As a teacher, your name and a link to your Profile page will show up in a Google search, but how much information this page displays is up to you. You can edit your Privacy settings to restrict most of your profile information from showing up to people who you are not connected with on Edmodo.
If your Privacy settings are set to "Only show profile to my connections," then people will only be able to see your name and profile picture. No other information will be displayed.
If this setting is not set, then your Profile will be public.
To learn more about your privacy settings, check out the screenshot below, and visit this page.