Critiquing Reasoning — I can critique someone else's reasoning and identify a specific error or missing justification. - Grade 9 Math

SECTION 1

Historical Context & Motivation

Mathematics has never been just about getting the right answer — it has always been about proving why an answer is right. Since ancient times, mathematicians have critiqued one another's work, searching for hidden assumptions, logical gaps, and outright mistakes. This tradition of peer critique is what separates math from guessing: every claim must stand up to scrutiny. When you learn to critique someone else's reasoning, you develop the same sharp eye that history's greatest mathematical minds relied on.

~300 BCE • Euclid's Elements

Euclid organized geometry into a logical system of axioms, definitions, and proofs. Every theorem had to follow logically from previous results — introducing the idea that reasoning must be justified step by step.

1637 • Descartes' Discourse on Method

René Descartes proposed that we should accept nothing as true unless we can verify it through careful reasoning. His method encouraged systematic doubt, which is the foundation of critiquing arguments.

1879 • Frege's Formal Logic

Gottlob Frege created the first system of formal logic, giving mathematicians precise tools to analyze whether an argument is valid. His work made it possible to detect exactly where reasoning breaks down.

2010 • Common Core Mathematical Practices

Mathematical Practice Standard 3 — "Construct viable arguments and critique the reasoning of others" — was adopted nationwide, recognizing that evaluating reasoning is a core mathematical skill for every student.

The central question this lesson addresses is: How do you read someone else's mathematical argument, identify the precise point where it goes wrong, and explain why it fails? Whether you're reviewing a classmate's proof, checking an answer on a test, or evaluating a claim in the real world, this skill is essential.

SECTION 2

Core Principles of Critiquing Reasoning

Critiquing reasoning is not about being negative or finding fault for its own sake. It is a constructive process of checking whether every step in an argument is logically valid and fully justified. To do this effectively, you need to understand the most common ways that reasoning can break down. The following principles form the foundation of mathematical critique.

Check Each Step

Read the argument one step at a time. Ask: "Does this step follow logically from the step before it?" A single unjustified leap can invalidate the entire conclusion.

Identify the Claim vs. the Evidence

Separate the conclusion (what the person says is true) from the reasoning (how they justify it). A correct conclusion reached through flawed reasoning is still a flawed argument.

Look for Hidden Assumptions

Some arguments rely on facts or rules that the author never states. These hidden assumptions may be wrong. If a step requires an assumption that isn't justified, the argument has a gap.

Test with Counterexamples

If someone claims something is always true, try to find one case where it fails. A single counterexample is enough to disprove a general statement.

Be Specific in Your Critique

Don't just say "that's wrong." Point to the exact step that fails and explain why. A good critique identifies the error type and suggests how the argument could be fixed.

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KEY TAKEAWAY

Think of critiquing reasoning like being a detective reviewing a case file. The conclusion is the suspect, and each step of reasoning is a piece of evidence. If any piece of evidence is fake, missing, or doesn't connect to the next one, the case falls apart — even if the suspect actually did it. Your job is to check every link in the chain.

SECTION 3

Anatomy of a Mathematical Argument

Every mathematical argument follows a chain-like structure: it starts with given information, moves through a series of logical steps (each supported by a justification), and arrives at a conclusion. The diagram below shows how a valid argument flows, and where common errors can occur.

The top row shows the chain of a valid argument: given information flows through justified steps to a conclusion. The bottom section shows the five most common failure points that a critique should look for.

When you critique an argument, you are essentially tracing this chain and testing every link. The five failure points shown in the diagram — computational error, missing justification, logical leap, wrong property, and hidden assumption — account for virtually all errors you'll encounter in grade 9 math. Getting comfortable with these categories makes your critiques more precise and useful.

SECTION 4

The Critique Framework — How It Works

Critiquing reasoning is a structured process, not just a feeling that something is "off." The framework below gives you a repeatable method you can apply to any mathematical argument. Think of it as a checklist you run through every time you evaluate someone's work.

The Four-Question Critique Framework

QUESTION 1 — VALIDITY

Does each step follow logically from the previous step?

Check that every algebraic manipulation, every geometric claim, and every logical inference is correctly derived from what came before.

QUESTION 2 — JUSTIFICATION

Is every step supported by a stated reason, rule, or property?

A step without a justification is a gap in the argument. Valid reasons include properties (commutative, distributive), definitions, theorems, or given information.

QUESTION 3 — ACCURACY

Are all computations and applications of rules performed correctly?

Even when the right rule is chosen, it can be applied incorrectly. Check arithmetic, sign handling, distribution, and simplification.

QUESTION 4 — COMPLETENESS

Does the conclusion actually answer the original question?

Sometimes the reasoning is valid but the final answer doesn't address what was asked, or the argument stops too early and leaves the conclusion unjustified.

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PRO TIP

When you write your critique, use this sentence template: "The error occurs in [step number], where the student [describes what they did]. This is incorrect because [states the rule or property that was violated]. A correct approach would be [suggests fix]."

SECTION 5

Classifying Common Errors

Not all errors are the same, and being able to name the type of error makes your critique clearer and more helpful. The diagram below classifies the five major error types with examples from algebra and geometry, which are the most common contexts in grade 9 math.

The five categories of reasoning errors, each with a concrete algebra or geometry example. When writing a critique, naming the error type helps both you and the person whose work you're reviewing.

Error Type	How to Spot It	Critique Phrase to Use
Computational	Redo the arithmetic or algebra yourself.	"In step __, the calculation should be ____ because ____"
Missing Justification	Ask "Why?" after each step. If there's no answer, it's a gap.	"Step __ lacks a justification; a valid reason would be ____"
Logical Leap	Check if the conclusion accounts for all possible cases.	"The conclusion doesn't follow because __ was not considered"
Wrong Property	Verify that the cited property actually says what they claim.	"The __ property was applied incorrectly; it actually states ____"
Hidden Assumption	Look for conditions required for a step to be valid.	"This step assumes __, which has not been established"

Error types with detection strategies and critique phrases

SECTION 6

Worked Example — Critiquing a Flawed Proof

Let's put the framework into action. Below is a student's attempt to solve an equation. Your task is to read the work, apply the four critique questions, and pinpoint exactly where the reasoning breaks down.

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THE STUDENT'S WORK

Solve: 2(x − 3) + 4 = 3x − 8 Step 1: 2x − 6 + 4 = 3x − 8 (Distribute the 2) Step 2: 2x − 2 = 3x − 8 (Combine like terms) Step 3: 2x = 3x − 6 (Add 2 to both sides) Step 4: −x = −6 (Subtract 2x from both sides — wait, I'll subtract 3x) Step 5: x = 6 (Divide by −1) The student claims: x = 6.

Critiquing the Student's Reasoning

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Step 1 — Read the original problem and restate the goal

The student needs to solve 2(x − 3) + 4 = 3x − 8 for x. We'll trace each step to check if it follows logically and is correctly computed.
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Step 2 — Check Student Step 1 (Distribution)

The student writes 2(x − 3) + 4 = 2x − 6 + 4. Using the distributive property: 2 × x = 2x and 2 × (−3) = −6. This is correct, and the justification ("Distribute the 2") is stated.

✓ Step 1 is valid and justified.
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Step 3 — Check Student Step 2 (Combining Like Terms)

The student simplifies 2x − 6 + 4 to 2x − 2. We compute: −6 + 4 = −2, so 2x − 2 is correct. The justification ("Combine like terms") is valid.

✓ Step 2 is valid and justified.
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Step 4 — Check Student Step 3 (Adding 2 to both sides)

The student goes from 2x − 2 = 3x − 8 to 2x = 3x − 6 by adding 2 to both sides. On the left: 2x − 2 + 2 = 2x ✓. On the right: 3x − 8 + 2 = 3x − 6 ✓. This step is correct.

✓ Step 3 is valid and justified.
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Step 5 — Check Student Step 4 (⚠️ ERROR FOUND)

The student writes "Subtract 2x from both sides — wait, I'll subtract 3x" and goes from 2x = 3x − 6 to −x = −6. Let's verify: if we subtract 3x from both sides, we get 2x − 3x = 3x − 6 − 3x, which is −x = −6. Mathematically this is actually correct, but the justification is confusing and incomplete — the student's crossed-out thought suggests uncertainty, and the stated operation changes mid-sentence. However, the result −x = −6 is valid.

⚠ Step 4 has unclear justification, but the math is technically correct.
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Step 6 — Check Student Step 5 and the Final Answer

From −x = −6, the student divides by −1 and concludes x = 6. This is correct. Now we verify the final answer by substituting x = 6 into the original equation: 2(6 − 3) + 4 = 2(3) + 4 = 6 + 4 = 10. And 3(6) − 8 = 18 − 8 = 10. Both sides equal 10, so x = 6 is correct.

✓ The answer is correct.
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Step 7 — Write the Critique

Our critique: "While the student arrived at the correct answer of x = 6, the justification in Step 4 is unclear. The student begins by stating 'subtract 2x' then changes to 'subtract 3x' without explaining why. A clear justification would be: 'Subtract 3x from both sides to isolate the variable on one side (Subtraction Property of Equality).' Unclear reasoning, even when the answer is correct, weakens the argument and could lead to errors in more complex problems."

Critique complete: error type = missing/unclear justification in Step 4

SECTION 7

Strengths and Limitations of Critique

Learning to critique reasoning is powerful, but like any skill, it has both strengths and limitations. Understanding these helps you use critique appropriately and avoid common pitfalls when reviewing others' work.

Strengths of Critiquing Reasoning	Limitations & Cautions
Deepens your own understanding by forcing you to think about why each step works.	A correct answer reached through flawed reasoning can be hard to critique — you might second-guess valid work.
Catches errors before they compound into larger mistakes in multi-step problems.	Critique only checks the reasoning shown — it can't catch errors in someone's mental math that they didn't write down.
Builds communication skills — explaining why something is wrong requires precise mathematical language.	Poorly phrased critique can feel like a personal attack rather than constructive feedback.
Prepares you for proof-based courses in Geometry and beyond.	Some errors are subtle and require advanced knowledge to detect (you may not always find every issue).

Strengths and limitations of mathematical critique

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KEY TAKEAWAY

Critiquing reasoning is like being a code reviewer for a software developer. A code review doesn't mean the program won't work — it means checking whether the code is clear, well-structured, and handles edge cases. Similarly, a good math critique checks not just whether the answer is right, but whether every step would hold up under any conditions.

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Connection to Formal Proof & Advanced Reasoning

The critiquing skills you're building now are the foundation for formal proof writing, which you'll encounter in Geometry and later in college-level mathematics. When you write a proof, you are essentially pre-critiquing your own reasoning: every step must be justified so that no one can find a gap. The habits of checking, questioning, and verifying that you develop through critique transfer directly to proof construction.

Critiquing (Grade 9)	Formal Proof (Geometry & Beyond)
Read someone else's argument and check for errors.	Construct your own argument with no errors from the start.
Identify where a justification is missing.	Provide explicit justification for every step (postulates, theorems, definitions).
Find one counterexample to disprove a claim.	Prove a claim is true for all cases (direct proof, proof by contradiction, etc.).
Use informal language: "This is wrong because..."	Use formal structure: Given, Prove, Statements & Reasons columns.

How critiquing reasoning evolves into formal proof writing

Beyond math class, the ability to evaluate reasoning is essential in science, law, business, and everyday decision-making. When you read a news article that cites statistics or hear someone make a logical argument, the same skills — checking each step, looking for hidden assumptions, testing with counterexamples — help you decide whether the reasoning is sound.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

A student claims: "I checked my answer by plugging it back into the equation and it worked, so my reasoning must be correct." Is this a valid way to verify that reasoning is sound? Explain why or why not.

Show answer

No, getting a correct answer does not guarantee that the reasoning is correct. A student might make two errors that cancel each other out, or might skip steps and arrive at the right answer by luck. Critiquing reasoning means checking that every step is logically valid and justified, not just that the final result is correct. Substitution checks the answer but does not check the process.

Question 2: BASIC CALCULATION

A student solves: 5x − 3 = 2x + 9. Step 1: 5x − 2x = 9 + 3 (Move terms). Step 2: 3x = 12. Step 3: x = 4. Identify whether there is an error. If the reasoning is correct, explain why each step is justified. If there is an error, name the step and the error type.

Show answer

The student's work is correct in this case. Step 1 subtracts 2x from both sides and adds 3 to both sides (using the Addition and Subtraction Properties of Equality), yielding 3x = 12. Step 2 divides both sides by 3 (Division Property of Equality), giving x = 4. However, the justification "Move terms" in Step 1 is informal. A stronger critique would note that while the answer is right, the student should cite the specific property of equality used, not just say "move."

Question 3: INTERMEDIATE

A student solves: (x + 5)² = 49. Step 1: x² + 25 = 49 (Square each term). Step 2: x² = 24 (Subtract 25). Step 3: x = √24 ≈ 4.9. Critique this reasoning. Identify each error, name its type, and provide the correct approach.

Show answer

There are two errors. First, in Step 1 the student incorrectly expands (x + 5)² as x² + 25. This is a Wrong Property error: the student appears to apply (a + b)² = a² + b², but the correct identity is (a + b)² = a² + 2ab + b², so (x + 5)² = x² + 10x + 25. Second, in Step 3 the student only gives the positive square root, which is a Logical Leap — when x² = c, both x = √c and x = −√c are solutions. The correct approach: take the square root of both sides first: x + 5 = ±7, giving x = 2 or x = −12.

Question 4: APPLIED

A classmate argues: "A cell phone plan charges $30 per month plus $0.10 per text. I sent 200 texts, so my bill is 30 + 200(0.10) = $50. Since my friend also paid $50, she must have sent 200 texts too." Critique this reasoning. What specific type of error does the classmate make, and why is the conclusion invalid?

Show answer

The classmate commits a Logical Leap. The computation 30 + 200(0.10) = 50 is correct for their own bill. However, concluding that the friend also sent 200 texts assumes the friend has the same plan with the same base rate and per-text charge. The friend might be on a different plan — for example, $40 per month plus $0.05 per text, in which case 200 texts would cost $50 as well, but so would many other combinations. The classmate's conclusion doesn't follow because they assumed, without justification, that both plans are identical. This is also a Hidden Assumption error.

Question 5: CRITICAL THINKING

A student writes: "Prove that all even numbers greater than 2 can be written as the sum of two prime numbers." Their proof: "4 = 2 + 2. 6 = 3 + 3. 8 = 3 + 5. 10 = 5 + 5. 12 = 5 + 7. Since all the examples I tried work, this proves the statement is true for all even numbers." Write a thorough critique of this argument. What type of error is present, and what would be needed to make this a valid proof?

Show answer

The student's argument is a Logical Leap. Showing that a pattern holds for five specific examples does not prove it holds for all even numbers. This is the difference between inductive evidence and deductive proof. To prove a statement about all even numbers, you would need a logical argument that covers every possible case, not just a few. In fact, this conjecture is known as Goldbach's Conjecture, and it has been verified for extraordinarily large numbers by computers, but no one has ever produced a complete proof. The student's examples provide supporting evidence, but a valid proof would require a deductive argument showing that for any arbitrary even number n > 2, primes p and q exist such that p + q = n. The student should acknowledge that checking examples can only suggest a pattern — it cannot establish truth for infinitely many cases.

SUMMARY

Lesson Summary

Summary

Critiquing mathematical reasoning means evaluating an argument step by step to determine whether every claim is logically valid and fully justified. The Four-Question Critique Framework guides you to check validity, justification, accuracy, and completeness. Errors fall into five categories: computational errors, missing justifications, logical leaps, wrong properties, and hidden assumptions.

A strong critique is always specific: it identifies the exact step where the error occurs, names the type of error, explains why it is incorrect, and suggests a correction. Remember that a correct final answer does not guarantee correct reasoning — and that counterexamples are one of the most powerful tools for disproving general claims. These skills prepare you for formal proof writing in Geometry and beyond, and for evaluating arguments in any context throughout your life.