You don’t need a new directory for your subdomain. Both your main URL (www.americanthought.org) and your subdomain URL (www.technology.americanthought.org) should point to the same directory. Multisite will serve the sites from the same set of WordPress files.
(I took a quick peek, and it appears that you’ve got your subdomain’s DNS entry pointing at an IP address, which is good. Your main site has a different IP address, but I assume that’s because you’re using CloudFlare.)
You will need to set up your .htaccess file correctly. According to the Codex, you probably want the following rules:
RewriteEngine On
RewriteBase /
RewriteRule ^index\.php$ - [L]
# add a trailing slash to /wp-admin
RewriteRule ^wp-admin$ wp-admin/ [R=301,L]
RewriteCond %{REQUEST_FILENAME} -f [OR]
RewriteCond %{REQUEST_FILENAME} -d
RewriteRule ^ - [L]
RewriteRule ^(wp-(content|admin|includes).*) $1 [L]
RewriteRule ^(.*\.php)$ $1 [L]
RewriteRule . index.php [L]
…but I can’t say with 100% certainty because I don’t know what your site’s setup looks like.