Use an Angular directive to generate html from an array

Your main issue is that the directive childDOB is never rendered. Even though your controller works because 1.2.x version of angular has global controller discover on. It will look for any public constructors in the global scope to match the controller name in the ng-controller directive. It does not happen for directive. So the absence of ng-app="appname" there is no way the directive gets rendered. So add the appname ng-app="myApp" and see it working. It is also a good practice not to pollute global scope and register controller properly with controller() constructor. (Global look up has anyways been deprecated as of 1.3.x and can only be turned off at global level.)

You would also need to add track by in ng-repeat due to the repeater that can occur due to increasing the length of the array based on textbox value. It can result in multiple array values to be undefined resulting in duplicate. SO:-

 ng-repeat="child in children track by $index"

Fiddle

Html

<div ng-app="myApp">
    <div ng-controller="KidCtrl">
        <form>How many children:
            <input ng-model="numChildren" ng-change="onChange()" />
            <br/>
            <ul>
                <li ng-repeat="child in children track by $index">{{$index}}
                    <child-dob></child-dob>
                </li>
            </ul>
        </form>
    </div>
</div>

Script

(function () {

    var app = angular.module('myApp', []);

    app.controller('KidCtrl', KidCtrl);

    KidCtrl.$inject = ['$scope'];
    function KidCtrl($scope) {
        $scope.numChildren = 2
        $scope.children = [{
            dob: "1/1/1990"
        }, {
            dob: "1/1/1995"
        }];
        $scope.onChange = function () {
            $scope.children.length = $scope.numChildren;
        }
    }

    app.directive('childDob', function () {
        return {
            restrict: 'E',
            template: 'Child {{$index+1}} - date of birth: <input ng-model="child.dob" required/>'
        }
    });
})();