Critiquing Reasoning and Error Analysis

In this lesson you will: Find the first line of a worked solution that does not follow from the line above it; Name the error rather than describe the symptom; Correct the work from that line on, and state the right answer.

Locate, name, correct

Ohio describes depth-of-knowledge level 3 tasks as including critiquing a statement and forming a conclusion, explaining or justifying a statement, and solving non-routine problems. An error-analysis question is all three at once, and it wants three things back.

Most lost points on these come from doing only the first. \"The student made a mistake\" locates nothing, names nothing, and corrects nothing.

Five errors that account for most of the damage

Errors in Algebra 1 are not evenly distributed. A small number of them show up again and again, each attached to one particular move, and each with a check that takes under a minute.

Learn them by name. Naming an error is what turns \"something went wrong here\" into a sentence a reader can verify.

The shift that goes the other way

The last row of that table deserves its own picture, because the rule sounds backwards and no amount of repeating it helps.

For g(x)=(x+5)^2, the inside is zero when x=-5. Whatever x^2 does at 0, g does at -5 — five units to the left. A plus inside the parentheses shifts left; a minus shifts right.

Do not memorize the direction. Find the input that makes the inside zero, and the vertex is sitting there.

Common mistake — Repairing the arithmetic below the error instead of the error

A student solves -4x+3<19, writes -4x<16, then x<-4. Told the answer is wrong, the student re-checks the division: 16 / (-4)=-4. Correct. Re-checks the subtraction: 19-3=16. Correct. Concludes the answer must be right after all.

Every individual computation really is correct, which is what makes this so hard to escape. The error was not an arithmetic slip; it was a missing rule — the symbol had to reverse when the divisor was negative.

Re-checking arithmetic can only find arithmetic errors. Rule errors survive any number of recomputations, because the rule was never applied in the first place and so there is nothing to recompute.

The correct solution is x>-4, and it differs from the student's answer on every single number.

How to catch it: Work down from line 1 and compare each line only to the line directly above it, asking what move was made and whether that move is permitted. The first line that does not follow is the error. Everything above it is fine; everything below it is irrelevant until that line is fixed.