Setting a Purpose.

Before planning a unit or lesson, it is important to consider the five strands of proficiency necessary for the teaching and learning of mathematics, recommended by the National Research Council (2001): conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition. According to Kobett et al. (2018), these provide the basis for the three types of mathematics lessons, which are conceptual understanding, procedural fluency, and transfer.
Conceptual understanding lessons provide opportunities for students to make sense of math and form connections between mathematical ideas (Williams, 2019). These lessons include opportunities for exploration, mathematical discourse, representing learning in multiple ways, and abstraction. Procedural fluency lessons also play a critical role in mathematics curriculum and instruction. Procedural fluency involves more than memorizing facts, formulas, and algorithms. DESE defines fluency as the ability to use an appropriate strategy in a reasonable amount of time, know multiple processes, and the ability to apply or adapt strategies to find a correct solution. While conceptual understanding and procedural fluency should play a balanced role in math teaching and learning, transfer is the ultimate goal. Transfer involves students’ ability to apply the math they have learned to open-ended problems with real-world connections. Students must have opportunities to engage in transfer lessons to ensure they can apply the math they are learning to problems they will encounter when they are no longer in school.

Launch.

According to Kobett et al. (2018), the Launch is a problem or situation posed to stimulate student thinking. It allows opportunities for students to notice, wonder, ask questions, and /or build interest in a task or lesson goal. The Launch ensures that all students have access to the content to explore that lesson’s big idea, concept, or skill.
During the Launch portion of the lesson, the teacher designs an opportunity to build students’ readiness and interest for the lesson based on the lesson’s purpose. The launch may include an introduction to the specific problem-solving task or a number sense routine to help students think and talk about numbers, equations, or computations. The launch may be tied directly to the big idea and learning intentions for the day, or it may be used to circle back to the big idea or a concept previously learned so that students continue to build understanding.

During the planning of the Launch, the following questions should be considered:

• What is the main purpose of the lesson? (conceptual understanding, procedural fluency, application)
• How does this launch connect to previous mathematics my students have studied in this investigation or unit?
• What do students need to know to understand the story and the challenge of the problem?
• What advantages or difficulties can I foresee?
• How can I keep from giving away too much of the problem?

Teachers may consider items from the list shown.

• Focus Question
• Compare/Contrast two approaches to a problem
• Entry Ticket
• Act 1 of a 3-Act Task
• Error Analysis
• Visual Thinking Strategies

Facilitate.

According to Miles et al. (2019), the purpose of the Facilitate phase of the lesson is for students to explore a rich problem or strategic series of problems which will enable them to analyze and generalize a concept or skill. In facilitate, the teacher continually encourages students and strategically uses questions to position students as the mathematical leaders in the classroom. Students may tackle the mathematical task individually, with a partner, with a small group or sometimes as a whole class, depending on the challenge and format of the problem or problems.

When a teacher is planning the Facilitate portion, these questions should be considered.

• How will students be organized to be best engaged in the exploration (individuals, pairs, groups, whole class, or a combination)?
• What materials should be made available?
• What mathematical practices do I anticipate students using?
• How will students organize their thinking?
• What questions will encourage conversation, prompt their thinking when frustration is high, or probe further?
• What kinds of strategies will facilitate the summary and how will I sequence them?

Teachers may consider items from the list shown.

• Act 2 of a 3 Act Task
• Vocabulary
• Cooperative learning structures
• Use of manipulatives and visual models
• Student discussion
• Student Self-Assessment

Close.

Close provides the opportunity to reorganize the information from a lesson in a meaningful way by asking them to summarize, review, and demonstrate an understanding of the big ideas from the lesson (Williams, 2019). Close is the final phase of the lesson when teachers orchestrate whole group student discourse about discoveries or connections students made during the Facilitate phase of the lesson. It is during the Close that the teacher guides the students to reach the mathematical goals of the lesson and to connect their new understanding to prior knowledge within and beyond the unit.

When teachers are planning the close, these questions should be considered.

• How can I help students make sense of different mathematical strategies?
• How can I facilitate student-led summary discussions so students share their thinking?
• What explicit mathematical processes should be emphasized?
• How will these ideas be recorded for future reference?
• How can we go beyond? What new questions might arise?
• What will I do to follow up, practice, or apply the ideas after summarizing?

Teachers may consider items from the list shown.

• Revisit focus question
• Act 3 of a 3-Act Task
• Connect vocabulary
• Classroom discourse
• Student reflection/self-assessment
• Exit ticket or other formative assessment

Number Sense Routines.

In Number Sense Routines students participate in a brief activity for promoting engagement, reasoning, and discourse. During Number Sense Routines, a prompt is provided with multiple solution paths. Students notice, wonder, and share their thinking. The purpose is to help develop number sense and reasoning over time.

When teachers are planning number sense routines, these questions should be considered:

• What is the goal of the routine?
• What routine makes sense with my lesson? Will it be connected to the lesson or separate?
• When will the number sense routine occur?
• What questions are appropriate?

Teachers may consider items from the list shown.

• Number Sense Routines (5-7 minutes)
• Number Talks (15 minutes)
• Daily Word Problems (10 minutes)

Spatial Relationships.

Spatial awareness develops as children begin to explore the relationship between their bodies and their environment. As they learn to navigate their environment, they learn about direction, perspective, distance, symbolization, and location.

Positional words describe spatial relationships and help children deepen their understanding of those relationships.
During the planning of instruction on spatial relationships, the following questions and considerations are made.

• Before instruction
  • Identify the DRDP measure
  • What materials can be provided based on students’ interests?
  • Provide a safe environment for exploration.
• During instruction
  • Ask questions that will encourage conversation, prompt thinking, and probe further
• After instruction
  • Reflect by reviewing the students’ experience as related to the determined objective to inform planning

Teachers often consider items from the following list of instructional practices to support learning about spatial relationships.

• Model and encourage the use of positional words
• Use maps or other representations to help them think spatially
• Encourage students to create shapes from different shapes
• Present shapes that differ in size and orientation
• Describe spatial relationships as children play (“You’re putting the horse inside the fenced area you made.”

Classification.

To match, compare, and sort according to one or more attribute.
Classification in early years first begins with sensory perceptions, develops into identifying particular criteria, then categories, and lastly subcategories. Classification provides order and control for young children. It is foundational for higher mathematical processes.
During the planning of instruction on classification, the following questions and considerations are made.

• Before instruction
  • Identify the DRDP measure.
  • What materials can be provided based on students’ interests?
  • How can I encourage students to match, sort, classify, and resort materials?
• During instruction
  • Ask questions that pertain to classification capturing both process and product.
• After instruction
  • Reflect by reviewing the students’ experience as related to the determined objective to inform planning

Teachers often consider items from the following list of instructional practices to support learning about classification.

• Intentionally describe objects using characteristics to build vocabulary
• Provide materials in pairs so students can match by attribute
• Provide materials that have a relationship (socks/shoes, fork/spoon)
• Model sorting and classification of materials with unique characteristics

Patterning.

A repeating sequence or arrangement of objects, numbers, actions and events that systematically follow a given rule. Children must demonstrate the ability to identify attributes and characteristics in order to pattern successfully.
During the planning of instruction on patterning, the following questions and considerations are made.

• Before instruction
  • Identify the DRDP measure.
  • What modeling is provided so students can recognize and complete patterns?
  • Are students able to recognize patterns in their environment?
• During instruction
  • Ask questions that will encourage conversation, prompt thinking, and probe further.
• After instruction
  • Consider what students demonstrated to guide future experiences.

Teachers often consider items from the following list of instructional practices to support learning about patterning.

• Identify patterns in daily routines and the environment
  • Sing songs, do finger plays, and read stories that involve patterns
  • Model and encourage children to describe patterns by attributes
  • Support children as they copy and extend patterns, beginning with simple repeating patterns and then growing patterns
  • Encourage children to compare patterns and to find similarities and differences among them.

Shapes.

Shapes are the forms of objects which have boundary lines, angles, and surfaces. Mastery of attributes and characteristics is necessary for children to discriminate between shapes. A child shows an increasing knowledge of shapes and their characteristics when given the opportunity to manipulate, draw, compare, describe, sort, and represent shapes in a variety of ways.
During the planning of instruction on shapes, the following questions and considerations are made.

• Before instruction
  • Identify the DRDP measure.
  • Provide exposure to shapes with real-world items.
• During instruction
  • Ask questions that will encourage conversation, prompt thinking, and probe further.
  • Utilize math vocabulary when working with shapes.
• After instruction
  • Consider what students demonstrated to guide future experiences.

Teachers often consider items from the following list of instructional practices to support learning about shapes.

• Guide explorations of shapes, discuss the attributes as children explore (surface)
  • Label shapes with the correct name as students use them (surface)
  • Present shapes that differ in size and orientation (deep)
  • Encourage children to create new shapes from other shapes (transfer of learning)
  • Model and describe how to make two- and three- dimensional shapes (transfer of learning)

Quantities.

A student can show an increasing knowledge of quantities as they learn the verbal number sequence, one-to-one correspondence, and cardinality.
During the planning of instruction on quantities, the following questions and considerations are made.

• Before instruction
  • Identify the DRDP measure.
  • Plan chants, songs, or rhymes to encourage counting and 1:1 correspondence.
• During instruction
  • Ask questions that will encourage conversation, prompt thinking, and probe further.
• After instruction
  • Consider what students demonstrated to guide future experiences.

Teachers often consider items from the following list of instructional practices to support learning about quantities.

• Read stories that include numbers to count
  • Use everyday routines and activities as opportunities to count (i.e. one student left the blocks center, now there are two students in blocks. Would anyone like to join?)
  • Problem solve with counting
  • Touch or point to each object as you count
  • Model counting strategies, show how to track the number of objects counted (i.e. moving objects to one side as you count)
  • Use multiple learning styles to count (i.e. touch, sight and sound)

Operations.

To quantify is to count or express something in numbers. Children identify the number of groups of objects in whole, when objects are added or are removed. Children learn concepts of more and less than, the position of sequence of something
During the planning of instruction on operations, the following questions and considerations are made.

• Before instruction
  • Identify the DRDP measure.
  • Provide materials for math exploration.
• During instruction
  • Ask questions that will encourage conversation, prompt thinking, and probe further.
  • Communicate the addition and/or subtraction when working with materials,
• After instruction
  • Consider what students demonstrated to guide future experiences.

Teachers often consider items from the following list of instructional practices to support learning about operations.

• Create a numerically rich environment
  • Use fingerplays, rhymes or songs about numbers
  • Practice counting using a variety of learning styles & representations
  • Model counting strategies
  • Make obvious mistakes so that children can identify the errors
  • Model comparing the number of objects in two sets
  • Use everyday situations to illustrate combining and separating

Measurement.

A student can show an increasing knowledge of measurement as they learn the concepts of measurable properties such as size, length, weight and capacity and how to quantify those properties.
During the planning of instruction on operations, the following questions and considerations are made.

• Before instruction
  • Identify the DRDP measure.
  • Provide standard and non-standard materials for exploration.
• During instruction
  • Ask questions that will encourage conversation, prompt thinking, and probe further.
• After instruction
  • Consider what students demonstrated to guide future experiences.

Teachers often consider items from the following list of instructional practices to support learning about measurement.

• Model measuring behaviors and language, including estimation vocabulary
  • Use daily experiences to discuss measurement concepts (surface)
  • Use self-talk as you measure
  • Offer a variety of measuring tools for investigation and encourage their use during activities and dramatic play
  • Offer opportunities that allow children to compare measuring with non-standard and with standard measures
  • Encourage measurement problem-solving activities