1. A worked example comes first
What we do
A new step starts with a worked example on the page.
What studies found
A U.S. Department of Education practice guide drew on many laboratory experiments and on classroom experiments from grade school through college. Students learned more when they alternated between studying a worked solution and solving a similar problem themselves. In one example, 8th and 9th graders learning algebra this way took less time and scored better on a later test. The guide rates the evidence "moderate."
What it doesn't show
Most of these studies were with older students, in subjects like algebra and science, and many were in laboratories. The studies alternated examples and problems throughout; our pages put the worked example at the start of a new step. None tested young children doing arithmetic at home, or our pages.
Pashler et al., 2007, Organizing Instruction and Study to Improve Student Learning, Recommendation 2. ies.ed.gov/ncee/WWC/Docs/PracticeGuide/20072004.pdf
2. Mixed practice
What we do
Practice mixes problem types, and your child moves on when a skill holds up over time.
What studies found
In a randomized trial in 54 seventh-grade math classes (787 students, 5 schools), some classes practiced with mixed problem types for about four months. On a surprise test one month later they scored 61%; classes that practiced one type at a time scored 38%.
What it doesn't show
These were 7th graders learning middle-school topics, not younger children learning arithmetic. The study tested how practice is ordered, not a rule for when to move on.
Rohrer, Dedrick, Hartwig & Cheung, 2020, Journal of Educational Psychology 112(1), 40-52. doi.org/10.1037/edu0000367
3. A slip and a wrong rule are different mistakes
What we do
A mistake that follows a wrong rule gets a short printed lesson, not a redo.
What studies found
Researchers built a model that explains many children's arithmetic mistakes as consistent wrong procedures ("bugs") rather than random slips, and used it to diagnose the work of more than 1,300 school students.
What it doesn't show
It shows that a mistake can follow a rule. It does not test how to correct one, and it does not test our lessons. Not redoing the same problems is our own reasoning.
Brown & Burton, 1978, Cognitive Science 2(2), 155-192. doi.org/10.1207/s15516709cog0202_4
What no study here tested
How we decide when to move on, and our printed lessons, are our own design. The studies above support the ideas behind them, not these choices.
