Whenever you do some form of operation outside of AngularJS, such as doing an Ajax call with jQuery, or binding an event to an element like you have here you need to let AngularJS know to update itself. Here is the code change you need to do:

app.directive("remove", function () {
    return function (scope, element, attrs) {
        element.bind ("mousedown", function () {
            scope.remove(element);
            scope.$apply();
        })
    };

});

app.directive("resize", function () {
    return function (scope, element, attrs) {
        element.bind ("mousedown", function () {
            scope.resize(element);
            scope.$apply();
        })
    };
});

Here is the documentation on it: https://docs.angularjs.org/api/ng/type/$rootScope.Scope#$apply