Elementary Mathematics Daily high-quality first instruction
Daily mathematics instruction is to be no less than 75 minutes. This time includes necessary transitions. 75 minutes provides for an opening routine (~10 min), core instruction, and closure. High-quality mathematics instruction has many components. The tabs below outline expectations and related resources for the components.
The goal of high-quality math instruction is to make it student centered. This means that students are the ones who are doing the thinking, talking, and sense making. Creating one's own understanding is essential for positive identities, student agency, and self-efficacy in mathematics. Explicit instruction is necessary. However, we want student to first engage, explore, and potentially develop their own understanding of concepts, skills, and procedures. Positioning students to try the math first, provides insight into strategies and misconceptions to address during explicit instruction. A process for a student-first approach is the Launch-Engage-Debrief-Explain (LEDE). LEDE builds on the 3-stage math lesson (Van de Walle, 2019), the LED model (SanGiovanni, et al, 2023), and traditional 5E lesson model. This LEDE process may play out once or possibly more than once during either a whole group or small group lesson structure.
Launch
Description:
Capture students’ interest
Encourage questions and critical thinking
Taps prior knowledge
Possible Actions:
Game or puzzle
Picture book
Video clip
Notice / Wonder (etc)
Real-world situation/connection
Where It Goes Wrong:
Takes too long
Math focus is distracted by activity
Implementation style doesn’t carry over to lesson
Engage
Description:
Hands-on/minds-on activity
Students work together
Possible Actions:
Cooperative learning task
Hands-on manipulative task
Investigation or problem solving task
Where It Goes Wrong:
Groups become idle when finished or stuck
Teacher gets hung-up with a group.
Debrief
Description:
Student explanations
Connections between representations and explanations
Focus on what makes sense and why
Possible Actions:
Poster sharing
Teacher recording student thoughts
Display and discuss
Where It Goes Wrong:
Discussion is random or focused only on correct
Sharing lingers (too many share, disengagement)
Student sharing cannot be heard/understood
Ideas are accepted without question
Explain
Description:
Teacher explains concept or process
Teacher may model a concept
Teacher connects logical approaches and/or points out flawed logic
Possible Actions:
Modeling
Recording
Demonstration
Where It Goes Wrong:
Student ideas are not connected or ignored
Stark shift to a specific way the teacher tells them they should do it
Selecting a Number Routine
Number routines are brief opportunities for students to engage in and discuss mathematics so that they can develop number sense, fluency, and reasoning. They are centered on the idea of a little bit of high-quality practice over the course of a year is highly beneficial. These routines take advantage of students' peak attention and interest during the first few minutes of class. They are used in place of going over homework or traditional bellringers or warm-ups. In fact, neither are appropriate, worthwhile or recommended for the start of the mathematics class. Use the Routines link in the upper lefthand side for examples and more information.
Keep in mind that number routines:
are intended to be a quick activity to develop number sense, fluency, and reasoning.
should take about 10 minutes.
should leverage think-pair-share and other cooperative strategies.
are intended to engage students in mental mathematics, reasoning, and discussion. Use of paper/pencil should be limited.
should be modified to meet the needs of your students.
should feature carefully selected topics.
should spotlight different strategies or solutions but are not intended for every student to share out or for every solution to be explored.
The Routine:
Engages students with a (typically) mental math activity.
Enables students to share their ideas with classmates.
Features a discussion facilitated by the teacher.
Is an opportunity for engaging practice of skills and concepts that were previously taught. It does not feature a topic that is currently being taught.
High-quality learning tasks have a significant impact on student learning. A low-quality task cannot be masked with a greater quantity of problems. In other words, 40 problems on a worksheet do not come close to the quality needed for deep learning of mathematics concepts. Additionally, learning tasks - regardless of rigor - must be aligned to the mathematics standards being taught.
Task Selection
Standards are linked module scope and sequence. Tasks are provided on each standard's page. Tasks are to align with the learning targets. Due to the nature of mathematics (and teaching and learning in general) tasks are likely to align with more than one learning target. Some tasks framed with Google slides. A good math task does not require slides. When slides are used, they should be modified for the needs of the students in your class.
High-quality Tasks
Align with relevant mathematics content standards.
Connect previous knowledge with new learning.
Encourage the use of representations.
Provide opportunities for students to develop and demonstrate the mathematical practices.
Promote reasoning and problem solving.
Allow for multiple entry points (All students can begin the task. Task can be extended.)
Allows for multiple solution approaches and strategies.
Engages students in explaining the meaning of the result.
Mathematics content can be represented in 5 distinct ways as shown in the graphic. It is important that students represent ideas in their own ways. It's also important that explicit connections are made between and within the types representations. An example of connections between types would be a physical representation of a fraction with fraction tiles should be connect to visual (drawing) and/or the symbolic recording. An example of within a type could be connections between a fraction shown with fraction tiles, Cuisenaire rods, fraction circles, and maybe pattern blocks.
The exchange of ideas is at the center of learning anything - especially mathematics. Math curriculum standards calls for students to make mathematical arguments and to critique the arguments of others. They call for students to talk about mathematics using mathematical language and to support or oppose the work of others as they actively engage in this practice. Discussions are a vital component of learning mathematics. They must be intentional, well planned, and student centered and must allow students to express and share their ideas freely yet purposefully. Discourse should be centered on the mathematical goal of the lesson, and you’ll want to be sure you’ve selected a task that not only meets the goal but also leaves plenty of room for exploration. Note that discourse also includes written and visual forms of expression.
In 5 Practices for Orchestrating Productive Mathematics Discussions, Smith and Stein (2011) describe five practices for effectively using student responses as a tool to facilitate student discussions.
Anticipate (while planning or preparing the lesson)
Monitor (while students engage with the task)
Select (while students engage with the task)
Sequence (the order of student ideas to build on one another; this begins during selection)
Connect (during the debrief, make connections between ideas)
Students should have numerous opportunities to engage in activities that promote active engagement in mathematical discourse throughout the lesson. Three instructional moves or tools to help are:
Using Turn and Talk or Think-Pair-Share liberally with good, thinking questions
Using Question Stems
Providing Sentence Starters to support discussion
Capability with any skill is rooted in understanding and realized through practice. Mathematics skills and concepts are no different. Students must have daily practice of instructional topics, especially computational and fact fluency. Quality, effective practice is just as important as the hours of practice or the volume of practice (Schwartz, 2017). High-quality fluency practice is focused, varied, processed, and connected (Bay-Williams & SanGiovanni, 2021). The amount of practice needed varies by student. Practice should be balanced between games, centers, and paper-and-pencil tasks.
Differentiated instruction is designed so that each and every student has access to first instruction. Basic examples of differentiated instruction in mathematics include:
Accuracy and Computation: Provide calculators, basic fact charts, or number charts for computation.
Discussion: Provide anchor charts with sentence frames to help students engage in productive discussions.
Vocabulary / Precision: Provide word cards and other tools for supporting vocabulary use and development.
Organization: Provide graphic organizers to help students organize their problem solving, sense making, or the development of their arguments.
Representation: Provide a variety of manipulatives and examples of how concepts can be drawn or diagramed. Make connections between the representations.
Group Structures: Use small group instruction with different tasks and scaffolds.
Open Tasks: Use open tasks so that students can enter (and solve) in a variety of ways. An example, the sum of two numbers is 51. What are those numbers? Or Make this statement true in as many ways as you can ___ + ____ = 73.
Collaboration: Promote partner and triad experiences for students to collaboratively compete.
Choice: Provide a variety of centers to students targeting different needs with skills and concepts.
Tiered Assignments and Parallel Tasks: Use Tiered Assignments when appropriate. Note that simply changing numbers can compromise the expectation of the grade level standard. In a parallel task, one group might be asked to create examples of fractions equivalent to 3/4, 2/3, and 4/5 whereas another group would make examples of any equivalent fraction
Extend and Enrich: Extend and enrich assignments for students who show mastery or proficiency.
Choice Boards: Provide choice boards for independent work time or homework.
Math Journals: Use writing prompts to know your students better and to incorporate reflection and summary.
Learning Spaces: Provide a variety of learning spaces for students to use so that they are comfortable and easily engaged. This could be a work area with a podium, a space on the floor, a space at a U-shape table, or anything else.
Time: Adjust or remove time limitations for tasks and related experiences.
Determining the Structure of the Mathematics Class
The 75-minute class should follow one of a few different specific structures listed below. The same structure should not be used every day. The structure should be aligned with the purpose of the lesson and the needs of your students. Classes are not required to use the same model everyday. The model for each day should be selected intentionally to support the needs of students and the goals of the lesson.
The purpose of the lesson and the desired outcome should drive structure of the mathematics class. Regardless of structure each lesson should begin with a number routine and end with closure.
Whole-Group, Collaborative Lessons
Whole-group, collaborative lesson structures are appropriate for any high-quality task. They are clearly the best option for 3-Act Tasks and similar activities. Tasks from the preferred resources tab are appropriate for this structure. This structure offers the most time for engaging, exploring, and discussing mathematics. These lessons typically feature three segments, including
Pose a problem or prompt
Engage pairs or triads in the problem (pausing the whole group for discussion as needed)
Debrief the task with the whole group discussion strategies and solutions
Small-Group Differentiated Lessons
Small-group lesson structures are appropriate for considerably differentiated lessons. During small group instruction, student groups should not experience the exact same task. It should be differentiated. Small-group lessons are good for extending or reteaching a concept. They should not be used merely for better classroom management. High-quality tasks require time and small-groups can jeopardize this. There should be no more than two-groups during the small-group lesson structure.
How is it differentiated?
Small-group or rotation-based lessons are perfect for differentiating a concept. Mathematics can be differentiated with the number of or type of representations used. It can be differentiated by the complexity of a task or the complexity of the numbers in the task. Collaborative discussions with co-teachers and coaches can help develop ideas for differentiating small-group instruction.
The Rotation Model is used to differentiate instruction. Student experiences in the two small groups learn about the same content in different ways.
This model is good for providing differentiated support to students. It begins with a whole group task. The teacher assesses student progress within the lesson. The teacher then pulls a small group to provide additional instruction while some students work on an independent task.
The National Council of Teachers of Mathematics (NCTM) Effective Teaching Practices are 8 evidence-based, research-informed instructional practices for equitable mathematics. The mathematics teaching practices are
Establish mathematics goals to focus learning. Effective teaching of mathematics establishes clear goals for the mathematics that students are learning, situates goals within learning progressions, and uses the goals to guide instructional decisions.
Implement tasks that promote reasoning and problem solving. Effective teaching of mathematics engages students in solving and discussing tasks that promote mathematical reasoning and problem solving and allow multiple entry points and varied solution strategies.
Use and connect mathematical representations. Effective teaching of mathematics engages students in making connections among mathematical representations to deepen understanding of mathematics concepts and procedures and as tools for problem solving.
Facilitate meaningful mathematical discourse. Effective teaching of mathematics facilitates discourse among students to build shared understanding of mathematical ideas by analyzing and comparing student approaches and arguments.
Pose purposeful questions. Effective teaching of mathematics uses purposeful questions to assess and advance students’ reasoning and sense making about important mathematical ideas and relationships.
Build procedural fluency from conceptual understanding. Effective teaching of mathematics builds fluency with procedures on a foundation of conceptual understanding so that students, over time, become skillful in using procedures flexibly as they solve contextual and mathematical problems.
Support productive struggle in learning mathematics. Effective teaching of mathematics consistently provides students, individually and collectively, with opportunities and supports to engage in productive struggle as they grapple with mathematical ideas and relationships.
Elicit and use evidence of student thinking. Effective teaching of mathematics uses evidence of student thinking to assess progress toward mathematical understanding and to adjust instruction continually in ways that support and extend learning.