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Test proctors at Arkansas State University spotted a woman wearing the cap while taking a national teacher certification exam under one name on a morning in June 2009 and then under another name that afternoon. A supervisor soon discovered that at least two other impersonators had registered for tests that day.

Ensuing investigations ultimately led to Clarence D. Mumford Sr., 59, who pleaded guilty on Friday to charges that accused him of being the cheating ring's mastermind during a 23-year career in Memphis as a teacher, assistant principal and guidance counselor.

Federal prosecutors had indicted him on 63 counts, including mail and wire fraud and identify theft. They said he doctored driver's licenses, pressured teachers to lie to the authorities and collected at least $125,000 from teachers and prospective teachers in Arkansas, Mississippi and Tennessee who feared that they could not pass the certification exams on their own.

Mr. Mumford pleaded guilty to two counts of the indictment, just a week after he rejected a settlement offer. At the time, he said that its recommended sentence of 9 to 11 years was "too long a time and too severe"; the new settlement carries a maximum sentence of 7 years.

Mr. Mumford appeared in Federal District Court here on Friday wearing a dark suit and a matching yellow tie and pocket handkerchief. He said little more than "Yes, sir" in answer to questions from Judge John T. Fowlkes.

Another 36 people, most of them teachers from Arkansas, Mississippi and Tennessee, have been swept up in the federal dragnet, including Clarence Mumford Jr., Mr. Mumford's son, and Cedrick Wilson, a former wide receiver for the Pittsburgh Steelers. (Mr. Wilson paid $2,500 for someone to take a certification exam for physical education teachers, according to court documents.)

In addition to the senior Mr. Mumford, eight people have pleaded guilty to charges stemming from the investigation into the ring, and on Friday, a federal prosecutor, John Fabian, announced that 18 people who confessed to paying Mr. Mumford to arrange test-takers for them had been barred from teaching for five years.

The case has rattled Memphis at a tumultuous time. The city's schools are merging with the suburban district in surrounding Shelby County, exposing simmering tensions over race and economic disparity. The state has also designated 68 schools in the city as among the lowest-performing campuses in Tennessee, and is gradually handing control of some of them to charter operators and other groups. And with a $90 million grant from the Bill and Melinda Gates Foundation, the district is overhauling how it recruits, evaluates and pays teachers.

District officials say that the test scandal does not reflect broader problems, and that none of the indicted teachers still work in the Memphis schools. (At least one teacher is working in Mississippi.) "It would be unfair to let what may be 50, 60 or 100 teachers who did some wrong stain the good work of the large number of teachers and administrators who get up every day and go by the book," said Dorsey Hopson, the general counsel for Memphis City Schools who this week was named the district's interim superintendent.

"A teacher's job is very hard. I know it is," said Threeshea Robinson, a mother who waited last week to pick up her son, a fourth grader at Raleigh-Bartlett Meadows Elementary School, where a teacher who has pleaded guilty taught until last fall. "But I would not want a doctor who did not pass all his tests operating on me."

The tests involved are known as Praxis exams, and more than 300,000 were administered last year by the nonprofit Educational Testing Service for people pursuing teaching licenses or new credentials in specific subjects like biology or history.

By and large, they are considered easy hurdles to clear. In Tennessee, for example, 97 percent of those who took the exams in the 2010-11 school year passed.

Robert Schaeffer, the public education director of FairTest, the National Center for Fair and Open Testing, said that the testing service had had problems with cheating before.

Ray Nicosia, the executive director of the testing service's Office of Testing Integrity, said episodes of impersonation were rare. "More than 99 percent of the people take the test honestly with no problems at all," he said.

In the case of the Mumford ring, the testing service spent a year conducting an inquiry before turning over its information to the Tennessee Bureau of Investigation, which worked with the Secret Service and referred the case to Edward W. Stanton III, the United States attorney for the Western District of Tennessee.

According to court documents, Mr. Mumford began soliciting teachers to take the Praxis exams for others in 1995, and continued to do so until at least 2010, when he retired from the Memphis schools and went to work as a guidance counselor at a small district in Arkansas. He paid the teachers from $200 to $1,000 per test. In Friday's hearing, Mr. Fabian said one client who paid Mr. Mumford $6,000 for multiple exams never got a successful result.

Mr. Mumford's personnel file in Memphis, which was reviewed by The New York Times, shows that he received above-average performance ratings. Not long after he began hiring the test-takers, a mother accused him of abusing her son with a paddle. (Corporal punishment is legal in Tennessee.) Mr. Mumford was suspended without pay for two years until a judge cleared him of the abuse charges, and he returned to Memphis as a guidance counselor in 1999. He applied for several jobs as a principal — describing himself in cover letters, misspellings and all, as an "experienced school administerator" — but he never rose any higher in the ranks.

Yet he continued to recruit teachers to take the Praxis exams, according to court documents. Felippia Turner-Kellogg, who until recently was an elementary school teacher in Memphis, made about $4,000 for taking tests over an 18-month period, according to the documents. Ms. Turner-Kellogg, who describes herself on her Twitter account as "on my way to millionaire status," told federal prosecutors that Mr. Mumford urged her to tell the authorities that he worked with test candidates because "he was tutoring teachers." Ms. Turner-Kellogg, who has entered a guilty plea and awaits sentencing, declined to comment.

In some instances, Mr. Mumford recruited teachers with troubled track records. Carlos Shaw, who has pleaded guilty to taking tests for others, received a string of weak performance evaluations and resigned in 2007 after he was reprimanded for writing inappropriate notes to a 17-year-old female student. Mr. Shaw did not respond to an e-mail seeking comment, and his lawyer did not return telephone calls.

Court papers portray Mr. Mumford as a con man who persuaded reluctant teachers to join his scheme. Shantell Shaw, a Memphis high school science teacher, told federal investigators that she initially turned down Mr. Mumford's offer, then agreed once he introduced her to another teacher who he said had failed a Praxis exam 11 times and needed help "so she could keep her job."

Ms. Shaw's lawyer declined to comment.

It was Ms. Shaw's pink ball cap that drew particular attention on that June morning in 2009.

Four people, including Ms. Shaw, arrived late at the testing site, arousing suspicions. Proctors did not know she was taking a test in another woman's name, but when they went to look for her — believing she would be taking another test in the same name — they found John Bowen, a Memphis substitute teacher, taking the examination instead. The police at Arkansas State confiscated counterfeit driver's licenses that Mr. Bowen was carrying and referred the matter to the testing service. "I kind of regret the people who are getting caught up in this," Mr. Bowen said in a brief telephone interview, "including myself."

Mr. Mumford declined requests for comment, but his lawyer, Coleman W. Garrett, said in an interview that "it never dawned on him that he was a criminal." "In his mind he was helping somebody," Mr. Garrett said.

Judge Fowlkes set Mr. Mumford's sentencing for May 13.