math writing assignmet

WRITING IN MATHEMATICS

As required by the Department of Mathematics, there will be at least two writing assignments graded in

each section of Calculus I. The nature of such assignments is up to your instructor, but typically the grading

scheme is not so much about “whether you arrived at the correct final answer and used appropriate methods”

as it is about how well you explained your work and the correctness of your “mathematical grammar”. This

is a new experience for most Calculus I students.

Attention to mathematical writing standards is not restricted to the writing projects! In mathematics

courses, you are to try to think (and write) more like a mathematician. Mathematics is sometimes called the

“queen of the sciences” because of its high standards for rigor, accuracy, and proof. Mathematicians pride

themselves on using mathematical symbols, words and terminology precisely. In Calculus I, we instructors

start imposing those standards upon students, and your task is to produce clear and logical work, sometimes

in complete sentences and paragraphs, always striving to use technical notation correctly. Even when it is

easy for your instructor to figure out what you meant, sloppy work may result in a lower score when

compared with more polished work. That’s only fair. Likewise, it is only fair to warn our students that such

standards may be applied, so consider this your fair warning! Also, note that your online WebAssign

homework answers will receive no credit from the electronic grading system if they are entered with even the

slightest error, such as missing one parenthesis! So, there’s even more reason for you to use math symbols

correctly.

For example, when performing basic arithmetic suppose that you wanted to add 10 to 8 and then

multiply the result by 4. You would add to get 18 and then multiply by 4 to get a final answer of 72. But

how would you show your work if for some reason you had to?

1st option: 8+10 = 18 . 4 = 72 ?

2nd option: 8+10 . 4 = 18 . 4 = 72 ?

3rd option: (8+10) . 4 = 18 . 4 = 72 ?

4th option: “I added 8 and 10 to get 18, and then multiplied by 4 to get 72.”

In the 1st option, the equals sign on the left has been misused! 8+10 and 18 . 4 are not equal. It is not

your instructor’s job to guess what you meant to say or what you were thinking and to fix it up for you for

free!

In the 2nd option, the rules of order of operations stipulate that the expression 8+10 . 4 refers to the

result of first multiplying 10 by 4, then adding 8 to the result. So, the first equals sign is misused, since

8+10 . 4 ≠ 18 . 4 . The 3rd and 4th options are correct.

Fortunately, your instructor doesn’t want to see your work broken down like this for such basic

computations. You can simply write (8+10) . 4 = 72 instead.

Here are examples that will matter more, but the writing principles are similar. You will be

asked various questions about limits, such as to evaluate

€

lim x→2

(3 + 8x) or

€

lim x→∞

5

x + 47 . Roughly, the 1st limit

refers to the limiting value of the output 3+8x when the input value for x is nearly 2 but not exactly 2.

Roughly, the 2nd limit refers to the limiting value of the output 5/(x+47) when x is positive and enormous.

If you were asked to “fill in the blanks for the values of these limits, with no work required”, then full

credit answers are

€

lim x→2

(3 + 8x) = 19 and

€

lim x→∞

5

x + 47 = 0. If asked instead to "use Limit Laws in order to

evaluate these limits, and show your work", then the 1st attempt below is better than the 2nd attempt.

1st attempt:

€

lim x→2

(3 + 8x) = 3 + 8(2) = 19, and

€

lim x→∞

5

x + 47 = 0

↑ ↑ direct substitution property form c/∞

2nd attempt:

€

lim x→2

(3 + 8x) = 3 + 8(2) = 19, and

€

lim x→∞

5

x + 47 =

5 ∞

= 0

In the 2nd attempt, no Limit Law is given as a reason for the first problem. And, for the second problem,

5/∞ is not a number (that might explain why there’s no ∞ button on your calculator!), so it isn’t okay to

divide 5 by ∞ and claim that the result equals 0, or that the result equals the limit in the problem.

Nevertheless, it is acceptable to refer to the “form” 5/∞ as indicating that we are dividing 5 by an enormous

positive number—we are not offering 5/∞ as a number in an equation. It is not any better to write

€

lim x→∞

5

x + 47 →

5 ∞ → 0. The arrows are a poor substitute (no better than commas), indicating no connection

between the three expressions they separate. There is no conventional definition for the meaning of the

arrows.

A third type of problem concerning limits is to "use the Definition of a Limit to prove

€

lim x→2

(3 + 8x) = 19". This is an entirely different situation, and the work above would receive little or no credit.

That’s because the technical meaning of the claim

€

lim x→2

(3 + 8x) = 19 is deeper than it might at first appear,

and the instructions force the solver to use the technical definition of the claim in order to solve the problem.

So, be advised that it is often not enough to just “get the final answers”. Each student is responsible for

learning what each set of instructions means. Each student is responsible for learning what standards their

instructor sets for the kinds of work and writing that receive full credit.

MORE TIPS ON WRITING IN MATHEMATICS

I. Goal The goal of mathematical writing in this course is to enhance thinking, comprehension and communication skills. It has been said, “If you can’t explain it, you don’t understand it.”

II. Expectations

You are expected to submit your work in organized, legible form. Use sentences that are mathematically and grammatically correct and logical. The ultimate test of what you wrote is this: Can someone learn from your paper, someone who hasn’t yet thought about the problem or topic you present? Remember that the reader will see only what you wrote, not what you meant to say.

III. Guidelines A. Keep your audience in mind.

Your audience is a person who has studied up to the current topic. Your grade reflects how well you address that audience; your instructor already knows the material! Think of explaining your solution to someone who needs help, perhaps a classmate who has been absent from class due to illness.

B. Use complete sentences.

The rules of grammar, spelling, and punctuation apply to mathematics as well as English. Mathematical grammar consists of the rules for combining symbols to make mathematical sentences. For clear communication, express one idea at a time in a complete sentence.

Example:

Nonsense Sentence slope zero, horizontal Since the slope is zero, the

line is horizontal.

m even, m = 2k If m is even, then m = 2k for some integer k.

C. Be careful with pronouns. Avoid using “it”, “that” or “this” when the equation, function or key term referred to is not perfectly clear.

Example: Explain how to find the slope of the line ax + by = c.

UNCLEAR BETTER

Solve it for y. Find the coefficients. Solve the equation for y. This gives the slope. The coefficient of x is the slope.

D. Use different letters for different things.

For example, if P and Q represent two points in the same plane, and you use (a, b) to denote the coordinates of P, then you must use different symbols to label the coordinates of Q, such as (c, d). Also, if you are asked to write an equation showing n as a multiple of 5, it is incorrect to write n = 5n. You must use a different letter such as k on one side: n = 5k.

E. Honor the Equal Sign.

Use an equal sign only when you mean to say two quantities are equal, and avoid it otherwise. The equal sign has a precise meaning; it is not a punctuation mark or an abbreviation for “this leads to” or “the solution is”.

Examples: 1. Solve the equation x2 – 5x + 4 = 0.

INCORRECT BETTER x2 – 5x + 4= 0 x2 – 5x + 4 = 0 (x-4) (x-1) (x-4) (x-1) = 0 4, 1 x = 4 or x = 1

2. Solve the equation 5x = 10 INCORRECT BETTER 5x = 10 = x = 2 5x = 10 x = 2

[10 does not equal 2. Here, the middle ‘equal sign’ has been used erroneously for “the solution is”.] F. Define terms.

When you introduce a new symbol or variable, be sure to define it precisely. A complete definition would include intended units. Be specific when you define symbols; it is wrong to say “n = people” when you actually mean “n = number of people”. Example: “The area of a parallelogram is A = xy.” [What are x and y? You must explain to the reader what parts of the parallelogram x and y actually represent]

G. Give reasons/show steps.

Keep the reader informed of what you are doing. Remember your audience. Justify every significant assertion you make with supporting reasons or work. Example: Solve the inequality. x2 – 4x – 5 < 0. Incomplete Solution:

x2 – 4x – 5 < 0 (x – 5) (x + 1) < 0 x ∈ (-1, 5)

[Here you have only shown some initial steps but skipped the significant ones at the end.] H. Answer the question.

When you finish a problem, go back and reread it. Be sure you have given a clear answer to the question asked. Example: What is the area of a square of perimeter 12 inches?

Non-answer: Let x = the length of one side in inches. Perimeter is 4x = 12 x = 3 in. [This solution does not answer the stated question.] I. Proofread to check how well you have addressed the intended audience.

Ideally, have another calculus student assess your work, pointing out where they get confused. REFERENCES

Ideas from the following sources have been used in the preparation of these notes. 1. L. Gillman, Writing Mathematics Well, Mathematical Association of America, Washington,

D.C., 1988. 2. D.E. Kullman, How to Study Mathematics, student handout, Miami University, Department of

Mathematics & Statistics, revised 1988. 3. J.J. Price, Learning Mathematics Through Writing: Some Guidelines, The College Math Journal

2(5), 1989, 393-401.