As an Occupational Therapist at Child and Family Development, I regularly assess and treat a child's fine motor abilities, including pre-handwriting and handwriting skills. This information is a useful summary of development.
Developmental Sequence of 10 different grasps (according to Schneck and Henderson, 1990). Below is a range of grasps that may be demonstrated over time. Not all will be seen. By ages 5-6 you expect to see a hand dominance and want your child to be using a grasp similar to numbers 7-10, number 10 being the most wanted.
1. Radial cross palmar grasp: pencil positioned across palm projecting radially, held with fisted hand, forearm fully pronated, full arm movement

2. Palmar supinate grasp: pencil positioned across palm projecting ulnarly, held with fisted hand, wrist slightly flexed and supinated away from midposition, full arm movement

3. Digital pronate grasp, only index finger extended: pencil held in palmar grasp with index finger extended along pencil toward tip, arm not supported on table, full arm movement

4. Brush grasp: pencil held with fingers with eraser end of pencil positioned against palm, hand pronated with wrist movement present, whole arm movement, forearm positioned in air

5. Grasp with extended fingers: pencil held with fingers, wrist straight and pronated with slight ulnar deviation, forearm moves as a unit

6. Cross thumb grasp: fingers fisted loosely into palm, pencil help against index finger with thumb crossed over pencil toward index finger, finger and wrist movement, forearm positioned on table

7. Static tripod grasp: pencils stabilized against radial side of third digit by thumb pulp with index pulp on top of shaft, thumb stabilized in full opposition, wrist slightly extended and hand moves as a unit, pencil rests in open webspace, forearm resting on table

8. Four finger grasp: pencil held with four fingers in opposition, wrist and finger movement, forearm positioned on table

9. Lateral tripod grasp: pencil stabilized against radial side of third digit with index pulp on top of shaft of pencil, thumb adducted and braced over or under anywhere along the lateral border of index finger, wrist slightly extended, fourth and fifth digits flexed to stabilize metacarpophalangeal arch and third digit, localized movement of digits of tripod and wrist movements on tall and horizontal strokes, forearm resting on table

10. Dynamic tripod grasp: pencil stabilized against radial side of third digit by thumb pulp with index pulp on top of shaft of pencil, thumb stabilized in full opposition, wrist slightly extended, fourth and fifth digits flexed to stabilize the metacarpophalangeal arch and third digit, localized movement of digits of tripod and wrist movements on tall and horizontal strokes, forearm resting on table

Examples of Inefficient Pencil Grips:

