Instructional Playbook
Essential Instructional Practice #6
Number Sense Routines
Early Math Instructional Playbook | 1
Essential Instructional Practices in Early Mathematics: Prekindergarten to Grade 5
Essential Instructional Practice #6
Introduction
Number sense routines are a powerful tool in the development of students’ mathematical thinking, problem solving and communication skills. All people, but especially our early learners, have an innate ability to ask questions and be curious. It is through number sense routines that we are able to tap into this curiosity, ask questions to push the collective learning edge and support students in deepening both conceptual and procedural understanding.
The following resource serves to articulate the common components of a high-quality number sense routine. Its intent is to provide guidance and clarity for the learning journey with the grand hope that all students in Michigan will be able to experience high-quality number sense routines. Similar to child learners, adult learners need to be met where they are in the learning process. Therefore, the playbook is most effective when regularly discussed with peers, broken down into the specific learning needs of the team or individual, practiced and reflected upon with colleagues. Let’s collectively use the playbook to bring joy back into mathematics.
Early Math Instructional Playbook | 1
The teacher poses a visual, verbal, and/or written mathematical prompt 3 Children think individually 4 Supported by the teacher, children share thinking, including non-fully formed ideas 5 The teacher notates and/or verbalizes the children’s strategies 6 The learning community discusses the thoughts that emerged and works to draw conclusions 7 mathessentials.org | #MiMathEssentials
Table of Contents
Essential 6
Engage children regularly in brief (5-15 minute) interactive number sense routines focused on developing mental strategies for seeing quantity and working flexibly with numbers.
Brief interactive number sense routines typically include the following steps:
the teacher poses a visual, verbal, and/or written mathematical prompt (e.g., “How many dots did you see?,” “How might you solve 32 x 5 using a mental strategy?”
children think individually;
supported by the teacher, children share thinking, including non-fully formed ideas;
the teacher notates and/or verbalizes the children’s strategies; and
the learning community discusses the thoughts that emerged and works to draw conclusions (e.g., “How do you know that doubling one factor and halving the other works? Does it always work?” and “Is this always an efficient strategy?”).
a. Use accessible prompts to engage children in conversations around purposefully crafted computation and/or quantitative reasoning problems to be solved mentally.
b. Encourage children to develop their own strategies, working toward solution strategies that make sense to them.
c. Elicit children’s thinking by asking them to share and explain their solution strategies, discuss the strategies of others, and make connections among multiple strategies.
Photo: Number sense routines.
Note: The example questions included in the description of typical number sense routines to the left were based on the student thinking shown in the photo above.
d. Support children in paraphrasing one another’s thinking to foster communication and language development.
e. Notate children’s strategies, as they collectively reason about numbers, to make ideas accessible to others and to encourage movement toward increasingly flexible and sophisticated mental strategies.
f. Emphasize sense making and de-emphasize speed.
g. Promote joy and curiosity by inviting children to share their mathematical ideas; strategically explore these ideas with excitement even though some may not yet be fully formed or correct.
Early Math Instructional Playbook | 2
The teacher poses a visual, verbal and/or written mathematical prompt
In One Sentence
X Teacher prompts are used to give children a mathematical problem to develop number sense, creativity, problem solving and communication.
Research
X John Hattie’s Research - Activating prior knowledge has an effect size of 0.98
X Tasks given orally drive groups to discuss what is being asked rather than trying to decode instructions on a page (Liljedahl pg. 22)
What’s the Point?
X Well situated prompts give space for creative problem solving inviting a range of strategies along a continuum of sophistication (low floor, high ceiling).
How prompts are used by teachers
X The teacher plans for a specific learning goal
X The teacher selects, often visual, prompts that are accessible to all children
SCAN QR CODE FOR VIDEO
Checklist: Posing Prompts
The teacher plans with a goal in mind (ie. math strategy, math concept, social interaction).
The teacher anticipates student strategies in the planning process.
Prior to launching the task, the teacher asks children what tools they can use to help them if they get stuck (fingers and anchor charts - ten frames, domino patterns, number lines, etc.).
The teacher invites children to solve the prompt using their own strategies.
X The teachers establishes a consistent classroom routine which includes the time of day, the location of students’ seats, the way the number-sense routine is launched
How prompts are used by children
X Children engage in sense-making by mentally exploring various strategies in pursuit of the solution to the prompt.
Early Math Instructional Playbook | 3
Children Think Individually
In One Sentence
X Children are given adequate, individual think-time to process connections and approaches to the prompt.
Research
X NCTM - Rethinking Wait Time: What can 3 seconds do?
What’s the Point?
X Giving children time to individually grapple with new ideas and prior concepts on their own deepens learning and potential connections.
How think-time is used by teachers
X Physically position your body to the side of the room or visual prompt so children are focused on the mathematics
X Observe ways children use the available tools (hands or classroom displays) such as fingers, hands, anchor charts, number lines, etc.
SCAN QR CODE FOR VIDEO
Checklist: Children Think
The teacher reminds children to use handsignals to express where they are in their thinking.
The teacher observes children during individual think time to ensure that children have enough time to productively engage in the prompt.
X Set norms for using hand signals to show when they are thinking or reveal how many strategies they have to solve the problem.
X Only move forward when all children have had adequate time to process a solution strategy.
How think-time is used by children
X Children make connections from what they already know and how it might connect to the prompt.
X They use the tools (ie. finger, anchor charts, number lines, etc.) to help them problem solve and be precise.
X They use hand signals to discreetly share where they are in their thinking.
Early Math Instructional Playbook | 4
Individually
Supported by the teachers, children share thinking, including non-fully formed ideas
In One Sentence
X After individual processing, the teacher invites children to speak, listen and question.
Research
X Teachers who consistently elicit student thinking during a lesson are better able to respond to the needs of students’ learning needs by using that evidence to adjust instruction. (Leahy et al. 2005).
X NCTM - Professional Noticing of Children’s Mathematical Thinking
What’s the Point?
X Allowing others to connect, question or expound on each others’ ideas deepens understanding of the concept or strategy.
How sharing thinking is used by teachers
SCAN QR CODE FOR VIDEO
Checklist: Children Share Thinking
The teacher uses structures to support all students in finding & engaging with a partner’s mathematical idea (ie. knee-to-knee, clap-and-turn).
The teacher provides a sentence stem (visually and with a verbal model) to encourage partner talk and to structure their language.
The teacher supports children in paraphrasing one another’s thinking.
The teacher positions themselves so they can listen and gather evidence of student thinking during partner talk.
After listening to the students’ partner talk, the teacher considers the sequence of strategies to be shared.
The teacher monitors talk time to allow the appropriate balance between partner and whole group dialogue.
X Provide sentence stems for sharing, responding and questioning.
X Use wait time to invite children to engage with each others’ ideas.
X Listen for key mathematical strategies in students’ responses.
Paraphrase or probe to emphasize these key strategies or ways of thinking (ie. unique approach, use of a math model, sequence to solve the problem, etc.).
X Intentionally consider diverse ways of thinking.
How sharing thinking is used by children
X Children show they are listening by looking, nodding and leaning-in.
X Children are curious and use sentence stems to share ways of thinking, respond to their classmates, paraphrase and question ideas.
Early Math Instructional Playbook | 5
The teacher notates
and/ or verbalizes the children’s strategies
In One Sentence
X The teacher honors children and their strategies by capturing, verbally or visually, the specific approach being offered and making the idea accessible to all students.
Research
X NCTM Taking Action - "[E]xternal representations give us greater access to our internal mental representations and allow us to examine, discuss, and explore mathematical ideas with others."
What’s the Point?
X Verbalizing and notating childrens’ strategies bridges the gap between their mathematical idea, the understanding of the class and more sophisticated ways of speaking, listening and using conventional symbols.
How verbalizing and notating is used by teachers
X Listen for and verbalize key strategies by paraphrasing in a way that is true to the students’ thinking.
SCAN QR CODE FOR VIDEO
Checklist: Teacher Notates and Verbalizes Children’s Strategies
Based on the goal and the evidence teachers have gathered, the teacher invites a child to share their strategy with the whole group.
The teacher pauses after a child shares.
The teacher inquires about a thought-process, mathematical idea or use of a tool so other children have access to it.
The teacher notates the child’s strategy using a mathematical model that exemplifies important mathematical ideas.
The teacher listens and invites class discussion.
The teacher asks the child if the summary was accurate.
X Notate strategies using purposefully chosen mathematical models that align with children’s thinking and support understanding.
X Through discussion, connect details of the model to students’ ideas to support sense-making.
X Label strategies to support connections between strategies and children’s thinking.
How verbalizing and notating is used by children
X Children share strategies using language, precise or imprecise, gestures, facial expressions and other means of communication.
X Children listen and make sense of their classmates’ thinking.
X Children encourage accurate portrayal of their strategy.
X Children make connections.
Early Math Instructional Playbook | 6
The learning community discusses the thoughts that emerged and works to draw conclusions
In One Sentence
X With the purpose of consolidation, the teacher invites students to make connections amongst the ideas that were shared and the goal of the routine.
Research
X Hattie Research - John Hattie’s research identifies Classroom Discussion as having an effect size on student achievement of 0.82.
What’s the Point?
X To consolidate important mathematical ideas from the lesson into a cohesive message connected to the learning goal.
How community discussion is used by teachers
X Model paraphrasing and encourage its use by students.
X Give students sentence stems.
SCAN QR CODE FOR VIDEO
Checklist: Teacher Notates and Verbalizes Children’s Strategies
The teacher intentionally plans for child discussion including partnerships and seating. Capture and communicate key mathematical ideas, both verbally and written, that align to the lesson goal.
The teacher and students paraphrase key mathematical ideas.
Sentence stems are considered to increase the sophistication of the class discussion.
The teacher consistently uses wait time
The teacher synthesizes the routine by connecting the goal with childrens’ strategies.
X Provide wait time.
X Apply reasoning to someone else’s idea - “Do you agree or disagree? Why?”
X Capture and communicate key mathematical ideas, both verbally and written, that align to the lesson goal.
How verbalizing and notating is used by children
X Students draw connections between the mathematical ideas of the lesson.
X Children paraphrase the thinking of others in their own words.
X Children ask their classmates questions about their thinking.
Early Math Instructional Playbook | 7