Early Childhood Math Assessment

The Counting Assessment Profile

The Counting Assessment Profile is a tool to diagnose a student’s counting abilities across several dimensions. The progression from “Limited” to “Exemplary” shows a clear developmental path from rote counting to flexible and meaningful enumeration.

Progression of Counting Skills

• Emerging: A student must have a stable number name sequence to at least 20 and a solid understanding of cardinality—that the last number counted represents the total quantity. One-to-one correspondence can be inexact but should be present.
• Proficient: The counting sequence must be accurate beyond 43. One-to-one correspondence and tracking of objects must be exact and deliberate, and the final count must be consistently accurate.
• Exemplary: The student meets all proficient criteria and shows advanced understanding by counting in groups (e.g., by twos or tens) and attending to both number and unit (e.g., “43 crayons”). This signals a transition toward a foundational understanding of multiplication. A 2025 study in the Journal of Experimental Child Psychology emphasizes that these early counting principles are strong predictors of later mathematical achievement.

Identifying Word Problem Structures

Understanding a word problem’s underlying structure helps students choose an appropriate strategy, often based on the Cognitively Guided Instruction (CGI) framework.

1. Problem 1: Join, Change Unknown. Jabul has 6 and needs 10. The joining action is clear, but the amount being added is unknown (6 + ? = 10).
2. Problem 2: Part-Part-Whole, Whole Unknown. The parts are known (5 rabbits, 2 squirrels, 8 foxes), and the question asks for the total (5 + 2 + 8 = ?).
3. Problem 3: Join, Result Unknown. A classic addition problem with a starting amount (3), a joining action (8 more), and an unknown result (3 + 8 = ?).
4. Problem 4: Compare, Difference Unknown. Two quantities ($5.28 and $11.12) are compared, and the question asks for the difference ($11.12 – $5.28 = ?).
5. Problem 5: Join, Start Unknown. The result (36) and the change (22) are known, but the starting amount is the unknown (? + 22 = 36).

Reflection on Mathematical Task Design

The article by Stein and Smith, “Mathematical Tasks as a Framework for Reflection,” is a foundational text in mathematics education. It argues that student learning quality is directly related to the cognitive demand of the tasks they perform.

Four Levels of Cognitive Demand

The framework outlines a thinking hierarchy:

1. Memorization: Recalling facts or definitions (e.g., “What is 2+2?”).
2. Procedures without Connections: Following an algorithm without understanding why it works.
3. Procedures with Connections: Using procedures with an understanding of the underlying mathematical concepts.
4. Doing Mathematics: Complex, non-algorithmic thinking that requires students to explore, conjecture, and reason.

Classroom Case Studies and Maintaining Demand

The case studies show how a high-demand task can decline during a lesson. A teacher might start with a great problem but then, to reduce student struggle, break it down into simple steps, lowering its cognitive demand. Factors causing this decline include giving too many hints, focusing only on the correct answer, or not allowing enough time to grapple with the problem. Maintaining high demand requires skillful teaching. For support in crafting high-level academic arguments, students can hire academic writers for essays and critiques.