Typically developing children naturally cross midline during play and functional activities. This is not a skill that parents usually teach, which is why you've probably never heard the term before. Children with dyspraxia other types of Sensory Processing disorders (SPD) may avoid crossing midline.
MIDLINE: is an imaginary vertical line running from the top of the head to the toes, that divides the body into left and right sides.
Seven-year-old Pedro reached for markers using whichever hand was closest to them, and then used that hand to color. He didn't attempt to stabilize the paper with his opposite hand. Observing this, the teacher, consulted with the school occupational therapist, Leila. Leila recommended that he practice forming large circles on a whiteboard. She offered him a marker positioned directly in front of him at, midline. He grasped it with his left hand, suggesting this hand might be dominant. Leila gently held his right arm at his side while he drew large circles on the board. Next, Leila placed her hand on top of his to guide his movements. With this help Pedro was able to trace over large diagonal crosses and horizontal figure-eights without switching hands.
Leila also recommended activities to help Pedro strengthen his hands, especially his fingers, so that he didn't switch the marker from hand to hand due to fatigue. His teacher tried the 1-2-3 PULL activity shown in the video below and realized that Pedro's left hand had better control than his right when pulling the rings. This observation reinforced the idea that he was left-hand dominant.
Source: Sensory Pull Bottle Helps Children with Autism or sensory Processing Disorders by RecyclingOT
Here are a few more activities that promote crossing midline:
1) Find two containers of different colors (such as red and blue) and bean bags to match. Place the red beanbags on the child's left and the red container on the child's right and vice-versa for the blue. Instruct the child to insert the red bean bags into the red container with the right hand, and the blue beanbags into the blue container with the left hand.
The girl in the photo has a basket on the other side of the horse so that she can sort the objects while working on crossing midline.
2) Ask the child to touch her right knee with her left hand. Repeat with various body parts. Reverse by showing her how to use the right hand to touch named body parts on the left side of her body.
Sometimes we can encourage crossing midline while children use both hands to move a larger object. The girl in the photo is grasping a vibrating cushion with body hands to touch it to her right leg. She is crossing midline with her left arm.
Suspend a ball so that the child can bat at it with a long tube. Initially have the child hold the tube with both hands. He will at times cross midline as the ball jumps around.
Position your hand while giving high fives, so that the child crosses midline. I did this repeatedly when performing hippotherapy.
The person inserting checkers into the Connect Four Board must stabilize it with one hand because it he doesn't it will fall over. When he began working from his left to right, he did not cross midline, but eventually he did as he worked his way across the board.
Another awesome game that will surely promote crossing midline is Twister.
Its also very easy to position materials so that the person reaches across midline with the dominant hand. The person in the video enjoys repetitive picture matching in the form board.
Source: Form Board Picture Activity for Individuals with Developmental Disabilities by RecyclingOT
Finally, the last video demonstrates an activity that uses either Scrabble tiles or alphabet blocks. This works on many visual and fine motor skills as I alternate hands to place the letters alphabetically.
Source: Crossing Midline Alphabet Sequencing for Children with Sensory Processing Disorders by RecyclingOT