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A teacher and toddler are playing with flashlights. The teacher decides to play a sort of hide-and-seek game with the spot of light by shining the flashlight on various surfaces. Treat the strategies of both teacher and child as a type of conversation. Notice that the boy pulls on the curtain instead of standing on the platform. Perhaps he thinks he can use the curtain to pull the light closer. Notice that after he slaps the light he looks behind the curtain. Indicating that he understands light moves around, he does not look on the platform where the light was just before.

The teacher hides a beam of light by placing the face of a flashlight flat on a toddler’s pant leg. She then invites him to find the light by asking, “Where’d it go?” The toddler answers by placing his flashlight upside down on his leg. It seems that the toddler is playing the game, 'Where should I place my flashlight?' while the teacher is playing, 'Where has the missing spot of light gone?' By degrees — sometimes imitating the child, sometimes thinking about the child’s intention — the teacher becomes more in sync with the game the child is playing.

The teacher and children began planning their doghouse on a previous school day. The discussion resumes as they try to decide what features their doghouse will include. Notice how the teacher involves the children by asking questions that are open-ended but focused on a specific task. Near the start of their discussion she asks, “What else is missing on our doghouse?” Observe what strategies the teacher uses to help frame the children’s debate.

Robbie is exploring a Jack-in-the-box toy with his caregiver. The caregiver patiently works to extend his understanding of how the Jack-in-the-box works. Notice that she invites Robbie to explore by demonstrating both the procedure (cranking the handle) and the effect (Jack pops up). She is careful to make the end-state clear. Robbie imitates the action modeled by his caregiver, but he is uncertain about how long he must wind the toy before Jack pops up and quickly becomes frustrated. Observe how the caregiver helps Robbie work through this source of tension.

Robbie and Alyssa are sitting in their caregiver’s lap when Robbie introduces the birthday theme. The caregiver supports the children’s theme by inviting them to sing the birthday song. Observe what strategies Alyssa uses in order to synchronize her song with Robbie's. Notice what the caregiver does to extend the birthday topic.

Chaislyn, a child living with Down’s syndrome, is putting together an eight-piece jigsaw puzzle. The teacher uses four basic strategies to help Chaislyn. 1) She works to reduce the number of ways that Chaislyn’s choice could go wrong. 2) She tells Chaislyn that her choice is not correct. 3) She orients Chaislyn to cues that will help her place the piece into the form board correctly. 4) She tells Chaislyn what to do to in general, but not exactly where to place a chosen piece. What purposes do you think these different strategies serve?

A girl walks her fingers, and then toy animals, up stairs constructed with blocks. She explores different ways to make her repeated actions correspond to the graduated steps. How is this math-play? The girl does not count, but she does create interesting variations on correspondence, a fundamental component of quantifying sets of objects. Tapping once on every step in sequence expresses her sense of a one to one correspondence (N+1 rule). Tapping every other step with alternating fingers or toys can be formalized as N+2 alternating with N+2.

Owen accidentally falls while pretending to cook playing cards in a pot. The fall surprises him and hurts a little because as he falls he bumps his left forearm on the edge of the coffee table. Owen wants to better understand what has happened, so a few seconds later he re-enacts the fall deliberately, as if in so doing he can figure out how he hit his forearm. Study how Owen re-enacts the fall. Owen acts just like a golfer who re-enacts a missed putt in an attempt to figure out what went wrong.

Jenna, a one-year-old girl, invites us to explore the beguiling ways of toddlers. Watch this clip and decide what intention to attribute to Jenna’s action. Would you say she is giving the cup to the infant? Getting rid of a cup she does not want to hold anymore? Seeing if the infant will toss it to the ground? Seeing if the infant will continue to hold the cup after it is passed to her? Pay particular attention to her utterances like, “Uh-oh”, her laughter, her body language, and the way she adjusts the placement of the cup before releasing it.

Three children and a teacher examine a box full of jewelry. One boy tries to pull the bracelet straight up from the teacher’s wrist as if it isn’t a closed loop. The teacher wiggles her fingers in hopes that he will move the bracelet down instead of pulling it straight out from her wrist. The girl on the floor adeptly places bracelets on her wrist, sometimes using only one hand and then expanding her fingers to seat the bracelet higher on her wrist. The children help us see that they are beginning to understand how these items are worn.