Mathematics Hl Core Worked Solutions (3rd Edition) Pdf Free
Vespasiano Jilg
Course Description:The class will meet 4 hours each week, roughly 50% on lectures and 50% on problem sessions.Because I believe strongly in active learning, the problem solving will be the core of the class. Most examples will be worked out during the problem sessions rather than in the lectures.The pace of the lectures will be brisk andthe students are expected to work hard outside the classroom.An average student should plan to study at least 9-12 hours per week to keep up with the class. (The time may vary depending on students' prior exposure to the subject.)
The current plan is to cover the entire book during the semester.Note however that the plan is tentative and the pace will ultimately depends on the class.
Qualifying exam requirements:This is one of the qualifying exam (prelim) classes. Prelim requires solid working knowledge on
Holomorphic functions, Cauchy-Riemann equations, Cauchy's Theorem, Cauchy's integral formula, Maximum principle, Taylor series for holomorphic functions, Liouville's theorem, Runge's Theorem. Normal families, isolated singularities, Laurent series, residue theorem, applications to compute definite integrals, Rouche's Theorem. Conformal mappings, examples, Schwartz lemma, isometries of the hyperbolic plane, Montel's theorem, Riemann mapping theorem. Infinite products, Weirstrass factorization theorem. Analytic continuation, monodromy. Elliptic functions. Picard's theorem.
These topics will form the core of this course. The class is designed in such a way that a student who does well in the classshould have no problem passing the complex analysis part of the analysis prelim.
Homework
Problem solving is vital for this class.Homework problems will be assigned during the lectures andposted at the class home page afterwards.Problem sessions will be held roughly every 1 to 2 weeks.Students will take turns to present their solutions during the problem sessions and turn in their work on Monday (or Wednesday if Monday is a holiday)after the conclusion of problem sessions.A student volunteer will organize the presentation of HW problemsaccording to students' own preferences.
How to do well in this class?The answer is straightforward and old-fashioned: Prepare for Class, Keep Up, and Do the Homework Problems. The exams will contain at least 70% from material covered in lecturs and homework problems, with little modification.A sure way to get a good grade is to study for the class, and dothe assignments as if you are taking the tests, without the helpof the book, notes and computers.It also helps a great deal to ask questions during and after the lectures,especially after you have already (p)reviewed the material.
Instructor's comments:The goal of this class is to have students learn the material well andthen to give them fair and accurate grades. To achieve this goal,the instructor belives in serious homework problems and hard exams. Serious problems make students learn more and better. Hard exams give a better evaluation of students' learning.In other words, if you are taking this class just to get a passing grade and with no intention to learn, consider taking another class.
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Problem solving is important to many activities, both in a learning context and in everyday tasks. We can be challenged to solve what later turn out to be very simple problems. By understanding more about these challenges and what occurs at a cognitive level during the problem-solving process, we can better support the development of problem-solving skills. Spatial ability has been shown to be related to the ability to develop accurate and schematic mental representations of problems during the problem-solving process. The purpose of this study was to examine the role of spatial ability in solving word problems in mathematics among a cognitively diverse sample of engineering students. A set of five word problems, a test of mental rotation and set of five questions testing the core mathematical competencies needed to solve the word problems were administered to 115 first year engineering students. Using a knowledge framework for problem solving, key aspects of representation were extracted from solutions to the word problems and combined to create a mental representation scale. A large and significant correlation was measured between mental rotation and problem representation, larger than the correlation between spatial ability and problem-solving. Mental representation was found to mediate the relation between spatial ability and word problem-solving. This relation was not found to be significantly moderated by core competency in mathematics. For high levels of core competency only, there was an interaction between spatial ability and core competency.
One line of research on this topic has focused on the process of representation during problem-solving. Having read or while reading the words in the problems, one begins to form a mental representation through visualization or the use of visual imagery (Hegarty and Kozhevnikov, 1999). While linguistic knowledge and verbal ability are required to correctly read the statement, spatial ability is needed to create the visualization. This process can vary in how successful it is at creating an accurate, complete and schematic representation of the problem (Boonen et al., 2014) which then impacts success in solving the problem.
The objective of this study is to contribute to our understanding of how spatial ability facilitates mental representation in problem-solving. We previously reported results from a study of problem-solving among engineering students which found large, significant correlations between spatial ability and success in both representing and solving word problems in mathematics (Duffy et al., 2020). We extend this analysis by excluding one of the problems from the set and by examining mediation and moderation effects. The problem we excluded was the only one of six problems which was solved through trial and error and the only problem not to have a significant correlation with spatial ability (Duffy et al., 2020). Below we describe the moderated mediation we applied to the data after excluding this problem. Our findings contribute not just to our understanding of the role of spatial ability in problem-solving, but also to our understanding of how spatial ability is related to performance in mathematics.
A problem can be made more difficult to solve by simply rephrasing it even though its mathematical structure and solution path remain unchanged. For example, consider the alternatively phrased statements of the same mathematical problem:There are 5 birds and 3 worms. How many more birds are there than worms?
When presented in word form, mathematical problems can be very difficult to solve even when the mathematical challenge is low. If success rate can be changed by rewording a problem without changing the mathematics, then success rate is determined by more than mathematical ability, assuming mathematical ability is defined in terms of skill in applying mathematical procedures. A major challenge lies in translating the words in the problem statement to a mathematical form and this challenge can be increased or decreased by rephrasing the problem.
The role of visualization in problem-solving was examined by Hegarty and Kozhevnikov (1999) among 6th grade students using a set of word problems in mathematics. A significant correlation was found between scores on the set of word problems and scores on each of the two spatial tests used in the study. Solutions were categorized as either pictorial or schematic based on sketches or verbal descriptions provided by the participants. The use of schematic representations was found to have a significant relationship with the math problem score. While a negative correlation was measured between math problem score and the use of a pictorial representation; it was not significant. Hegarty and Kozhevnikov (1999) also measured a significant correlation between visualization quality and the Block Design spatial test, but not the Primary Mental Abilities Space subtest, a speeded rotation test. They also found that the likelihood of producing a poor-quality pictorial representation was negatively correlated with both spatial tests, but not to a significant extent. In another study, Edens and Potter (2008) found a much larger correlation between schematic representation and drawing skill than with spatial ability. While findings are mixed, visualization has been found to have an important bearing on problem representation.
Spatial ability also appears to be associated with the quality of the pictorial representations created by students as they are solving problems. Boonen et al. (2014) administered the same problems used by Hegarty and Kozhevnikov (1999) to a larger sample of 6th grade students. They further categorized schematic visualizations as either inaccurate or accurate and found that high spatial students created accurate, schematic visualizations more frequently than low spatial students; those who used an accurate visual-schematic representation were six times more likely to solve a word problem than those who did not. What emerges from these studies is that success in problem solving is more likely for those who can produce accurate, visual-schematic representations and the ability to create such representation is related to spatial ability.
An alternative classification is based on a 22 typology in which spatial factors are viewed as tasks and these require visualization that is (i) either object intrinsic or extrinsic and (ii) static or dynamic (Uttal and Cohen, 2012). In this case, finding a figure embedded in a picture is an intrinsic-static task, mental rotation is an intrinsic-dynamic spatial task, orienting with respect to a frame of reference (e.g., map reading, water levels test) is an extrinsic-static task and re-orienting with respect to a frame of reference (e.g., 3 mountains task) is an extrinsic-dynamic task.
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