Custom Contact Form with simple Math Captcha

I have a well working contact form on my site. Now i try to include a math captcha to the from. Unfortunatly i was not able to make it work. Any idea how (where) to include the captcha codes? I tried a lot of things - but id does not work proper. Maybe the code is not the best :) I already searched for a solution in stackoverflow and google - but no luck.

My working contact form at the moment:

<?php
if(isset($_POST['submitted'])) {
if(trim($_POST['contactName']) === '') {
    $nameError = sprintf( __( 'Please enter your name.', 'test_theme' ) );
    $hasError = true;
} else {
    $name = trim($_POST['contactName']);
}

if(trim($_POST['email']) === '')  {
    $emailError = sprintf( __( 'Please enter your email address.', 'test_theme' ) );
    $hasError = true;
} else if (!preg_match("/^[[:alnum:]][a-z0-9_.-]*@[a-z0-9.-]+\.[a-z]{2,4}$/i", trim($_POST['email']))) {
    $emailError = sprintf( __( 'You entered an invalid email address.', 'test_theme' ) );
    $hasError = true;
} else {
    $email = trim($_POST['email']);
}

if(trim($_POST['comments']) === '') {
    $commentError = sprintf( __( 'Please enter a message.', 'test_theme' ) );
    $hasError = true;
} else {
    if(function_exists('stripslashes')) {
        $comments = stripslashes(trim($_POST['comments']));
    } else {
        $comments = trim($_POST['comments']);
    }
}

if(!isset($hasError)) {
    $emailTo = get_option('tz_email');
    if (!isset($emailTo) || ($emailTo == '') ){
        $emailTo = get_option('admin_email');
    }
    $subject = sprintf( __( 'Contact Form', 'test_theme' ) );
    $body = "Name: $name \n\nMail: $email \n\n $comments";
    $headers = 'From: '.$name.' <'.$emailTo.'>' . "\r\n" . 'Reply-To: ' . $email;

    wp_mail($emailTo, $subject, $body, $headers);
    $emailSent = true;
}

} ?>

                    <?php if(isset($emailSent) && $emailSent == true) { ?>
                        <div class="thanks">
                            <p><?php _e( 'Thanks, your email was sent successfully.', 'test_theme' ); ?></p>
                        </div>
                    <?php } else { ?>
                        <?php if(isset($hasError) || isset($captchaError)) { ?>
                            <p class="error-conform"><?php _e( 'Sorry, an error occured.', 'test_theme' ); ?><p>
                        <?php } ?>

                <form action="<?php the_permalink(); ?>" class="author-description" id="contactForm" method="post">
                            <h2><?php _e( 'Contact Us', 'test_theme' ); ?></h2><br />
                            <p><label for="contactName"><?php _e( 'Your Name', 'test_theme' ); ?> <span>*</span>
                            <?php if($nameError != '') { ?>
                                <span class="error-conform"><?=$nameError;?></span>
                            <?php } ?><br />
                            <input type="text" name="contactName" id="contactName" value="<?php if(isset($_POST['contactName'])) echo $_POST['contactName'];?>" /></label></p>

                            <p><label for="email"><?php _e( 'Your Email Adress', 'test_theme' ); ?> <span>*</span>
                            <?php if($emailError != '') { ?>
                                <span class="error-conform"><?=$emailError;?></span>
                            <?php } ?><br />

                            <input type="text" name="email" id="email" value="<?php if(isset($_POST['email']))  echo $_POST['email'];?>" /></label></p>

                            <p><label for="commentsText"><?php _e( 'Your Message', 'test_theme' ); ?> <span>*</span>
                            <?php if($commentError != '') { ?>
                                <span class="error-conform"><?=$commentError;?></span>
                            <?php } ?><br />                                
                            <textarea type="text" name="comments" id="commentsText" rows="20" cols="30"><?php if(isset($_POST['comments'])) { if(function_exists('stripslashes')) { echo stripslashes($_POST['comments']); } else { echo $_POST['comments']; } } ?></textarea></label></p>

                        <p><input type="submit"></p>

                    <input type="hidden" name="submitted" id="submitted" value="true" />
                </form>
            <?php } ?>  

Captcha Code that should included in the contact form above:

Part 1:

if(!isset($_POST['submitted'])) {
    session_start();
    $digit1 = mt_rand(1,6);
    $digit2 = mt_rand(1,6);
    $math = "$digit1 + $digit2";
    $_SESSION['answer'] = $digit1 + $digit2;
}

Part 2:

    if ($_SESSION['answer'] != $_POST['answer'] ) {
    $mathError = 'Please answer the math question.';
    $hasError = true;
  }

Part 3:

    <span class="error-conform"><?=$mathError;?></span>
    <li><label>What's <?php echo $math; ?> = </label><input name="answer" type="text" /></li>