All I know about Garfield and math was that he made an original proof of the Pythagorean theorem. Did he make any other mathematical advancement (big or small)?
No. The library of Congress has a well-organized website of Garfield's papers, and he did not publish anything on mathematics other than that one note on the Pythagorean theorem in April 1, 1876 issue of the New-England Journal of Education. Perhaps in a gesture to the publication date, the editor mislabeled it pons asinorum (the bridge of asses), the medieval Latin nickname for the isosceles triangle theorem whose Euclid's proof had a reputation for getting students stuck (the diagram resembled a bridge they could not "cross").
However, Garfield did have oversized enthusiasm for statistics and pushed it everywhere as a public official. He advocated it for both informing public policy and establishing a more egalitarian picture of history. See "James A. Garfield, Historian" by Allan Peskin:
Garfield's fascination with quantifiable data was surprising, considering that his formal education had all but neglected mathematics. Yet his passion for statistics was so intense that he was accused of having "gone mad" on the subject. It was charged that "he would have the Congress and the officers of all Departments of the Government constantly running up and down the country gathering statistics." There was some truth to this. Garfield did propose the establishment of a "Bureau of Statistics" in the Treasury Department, and when he was chairman of the House Committee on the Census he overloaded the schedule with questions designed to uncover all sorts of miscellaneous statistical data. Some thought these questions trivial but Garfield defended their utility. "This is the age of statistics..." he insisted. "When we propose to legislate for great masses of people, we must first study the great facts relating to the people - their number, strength, length of life, intelligence, morality, occupations, industry and wealth."
These "great facts" had a further application: "The developments of statistics are causing history to be rewritten," Garfield declared.
Till recently, the historian studied nations in the aggregate, and gave us only the story of princes, dynasties, sieges and battles. Of the people themselves - the great social body, with life, growth, forces, elements, and laws of its own - he told us nothing. Now, statistical inquiry leads him into the hovels, homes, workshops, mines, fields, prisons, hospitals, and all other places where human nature displays its weaknesses and its strength. [...]