Introduction Strunk & White A mathematical version Student reactions Conclusion

Elements of Style for Proofs

Anders O. F. Hendrickson

Concordia CollegeMoorhead, MN

MathFest, PittsburghAugust 5, 2010

Introduction Strunk & White A mathematical version Student reactions Conclusion

Outline

1 Introduction

2 Strunk & White

3 A mathematical version

4 Student reactions

5 Conclusion

Introduction Strunk & White A mathematical version Student reactions Conclusion

The problem

Actual conversation after Calculus II exam(edited for dramatic effect)

3. Does this series converge or diverge? Why?

Student. Why did I get no partial credit on this problem?Me. Because you got it entirely wrong.

Student. But I said it converged, and that’s the right answer.Me. Yes, but look! Every single line of your argument

contains at least one massive logical flaw.Student. But I thought this was math, where all that matters

is getting the right answer.Me. You thought that, and you were wrong.

Introduction Strunk & White A mathematical version Student reactions Conclusion

The problem

We all recognize the problem:For 14 years, math is about finding “the answer.”“Showing your work” means leaving the cryptic residue ofyour algorithms on the paper.Then suddenly our students need to learn to write proofs.

A proof-centered mathematics course is more like anEnglish course, requiring

Complete sentencesCompelling argumentsLinguistic conventions

Then why not teach it like an English course?

Introduction Strunk & White A mathematical version Student reactions Conclusion

The “little book”

1918 self-published by WilliamStrunk of Cornell University foruse in his classroom1959 revised by Strunk’sstudent E.B. White

Introduction Strunk & White A mathematical version Student reactions Conclusion

The “little book”

1918 self-published by WilliamStrunk of Cornell University foruse in his classroom1959 revised by Strunk’sstudent E.B. White

Introduction Strunk & White A mathematical version Student reactions Conclusion

Strunk & White: contents

The contents of Strunk and White fall into two categories:Rules of grammar and punctuation (mechanics)10. Use the proper case of pronoun.11. A participial phrase at the beginning of a sentence must refer to

the grammatical subject.Rules for clear, vigorous, and memorable prose (style)16. Use definite, specific, concrete language.17. Omit needless words.

Introduction Strunk & White A mathematical version Student reactions Conclusion

Strunk & White: samples

16. Use definite, specific, concrete language.

Prefer the specific to the general, the definite to the vague, theconcrete to the abstract.

A period of unfavorableweather set in.

It rained every day for aweek.

He showed satisfaction ashe took possession of hiswell-earned reward.

He grinned as he pock-eted the coin.

If those who have studied the art of writing are in accord on any onepoint, it is on this: the surest way to arouse and hold the attention ofthe reader is by being specific, definite, and concrete. The greatestwriters—Homer, Dante, Shakespeare—are effective largely becausethey deal in particulars and report the details that matter. Their wordscall up pictures.

Introduction Strunk & White A mathematical version Student reactions Conclusion

Strunk & White: samples

12. Choose a suitable design and hold to it.

A basic structural design underlies every kind of writing. The writerwill in part follow this design, in part deviate from it, according to hisskill, his needs, and the unexpected events that accompany the act ofcomposition. Writing, to be effective, must follow closely the thoughtsof the writer, but not necessarily in the order in which those thoughtsoccur. This calls for a scheme of procedure. In some cases the bestdesign is no design, as with a love letter, which is simply anoutpouring, or with a casual essay, which is a ramble. But in mostcases planning must be a deliberate prelude to writing. The firstprinciple of composition, therefore, is to foresee or determine theshape of what is to come and pursue that shape.

Introduction Strunk & White A mathematical version Student reactions Conclusion

Strunk & White: samples

12. Choose a suitable design and hold to it.

A basic structural design underlies every kind of writing. The writerwill in part follow this design, in part deviate from it, according to hisskill, his needs, and the unexpected events that accompany the act ofcomposition. Writing, to be effective, must follow closely the thoughtsof the writer, but not necessarily in the order in which those thoughtsoccur. This calls for a scheme of procedure. In some cases the bestdesign is no design, as with a love letter, which is simply anoutpouring, or with a casual essay, which is a ramble. But in mostcases planning must be a deliberate prelude to writing. The firstprinciple of composition, therefore, is to foresee or determine theshape of what is to come and pursue that shape.

Introduction Strunk & White A mathematical version Student reactions Conclusion

Strunk & White: structure

Key points about the structure of Strunk & White:An eclectic, mostly unsystematic list of rulesEach rule is brief, suitable for memorization.Each rule is explained at length, with examples.Each rule is bold and imperious.(Qualifications and exceptions are reserved for the explanation.)

Introduction Strunk & White A mathematical version Student reactions Conclusion

Strunk & White: structure

Key points about the structure of Strunk & White:An eclectic, mostly unsystematic list of rulesEach rule is brief, suitable for memorization.Each rule is explained at length, with examples.Each rule is bold and imperious.(Qualifications and exceptions are reserved for the explanation.)

Introduction Strunk & White A mathematical version Student reactions Conclusion

Strunk & White: structure

Key points about the structure of Strunk & White:An eclectic, mostly unsystematic list of rulesEach rule is brief, suitable for memorization.Each rule is explained at length, with examples.Each rule is bold and imperious.(Qualifications and exceptions are reserved for the explanation.)

Introduction Strunk & White A mathematical version Student reactions Conclusion

Strunk & White: structure

Key points about the structure of Strunk & White:An eclectic, mostly unsystematic list of rulesEach rule is brief, suitable for memorization.Each rule is explained at length, with examples.Each rule is bold and imperious.(Qualifications and exceptions are reserved for the explanation.)

Introduction Strunk & White A mathematical version Student reactions Conclusion

Strunk & White: reception

The “little book” has many devoted followers and a place incountless classrooms.Even its critics mainly object that it is a starting point, notthe final word.The mean-spirited call it a list of “pet peeves”—exactlywhat a teacher needs.

Introduction Strunk & White A mathematical version Student reactions Conclusion

Strunk & White: reception

The “little book” has many devoted followers and a place incountless classrooms.Even its critics mainly object that it is a starting point, notthe final word.The mean-spirited call it a list of “pet peeves”—exactlywhat a teacher needs.

Introduction Strunk & White A mathematical version Student reactions Conclusion

A mathematical Strunk & White

My projectI set out to create and use an “Elements of Style” handout tohelp teach proof-writing skills in a first-semester algebra course.

Introduction Strunk & White A mathematical version Student reactions Conclusion

Sample rules

1. The burden of communication lies on you, not on your reader.

It is your job to explain your thoughts; it is not your reader’s job toguess them from a few hints. You are trying to convince a skepticalreader who doesn’t believe you, so you need to argue with airtightlogic in crystal clear language; otherwise he will continue to doubt. Ifyou didn’t write something on the paper, then (a) you didn’tcommunicate it, (b) the reader didn’t learn it, and (c) the grader has toassume you didn’t know it in the first place.

Introduction Strunk & White A mathematical version Student reactions Conclusion

Sample rules

5. Don’t interchange = and⇒.

The equals sign connects two objects, as in “x2 = b”; the symbol “⇒”connects two statements, as in “ab = a⇒ b = 1.” And please, pleasedon’t just use a generic “→” to connect two lines. That symbol has nomeaning in that context; it doesn’t tell the reader anything.

Introduction Strunk & White A mathematical version Student reactions Conclusion

Sample rules

7. Use multiple sheets of paper.

Some people write in tiny letters, trying to cram everything onto asingle sheet of paper, with the result that their proofs are so terse asto be incomprehensible. An extra tree will gladly sacrifice its life tohelp your proofs be legible and understandable.

Introduction Strunk & White A mathematical version Student reactions Conclusion

Categories of rules

Principles of writing1. The burden of communication lies on you, not on your

reader.3. Use complete sentences.9. Show the logical connections among your sentences.

10. Say exactly what you mean.Logical principles13. Don’t beg the question.19. Don’t “prove by example.”21. Make your counterexamples concrete and specific.Grammatical notes

5. Don’t interchange = and⇒.12. Write strings of equalities in the proper order.20. Write “Let x = . . .,” not “Let . . . = x .”Manuscript preparation

6. Use whitespace.7. Use multiple sheets of paper.8. Use scratch paper.

Introduction Strunk & White A mathematical version Student reactions Conclusion

Categories of rules

Principles of writing1. The burden of communication lies on you, not on your

reader.3. Use complete sentences.9. Show the logical connections among your sentences.

10. Say exactly what you mean.Logical principles13. Don’t beg the question.19. Don’t “prove by example.”21. Make your counterexamples concrete and specific.Grammatical notes

5. Don’t interchange = and⇒.12. Write strings of equalities in the proper order.20. Write “Let x = . . .,” not “Let . . . = x .”Manuscript preparation

6. Use whitespace.7. Use multiple sheets of paper.8. Use scratch paper.

Introduction Strunk & White A mathematical version Student reactions Conclusion

Categories of rules

Principles of writing1. The burden of communication lies on you, not on your

reader.3. Use complete sentences.9. Show the logical connections among your sentences.

10. Say exactly what you mean.Logical principles13. Don’t beg the question.19. Don’t “prove by example.”21. Make your counterexamples concrete and specific.Grammatical notes

5. Don’t interchange = and⇒.12. Write strings of equalities in the proper order.20. Write “Let x = . . .,” not “Let . . . = x .”Manuscript preparation

6. Use whitespace.7. Use multiple sheets of paper.8. Use scratch paper.

Introduction Strunk & White A mathematical version Student reactions Conclusion

Categories of rules

Principles of writing1. The burden of communication lies on you, not on your

reader.3. Use complete sentences.9. Show the logical connections among your sentences.

10. Say exactly what you mean.Logical principles13. Don’t beg the question.19. Don’t “prove by example.”21. Make your counterexamples concrete and specific.Grammatical notes

5. Don’t interchange = and⇒.12. Write strings of equalities in the proper order.20. Write “Let x = . . .,” not “Let . . . = x .”Manuscript preparation

6. Use whitespace.7. Use multiple sheets of paper.8. Use scratch paper.

Introduction Strunk & White A mathematical version Student reactions Conclusion

Categories of rules

Principles of writing1. The burden of communication lies on you, not on your

reader.3. Use complete sentences.9. Show the logical connections among your sentences.

10. Say exactly what you mean.Logical principles13. Don’t beg the question.19. Don’t “prove by example.”21. Make your counterexamples concrete and specific.Grammatical notes

5. Don’t interchange = and⇒.12. Write strings of equalities in the proper order.20. Write “Let x = . . .,” not “Let . . . = x .”Manuscript preparation

6. Use whitespace.7. Use multiple sheets of paper.8. Use scratch paper.

Introduction Strunk & White A mathematical version Student reactions Conclusion

Use in the classroom

Printed on neon red paper,distinct from all other course materialsHanded out the first day of courseWent over one rule at beginning of each classRefer to rules by number when grading

Forces students to read the rulesSaves immense time in grading

Introduction Strunk & White A mathematical version Student reactions Conclusion

Student reactions

Modern Algebra I,end-of-semester questionnaire

“Did the Elements of Style handout help you? If so, how?If it was not helpful, why not? How should I improve it?”

“Helpful. That handout shows your class the expectationsyou have for their homework in general. Also, it forcesstudents to ‘normalize’ their proofs. That makes thembetter, hopefully.”

“I think I quickly mastered the basic format of proof makingthanks to this handout. Perhaps spend more time in thecourse introduction emphasizing particular elements of thissheet.”

“Helpful right away in the semester to help getting back intothinking proof-ishly.”

Introduction Strunk & White A mathematical version Student reactions Conclusion

Student reactions

Modern Algebra I,end-of-semester questionnaire

“Did the Elements of Style handout help you? If so, how?If it was not helpful, why not? How should I improve it?”

“Helpful. That handout shows your class the expectationsyou have for their homework in general. Also, it forcesstudents to ‘normalize’ their proofs. That makes thembetter, hopefully.”

“I think I quickly mastered the basic format of proof makingthanks to this handout. Perhaps spend more time in thecourse introduction emphasizing particular elements of thissheet.”

“Helpful right away in the semester to help getting back intothinking proof-ishly.”

Introduction Strunk & White A mathematical version Student reactions Conclusion

Student reactions

Modern Algebra I,end-of-semester questionnaire

“Did the Elements of Style handout help you? If so, how?If it was not helpful, why not? How should I improve it?”

“Very helpful. Especially easy to find out exactly what youhad points taken off for.”

“I think it made it easier for us to realize if we kept makingthe same type of mistake.”

“I think you should give these handouts in linear [algebra]because it is often the first time students encounterabstract math / in-depth proofs.”

Introduction Strunk & White A mathematical version Student reactions Conclusion

Student reactions

Modern Algebra I,end-of-semester questionnaire

“Did the Elements of Style handout help you? If so, how?If it was not helpful, why not? How should I improve it?”

“I didn’t use them as reference tools personally.”

“To be honest I just didn’t use it much. It was good to see itthe first time but I don’t think I looked at it after that. I wouldstill hand it out though.”

Introduction Strunk & White A mathematical version Student reactions Conclusion

Student reactions

Summary:8 of the 11 students in class responded.6 positive responses2 neutral responses (“didn’t use it”)

Introduction Strunk & White A mathematical version Student reactions Conclusion

Benefits

Benefits of Elements of Style handout:Students are learning a new language;this gives them a reference manual.Rules are pithy, can be memorized.Encourages students to think of proof-writinglike the writing exercises of their other coursesSpeeds grading and gives better andmore consistent feedback—“sermon in a can”

Introduction Strunk & White A mathematical version Student reactions Conclusion

Future changes

Possible improvements for the future:Organize rules (perhaps by specificity).Give more examples, contrasting bad proofs with good.Add a glossary of words commonly misused in proofs.Seek more positive formulations of rules.Eventually spiral-bind it to give it more authority.But don’t make it so long that students won’t read it.

Introduction Strunk & White A mathematical version Student reactions Conclusion

Thank you!

Full list of rules

1 The burden of communication lies on you, not on yourreader.

2 Use English words.3 Use complete sentences.4 Know the difference between statements and objects.5 Don’t interchange = and⇒.6 Use whitespace.7 Use multiple sheets of paper.8 Use scratch paper.9 Show the logical connections among your sentences.

10 Say exactly what you mean.11 Don’t write anything unproven.12 Write strings of equalities (or inequalities) in the proper

order.

Full list of rules, cont.

13 Don’t beg the question.14 Don’t write the proof backwards.15 Be concise.16 Avoid weasel words.17 Introduce every symbol you use.18 Use a symbol to mean only one thing.19 Don’t “prove by example.”20 Write “Let x = . . . ,” not “Let · · · = x .”21 Make your counterexamples concrete and specific.22 Tell the reader what you’re proving.23 Shun pronouns, especially “it.”24 Don’t use undefined words.