Whole Number Foundations Level 2

Whole Number Foundations Level 2 (WNF-2) is designed to accelerate the mathematics achievement of second-grade students who are at risk for mathematics difficulties by focusing on whole number understanding within 1000 and related skills. Topics include place value and base ten concepts within 1000, number combinations to 20, multi-digit addition and subtraction within 1000 (with and without regrouping), word problem solving, and foundations for multiplication. 

• Tier 2 small-group intervention provided in addition to core (Tier 1) instruction
• 70 lessons delivered three to five times per week in 30-minute sessions
• Instructional objectives aligned with the Common Core State Standards for Grade 2 (specifically the Operations & Algebraic Thinking and Number & Operations in Base 10 Domains)
• In our 2023-2024 Pilot Study conducted with 124 students, we found that students randomly assigned to receive WNF-2 experienced significantly greater gains compared to a business-as-usual control group on a math curriculum-based measure (ASPENS; g = 0.56), the WNF-2 Proximal Assessment (g = 0.42), and a Mathematics Vocabulary measure (g = 1.23).

Conceptual Framework

The WNF-2 Conceptual Framework is composed of three elements: (1) Understanding of Whole Number Concepts, Skills, & Vocabulary, (2) High-Quality Instructional Interactions, and (3) Self-Regulated Learning. Figure 1 shows the WNF-2 Conceptual Framework and alignment with the 8 Standards for Mathematical Practice.

Understanding of Whole Number Concepts, Skills, & Vocabulary

• WNF-2 focuses on critical math content for students to master in second grade. This includes teaching students to use flexible problem-solving strategies, engage in multidigit computation, and establish a deep understanding of place value and the base ten system. Students also learn to identify and solve three types of word problems (part-whole, change, and difference problems).
• Targeted Mathematical Practice Standards:
  • (MP 2) Reason abstractly & quantitatively: In WNF-2, students write equations to represent and solve word problems. They identify key components of word problems, explain the meaning of each number, and reason to determine the problem type and operation needed to solve. Students also reason about effective strategies to solve number combinations within 20 based on their unique characteristics.
  • (MP 4) Model with mathematics: In WNF-2, students model real-life situations using manipulatives, pictorial models, and equations. Students have access to a variety of mathematical models which they use as they learn new skills to build their conceptual understanding.
  • (MP 5) Use appropriate tools strategically: In WNF-2, students use mathematical models, such as base-ten blocks, counters, number lines, and place value charts to represent and solve mathematical problems. Students also use these models to demonstrate mathematical ideas, including place value, regrouping, multidigit computation, skip counting, even and odd numbers, and strategies for solving number combinations.
  • (MP 8) Look for & express regularity in repeated reasoning: In WNF-2, students are taught to look for structures to help them solve problems. To solve word problems, students learn to identify the problem type and use the underlying problem structure to help them write an equation. Students also learn strategies to solve different types of number combinations, then apply those strategies to build math fact fluency. Opportunities to promote generalization are provided in Math Extension Sheets.

High-Quality Instructional Interactions

• WNF-2 is designed to facilitate frequent and high-quality interactions between students and teachers around key mathematical content. Teachers engage students in varied response opportunities designed to maximize practice and increase their capacity to think mathematically and justify their reasoning:
  • Group Response Opportunities: Students respond in unison. Group response opportunities are designed to maximize practice throughout a lesson.
  • Individual Response Opportunities: A single student demonstrates a mathematical idea or concept. These are used to check for understanding, or have students illustrate an example for other students.
  • Partner or Whole Group Discussion Opportunities: Students work with their partner or as a group to solve a problem, discuss a mathematical idea, or share their thinking and reasoning. Partners then come together and share with the group as the teacher monitors and builds on student responses.
• Targeted Mathematical Practice Standards:
  • (MP 3) Construct viable arguments and critique the reasoning of others: In WNF-2, students use their knowledge of mathematical properties and patterns to identify efficient problem-solving strategies and generate solutions. Students engage in discussion to share different strategies to solve problems, and are encouraged to think about similarities and differences between their own approach and those of other students.
  • (MP 7) Look for & make use of structure: During discussion, students identify patterns and structures in the number system. Examples include regrouping to compose or decompose a new unit, identifying number patterns when skip counting, and using part-whole relationships to understand the inverse relation between addition and subtraction.

Self-Regulated Learning (SRL)

• Students with mathematics difficulties often struggle with where to start and how to track their steps as they approach and solve complex math problems. WNF-2 incorporates self-regulated learning (SRL) guides, called Student Strategy Guides, to help students remember the steps to approach and solve problems. Strategy Guides focus on word problem solving, multi-digit addition and subtraction, and comparing 2- and 3-digit numbers. Students are encouraged to self-monitor, track their progress, and check their work to ensure accuracy and completeness.
• Targeted Mathematical Practice Standards:
  • (MP 1) Make sense of problems & persevere in solving them: In WNF-2, students learn entry points for approaching complex math problems, as well as steps to monitor their progress toward a solution. Students apply and generalize learned problem-solving strategies to new mathematical situations.
  • (MP 6) Attend to precision: In WNF-2, students monitor their decision-making as they engage in mathematical problem solving. After completing a problem, students check their work to ensure their calculations are correct and ask themselves whether their answer makes sense.

Whole Number Foundations Level 2 includes 70 lessons divided into two Teacher Books, Math Extension Sheets (1 per lesson), Student Problem-Solving Notebooks (1 per student), Program Support Materials, and a manipulative list.

Materials are available in two different formats:

• Convenient, downloadable PDF format. Once the PDFs are purchased, copies may be made according to the Distribution License Agreement.
• Pre-printed by the University of Oregon Printing Services Department.

Sample Lessons

Whole Number Foundations Level 2 Sampler (pdf)

Lesson Content

The WNF-2 program covers critical whole number content across five content strands (see Figure 1). Content is aligned with the Common Core State Standards for Grade 2.

• Number Combinations (NC): Students learn strategies to add and subtract within 20 and build fluency with math facts.
• Place Value (PV): Students count and work with numbers to 1000, building understanding of place value concepts and applying that knowledge to compare numbers, add, and subtract.
• Problem Solving (PS): Students represent and solve story problems involving addition and subtraction, including part-whole, change, and difference problems. Starting in Lesson 15 (PS), every fifth lesson is focused exclusively on word problem-solving. These lessons are designed to be taught as part of the WNF-2 curriculum, but may also be taught as a standalone supplemental intervention for students with word problem solving difficulties.
• Multi-Digit Computation (MD): Students use concrete models, place value strategies, and properties of operations to add and subtract multi-digit numbers.
• Multiplication (MUL): Students build foundational knowledge for multiplication by forming pairs and identifying numbers as even or odd.

Figure 1. WNF-2 mathematical content strands.

Teacher Books

Lessons are divided into two teacher books. Book 1 includes the Teacher’s Guide and Lessons 1-35. Book 2 includes Lessons 36-70 and an optional problem solving extension lesson (Lesson Extension PS). Major objectives and their corresponding content strands for each book are listed below.

Book 1 (Lessons 1-35)

• Learn and fluently solve number combination strategies within 20 (new: add/subtract 1, combinations that equal 10, solve doubles facts, add/subtract by making a ten, and decompose teen numbers) (NC)
• Represent numbers 1-1000 on the place value chart (PV)
• Compare 2-digit numbers (PV)
• Count by 1s, 2s, 10s, and 100s (PV)
• Solve word problems (new: part- whole and change problems) (PS)
• Add 2-digit numbers with and without regrouping (MD)
• Subtract 2-digit numbers without regrouping (MD)
• Understand the concept of a pair (MUL)

Book 2 (Lessons 36-70)

• Learn and fluently solve number combination strategies within 20 (new: add 9) (NC)
• Write numbers 1-1000 and represent 3-digit numbers using expanded notation (PV)
• Compare 3-digit numbers (PV)
• Count by 5s (PV)
• Mentally add/subtract 10 and 100 from a number (PV)
• Solve and identify story problems of different types (new: difference problems) (PS)
• Add and subtract 3-digit numbers with and without regrouping (MD)
• Identify numbers as even or odd; write an addition equation to represent an even number as the sum of two equal addends (MUL)

Math Extension Sheets:

• Math Extension Sheets are included at the end of most WNF-2 lessons. They are designed to provide teacher-guided individual practice opportunities to help students build on what they learned in the lesson, review previously learned content, and promote generalization.
• On each sheet, a “Note Home” is included in both English and Spanish, fostering home-school connecting and providing activities for adults to engage their child in additional practice on key concepts and skills covered in the lesson.
• Math Extension Sheets may be copied for implementation of WNF-2.

Student Problem-Solving Notebooks:

• Student Problem-Solving Notebooks are used in most problem-solving (PS) lessons instead of Math Extension Sheets. The student Problem-Solving Notebooks are similarly designed to provide teacher-guided individual practice opportunities. The notebooks are also designed to be a reference and reflection resource for students, as targeted by the Mathematical Practice Standards.
• Program Support Materials:
  
  • Program Support Materials include reproducible masters of the following materials:
    
    • Hundreds Charts (0-100 up to 900-1000)
    • Place Value Charts (with 1 or 2 ten frames)
    • Expanded Notation Practice Sheet
    • Math cards (for Number Combinations to 20)
    • Story Problem Cards
    • Story Problem Diagrams (Part-Whole, Change, and Difference)
    • Teacher Strategy Guides
    • Student Strategy Guides
  • These are not sold separately.
• Manipulative list: Manipulatives are required for implementation of the program and are NOT included. These manipulatives are commonly available in second grade classrooms, or may be purchased from educational retailers.

These products can be purchased together as an Intervention Kit. Additional and/or replacement Teacher Books, Math Extension Sheets, and Student Problem-Solving Notebooks may also be purchased separately.

Our Research to Your Classroom

Whole Number Foundations Level 2 was developed and evaluated with funding from the National Science Foundation (NSF). An initial study conducted in the 2022-2023 school year demonstrated that teachers could feasibly implement the WNF-2 intervention in schools. A follow-up pilot study involved 124 at-risk students in 15 second grade classrooms in Oregon (Clarke et al., in preparation). Eligible students were randomly assigned to the WNF-2 intervention (n = 65) or a control condition (i.e., standard district mathematics instruction; n = 64). Several measures of mathematics achievement were collected at pre-test and post-test. Results suggested that WNF-2 students made greater gains than control peers on  a math curriculum-based measure (ASPENS; g = 0.56), the WNF-2 Proximal Assessment (g = 0.42), and a Mathematics Vocabulary measure (g = 1.23).

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