Summarize data into occurrences per month

here is one way, assuming you put 6/1/2018 in G1 and 7/1/2018 in H1 etc going toward the right then putting this in G2 and dragging right...

=COUNTIFS($A$2:$A$11,">="&G$1,$A$2:$A$11,"<"&H$1,$B$2:$B$11,"present")+COUNTIFS($A$2:$A$11,">="&G$1,$A$2:$A$11,"<"&H$1,$C$2:$C$11,"present")+COUNTIFS($A$2:$A$11,">="&G$1,$A$2:$A$11,"<"&H$1,$D$2:$D$11,"present")

this also assumes your data is in columns A, B, C and D beginning in A1 as shown above.
A sumproduct would also likely work but I didn't work that one out yet.
EDITED, something copied wrong so I corrected the formula.

Although it might take a bit more work on data entry, if you enter the data in in normalized format as shown in the attached spreadsheet, you can use it in a pivot table and get it in almost any format imaginable.

One of the advantages of pivot tables is that they adjust automatically to new data such as a new student. As the data are organized now, you would have to write a special formula for each student or month depending on how you want to display the data. Look up COUNTIF.

I suggest you try out some pivot table tutorials on line. They explain it better than I can.

this sumproduct would work too assuming the same locations and is a little simpler...
=SUMPRODUCT(($B$2:$D$11="present")*($A$2:$A$11>=G1)*($A$2:$A$11<H1))
again, G1 is 6/1/2018, H1 is 7/1/2018, I1 would be 8/1/2018 etc

check my post #4, it worked for data in B through D by month.

if you want to use "month" in the formula this also works and you don't have to point it at G1 or 6/1/2018, the "month" will be equal to the column value which for F1 is 6

=SUMPRODUCT(($B$2:$D$11="present")*(MONTH($A$2:$A$11)> =COLUMN(F$1))*(MONTH($A$2:$A$11)< COLUMN(G$1)))
dragged right it indexes to column G1 and H1 which would represent months 7 and 8

You're welcome. And if you are so inclined a bump to the rep is always welcome.