Research the identified deficits. (1) List the deficits. (2) Research best practices to remediate. (3) Create a lesson plan for each deficit.(4) Explain rational for each.
An Action Plan for Remediating Key 2nd-Grade Mathematical Deficits at Marlinton Elementary
Section 1: Executive Summary and Overview of Deficits
1.1 Introduction to the Report
This report provides a comprehensive, research-based framework for addressing five specific mathematical deficits identified in 2nd-grade students at Marlinton Elementary. The purpose of this document is to move beyond mere identification of weaknesses and to equip educators with a deep understanding of the underlying conceptual challenges, evidence-based remediation strategies, and practical, ready-to-implement lesson plans. The pedagogical approach presented herein is rooted in the core principles of modern mathematics education: that genuine understanding is built through hands-on exploration, that conceptual knowledge must precede procedural fluency, and that the ultimate goal is to foster flexible mathematical reasoning. By focusing on targeted, high-impact instructional practices, this action plan aims to remediate the identified deficits and cultivate a more robust and resilient mathematical foundation for all learners.
1.2 Identified Deficits
Analysis of the provided student performance data indicates five key areas of concern where students are not meeting grade-level expectations. These deficits represent critical foundational skills and concepts within the 2nd-grade mathematics curriculum. The specific deficits to be addressed are:
Identify appropriate unit (of measurement)
Identify components of figures (geometry)
Identify congruent figures (geometry)
Find a date on a calendar (measurement and data)
Recognize commutative property (operations and algebraic thinking)
1.3 Summary of Remediation Approaches
To effectively address these distinct yet interconnected challenges, a strategic approach is required for each deficit. The following table provides a high-level overview of the core conceptual hurdle students face in each area and the primary, research-supported strategy that will be employed for remediation. This summary serves as an advance organizer for the detailed analysis and lesson plans that follow.
Table 1.1: Summary of Identified Deficits and Core Remediation Strategies
Section 2: Deficit 1 - Identifying Appropriate Units of Measurement
2.1 Analysis of the Deficit
The deficit in identifying appropriate units of measurement extends far beyond the rote memorization of facts (e.g., "a book is measured in inches"). At its core, this challenge reflects a lack of what is often called "measurement sense"—an intuitive, flexible understanding of the size of standard units and the logic of selecting a unit that is appropriate to the scale of the object being measured. Second-grade students struggling with this concept often exhibit several underlying misconceptions. They may not fully grasp why a standard, consistent unit is necessary, leading to confusion when transitioning from non-standard units (like their own hand spans) to standard units (like inches).
A more profound conceptual hurdle is the failure to understand the inverse relationship between the size of a measurement unit and the number of those units required to measure an object.1 A student who understands this principle knows that measuring a long hallway will require a very large number of small units (like inches), making a larger unit (like feet or yards) a more efficient and logical choice. Without this understanding, the selection of a unit becomes an arbitrary guess rather than a reasoned decision. Furthermore, many students lack concrete mental benchmarks for the size of an inch, a foot, a centimeter, or a meter, rendering them unable to make reasonable estimates—a skill that is inextricably linked to choosing an appropriate unit.1
2.2 Research-Informed Best Practices for Remediation
Effective remediation must build measurement sense from the ground up, focusing on hands-on experiences that allow students to construct their own understanding.
Progression from Non-Standard to Standard Units: Before students can appreciate the need for a ruler, they must first internalize the fundamental concepts of measurement. This is best accomplished by using non-standard units, such as paper clips, linking cubes, or even their own shoes, to measure objects.3 This process teaches the critical principles of measurement: starting at the beginning of an object, iterating a unit without leaving gaps or creating overlaps, and understanding that the final count represents the length in those specific units.1 This experience provides the "why" behind the "how" of standard measurement.
Establishment of Personal Benchmarks: The abstract nature of a "foot" or a "centimeter" becomes tangible when students can connect these units to their own bodies. Guiding students to discover that the distance between two knuckles is about an inch, the width of their pinky finger is about a centimeter, or the length of their forearm is about a foot provides them with accessible, portable reference points.1 These personal benchmarks are the bedrock of estimation, allowing a student to look at an object and think, "That looks like it's about three of my forearms long, so it's probably about three feet."
Engagement in Sorting and Comparison Activities: To practice the logic of unit selection, students should engage in activities that require them to make choices and justify them. A highly effective method is a whole-group pocket chart sort, where students categorize pictures of objects under the headings "Inches," "Feet," and "Yards".5 This can also be structured as a measurement station activity where small groups sort physical objects.6 The critical component of these activities is the accompanying discussion, where students must explain why they chose a particular unit.
Emphasis on Estimation and Reasonableness: Instruction should consistently prioritize estimation before measurement.2 This simple step transforms a procedural task into a cognitive one, forcing students to access their mental benchmarks and apply logical reasoning. The instructional focus should be on questions that promote critical thinking, such as, "Would it be reasonable to measure your pencil in feet? Why or why not?" These discussions are often more valuable for building measurement sense than the act of measuring itself.
Utilization of Hands-On, Kinesthetic Learning: Measurement is an active process, and instruction should reflect this. Learning is most effective when students are moving and doing. This can include outdoor measurement scavenger hunts where students find items that are "about a foot long" 6, creative projects like building "Measurement Monsters" and then measuring their various parts 6, or using playdough to roll out "snakes" of different lengths to be estimated and then measured.2
2.3 Detailed Lesson Plan: The Right Tool for the Job
Objective: Students will be able to select and justify the use of an appropriate standard unit of length (inches, feet, yards) to measure common classroom objects.
Grade Level: 2
Time: 45 minutes
Materials: Rulers (with inch and foot markings), yardsticks, measuring tapes, a collection of classroom objects of varying sizes (e.g., pencil, book, student desk, whiteboard, paperclip, classroom door), chart paper, markers, a set of "The Right Tool" sorting cards (pictures of various objects like a key, a car, a football field, a crayon, a bed), pocket chart with "Inches," "Feet," and "Yards" header cards.5
Procedure
Hook (5 minutes): Gather students and present a humorous, paradoxical scenario. "Class, I have a very important job. I need to measure our big classroom whiteboard to see if a new poster will fit. I've chosen the perfect tool for the job: this tiny paperclip!" Begin to measure the whiteboard with the paperclip, making it look difficult and slow. Ask the students: "What do you think of my plan? Is this a good tool for this job?" Elicit student responses, guiding them to conclude that the paperclip is too small and the job would take too long. This leads to a discussion about needing different tools (and units) for different-sized jobs.
Mini-Lesson & Benchmark Anchor Chart (10 minutes): Formally introduce the measurement tools: a ruler (highlighting both the inch and the 12-inch foot) and a yardstick. Guide students in a discovery process to find personal benchmarks for each unit.
Inch: "Let's find an inch on our bodies. Look at your pointer finger. The distance from the tip to your first knuckle, or between your first and second knuckle, is often about one inch." Have them check it against a ruler.5
Foot: "A foot is 12 of these inches. What in our classroom is about a foot long?" (e.g., a textbook, a large shoe, their forearm).
Yard: "A yard is three whole feet! A big step for an adult is often about a yard."
As a class, create an anchor chart titled "Measurement Benchmarks" with drawings and labels for each unit and its corresponding personal benchmark.1
Guided Practice - Pocket Chart Sort (15 minutes): Direct students' attention to the pocket chart with the "Inches," "Feet," and "Yards" headers. Hold up a picture card (e.g., a key). "Let's be detectives. What is the best unit to measure a key? Should we use inches, feet, or yards? Talk to your partner." After a brief discussion, call on students to share their answers. Crucially, require them to justify their choice using the new benchmarks: "I would use inches because a key is small, maybe just a couple of my knuckles long. It would be silly to use a big yardstick".5 Continue with several more cards, including some that are less obvious (like a bed, which could be feet or yards) to spark discussion.
Partner Practice - Classroom Scavenger Hunt (15 minutes): In pairs, students receive a recording sheet with three columns labeled "Objects to Measure in Inches," "Objects to Measure in Feet," and "Objects to Measure in Yards." Explain the task: "Your mission is not to actually measure anything yet. Your job is to be a measurement planner. With your partner, you will walk around the room and find at least two objects that would be best measured by each unit. Write the name of the object in the correct column".7 This task forces them to apply their reasoning skills without getting bogged down in the procedural mechanics of measuring.
Closure & Share-Out (5 minutes): Bring the class back together. Have a few pairs share one object from each of their columns. Ask probing questions to reinforce the core concept: "The Blue Team chose to list the classroom door under 'feet.' Could you have measured it in inches?" (Yes). "So why are feet a better choice?" (Because it's a big object, and using inches would take a long time and result in a very big number). This final discussion solidifies the concept of efficiency and appropriateness in unit selection.
2.4 Rationale
This lesson is meticulously designed to build conceptual understanding rather than rote procedural skill. It directly confronts the core deficit of "measurement sense" by prioritizing the why of unit selection over the how of using a measurement tool.
The lesson begins by establishing a clear and relatable context—the inefficiency of using the wrong tool for a job. This framing immediately engages students in thinking about the logic of measurement. The subsequent development of physical, personal benchmarks is a critical step that grounds the abstract concepts of "inch," "foot," and "yard" in the students' own lived experience, making these units tangible and meaningful.1
The central activities of the lesson—the pocket chart sort and the scavenger hunt—are intentionally designed as decision-making tasks. They require students to engage in higher-order thinking skills such as analysis, comparison, and justification. By asking students to choose and defend their choice of unit, the lesson promotes mathematical discourse and reasoning. The final discussion explicitly draws out the inverse relationship between unit size and the number of units needed, solidifying one of the most critical and often-missed concepts in primary measurement instruction. This focus on reasoning and justification ensures that students are not just learning a procedure, but are truly building a lasting and flexible understanding of measurement.
Section 3: Deficit 2 - Identifying Components of Geometric Figures
3.1 Analysis of the Deficit
The deficit in identifying the components of geometric figures signifies a critical bottleneck in a student's geometric development. According to the van Hiele model of geometric thought, young children typically begin at Level 0 (Visualization), where they recognize shapes by their holistic appearance—a square looks like a window, a triangle looks like a roof. The 2nd-grade curriculum aims to move students to Level 1 (Analysis), where they begin to see shapes as collections of their properties. This deficit indicates that students are struggling to make this crucial transition. They have not yet learned to deconstruct shapes into their component parts: sides, angles, and vertices for two-dimensional (2D) figures, and faces, edges, and vertices for three-dimensional (3D) figures.9
Common manifestations of this deficit include confusing the terms (e.g., calling an edge a face), being unable to accurately count the components of a given shape, or being unable to use the components to define a shape (e.g., "A triangle is a shape with three sides and three vertices"). This analytical skill is not trivial; it is the absolute foundation for all future geometric reasoning, including the ability to classify shapes, understand the relationships between different shapes (e.g., a square is a special type of rectangle), and, as will be discussed later, determine congruence.
3.2 Research-Informed Best Practices for Remediation
To move students from holistic recognition to analytical understanding, instruction must be explicit, tactile, and focused on the properties of shapes.
Prioritize Properties Over Names: The instructional focus must shift from a naming game ("What is this shape called?") to an analytical investigation ("What are the properties of this shape? What can you tell me about its parts?").11 A key classroom tool for this is an anchor chart that is co-constructed with students, listing various shapes alongside a clear tally of their sides, vertices, and angles (for 2D) or faces, edges, and vertices (for 3D).11
Utilize a Wide Variety of Examples and Non-Examples: To build a robust and flexible understanding of a shape's properties, students must be exposed to a wide range of examples and non-examples.12 This means intentionally showing them non-prototypical shapes: long, skinny scalene triangles in addition to equilateral ones; trapezoids and rhombuses in addition to squares; and irregular pentagons that look nothing like the classic "house" shape.11 It also means showing them non-examples, such as open figures or shapes with curved sides, and discussing why these do not fit the definition of a polygon. This practice forces students to rely on the defining properties rather than on a fragile visual memory of a single prototype.
Embrace a Constructivist Approach: Build It to Understand It: The most potent strategy for making the abstract vocabulary of geometry concrete is to have students physically construct the shapes themselves. The classic activity of using toothpicks to represent edges and mini-marshmallows or balls of play-doh to represent vertices is exceptionally effective for teaching 3D shape components.15 As students build a cube, they are not just learning the definitions of "edge" and "vertex"; they are internalizing them through a memorable, kinesthetic experience.
Explicitly Connect 2D and 3D Shapes: Students need to see that the 2D shapes they know (squares, triangles, rectangles) are the building blocks for the 3D shapes they are learning about. The "faces" of a cube are squares; the faces of a pyramid are triangles and a square (or other polygon).16 A powerful visual tool for demonstrating this relationship is a "net"—a 2D pattern that can be folded to form a 3D shape. Unfolding a cardboard box (a rectangular prism) to show its net of six rectangles is a simple but profound demonstration.16
Incorporate Movement and Gamification: Learning should be active and engaging. A "Shape Scavenger Hunt" where students search the classroom or school for real-world examples of cubes, cylinders, vertices, and edges connects geometry to their environment.15 A "scoot" activity, where task cards depicting different shapes are placed around the room and students rotate through them, identifying and recording their components, provides practice in a dynamic format.11
3.3 Detailed Lesson Plan: Shape Detectives - Deconstructing 2D and 3D Figures
Objective: Students will be able to identify, count, and name the number of sides, vertices, and angles of 2D shapes, and the faces, edges, and vertices of 3D shapes.
Grade Level: 2
Time: 50 minutes
Materials: A set of common geometric solids (cube, rectangular prism, square pyramid, cylinder, cone), small bags for each student containing approximately 20 toothpicks and 10 mini-marshmallows (or small balls of play-doh), chart paper, markers, a set of 2D shape cards (including various triangles, quadrilaterals, pentagons, and hexagons, with a focus on non-prototypical examples), student whiteboards and markers.
Procedure
Hook (5 minutes): Hold up a wooden or plastic cube for the class to see. "We all know this is called a cube. But today, we are going to be shape detectives. A detective doesn't just look at something; they take it apart to see what it's made of. If we could take this cube apart, what pieces would we find?" Guide students to use their own language to describe the parts: "the flat parts," "the straight lines," and "the pointy corners." Write their informal terms on the board.
Mini-Lesson - Creating a Vocabulary Anchor Chart (15 minutes): "Mathematicians have special names for these parts." Introduce the formal vocabulary, linking each term to the students' informal descriptions and the physical cube.
Faces: "What you called the 'flat parts,' we call faces." Use a dry-erase marker to number the faces of the cube as the class counts them aloud (1 to 6).
Edges: "What you called the 'straight lines,' we call edges. An edge is where two faces meet." Count the 12 edges together.
Vertices: "What you called the 'pointy corners,' we call vertices. A vertex is a point where the edges meet." Count the 8 vertices together.
Co-construct a "3D Shape Components" anchor chart, drawing a cube and labeling its face, edge, and vertex. Record the final counts: A cube has 6 faces, 12 edges, and 8 vertices.11
Guided Practice - Build a Shape (20 minutes): Distribute the bags of toothpicks and marshmallows. "Now you get to be shape engineers! Your mission is to build your own cube. The toothpicks will be the edges, and the marshmallows will be the vertices." Circulate as students work, providing support and reinforcing the vocabulary. This activity provides a powerful, tactile experience that solidifies the meaning of the terms.16 Once most students have completed their cube, have them verify the numbers on the anchor chart by counting the components of their own creation. "How many marshmallows, or vertices, did you use?" (8). "How many toothpicks, or edges, did you use?" (12).
Partner Practice - 2D Shape Sort (10 minutes): Transition to 2D shapes. "Now let's use our detective skills on flat shapes." In pairs, give each group a set of varied 2D shape cards. "With your partner, sort these shapes into groups based on how many sides and vertices they have." As they work, encourage them to notice that for any given polygon, the number of sides is always equal to the number of vertices.11 Have them record their findings on their whiteboards (e.g., drawing a pentagon and labeling it "5 sides, 5 vertices").
Closure & Exit Ticket (5 minutes): Hold up a cylinder and a cone. Ask, "How are these shapes different from the cube and pyramid we looked at? What components do they have? What are they missing?" Guide a brief discussion about curved surfaces versus flat faces. For an exit ticket, draw an irregular hexagon on the board and ask students to write down the number of sides and vertices it has.
3.4 Rationale
This lesson is strategically structured to follow a concrete-to-representational learning progression, a highly effective sequence for teaching abstract mathematical concepts. The hands-on, tactile experience of building a 3D shape (the concrete phase) provides a memorable, physical anchor for the abstract vocabulary of "face," "edge," and "vertex".16 Students are not merely told what a vertex is; they physically create one by joining toothpick "edges" into a marshmallow. This act of construction builds a deep, intuitive understanding that passive observation cannot replicate.
By having students verify the component counts on their own creations, the lesson reinforces the concepts through self-discovery. The subsequent transition to sorting 2D shapes (the representational phase) allows students to apply their newly solidified understanding in a different context. The intentional inclusion of non-prototypical shapes in the sort is a critical design feature; it compels students to rely on an analytical count of the properties rather than a simple visual match, directly addressing the core objective of moving them into the analytical stage of geometric thought as described by the van Hiele model. This lesson prioritizes the student as an active constructor of knowledge, which is essential for developing lasting geometric understanding.
Section 4: Deficit 3 - Identifying Congruent Figures
4.1 Analysis of the Deficit
Congruence is a fundamental geometric concept signifying that two figures are identical in every respect—they have the exact same size and the exact same shape.18 The primary cognitive challenge for 2nd-grade students is to develop "object constancy," the understanding that a shape's intrinsic properties (its side lengths and angle measures) do not change when its position or orientation in space is altered. Students with this deficit often operate under the misconception that a rigid transformation—a slide (translation), a flip (reflection), or a turn (rotation)—changes the shape itself.20 For example, they may see an isosceles triangle and an identical but upside-down isosceles triangle and declare them to be different shapes, and therefore not congruent.
This difficulty is rooted in a reliance on visual perception over analytical reasoning. The student's brain perceives the rotated shape as "different" because it is not oriented in the familiar, prototypical way. Another common point of confusion is the distinction between congruence and similarity. Students may incorrectly identify two shapes of the same type but different sizes (e.g., a large square and a small square) as congruent, because they are focusing only on the "same shape" aspect and ignoring the "same size" requirement.20 Remediating this deficit requires providing students with concrete methods to override their visual intuition and prove whether two shapes are, in fact, geometrically identical. It is also important to recognize that a student cannot successfully determine congruence without first being able to identify and compare the components of a figure (Deficit 2), as verifying congruence ultimately relies on confirming that all corresponding sides and angles are equal.
4.2 Research-Informed Best Practices for Remediation
Effective remediation strategies for congruence focus on making the abstract idea of "same size, same shape" a concrete, verifiable reality through physical interaction.
Prioritize Physical Manipulation and Superposition: The most direct and intuitive way to teach congruence is through action. Students should be given physical shapes, such as pattern blocks, tangrams, or simple paper cut-outs, that they can physically slide, flip, and turn.20 The ultimate test of congruence is superposition—the act of placing one shape directly on top of the other to see if it covers it exactly, with no parts sticking out.19 This hands-on test provides irrefutable proof that two shapes are, or are not, congruent.
Leverage Tracing Paper as a Key Tool: Tracing paper is an exceptionally powerful and accessible tool for verifying congruence, especially when physical cut-outs are not available or practical. Students can carefully trace one figure, then move the tracing paper over the second figure. By sliding, flipping, and turning the tracing paper, they can physically attempt to map the first figure onto the second. A perfect fit demonstrates congruence.20 This method elegantly bypasses the orientation problem, as it allows the student to manipulate the traced image without changing its size or shape.
Focus on Identifying and Matching Corresponding Parts: Instruction should explicitly guide students to check for congruence by matching up the corresponding parts of the two figures—the matching sides and the matching angles.21 This practice directly connects to and reinforces the skills developed in remediating Deficit 2. Students should be encouraged to use language like, "This side on the first shape is the same length as this side on the second shape, and this corner feels just as pointy as this corner."
Utilize Geoboards and Dot Paper for Structured Practice: Geoboards and dot paper provide a structured environment for exploring congruence and transformations. The grid of pegs or dots acts as a scaffold, making it easier for students to create shapes with specific side lengths and to replicate them accurately.21 An effective activity is to have one student create a shape on a geoboard and challenge their partner to create a congruent shape that is in a different location or has been flipped or rotated. This gamified approach encourages experimentation and solidifies the concept of rotational invariance. The underlying grid structure provides a hidden but powerful support system, removing some of the cognitive load of visual estimation and allowing students to focus solely on the properties of congruence.
4.3 Detailed Lesson Plan: Twin Shapes - A Congruence Investigation
Objective: Students will be able to identify congruent figures by demonstrating that they are the same size and shape, regardless of their orientation, using methods of superposition and tracing.
Grade Level: 2
Time: 45 minutes
Materials: For each pair/group: a baggie containing several pairs of paper shapes (e.g., two identical triangles with one rotated, two identical L-shapes with one flipped, two squares of different sizes, a rectangle and a parallelogram). Also include tracing paper, student whiteboards, and sets of pattern blocks.
Procedure
Hook (5 minutes): Display a photograph of identical twins standing side-by-side. Ask, "What can you tell me about these two people?" Guide the discussion to the idea that they are "the same" or "identical." Then, show a second picture where one twin is doing a handstand or is upside down. Ask, "Are they still the same people? Did the person change just because they turned upside down?" Establish that they are still the same person, just in a different position. "In math, shapes can have twins, too! We call them congruent shapes. Congruent means they are exactly the same size and exactly the same shape, even if one of them is flipped or turned around."
Mini-Lesson - How to Prove It! (15 minutes): Hold up two congruent paper triangles, but with one rotated 90 degrees. "A student told me these are not congruent because one is 'pointy-up' and the other is 'pointy-sideways.' Are they right? How can we prove it?" Model three distinct strategies for the class.
The Flip, Slide, and Turn Test (Superposition): Take one of the cut-outs and physically move it. "Let's see if I can make it fit on top of the other one. I can slide it over... that doesn't work. I can flip it... still not right. Let me try to turn it... Look! It's a perfect match!".19
The Tracing Paper Test: Place a piece of tracing paper over one triangle and carefully trace it. Then, move the tracing paper over to the second triangle. "Now I can move my tracing and see if it matches. I'll turn my paper until... yes! A perfect fit. This proves they are congruent".20
The "Check the Parts" Test (Briefly): Mention, "We could also check to make sure all the matching sides are the same length and all the matching corners are the same size." This connects to prior learning about shape components.
Guided Partner Practice - Congruent or Not? (15 minutes): Distribute the baggies of shape pairs to each group. "You are now congruence detectives. Your job is to sort the pairs of shapes in your bag into two piles: a 'Congruent' pile and a 'Not Congruent' pile. For every pair you say is congruent, you must be able to prove it to me using either the 'Flip, Slide, and Turn Test' or the 'Tracing Paper Test'." Circulate as students work, listening to their discussions and asking them to justify their sorting decisions. Pay special attention to how they handle the similar shapes (same shape, different size).
Independent Practice - Pattern Block Challenge (10 minutes): Each student receives a small set of pattern blocks. On their whiteboard, they create a simple design or composite shape using 3-4 blocks. They then carefully slide their whiteboard to their partner. The partner's challenge is to build a congruent design next to the original, proving it by matching the blocks one-to-one. They then switch roles. This activity reinforces the concept in a creative, constructive context.
Closure & Exit Ticket (5 minutes): On the main board, display one target shape (e.g., a trapezoid). Below it, display three other shapes labeled A, B, and C: one that is congruent but flipped, one that is similar (smaller), and one that is a different shape entirely (e.g., a pentagon). Ask students to write on their whiteboard the letter of the shape that is congruent to the target shape.
4.4 Rationale
This lesson is fundamentally grounded in kinesthetic and visual learning, directly addressing the core reason students struggle with congruence: a conflict between abstract rules and visual intuition. The abstract concepts of "same size and shape" and "rigid transformations" are made undeniably concrete through the physical acts of moving, turning, flipping, and tracing shapes. The twin analogy in the hook provides a powerful and memorable real-world anchor for the mathematical term "congruent."
By providing students with both examples (congruent pairs) and non-examples (similar shapes, different shapes), the lesson compels them to move beyond a superficial understanding and refine their definition of congruence to include both "same shape" and "same size." The partner activities are designed to foster rich mathematical discourse, requiring students to articulate their reasoning and justify their conclusions to a peer. This process of proving and explaining moves them from being passive observers of shapes to active geometric investigators, building a durable and flexible understanding of this foundational geometric principle.
Section 5: Deficit 4 - Finding a Date on a Calendar
5.1 Analysis of the Deficit
Difficulty with finding a date on a calendar is a multifaceted challenge that requires the integration of several distinct skills. Unlike a purely conceptual deficit, this reflects a lack of fluency with a specific, culturally-defined tool for organizing and representing time. A calendar is not a simple number line; it is a two-dimensional grid, and students with this deficit struggle to navigate its structure. The core challenges include:
Sequencing: Students must have a firm grasp of the sequence of the days of the week and the months of the year.
Grid Navigation: They must understand the calendar's layout, where rows typically represent weeks and columns represent specific days of the week (e.g., all the Tuesdays are in one column).22 Locating a date such as "the third Tuesday of the month" requires a student to first identify the correct column ("Tuesday") and then count down three rows.
Temporal Language: Students may struggle with the vocabulary of time, such as "yesterday," "tomorrow," "a week from now," or "last Thursday."
A student with this deficit may be able to count to 31 but cannot apply that skill within the calendar's structure. They may not understand the cyclical nature of weeks or the fact that months have varying lengths. This skill is essential not only for mathematics but also for developing organizational skills and understanding the structure of their daily lives.
5.2 Research-Informed Best Practices for Remediation
The most effective pedagogy for teaching calendar skills is not a single, intensive unit but rather consistent, distributed practice integrated into the daily life of the classroom.
Establish a Consistent Daily Routine: The cornerstone of calendar instruction is brief, repeated exposure every single day.23 A 5- to 10-minute "calendar time," typically as part of a morning meeting, is far more effective for building fluency than a week-long unit on calendars.25 This distributed practice allows students to gradually internalize the patterns and structure of the calendar.
Utilize an Interactive, Large-Format Calendar: The classroom calendar should be a central, dynamic tool, not a static poster. A large pocket chart calendar or a digital, interactive whiteboard calendar allows for daily manipulation.22 Assigning a "Calendar Helper" as a daily classroom job gives students ownership and provides regular, low-stakes practice in leading the routine.25
Ask Purposeful, Progressively Complex Questions: The daily routine must transcend simply stating the date. The power of calendar time lies in the teacher's questioning, which should be designed to make students navigate the calendar and think about temporal relationships.22 The questions should gradually increase in complexity throughout the year:
Beginning of year: "What is today's date? What day of the week is it? What was yesterday?"
Middle of year: "Our field trip is in 5 days. What will the date be? What day of the week will it be?"
End of year: "What is the date of the fourth Monday of this month? Our project is due two weeks from yesterday. What is the due date?"
Connect the Calendar to Real-World, Meaningful Events: The calendar becomes relevant and purposeful when it is used to track events that matter to students. Marking students' birthdays, school holidays, classroom parties, library days, and assignment due dates provides a genuine context for using the calendar as an organizational tool.23 This transforms it from an abstract math exercise into a functional part of their lives.
Integrate a Broad Spectrum of Math Skills: Calendar time is an exceptionally efficient vehicle for reinforcing a wide array of mathematical concepts. The daily date can become the "Number of the Day," used for practice with place value (e.g., the 16th is 1 ten and 6 ones), even/odd identification, or generating equations.22 A "Days in School" chart builds understanding of place value up to 100 and beyond. Tracking daily weather on the calendar provides a month's worth of data that can be used to create bar graphs, connecting calendar skills to data analysis.25 This "Trojan Horse" approach makes calendar time one of the most instructionally dense routines in an elementary classroom.
5.3 Detailed Lesson Plan: Calendar Quest - A Week of Daily Routines
Note: This is not a single, monolithic lesson but a structured plan for implementing a 10-minute daily routine to be integrated into a morning meeting. The goal is to establish the practice and progressively introduce more complex tasks over the course of one week.
Objective: Over the course of the week, students will be able to identify the current day, date, month, and year, and begin to use a calendar to determine the date of future and past events within a two-week span.
Materials: Large, interactive classroom wall calendar with removable number cards and day/month labels; a pointer; markers to note special events; individual student copies of the current month's calendar page.25
Daily Routine Procedure (10 minutes each day)
Day 1: Establishing the Basics.
The "Calendar Helper" for the day comes to the calendar.
The teacher guides the helper to identify and state the full date: "Today is Monday, October 16th, 2023." The class repeats the full date.
The helper places the number card "16" in the correct spot on the calendar grid.
The teacher asks, "If today is Monday, what day was yesterday?" and "What day will tomorrow be?" The helper points to the corresponding days on the calendar.22
Students find and circle the number 16 on their personal calendars.
Day 2: Counting Forward.
Repeat the basic routine from Day 1.
Pose a new question: "We have our school assembly in two days. Let's be time travelers and figure out the date. If today is the 17th, what will the date be in two days?"
Model using the pointer to "hop" one space to the 18th ("one day") and a second space to the 19th ("two days"). State the conclusion: "The assembly will be on the 19th."
Students practice by hopping on their own calendars.
Day 3: Counting Backward.
Repeat the basic routine.
Pose a question about the past: "We had music class two days ago. What was the date?"
Model hopping backward two spaces from the current date to find the answer.
Connect to vocabulary: "Two days ago is the day before yesterday."
Day 4: Using the Structure of a Week.
Repeat the basic routine.
Pose a question involving a week: "Our spelling test is always on Friday. If this coming Friday is the 20th, what will the date be next Friday, exactly one week later?"
Model how to find the 20th and then look directly down one row in the same column to find the date for the following week (the 27th). Explain that "one week later" is always the number directly below on the calendar.
Day 5: Navigating the Grid for Ordinal Dates.
Repeat the basic routine.
Pose a more complex navigational challenge: "We have a special visitor coming on the second Wednesday of this month. Let's use our detective skills to find the exact date."
Model the two-step process: First, find the column labeled "Wednesday." Then, count down the boxes in that column: "This is the first Wednesday... and this is the second Wednesday." Identify the corresponding number.22
Have students find the "third Thursday" on their own calendars as a check for understanding.
5.4 Rationale
Calendar proficiency is a skill of fluency and familiarity, not a singular conceptual breakthrough. Therefore, the pedagogical approach must be built on the principle of distributed practice rather than massed practice. By integrating calendar work into a brief, predictable, and engaging daily routine, this plan provides the repeated exposure necessary for students to internalize the complex structure and function of a calendar. The abstract nature of time is made visible and concrete through the calendar's spatial grid, and actions like "hopping" along the days translate temporal relationships into spatial ones, which are easier for young learners to process.
The questioning strategy is deliberately scaffolded, designed to gradually increase in complexity over the week and throughout the school year. This progression moves students from simple identification ("What is today's date?") to relational thinking ("What will the date be in one week?") and finally to complex navigation ("Find the third Friday"). This approach ensures that all students can experience success while being consistently challenged to deepen their understanding of how we organize and measure time.
Section 6: Deficit 5 - Recognizing the Commutative Property
6.1 Analysis of the Deficit
The commutative property of addition is a foundational principle of mathematics and early algebraic thinking. It states that changing the order of the addends in an addition equation does not alter the sum (formally, $a + b = b + a$).27 A student exhibiting a deficit in this area has not yet grasped this underlying structure of addition. They likely perceive $4 + 7$ and $7 + 4$ as two distinct, unrelated facts that must be memorized separately. This lack of understanding has significant consequences.
First, it dramatically increases the cognitive load of learning basic addition facts; a student who understands commutativity effectively cuts the number of required memorized facts nearly in half. Second, and more critically, it prevents them from developing flexible and efficient computational strategies. The powerful "count on from the larger number" strategy is entirely dependent on an intuitive understanding of the commutative property.29 A student who sees the problem $3 + 18$ and begins counting on from 3 is demonstrating a lack of this understanding. They are unable to mentally transform the problem into the much easier equivalent, $18 + 3$. This deficit, therefore, is not merely a gap in vocabulary but a barrier to computational fluency and the development of number sense.
6.2 Research-Informed Best Practices for Remediation
Teaching an abstract property like commutativity requires a deliberate and structured approach that moves from the concrete to the abstract. Simply telling students the "rule" is ineffective and leads to shallow, easily forgotten learning.
Strict Adherence to the Concrete-Representational-Abstract (CRA) Sequence: This instructional sequence is paramount for teaching abstract mathematical properties.
Concrete Phase: Students must first physically experience and manipulate the property. This involves using hands-on materials like two-color counters, linking cubes, or buttons. The teacher guides them to create a representation of an equation (e.g., 3 red cubes and 5 blue cubes), find the total, and then physically "commute" or rearrange the groups (5 blue cubes and 3 red cubes) to prove to themselves that the total remains unchanged.30
Representational Phase: After extensive concrete practice, students move to drawing pictures that represent the manipulatives. They might draw 3 red circles and 5 blue circles to represent $3 + 5$, and then below it, draw 5 blue circles and 3 red circles to represent $5 + 3$, writing the matching equation next to each drawing.31 This phase acts as a crucial bridge between the physical objects and the abstract symbols.
Abstract Phase: Only after students have demonstrated a solid understanding at the concrete and representational levels should they work solely with the numbers and symbols (e.g., completing equations like $4 + 9 = 9 + \_$).
Emphasis on Student Discovery, Not Teacher Proclamation: The most powerful learning occurs when students discover the property for themselves. The instructional design should guide students through several concrete examples and then prompt them with questions like, "What do you notice? What is happening every time we switch the groups? Is there a pattern here?" This allows them to articulate the "rule" in their own words before the teacher provides the formal mathematical name.31
Use of Storytelling and Real-World Contexts: Framing the property within simple, relatable narratives helps students see its relevance outside of the math classroom. Stories about combining groups of objects—"I have 2 apples and 6 oranges in my grocery basket. How many pieces of fruit do I have? What if I had put the 6 oranges in first and then the 2 apples? Would the total number of fruit in my basket change?"—make the concept intuitive and logical.31
Explicitly Address Non-Examples: A robust understanding requires knowing not only where a property applies but also where it does not. After students have a firm grasp of commutativity in addition, it is critical to investigate whether the same "rule" applies to subtraction. Guiding them to discover with manipulatives that $5 - 2$ is not the same as $2 - 5$ prevents over-generalization and deepens their understanding of the unique properties of each operation.27
6.3 Detailed Lesson Plan: Flip-Flop Facts - The Order Doesn't Matter!
Objective: Students will be able to demonstrate and explain the commutative property of addition by showing that addends can be added in any order to arrive at the same sum.
Grade Level: 2
Time: 45 minutes
Materials: For each student or pair: a cup of approximately 15 two-color counters (e.g., red/yellow), a personal whiteboard, and a marker.
Procedure
Hook (5 minutes): Tell a simple, relatable story. "Yesterday at the library, I checked out 2 mystery books and 5 animal books. How many books did I get in total?" Write $2 + 5 = 7$ on the board. "Today, my friend went to the library. She checked out 5 animal books and 2 mystery books. How many books did she get?" Write $5 + 2 = 7$ on the board. Pause dramatically. "Wait a minute... look at my equation and my friend's equation. Talk to your partner. What do you notice?" Guide the discussion to the observation that the numbers are the same but "flip-flopped," and the answer is the same.
Concrete Exploration with Counters (20 minutes): Distribute the counters and whiteboards. "We are going to be math scientists and test if this 'flip-flop' idea is always true. Your first experiment: On your desk, show me 3 red counters and 4 yellow counters. How many do you have in all?" (7). "On your whiteboard, write the addition sentence that matches." (3+4=7). "Now, don't add any more counters or take any away. Just flip your counters over so you have 4 red and 3 yellow. How many do you have now?" (Still 7). "Write the new addition sentence on your board." (4+3=7).
Lead them through this process for two more number pairs (e.g., 6 and 2, 5 and 1). Encourage them to work with a partner and talk about what they are seeing.Guided Discussion - Discovering the "Rule" (10 minutes): Bring the class together for a discussion. "Math scientists, what did you discover in your experiments? What happened every single time you 'flip-flopped' the numbers? Did the total, or the sum, ever change?" Guide students to articulate the property in their own words: "It doesn't matter which number comes first, the answer is always the same!"
"You just discovered a very important rule in math! It has a big, fancy name to show how smart you are. It's called the Commutative Property. Can you say that? Commutative. It just means the numbers can 'commute' or move, and it's okay!".31Representational Practice - Whiteboard Drawings (10 minutes): "Now let's prove it with drawings." Give students a problem: "Draw 6 triangles and 3 squares. Write the addition sentence for your drawing." ($6 + 3 = 9$). "Now, right below it, draw 3 triangles and 6 squares. Write the addition sentence for that drawing." ($3 + 6 = 9$). This step transfers the concrete understanding from the counters to a pictorial representation, bridging the gap to the abstract symbols.31
Closure & Strategic Application (5 minutes): Pose a final problem that highlights the utility of the property. "If you know that $19 + 2 = 21$, what other addition fact do you know for free because of our 'Flip-Flop Rule'?" Elicit the answer $2 + 19 = 21$. Then ask, "Which one is easier to figure out in your head? Starting with 19 and counting up 2, or starting with 2 and counting up 19?" This final question frames the property not just as a fact, but as a useful strategy for solving problems more easily.29
6.4 Rationale
This lesson is meticulously designed to align with the Concrete-Representational-Abstract (CRA) instructional framework, which is widely recognized as the gold standard for teaching abstract mathematical principles to young learners. The lesson's power lies in its refusal to simply state a rule for memorization. Instead, it positions students as "math scientists" who discover the property for themselves through experimentation.
By starting with concrete manipulatives, the lesson allows students to physically experience and prove the commutative property. The act of taking a group of counters, noting the total, physically reconfiguring them, and seeing that the total remains constant provides an irrefutable, kinesthetic proof that is far more powerful than a teacher's explanation. The subsequent representational phase ensures that students can connect this physical experience to the mathematical drawings and symbols, preventing a disconnect between the hands-on activity and the academic task. Finally, the closure explicitly frames the property as a metacognitive tool—a strategy that students can use to make computation easier. This approach builds deep, lasting conceptual understanding and empowers students to become more flexible and efficient mathematical thinkers.
Section 7: Conclusions and Recommendations
The analysis of the five identified deficits at Marlinton Elementary reveals a common thread: students are struggling at the critical juncture where concrete understanding must transition to abstract reasoning. The challenges in measurement, geometry, and operations are not indicative of an inability to perform procedures, but rather a fragile grasp of the underlying mathematical concepts. To address these issues effectively, a shift toward more conceptually-focused, hands-on, and discourse-rich instructional practices is essential.
Based on the comprehensive analysis, the following recommendations are proposed:
Prioritize Manipulative-Based Instruction: For all new and remediated mathematical concepts, instruction should systematically follow the Concrete-Representational-Abstract (CRA) sequence. The school should ensure that classrooms are well-stocked with essential manipulatives such as linking cubes, two-color counters, geometric solids, and measurement tools. Professional development should focus on training teachers to use these tools not as optional add-ons, but as the foundational first step in instruction, particularly for abstract concepts like the properties of operations and geometric attributes.
Implement a Robust, Daily Mathematical Routine: The success of remediating the calendar deficit highlights the power of distributed practice. It is strongly recommended that a brief (10-15 minute) daily math routine, such as "Calendar Time" or "Number of the Day," be implemented in all K-2 classrooms. This routine should be intentionally designed to be a "Trojan Horse" for spiraled review of a wide range of skills, including place value, computation, time, money, and data analysis. This provides the consistent reinforcement necessary to build fluency and maintain skills throughout the year.
Emphasize Mathematical Discourse and Justification: Across all deficits, a key remediation strategy is to require students to explain their thinking. Instruction should consistently feature opportunities for partner talk and whole-group discussion where students are asked "Why?" and "How do you know?" Sentence stems should be provided to support students in articulating their reasoning (e.g., "I chose to use inches because... ," "I know these shapes are congruent because..."). This focus on discourse shifts the classroom culture from one of answer-getting to one of sense-making.
Adopt a Sequential and Interconnected Approach to Remediation: The analysis revealed clear dependencies between the identified deficits. For example, a student must be able to identify the components of a figure (Deficit 2) before they can meaningfully determine if two figures are congruent (Deficit 3). Remediation efforts should be sequenced logically, addressing foundational skills first. Furthermore, instruction should explicitly highlight the connections between mathematical ideas, such as how the commutative property is a tool for making computation easier.
By implementing these recommendations, Marlinton Elementary can move beyond addressing isolated skill gaps and begin to cultivate a more profound and durable mathematical understanding in its students. The focus on building concepts from the ground up through exploration, discourse, and strategic practice will not only remediate the current deficits but will also better prepare students for the increasing complexities of mathematics in the grades to come.
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