Creating Time and Space for Students to Develop Foundational Mathematical Ideas

By Maureen D. O’Connell

“Slow down, you’re moving too fast, you got to make the morning last…”

When we consider early childhood mathematics this familiar song comes to mind. In our hurried society where more is more, childhood expectations have been pushed to earlier and earlier ages. In this environment, the time and space to allow early mathematics to grow deeply is endangered. Structuring opportunities for exploration and discovery of number concepts is the work of master teachers using an artful mathematical story. 

Too often, administrators, colleagues and parents pass kindergarten rooms and see the work as “just play,” when in reality, we know that a critical mathematical foundation is being purposefully explored and established. Due to this perception, as children learn to count to five or 10, we feel the need to add additional concepts to challenge them. These challenges often take the shape of using larger numbers or learning procedures before developing a deep conceptual understanding, when there are so many subtleties in the foundational work IM K–5 Math. 

The National Research Council elaborates on these subtleties. 

To count means to list the counting numbers in order, usually starting at 1, but sometimes starting at another number, as in 5, 6, 7, . . . . (Other forms of counting include “skip counting,” in which one counts every second, or third, or fourth, etc., number, such as 2, 4, 6, . . . , and counting backward, as in 10, 9, 8, 7, . . . .) 

Although adults take it for granted because it is so familiar, the connection between the list of counting numbers and the number of items in a set is deep and subtle. It is a key connection that children must make. There are also subtleties and deep ideas involved in saying and writing the number list, which adults also take for granted because their use is so common. Because of the depth and subtlety of ideas involved in the number list and its connection to cardinality, and because these ideas are central to all of mathematics, it is essential that children become fluent with the number list.

(Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity, 2009)

Throughout this report, the authors stress that our goals for early mathematics must go beyond “ideas, facts, and skills. Processes and attitudes are also essential outcomes for all children.” (p.5) Students need experience using the 8 Standards for Mathematical Practice and benefit from lessons that incorporate their work. Students see themselves as mathematicians when they share their work and explain their thinking about a topic.

So, let’s take a closer look at what this could look like in terms of the mathematics students are exploring. 

When a kindergarten or grade 1 student sees an image like this one they are asked to explain, How many do you see? and How do you see them? This develops number relationships, in addition to ability to know how many without counting. When they describe counting by 2s, or seeing 6 and then adding 2 more, they are experiencing the joy of learning math by doing math.

When children respond to a Which One Doesn’t Belong like this

they recognize that, depending on the attribute they choose, there are multiple reasons why one might not belong with the other three. They notice, engage in purposeful math discourse, defend their thinking and consider the thinking of others. They engage in the habits of mathematicians.

When teaching grade 2, I could surmise within the first days of school which students had summer birthdays. These newly minted 7-year-old students were bright and capable, and also showed marked differences in their world views learning side by side with their turning 8 peers. Imagine at that age, the fraction of life that 10 or 11 months represent in a child’s experience and development. These students could have been described as “behind” or in need of additional support, when in fact all they needed were additional experiences with numbers and shapes.

As I moved to my current role as math specialist I found the referrals for intervention groups were filled with these young-in-their-grade students. Intervention lists were also populated by students who were not the youngest in their grade, but needed more experiences with the material for one reason or another. Other than cases of an underlying math disability, these students made nice gains and closed the gap in benchmarks with their near aged peers. 

Intervention was filled with opportunities to explore combinations of 5 or 10, to consider magnitude, to say what number is one or two more or less. If our math program had gone deep into these areas of number sense and provided ample opportunity to explore these ideas for learners at all stages of development, what might happen?

Reflecting on how we as a district contributed to this growth in intervention referrals, we realized our math curriculum was not aligned with the common core standards for grade-level expectations. It offered rigor by pushing concepts down to lower grade levels, but for students with fewer months of development or fewer opportunities to experience and interact with mathematics in the world, this created a disconnect. 

Considering the vast differences in experience with math that students bring to the early childhood math classroom, a purposeful progression of concepts is critical. The research shows that those who begin school with the lowest math achievement levels show the slowest growth from kindergarten to grade 3. (Clements, 28) 

Clements et al. state:

This is especially alarming because children’s early knowledge of math strongly predicts their later success in math (30), even into high school (5, 25, 31). Persistent problems with mathematics is the best predictor of failing to graduate from high school or enter college (32). More surprising is that it also predicts later reading achievement, even better than early reading skills (33–35).

IM K–5 Math 

Last year we participated in the alpha pilot of IM K–5 Math in our primary grades. We did so without intervention groups. How was this possible? 

IM K–5 Math offers a standards-based and developmentally appropriate kindergarten and early grade curriculum where the long term significance of developing foundational mathematics is valued. IM K–5 welcomes kindergarten students to a curriculum that makes time to explore counting, cardinality, do I have enough?, magnitude, one more or one less.

From the very first day, students were welcomed into the grade with a thoughtful, developmentally appropriate curriculum. Each unit began with invitational content. Each lesson began with an accessible warm-up. These 10 predictable math content routines provided a scaffold for deep exploration of important grade level math content. They supported the cognitive demand of grade level math by removing the struggle or anxiety of “what am I supposed to do now?” The Standards for Mathematical Practices and 5 Practices for Math Discussion were woven seamlessly into each lesson and teaching plan. Centers and explorations provided the extra time students needed to grow their number sense and primary work of the grade. Teachers had the resources to support those students who needed more without leaving the room.

Our alpha pilot year was cut short with school closings due to COVID-19 in March. This fall we are back, enjoying the beta pilot, meeting students where they are with the support of IM K–5’s Adaptation Pack, available at the IM Community Hub. These extra documents for each grade level offer a few “just in time” lessons from the prior grade to create an accessible, shared experience as an on-ramp to the new grade-level content. What a difference this has made. Students have begun the year with smiles and confidence. Teachers can focus on what students know and are able to do now. The curriculum offers questions and components to move them to their next steps in the journey with the big ideas of mathematics education. 

Because we are in a hybrid model we also rely upon IM K-5’s Section Level Planning Guide. This helps us prioritize the lesson components that make the most sense to address when we have synchronous learning time, either in person or on a screen.

“Slow down, you’re moving too fast, you got to make the morning last…”

I encourage you to slow down and take the time to listen as students answer the questions, What do you notice? What do you wonder?” These might be the two most important questions in a primary classroom to support us in learning more about each student’s experiences and building on what they know. Based on this, we can take time to deeply engage with mathematical concepts. 

Thankfully IM K–5 gives us an aligned, engaging curriculum that creates the space to grow mathematicians who know, use, and enjoy their math. It will be exciting to see these mathematicians bring their deep knowledge and positive experiences to the world.

Next Steps

How can you help yourself and other educators slow down to explore student thinking? How can you build in time for all students to develop foundational math ideas?

The K–5 Adaptation Packs and Section Level Planning Guides can be found on the IM Distance Learning page