*By **Jennifer S. McCray**, Assistant Research Scientist and Director of the **Early Math Collaborative at Erikson Institute*

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 [1] 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 [2], 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 [4] on the need for improved pedagogical content knowledge, and by Liping Ma [3] 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.

**Conclusion**

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.

**References**

[1] 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.

[2] 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.

[3] Ma, Liping. *Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States*. Routledge, 1999.

[4] 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.

THIS is what is wrong with all those FB posts about the crazy “new math” in schools these days! Everyone thinks early math is simple and soooo easy to teach well. NOT. The people writing those need to come at the problems from the perspective of a very young, very uneducated child. I always suggest that they need to try to learn basic math in a different base to remember just how complex it is to learn math at that age.

I’m a current student at the University of Illinois at Urbana-Champaign studying math education. I’m curious about the “new math” in schools that you’ve been hearing about through FB. I haven’t seen anything on it. Can you enlighten me?

From what I see, I agree with you that it’s hard for people to really put into perspective of what it’s like to teach a younger child. Even I, as someone who will be student teaching in a high school next semester, understand that difficulty and don’t know how to see it in that way for the much younger children. The study of mathematics really is such an abstract idea to younger children and it would benefit them to really be taught right.

Also, I’m curious on your thoughts, and others who come across this, about their opinions on Lockhart’s Lament or also known as A Mathematician’s Lament that talks about how education with math should be. I know it may not be the early math being mentioned here, but it does relate in the sense of bringing up the abstractness of math and introducing it much earlier to students in math.

Very informative post! I liked your take on the importance of understanding that early math concepts are much more abstract than most adults believe that they are. This has helped lead to some of the hysteria that parents have when they look at the ways that math is being taught to their children today because it can’t be related to the rote methods from which they were taught. Your example of sorting and organizing the different types of fruits in as many different ways possible is a great way to get children to think about some of the fundamental ideas of sets and organization of objects. I thought it was great that you explained how these types of skills can help children in areas such as algebra and improve their logical thinking. When put into these terms it is easy for one see how important those early childhood years can be in one’s education.

“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.” The first sentences of this blog post are exactly what we have been working on all semester! Trying to develop a deeper understand as to why we do the thing we do when working with math problems so we are able to better explain and teach the topics to our students. Going away from the ways we were taught and really understanding the process. I can relate a lot to this entry because I did my practicum semester in a 4K classroom. The role of “active learning” is so important for students at this age because they are so young and they need time to have choice and a curriculum that is not based off worksheets and homework.

o Example idea: “any collection of objects can always be sorted in more than one way.” This concept helps with the foundational thinking of sets. It is a great way to introduce grouping to students at a young age and we have talked about this in our methods course. This could be a gateway to multiplication in older years.

o What happens in these early stages greatly effects how students will preform in later years. We have related back to this idea in our methods class also. Developing the building blocks is so important for students. They need this ideas and concepts to be concrete before they can fully move on and understand more complex ideas in math.

Hello – I am a senior studying math education at the University of Illinois. I completely agree that the math taught at early ages – even preschool – can have a lasting impact on a student’s education. For example, I really like your example about having the students sort the creatures using binary. While this may not initially seem like a math idea to many people, in future years an activity like this could, say, help students in an Algebra class classify numbers as integers, natural numbers, rational/irrational numbers, real numbers, and complex numbers. Even in a more general sense, an activity like this starts getting students used to the idea of comparing and contrasting, a topic that is heavily emphasized as students get older. I am curious – are there any numerical activities/numerical puzzles that you have exposed children to at this age (maybe in your Research Lessons) whose ideas provide a foundation for later mathematics?

Thank you for sharing . Very informative post! I liked your take on the importance of understanding that early math concepts are much more abstract than most adults believe that they are.

Nice information and Thanks for sharing it with all of us. I think the early math is much useful and their concepts are easier to understand.

Nice article,Thanks for posting this much of information.