Starting at

if ( is_page('Sports') ) { 

$cat = 6; //sports category id

$children = get_term_children( $cat );

if( is_array( $children ) && !empty( $children ) ) {

  foreach( $children as $child ) {

    query_posts( 'cat=' . $child );

    if( have_posts() ) { ?>

        <h4><a href="<?php echo get_category_link( $child ); ?>"><?php echo get_cat_name( $child ); ?></a></h4>

<ul>

<?php while( have_posts() ) : the_post(); ?>

<li><a href="<?php the_permalink(); ?>"><?php the_title(); ?>



  			<?php // This is the implementation of custom write panels.



 			$data = get_post_meta( $post->ID, 'module', true ); ?>



		 	<span class="listing-entry organization"><?php echo $data[ 'organization' ]; ?></span>



		 	<span class="read-more">(read more)</span>



		 	</a></li> 

<?php endwhile; ?>

</ul>

<?php

    }

  }

}

?>

The post limit on this will be whatever is set in your reading settings for the number of posts to show. If you want to show just 4 post titles per category then add

showposts=4

to the query_posts.