using COUNTIF to check against two conditions

I have a spreadsheet (see attached) where I am using the COUNTIF function to let me know the number of times a persons name appears in each of three (3) columns. I also have a column showing the date and a column next to it showing the day of the week.

The COUNTIF function is doing exactly what I want, but now I want to also be able to make another count against 2 different criteria:

In addition to knowing the total number of times a name appears in a particular column I want to separately display the number of times a person's name appears only in those instances for which the day of the week happens to be a Saturday and/or Sunday.

Thanks in advance for any advice on setting up a formula/function that does this.