Effective early childhood math teaching is much more challenging than most people anticipate. Because the math is foundational, many people assume it takes little understanding to teach it, and unfortunately this is distinctly not the case. In fact, the most foundational math ideas — about what quantity is, about how hierarchical inclusion makes our number system work, about the things that all different shapes and sizes of triangles have in common — are highly abstract ones. While we should not expect or encourage young children to formally recite these ideas, they are perfectly capable of grappling with them. Further, they need to do so to develop the kind of robust understanding that will not crumble under the necessary memorization of number words and symbols that is to come in kindergarten. In preschool, before there is really any opportunity for “procedural” math, it is important that we give children ample opportunity to think about math conceptually. In this essay I will discuss several profound ideas from early childhood mathematics, including examples of effective early math classrooms. Along the way I will share some of the resources that my colleagues and I have developed to help early childhood educators develop as skillful teachers of early mathematics.
About Early Math
As a doctoral student, I first got interested in early mathematics by way of cognitive science. I fell in love with the precise and thoughtful cognitive developmental work that built on what Jean Piaget had begun. Through clever experimental designs and a careful parsing of concepts over the last 40 years, developmental psychologists have made enormous strides in understanding how the mind develops during childhood. Many of their findings have profound implications for mathematics, and since my degree was to be in applied child development, early math education provided a way to make studying cognitive development useful to me.
As it turned out, early math was a useful place to put energy for far more important reasons. In a now-landmark study in 2007  using six longitudinal data sets, Duncan et al. found that math concept understanding at kindergarten entry predicted not only later math achievement, but also later reading achievement; reading at kindergarten entry, however, did not predict later math. This finding was replicated in a large-scale Canadian study in 2010 , which found that early math skills were stronger predictors of general academic success than either reading skills or social-emotional skills at school entry. We don’t yet know for certain why this association is so strong, but it is at least clear that early math is important. It is also true that the differences we observe in math achievement at kindergarten entry tend to fall along socio-economic lines, so alleviating those differences relates to issues of educational equity.
Early math was also a useful focus because of the pronounced need (in the U.S. especially) for improved instruction in mathematics in preschool and early elementary settings. Years after the seminal work by Deborah Ball  on the need for improved pedagogical content knowledge, and by Liping Ma  on the lack of a “profound understanding of fundamental mathematics” among later-grade elementary teachers, math educators turned their lens to those teaching our youngest students. It turns out that students of teacher education who “love kids but hate math” are commonly directed by faculty to teach in the younger grades. This has left us with a preponderance of preschool and primary teachers who are both underconfident and underprepared in mathematics teaching.
Teaching Early Math
So what does mathematics teaching look like in a preschool classroom? Recall first that preschool means children between the ages of 3 and 5, and that their range of normative development is exceedingly wide. In this group of kids there will be children who are not “potty-trained” alongside children who have begun to read, so teachers have to cast a very wide net. Further, for this age group, “teaching” is something that is often done only when all the heavy lifting of being sure everyone is comfortable, rested, and not in tears is complete. Sit-and-listen techniques are effective only when the content is exceedingly entertaining — as in a story is being read — and the children have very limited capacity for absorbing information directly from text, and less-limited but still primitive abilities to communicate their own ideas.
For these reasons, learning in early childhood classrooms consists almost entirely of “active learning.” In fact, early childhood has a long and proud connection to the type of teaching that emphasizes student-directed/teacher-facilitated activities. Child choices and the use of prepared “centers” are favored, with limited time spent on whole group activities of any kind (“circle time” being the exception), and small groups being occasionally led by a teacher. This is not an environment that is amenable to worksheets, and for that, early childhood teachers are generally extremely grateful. It also means, however, that whatever content is introduced comes fairly directly from the intentions and understandings of the teacher, who designs and facilitates experiences that lead children to construct new thinking.
Some Useful Interventions
Given this learning environment, my colleagues and I decided to focus our work on improving teachers’ understanding of the early math content they should be working into their interactions with young children. By studying the cognitive developmental and early math education literatures, we developed 26 Big Ideas that we wanted to be sure early childhood teachers understood well and knew how to address. One example is the idea that “any collection of objects can always be sorted in more than one way.” While this is not a conventional mathematical idea, it is foundational to the types of thinking that underlie our experience of sets (there are 6 pieces of fruit; there are 2 apples, 2 lemons, and 2 bananas; there are 2 red pieces of fruit and 4 yellow pieces of fruit) and therefore an important understanding for young children to see, explore, and experience. It has generative implications for understanding number and algebra in later life, and helps children flex and develop their logical thinking skills.
To help teachers make such an idea come to life, we developed what we call “Research Lessons.” These are skeleton lesson plans for activities teachers can use over the period of a month or more (through slightly altered iterations). For the Big Idea above, we ask teachers to conduct a read-aloud of a beautifully illustrated children’s book called Five Creatures by Emily Jenkins. In the book, a family of two adults, one child, and two cats is described differently from page to page, as in “In my family, there are five creatures…three who like milk, one who does not, and one who only drinks it in coffee…three with orange hair (child, one adult, one cat), one with gray hair, and one with stripes…” This book is read several times over a period of days, with lots of discussion. At some point, the teacher introduces two large circles, drawn out on the rug with tape: half the class are the “creatures” and half are the audience. Together, teacher and audience sort the “creatures” using binary (A/B) sorting to place them inside the circles, as in “the creatures with long hair and the creatures with short hair” or “the creatures with white in their shirts and the creatures without white in their shirts.” This leads to useful discussions about shared definitions for categories and sometimes generates the (exciting!) need for a third circle.
While it often goes unrecognized, the need for strong early math skills among children and early childhood educators is strong. Early math is highly abstract, and is a key indicator of later school success. What happens in preschool and early elementary classrooms has a direct impact on students for the rest of their educational experiences, from elementary school through postsecondary work. Our early childhood teachers need better preparation and in-service training to understand their crucial role in mathematics education. We will best be able to rise to the challenges of early math education through collaborative efforts involving teachers, teacher educators, and mathematical scientists.
 Duncan GJ, Dowsett CJ, Claessens A, Magnuson K, Huston AC, Klebanov P, Pagani LS, Feinstein L, Engel M, Brooks-Gunn J, Sexton H, Duckworth K, Japel C. “School readiness and later achievement.” Dev Psychol. 2007 Nov;43(6):1428-46.
 Pagani, Linda S. et al. “School Readiness and Later Achievement: A French Canadian Replication and Extension.” Developmental Psychology. Vol. 46(5): 984-994. September 2010.
 Ma, Liping. Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States. Routledge, 1999.
 Ball, D.L. (1988). Knowledge and reasoning in mathematical pedagogy: Examining what prospective teachers bring to teacher education. Unpublished doctoral dissertation, Michigan State University, Lansing.