By Dr. Seanyelle Yagi and Dr. Linda Venenciano
“It’s equal!” squealed the Kindergarten student who poured pebbles from a tall, narrow container into a short, wide container. As many progressive educators know, young mathematicians find joy when they have the freedom to explore and observe their world. They also appreciate that teachers have opportunities to leverage such moments to introduce or confirm math concepts and terms. For example, children often recognize ”same” or “different”, but they may not know how to articulate what exactly they see as the same or different. In the illustration above, the two containers might be different in height, shape, or even color. Yet when they discover that the same quantity of pebbles that filled one container also fills the other, this opens into a teachable moment about equality and proof, albeit within a context that young mathematicians can grapple with.
Through research conducted at the Curriculum Research & Development Group, we (the authors of this blog) have identified five big ideas in early childhood mathematics that can be helpful for progressive educators as they look to seize teachable moments with young math learners. They are as follows,
(1) equality and the equal sign;
(2) properties of number and operation;
(3) the concept of variable;
(4) the ability to generalize; and
(5) the ability to think flexibly.
These big ideas are foundational for students’ understanding of number concepts throughout the elementary years and into algebra. Beginning number concepts include an understanding of relationships between quantities (Dougherty et. al., 2010) and can be developed without the use of numbers (Dougherty & Karp, 2019). Comparing (or measuring) attributes of objects (such as two different lengths, or equal volumes) helps students see relationships between quantities that are key for understanding numbers. Students can compare quantities of volume, length, weight or mass and represent the comparisons in different ways. To follow, we take some time in this blog to expand on each of these big ideas in early childhood mathematics.
Equality and the Equal Sign
The concept of equality is easily noticeable and builds upon students’ natural tendency to make observations and relate objects to each other. It is important for students to understand what it means to be equal at a deep level. After many repeated experiences writing equations from left to right, with an operation on the left followed by the result on the right, students can develop an operational understanding of the equal sign, as the place for an answer to an equation. In early mathematics, we want students to develop a relational understanding of the equal sign, as a symbol that expresses an equal relationship between two quantities. Encouraging students to communicate about the specific quantities (length, volume, mass) they are comparing and their reasoning behind their observations, helps students to think about the relationship between the quantities. Using the word, equal, to describe how the volumes of two containers compare along with representing the relationship with concrete (physical), pictorial, diagrammatic (semi-concrete) and symbolic representations also supports students in understanding the concept. The physical volumes can be labeled collaboratively with the students as volume A and volume B, and students can record line segments and equations to represent the equal relationship (See Figure 1).
Figure 1
Properties of Number and Operations
In the line segments and equations in Figure 1, one might observe that the quantities of A and B were switched in the equations. In the first equation, A is equal to B, and in the second, B is equal to A. This illustrates the reflexive property of equality that is true for all numbers – if A is equal to B, then B will be equal to A. This relationship is commonly observed in equations with numbers, for example:
3 + 2 = 5 5 = 3 + 2
Writing equations using the reflexive property of equality is based on the understanding that there are equal quantities on the right and left sides of the equal sign, and hence, the equations can be written in both ways and still be true. The physical containers can be switched to show volume A is equal to volume B and volume B is equal to volume A. The quantities of volumes have not changed, thus, they are still equal.
The properties of number and operations build upon the concept of equality and are based on patterns and characteristics of numbers and operations. In addition, the properties describe relationships between numbers and their operations. For example, a property of equality states that for any two numbers, only one of the following statements is true. This provides the basis for all numbers (Davydov, 1975):
A = B A < B A > B
Figure 2
Students can compare different lengths, masses, and volumes of objects to decide if they are equal or not equal (see Figure 2). If they are not equal, students can describe them more specifically as greater, more, less, or fewer. Working with representations of continuous quantities (length, mass, volume), where counting objects is not possible, students focus on the relationship between the quantities and notice their magnitudes. Students are introduced to the < and > symbols within this context, through repeated experiences with comparing and representing quantities in multiple ways. The symbols < and > are written in statements along with variables to represent the comparisons. Writing the symbolic statements directly with concrete (physical) and semi-concrete (diagrams) representations supports students in developing conceptual understanding underlying the symbolic statements (see Figure 3). In Figure 3, students compared containers of physical volumes and recorded the comparison relationships with letter names and pictures of the volumes. They placed the volumes in order from least to greatest and recorded symbolic statements to represent how the volumes compared.
Figure 3
Concept of Variable
Figure 4
Students can use letters to name an amount or as placeholders for a quantity. Using letters to represent a quantity allows students to communicate their observations about number relationships (see Figure 4). Choosing letters randomly to name a quantity supports students in developing an understanding of concepts of variables. They find that variables are easier to work with when reasoning about quantities and relationships. In Figure 4, first grade students work to label generalized points and an expression on a number line. Students discuss the variables as representing any number on the number line.
The Ability to Generalize
Students enjoy noticing and exploring patterns in their world. They might see that the wings of a butterfly have a symmetrical pattern, or notice the equal lengths of wire between posts on a fence. Observing and exploring patterns provides opportunities for students to ask questions, reason, and make predictions about what may or may not be true. Generalizations are big mathematical ideas that emerge from noticing commonalities across multiple examples. They include the processes of justifying, reasoning, and representing (Karp, Doughery, & Bush, 2021). For young mathematicians, this means linking concrete (physical) representations with a variety of other representations (concrete, semi-concrete, symbolic) and developing their communication and reasoning skills. Conjecturing why they think something might be true for multiple cases based on their observations and representations encourages students to make generalizations based on evidence. Generalizations emerge through students’ active use of representations and reasoning and have meaning for the student.
The Ability to Think Flexibly
Thinking flexibly involves the ability to shift one’s thinking and solve a problem in more than one way. This also includes using what one learned from solving one problem to solve another problem. Representing relationships with length, volume, and mass can help students apply what they learned in one context to another and develop a more rigorous understanding of the concept. For example, when students represent one volume as greater than another volume, and then use that experience to solve a problem in which they determine how two masses compare (i.e. one mass is greater than another), they develop a stronger understanding of what it means for one quantity to be greater than another (see Figure 5). Thinking flexibly allows students to be adaptable problem solvers who can use the mathematics they know in different contexts to solve a range of problems.
Figure 5
Summary
Young children have many opportunities, both in formal contexts and informally in their worlds outside of school to think mathematically. The adults with whom they interact can enhance their mathematical thinking through exploration, investigation, and making observations and connections between their intuitive thinking and formalized mathematical ideas. Encouraging young children to notice different measurable attributes in their world is an excellent place to start. In Venenciano et al. (2021), we shared findings on how these types of tasks and learning experiences supported first-grade students in learning how to meaningfully use representations to convey their thinking about relations among measurable quantities. These learning experiences have also been characterized as laying a path towards developing algebraic reasoning (Venenciano et al., 2019; Carraher and Schliemann 2018). Our experiences working with early elementary students showed us that young children are able to develop relational thinking and can verbally and symbolically (e.g., <, >, =) communicate their emerging ideas about relationships between and among quantities.
To ensure that more early childhood educators have the opportunity to leverage these big ideas in mathematics as powerful tools for integrating math education into the everyday lived experiences of children, the Hanahau‘oli School Professional Development Center and the Samuel N. and Mary Castle Foundation are excited to offer a new locally developed Mathematics workshop series for early childhood educators this summer. Big Ideas in Early Childhood Mathematics is designed for teachers across the state of Hawai‘i who work with children in preschool through grade 2, and will take place over the course of three face-to-face meetings in June 2025, and two online follow up sessions in Fall 2025. To learn more, click here.
References
Carraher, D. & Schliemann, A. (2018). Cultivating Early Algebraic Reasoning. In C. Kieran (Ed.), Teaching and learning algebraic thinking with 5-12 year-olds. The Global Evolution of an Emerging Field of Research and Practice. ICME-13 Monographs. Chai, Siwtzerland: Springer.
Dougherty, B., Flores, A., Louis, E., Sophian, C., & Zbiek, R. M. (2010). Developing essential understanding of number and numeration for teaching mathematics in preK–2. National Council of Teachers of Mathematics.
Dougherty, B. & Karp, K. (2019). Putting essential understanding into practice: Number and numeration PK–2. National Council of Teachers of Mathematics.
Davydov, V. V. (1975). Logical and psychological problems of elementary mathematics as an academic subject. In Steffe, L. P. (Ed.), Children’s capacity for learning mathematics: Soviet studies in the psychology of learning and teaching mathematics, Vol. VII. University of Chicago.
Karp, K., Dougherty, B., & Bush, S. (2021). The Math Pact: Achieving Instructional Coherence Within and Across Grades. Corwin.
Venenciano, L. C., Yagi, S. L., Zenigami, F. K., & Dougherty, B. J. (2019). Supporting the development of early algebraic thinking, an alternative approach to number. Investigations in Mathematics Learning, 38–52, 12(1). DOI: 10.1080/19477503.2019.1614386
Venenciano, L. C., Yagi, S. L., Zenigami, F. (2021). The development of relational thinking: A study of Measure Up first-grade students’ thinking and their symbolic understandings. Educational Studies in Mathematics, 413–428.
ABOUT THE AUTHORS:
Seanyelle Yagi is an associate professor in the School of Teacher Education at the University of Hawaiʻi at Mānoa. She has over 20 years’ experience in teaching mathematics at the elementary and middle school levels and as a mathematics coach and mathematics resource teacher in the Hawaii Department of Education. She has provided professional learning sessions for teachers in a range of contexts, including those who teach mathematics in preschool through high school. Currently she teaches mathematics methods to preservice teachers at the undergraduate and graduate levels, and serves as a cohort coordinator, and field supervisor. Dr. Yagiʻs focus is on supporting teachers in providing meaningful learning experiences in mathematics for elementary students.
Linda Venenciano is the Interim Director of Pacific University's School of Learning and Teaching. Prior to starting at Pacific in 2022, Dr. Venenciano was a member of the mathematics team at the Curriculum Research & Development Group at the University of Hawai‘i. Dr. Venenciano has taught students from first grade through undergraduate levels and delivered numerous professional development workshops, both locally and nationally. Dr. Venenciano currently serves on the editorial board of Investigations in Mathematics Learning and conducts research on equity, identity, and a sense of belonging in the mathematics education community. Dr. Venenciano has taught students from first grade through undergraduate levels and delivered numerous professional development workshops, both locally and nationally. Dr. Venenciano currently serves on the editorial board of Investigations in Mathematics Learning and conducts research on equity, identity, and a sense of belonging in the mathematics education community.
