[PPT] - Teaching Students to Communicate as Mathema3cians November 27, 2018 PowerPoint Presentation

SLIDE 1

Teaching Students to Communicate as Mathema3cians

November 27, 2018

Susan Ruff

Lecturer II Wri3ng, Rhetoric, and Professional Communica3on Department of Compara3ve Media Studies/Wri3ng MassachuseIs Ins3tute of Technology

SLIDE 2

Teaching Students to Communicate as Mathema3cians

November 27, 2018

Susan Ruff

Lecturer II Wri3ng, Rhetoric, and Professional Communica3on Department of Compara3ve Media Studies/Wri3ng MassachuseIs Ins3tute of Technology

15 communica3on- intensive subjects:

4 introductory-level

subjects (mostly proof wri3ng)

10 undergraduate

seminars

Project laboratory

in mathema3cs

SLIDE 3

Why have students communicate mathema/cs?

Wri3ng to enable assessment
Wri3ng to learn mathema3cs
Learning to write mathema3cs

SLIDE 4

Why have students communicate mathema/cs?

Wri3ng to enable assessment
Wri3ng to learn mathema3cs
Learning to write mathema3cs

Q: Why do you have students communicate mathema/cs?

SLIDE 5

Why have students communicate mathema/cs?

Wri3ng to enable assessment
Wri3ng to learn mathema3cs
Learning to write mathema3cs

Today’s focus: Learning to communicate effec3vely as mathema3cians

SLIDE 6

Why have students communicate mathema/cs?

Wri3ng to enable assessment
Wri3ng to learn mathema3cs
Learning to write mathema3cs

Today’s focus: Learning to communicate effec3vely as mathema3cians Q: What does it mean to communicate effec/vely as a mathema/cian?

SLIDE 7

Entering a community of practice requires knowledge of various domains

Communica/ng effec/vely as a mathema/cian requires command

f various domains.

SLIDE 8

Entering a community of practice requires knowledge of various domains

Communica/ng effec/vely as a mathema/cian requires command

f various domains.

Genre system for research Funding proposal Notebook Mee3ngs and emails with collaborators Colloqium talk

Conf. presenta3on

arXiv preprint Referee report Journal ar3cle Expository ar3cle

SLIDE 9

Entering a community of practice requires knowledge of various domains

Communica/ng effec/vely as a mathema/cian requires command

f various domains.

Q: Which domains challenge your students most?

SLIDE 10

How can students learn to communicate effec/vely as mathema/cians?

? ? ? ?

Students Community of Mathematicians

SLIDE 11

How can students learn to communicate effec/vely as mathema/cians? Q: How did you learn to communicate as a mathema/cian?

? ? ? ?

Students Community of Mathematicians

SLIDE 12

How can students learn to communicate effec/vely as mathema/cians? Appren3ces learn via legi3mate peripheral par3cipa3on in the community of prac3ce… (Lave & Wenger)

? ? ? ?

Students Community of Mathematicians

SLIDE 13

How can students learn to communicate effec/vely as mathema/cians? Appren3ces learn via legi3mate peripheral par3cipa3on in the community of prac3ce. (Lave & Wenger) My takeaway: as much as feasible, have students communicate as mathema(cians …and “read.”

? ? ? ?

Students Community of Mathematicians

SLIDE 14

Project laboratory in mathema/cs Teams of 3 research open-ended problems Three projects during term Write a paper for each project Present one project to classmates

Image courtesy of MIT Open Courseware

SLIDE 15

Undergraduate seminars Students lecture to each other following a book or on topics of interest. Write expository paper.

SLIDE 16

arXiv:1605.09223v1 [math.CO] 30 May 2016

O N L A R G E S U B S E T S O F F

n q

W I T H N O T H R E E

T

E R M A R I T H M E T I C P R O G R E S S I O N

JORDAN S. ELLENBERG AND DION GIJSWIJT

Abstract. In this note, we show that the method of Croot, Lev, and Pach can be used to

bound the size of a subset of Fn q with no three terms in arithmetic progression by cn with c < q. For q = 3, the problem of finding the largest subset of Fn 3 with no three terms in arithmetic progression is called the cap problem. Previously the best known upper bound for the affine cap problem, due to Bateman and Katz [BK12], was on order n−1−3n.

T h e p r

b

l e m

f

fi n d i n g l a r g e s u b s e t s

f

a n a b e l i a n g r

u

p G w i t h n

t

h r e e

t

e r m a r i t h m e t i c p r

g

r e s s i

n

,

r
f

fi n d i n g u p p e r b

u

n d s f

r

t h e s i z e

f

s u c h a s u b s e t , h a s a l

n

g h i s t

r

y i n n u m b e r t h e

r

y . T h e m

s

t i n t e n s e a t t e n t i

n

h a s c e n t e r e d

n

t h e c a s e s w h e r e G i s a c y c l i c g r

u

p Z / N Z

r

a v e c t

r

s p a c e ( Z / 3 Z )

n

, w h i c h a r e i n s

m

e s e n s e t h e e x t r e m e s i t u a t i

n

s . W e d e n

t

e b y r

3

( G ) t h e m a x i m a l s i z e

f

a s u b s e t

f

G w i t h n

t

h r e e

t

e r m a r i t h m e t i c p r

g

r e s s i

n

. T h e f a c t t h a t r

3

( ( Z / 3 Z )

n

) i s

(

3

n

) w a s fi r s t p r

v

e d b y B r

w

n a n d B u h l e r [ B B 8 2 ] , w h i c h w a s i m p r

v

e d t

O

( 3

n

/ n ) b y M e s h u l a m [ M e s 9 5 ] . T h e b e s t k n

w

n u p p e r b

u

n d , O ( 3

n

/ n

1 +

)

, i s d u e t

B

a t e m a n a n d K a t z [ B K 1 2 ] . T h e b e s t l

w

e r b

u

n d , b y c

n

t r a s t , i s a r

u

n d 2 . 2

n

[ E d e 4 ] . T h e p r

b

l e m

f

a r i t h m e t i c p r

g

r e s s i

n

s i n ( Z / 3 Z )

n

h a s s

m

e t i m e s b e e n s e e n a s a m

d

e l f

r

t h e c

r

r e s p

n

d i n g p r

b

l e m i n Z / N Z . W e k n

w

( f

r

i n s t a n c e , b y a c

n

s t r u c t i

n
f

B e h r e n d [ B e h 4 6 ] ) t h a t r

3

( Z / N Z ) g r

w

s m

r

e q u i c k l y t h a n N

1 −

f
r

e v e r y

>

. T h u s i t i s n a t u r a l t

a

s k w h e t h e r r

3

( ( Z / 3 Z )

n

) g r

w

s m

r

e q u i c k l y t h a n ( 3 −

)

n

f

r

e v e r y

>

. I n g e n e r a l , t h e r e h a s b e e n n

c
n

s e n s u s

n

w h a t t h e a n s w e r t

t

h i s q u e s t i

n

s h

u

l d b e . I n t h e p r e s e n t p a p e r w e s e t t l e t h e q u e s t i

n

, p r

v

i n g t h a t f

r

a l l

d

d p r i m e s p , r

3

( ( Z / p Z )

n

)

1 / n

i s b

u

n d e d a w a y f r

m

p a s n g r

w

s . T h e m a i n t

l

u s e d h e r e i s t h e p

l

y n

m

i a l m e t h

d

, i n p a r t i c u l a r t h e u s e

f

t h e p

l

y n

m

i a l m e t h

d

d e v e l

p

e d i n t h e b r e a k t h r

u

g h p a p e r

f

C r

t

, L e v , a n d P a c h [ C L P 1 6 ] , w h i c h d r a s t i c a l l y i m p r

v

e d t h e b e s t k n

w

n u p p e r b

u

n d s f

r

r

3

( ( Z / 4 Z )

n

) . I n t h i s c a s e , t h e y s h

w

t h a t a s u b s e t

f

G w i t h n

t

h r e e

t

e r m a r i t h m e t i c p r

g

r e s s i

n

h a s s i z e a t m

s

t c

n

f

r

s

m

e c < 4 . I n t h e p r e s e n t p a p e r , w e s h

w

t h a t t h e i d e a s

f

t h e i r p a p e r c a n b e e x t e n d e d t

v

e c t

r

s p a c e s

v

e r a g e n e r a l fi n i t e fi e l d . R e m a r k 1 . T h e i d e a s

f

t h i s p a p e r w e r e d e v e l

p

e d i n d e p e n d e n t l y a n d e s s e n t i a l l y s i m u l t a n e

u

s l y b y t h e t w

a

u t h

r

s . S i n c e t h e a r g u m e n t s

f
u

r t w

p

a p e r s w e r e e s s e n t i a l l y i d e n t i c a l , w e p r e s e n t t h e m a s j

i

n t w

r

k . W e b e g i n w i t h a s l i g h t g e n e r a l i z a t i

n
f

L e m m a 1

f

[ C L P 1 6 ] . L e t F

q

b e a fi n i t e fi e l d a n d l e t n b e a p

s

i t i v e i n t e g e r . L e t M

n

b e t h e s e t

f

m

n
m

i a l s i n x

1

, . . . , x

n

w h

s

e d e g r e e i n e a c h v a r i a b l e i s a t m

s

t q − 1 , a n d l e t S

n

b e t h e F

q

v

e c t

r

s p a c e t h e y s p a n .

Date: 27 May 2016. The first author is supported by NSF Grant DMS-1402620 and a Guggenheim Fellowship. We thank Terry Tao, Tim Gowers, and Seva Lev for useful discussions during the production of this paper. 1

Introductory classes

?

Large faculty-led lectures on topic (e.g., discrete mathema3cs) Smaller, more ac3ve “recita3ons” Wri3ng = proving assigned statements Legi3mate peripheral par3cipa3on? Valuable? Example: teaching “audience” via hypothe3cal scenarios vs. published versions of same result: research ar3cle, Quanta ar3cle, blog post

SLIDE 17

arXiv:1605.09223v1 [math.CO] 30 May 2016

O N L A R G E S U B S E T S O F F

n q

W I T H N O T H R E E

T

E R M A R I T H M E T I C P R O G R E S S I O N

JORDAN S. ELLENBERG AND DION GIJSWIJT

Abstract. In this note, we show that the method of Croot, Lev, and Pach can be used to

bound the size of a subset of Fn q with no three terms in arithmetic progression by cn with c < q. For q = 3, the problem of finding the largest subset of Fn 3 with no three terms in arithmetic progression is called the cap problem. Previously the best known upper bound for the affine cap problem, due to Bateman and Katz [BK12], was on order n−1−3n.

T h e p r

b

l e m

f

fi n d i n g l a r g e s u b s e t s

f

a n a b e l i a n g r

u

p G w i t h n

t

h r e e

t

e r m a r i t h m e t i c p r

g

r e s s i

n

,

r
f

fi n d i n g u p p e r b

u

n d s f

r

t h e s i z e

f

s u c h a s u b s e t , h a s a l

n

g h i s t

r

y i n n u m b e r t h e

r

y . T h e m

s

t i n t e n s e a t t e n t i

n

h a s c e n t e r e d

n

t h e c a s e s w h e r e G i s a c y c l i c g r

u

p Z / N Z

r

a v e c t

r

s p a c e ( Z / 3 Z )

n

, w h i c h a r e i n s

m

e s e n s e t h e e x t r e m e s i t u a t i

n

s . W e d e n

t

e b y r

3

( G ) t h e m a x i m a l s i z e

f

a s u b s e t

f

G w i t h n

t

h r e e

t

e r m a r i t h m e t i c p r

g

r e s s i

n

. T h e f a c t t h a t r

3

( ( Z / 3 Z )

n

) i s

(

3

n

) w a s fi r s t p r

v

e d b y B r

w

n a n d B u h l e r [ B B 8 2 ] , w h i c h w a s i m p r

v

e d t

O

( 3

n

/ n ) b y M e s h u l a m [ M e s 9 5 ] . T h e b e s t k n

w

n u p p e r b

u

n d , O ( 3

n

/ n

1 +

)

, i s d u e t

B

a t e m a n a n d K a t z [ B K 1 2 ] . T h e b e s t l

w

e r b

u

n d , b y c

n

t r a s t , i s a r

u

n d 2 . 2

n

[ E d e 4 ] . T h e p r

b

l e m

f

a r i t h m e t i c p r

g

r e s s i

n

s i n ( Z / 3 Z )

n

h a s s

m

e t i m e s b e e n s e e n a s a m

d

e l f

r

t h e c

r

r e s p

n

d i n g p r

b

l e m i n Z / N Z . W e k n

w

( f

r

i n s t a n c e , b y a c

n

s t r u c t i

n
f

B e h r e n d [ B e h 4 6 ] ) t h a t r

3

( Z / N Z ) g r

w

s m

r

e q u i c k l y t h a n N

1 −

f
r

e v e r y

>

. T h u s i t i s n a t u r a l t

a

s k w h e t h e r r

3

( ( Z / 3 Z )

n

) g r

w

s m

r

e q u i c k l y t h a n ( 3 −

)

n

f

r

e v e r y

>

. I n g e n e r a l , t h e r e h a s b e e n n

c
n

s e n s u s

n

w h a t t h e a n s w e r t

t

h i s q u e s t i

n

s h

u

l d b e . I n t h e p r e s e n t p a p e r w e s e t t l e t h e q u e s t i

n

, p r

v

i n g t h a t f

r

a l l

d

d p r i m e s p , r

3

( ( Z / p Z )

n

)

1 / n

i s b

u

n d e d a w a y f r

m

p a s n g r

w

s . T h e m a i n t

l

u s e d h e r e i s t h e p

l

y n

m

i a l m e t h

d

, i n p a r t i c u l a r t h e u s e

f

t h e p

l

y n

m

i a l m e t h

d

d e v e l

p

e d i n t h e b r e a k t h r

u

g h p a p e r

f

C r

t

, L e v , a n d P a c h [ C L P 1 6 ] , w h i c h d r a s t i c a l l y i m p r

v

e d t h e b e s t k n

w

n u p p e r b

u

n d s f

r

r

3

( ( Z / 4 Z )

n

) . I n t h i s c a s e , t h e y s h

w

t h a t a s u b s e t

f

G w i t h n

t

h r e e

t

e r m a r i t h m e t i c p r

g

r e s s i

n

h a s s i z e a t m

s

t c

n

f

r

s

m

e c < 4 . I n t h e p r e s e n t p a p e r , w e s h

w

t h a t t h e i d e a s

f

t h e i r p a p e r c a n b e e x t e n d e d t

v

e c t

r

s p a c e s

v

e r a g e n e r a l fi n i t e fi e l d . R e m a r k 1 . T h e i d e a s

f

t h i s p a p e r w e r e d e v e l

p

e d i n d e p e n d e n t l y a n d e s s e n t i a l l y s i m u l t a n e

u

s l y b y t h e t w

a

u t h

r

s . S i n c e t h e a r g u m e n t s

f
u

r t w

p

a p e r s w e r e e s s e n t i a l l y i d e n t i c a l , w e p r e s e n t t h e m a s j

i

n t w

r

k . W e b e g i n w i t h a s l i g h t g e n e r a l i z a t i

n
f

L e m m a 1

f

[ C L P 1 6 ] . L e t F

q

b e a fi n i t e fi e l d a n d l e t n b e a p

s

i t i v e i n t e g e r . L e t M

n

b e t h e s e t

f

m

n
m

i a l s i n x

1

, . . . , x

n

w h

s

e d e g r e e i n e a c h v a r i a b l e i s a t m

s

t q − 1 , a n d l e t S

n

b e t h e F

q

v

e c t

r

s p a c e t h e y s p a n .

Date: 27 May 2016. The first author is supported by NSF Grant DMS-1402620 and a Guggenheim Fellowship. We thank Terry Tao, Tim Gowers, and Seva Lev for useful discussions during the production of this paper. 1

Introductory classes

Large faculty-led lectures on topic (e.g., discrete mathema3cs) Smaller, more ac3ve “recita3ons” Wri3ng = proving assigned statements Legi3mate peripheral par3cipa3on? Valuable? Example: teaching “audience” via hypothe3cal scenarios vs. published versions of same result: research ar3cle, Quanta ar3cle, blog post Q: How do/could you demys/fy for your students what it means to communicate as a mathema/cian?

SLIDE 18

How do these concepts inform teaching?

? ? ? ?

Students Community of Mathematicians

SLIDE 19

18.642 recitations address each domain

10/20 Rhetorical context in industry (Vasily Strela)  11/17 Reproducible research 9/8 linear algebra 9/15 fixed income  assets 9/22 forward rate  agreements 10/13 quiz review 10/27 regulation 11/3 statistics 9/8 Sweave 9/22 Tools for figures 12/1 Informal peer   critique 10/27 Model documentation   (Michal Kowalik) 11/3 Statistical analyses 9/8 Context/Summary 9/15 Development 9/22 Figure Design 10/6 Flow 12/1 Elevator Pitch

Designing Curriculum

Example is from Topics in Math with Applica3ons to Finance

SLIDE 20

Teaching Paper Wri/ng Reading Assignment + Discussion

Choose a published paper that reinforces course content. What is the purpose of the paper? What strategies does the author use to – convince readers? – help readers understand? – interest readers? Which conven/ons noted on “Maximum Overhang” does the author follow? Are these choices effec/ve? Summarize own process.

M a x i m u m O v e r h a n g

M i k e P a t e r s

n

, Y u v a l P e r e s , M i k k e l T h

r

u p , P e t e r W i n k l e r , a n d U r i Z w i c k

1. INTRODUCTION. How far can a stack of n identical blocks be made to hang
ver the edge of a table? The question has a long history and the answer was widely

believed to be of order log n. Recently, Paterson and Zwick constructed n-block stacks with overhangs of order n1/3, exponentially better than previously thought possible. We show here that order n1/3 is indeed best possible, resolving the long-standing overhang problem up to a constant factor. This problem appears in physics and engineering textbooks from as early as the mid-19th century (see, e.g., [15], [20], [13]). The problem was apparently first brought to the attention of the mathematical community in 1923 when J. G. Coffin [2] posed it in the “Problems and Solutions” section of this MONTHLY; no solution was presented

there. The problem recurred from time to time over subsequent years, e.g., [17, 18,

19, 12, 6, 5, 7, 8, 1, 4, 9, 10], achieving much added notoriety from its appearance in 1964 in Martin Gardner’s “Mathematical Games” column of Scientific American [7] and in [8, Limits of Infinite Series, p. 167].

1 11 12 ≃ 0.916667 25 24 ≃ 1.04167 15−4 √ 2 8 ≃ 1.16789 Figure 1. Optimal stacks with 3 and 4 blocks, compared to the corresponding harmonic stacks. The 4-block solution is from [1]. Like the harmonic stacks it can be made stable by minute displacements.

Most of the references mentioned above describe the now-classical harmonic stacks in which n unit-length blocks are placed one on top of the other, with the ith block from the top extending by

1 2i beyond the block below it. The overhang achieved by

such stacks is 1

2 Hn = 1 2

n

i=1 1 i ∼ 1 2 ln n. The cases n = 3 and n = 4 are illustrated at

the top of Figure 1 above, and the cases n = 20 and n = 30 are shown in the back- ground of Figure 2. Verifying that harmonic stacks are balanced and can be made stable (see definitions in the next section) by minute displacements is an easy exercise. (This is the form in which the problem appears in [15, pp. 140–141], [20, p. 183], and [13, p. 341].) Harmonic stacks show that arbitrarily large overhangs can be achieved if sufficiently many blocks are available. They have been used extensively as an introduction to recurrence relations, the harmonic series, and simple optimization problems (see, e.g., [9]).

doi:10.4169/000298909X474855

November 2009]

MAXIMUM OVERHANG

763 An article's introduction should 1) indicate the article's main result(s) 2) indicate why the results are important--this is usually accomplished by summarizing how the results further research within the field 3) be worded to be understood by the target audience while remaining relatively nontechnical 4) preview the paper's structure. The first and third goals often conflict with each other. This paper does a very nice

f addressing goals 1-3 within

the first paragraph, so readers immediately know the focus relevance of the paper. To indicate how a paper's results further research within the field, the introduction usually includes a literature

review. A well-written review

gives readers confidence that the authors are familiar with the relevant literature. Furthermore, because the state of the field constantly evolves, the literature reviews in introductions often provide the primary means for those new to a field (e.g., graduate students) to get to know the field. But the primary purpose of the review is to indicate the significance of the paper's results, so only directly relevant literature should be included in the review. The literature review can also be used as a vehicle for introducing concepts that will be needed to understand the statement or the significance of the paper's main result(s). literature review Citation styles vary by journal, but the style shown here is common in mathematics. Including the page number helps readers to find information if the source is a book. BibTeX can handle the citation style for you and is particularly useful if you have many references or are writing multiple papers. The introduction should remain relatively nontechnical, so delay formal definitions to the body of the paper if possible. In 2011, this paper received the David P. Robbins Prize, one of the MAA's Writing Awards. The annotations presented here provide tips to students for how to write a mathematics paper. Annotations by S. Ruff. This annotated paper bears Creative Commons license Attribution-NonCommercial-ShareAlike with attribution "MAA Mathematical Communication (mathcomm.org)"

R e p r i n t e d w i t h c

r

r e c t i

n

s f r

m

T h e B e l l S y s t e m T e c h n i c a l J

u

r n a l , V

l

. 2 7 , p p . 3 7 9 – 4 2 3 , 6 2 3 – 6 5 6 , J u l y , O c t

b

e r , 1 9 4 8 .

A M a t h e m a t i c a l T h e

r

y

f

C

m

m u n i c a t i

n

By C. E. SHANNON INTRODUCTION HE recent development of various methods of modulation such as PCM and PPM which exchange bandwidth for signal-to-noise ratio has intensified the interest in a general theory of communication. A such a theory is contained in the important papers of Nyquist1 and Hartley2 on this subject. In the will extend the theory to include a number of new factors, in particular the effect of noise vings possible due to the statistical structure of the original message and due to the the information. unication is that of reproducing at one point either exactly or ap-

nt. Frequently the messages have meaning; that is they refer

tain physical or conceptual entities. These semantic

lem. The significant aspect is that the actual

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Mathcomm.org

SLIDE 21

Teaching Paper Wri/ng Reading Assignment + Discussion

Choose a published paper that reinforces course content. What is the purpose of the paper? What strategies does the author use to – convince readers? – help readers understand? – interest readers? Which conven/ons noted on “Maximum Overhang” does the author follow? Are these choices effec/ve? Summarize own process.

M a x i m u m O v e r h a n g

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, Y u v a l P e r e s , M i k k e l T h

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1. INTRODUCTION. How far can a stack of n identical blocks be made to hang
ver the edge of a table? The question has a long history and the answer was widely

believed to be of order log n. Recently, Paterson and Zwick constructed n-block stacks with overhangs of order n1/3, exponentially better than previously thought possible. We show here that order n1/3 is indeed best possible, resolving the long-standing overhang problem up to a constant factor. This problem appears in physics and engineering textbooks from as early as the mid-19th century (see, e.g., [15], [20], [13]). The problem was apparently first brought to the attention of the mathematical community in 1923 when J. G. Coffin [2] posed it in the “Problems and Solutions” section of this MONTHLY; no solution was presented

there. The problem recurred from time to time over subsequent years, e.g., [17, 18,

19, 12, 6, 5, 7, 8, 1, 4, 9, 10], achieving much added notoriety from its appearance in 1964 in Martin Gardner’s “Mathematical Games” column of Scientific American [7] and in [8, Limits of Infinite Series, p. 167].

1 11 12 ≃ 0.916667 25 24 ≃ 1.04167 15−4 √ 2 8 ≃ 1.16789 Figure 1. Optimal stacks with 3 and 4 blocks, compared to the corresponding harmonic stacks. The 4-block solution is from [1]. Like the harmonic stacks it can be made stable by minute displacements.

Most of the references mentioned above describe the now-classical harmonic stacks in which n unit-length blocks are placed one on top of the other, with the ith block from the top extending by

1 2i beyond the block below it. The overhang achieved by

such stacks is 1

2 Hn = 1 2

n

i=1 1 i ∼ 1 2 ln n. The cases n = 3 and n = 4 are illustrated at

the top of Figure 1 above, and the cases n = 20 and n = 30 are shown in the back- ground of Figure 2. Verifying that harmonic stacks are balanced and can be made stable (see definitions in the next section) by minute displacements is an easy exercise. (This is the form in which the problem appears in [15, pp. 140–141], [20, p. 183], and [13, p. 341].) Harmonic stacks show that arbitrarily large overhangs can be achieved if sufficiently many blocks are available. They have been used extensively as an introduction to recurrence relations, the harmonic series, and simple optimization problems (see, e.g., [9]).

doi:10.4169/000298909X474855

November 2009]

MAXIMUM OVERHANG

763 An article's introduction should 1) indicate the article's main result(s) 2) indicate why the results are important--this is usually accomplished by summarizing how the results further research within the field 3) be worded to be understood by the target audience while remaining relatively nontechnical 4) preview the paper's structure. The first and third goals often conflict with each other. This paper does a very nice

f addressing goals 1-3 within

the first paragraph, so readers immediately know the focus relevance of the paper. To indicate how a paper's results further research within the field, the introduction usually includes a literature

review. A well-written review

gives readers confidence that the authors are familiar with the relevant literature. Furthermore, because the state of the field constantly evolves, the literature reviews in introductions often provide the primary means for those new to a field (e.g., graduate students) to get to know the field. But the primary purpose of the review is to indicate the significance of the paper's results, so only directly relevant literature should be included in the review. The literature review can also be used as a vehicle for introducing concepts that will be needed to understand the statement or the significance of the paper's main result(s). literature review Citation styles vary by journal, but the style shown here is common in mathematics. Including the page number helps readers to find information if the source is a book. BibTeX can handle the citation style for you and is particularly useful if you have many references or are writing multiple papers. The introduction should remain relatively nontechnical, so delay formal definitions to the body of the paper if possible. In 2011, this paper received the David P. Robbins Prize, one of the MAA's Writing Awards. The annotations presented here provide tips to students for how to write a mathematics paper. Annotations by S. Ruff. This annotated paper bears Creative Commons license Attribution-NonCommercial-ShareAlike with attribution "MAA Mathematical Communication (mathcomm.org)"

Mathcomm.org

SLIDE 22

Designing Assignments: Seminar presenta/ons + associated genre system

Process is scaffolded into assignment sequence:

Content review with course lead
Prac/ce presenta/on with me
Write presenta/on abstract for classmates
Present to classmates
Classmates provide feedback
Write lecture notes for classmates

SLIDE 23

Providing Feedback

It’s conven/onal to write the introduc/on as though readers haven’t seen the abstract. Add to your edi/ng checklist: check signs throughout I don’t understand: are you working leU-to-right or right-to leU? You could write the point of each paragraph in the margin to create a “retroac/ve outline” that’s likely to reveal ways to restructure. Has your audience seen this approach before? If not, will they need to see a concrete example,

r is it obvious enough that cita/on is sufficient?

Why can we swap the integrals here?

We’re teachers not editors

You keep making sign errors. Discussing this baby case does a good job of achieving the purposes of an introduc/on while avoiding technicali/es. Insert “right-to-leU”

SLIDE 24

Grading

Grading Rubric for 18.821 Papers (20 points total)

Spring, 2018 Mathematical Correctness and Vision (10) 9–10 The students discovered something remarkable and provided exceptionally elegant explanations of the phenomena they identified. 7–8 The students discovered something substantial and explained convincingly the phenomena they found (i.e., proofs are rigorous; conjectures are supported with convincing evidence). 5–7 The students made substantial progress and offered explanations for the phenomena they identified (i.e., claims are rigorously stated and support goes beyond a few specific examples). 3–5 The students gave a good expository description of the problem and of the most interesting aspects of the phenomena they found (e.g., conjectures are stated). 1–3 The students described the problem and found some immediately apparent aspects of it. Exposition (6) 6 The paper is exceptionally interesting and engaging. 5 The paper is easy to read and understand and is well suited to the target audience (peers

f the authors). The paper is consistent and cohesive (not just 3 parts pasted together); the

paper is focused and structured and the structure is communicated to readers; new ideas are introduced efficiently and with proper motivation; displays and examples are well chosen to aid understanding; mathematical language and notation are used appropriately; citations clearly acknowledge any sources used; writing is accurate, appropriately concise, and carefully proofread. 4 Many of the criteria for a grade of 5 are met. The paper is sufficiently clear that peers can easily discern what was intended whenever expository roughness is encountered. 3–4 Peers must expend some effort to discern what was intended when expository roughness is encountered. 1–3 Substantial effort is needed to discern what was intended. Research and Writing Process (4) All teammates contributed substantively to the research and to the writing and attended all meetings. The draft was complete and carefully written. The revision took into account but was not limited to the feedback of course staff and of teammates.

Process

Genre/ Discourse

Rhetoric Content Mathcomm.org

SLIDE 25

Teaching Students to Communicate as Mathema/cians

Demys3fy communica3on of mathema3cians, e.g., via legi3mate peripheral par3cipa3on in a community of mathema3cians …and reading. The knowledge domains can inform design of curriculum, instruc3on, assignments, feedback, and grading. Some Resources MAA Mathema(cal Communica(on mathcomm.org Bahls, Student Wri(ng in the Quan(ta(ve Disciplines Gopen & Swan, “The Science of Scien3fic Wri3ng” American Scien(st, 1990. Wolfe, Team Wri(ng: A Guide to Working in Groups

SLIDE 26

Thank you

SLIDE 27

CraUing feedback takes /me

priori3ze (for yourself & students)
give class-wide feedback
meet with students
be kind (to yourself & students)
get help

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Strong introduction. Gets right to the point. Good abstract. Student Name removed This is a strong start, and you’ve clearly put a great deal of care into the mathematical content of this

paper. I particularly appreciate that these proofs, which involve some fairly messy algebra, are presented

in a way that is accessible to the reader. Your formatting is curious; are you using the section and theorem LaTeX environments to set apart things clearly? I also see a need for more guiding text to clarify exactly where sections of proofs begin and end, and what strategies you will be using. Please let me know if you have any questions. I’m happy to meet to discuss your writing. See note P4.

(wri3ng center) wri3ng center If audience is peers, peers can comment on effec3veness for audience tutors