3 — Construct Viable Arguments and Critique the Reasoning of Others
Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose.
Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen to or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.
New: Vignettes and videos
Grade 1: Related Story Problems
A major emphasis of the primary grades involves learning how to add and subtract using a variety of representations for the different kinds of contexts that are modeled by addition and subtraction. Read more (PDF)
Grade 2: Meg’s Balloons
Although students may have methods to calculate and to solve different kinds of story problems, it is a different skill to look across related problems to notice generalizations about the behavior of the operations involved. Read more (PDF)
See the related video: Meg's Balloons
Grade 2: How Do You Know That 23 + 2 = 2 + 23?
In this example, many of the students are focused on the regularity they notice: that addition is commutative no matter what numbers they use, and subtraction is not. Read more (PDF)
See the related video: Why Does 23 + 2 = 2 + 23?
Grade 3: Adding 1 to an Addend, Adding 1 to a Factor
This extended example presents a sequence of eight lessons in which students 1) identify regularities they notice in pairs of related problems, 2) articulate a generalization about the behavior of an operation, 3) explore that generalization, and 4) develop arguments to prove that the generalization is true for all whole numbers. Read more (PDF)
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Grade 3/4: Crazy Cakes
Students in a fourth grade class worked in pairs to divide shapes such as those shown below, "crazy cakes," into two parts of equal area. Read more (PDF)
See the related video: Crazy Cakes
Grade 4: Joey’s Run
A fourth-grade class has been working on a series of story problems that involve multiplying a whole number times a fraction. Read more (PDF)
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Grade 4: Finding Equivalent Fractions
This classroom example begins as students are trying to articulate a conjecture about the relationship between the numerator and denominator in fractions equivalent to unit fractions (MP6). Once they have articulated their conjecture, one of the students offers a representation to explain why the relationship they have noticed must work. Read more (PDF)
Grade 5: Filling Boxes
Understanding the volume of a rectangular solid is not simply a matter of memorizing and applying the formula $l \times w \times h$. Rather, students must be able to view rectangular prisms as decomposed into layers of arrays of cubes and derive their strategy for finding volume from such a decomposition. Read more (PDF)
Illustrations