Professional Learning Modules

Leader Resources

Overview

The leader resources provide district, school, and early childhood center leaders with tools and recommendations to implement the Teaching Math to Young Children Toolkit in their setting. The toolkit’s professional learning modules help teachers strengthen early math instruction using evidence-based practices; these leader resources ensure the structures, schedules, and supports are in place so teachers can successfully complete the professional learning and adopt and sustain evidence-based instructional practice.

These resources help leaders:

• Understand the purpose and structure of the professional learning modules.
• Assess and improve system-level supports for ongoing professional learning.
• Observe and discuss early math instructional practice.
• Coordinate implementation across classrooms or programs.
• Support teachers in building long-term, evidence-based math instruction.

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Download the Supporting Evidence-Based Practices for Teaching Math to Young Children Guide to access all leader resources and get started!

Supporting Evidence-Based Practices for Teaching Math to Young Children Guide

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Assessment of System Supports for Evidence-Based Teaching

The Assessment of System Supports for Evidence-Based Teaching (ASSET) is a diagnostic and monitoring tool that helps leaders determine whether the organizational structures needed to support high-quality early math instruction are in place. Built to accompany the Teaching Math to Young Children Toolkit, the ASSET outlines specific system supports, such as planning time, instructional time for math, and access to materials and helps leaders identify strengths and gaps across these supports. With this information leaders in early childhood centers, schools, and districts can put conditions in place that help teachers successfully implement and sustain the toolkit’s evidence-based instructional practices.

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Download the ASSET to get started!

Assessment of System Supports for Evidence-Based Teaching

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Early Math Teaching Observation Tool

The Early Math Teaching Observation Tool (EMTOT) measures teacher implementation of key instructional practices to develop children’s early math skills that are supported by research. It is designed for use by leaders who support teachers’ early math instruction, such as instructional coaches, principals, or early childhood directors.

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The EMTOT includes multiple resources.

• EMTOT (342 KB): The Excel-based EMTOT includes information on how to use the tool, four timepoint tabs to collect observation data, a results summary tab, and a tab with professional learning resources that support the observed practices.
• EMTOT Observer Form (425 KB): This form is an optional companion to the EMTOT that can be easily printed or completed electronically during classroom observations.
• Teacher and Leader Professional Dialogue Protocol (554 KB): This protocol is designed for use following an EMTOT observation. It supports a strengths-based dialogue geared toward professional learning.

How the EMTOT supports early math instruction

Using the EMTOT helps:

• Provide a common language and shared understanding about key math instructional practices in early childhood classrooms.
• Generate instructional practice data to inform conversations between leaders and teachers about teachers’ areas of strength and where teachers might need professional learning support.
• Identify ongoing professional learning goals and associated resources for professional learning.

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Glossary

The   Glossary (61 KB) defines key terms used in the toolkit. Access a downloadable version or review terms below.

Abstraction principle

The understanding that anything can be counted; for example, cats, ideas, pairs of things, sounds, and dozens can all be counted.

Advanced counting strategies

Efficient ways children use counting to solve addition or subtraction problems, rather than needing to start from one each time or use objects to represent every quantity. For example, when given a set of three green beads and four purple beads and asked “how many beads all together?” the child represents three mentally and counts on as “...four, five, six, seven.” Or a child skip counts by two to determine how many shoes are in the classroom cubbies.

Basic addition and subtraction facts (also called “basic facts”)

Single-digit (0–9) addition and subtraction combinations or problems. Basic addition facts range from 0 + 0 = 0 to 9 + 9 = 18. Basic subtraction facts range from 0 - 0 = 0 to 18 - 9 = 9.

Basic fact recall (also called “basic fact fluency”)

Ability to retrieve basic addition and subtraction facts from memory without counting.

Cardinal amount

The total quantity in a set; also the last counting word used in a counting procedure.

Cardinality

The property of a number word that tells “how many” items are in a set. For example, the “three-ness of three”—as opposed to that word’s use as a label in the counting process, “one, two, three, four... there are four.”

Cardinality chart

A visual display that demonstrates how each number in the count sequence relates to the total number of objects it represents and illustrates how quantity increases by one with each count word.

Cardinality principle

The understanding that when a set of objects is counted, the last number word spoken represents the total number of objects in the set.

Conceptual subitizing

Instantly seeing (without counting) multiple small, perceptually subitizable quantities, and recognizing the whole quantity they combine to create.

Coordination errors

Counting errors in which a child fails to keep number words and objects in one-to-one correspondence, such as labeling an object with more than one number word or pointing to an object without saying a number word for it.

Counting all

The most basic of direct modeling strategies for addition where a child starts from one and counts every item in all sets of objects to find the total. For example, when adding 3 and 3, holding up three fingers on each hand and then counting the fingers held up on both hands as “One, two, three, four, five, six. Three and three make six.”

Counting on

A type of advanced counting strategy for adding two sets by starting with the counting word for the quantity of one set and then counting upward to account for the second set. Young children may use fingers (or other objects) to represent the quantities so they can track when to stop counting. For example, when adding 5 and 2, a child holds up five fingers at once and says “five...” and then holds up two fingers and counts them as “...six, seven... the answer is seven.” As children advance in number sense, they may use the counting on strategy mentally, without the use of fingers or objects.

Counting out

Creating a new, smaller set of an exact size from a larger set by counting each object as it is moved and stopping when the desired number is reached. This requires the counter to enact the counting process and organize materials while simultaneously “holding” the amount desired for the new set in their mind so they can stop their process at the right time.

Counting procedure

A coordinated process to determine quantity, it combines the use of counting words, a stable order or sequence for those words, one-to-one tagging between the words and the objects to be counted, and the use of the last counting word to name the quantity.

Counting words

The sequence of words we use to tag items when counting and denote quantities. For example, “one, two, three...” and so on.

Count sequence

The order of the number words that must be used when counting to produce an accurate total quantity.

Developmental progression

An ordered sequence of increasingly sophisticated phases that children typically move through as they build their skills and understanding of a particular math topic.

Direct modeling strategies

When a child uses real objects or representations of objects (such as fingers, counters, blocks, or tally marks) to “act out” a math problem such as addition, subtraction, or comparing two sets. For example, to solve two bears and two more bears, a child holds up two fingers on one hand and two fingers on their other hand, and then counts all the standing fingers to get a total of four. Or, when comparing sets of five and three blocks, a child builds two towers of blocks, one for each set, and places them side by side to see which is taller and by how many blocks.

Five counting principles

The core principles that underlie the counting process and define an adult understanding of what counting is and can do. These are 1) the stable order principle, 2) the one-to-one principle, 3) the cardinality principle, 4) the abstraction principle, and 5) the order irrelevance principle.

Increasing magnitude principle

The understanding that as the number word sequence continues, each successive number represents a larger quantity than the one before it; that is, their magnitude increases.

Keeping track errors

Counting errors in which a child loses track of which objects have already been counted, often resulting in recounting or skipping items.

Key instructional practices

A specific set of teaching actions that embody the recommendations the toolkit is designed to help teachers implement.

Magnitude

The size of a quantity; how relatively large or small it is. That is, seven has greater magnitude than five.

Meaningful object counting

The ability to accurately apply the counting procedure to a set of objects and understand that the final number word represents the total amount; see also cardinality principle.

Mental number list

A child’s internal, organized representation of the number word sequence, in which each number has a stable position and represents a specific, increasing quantity. This mental number list can become a tool for determining which quantity is bigger and which is smaller, or for counting on to solve addition or subtraction problems.

Nonverbal addition/subtraction (also called mental addition or subtraction)

Solving basic addition or subtraction problems by producing the solution set using mental representations of the quantities, often within the context of “hidden results” problems. For example, a child is shown two blocks, and then those blocks are hidden from view. Next, the child is shown another block being added to the hidden set. Without being able to see all the blocks, the child is asked to show how many blocks are hidden using another set of blocks.

Number

A symbol or word that represents a specific quantity; it can refer to the word (“three”), the written numeral (3), the actual quantity of objects being represented, or the concept of “three-ness.”

Number-after knowledge

The ability to state the number word that follows a given number in the count sequence without having to count from one. This skill contributes to children’s ability to “count on” and use other advanced counting strategies.

Number and operations

The area of mathematics that focuses on quantities and the relationships among them, including how numbers can be combined, separated, and compared.

Number path

A visual model that represents the count sequence in order, often presented as a continuous row of connected boxes (or spaces), each labeled with a numeral to show the progression of numbers.

Number word sequence

The verbal list of number words that children learn to say in order; knowing which word comes next and in what order. Young children may know the number word sequence but not be able to apply it accurately when counting objects.

Numeral

A visual symbol for referring to a quantity.

One-to-one correspondence

The understanding that each object in one set is paired with exactly one object in another set, showing that the two sets have the same quantity. For example, the fact that three pencils are enough for three children to each have one.

One-to-one tagging

The assignment of one and only one count word to each object in a set to be counted. May involve touching objects or moving them, or may be enacted entirely with words.

Order irrelevance principle

The understanding that the objects in a set can be counted in any order.

Perceptual subitizing

Instantly seeing and recognizing small quantities, up to four or five, without counting.

Progress monitoring

An ongoing process in which a teacher tracks children’s developing skills and understanding in a math topic in order to anticipate and plan for future learning needs.

Quantity

The measurable or countable attribute of something; its cardinal amount. Quantities can be discrete (individual items you can count, like five apples) or continuous (something you can measure, like five inches of string).

Skimming errors

Counting errors in which a child gestures vaguely over a group of objects and moves their finger across the entire set without establishing one-to-one correspondence.

Stable order principle

The understanding that number words must always be said in the same fixed order when counting to produce an accurate quantity.

Subitizing

Recognizing a quantity without counting. For example, glancing at * * * and knowing that there are three asterisks without enacting a counting procedure (that is, without saying “one, two, three... there are three”). See also conceptual subitizing and perceptual subitizing.

Successor principle

The understanding that each counting word is exactly one more than the number it follows, and one less than the number it precedes.

Visual structure

The organization of visual elements that can support children’s ability to subitize. For example, the way dots on the five face of a die are organized provides visual structure as a recognizable pattern.