ADDRESSING FIRST YEAR UNIVERSITY MATHEMATICS AND THE TRANSITION FROM HIGH SCHOOL AT SIMON FRASER UNIVERSITY

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1 12 th International Congress on Mathematical Education Program Name XX-YY-zz (pp. abcde-fghij) 8 July 15 July, 2012, COEX, Seoul, Korea (This part is for LOC use only. Please do not change this part.) ADDRESSING FIRST YEAR UNIVERSITY MATHEMATICS AND THE TRANSITION FROM HIGH SCHOOL AT SIMON FRASER UNIVERSITY Randall Pyke Department of Mathematics, Simon Fraser University [email protected] Simon Fraser University (SFU) has developed a number of initiatives to deal with the delivery of first year mathematics courses and to address the challenges students face in making the transition from high school to university. Demand for university admittance continues to grow, while at the same time the level of student preparation and training in high school and their career goals are constantly changing. The Department of Mathematics at SFU has implemented a variety of strategies to address these issues. These strategies can be broadly grouped into three categories; outreach to high school students, the transition to university, and facilitating learning mathematics at the university. Key words: Classroom technology, Online assessment, Transition to university INTRODUCTION Simon Fraser University (SFU) was established in 1965 on Burnaby Mountain near Vancouver, British Columbia, Canada. Currently, it has a student population of more than 30,000 undergraduates, 5000 graduates, and 950 faculty spread across three campuses (the additional campuses being located in downtown Vancouver and in Surrey, just east of the city) with over 100,000 alumni. SFU has consistently been rated as one of Canada's best comprehensive universities and is well known for its emphasis on teaching. Most undergraduates come from the Vancouver area and interior regions of British Columbia, but there is a sizable international student component, mainly from Asia, that continues to grow. There are 8 faculties at SFU: Applied Science; Arts and Social Sciences; The Beedie School of Business; Communication, Art and Technology; Education; Environment; Health Sciences; and Science. This article describes efforts by members of the Department of Mathematics at SFU to enhance the learning experience of students in first year mathematics courses. Particular emphasis is placed on using new technologies to manage courses, produce online assessments and resources, and enrich the classroom lectures. Much effort is also directed to helping students make the transition from high school to university mathematics. The work presented here is mainly due to Malgorzata Dubiel, Justin Gray, Veselin Jungic, Natalia Kouzniak, Petra Menz, Jamie Mulholland, Nadia Nosrati, and Randall Pyke, in addition to the invaluable help of other members of the Department of Mathematics, SFU. abcde

2 OUTREACH TO HIGH SCHOOLS Since 2004 SFU has hosted regular weekend lectures on topics in mathematics for high school students and high school teachers. These A Taste of Pi 1 events have covered a wide range of topics and continue to promote interest in mathematics among high school students and bring together the university and high school communities. Communication between university faculty and high school teachers helps build awareness and understanding of the issues faced by both communities. Each summer the Department of Mathematics hosts a math camp for high school students 2. This camp lasts for several days and is comprised of a variety of activities ranging from problem sessions, invited lectures, to video presentations. In addition to the student camp we now host a math camp for high school math teachers that runs parallel to the student camps 3. These latter theme oriented camps are based on mathematical activities that are connected to the high school curriculum, teaching, and assessment. Experiences and questions are shared during discussions with colleagues. A number of video productions directed to the K-12 audience have been created that are available on the internet. These include animations that discuss mathematics topics in a more entertaining way 4 or in a context that is culturally sensitive 5. TRANSITION TO UNIVERSITY The transition from high school to university, with all the new demands and expectations placed on students, is often the most challenging experience a student faces while at university. This makes the first year of study especially crucial for students, and for the faculty that are responsible for the delivery of first year courses. Instructors of those courses realize that it is not enough to address this issue only when students have arrived on the university campus, but efforts must be made to reach out to high school students and teachers to convey the importance of this transition and take measures to help students prepare while they are still in high school (for example, by developing good study habits, working responsibly on home work, and managing their time effectively). To give senior high school students an idea of what to expect as university students, two outreach initiatives take place at SFU throughout the academic year. One is the Math Ambassador visits to local high school math classes 6. These Ambassadors are first or second Abcde+3 ICME-12, 2012

3 year university students who have completed at least one mathematics course at the university. They visit high school math classes (often the classes of teachers they themselves attended when in high school one year earlier) to give the high school students their impressions of university and how different it is from high school. This peer-to-peer interaction is highly effective in communicating to high school students the challenges they will face as they approach the transition to university. The second initiative is the Meet and Greet Math events held at the University 7. Here mathematics classes from high schools are invited to visit SFU for a day during the regular semester and to attend (first year) university mathematics classes. These high school students see with their own eyes the pace and expectations that these classes place on students. They also have an opportunity to meet with university students and faculty during their visit. Both of these events give high school students the opportunity to witness for themselves the environment of the university and it gives them an idea of what will be expected of them once they enter university. But it also gives high school teachers and university instructors an opportunity to share experiences and insights on teaching mathematics. Just as important it is for high school teachers to be aware of what goes on in university mathematics courses and how they are structured (the role of homework, how tests and evaluations are administered, etc.), university instructors also need to be aware of how mathematics is taught in high school and the issues faced by teachers there. This helps university instructors foresee the difficulties many of these students will face when they enter university and therefore to adjust the delivery of entry-level courses accordingly. To further the collaboration between university and secondary school teachers, many SFU instructors participate in the annual mathematical education conference Changing the Culture 8 which brings together university instructors and teachers to discuss issues related to teaching and learning mathematics in elementary school, high school and university. Once high school students have entered university they often need extra attention during the first crucial year, beginning with guidance on which mathematics course to register in. SFU has a series of entry-level mathematics courses (each course lasting one semester; 13, 3-hour lecture weeks, or 39 lecture hours in total 9 ) to accommodate various backgrounds, program interests, and levels of preparedness of students. These courses range from a quantitative numeracy course (Foundations of Analytical and Quantitative Numeracy, or FAN for short), precalculus (college algebra), to three streams of calculus (physical/computing sciences, business/economics, and health sciences) Hosted by the Pacific Institute for the Mathematical Sciences. See 9 One lecture 'hour' is actually 50 minutes. The exception to this is Math 150, an introductory calculus course with extra review, which has 4 lecture hours per week and is discussed below. 10 There are other entry-level courses such as linear algebra and discrete mathematics, but most students who are required to take mathematics courses in their first year will take the courses mentioned above. ICME-12, 2012 abcde+2

4 The FAN course is one of two Foundations Courses 11 that all students admitted to SFU must either pass in their first year or else demonstrate competency in these areas from previous studies. The Foundations Courses are the beginning of the Writing, Quantitative, and Breadth requirements that all students at SFU are required to satisfy, regardless of the area of study they are in. The FAN course, then, will include students from disciplines outside of the sciences who may not have had much mathematical exposure in high school. In addition, FAN accommodates students who have been outside school for a number of years and more generally students who do not meet the entry requirements for the more advanced courses (precalculus and calculus). This multi-section course has a class size limit of 40 students, and instructors collaboratively prepare teaching material that is problem based and group oriented. Students are expected to engage with the problems in order to explore and discover underlying mathematical concepts. The idea is that these self-discovery activities will help the students retain the concepts and to apply it to more sophisticated mathematical situations. The next course in increasing sophistication is the precalculus course which aims to prepare students for calculus by giving them a thorough review of basic algebra and functions. Students succeeding in this course will be prepared to enter one of the calculus courses. Students may still enter these calculus courses directly from high school if they have sufficiently high grades in certain high school mathematics courses, but students who do not meet these entry requirements for calculus are required to take precalculus. Precalculus as well as the calculus courses are large classes with up to hundreds of students and are taught in a more traditional way than the smaller FAN classes. The delivery of the precalculus and calculus courses is discussed below. One of the main reasons students perform poorly in first year calculus is their weak basic algebraic skills that underlie the dialog of calculus (another important reason is their lack of study and time management skills which is brought up below). To address this the Department of Mathematics at SFU has developed two initiatives. The first is a new calculus course which runs parallel to the standard calculus course for the physical sciences called Math 150. This course has an additional hour of lecture each week which allows the instructor to include review of basic algebra within the course material. It also slows down the pace of coverage of new material which some students find more accommodating 12. Math 150 has slightly lower entrance requirements than the standard calculus course. Enrolment in this Calculus with Review course continues to grow, and even attracts more and more students that satisfy the entrance requirements to the standard calculus course. The second is supplementary support that has been developed over the past three years for all three calculus streams. At the beginning of term all calculus students take a diagnostic test to ascertain their competence in basic algebraic skills, and those that are weak are required to attend Calculus Support Sessions throughout the term. These support sessions are tutorials 11 The other Foundations Course is Foundations of Academic Literacy. 12 Math 150 covers exactly the same curriculum and has the same final examination as the regular calculus course. Abcde+3 ICME-12, 2012

5 that run parallel to the calculus classes (one 50 minute session per week) and provide students with a review of mathematical skills required for the techniques they are learning in the calculus classes. We have found that among students who failed the diagnostic test, those that participated fully in these review sessions ended up with significantly higher final grades (on average 20% higher) than those students whose attendance in the review sessions was poor 13. The pace of courses at university require good study and time management skills on the part of the student which they may not have acquired in high school. In addition, high school students in Canada are often faced with different learning expectations and evaluation methods (at least in mathematics) compared to what they will experience in university. Thus when these students enter university they often do not have the effective study skills needed to succeed in their studies which compounds the difficult task of learning the material, and they may not be accustomed to what is expected of them in terms of understanding the material. To help counter this we offer information sessions on effective study habits and provide this material online. Instructors of entry-level courses try to incorporate some of this material on study habits into their lectures so that students receive not only the mathematical content of the subject during lectures but also guidance on how to study this material on their own. Frequent (and early) feedback 14 during the course provides students indicators of what is expected of them. Most first year mathematics courses at SFU have two 50-minute tests during the term in addition to a comprehensive final examination at the end of the course. The first midterm usually takes place in the fifth week of the (13 week) course. For a student to receive their first substantial feedback on their performance at this point in the course makes it difficult for them to make a significant change, if needed, to the way they approach the course and recover before the end of the term. Therefore it is important to have other means by which students receive feedback that begins near the start of the course. Regular, weekly, hand written assignments are the traditional way of providing frequent feedback to students but this has its drawbacks, as discussed below. In addition to this type of homework we administer frequent online assessments and in-class quizzes which provide more significant feedback to students as they progress through the course. Regular feedback is an important component of the learning process (Brown, Bull, Pendlebury, (1997)). All first year mathematics courses at SFU have a homework 15 component where problems are assigned at regular intervals (for example, weekly) and are designed to assist students in learning the material. Although it is clear to educators that it is essential for students to work through problems (on their own) in order to learn mathematics, many students don't follow 13 Based on results of analysis by Justin Gray. 14 Feedback consists of marked assignments, online problems, quizzes and tests. Quizzes, tests and exams are written individually by the students with no aids; quizzes are short with one or two questions and a time requirement of 10 to 15 minutes, tests last 50 minutes, and exams, given at the end of the course, last up to 3 hours. 15 The terms home work and assignments are synonymous; they both refer to work the student is requested to do outside of class time. ICME-12, 2012 abcde+2

6 this principle and either copy homework answers from other students or find them on the internet 16. That students do not appreciate the purpose of homework is corroborated by frequently heard comments implying that they are doing the homework "for the instructor" and not for themselves. This is often a remnant of students high school experience so it is most important to address this in the first year classes (once students have been initiated to the real purpose of homework they will have a better appreciation for it in subsequent courses). In addition, due to the large class sizes of first year courses we only have resources to mark a fraction (typically 2 out of 12 problems) of the assigned homework and with these very little in the way of written comments are added. Consequently, students receive little feedback about their homework (we do provide complete solutions once the homework is handed in for marking). Recently, we have revised the way homework is administered in first year classes to increase the amount of feedback students receive and to impress upon the students that homework is there to help them learn the material. Now instead of asking students to hand in homework for credit, we assign homework without requiring it to be handed in for grading. This removes any semblance the homework may have as a need for the instructor, and makes it clearly the prerogative of the student. To motivate students to learn the material on the assignment (for themselves and not for the instructor ) we administer weekly quizzes in class on topics covered in the most recent assignment. These quizzes typically contain two problems similar to problems in the recent homework and students have 10 minutes to complete it. They contribute to the students' final grade in the course as much as the assignments did (approximately 10%). We have followed up on this change to the way assignments are administered by looking at how students' marks in the quizzes are correlated to their grades in tests (a test is written without any aids to the student and takes places during the entire lecture hour). We have compared recent data (under the new home work system) to previous years data (where assignments were handed in for marking and there were no quizzes) 17. We found that in the past there was little correlation between the marks students received on their assignments and the marks they were receiving on tests. In particular, many students were regularly receiving high marks on their assignments but poor marks in tests 18. Under the new approach of having weekly quizzes instead of handed in homework we have seen a stronger correlation. Now 16 This is based on our experience of finding copied assignments among the submitted homework of students and students' own admissions of copying or searching for solutions on the internet. In fact it is not hard to find complete solutions for practically any textbook problem (or any problem from any course) on the internet and the sophistication of online mathematical oracles such as Wolfram Alpha ( makes creating homework problems immune to these duplicitous tactics more and more difficult. 17 This analysis prepared by Justin Gray. 18 Typically, most students were receiving grades between 70%-100% on their assignments but the grades of these students in tests were spread from 40% - 90%. Abcde+3 ICME-12, 2012

7 students that are doing well in the quizzes are most likely the same students that are doing well in the tests, and similarly students who do poorly on tests are typically the students who do poorly on the quizzes. The early and frequent (and accurate) feedback the quizzes provide give students a better idea of where they stand in terms of learning expectations before they obtain results of the first substantial test. Good performance in the quizzes is due to better understanding of the homework material. So the message that homework matters is getting through to the students. To further guide students in their first year, and to provide another means for students to receive feedback, we have recently introduced a homework journal system in some courses. Here students are required to maintain a homework journal (either on paper or electronically) which contains, among other things, the course assignments and the students solutions so they can record their learning progress through the course. The journal is annotated to provide a structure for the student to organize material covered in the course, and hence helps them understand how the material fits together. For example, the journal template (the blank journal with only course information and assigned problems) has a preamble describing to the student strategies for solving problems, how solutions should be presented, hints and selected answers to problems, a check list to assure students re-do incorrect solutions (or ask the instructor or teaching assistant 19 for help) and review problems before tests and exams, and a categorization of questions as routine, time challenge, concepts and definitions, explorations and higher level understanding which elucidate to the student the various types of problems that arise in the course. A students journal also assists instructors and teaching assistants of the course in identifying difficulties the student is facing because when a student seeks help from the instructor or teaching assistant the student is required to present their journal showing that they are maintaining it. Overall, the homework journal brings to the fore the students responsibility in taking an active role in learning. LEARNING MATHEMATICS AT UNIVERSITY In addition to the variety of entry-level mathematics courses available and extra support for students in first year calculus as mentioned above, faculty are using innovative teaching methods and technology to facilitate students learning experience. All courses are managed online where students have instant and continual access to course information, lecture materials, homework assignments and solutions, practice problems, grades, links to resources, and discussion groups. Furthermore, we use extensively an open-source distributed learning content management and assessment system, LON-CAPA 20 to create and administer online assignments and other assessments. For example, all first year (and some higher level) courses have a complete set of LON-CAPA problems that students access throughout the 19 A teaching assistant is a graduate or advanced undergraduate student in mathematics. 20 LectureOnlineNetwork and Computer-Assisted Personalized Approach; the Learning Online Network with CAPA: ICME-12, 2012 abcde+2

8 course. Typically, a few (short) problems will be posted for students to work on after each lecture and will be due before the next lecture. The topic of the problems would be that introduced in the immediately preceding lecture. This motivates students to think about and work with the concepts before the next lecture, and helps keep them current in the lecture material. We have surveyed students in these courses and the majority have found these frequent online assignments very helpful 21. We are producing virtual (video) lectures for the first year mathematics courses. These will cover the range of topics within the curriculum of these courses and will supplement the live lectures allowing students to review topics discussed in class and to prepare ahead for upcoming lectures. And being available online students will be able to access them at their convenience, as often as they like. At this preliminary stage we are finding that many students are accessing these virtual lectures in addition to attending the regular (live) lectures. Many instructors at SFU use mathematical and analytical software to supplement traditional lectures. This includes MAPLE and GeoGebra to produce figures, solve equations, evaluate functions and integrals, etc, and create applets illustrating mathematical ideas. In addition to providing visual information, the use of software in the courses encourages students to become familiar with software tools themselves (which they will often be required to use in subsequent courses and in their career). Some instructors have been using classroom response systems, such as clickers, to assess student's responses to questions posed by the instructor in class (Jungic, Menz, and Wiebe, 2009). This provides fast, visual, accurate and anonymous feedback to the instructor. The use of clickers allows instructors to address misconceptions and provides on-going feedback to the students. Our experience corroborates other studies on the use of clickers in class in that most students appreciate clicker activities and are kept engaged with the lecture material. In addition to the classroom lectures that comprise the main delivery of course material, a central role in the delivery of mathematics courses at SFU are the Mathematics Workshops. The Workshops are dedicated spaces where students can drop in at their convenience and find mathematics teaching assistants and/or instructors who can answer questions about the material being covered in class. There are four Workshops that cater to different groups of courses (pure calculus, applied calculus, algebra, and introductory mathematics). The drop-in arrangement is flexible for students and better meets their particular mathematical needs than traditional tutorial based support. In the Workshop the onus is placed on the student to locate and explain to the teaching assistant or instructor where their difficulty in understanding the material lies. This contrasts to traditional tutorials where, typically, a teaching assistant would do prepared problems and answer questions, thus allowing students to take a less participatory role. The Workshops also provide a sense of community among the students, teaching assistants, and instructors. Students find these open work areas convenient places to 21 In one such survey 82% of students agreed or strongly agreed with the statement, "Online assignments helped me learn the course material better". Abcde+3 ICME-12, 2012

9 study with fellow students and so encourages them to discuss the ideas being covered in the class among themselves. Many universities, including SFU, have to accommodate large classes in some courses. These are classes with 50 or more students. This is very far from what students were used to in high school and poses extra challenges for the instructor. In particular, it is especially demanding for the instructor to engage with students and create an atmosphere of interaction. Mathematics faculty at SFU who have experience teaching large classes use a variety of approaches to enhance the learning experience of students in these classes, and have developed methods specific for instruction of these courses (Jungic, Kent, and Menz, 2006). For example, the use of clickers, as mentioned above, is one such method to help instructors interact with students in large classes, as well as utilizing web-based course support and assessment. CONCLUSION In this article we have summarized some of the initiatives developed in the Department of Mathematics at Simon Fraser University to enhance the learning experience of first year mathematics students and to help high school students make the transition to university. These initiatives begin while students are still in high school, through visits to high schools by faculty and university students, offering weekend mathematical lectures for high school students, holding math camps at the university in the summers, and continues into their first year at university by offering a variety of entry-level mathematics courses, administering diagnostic tests and associated remedial tutorials, exploring ways to augment traditional lectures with technology, making use of web-based assessment tools, providing ample opportunity for personal contact with an instructor or teaching assistant, judicious design of homework, and providing guidance on developing good study habits. The area is constantly changing with the development of new technologies, changing curricula (both in the high school and at the university), and new ideas about teaching mathematics. References Brown, G., Bull, J., and Pendlebury, M. (1997). Assessing Students' Learning in Higher Education. London, United Kingdom: Routledge. Jungic, V., Kent, D., and Menz, P. (2006). Teaching Large Math Classes: Three Instructors, One Experience. International Electronic Journal of Mathematics Education (Vol. 1, No. 1, October 2006). Jungic, V., Kent, D., and Menz, P. (2009). On Online Assignments in a Calculus Class. Conference Proceedings: The Mathematics Education into the 21st Century Project, 10 th International Conference Models in Developing Mathematics Education. Dresden, Germany. ICME-12, 2012 abcde+2

10 Jungic, V., Menz, P., Wiebe, A. (2009). Instructor and Student Feedback on Clickers in Large Calculus Courses. Conference Proceedings: 34th International Conference on Improving University Teaching (IUT). Vancouver, Canada. Jungic, V., and Menz, P. (2009). Using Technology to Motivate and Assess Students Understanding. Northwest Mathematics Conference. Whistler, Canada. Abcde+3 ICME-12, 2012