Learning theorist Jean Piaget (1896-1980) was one of the first people who perceived children’s minds as more than empty vessels to fill with knowledge. He theorized that knowledge was constructed from within as a child matured and processed experiences. In 1920, Piaget began his career studying children at the Binet Laboratory in Paris, which is credited with development of the first IQ test. However, his interests soon began to focus on how each child thought about things and not whether one child was more intelligent than another. Furthermore, he found their wrong answers much more fascinating than correct ones, noticing that children of the same age frequently thought of the same wrong answers. From this vantage point based on appreciation and respect for the thinking capabilities of young children, Piaget began his life’s work of interviewing children of varying ages about a particular task and writing volumes analyzing the ideas which children reported to him. 
Now, let’s move ahead in time 50 years (or half of a century) to our current age of technological advancements, none of which were available to Piaget. Today, numerous groups of researchers are studying infants and toddlers through brain-imaging techniques. These studies are constantly revealing to us newly discovered and surprising capacities possessed, even by infants for making sense of the world. In general, brain-imaging research involves non-invasive techniques, which track brain activity on a computer screen while the child has certain visual and auditory experiences provided by researchers. In the lab of Harvard professor, Elizabeth Spelke, the length of time a baby stares at a scene is used in conjunction with screen images of the child’s brain structure and functioning as keys to unlocking the baby’s thoughts. Spelke’s studies have discovered innate abilities which are mathematical in nature including perception of objects as discrete physical units; a quantitative sense of more or less; and geometric navigational abilities. 
Humans share with other animals capabilities for quantitative perceptions, as well as navigational skills, but it is the human capacity to understand and use language which allows only our species to think abstractly. One crucial competency for language acquisition found in infants by Spelke and colleagues is the connection of words with things to which they refer.  Furthermore, humans are the only species able to master systems of explicit symbols, including the words of spoken language and symbols used to express mathematical ideas. An additional example of abstract thinking is the use of what Lev Vygotsky (1896-1934) and his followers called “tools of the mind.” The most important mental tool humans possess is, of course, language — both spoken and written, but other tools, both mental and visual, allow abstract thought to be structured and remembered. The brain capabilities of very young children allow them to order of a day’s events or pages in a book; to nest a set of boxes by size; or to find the correct fit for a puzzle piece or the correct hole in a shape-sorting toy. What are some other mental tools do you see your infants and toddlers using spontaneously?
Because children are understanding so many things within their brains even before they can express their thoughts verbally, we should not miss these early opportunities to use spoken words to identify objects; describe actions or locations; as well as explain other experiences and perceptions present in our surroundings. Thus, I join Professor Herbert Ginsberg of Columbia University in advocating that adults incorporate math ideas into these one-way conversations with their “baby-mathematicians.”  And when our infants and toddlers begin to make gestures, as well as touch and point to things, we can transform one-way conversations into continuously enriching dialogues. Then, possibilities for learning will move ahead at light speed if we continually take advantage of young children’s amazing brains.
 Details of Piaget’s life and work appear in the 1988 book by Herbert Ginsberg and Sylvia Opper, Piaget’s Theory of Intellectual Development, p 1 – 12.
 I found two interesting articles from the New York Times by Natalie Angier which focused on Spelke’s work: “Insights from the Youngest Minds” (4/30/2012) and “From the Minds of Babes” (5/1/2012). The Elizabeth Spelke link accesses the first article and is prefaced by a video of Dr. Spelke talking about her work.
 See Dehaene-Lambertz & Spelke (2015), “The Infancy of the Human Brain.”
 If you haven’t already, watch the 10-minute video of Herbert Ginsburg’s presentation “Preparing Children for School: Literacy and Numeracy during the Early Years” as he asks his listeners to pretend that he is an unhappy baby mathematician – unhappy because his caregivers are not teaching him about math. https://www.youtube.com/watch?v=dderI82YpN0 (This video was also cited on the parent page “It’s 2018…Start Now!”)