Case Study 9.1: The Math Class That Scrambled Its Homework
When a Teacher Rearranges Problem Sets and Changes Everything
At a suburban high school in the American Midwest, a 10th-grade geometry teacher named Mr. Vasquez was getting tired of a pattern he saw every year.
Students would perform reasonably well on chapter tests — each chapter focused on one type of problem, and if you'd studied the chapter, you could handle it. But on unit exams (which covered three or four chapters) and on the end-of-year state assessment (which covered everything), students struggled badly. They couldn't apply concepts they'd demonstrated understanding of only weeks before.
He'd tried many solutions: more review sessions, extra practice tests, more thorough in-class explanations. None produced lasting change.
Then, preparing for a math pedagogy workshop, he read a summary of Rohrer and Taylor's interleaving research.
The Experiment
Mr. Vasquez decided to try a modification to his homework structure. He would keep his teaching structure exactly the same — introducing one topic at a time in class, using examples and explanations in order. What he would change was the homework.
Instead of assigning problem sets organized by type (Chapter 5 homework = all Chapter 5 problems), he would assign mixed homework sets — problems drawn randomly from the current chapter and all previous chapters, with no label indicating which chapter each problem came from.
The first time he implemented this was with his spring semester geometry class, mid-unit. He handed out the new-format homework on a Thursday.
By Friday morning, he had a line at his door.
The Initial Resistance
The students were frustrated. Several were upset.
"These problems are from all different chapters," one student said. "We don't know how to do some of these yet."
"All the problems are from chapters we've covered," Mr. Vasquez clarified. "They're just mixed together."
"But how are we supposed to know what method to use?"
"That," he said, "is the question I want you to figure out."
He wasn't unsympathetic. The frustration was real, and he recognized it. With blocked homework, students enter the problem set knowing what tool they need — they just need to apply it. With mixed homework, they face a problem and must first identify what it is before they can begin. That extra cognitive step is real extra work.
He added something to help: at the top of each problem, students were asked to write (before attempting to solve): "This problem type is: ___." Identifying the problem type explicitly before solving it was a metacognitive step that helped students develop the discrimination skill consciously.
Within two weeks, the complaints dropped significantly. By week four, students were doing the identification step automatically — they barely thought about it. They were performing the skill.
The Midterm Results
The midterm exam covered five chapters of geometry: basic proofs, triangle congruence, parallel lines and angles, quadrilaterals, and similarity.
The midterm was naturally interleaved — problems from all five chapters in random order, no labels.
Mr. Vasquez compared his current class performance to the prior year's class performance on the same exam. The prior year's class had used blocked homework throughout.
The current class (interleaved homework) scored an average of 8 percentage points higher on the midterm than the previous year's class.
This is a real but modest improvement. More telling was the score distribution: the prior year had a long tail of students who did well on individual chapter tests but collapsed on the unit exam. The current class showed less of that pattern — students who demonstrated competence during the chapter were more likely to demonstrate it on the mixed-format exam.
The distinction between "can do it when prompted" and "can do it when I have to identify it myself" had been partially closed.
The End-of-Year Assessment
Mr. Vasquez tracked his students' performance on the state geometry assessment, which covers the full year's curriculum in an interleaved format.
Compared to his previous three years of classes (which used blocked homework throughout), the interleaved-homework class performed: - 6 percentage points higher on average - With meaningfully better performance specifically on problems from early-year chapters (which had been "forgotten" more by previous classes)
The early-chapter advantage suggests what the research predicts: interleaved practice, by regularly revisiting earlier material in the context of mixed problem sets, provides a natural form of distributed review. Earlier topics don't get abandoned when new topics arrive — they stay in the rotation.
Student Feedback at Year's End
At the end of the year, Mr. Vasquez asked his students to reflect on the homework format. Some edited excerpts:
"At first I hated it. It was so much harder than just doing the problems from whatever we'd just learned. But by the second semester, figuring out what kind of problem it was first actually felt natural. On the state test, I wasn't confused by the order — I just did them."
"I feel like I actually remember geometry now? Like, I could still do problems from September. With my other classes, I usually feel like I've forgotten the early stuff by May."
"It was frustrating but I think it's better. I can't explain exactly why."
The last quote is worth sitting with. Students who benefit from interleaving often can't fully articulate why — they just notice that transfer, retention, and performance in novel situations are better. The mechanism isn't transparent to the learner while it's happening.
What Mr. Vasquez Did Differently (and What Stayed the Same)
For those interested in replicating this approach, the specific changes were minimal:
Changed: - Homework problem sets: reorganized from chapter-specific to mixed (current chapter + all previous chapters) - Added the "identify problem type before solving" metacognitive prompt
Unchanged: - Teaching sequence (topics introduced in order) - In-class examples (still organized by topic) - Number of problems assigned - Total homework time expected
The minimal disruption to teaching structure made this achievable without institutional resistance. Mr. Vasquez didn't need approval to change his problem sets. He didn't need new curriculum materials. He needed a better understanding of what the homework was for — and the answer was not practice, but transfer.
Principles Illustrated by This Case Study
- The test of real learning is not "can you do it when prompted" but "can you do it when you have to figure out what's needed." Interleaved practice trains the latter.
- Initial student resistance is expected and temporary. The cognitive load of identifying problem types drops substantially after consistent practice.
- The metacognitive step helps. Explicitly requiring students to identify the problem type before solving it accelerates the development of discrimination skill.
- Long-term retention of early material is an underappreciated benefit of interleaved practice. Mixed homework sets keep earlier topics active, preventing the typical forgetting of early-year material.
- Minimal changes to teaching structure can produce meaningful learning improvements. Teachers don't need to overhaul their curriculum to implement interleaving — just the structure of the practice.