An Eight-Item Quiz, an Inverted Ratio, and Three Items Nobody Taught
Ms. Okafor teaches seventh-grade mathematics. Her second-period class of 26 students took an eight-item quiz on proportional reasoning; the quiz counts 20 percent of the quarter grade. The item analysis shows items 1 and 2 (unit rates) at p = .88 and .85, item 3 (equivalent ratios) at p = .81, and item 5 (setting up a proportion from a word problem) at p = .19, with 14 of 26 students choosing distractor C, which reflects an inverted ratio. Items 6, 7, and 8 have p values of .12, .08, and .15 and test a standard the team has not yet taught; it is scheduled for two weeks later, but a colleague on the team shared the quiz. Two students, both recent arrivals at WIDA level 2, scored 2 out of 8, although their scratch work on items 5 through 8 shows correct proportional setups; the word problems use passive voice and two-clause sentences. The pacing calendar puts the proportional-reasoning unit test in six school days. Ms. Okafor's principal expects every teacher to enter the quiz score in the online gradebook by Friday.
1. Analyze the item and distractor data. What does the pattern on item 5 tell you about students' understanding, and what does it not tell you? Include what additional evidence you would collect.
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The item 5 pattern is a systematic error, not noise. Fourteen of 26 students chose the same distractor, and that distractor encodes one specific mistake: inverting the ratio, most likely by dividing in the wrong order or by setting up part-to-part where the problem requires part-to-whole. When a majority converges on a single wrong answer, the class has learned a procedure without the underlying relationship, and the procedure has a consistent flaw. That is a reteaching target, not a discipline or effort problem.
What the data do not tell her:
- Whether students can reason about ratios qualitatively, using a ratio table or a bar model, when no proportion setup is required.
- Whether the setup failure is a reading problem or a mathematics problem.
- Whether the two WIDA level-2 students missed items because of the passive voice and multi-clause sentences, because of computation, or both. Their correct setups suggest their proportional reasoning is stronger than their score.
Additional evidence to collect before planning: pull the scratch work for item 5 and sort it by error type; interview four or five students, including the two multilingual learners, asking them to talk through one word problem aloud; and administer two quick items, one with a diagram and minimal text and one requiring a written proportion, to separate representation from language.
That combination distinguishes a conceptual gap from a language barrier from a computational slip, which determines whether the reteach is about meaning, about sentence structure, or about arithmetic.
2. What is the appropriate action regarding grading and the pacing calendar for items 6 through 8? Explain your reasoning and what you would say to the team and to the principal.
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Items 6 through 8 test a standard the class has not been taught, so they cannot count toward a grade. A grade must represent achievement of taught content; including untaught material makes the score uninterpretable, penalizes students for the pacing calendar, and would be hard to defend to a parent or an administrator. Ms. Okafor should rescore the quiz out of 5, enter that score by Friday, and tell students which items were set aside and why. The three items are still useful: they are a free pre-assessment for the standard coming in two weeks, so she should keep the responses and analyze them rather than discard them.
To the team: raise the quiz-sharing problem as a calibration issue, not an accusation. Propose that the team agree on which items are eligible for a grade, that quizzes be labeled with the standards they assess, and that anything ahead of pace be marked as pre-assessment. That norm protects everyone.
To the principal: a brief, factual note. The quiz included three items from an untaught standard because of a team mix-up; the graded score reflects only the five taught items; the remaining items will be used to plan the upcoming unit. Giving the principal the reasoning before a parent asks about a low quiz score is far better than explaining afterward.
What she should not do is enter the 8-item score to meet the Friday deadline and then 'curve' it. Curving conceals the validity problem and teaches students that grades are negotiable.
3. Design the next six days. Specify the reteaching for item 5, how to support the two multilingual learners without lowering cognitive demand, and what formative checks to run before the unit test.
Read the worked answer
Days 1 and 2, ratio meaning before procedure. Start with a ratio table and a bar model for the same context, then move to the proportion as a record of the relationship. Make the ordering question explicit: 'Which quantity goes with which?' Have students label both sides of every proportion in words before writing numbers, and compare a correct setup with the inverted one side by side so the error is visible rather than merely corrected.
Day 3, faded scaffolds. Students sort eight pre-written setups into correct and inverted, then write their own for two contexts. Pair work with a sentence frame: 'The ratio of ___ to ___ is ___ , and the ratio of ___ to ___ must be the same because ___ .'
Day 4, word-problem structure. Rewrite the passive-voice items into active voice for practice and teach students to identify the two quantities and the comparison before computing. Amara and the other multilingual learner get a pre-taught picture glossary, a paraphrased version of each problem with the same numbers and the same reasoning demand, and the option to explain a setup orally. The mathematics is not simplified; only the language load is reduced.
Day 5, mixed practice with two non-routine items requiring a diagram, not a proportion.
Day 6, short formative quiz of four items at the same cognitive demand as the unit test.
Formative checks: a one-item exit ticket each day scored for setup only; a sorting task on day 3; and a verbal think-aloud from four target students on day 5. If setup accuracy is below about 70 percent by day 5, delay the unit test rather than test anyway.