning guide16UNIT 1Numbers and operations17UNIT 2Algebra81UNIT 3Geometry and geometric measurement124UNIT 4Information handling195Resources222Appendix:Professional standards for teaching mathematics251

About the course guideThis Course Guide includes a syllabus and planning guide. The syllabus is primarilyfor Student Teachers. The planning guide is intended for faculty teaching the course.The planning guide provides suggestions, resources, and links to resources forteaching the course.The planning guide and course is divided into four units. Each unit is further brokendown into weeks.For each week, the planning guide provides faculty notes that introduce anddiscuss the content for teaching and learning. After the faculty notes, threesession plans are provided.Each session plan is organized into five parts: What do Student Teachers need to know? How do children think about these concepts? What is essential to do with Student Teachers? Activities with Student Teachers AssignmentsSession plans contain links to useful resources for faculty and Student Teachers. Someof the session resources are also reproduced in the planning guide.The planning guide is a guide only. Faculty should use their professional judgement todecide if and how to use the planning guide and to modify and adapt the content fortheir context.The writers of the course aimed to respect copyright and have sought permission touse copyrighted material when necessary. Please contact EDC in case of questions orconcerns about any of the materials used, at:ØØ www.edc.org07ABOUT THE COURSE GUIDE

SyllabusMATHEMATICS I:GENERAL MATHEMATICS

GENERAL MATHEMATICSSubjectGeneral MathematicsCredit value3 creditsPrerequisitesSSC MathematicsCourse DescriptionThis course is designed to prepare Student Teachers for teaching mathematics inelementary grades. It provides opportunities for Student Teachers to strengthen theirmathematical knowledge and skills and to gain confidence in their understanding ofmathematics. An important outcome of this course is for Student Teachers to be ableto teach mathematics successfully in the primary, elementary, and middle grades.Research-based knowledge about good maths instruction provides a solid base ofinformation for educators to use as they identify mathematics skills that StudentTeachers need to develop, as well as teaching strategies and instructional approachesthat best support the development of these skills. The course design is based on whatresearch tells us about good maths instruction.The overall organization of the course is divided into four units:1) Numbers and operations2) Algebra3) Geometry and geometric measurement4) Information handlingEach unit of study has a consistent design or organization and is meant to maximizeStudent Teachers’ time for learning.09COURSE SYLLABUS: Mathematics I (General Mathematics)

ContentMost one-hour sessions will begin by working on a maths problem. Student Teacherswill engage in solving and discussing a maths problem and sharing approaches andsolutions. The content has been developed so that Student Teachers will engage inmathematics in depth to help them connect concepts within and across the four units.PedagogyIn each lesson, Student Teachers will actively engage in doing mathematics in orderto experience approaches to teaching and learning maths that they can use whenthey teach. They will recognize that there are often multiple ways of approachinga problem and, in some instances, more than one correct answer. The Instructorwill present questions that stimulate curiosity and encourage Student Teachers toinvestigate further by themselves or with their classmates.The course will also examine how children learn and develop mathematicalunderstanding and skills and how the way children think influences the teaching ofmathematics in elementary grades.AssignmentsStudent Teachers are expected to continue learning about maths and the teaching ofmaths after class. There will be assignments to stretch their content knowledge andto learn more about teaching maths. Assignments will take many forms, includingindependently solving maths problems and school-based tasks.In summary, the General Mathematics course is a comprehensive effort to build anddeepen maths content knowledge, learn and use high-quality instructional practices,and study ways in which children approach and learn mathematics.Course objectivesStudent Teachers will be able to: increase their mathematical content knowledge for numbers and operations,algebra and algebraic thinking, geometry and geometric measurement, andinformation handling for teaching in elementary grades increase their confidence, competence, interest, and enthusiasm for mathematicsby exploring and doing mathematics deepen an understanding of how children learn mathematics build a variety of instructional techniques with clear purposes enhance their use of questioning techniques to elicit children’s understanding learn ways to engage children in mathematical thinking through interactive activities.ASSOCIATE DEGREE IN EDUCATION/B.ED. (HONS) ELEMENTARY10