A brief overview of the development of counting in young children. Counting is more complex than usually assumed and learning it can be challenging. Children learn some aspects of counting in their everyday lives, but benefit from intentional teaching too.
Dr. Herb Ginsburg draws on his expertise on the development of mathematical thinking to detail the importance of counting for young children, the complex nature of counting, and the necessity of attending to children’s thinking as they count.
Counting in preschool and the early elementary years supports the development of a variety of other mathematical abilities. Patterning, part-whole relationships, place value, composition and decomposition, equivalence, operations, and magnitude are all important mathematical concepts that use counting as a foundation.
This account describes how number develops in the preschool range, from about 3-5 years, and provides a detailed overview of the various skills and understandings that children need in order to achieve competence in counting.
A thinking story about one child’s counting from one to 99 (she says she has to ask her mother about what comes after that). A thinking story is a written narrative presented along with embedded and carefully selected video clips–a kind of multimedia case study–that attempts to bring to life the drama of a child’s mathematical thinking.
In this activity prospective or practicing teachers attempt to learn to count in a foreign language (Farsi). The experience is designed to help them understand how children feel when they learn to count, and what kind of learning is involved in mastering the counting words.
This activity was developed for our graduate students (prospective early childhood teachers). It helps adults to understand how determining how many can involve many different strategies and ideas.
by Herbert P. Ginsburg & Rita Sanchez
This activity for teacher educators will help practicing and prospective teachers look carefully at the processes of interviewing and interpretation. Each of the following videos includes a voice-over commentary of a teacher conducting a clinical interview with a four-year-old girl, Angela.
Below are a selection of tasks you can use to determine what children know about counting. The prompts are just suggestions; consider how follow up questions requiring explanation inform your understanding of the child.
by Colleen Oppenzato & Herbert P. Ginsburg
Counting Collections is a set of activities that provides support for connecting the content of counting, children’s thinking in counting, and pedagogy in classroom practice.
by Megan Franke
This guide discusses general principles of reading and analyzing storybooks and offers brief descriptions of storybooks that children enjoy and that can be used to teach counting.
by Colleen Uscianowski, Colleen Oppenzato, & Herbert P. Ginsburg
This is the story of two teachers who learn how to blend their insistence on high quality support for early math learning with an approach that uses children’s natural engagement in and enthusiasm for mathematics, to guide lesson planning and classroom interactions.
This activity accompanies the “Play-Based? Math?” vignette to address beliefs and attitudes about mathematics teaching and learning in the early childhood classroom. This activity will help teachers examine their role in mathematics instruction within a play-based environment or philosophy.
This activity addresses beliefs and attitudes about teachers’ own mathematical skills and their effects on classroom teaching. Beliefs and attitudes directly affect how teachers support mathematical development in their classrooms. These beliefs and attitudes include fear and anxiety around their own competence in mathematics and perceived lack of knowledge about mathematical development and how to support it, and beliefs about how children learn mathematics and the teacher’s role in mathematics instruction in the early childhood classroom.
This activity addresses beliefs and attitudes about mathematics teaching and learning in the early childhood classroom. Although unseen, beliefs and attitudes directly affect how teachers support mathematical development in their classrooms. This activity helps teachers examine their own beliefs and practices concerning the roles of children and teachers in the process of learning.
Why early childhood education requires attention to mathematics learning in addition to work on literacy, science, and socio-emotional development.
by Alan H. Schoenfeld and Deborah Stipek
This article asserts that flexible assessment plays a central role in successful education. Flexible assessment includes careful attention to each child’s productions, close observation of their behavior, and flexible interviewing of individual children.
This article shows how different aspects of shape and space are fundamental to children’s lives. From the earliest years, as children navigate around their environment, they must learn about near and far, up and down, round and straight, and much more. Adults can help to extend and make explicit their children’s ideas about spatial relations, which is a more complex and challenging topic than usually assumed. Educators can use many different activities and approaches, including intentional teaching, to foster learning of spatial relations.
Although children accurately perceive shape and space in their everyday environments, preschool children need to learn to think about these topics. Our main goal should be to promote understanding of basic geometry.
by Herbert P. Ginsburg and Colleen Oppenzato
This 40-second video shows a young child’s understanding of spatial concepts such as in front of and behind. The video demonstrates that taking the child’s point of view can lead the adult to appreciate how an apparently wrong answer can result from sound ideas about spatial relations. The teacher needs to dig beneath the surface behavior of right and wrong answers to undercover the child’s interesting ideas.
Different approaches to learning what children know about shape and space. This article shows how participants can use observation, drawing, etc. to learn about children’s knowledge and language of geometric figures, physical and mental transformations, and spatial relations.
These tasks are designed to help you assess children’s concepts of shape. The protocol begins with relatively simple concepts like attributes of two dimensional shapes and ends with the analysis of solid shapes.
by Colleen Oppenzato and Herbert P. Ginsburg
These tasks are designed to help you assess a child’s understanding of spatial topics. The protocol begins with relations like over, near, next to, and the like and progresses to a child’s understanding of symmetry.
by Colleen Oppenzato and Herbert P. Ginsburg
This article introduces the vital perspective that spatial relations involve both ideas and language to express and understand them. Teachers who engage with dual language learners need to be sensitive to the need to help them understand both the spatial concepts and the associated language. Integrating spatial language and concepts is a challenge for all children, but poses distinctive challenges and opportunities for dual language learners (DLLs).
To teach well, you need to understand the individual child’s math knowledge. This document provides guidelines for conducting conversations that can reveal a child’s math thinking and guide your teaching. Math thinking conversations do not need to take a long time. They may, for example, occur in the context of working with a small group of children. While the other children are working, the teacher can take a few minutes to engage a child in a conversation related to the task.
Video used in teacher learning settings is a powerful tool for engaging participants in understanding children’s mathematical thinking. Yet by itself the video, no matter how wonderful, is not sufficient. This example of using video in a preservice course illustrates how teacher educators can engage participants around carefully selected video clips to promote their understanding of children’s thinking and how to assess it.
This activity supports facilitating Common Core Mathematical Practices in the classroom. It provides teachers with an opportunity to explore how the Mathematical Practices work in their own lives, and how they support facilitation of the Practices with their own students.
Supporting Dual Language learners (DLLs) in our classroom is a valuable opportunity to focus on the importance of language, think beyond the math, and provide for mathematical sense making in meaningful ways. Here are a few suggestions to make math meaningful in the classroom, not only for DLLs, but for all children.
As a professional developer working with preschool teachers who have had few opportunities to engage in work with others on their teaching of mathematics, I have three main goals: (1) to support them to see their strengths as teachers, so they can draw upon them when teaching mathematics; (2) to provide opportunities to understand children’s mathematical thinking; and (3) to help them learn how to “do” the work of engaging students in understanding mathematics.
As a teacher educator working with preservice participants who are just beginning their preschool teaching careers, I have three main goals: (1) to help students draw on their own experiences in mathematics, good and bad, to create supportive mathematical learning environments for preschool children, (2) to support students in gaining knowledge about early mathematical development, and (3) to help students learn how to engage young children in rich mathematical activities.
As professional developers we often have little opportunity ourselves to learn more about preparing and supporting preschool teachers teach mathematics. The materials in these modules are to support you in learning more about supporting preschool teachers in mathematics. We also provide resources to use for your support. This case lays out one way you might use the materials to prepare yourself for professional development around preschool mathematics.
A short session with preschool teachers can spark their interest in how upcoming professional development work can support them in improving their practice. The session will help them see what is possible in preschool mathematics and see how possible it is for them to engage in this work.
This story is written not for children, but for prospective and practicing teachers. It takes the child’s point of view about counting and presents the adventure in an amusing manner that will draw in adults who may have some anxiety regarding mathematics.
This story is written not for children, but for prospective and practicing teachers. It takes the child’s point of view about shapes and presents the adventure in an amusing manner that will draw in adults who may have some anxiety regarding mathematics.
Imagine that you, as a teacher, are preparing to read a book to a child or group of children and you want to use the story as an opportunity to develop understanding of shapes or spatial relations. This document describes what you might do.
In the activity, children are asked to view a picture (simple drawing, storybook page, complex geometrical shape, etc.), and describe what they notice in the picture. Then they draw what they see. This activity enables children with varying understandings of spatial relations to learn.
This resource provides an overview of how teacher educators can engage participants in Describe, Draw, Describe (DDD). The components have been developed to provide support for connecting children’s ideas about spatial relations and pedagogy.
This resource is designed to engage your participants in learning about shape. The basic idea is to help your participants gain some insight into the challenges children face in learning about shape.
by Colleen Oppenzato and Herbert P. Ginsburg
This fictionalized but very real story of actual teacher experiences tells the story of a team-teaching effort to incorporate meaningful and important concepts in geometry and spatial relations into a project-based preschool curriculum.
The story of a preschool teacher leveraging children’s ideas to engage in geometry. It addresses anxiety teachers experience when working beyond simple shapes, offering a perspective that engages participants and the children they teach.
This activity provides participants with an opportunity to explore the attributes of 2-D geometric figures (similar to how one would engage children), and to explore their own beliefs and attitudes about supporting children’s development in geometry and spatial relations.
A short session around spatial relations can support moving beyond thinking about spatial relations as identifying the shapes, expanding their conceptualization of spatial relations and helping them see opportunities to support learning.
Children often describe characteristics of shapes using their informal out-of-school language. We want to encourage participants to elicit children’s informal knowledge and language around the shapes and move them to being as specific as possible.
Often in professional development settings it is productive to anchor the content of the day around one activity designed for children. Within the context of the activity, we can engage participants as learners themselves, explore ideas of children’s mathematical thinking within specific content areas, and discuss how the activity can be used in classrooms. This one-day session for practicing teachers explores spatial relations through the activity Describe, Draw, Describe.
Multiplication and division are often considered to be too mathematically complex for young children. However, we see that young children engage in multiplicative reasoning as they play and that their early engagement in grouping provides a foundation for operations.
This story is written not for children, but for prospective and practicing teachers. It takes the child’s point of view about operations and presents the adventure in an amusing manner that will draw in adults who may have some anxiety regarding mathematics.
One of our main goals is to help participants understand the importance of assessing children’s thinking. The activity described here engages participants in an analysis of their own thinking as they solve math problems.
One of our main goals is to help participants understand the importance of assessing children’s thinking. This activity engages participants in bug detection and analysis; that is, figuring out the thinking that may underlie incorrect answers.
Imagine that you, as a teacher, are preparing to read a book to a child or group of children and you want to use the story as an opportunity to develop understanding of addition and subtraction. This document describes how you might proceed.
by Colleen Uscianowski and Herbert P. Ginsburg
Mathematics, specifically number and operations, is at the heart of the intensely social interactions of fair-sharing. In attempts to make our classrooms engaging but calm, we frequently underestimate children’s abilities in both mathematics and social relations.
This activity provides participants with an opportunity to explore children’s early abilities in fair-sharing and to explore their own beliefs and attitudes about supporting children’s mathematical and socio-emotional development.
Attending to the nuance of children’s responses enables teachers to figure out what children understand and what questions and supports can help them extend their thinking. Figuring out which questions to ask, which tasks to pose, and which tools to offer is often the most challenging aspect of teaching mathematics. This activity engages participants in both attending to the details of children’s ideas and working on how to support children to think more deeply.
Here is a brief account of how an understanding of the operations develops in the preschool years. Young children’s understanding of the operations develops out of their understanding of counting. For a complete account of what children need to learn about counting, see What Young Children Know and Need to Learn about Number.
Getting started with solving addition, subtraction, multiplication, and division problems begins by asking teachers to rethink their existing ideas of teaching the operations and providing opportunities for teachers to see that children can solve problems without being taught the symbols, number facts, or number sentences.
Children are interested in order at an early age. When order has regularity (that is, size increases by a specific amount each time), the order becomes a pattern. This handout illustrates examples of this particular kind of pattern, called growing patterns.
The environment constantly exposes young children to regularities of various types: in music, shirts, dance, shape, and spatial relations. Given these experiences, children ages three through six years old develop an intuitive understanding of pattern. The goal of this article is to show how children develop intuitive pattern knowledge in everyday life.
This article explains the basic math underlying the patterns that children encounter in their everyday lives and in preschool. . One key concept is the generalization of patterns: the ability to apply a pattern (e.g., ABAB) to multiple materials and contexts. Other concepts covered include copying, extending, and creating patterns.
This story is written not for children, but for prospective and practicing teachers. It takes the child’s point of view about patterns and presents the adventure in an amusing manner that will draw in adults who may have some anxiety regarding mathematics.
This resource is designed to engage your participants in learning about patterns and algebraic thinking. The activities are similar to those your participants can use in teaching children, but are more complex and demanding. The basic idea, (one often used in teacher workshops) is to help your participants gain some insight into the challenges children face in learning about patterns and algebra, as well as to explore different representations of the same pattern.
These tasks are designed to help you assess an individual child’s understanding of patterns. The protocol begins with relatively simple concepts such as recognizing and completing repeating patterns and moved on to more advanced concepts like growing patterns.
Supporting children to notice and describe what they notice is a productive way for teachers to leverage the ways in which children naturally find pattern in the world around them. This resource provides an overview of how teacher educators can engage teachers in this idea.
These tasks are designed to help you use the clinical interview to assess an individual child’s understanding of measurement. The protocol begins with relatively simple measurement tasks that rely on perception and advances to more complex tasks requiring measuring in units, although you can pick and choose among them. They do not need to be given in order.
Measurement in early childhood involves important mathematical concepts that can be supported in the preschool classroom through children’s everyday problem solving. This handout provides insights into children’s development in this important domain and gives examples of classroom applications.
The following handout presents an accessible overview of the mathematics of measurement. It provides important information on the foundations of measurement and ideas for supporting development of measurement skills in the preschool classroom.
What is measurement? It is the process of accurately determining the extent, amount, or quantitative value of an attribute such as length, weight, volume, or area. When you think of measurement, you probably think about measuring in feet or pounds. But measurement is more than that. In this handout, we first consider the basic concepts that form the foundation of measurement, then we examine exact measurement, and finally we discuss estimation. We focus on length, which is very salient for young children, but also touch upon temperature, distance, and time. Many of the concepts we discuss will not be fully mastered in preschool, but preschool children can develop the foundational knowledge that they will continue to build on through the elementary school grades.
This story is another approach to explaining the mathematics of measurement. It is written not for children, but for prospective and practicing teachers. It presents the adventure in an amusing manner that will draw in adults who may have some anxiety about or disinterest in mathematics.
Children need to understand some basic mathematical ideas in order to measure accurately. First, they need to know what Goldilocks knows (as revealed by my extensive clinical interview with her), namely that Baby Bear is smaller than Momma Bear and Momma Bear is smaller than Poppa bear; and also that Momma is at the same time smaller than Poppa Bear and larger than Baby Bear. Children also need to learn what Goldilocks does not yet know, namely how to measure the bears exactly. How much smaller is Baby Bear than Momma Bear? And what is this smallness anyway? Is Baby Bear shorter, or less voluminous (skinnier), or lighter, and by how much exactly?.
You’ve been counting regularly in your classroom and you’ve decided to start including activities that encourage children to delve into the exciting world of measurement. This handout discusses some important differences between counting and measuring, and how to talk about measurements that aren’t exact.
Imagine that you, a teacher, are preparing to read a book to a child or group of children and you want to use the story as an opportunity to develop their understanding of measurement. This document describes how you might proceed.
This activity explores the uses of calendar activities with preschoolers. It will help teachers examine the appropriateness of calendar activities in meeting developmental and learning goals, as well as addressing children’s interests.
What do children need to know in order to use data? This handout describes:
1. The foundational concepts of data,
2. The importance of context in understanding the questions and problems that data can solve, and
3. how young children can represent and interpret data.
Children compare and measure all the time in the classroom. How can we assess their knowledge in this important area of mathematics? This handout provides suggestions for improving our awareness of children’s understanding of measurements.
This resource is designed to engage your participants in learning about measurement. Notions of big and little and biggest and littlest permeate children’s play and preschool learning. However, deciding which object is biggest or littlest depends on which attribute (height, width, capacity, or weight) a child is considering. In this activity, participants are given various objects and asked to make a case that their object is “the greatest” in some way.
Although they perceive length at an early age, children have difficulty measuring it. To do so, they need to learn several basic mathematical concepts, like the idea of a constant, conventional unit of measurement.This handout describes challenges children face as they learn to measure lengths using non-conventional units (in this example, connecting cubes), and those faced by interviewers in trying to determine what children understand about measurement.
Often when working with data in preschool, teachers collect or ask children to collect survey data (months of birthdays, favorite color). The activity described here will help teachers to consider how to use observation data to promote engagement and learning in mathematics in ways that are authentic, purposeful, and interesting for students.
The initial 2-hour session focuses on supporting participants to notice patterns in their environment and consider what young children may notice. The second 2-hour session focuses on deepening understanding of what a pattern is, and what we want preschool children to understand about pattern.
As a pre-service teacher educator working with students who are just beginning their preschool teaching careers, I have three main goals: (1) help students draw on their own experiences in mathematics, good and bad, to create supportive mathematical learning environments for preschool children; (2) support students in gaining knowledge about early mathematical development; and (3) help students learn how to engage young children in rich mathematical activities
As a pre-service teacher educator working with students who are just beginning their preschool teaching careers, I have three main goals: (1) help students draw on their own experiences in mathematics, good and bad, to create supportive mathematical learning environments for preschool children; (2) support students in gaining knowledge about early mathematical development; and (3) help students learn how to engage young children in rich mathematical activities.
We describe a way to engage participants in learning about patterns and algebraic thinking in Teaching Algebraic Thinking to Young Children. Here is a description of how we used the activity as part of a university course.
Here is a detailed description of one round of the Shape Touch activity described in Teaching about Shape.
This resource explores connections among executive functioning (EF) and geometry and spatial relations and provides tips for early childhood educators to support EF while engaging in and geometry and spatial relations.
This resource explores three projects that can help your participants understand and build home-school connections into their mathematics teaching practice. Engaging in these projects might also help your participants uncover potential biases and incorrect assumptions they might have about children’s ability to learn math, as well as their families’ capacity to support them.
This page consists of video clips that are designed to illustrate assessing students using the pattern and algebra assessment protocol. Please visit the Pattern and Algebra Assessment Protocol for more details.
There are several ways in which you can use Thinking Stories about children’s mathematical thinking and/or corresponding unedited videos in your teaching, in a live class or on-line. Here are some simple exercises to accompany unedited videos either before examining them in class or before reading the corresponding Thinking Story.
Support early math learning in your preschool classroom by understanding practical strategies that can be used to support children’s success in math, while gaining confidence for yourself as a capable math teacher!
Drawing attention to supporting the math learning of dual language learners (DLLs) provides an opportunity for teacher educators to engage participants in exploring in-depth the relationship between language and mathematics in early learning settings. This resource highlights DREME TE materials that can be used in a variety of ways to support a focus on language and math.
By Angela Chan Turrou and Karen Recinos