"Do not worry about your difficulties in mathematics, I can assure you mine are still greater." - Einstein

MATH SURVIVAL GUIDE FOR FIRST YEAR STUDENTS

UTSCMATH & STATS HELP CENTRE

Compiled and edited by Geanina Tudose

CONTENTS

1. What is university math like?2. How to be a great math student3. Problem Solving4. Writing mathematics (homework and tests)5. Preparing and taking a math test. Dealing with

anxiety6. Getting Help7. Appendices: FAQ (Common Student Concerns) TLS Support Additional Readings

Teaching and Learning ServicesMath & Stats Help CentreUniversity of Toronto at Scarborough©2004 TLS

1. What is university math like?

What is new and different in university? Well, almost everything: new people (your peers/colleagues, teaching and lab assistants, instructors, administrators, etc.), new environment, new social contexts, new norms, and – very important - new demands and expectations. Think about the issues raised below. How do you plan to deal with it? Read tips and suggestions, and try to devise your own strategies.

First-year lectures are large – you will find yourself in a huge auditorium, surrounded by 300, 400, or perhaps even more students. Large classes create intimidating situations. You listen to a professor lecturing, and hear something that you do not understand. Do you have enough courage to rise your hand and ask the lecturer to clarify the point? Keep in mind that you are not alone – other students feel the same way you do. It’s hard to break the ice, but you have to try. Other students will be grateful that you asked the question – you can be sure that lots of them had exactly the same question in mind.

Lectures move at a faster pace. Usually, one lecture covers one section from your textbook. Although lectures provide necessary theoretical material, they rarely present sufficient number of worked examples and problems. You have to do those on your own.

Certain topics (trigonometry, exponential and logarithm functions, vectors, matrices, etc.) will be reviewed in your first-year calculus and linear algebra courses. However, the time spent reviewing in lectures will not suffice to cover all details, or to provide

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sufficient number of routine exercises – you are expected to do it on your own.

You have to know and be proficient with the material from Basic Algebra Basic Formulas from Geometry Equations and Inequalities Elements of Analytic Geometry. For instance, computing common denominators, solving equations involving fractions, graphing the parabola y=x2, or solving a quadratic equation will not be reviewed in lectures.

In university, there is more emphasis on understanding than on technical aspects. For instance, your math tests and exams will include questions that will ask you to quote a definition, or to explain a theorem, or answer a ‘theoretical question.’ Here is a sample of questions that appeared on past exams and tests in the first-year calculus course:

Is it true that f’(x)=g’(x) implies f(x)=g(x)? Answering ‘yes’ or ‘no’ only will not suffice. You must explain your answer.

State the definition of a horizontal asymptote. Given the graph of 1/x, explain how to construct

the graph of 1+1/(x-2). Using the definition, compute the derivative of

f(x)=(x-2)-1.

Mathematics is not just formulas, rules and calculations. In university courses, you will study definitions, theorems, and other pieces of ‘theory.’ Proofs are integral parts of mathematics, and you will meet some in your first-year courses. You will learn

how to approach learning ‘theory,’ how to think about proofs, how to use theorems, etc.

Layperson-like attitude towards mathematics (and other disciplines!) - accepting facts, formulas, statements, etc. at face value - is no longer acceptable in university. Thinking (critical thinking!) must be (and will be) integral part of your student life. In that sense, you must accept the fact that proofs and definitions are as much parts of mathematics as are computations of derivatives and operations with matrices.

©Mathematics Review Manual, Miroslav Lovric, McMaster University, 2003.

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2. How to be great Math StudentThese remarks are provided to assist you, the first year student, in making the transition from high school to university. For a student with intellectual curiosity who is determined to work regularly from the beginning of the term, a first year mathematics course can be remarkably rewarding and stimulating. However, the unwary student may fall into difficulties and have a poor experience instead. These following are intended to help you avoid that.

1. In all mathematics courses, the key to success can be summarized briefly:

DEVELOP REGULAR WORK HABITS SO YOU DO NOT FALL BEHIND!

This will ensure that you develop the depth, breadth and maturity of your knowledge. It means: attend lectures and tutorials, do assignments and enough extra problems to master the material. If you attend lectures, but don't do exercises, you may get lulled into a false sense of accomplishment and can expect a rude shock. In mathematics a thorough knowledge of the previous material is essential to reach an understanding of new material. Hence, falling behind tends to be cumulative and is one of the most frequent causes of failure. Understanding grows with time and experience. Do not expect to follow the

mathematics completely, right away; you will have to think about it, and it may not be until later work is covered that you can appreciate the full significance of earlier material.

2. Some of the ideas in many first year courses, such as differentiation, have been introduced in high school. This does not mean the course is a review. New and more sophisticated concepts will be introduced and must be mastered at a new and higher level of thoroughness and understanding.

3. Learn from doing badly. If you receive a poor grade on early tests or assignments, that is an important signal that you are not mastering the material at an appropriate level. You can deal with this by working harder and consulting about problems with your TA or instructor.

4. If you are having difficulty, first consult your TA; then if the problems persist, your instructor. Professors have regular office hours and are generally willing to meeting with students outside these times by appointment. It should be emphasized that it is your responsibility to seek help if difficulties arise.

5. The Math & Stats Help Centre AC320 and the Math Aid Room S506F is open for extended periods and staffed by faculty and TAs who will assist you. The Math & Stats Help Centre offers tutoring, study groups, and workshops on study techniques and seminars on various mathematics topics. More detailed information can be found on the centre’s website.

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6. Do not delay asking for assistance until the day before the exam. It is impossible to cram mathematics at the last minute. Just as with playing a musical instrument, learning mathematics involves a development of skills and understanding that must be consolidated over a period of time.

7. One of the main differences between high school and university is that, at the university, you are expected to be responsible for mastering course material. Considerable help is offered--lectures, tutorials, mathematics assistance centres and personal help--but it's your responsibility to utilize it.

8. If, nevertheless, you find that you have fallen behind in your coursework, speak with your instructor. He or she can advise you on what to do next.

3. Problem SolvingProblem Solving (Homework and Tests)

The higher the math class, the more types of problems: in earlier classes, problems often required just one step to find a solution. Increasingly, you will tackle problems which require several steps to solve them. Break these problems down into smaller pieces and solve each piece divide and conquer!

Problem types:

1. Problems testing memorization ("drill"), 2. Problems testing skills ("drill"), 3. Problems requiring application of skills to

familiar situations ("template" problems), 4. Problems requiring application of skills to

unfamiliar situations (you develop a strategy for a new problem type),

5. Problems requiring that you extend the skills or theory you know before applying them to an unfamiliar situation.

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In early courses, you solved problems of types 1, 2 and 3. By College Algebra you expect to do mostly problems of types 2 and 3 and sometimes of type 4. Later courses expect you to tackle more and more problems of types 3 and 4, and (eventually) of type 5. Each problem of types 4 or 5 usually requires you to use a multi-step approach, and may involve several different math skills and techniques.

When you work problems on homework, write out complete solutions, as if you were taking a test. Don't just scratch out a few lines and check the answer in the back of the book. If your answer is not right, rework the problem; don't just do some mental gymnastics to convince yourself that you could get the correct answer. If you can't get the answer, get help.

The practice you get doing homework and reviewing will make test problems easier to tackle.

Tips on Problem Solving

Apply Pólya's four-step process:

1. The first and most important step in solving a problem is to understand the problem, that is, identify exactly which quantity the problem is asking you to find or solve for (make sure you read the whole problem).

2. Next you need to devise a plan, that is, identify which skills and techniques you have

learned can be applied to solve the problem at hand.

3. Carry out the plan. 4. Look back: Does the answer you found seem

reasonable? Also review the problem and method of solution so that you will be able to more easily recognize and solve a similar problem.

Some problem-solving strategies: use one or more variables, complete a table, consider a special case, look for a pattern, guess and test, draw a picture or diagram, make a list, solve a simpler related problem, use reasoning, work backward, solve an equation, look for a formula, use coordinates.

"Word" Problems are Really "Applied" Problems

The term "word problem" has only negative connotations. It's better to think of them as "applied problems". These problems should be the most interesting ones to solve. Sometimes the "applied" problems don't appear very realistic, but that's usually because the corresponding real applied problems are too hard or complicated to solve at your current level. But at least you get an idea of how the math you are learning can help solve actual real-world problems.

Solving an Applied Problem

First convert the problem into mathematics. This step is (usually) the most challenging part of an applied problem. If possible, start by drawing a picture. Label it with all the quantities

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mentioned in the problem. If a quantity in the problem is not a fixed number, name it by a variable. Identify the goal of the problem. Then complete the conversion of the problem into math, i.e., find equations which describe relationships among the variables, and describe the goal of the problem mathematically.

Solve the math problem you have generated, using whatever skills and techniques you need (refer to the four-step process above).

As a final step, you should convert the answer of your math problem back into words, so that you have now solved the original applied problem.

©Source: Department of Mathematics and Computer Science SAINT LOUIS UNIVERSITY

4. Writing MathematicsMathematics is a language, and as such has standards of writing which should be observed. In a writing class, one must respect the rules of grammar and punctuation, one must write in organized paragraphs built with complete sentences, and the final draft must be a neat paper with a title. Similarly, there are certain standards for mathematics assignments.

Write your name and class number clearly at the top of at least the first page, along with the assignment number, the section number(s), or the page number(s).

Use standard-sized paper (8.5" x 11"), with no "fringe" running down the side as a result of

the paper’s having been torn out of a spiral notebook.

Attach your pages with a paper clip or staple.  Do not fold, tear, or otherwise "dog-ear" the pages

Clearly indicate the number of the exercise you are doing.  If you accidentally do a problem out of order, or separate part of the problem from the rest, then include a note to the grader, referring the grader to the missed problem or work.

Write out the problems (except in the case of word problems, which are too long).

Do your work in pencil, with mistakes cleanly erased, not crossed or scratched out. If you work in ink, use "white-out" to correct mistakes.

Write legibly (suitably large and suitably dark); if the grader can't read your answer, it's wrong. Write neatly across the page, with each succeeding problem below the preceding one, not off to the right. Please do not work in multiple columns down the page (like a newspaper); your page should contain only one column.

Keep work within the margins. If you run out of room at the end of a problem, please continue onto the next page; do not try to squeeze lines together at the bottom of the sheet. Do not lap over the margins on the left or right; do not wrap writing around the notebook holes.

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Do not squeeze the problems together, with one problem running into the next. Use sufficient space for each problem, with at least one blank line between one problem and the next.

Do "scratch work," but do it on scratch paper; hand in only the "final draft." Show your steps, but any work that is scribbled in the margins belongs on scratch paper, not on your homework.

Show your work. This means showing your steps, not just copying the question from the assignment, and then the answer from the back of the book. Show everything in between the question and the answer. Use complete English sentences if the meaning of the mathematical sentences is not otherwise clear. For your work to be complete, you need to explain your reasoning and make your computations clear.

Do not invent your own notation and abbreviations, and then expect the grader to figure out what you meant.  For instance, do not use "#" in your sentence if you mean "pounds" or "numbers". Do not use the "equals" sign ("=") to mean "indicates", "is", "leads to", "is related to", or anything else in a sentence; use actual words. The equals sign should be used only in equations, and only to mean "is equal to".

Do not do magic. Plus/minus signs, "= 0", radicals, and denominators should not disappear in the middle of your calculations, only to

mysteriously reappear at the end. Each step should be complete.

If the problem is of the "Explain" or "Write in your own words" type, then copying the answer from the back of the book, or the definition from the chapter, is unacceptable. Write the answer in your words, not the text's.

Remember to put your final answer at the end of your work, and mark it clearly by, for example, underlining it. Label your answer appropriately.  If the answer is to a word problem, make sure to put appropriate units on the answer.

In general, write your homework as though you're trying to convince someone that you know what you're talking about.

http://www.purplemath.com/guidline.htmCopyright © 1990-2004 Elizabeth Stapel, Used By Permission

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5. Preparing and taking a math test. Dealing with anxietyEveryday Study is a Big Part of Test Preparation

Good study habits throughout the semester make it easier to study for tests.

Do the homework when it is assigned. You cannot hope to cram 3 or 4 weeks worth of learning into a couple of days of study.

On tests you have to solve problems; homework problems are the only way to get practice. As you do homework, make lists of formulas and techniques to use later when you study for tests.

Ask your Instructor questions as they arise; don't wait until the day or two before a test. The questions you ask right before a test should be to clear up minor details.

Studying for a Test

1. Start by going over each section, reviewing your notes and checking that you can still do the homework problems (actually work the problems again). Use the worked examples in the text and notes - cover up the solutions and work the problems yourself. Check your work against the solutions given.

2. You're not ready yet! In the book each problem appears at the end of the section in which you learned

how do to that problem; on a test the problems from different sections are all together.

Step back and ask yourself what kind of problems you have learned how to solve, what techniques of solution you have learned, and how to tell which techniques go with which problems.

Try to explain out loud, in your own words, how each solution strategy is used (e.g. how to solve a quadratic equation). If you get confused during a test, you can mentally return to your verbal "capsule instructions". Check your verbal explanations with a friend during a study session (it's more fun than talking to yourself!).

Put yourself in a test-like situation: work problems from review sections at the end of chapters, and work old tests if you can find some. It's important to keep working problems the whole time you're studying.

3. Also:

Start studying early. Several days to a week before the test (longer for the final), begin to allot time in your schedule to reviewing for the test.

Get lots of sleep the night before the test. Math tests are easier when you are mentally sharp.

TAKING A MATH TEST

Test-Taking Strategy Matters

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Just as it is important to think about how you spend your study time (in addition to actually doing the studying), it is important to think about what strategies you will use when you take a test (in addition to actually doing the problems on the test). Good test-taking strategy can make a big difference to your grade!

Taking a Test

First look over the entire test. You'll get a sense of its length. Try to identify those problems you definitely know how to do right away, and those you expect to have to think about.

Do the problems in the order that suits you! Start with the problems that you know for sure you can do. This builds confidence and means you don't miss any sure points just because you run out of time. Then try the problems you think you can figure out; then finally try the ones you are least sure about.

Time is of the essence - work as quickly and continuously as you can while still writing legibly and showing all your work. If you get stuck on a problem, move on to another one - you can come back later.

Work by the clock. On a 50 minute, 100 point test, you have about 5 minutes for a 10 point question. Starting with the easy questions will probably put you ahead of the clock. When you work on a harder problem, spend the allotted time (e.g., 5 minutes) on that question, and if you have not almost finished it, go on to another

problem. Do not spend 20 minutes on a problem which will yield few or no points when there are other problems still to try.

Show all your work: make it as easy as possible for the Instructor to see how much you do know. Try to write a well-reasoned solution. If your answer is incorrect, the Instructor will assign partial credit based on the work you show.

Never waste time erasing! Just draw a line through the work you want ignored and move on. Not only does erasing waste precious time, but you may discover later that you erased something useful (and/or maybe worth partial credit if you cannot complete the problem). You are (usually) not required to fit your answer in the space provided - you can put your answer on another sheet to avoid needing to erase.

In a multiple-step problem outline the steps before actually working the problem.

Don't give up on a several-part problem just because you can't do the first part. Attempt the other part(s) - if the actual solution depends on the first part, at least explain how you would do it.

Make sure you read the questions carefully, and do all parts of each problem.

Verify your answers - does each answer make sense given the context of the problem?

If you finish early, check every problem (that means rework everything from scratch).

TEN WAYS TO REDUCE MATH ANXIETY

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1. You are not alone! Relax. Many people dislike and are nervous about math. Even mathematicians are unsure of themselves and get that sinking, panicky feeling called "math anxiety" when they first confront a new problem.

2. If you have math anxiety, admit it. If you pretend not to have it, you will not learn to overcome it or manage it.

3. If you're having math trouble, practice a little math each day.

4. Ask questions. Some people think asking questions is a sign of weakness. It's not. It's a sign of strength. In fact, other students will be glad. (They have questions, too.)

5. Do math in a way that's natural for you. There's often more than one way to work a math problem. Maybe the instructor's way stumps you at first. Don't give up. Work to understand it your way. Then it will be easier to understand it the instructor's way. Remember, "each mind has it's own method."

6. Notice your handwriting when you do math. The sloppier it gets, the more confused or angry you probably are. When it gets really sloppy, STOP. Look away for a few seconds. Then erase the messy parts. Start again. Try not to let your attitude interfere with learning math.

7. Know the basics. Be sure you know your math from earlier grades. Maybe you missed something when you moved to a new high school. Face it: Math

builds on itself. You have to go back and relearn that stuff.

8. Don't go by memory alone. Try to understand your math. Memorizing is a real trap. When you're nervous, memory is the first thing to go.

©Source: Department of Mathematics and Computer Science SAINT LOUIS UNIVERSITY6. Getting HelpGet help as soon as you need it. Do not wait until the test is near. The new material builds upon the previous one, so anything that you do not understand will make future material even harder.Resources:

1. Ask questions in class. Do not be intimidated by the size of the class, many students will be grateful you ask.

2. Visit your Professor’s office hours. We love to see students and talk to them.

3. Go to your tutorial and ask your Teaching Assistant (TA) questions.

4. Come to the Math & Stats Help Centre AC 217 and Math Aid Room S506F. The schedule is listed at http://tls.utsc.utoronto.ca/data_interpretation/default.htm

5. Come to the regular Math Workshops we offer.

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6.7. Form a Study Group and regularly meet.

There are many places on campus where you can do that: the Math Help Centres, Study Rooms, Booked Rooms in the ARC, other classrooms not in sessions, etc.

8. Find a private Tutor. Check for ads in the student union area.

APPENDIX 1FAQ

1.``I really know this material, but I just don't do well on tests.''

This is a common complaint. But there is a distinction between knowing something and having seen it before. Sometimes you may recognize the correct answer; but with real knowledge, you can construct solutions and even reconstruct the theory with your pencil. While most instructors will say that students eventually mature into effective ways of learning, we have very little to guide students in this direction, particularly as reading texts, listening to lectures, and reading notes may tend to reinforce that learning is recognition. To get to the bottom of the problem you may have to reconsider how you study, and find more ways to make studying active rather than passive. Just as sports or music or theater performance require lots of practice before you can ``pull it off under the gun'', mathematics takes a lot of practice and drill -- and adrenalin is not beneficial on mathematics tests. On a practical level, one can, with experience, learn how to

anticipate tests. Rewrite your notes, make up review sheets, join a study group, and really study for tests (even if you didn't have to in high school). Your instructor teaches what is important and tests on it; but you must come to tests over-prepared. If you have a serious anxiety problem, discuss it with your instructor; there may be anxiety workshops and specially trained counselors who can help, or your instructor may suggest another solution.

2. ``The test is too long; if I'd had more time I could have done really well.''

Speed in mathematics is actually an important measure of how well we understand the subject. In the working world, doctors, police officers, and airplane pilots all have to make fast, accurate decisions. The writing you did on the test probably did not take more than about ten minutes, so we have to ask how the rest of the time was spent. Perhaps you are still feeling your way over subject matter that requires a quick reaction. Do you write a lot in hopes of partial credit? A practical suggestion is to browse through the test, allot your time, and simplify answers last. Read the directions carefully, limit answers to precisely what is asked, and always check your work line by line, to avoid going off in disastrously wrong directions.

3. ``The tests aren't like the homework.'' We do feel a definite security in seeing problems identical to what we did before, but to emphasize these problems would be to validate rote learning alone. In mathematics, we do homework for the purpose of learning the material, not the other way around. Use the class period as a guide to what your instructor thinks is important, and be sure to read the

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text. As you work homework problems, try to see the bigger picture: why am I being given this problem, and how does it reinforce and relate to the theory? Also, the test problems may be closer to the homework than you recognize, but you may be falling into a rote mode as you do the homework. If your instructor does not use the precise wording of the book, or blends several problems, you may then feel lost. Scramble the order of the problems you work as you study for tests. And browse through other books in your library, and study with friends so you can verbalize the material as much as possible, in many different ways.

4. ``Careless mistakes keep killing me. I make a lot of stupid mistakes''

If this is a chronic problem, your mistakes may not be careless. There is a type of mistake that will disappear and a type that is related to more fundamental problems of understanding. But careless mistakes are nevertheless a problem; for example, careless mistakes are not permitted of bank tellers, construction workers, airplane pilots, or neurosurgeons. Check all answers for accuracy and reasonability, backtracking line by line; and reserve time on tests for a final check. If you practice being careful as you work homework problems, you can overcome the problem of ``careless'' or ``stupid'' mistakes. But it is interesting that many students would prefer to blame their intelligence or their carelessness before their effort becomes the variable.

5. ``Why didn't I get more partial credit?'' Sometimes students see knowledge as something that generates grades, and feel that their partial knowledge should be rewarded accordingly. However, a lot of

partial knowledge on many topics does not add up to real knowledge, and to learn for partial knowledge can eventually lead to a ``mathematical shut-down'' in understanding. An instructor naturally does not want to encourage learning for partial knowledge. What may seem to you a halfway answer would probably not be accepted in most careers in the real world where small errors could send an astronaut on the wrong orbit or produce other disasters. On a practical level, neater, more organized work will help you stay under control while working a problem. An instructor is more likely to assign partial credit if you appear to be in control of the problem, rather than flailing; and the way you present the mathematics on your test (do you work down the page or scribble all over?) may affect this perception more than you realize.

6. ``I didn't know what you wanted'', or ``What do you want here?''

This question can cause an instructor to feel put in the role of a demagogue whose ``wants'' are mysterious to you. They may answer, ``I want the correct answer!'' If the question is incomplete or ambiguous, your instructor will not mind clarifying the question, and you should make it your responsibility to come forward (but don't ask this question if you only want to know if your solution is correct). You may have a better understanding of the question than you say, but you may just not be able to solve the problem. Sometimes you can explain on your test how you are interpreting the question, and respond accordingly``I just don't use the book; I can't understand it at all.'' The text is definitely necessary for this subject, just as a racquet is to a game of tennis or a violin to a violinist; the text is the main tool for the course! But

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we can't expect simply to read, track, and understand a mathematics text. Skim the book, look at the problems. and see what is needed for a good understanding. This decoding process does not happen in one pass; you may need to reread some sections many times. You can learn to make a text work for you, especially if you read it before coming to class and then again, after class. Try rewriting sections of the text, synopsizing it in your own words. And there are many other texts in the library that you can refer to. Frequently rereading a particular passage simply is no help to us, while all we need is another author's language.

7. ``Why do I have to memorize this? Memorization isn't learning. Besides I know I'm

going to forget it anyway.'' We memorize in order to facilitate learning, so we can function with the demands of the field. Memorization is not an end in itself, and it does not constitute learning. But when you use this information, you won't forget it. Every field requires memorization, and most fields -- biology, history, physics, political science, languages -- require far more. We are only able to solve problems if we are familiar with the necessary terms and laws. To improve your memory, don't trust your recognition memory when it comes to a test. Practice writing out the definitions and theorems, and make outlines of the major points of the theory. Check back to your text for accuracy. It is easy to think we know something until we attempt to put it in writing. By practicing studying continuously in this way, rather than cramming at the last minute, you will find memorization will feel more naturally like part of the learning process.

8. ``Why should we do these long problems -- they won't be on the test away''

These problems are useful because they synthesize the material and get us beyond rote skills. In track, for example, runners may lift weights in practice, although they won't be doing this in the tournament. These problems push your ability to manipulate and control the mathematics by engaging you in multi-step reasoning, and they train you to recognize where the skills you are learning can be useful.

``I've always been good at mathematics until this course.''

Mathematics courses are built on previous courses, but unfortunately, our performance in one course does not guarantee our success in another. Mathematics is an extremely complex field, and every mathematics course has new challenges and introduces new ways of thinking. There are things that are important that we aren't learning in this course, but what we are learning is important. Also, different instructors of mathematics may stress different things. Perhaps you should discuss with your instructor what it is that is not meeting the instructor's standards.

9. ``I can never understand my class notes; I don't read them. I didn't follow you that day.''

Sometimes students write down material they don't understand, feeling that in writing it down, understanding will come. But in class, instructors may present the theory, work examples, go over troublesome homework problems, give insights into the material, respond to questions or ask probing questions. With all this on an instructor's agenda, your notes indeed may not seem too clear! Ask for clarifications in class at the time. If you read the text

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before class, you may recognize material from the book, and where you do not need to take notes; but jot down what topics the instructor discussed. Reading the text beforehand will also help you focus your questions in class in ways that your instructor will probably appreciate. Bring your text to class. And rewrite your notes, incorporating material from the book and problems. You will have created an excellent study guide for your use.

10. ``I couldn't make it to class yesterday; did I miss anything important?''

You may be asking if you missed something with a grade attached. But in any case, the particular information is indeed important (or we might have been tempted to miss ourselves!), but the most important thing you missed is the practice of seeing and doing things with new material.

11. ``Where are we ever going to use this stuff?'' People who don't learn or understand this material probably won't use it, but people who do may be surprised to find where it is useful. This applies not just to the content of the course, but to its association with careful, creative thinking. It will probably be up to you to find places where you can use this mathematics. But depending on your career, you may find that things that are now obvious to you are not known to others; or on the other hand, you may find it taken for granted that you know this material and much more. But most likely, you may actually use the subject of this course and the skills you've gained, without even realizing it. In reality, the questions and complaints mentioned above are all too frequently tacit, and it may be that

much more difficult to bring these issues to a point of real discussion. Sometimes these complaints only show up on instructors' end-of-term evaluations. There are certainly more useful responses for individual students in individual situations than those offered here.

APPENDIX 2 Teaching and Learning Services Support http://tls.utsc.utoronto.ca/

1. Math and Statistics Help CentreGeanina Tudose, Coordinator 287.5667

2. Academic Learning Services for Students Martha Young, Coordinator 287.7557

3. English Language Development Elaine Khoo, 287.7562

4. The Writing CentreSarah King, Coordinator 287.7480

5. Presentation Skills InstructionSaira Mall, 287.5666

6. Research Skills InstructionFrances Sardone, 287.7502

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APPENDIX 3 Additional Readings

1. What is Mathematics?, R. Courant and H. Robbins; Oxford, 1941.

2. Number, The Language of Science, T. Dantzig; Anchor, 1956.

3. The Mathematical Experience, Davis, P.J. and R. Hirsch; Birkhauser, Boston, 1981.

4. Art and Science, Escher, M.C.; (H.S.M. Coxeter, M. Emmer, R. Penrose and M.L. Trewber, Editors); North Holland, 1985.

5. Great Moments in Mathematics (2 vols.), H. Eves; Mathematical Association of America, 1983.

6. A Mathematician's Apology, G.H. Hardy; Cambridge, 1940.

7. Geometry and the Imagination, D. Hilbert and S. Cohn-Vossin; Chelsea, 1952.

8. Godel, Escher, Bach, D. Hofstader; Basic Books, New York, 1979.

9. The World of Mathematics (4 vols.), J.R. Newman; Simon and Schuster, New York, 1956.

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