puedes probar este código
<?php
if(!empty($_POST) && isset($_POST["submit-btn"])){
$gender = isset($_POST["student_gender"]) ? $_POST["student_gender"] : "";
$student_session_one_preference = isset($_POST["student_session_one_preference"]) ? $_POST["student_session_one_preference"] : "";
// do the rest of the elements
echo $student_session_one_preference;
}
?>
<form method="POSt">
<input type="radio" name="student_gender" value="Male">Male<br>
<input type="radio" name="student_gender" value="Female">Female<br>
<select name="student_session_one_preference">
<option value="Session 1">Session 1: June 30 - July 21</option>
<option value="Session 2">Session 2: July 28 - August 18</option>
</select> <br/>
Preference 2:
<select name="student_session_two_preference">
<option value="Session 1">Session 1: June 30 - July 21</option>
<option value="Session 2">Session 2: July 28 - August 18</option>
</select>
<input type="checkbox" name="waldron_scholarship" value="Waldron Scholarship">Waldron Scholarship<br/>
<input type="checkbox" name="jesse_van_anterp_huyck_scholarship" value="Jesse Van Anterp Huyck Scholarship">Jesse Van Anterp Huyck Scholarship<br/>
<input type="checkbox" name="edward_raney_scholarship" value="Edward Raney Scholarship">Edward Raney Scholarship<br/>
<select name="school_type">
<option value="Public">Public</option>
<option value="Private">Private Parochial</option>
<option value="Other">Other</option>
</select>
<input type="radio" name="waiver" value="">I waive any right of access that I may have to information submitted by my referee<br>
<input type="radio" name="waiver" value="">I do not waive any right of access that I may have to information submitted by my referee<br>
<input type="checkbox" name="student_statement_consent_check_box" value="Agree">By checking this box, the student acknowledges <br/> that they have read the statement <br/> of consent and agrees with it.<br>
<input type="checkbox" name="guardian_statement_consent_check_box" value="Agree">By checking this box, the guardian acknowledges <br/> that they have read the statement <br/> of consent and agrees with it.<br>
<input type="submit" value="Submit" name="submit-btn">
</form>
este código asume que el formulario se envía a la misma página.
Espero que esto te ayude.