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Born in Chicago, Illinois, Kocialkowski was orphaned at an early age. He was educated in private schools, which he supplemented by a business course. He worked in various capacities in several business houses in Chicago. He engaged in tax appraisal and delinquent tax supervision in Cook County, Illinois from 1916 to 1932. He served as delegate to the Democratic National Convention in 1928.
Kocialkowski was elected as a Democrat to the Seventy-third and to the four succeeding Congresses (March 4, 1933-January 3, 1943). He served as chairman of the Committee on Insular Affairs (Seventy-fourth through Seventy-seventh Congresses). He was an unsuccessful candidate for renomination in 1942. He served as member of the Civil Service Commission of Cook County, Illinois from 1945 to 1949. He died in Chicago, Illinois, September 27, 1958. He was interred in St. Adelbert Cemetery.
This article incorporates public domain material from websites or documents of the Biographical Directory of the United States Congress.