11.3 Reasonableness of Results & the Validity of Student Arguments

11.3 Reasonableness of Results & the Validity of Student Arguments

Skill 3 of Competency 5 has two halves. The first is about students checking their own results; the second is about you judging whether a student's argument actually establishes what it claims.

Building reasonableness checking into instruction

The reason students accept absurd answers is that they never formed an expectation. Estimating before computing creates a benchmark against which the computed value can be judged.

Before computing 18.7 × 4.2, estimate 19 × 4 ≈ 76. A calculated answer of 7.85 or 785 is then visibly a decimal-placement error.

Useful estimation techniques:

• Rounding to compatible numbers. 348 ÷ 6.9 ≈ 350 ÷ 7 = 50.
• Front-end estimation. 4,215 + 3,890 ≈ 4,000 + 3,000 = 7,000, then adjust upward.
• Benchmark fractions. 7/15 is a little under 1/2, so 7/15 of 60 is a little under 30.
• Order-of-magnitude checks. Is the answer in the tens, hundreds, or thousands?

Structural expectations are the other half of reasonableness, and they are what the fraction-division student in section 11.2 lacked:

Contextual reasonableness is the third layer. An answer of 7.1 buses, a negative length, a probability of 1.4, or a student height of 12 feet each signal an error regardless of the arithmetic. Building a habitual final question — "does this answer make sense for this situation, in these units, at this size?" — is exactly the opportunity the skill asks teachers to create.

What makes an argument valid

A valid mathematical argument establishes a general claim through reasoning that covers every case in the claim. Verifying examples does not.

This is the single most tested idea in the skill.

Claim: The sum of two odd numbers is always even. Student A: "3 + 5 = 8, 7 + 11 = 18, and 9 + 13 = 22. So it's always even." Student B: "An odd number can be written as 2n + 1. Adding two of them gives (2n + 1) + (2m + 1) = 2n + 2m + 2 = 2(n + m + 1), which has a factor of 2, so it is even."

Student A has checked three cases out of infinitely many; the argument is incomplete, not wrong. Student B's argument uses a general representation that covers every odd number, so it is valid. Items present exactly this pair and ask which argument is complete and why.

Examples do have a legitimate role: they build conjectures, illustrate reasoning, and can disprove a claim. What they cannot do is establish a universal statement.

Counterexamples

A single counterexample disproves a universal claim. No number of confirming examples proves one.

The asymmetry is fundamental. To refute "every prime number is odd," produce 2. To refute "the square of a number is always greater than the number," produce 1/2, since (1/2)² = 1/4 < 1/2. To refute "if a quadrilateral has four congruent sides it must be a square," produce a non-square rhombus.

Constructing a counterexample is a frequent item task, and the productive habit is to test boundary and special cases: zero, one, negative numbers, fractions between 0 and 1, and degenerate figures. Those are where over-general claims usually break.

Inductive and deductive reasoning

Inductive: measuring the angles of several triangles, finding sums near 180°, and conjecturing that every triangle sums to 180°. Deductive: proving it from the parallel postulate, establishing it for all triangles at once.

Inductive reasoning can mislead. The expression n² + n + 41 produces a prime for n = 0 through 39, which is 40 confirming cases — and then fails at n = 40, where the value is 41², not prime. Items use examples like this to show why patterns must be proved rather than trusted.

A good lesson sequence uses both: students explore inductively to form a conjecture, then reason deductively to justify it.

Evaluating a student's argument

Judge the reasoning, not only the answer. Four outcomes are possible:

1. Correct answer, valid reasoning — full understanding.
2. Correct answer, flawed reasoning — the dangerous case. A student who says "I divided because the word 'each' was there" may be right this time and wrong the next.
3. Incorrect answer, sound reasoning with a computational slip — the understanding is present; the arithmetic needs attention.
4. Incorrect answer, flawed reasoning — conceptual reteaching is required.

Distinguishing case 2 from case 1, and case 3 from case 4, is precisely what these items assess. It is also why a well-designed assessment asks students to show or explain their work: an answer-only response makes cases 1 and 2 indistinguishable.

When responding to an incomplete-but-not-wrong argument like Student A's above, the productive move is to press on generality: "You've shown it works for those three pairs. How could we be sure it works for every pair of odd numbers, including ones nobody has tried?" That question moves the student from verification toward justification without telling them their work was worthless.