“Two, four, six, eight, who do we appreciate?” is a popular cheer. This related counting-out rhyme was printed in the Aberdeen (Scotland), Journal on March 11, 1880:
 
“Ane, twa, three, four,
Mary at the cottage door
Eating cherries off a plate,
Five, six, seven, eight.”
 
This was printed in Notes and Queries , with Answers in January 1885:
 
“1, 2, 3, 4, 5, 6, 7, 8,
Mary at the cottage gate,
Eating grapes off a plate,
1, 2, 3, 4, 5, 6, 7, 8.”
   
The popular modern form was printed in the Elmira (NY) Star-Gazette on April 24, 1920:
 
“Two, four, six, eight
Who do we appreciate,
Mr. McCombe.”
   
   
Newspapers.com
11 March 1880, Aberdeen (Scotland), Journal, pg. 7, col. 5:
FOLK-LORE.
“COUNTINGS OUT.”
(...)
Ane, twa, three, four,
Mary at the cottage door
Eating cherries off a plate,
Five, six, seven, eight.
 
Google Books
January 1885, Notes and Queries , with Answers, pg. 485:
1, 2, 3, 4, 5, 6, 7, 8,
Mary at the cottage gate,
Eating grapes off a plate,
1, 2, 3, 4, 5, 6, 7, 8.
   
This is given also “plums” in place of “grapes.” and “garden gate” for “cottage gate.” When “cottage door,” ends the second line the counting stops at “four” to satisfy the rhyme.
 
Newspapers.com
1 April 1885, The Constitution (Atlanta, GA), pg. 4, col. 6:
COUNTING-OUT RHYMES.
Specimens of the Quaint Doggerels Used by Children in Their Play.
(...)
1, 2, 3, 4, 5, 6, 7, 8,
Mary at the cottage gate,
Eating grapes off a plate,
1, 2, 3, 4, 5, 6, 7, 8.
 
This is given, also, “plums” in place of “grapes” and “garden gate” for “cottage gate.” When “cottage door,” ends the second line the counting stops at “four” to satisfy the rhyme.
     
Newspapers.com
19 March 1905, The Daily Arkansas Democrat (Little Rock, AR), pg. 14, col. 6:
Good Counting-Out Rhymes.
(...)
One, two, three, four,
King at the cottage door,
Eating plums off a plate
Five, six, seven, eight!
 
Newspapers.com
24 June 1905, New York (NY) Times, “A Persian Town,” pg. 415, col. 2:
“The commoner form is with a rosary, rather on a plan of ‘Two, four, six, eight, Mary at the cottage gate.’”
   
Newspapers.com
24 April 1920, Elmira (NY) Star-Gazette, pg. 13, col. 3:
Several of their yells were very unique, and one which brought forth considerable applause was:
 
“Two, four, six, eight
Who do we appreciate,
Mr. McCombe.”
   
5 February 1921, Evansville (IN) Courier, “The Jingle Contest Close January 31st,” pg. 9, col. 3:
2! 4! 6! 8!
Who do we appreciate,
Are they in it, well I guess
The Standard Evansville Realty Co.
Yes! Yes! Yes!
MRS. META HOOE
1501 Third Ave.
   
Newspapers.com
19 March 1921, The Evening News (Harrisburg, PA), “The School Evening News,” pg. 8, col. 4:
SCHOOL YELL
Two-four-six-eight!
Who do we appreciate?
Mr. Senseman! Mr. Senseman!
Rah-rah-rah!
By ERMAN REAM.
Eighth Grade, Penbrook School.
 
Newspapers.com
3 May 1921, The Evening News (Harrisburg, PA), “The School Evening News,” pg. 18, col. 5:
TWO! Four, Six Eight!
Who do we appreciate?
Orchestra! Orchestra!
ORCHESTRA!
   
9 July 1921, New York (NY) Times, “Big City Not Wicked, Endeavorer Finds,” pg. 6, col. 2:
The junior delegates (to the Junior Christian Endeavor Convention in New York City—ed.) made the armory ring with a cheer for each one of the favorites among their officers. Thus:
 
“Two, four, six, eight! Two, four, six eight!
“Who do we appreciate?
“Dr. Clark! Dr. Clark! Dr. Clark!”
 
Newspapers.com
22 September 1921, Chicago (IL) Daily Tribune, “A Line O’ Type Or Two,” pg. 8, col. 3:
Here is one, A. A. writes, that was used in England in the ‘60s:
 
Two, four, six, eight,
Mary at the cottage gate.
Eating cherries off a plate;
Two, four, six, eight.
   
Newspapers.com
23 July 1923, Daily News (New York, NY), “Youngsters Ride Chain of Thrills at Steeplechase” by Sally Joy Brown, pg. 18, col. 5:
Throughout the entire afternoon the boys and girls attracted a great deal of attention by keeping up their newly acquired yell: “Two, four, six, eight, who do we appreciate? SALLY!”
 
Newspapers.com
6 March 1925, Raleigh (NC) Student, pg. 3, col. 4:
Yells
2, 4, 6, 8, who do we appreciate?
Team, team, team.
 
Slate
Two, Four, Six, Eight, Who Do We Appreciate?
A modern history of childhood, in one postgame cheer.
By REBECCA ONION
JUNE 04, 2018 9:00 AM
(...)
The “Two, Four, Six, Eight” chant seems to have been a counting-out rhyme first—one meant, like the more familiar “Eeny, Meeny, Miny, Moe,” to be used as a tool to pick a child out of a group, either to eliminate players or to select someone to act as “it” in a subsequent game. In a 1969 book, the Opies recorded a variation of a “Two, Four, Six, Eight” counting-out rhyme—“Two, four, six, eight/ Mary’s at the cottage gate/ Eating cherries off a plate/ Two, four, six, eight”—that they dated back to the early 19th century. Later, folklorists found 20th-century uses of the “two, four” structure in jump-rope chants. Collected in Pennsylvania in 1959: “Two, four, six, eight/ Don’t make love at the garden gate/ ’Cause love is blind/ But the neighbors ain’t.” Collected in 1956: “Two, four, six, eight/ Papa caught a rattlesnake/ The snake it died/ And Papa cried/ Two, four, six, eight.”
(...)
It’s in the interwar years, as recreation directors, coaches, supervisors, and teachers struggled to harness kids’ restless physical energy for educational purposes, that the string of words two, four, six, eight begins to appear in databases of newspapers and books alongside the phrase who do we appreciate? In 1921, a New York Times reporter writing a cute color piece about the meeting of the Junior Christian Endeavor Convention in New York City wrote that the “junior delegates” made the room ring by cheering their adult leaders: “Two, four, six, eight! Who do we appreciate? Dr. Clark! Dr. Clark! Dr. Clark!”