Math Survival Guide for First Year Students

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

MATH SURVIVAL GUIDE FOR FIRST YEAR STUDENTS

UTSC

MATH & STATS HELP CENTRE

Compiled and edited by Geanina Tudose

CONTENTS

What is university math like?
How to be a great math student
Problem Solving
Writing mathematics (homework and tests)
Preparing and taking a math test. Dealing with anxiety
Getting Help
Appendices:

FAQ (Common Student Concerns)
TLS Support
Additional Readings

Teaching and Learning Services

Math & Stats Help Centre

University of Toronto at Scarborough

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 presentsufficient 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 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.

2.How to be great Math Student

These 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.

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 followthe 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.

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.
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.
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.
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.
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.
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.
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 Solving

Problem 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:
Problems testing memorization ("drill"),
Problems testing skills ("drill"),
Problems requiring application of skills to familiar situations ("template" problems),
Problems requiring application of skills to unfamiliar situations (you develop a strategy for a new problem type),
Problems requiring that you extend the skills or theory you know before applying them to an unfamiliar situation.

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:

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).
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.
Carry out the plan.
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 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.

4. Writing Mathematics

Mathematics 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.
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.

5. Preparing and taking a math test. Dealing with anxiety

Everyday 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: