# of Arnold Ross and his Program.

by Daniel Shapiro

Arnold Ephraim Chaimovitch was born in Chicago on August 24, 1906. In 1909 Arnold and his mother moved home to Odessa in the Ukraine. As a talented teenager he worked closely with Professor Samuil Shatunovsky, a math professor at Odessa University. Since he was an American citizen, Chaimovitch was allowed to move to Chicago around 1922. After a year working and learning English, he attended the University of Chicago where we was deeply influenced by the teaching methods of E. H. Moore. He continued into graduate study at Chicago and did research under the direction of L. E. Dickson, completing his PhD in 1931. Around 1928 he changed his surname to “Ross” since Americans had trouble pronouncing “Chaimovitch.”

** **In 1931-1933 Ross worked as a postdoc at Caltech with E. T. Bell. He returned to Chicago and became involved with People’s Junior College for a couple of years before taking a position at St. Louis University in 1935. During World War II he worked as a research mathematician in a team developing proximity fuzes for the US Navy. In 1946 he was hired as the head of the mathematics department at the University of Notre Dame. Ross soon established a summer program for in-service high school teachers, showing them how to implement the style of “discovery” teaching that Ross had experienced with Shatunovsky and Moore. In 1957 he invited some high school students to be a model class for that summer program, and within a few years, Ross transformed his program to focus more on the young students rather than teachers.

Arnold Ross left Notre Dame in 1963 to become chair of the department of mathematics at the Ohio State University. In the summer of 1964 his program for high school students also moved to OSU and has thrived there every summer since then, except for four sessions at the University of Chicago (1975 – 1978). Ross retired from Ohio State in 1976 but continued to lead his program every summer until 2000, when he stepped down after suffering a stroke at age 93. Since then, Daniel Shapiro has been director and has tried to maintain the Ross teaching style.

From the beginning, Dr. Ross used Number Theory as the vehicle for his summer teaching. As Ross wrote, “Number theory proper and its rich environment are a fertile ground for exploration and are a valuable source for nontrivial but accessible problems.” [Footnote: From his response to the AMS 1998 Citation for Public Service.] Ross Program participants work on intriguing problems about numbers and share results of their observations and explorations with others. First-year students live in a college dormitory next to returning students and counselors. Participants in this multi-level program form a community of young scholars in which a vivid exchange of ideas between newcomers and program veterans enriches their lives. This method exemplifies Ross’s motto:

“Think deeply about simple things.”

Those challenging summer experiences provide long-lasting intellectual benefits. Over the years alumni have submitted reports that verify the value of this sort of training.

In 2002 we posted a short biography of Dr. Ross, along with a page of photographs of Arnold Ross at various ages. Some published articles about Ross and his summer program are mentioned at the end of this web page.

# More detailed history.

The next section is a narrative that includes memories of the stories that Arnold Ross told about himself and about the origins of his summer program. *We welcome your comments, suggestions, and corrections.*

### Years in Russia.

Arnold Ephraim Chaimovitch was born in Chicago in 1906 to Isaac Chaimovitsch and Clara Greenberg, immigrants from Odessa, Russia, who married in Chicago on May 3, 1905. [N.B. English spelling differs between Arnold’s and Isaac’s documents.] His mother returned to Odessa with little “Noli” in 1909, moving back to live with her parents. Arnold’s uncle was a well-respected doctor who helped support the family, but there were major social and economic troubles during those years, centering on World War I, the Russian Revolution, and the start of the USSR.

Since the Soviet economy was so bad in the early 1920s, even research professors took on tutoring work. Professor Samuil Shatunovsky, a distinguished mathematician at Odessa University, was hired to train the doctor’s son (Arnold’s cousin), and Arnold was part of a small group of local boys eager to learn some mathematics from the professor. Since no cash was available, Sahtunovsky was paid with hard candy, a standard replacement for currency at that time.

One of the other boys in that group was Felix Gantmacher (or Feliks Gantmakher, nicknamed Fele). He later became a distinguished Soviet mathematician who wrote a classic treatise on linear algebra.

Ross seldom mentioned his Jewish roots, but he did tell one story. In grade school he saw that Jewish boys with Yiddish accents were harassed and bullied. This motivated Arnold to concentrate on learning to speak perfect Russian, with no accent, even though his family spoke Yiddish at home. He succeeded in avoiding the typical harassments.

### Leaving Soviet Russia.

Getting permission to leave Soviet Russia in 1922 was not easy. Native born Russians were not allowed to leave the country, but Arnold could claim American citizenship because he was born in Chicago. Since the United States had no diplomatic relations with the new USSR, Arnold could not get an American passport in Odessa. His father had arranged for Arnold’s passport to be available at the US consulate in Constantinople (now Istanbul), across the Black Sea. The Soviet bureaucrat in the Odessa mayor’s office refused to give Arnold permission to leave without that passport. Ross often told us the story of how he went to that bureaucrat’s office every day, repeatedly asking for permission to leave. After many days of doing this, that official finally called 16-year-old Arnold into the office, told him to keep quiet about this situation, and gave him a document permitting him to leave.

Arnold got a ticket on a ship to Constantinople but couldn’t travel as scheduled because of some illness … and that ship sank! He traveled some weeks later, and arrived in Constantinople. While walking from the dock to the American consulate, he was bothered by a whirling dervish who followed him for several blocks. [Ross never mentioned further details about that religious sect.] He got the passport, sailed to New York, and took a train to Chicago to live with his father.

### In Chicago.

On his first night there he met Bertha Halley Horecker, nicknamed Bee, the daughter of some family friends. (She is related to the Edmund Halley of comet fame. ) Bee was well educated and loved to sing classical music. The story is that they quickly fell in love, but postponed marriage until 1931, soon after he earned his PhD.

His engineer father told Arnold that he would pay for his college education provided he studied Engineering, but not if he studied Math. Arnold soon moved away from his father, finding a room in a boarding house. That year he took a course in English at Stevens Institute and supported himself by working in a book bindery.

Shatunovsky had given Arnold a letter of introduction to E. H. Moore, a prominent math professor at the University of Chicago. With Moore’s help, Arnold Chaimovitch enrolled there as an undergraduate.

Arnold greatly admired Professor Moore, taking math classes from him and appreciating his Socratic teaching style. In later years, Ross repeatedly emphasized that he was inspired by the “inquiry-based” teaching methods of E.H. Moore, not by the more extreme methods of R.L. Moore.

In 1928 Arnold decided to change his surname because many Americans had difficulty pronouncing Chaimovitch. He and his friend Gordon Pall decided on “Ross”. Dr. Ross told us this story with a laugh, apparently indicating that this name was a joke of some kind. No further details were explained. The date of this name change was probably during 1928, as evidenced by the introduction to Ross’s translation of Nazimoff’s book on Elliptic Functions.

After getting an undergraduate degree, Ross decided to work on his PhD under the direction of L. E. Dickson at Chicago, since Moore was near retirement. Dickson’s 3-volume treatise on the history of number theory had been recently completed. As a Research Assistant in the University of Chicago, Ross helped revise three chapters and read proof sheets for Dickson’s next book, *Studies in the Theory of Numbers,* published in 1930.

Ross took some courses from Professor G.A. Bliss who also helped him navigate through college. Gordon Pall and A. Adrian Albert were fellow math graduate students, although a few years older. Ross was close friends with Pall, but viewed Albert as overly formal and pretentious. However, Bee and Albert’s wife Frieda were close friends.

### After graduating.

Ross married Bee soon after graduating in 1931. They were very close for more than fifty years, but did not have children.

Ross spent 1931 – 1933 at Caltech as a postdoc working with E. T. Bell. He received partial support from a National Research Fellowship. When asked, Ross said that he greatly admired Bell but gave no further details of his time in California. Arnold and Bee returned to Chicago in the mid-1930s when paying jobs were scarce. He helped create People’s Junior College in Chicago, an experimental venture put together by a group of unemployed postdocs representing English, foreign languages, sciences, and economics. Ross taught math classes to economics students who paid for their schooling in trade (food or labor) rather than cash. This institution lasted only a few years, and the name suggests that some left-wing political sympathies were involved.

### Saint Louis University.

Ross began working at Saint Louis University, a catholic, Jesuit college. He was an Instructor in 1935, Assistant Professor in 1939, Associate Professor in 1944, and left St Louis for Notre Dame in 1946. Ross directed two PhD students at SLU: Margaret Frances Willerding (1947) and L. Thomas Matthews (1948), graduating a year or two after Ross had departed. Both of their dissertations involved representations of integral quadratic forms. In 2001 I telephoned Professor Willerding, a retired professor at San Diego State University, and asked about her experiences with Arnold Ross. She said that he took a deep personal interest in all of his students, and [to her annoyance] he even tried to arrange a marriage for her to his friend Eugene Guth. She was a bit embarrassed to admit that she thought that Ross was a terrible teacher, giving unorganized lectures and never keeping to the topic. Willerding also remarked that Bee Ross accompanied Arnold to all of his lectures, sitting quietly in the audience. (Twenty years later, in the 1960s at Ohio State, Bee never attended Arnold’s lectures.)

### War Work.

During the war years, Ross took leaves from Saint Louis University and worked for the U.S. Navy doing applied mathematical work associated with proximity fuzes. Ross met many other mathematicians doing war work during those years, and built an extensive network of mathematical friends and acquaintances.

### Notre Dame.

The story is that Ross was a close friend of Notre Dame physicist Eugene Guth, who urged university president Kavanaugh to hire Ross to succeed Karl Menger as Head of the Math Department. In 1946 Ross went to Notre Dame as department head and within a year he started a summer program for in-service high school math teachers. Creating this program took some effort because most of the math teachers were women, and no females were allowed on the Notre Dame campus in those days. Ross was able to arrange a special dispensation to allow women to attend classes during those summer weeks. The teacher program grew larger with support from the National Science Foundation, peaking at more than 200 participants for the seven-week session. Ross prepared some reports on the success of this program for teachers. Here are samples from **1959** and **1963**. That 1959 report mentions his experiments in using closed-circuit television as a teaching tool.

As department head, Ross ruled with a sure hand and had strong opinions about what issues were important. Sometimes he made decisions without following the preferences of the faculty. When one of his projects didn’t get approval from the dean, Ross would do everything he could to get around the dean’s decision, occasionally going directly to Father Hesburgh (the university president).

Ross directed one Ph.D. student at Notre Dame: Paul J. McCarthy earned his doctorate in 1955 with a thesis about integral quadratic forms of class number one.

Ross was an advisor and mentor to several bright young math students at Notre Dame during the 1950s, helping them navigate through college requirements, getting them permission to enroll in advanced math courses, etc. Some of those students were:

- Charles Misner (physics professor at Maryland working in general relativity). In fact Ross is listed as a co-advisor for Misner’s PhD thesis at Princeton (1957).
- Thomas Banchoff (math professor emeritus at Brown, served as MAA president),
- John Riedl (OSU math professor, dean and director of OSU’s Mansfield Campus).
- John Polking (Rice University math professor).
- Manuel Berriozábal (University of Texas at San Antonio math professor and founder of TexPREP).

### Start of the Ross Program.

In the mid-1950s, some teachers participating in Ross’s summer program insisted that young students were not mentally capable of learning abstract ideas. [Piaget’s work on cognitive development suggests that teenagers are unable to comprehend abstractions.] To prove them wrong, Ross arranged for several teenage students to attend his class the following summer, while the high school teachers observed them. This story may bring various images to mind, but many years later a math professor who was one of those boys said: “Think of a classroom with a dozen Jewish boys in the middle, surrounded by nuns.”

Ross found that he enjoyed working with bright young students, and the teacher component of his program slowly dwindled away during the next several years. The earliest written report about the student program dates from 1961. By the late 1950s the teacher program, supported by the National Science Foundation, involved hundreds of math teachers every summer, with some of them continuing with an Academic Year Institute to earn a Masters Degree. Ross was pleased that he was able to convince many distinguished colleagues to teach classes for those teachers (with some of the more advanced high school students attending). Those instructors included S. Chowla, Max Dehn, H. D. Kloosterman, Walter Ledermann, Joseph Landin, Kurt Mahler, E.J. McShane, Norman Oler, W. W. Rogosinski, Helmut Röhrl, Thoralf Skolem, and Ivo Thomas. Short term visitors, each presenting one or two lectures, included people like Adrian Albert, Alfred Brauer, Richard Brauer, Paul Erdös, Hans Jonas, Solomon Lefschetz, Wilhelm Magnus, Marston Morse, John Todd, and Harry Vandiver.

Ross had close connections with Paul Erdös, as mentioned in his article *Remembering Paul. *He offered Erdös a professorship at Notre Dame, but Erdös chose to travel without keeping a home base at any one university. We found several handwritten Erdös letters to Ross, and one of them mentioned the “epsilons” that Ross was teaching every summer.

### Move to Ohio State.

In Autumn 1963 Arnold Ross was appointed chair of the Department of Mathematics at Ohio State. He did not tell stories about the reasons for this move, but other people who were at Notre Dame at that time told me that a new dean was appointed at Notre Dame in the early 1960s. That dean resisted Ross’s practice of avoiding proper channels and doing things without working through the dean’s office. In addition, several math faculty members at Notre Dame were disgruntled by Ross’s authoritative style and complained about their lack of input. Those pressures induced Ross to look for employment elsewhere.

Many American universities began major expansions in the mid 1960s. In that climate Ross built a strong department at Ohio State, attracting several promising young mathematicians while jobs were plentiful.

Ross moved his summer program to OSU as well. In 1964 and 1965 students lived in South Campus dorms (with no air conditioning). In 1966 they moved to the newly built North Campus area, staying in air-conditioned dorms like Nosker, Norton, Blackburn, Scott, Drackett, Halloran, and Houck.

Several mathematicians at Ohio State were involved with Prof. Ross’ summer program in various ways during the 1960s and 1970s. Some presented advanced courses, while others ran a problem seminar, or just gave a couple of lectures. For instance, Kurt Mahler ran courses in the Geometry of Numbers for two or three summers, and Ivo Thomas (visiting from Notre Dame) led logic courses for several summers. Hans Zassenhaus assisted Ross both at Notre Dame and at OSU, and taught courses on “experimental number theory” during the late 1960s. Charles Saltzer ran analysis courses, Jill Yaqub taught projective geometry, and R. P. Bambah, Harold Brown, Surinder Sehgal, and Alan Woods, were active with problem seminars. Other Ross Program instructors in the 1960s include Paul Turán and Joan Leitzel.

The National Science Foundation funded many SSTPs (Summer Science Training Programs) in the 1960s, as part of an American response to the Soviet space program. For more than two decades Dr. Ross supported his SSTP (later called a Young Scholars Program) almost entirely from NSF grants. Funding was sometimes restricted to rising seniors (students entering their senior year of high school), and the NSF did not provide funds for a student to return to the same program for a second time. Ross considered those restrictions to be misguided, and insisted on a multi-level program that involved students at several ages and mathematical levels. He found other sources of support for younger and older participants.

### Ross Retirement and moving the Program.

Ross stepped down as professor and department chair in 1976 because of mandatory retirement policies at Ohio State. During his last years of active employment he tried to find ways for his Program to survive. He worked with his friend Felix Browder to move the Ross Program to the University of Chicago. The plan was that Ross would run the Program there for a few years and then others would take over, using the same teaching methods. Ross led his program in Chicago for four summers, 1975 – 1978 and then left it for others to run.

During those summers in Chicago, Dr. Ross had close contacts with faculty members Paul Sally and Charles Fefferman. He had many discussions with them about effective ways to teach math to various types of students.

### Restarting at Ohio State.

In the Spring of 1979 Ross learned that the mathematicians running the Chicago program were not going to use an “inquiry-based” teaching style. Instead the Program would be taught as a standard number theory course, without the open-ended problem sets and without much pressure on students to immerse themselves in math. Ross felt obliged to re-create his program at Ohio State, starting it on short notice, with a tiny budget, using volunteer instructors. He was able to attract only one or two of the trained counselors from the Chicago program. Ross recruited Gloria Woods (wife of OSU Professor Alan Woods) to help organize things, and during the next few years she became deeply involved with running the Program. The students they were able to recruit during the next few years were younger that those typically attending the Program in earlier summers. Ross realized that his Program was still effective for bright students who were 13 or 14 years old. After 1980 he continued to allow such young students to enroll, even though their social immaturity sometimes caused behavior problems in a college dormitory setting.

Since Gloria Woods worked so closely with Dr. Ross, he named her as Assistant Director and strongly encouraged her to get a PhD in Math Education. Her 1981 doctoral dissertation, supervised by Prof. Harold C. Trimble, analyzed the teaching methods used in the Ross Program.

During the 1980s the Ross Program slowly grew in size, and the best of the students returning for several summers as counselors. However, in the early 1980s Arnold Ross went through serious personal difficulties because of Bee’s cancer diagnosis and treatment. She had some painful years of chemotherapy and operations, finally passing away in 1983. Since Arnold invested so much of his energy in her care, he was not very functional during those years and for several years afterwards. He seemed old and frail, and I remember students in 1986 asking whether I thought Dr. Ross would last through the summer.

Although he was able to find energy to give daily number theory lectures every summer, during the 1980s Ross was on campus only in the mornings during the eight weeks of the Program. Most of the Program’s administration (including grant applications and much of the student admission process) was done by Gloria Woods. The Ross Program survived those years mostly because of her efforts.

Things changed dramatically in 1990: Early in the summer Dr. Ross surprised the counselors by introducing the new Mrs. Ross. He had met Madeleine Green a year or so earlier, had a whirlwind romance (even though he was in his mid-80s) with a wedding that Spring. During the 1990s Dr. Ross was healthier and became more involved with administrative details that he had abandoned earlier. Gloria was happy to step back and let him take the reins again.

Ross continued giving daily lectures in Number Theory every summer. During the 1990s his lectures gradually became more disjointed and rambling, but that style fit well with the Ross Program’s teaching methods! Gaps and errors were viewed as “podasips,” routinely salvaged by the students.

Three Problem Seminars supported the Number Theory lecture every year. From 1986 to 2000, almost all of those seminars were taught by Professors Bogdan Baishanski, Ranko Bojanic, and me (Daniel Shapiro). During those years several other OSU faculty members led advanced courses for the more experienced students and counselors. That list includes Professors Dijen Ray Chaudhuri, Gerald Edgar, Dan Burghelea, and Ken Supowit.

In the last week of the Program in August 2000, Arnold Ross had a stroke. He was hospitalized and later moved to a nursing home on the north side of Columbus, a few miles from his home in Worthington. Madeleine took good care of him, but he never returned home, and died in that nursing home in 2002.

I was a first-year student in the Ross SSTP in 1966 and a counselor in 1967, 1968, and 1970.

I was hired as an assistant professor at Ohio State in 1974, and have taught in the Ross program every summer since 1985. I became Ross Program Director in 2000 and have tried to continue promoting the teaching style that Arnold Ross established. During the past 16 years the Ross number theory course has been led by three of us: Jim Fowler, Warren Sinnott, and me. Some details about Ross Program personnel since 2000 are posted here.

### Awards and honors.

Professor Ross earned many honors for teaching and service, both from OSU and from national organizations like the MAA (Award for Distinguished Service, 1986) and the AMS (1998 Citation for Public Service). Ross also received the Ohio State University’s 1981 Distinguished Service Award, and an honorary degree from Denison University. The Arnold Ross Lecture Series, run by the American Mathematical Society, is a wonderful program in which outstanding mathematicians present lectures to audiences of high school students. The series was initiated by Paul Sally and named after Ross in 1993, in honor of his many contributions to the development of mathematical talent. The lectures are presented every year or so in different cities across the United States.

In addition to his accomplishments and awards, Dr. Ross is remembered for his deep insights into current trends in education, for nurturing the talents of his students and colleagues, for inspiring others by his words and example, and for his strength of personality.

### Ross problem sets.

In the 1960s the problem sets were handwritten and duplicated from ditto masters. In the early years Dr. Ross made changes in the sets every year, sometimes just tweaking a few problems but other times substituting new topics for old. In the 1990s Daniel Bernstein (one of the counselors) typed up the problem sets in TeX. After that the sets seemed to have stayed nearly unchanged until 2001. Students and counselors across several years often referred to particular problems by their location rather than content. (“I struggled with

P7 set 10, … ”)

The problems in each set are grouped into categories like Terminology, Numerical, Exploration, and “podasip”: Prove or Disprove and Salvage if Possible. (I think the acronym PODASIP was first used in the 1990s.) Every summer, students require some practice before understanding the idea of podasips, but they soon begin to enjoy struggling with those somewhat open-ended problems.

After becoming Director, I asked some former counselors to help rework the sets for the 2001 session. Anna Medvedovsky and Noah Snyder worked many hours to reorganize the problem sets and type them in LaTeX. They traced all the threads of ideas in the original sets and rewove them to slow down the pace in the early weeks and speed it up later on. Over the next few years several small changes in the sets were made by Shapiro and Warren Sinnott, who led the Ross Number Theory class during 2006 – 2008.

### Switch to a six-week format.

The Ross Program ran for eight weeks every summer from 1957 to 2011. For various political reasons, the Ohio State University changed from the quarter-system to the semester-system in 2012. Instead of having Autumn Quarter classes start in mid September, they decided to begin Autumn Semester classes in mid August. This calendar change forced the Ross Program to end before August 1, and it suddenly became a six-week program.

Since 2012, we have edited the sets to fit all those great topics more neatly into the shorter time frame. (Some students suggested that we simply distribute problem sets on Saturdays and Sundays to accommodate the two missing weeks’ worth of problems. That didn’t seem feasible.)

### Ross Program genealogy project.

Many years ago some Ross counselors began tracing their “descendants.” If first-year students in a counselor’s family later became counselors themselves, their students would be the grand-students of the original counselor. Some charts have been made of those lines of descent, but information before 2000 is relatively scarce. If there is interest we might post the descent lines that are recorded.

### Sister Programs.

After his retirement in the 1970s, Arnold Ross established some related programs at other universities, spurred by the worry that his program at Ohio State would get little support from subsequent chairs of the math department. In addition to moving his program to the University of Chicago in 1975, he convinced colleagues to establish programs in other countries during those years.

At the University of Heidelberg (Germany) he founded a program with help from Professor Peter Roquette.

At Bangalore University (India) he created a program, with input and help from Professor

R. P. Bambah.

Starting in January 1975, Ross contributed his ideas and methods to an already existing high school talent search program in Australia: the National Mathematics Summer School at the Australian National University in Canberra, a two-week program that still runs every January. Ross worked closely with its founder, Professor Larry Blakers.

Perhaps some alumni can fill in further details here, providing dates and durations of those programs in Germany, India, and Australia, along with lists of names of instructors and participants.

In the late 1980s a few alumni realized that they could continue the Ross tradition by creating their own programs.

**PROMYS** was founded in 1989 by Professors Glenn Stevens and David Fried at Boston University. Several other Ross alumni have been involved with PROMYS over the years, including Steve Rosenberg, Marjory Baruch, Tom Roby, Keith Conrad, Robert Pollack, and Jared Weinstein.

**Honors Summer Math Camp** was founded in 1988 by Professor Max Warshauer at Texas State University in San Marcos (formerly Southwest Texas State University).

Dr. Ross was deeply gratified to see the success of those two programs founded by alumni of his program. In recent years, some other Ross alumni have mentioned that they would like to create their own summer math programs, following the inspiration of Arnold Ross.

In 2016 we established Ross/Asia , a version of the Ross Program held near Nanjing, China. That five-week session was based on a number theory course with daily problem sets, closely following the Ross model. Students there were mostly from China but almost all the junior counselors, counselors, and instructors were American, and all classes were held in English. We will run a similar session in 2017 in Huangshan City.

# Bibliography of Arnold Ross.

1. On certain universal and zero indefinite ternary quadratic forms, *Bull. Amer. Math. Soc*. **36** (1930), 364. Abstract #276

2. On universal indefinite quaternary quadratic forms $\phi = f(x, y, z) + \alpha w^2$, *Bull. Amer. Math. Soc*. **36** (1930), 364. Abstract #277

3. *On Representation of Integers by Indefinite Ternary Quadratic Forms.* Ph.D. Thesis, The University of Chicago. 1931.

4. On criteria for universality of ternary quadratic forms, *Bull. Amer. Math. Soc.* **38** (1932), 632. Abstract #192.

5. On representation of integers by ternary forms, *Bull. Amer. Math. Soc.* **38** (1932) 636-637. Abstract #208.

6. A note on three equivalent theorems, *Bull. Amer. Math. Soc. ***39** (1933), 204. Abstract #100.

7. On criteria for universality of ternary quadratic forms, *Quart. J. Math.* **4** (1933) 147–158.

8. On representation of integers by indefinite ternary quadratic forms of quadratfrei determinant, *Amer. J. Math.* **55** (1933) 293-302.

9. On representation of integers by quadratic forms, *Proc. Nat. Acad. Sci. U.S.A.* 18 (1932) 600-608.

10. Positive quaternary quadratic forms representing all but a finite number of integers, *Bull. Amer. Math. Soc.* **39** (1933) 510. Abstract #228.

11. A theorem on simultaneous representation of primes and its corollaries, *Bull. Amer. Math. Soc.* **45 **(1939) 899–906. MR0000407

12. On a problem of Ramanujan, *Amer. J. Math.* **68** (1946) 29–46.

13. (with Gordon Pall) An extension of a problem of Kloosterman, *Amer. J. Math.* **68** (1946). 59–65. MR0014378

14. The Notre Dame Mathematics Teacher Training Program, Summer 1959. (unpublished).

15. Notre Dame’s 1960 summer program for gifted high school children, *Math. Teacher* **54** (1961) 440-443.

16. “To teach how to teach how to do”, The Notre Dame Mathematics Teacher Training Program, Summer 1963. (unpublished).

17. Proposal for a plan to upgrade the performance of graduate student assistants at The Ohio State University 1965 (unpublished).

18. Horizons Unlimited, Preprint 1969.

19. The Shape of Our Tomorrows, *Amer. Math. Monthly* **77** (1970) 1002-1007.

20. Nature or Nurture; December 1973. A report on the student science-training summer program sponsored by the National Science Foundation and by the Ohio State University.

21. Fostering Scientific Talent, Chapter V of: *Science and Technology Policies,* Ballinger Publ. Co, Cambridge, MA (1973) 71-77.

22. Chairman’s Report, The National Research Council, Committee on Undergraduate Education (1974).

23. Towards the Abstract, *Math. Spectrum* **10** (1977/78) 88-95.

24. Talent Search and Development, *Math. Scientist* **3** (1978) 1-7.

25. What Mathematics for Gifted Young People: the Problems of Selection of Content and of Bringing about Deep Student Involvement, *Proceedings of the International Congress on Mathematical Education* **4** (1980) 696-699.

26. New-math note, *The Math. Teache*r **78** (1985) 496, 498.

27. The Teacher’s Role in Fostering Creativity, manuscript 1989.

28. 1998 Citations for Public Service, *Notices of the Amer. Math. Soc.* **45** (1998) 513-516.

29. Remembering Paul, in: *Paul Erdős and his mathematics, I (Budapest, 1999), *35–37, Bolyai Soc. Math. Stud., **11**, János Bolyai Math. Soc., Budapest, 2002.

30. On the logistics of talent, *Proceedings of the Ramanujan Centennial International Conference (Annamalainagar, 1987), *161–167, RMS Publ. **1** Ramanujan Math. Soc., Annamalainagar, 1988.

31. Creativity: Nature or Nurture, *CBMS Issues in Mathematics Education*, v. 2, (1989-90) 39-56.

32. Teacher as a Role Model, *CBMS Issues in Mathematics Education*, v. 2, (1989-90) 80-84.

33. Creativity in the Mathematical Sciences, *AAAS Symposium*, San Francisco (January 18, 1989).

34. Windmills or Stepping Stones?, in: *A Century of Mathematical Meetings*, Bettye Anne Case (ed.), Amer. Math. Soc., 1996.

35. Unfulfilled tomorrows, *Notices of the Amer. Math. Soc.* **43** (1996) 1147-1150.

36. Response to: 1998 Citations for Public Service, *Notices of the Amer. Math. Soc.* **45** (1998) 513-516.

37. Preachment versus Apprenticeship, unpublished manuscript.

38. Quo Vadis America, in: *Intellectual Talent: Psychometric and Social Issues*. Camilla Persson Benbow and David Lubinski, eds. The Johns Hopkins Univ. Press, 1996, Chapter 13, pp. 221-224. The Davidson Institute also posted that article.

**Articles about Arnold Ross or his summer program:**

Browder, Felix; article about the 1975 Chicago program.

Zassenhaus, Hans; Arnold E. Ross, pp. xxvi–xxxiii in: *Number theory and algebra.* Collected papers dedicated to Henry B. Mann, Arnold E. Ross, and Olga Taussky-Todd. Edited by Hans Zassenhaus. *Academic Press,* 1977.

Woods, Gloria; *A Study of Nurture of Mathematically Talented High School Children*, Doctoral Dissertation, the Ohio State University, 1981.

Shapiro, Daniel; A Conference Honoring Arnold Ross on His Ninetieth Birthday, *Notices of the Amer. Math. Soc.* **43** (1996) 1151-1154.

Jackson, Allyn; Interview with Arnold Ross. *Notices Amer. Math. Soc.* **48** (2001) 691–697.

Jackson, Allyn and Shapiro, Daniel; Arnold Ross (1906–2002). With contributions by: Prakash Bambah, Thomas Banchoff, Felix Browder, David Pollack, Peter Roquette, Karl Rubin, Paul J. Sally Jr., James Schultz, Alice Silverberg, Glenn Stevens, Bert K. Waits, Max Warshauer and Gloria Woods. *Notices Amer. Math. Soc.* **50** (2003) 660–665.

Goss, David; Arnold Ephraim Ross (1906–2002), *J. Number Theory* **110** (2005), 1-2.