We help teachers, coaches, and administrators to define and work towards a common vision of what it means to know, learn, and teach mathematics.

We provide teachers with the knowledge and tools to engage all students in rigorous mathematics instruction and implement the kinds of cognitively demanding tasks that are called for by the National Council of Teachers of Mathematics and 21st century standards.

We understand the power of shared experiences. We ask participants to implement common tasks, lessons, or units with their students and bring back artifacts of practice for discussion, reflection, and further planning.

Math Planning Essentials

Each Math Planning Essentials guide (MPEs) pulls together key elements of lesson planning that are organized by a set of essential understandings-the underlying mathematical truths worth exploring for a given concept of study. MPEs can help make sense of these truths by providing information about how they might playout as students' conceptual understanding develops.

Each MPE provides:

  • Essential Understandings that name what students need to know for a mathematical concept.
  • Tasks to contextualize the mathematics.
  • Ways students may Use & Explain Representations in response to the tasks, paired with suggestions to further elicit student thinking.
  • Prompts and questions that
    • ask students to Make Connections among differentrepresentations shown in and across student responses and
    • require students to Explain Mathematical Reasoning about the ideas articulated in the essential understandings.
  • A generalized list of what Students Should be Able to Do and Say, if they have made sense of the essential understandings.
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  • Multiplication and Division
  • Understanding the Meaning of Multiplication
  • Understanding and Using Multiplication Properties of Operations
  • Understanding the Meaning of Division
  • Fractions
  • Understanding Fractions as Numbers
  • Comparing Fractions
  • Adding and Subtracting with Fractions
  • Multiplying with Fractions
  • Dividing with Fractions
  • Negative Numbers
  • Understanding, Ordering, and Comparing Negative Numbers
  • Adding and Subtracting with Negative Numbers
  • Multiplying and Dividing with Negative Numbers
  • Ratios & Proportional Relationships
  • Understanding Different Types of Ratios
  • Recognizing and Generating Equivalent Ratios
  • Identifying and Using Proportional Relationships
  • Expressions, Equations, & Inequalities
  • Writing, Interpreting, and Evaluating Expressions
  • Solving One-Variable Equations
  • Solving Two-Variable Equations
  • Analyzing Linear Relationships
  • Solving Systems of Linear Equations
  • Solving Inequalities

Professional Development

Shapes discussion elementary girls

Our professional development courses help mathematics educators

  • Select cognitively demanding tasks that are consistent with 21st century standards
  • Modify tasks to increase the cognitive demand
  • Engage students in cognitively demanding tasks in ways that maintain the cognitive demand
  • Use targeted questions to assess and advance student learning so that all students have the opportunity to advance in their thinking
  • Engage students in discussions to make sense of mathematical concepts and make connections between and among mathematical representations in order to form mathematical generalizations
  • Use quick writes to formatively assess student growth
  • Study student work to identify what students know and can do and determine instructional next steps

Content Overlapping Verbal-language and Efficacy Learning

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COVE is designed to support educators in assessing and advancing children's thinking and reasoning in mathematics and was developed by IFL mathematics fellows based on years of research about teaching and learning in early childhood. COVE helps educators deliver instruction that encourages engagement and sense-making. These materials provide educators insights about how to leverage students' natural curiosity and interests related to foundational mathematical ideas (sorting and categorizing, conservation, and counting and cardinality) and offers educators opportunities to deepen their understanding of mathematical content, content pedagogy, and student thinking. COVE sets clear expectations that students should communicate their thinking and reasoning about mathematical concepts through actions and expressive language.

The Content Overlapping Verbal-language and Efficacy product is based on the understanding that early mathematical learning experiences shape not only students' mathematical thinking, but also their thinking about what mathematics is. Educators explore the progressions through which students move while developing mathematical understanding and reflect on the ways in which students engage with and make sense of mathematics (efficacy), and communicate their mathematical thinking (verbal language).

Through the use of this product, educators learn about strategies for increasing opportunities for young students to share their thinking and reasoning related to mathematical ideas.

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Supporting Ongoing Achievement Responsively

Mathematics materials
SOAR tools and activities are designed to identify and support students' mathematical understanding that is in need of further inquiry and development

The Supporting Ongoing Achievement Responsively (SOAR) learning series provides educators an opportunity to explore topics related to supporting students as they solidify their areas of unfinished learning, while making use of the IFL's SOAR materials. The SOAR materials comprise two main parts: math survey and instructional materials. The SOAR survey and materials can be used at all levels of tiered support.

SOAR Survey Tools

The survey tools are comprised of three parts: a set of survey questions, a monitoring checklist, and a tiered survey summary. The survey can be used as a stand-alone tool or as a follow-up to universal screeners that give only broad areas of need. Teachers gain valuable data on their students' ability to make send of, represent, solve, and explain mathematical concepts.

SOAR Activities

The intervention activities are designed and sequenced to support students across tiers of intervention. These activities are aligned to the survey tool and are designed to support student learning in making sense, creating and using representations, using strategies, and providing explanations. Each activity has teacher facing materials that outline the mathematical foci, identify the essential mathematical understandings, as well as provide questions to target those understandings.

SOAR tools and activities are available for the following concepts:

  • Number and Operations in Base Ten: (Grades 1 and Up)
  • Operations and Algebraic Thinking: Addition and Subtraction (Grades 1 to 5)
  • Operations and Algebraic Thinking: Multiplication and Division (Grades 3 and Up)
  • Number and Operations: Fractions (Grades 3 to 6)
  • Ratios and Proportional Relationships (Grades 6 and Up)
  • Expressions and Equations (Grades 6 and Up) JANUARY 2020

Contact the IFL to host a workshop in your district.


Instructional Materials

Our professional development courses range from short, targeted workshops to courses that are embedded in multi-year school, district, or state partnerships. Our courses are most effective when teachers work side-by-side with professionals who support classroom instruction, such as coaches, administrators, and central office educators. Additional courses are available for leaders to support mathematics teaching and learning.

The IFL's Sets of Related Lessons include eight high-level tasks designed around key mathematics concept, and accompanying lesson guides. Sets of Related Lessons can be strategically integrated into existing curricula.

A Set of Related Lessons has two types of tasks: developing understanding tasks and solidifying understanding tasks. Tasks are sequenced to provide students with opportunities to investigate concepts and skills in a coherent way. First, to make discoveries and then to form mathematical generalizations related to the concepts under study.

Math SOAR bookletsSets of Related Lessons include:

  • an outline of the concepts students will study;
  • an articulation of strategies and representations for students to learn and/or use;
  • multiple and varied forms of formative assessment, including opportunities for students to write about their mathematical understanding; and
  • structures and routines to support students' engagement with tasks, including information about:
    • setting up the task so students have an entry point;
    • supporting students in exploring the mathematical ideas developed in the task both individually and in small groups; and
    • engaging students in teacher-orchestrated discussions to deepen and solidify their understanding of the key mathematical ideas.

A unique aspect of the Sets of Related Lessons is the identification of essential understandings (EUs)-underlying mathematical truths-within the lessons. EUs drive the sequence of the solution paths and the discussion questions outlined in the lesson guides.

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Performance Based Assessments

The IFL's math performance-based assessments (PBAs) are comprised of open-ended, high-level tasks designed to assess students' learning of Standards for Mathematical Content and Standards for Mathematical Practice as outlined in the Common Core State Standards. PBAs require students to apply what they have learned related to a CCSS concept or domain.

The IFL math PBAs can be flexibly inserted into existing curricula, or they can be used in conjunction with the IFL's Sets of Related Lessons. Because the IFL's PBAs assess students' learning of concepts and practices specific to IFL math Sets of Related Lessons, assessments can be used strategically during the implementation to make sure learning is on target. Each PBA comes with a scoring guide and annotated student work. PBAs that are related to a Set of Related Lessons are indicated in the description.

Contact the IFL ( for more information about Project-Based Assessments.