Plagiarism in Mathematics

Plagiarism is a common problem, primarily among students. Merriam-Webster Dictionary defines plagiarism as follows:

Definition“To steal and pass off (the ideas or words of another) as one’s own: to use (another’s production) without crediting the source”

Many people think plagiarism is stealing someone else’s words, but it also means stealing someone else’s ideas. In other words, even if you paraphrase a text, the idea still needs to be cited. The phrase “without crediting the source” means that it’s okay to use the words and ideas of others, but you have to cite the source to avoid committing plagiarism.

Types of plagiarism

Plagiarism takes various forms. It ranges from reusing an entire document to rewriting a single paragraph. In the end, all types of plagiarism come down to passing off someone else’s ideas or words as your own.

Copy-and-paste plagiarism	Copy-and-paste plagiarism, also known as direct plagiarism, means using a paragraph from another source without a citation. If you really want to include a passage from another source word for word, you should learn how to quote it.
Mosaic plagiarism	Copying and pasting different pieces of text together to create a kind of “mosaic” or “patchwork” of other researchers’ ideas is plagiarism. Although the result is a completely new piece of text, the words and ideas aren’t new.
Self-plagiarism	When you use parts of your previous work (e.g. a paper, a literature review or a dataset) without properly citing it, you commit what’s called self-plagiarism. Although it sounds a bit crazy to be penalized for plagiarizing your own work, you should know that it is done because it goes against the expectations of the readers of your paper. They expect the work to be original.
Global plagiarism	When you use someone else’s paper, you are committing plagiarism because you are pretending that the words and ideas are yours. Using someone else’s work includes, for example, having a friend or family write the text for you or buying an essay from a so-called essay mill.

Consequences of plagiarism

The consequences of plagiarism depend on the type of plagiarism and whether you’re a first-year student, an experienced academic or a working professional.

These are some possible consequences of plagiarism:

• Failing the course

• Expulsion or suspension from your university

• Copyright infringement

• Ruined reputation and potentially the end of your career

Statements about plagiarism from universities and journals

The consequences of committing plagiarism vary according to the university or journal. Below, you can find statements from American University and the American Marketing Association (AMA). Always check the editorial policies and academic integrity code of your institution.

American University“Sanctions for code violations [plagiarism] include loss of credit for the assignment, a failing grade for the course, a permanent notation on the transcript and dismissal from the university.”American Marketing Association“The penalty will be dictated by the nature of the offense and will likely include a ban on submitting to any journal published by the AMA for a period of time. All sitting Editors of AMA journals will be informed. … In extreme circumstances, the committee reserves the right to inform an author’s institution, depending on the seriousness of the offense.”

What is plagiarism in the context of mathematics? 

What is plagiarism in the context of mathematics? It can be confusing to understand what is meant by plagiarism and paraphrasing in the context of mathematics, since mathematics is built on a precision of language that often does not give many choices for variation. Also, it is sometimes hard to tell what might be considered as general knowledge and what information requires a reference. To try to make this clearer, I have created some examples below. 

Paraphrasing and not paraphrasing a text that deals with a rigidly defined mathematical concept Here is an example of a text that appears in Wikipedia (“Binomial Coefficients,” 31/1/2011) : 

Here is an example of a paraphrase of the above text: The positive integers that occur as coefficients in the binomial formula are called binomial coefficients. Binomial coefficients are given by two numbers, n and k, and are written as . This symbol represents the number that appears as the coefficient of the xk term in the polynomial expansion of (1+x)n . If we arrange all of the binomial coefficients into rows where each row corresponds to a fixed value of n, and where k goes from 0 to n in the nth row, we get a triangle of numbers called Pascal’s Triangle. Binomial coefficients occur in various areas of mathematics, especially combinatorics. If a set has n elements, then there are ways to choose a subset of size k. This is why this symbol is often read as “n choose k”. Notice that in this example, 1) The sentences of the paraphrase correspond almost directly to the sentences in the original document, although the wording has been changed. 2) The organization is exactly the same as the original. This text would still be a form of plagiarism if a reference to the Wikipedia article were made at the end. Here is an example of different way of explaining the same idea: The binomial coefficients are a set of positive integers that describe the coefficients of x in powers of the binomial (x+1). They are denoted by the symbol , which is pronounced “n choose k”. Specifically, denotes the kth coefficient in the expanded form of (x+1)n , that is, the coefficient of xk . These coefficients have many applications in combinatorics and probability. Notably, is the number of subsets of size k of a set of size n. Binomial coefficients have many interesting and useful properties, such as fitting into a diagram called Pascal’s Triangle, and are related to many other important sets of numbers, such as the Fibonnaci sequence. Note that although the definition of the binomial coefficients themselves is the same as in the Wikipedia text, as, in fact, it must be, the sentences do not match up, the paragraph is organized differently, and there is an additional piece of information (of course, the Wikipedia article was also much longer and contained more than is quoted above).

Common knowledge and knowledge requiring citation First of all, historical background is almost never common knowledge. In the continuation of the Wikipedia article on binomial coefficients, two of the three given pieces of historical information are given references. Really, the third piece (about Halayudha) also needs a reference, and this is noted on the linked page about him. 

Of course, there are some historical pieces of information that are common knowledge, such as that Newton and Leibnitz developed calculus. But when in doubt, put in a reference. Whether a piece of mathematical knowledge is considered to be common or not depends on the audience for your work. If one purpose of references is to give credit where credit is due, an equal purpose is to give your reader information on where to look for more on your topic. So, if it seems likely that your reader (who, for this assignment is a third year undergraduate in maths who may not have had any given optional module) will not yet be familiar with a certain piece of maths, give a reference where he or she could read more. It is generally better form to refer to a textbook than to Wikipedia or other internet sources for such things. For example, consider the text on the next page from the Wikipedia article about group representations (accessed 1/2/2011). Assume that you were going to write about group representations for your report. Since your fellow students will all have had abstract algebra and linear algebra, there is no need to cite any reference for the definition of group, homomorphism, invertible matrix, vector space isomorphism, basis, or injective. However, you don’t know that all students have had Vector Spaces, so you do need to cite a reference for the concept of a field and a vector space over a field, you need to say what GL(V) is and provide a reference, and you need to explain the idea of a topological vector space. You of course also need to provide references for all of the new ideas of representation theory that you take from this article, such as representation, representation space, continuous representation, kernel of a representation, etc. If you have used the same source for all of the information in a paragraph (but not paraphrased that source!) it is okay to simply give a single reference at the beginning of the paragraph, such as, “the material in this paragraph not otherwise indicated is based on (…..).” It is even better, though, to find a couple or three different sources, base your representation on the understanding you get from looking at all of them, and then include a sentence at the beginning of the paragraph or section that says, “The ideas in this section are drawn from the sources [1], [2], and [3], and further relevant material can be found there.”

Appropriate citation format can be found on the Learn page. It doesn’t matter which style of citation you choose (Harvard, MLA, etc.), but make sure all of your references follow the same formatting style. 

Case study of plagiarism in Mathematics

Calcutta University mathematician Mahimaranjan Adhikary has been accused of plagiarism by his home institution after reviewers writing in Math Reviews found three of his papers to be tainted. Math Reviews is a publication of the American Mathematical Society.

According to the reviewers, substantial parts of Adhikary's papers had been lifted from other people's work, often word for word. The reviewers merely pointed this out and did not use the term "plagiarism" to describe the misdeeds.

Readers of the reviews, however, could infer what the reviewers were getting at. So did Adhikary's home institution. A group of scholars at his university then determined that Adhikary had, indeed, plagiarized material.

Adhikary holds prestigious positions in Indian mathematics and is the author of several textbooks. His specialty is algebra, and he is best known for his work on graph theory, the topic of the disputed papers.

Regarding the paper titled "The connectivity of squares of box graphs,” the reviewer noted that Adhikary had "copied verbatim from mathematicians like Simoes-Pereira, D. Bauer and R. Tindell,” and suggested that the paper should not have been published.

The other two papers in question are titled "On edge-connectivity of inserted graphs" and "Factors of inserted graphs."

Queried about the charges, Adhikary told The Times of India, "I'm not willing to comment. I'll have to go through the AMS reviews to see if what you are saying is true."

"The charges against Adhikary are true," said Manjusha Majumdar, head of Calcutta University's department of mathematics, who led an inquiry.

“We are checking if Adhikary's other papers, including his thesis, were copied too,” said CU registrar Samir Bandyopadhyay. "In the past, when such a case was proved against a principal of an affiliated college, his thesis was cancelled."

The university’s Syndicate will decide the punishment to be meted out to Adhikary, who worked for over three decades at CU and had recently been recalled from retirement to continue his studies.

Source: Times of India, Jan. 2, 2008

References

1)www.scribbr.com

2)https://cdn.mathscareers.org.uk/wp-content/uploads/2018/03/Hunsicker_plagiarism.pdf

3) Wikipedia

4) https://www.maa.org/news/math-news/prominent-mathematician-in-india-accused-of-plagiarism