Placement, Repair and Retention: Keeping Mathematics Alive

Finishing a course does not mean you can use what it taught, and failing a test does not mean you must start over. This page is about placing yourself honestly, repairing the specific skills that fail, and keeping what you learn alive over months. It is the practical companion to Mathematics: Start Here.

Readiness versus completion

A platform progress meter, a list of finished chapters and hours logged all measure activity. Readiness is a different thing: can you solve fresh problems on the prerequisites of the next subject, without help, and explain why the methods work? A meter can stand at a high percentage while a whole domain is weak, because it samples skills rather than covering all of them. Treat meters as signals, not exit requirements.

Pretest first, or start at the beginning

• Familiar material: take a diagnostic before any lessons. If you pass, credit those skills and move on. Watching explanations of things you already know feels productive and teaches little.
• Unfamiliar material: start at unit 1. A course-wide pretest on content you have never seen is discouraging and tells you nothing you did not already know.

Working score bands

These are working standards for self-study, not official standards of any course or platform.

Round required counts upward on short tests (for example 18 of 20 or 9 of 10). Provisional The numbers are conventional thresholds chosen for practical use; the research below supports the principles of testing and spacing, not these exact cut-offs.

What counts as independent

Only a fresh, independent attempt counts: no hints, no copied solution, no lucky guess you could not defend. Record the help used on each attempt as none, hint, worked example or checked solution. Then schedule a delayed check at least seven days later on new problems. Success on both the first and the delayed check is what "durably passed" means. The delay is a gap between checks, not a week of waiting or repeating lessons.

Reading a sample test honestly

Any test samples. A 20-question check across five domains gives only about four questions per domain. A learner who truly succeeds 70 percent of the time on a domain will still get four out of four about one time in four (0.7 to the fourth power is about 0.24), so a clean score on a few questions can hide a weak domain. Likewise, a learner whose true accuracy is 80 percent will score at least 18 of 20 about one time in five. Demonstrated by the binomial arithmetic. Practical consequences: cover every required domain, add questions where a test omitted one, and never let a high total excuse a failed domain.

Diagnosing an error

"Careless" is not a diagnosis. For each wrong answer write the incorrect step and the corrected reason, then label the error:

1. Concept: you misunderstood what something means.
2. Method selection: you chose the wrong tool.
3. Execution: the method was right but a step went wrong.
4. Notation or attention: you misread, dropped a sign, or mis-copied.

Keep these in an error log. Recurring entries show where the real repair belongs. Concept errors need an explanation and example; execution errors need slower, checked practice. See Algebra: The Skills That Carry Everything for the reasons behind common algebra rules.

Forgetting is normal; retrieval beats rereading

Forgetting over time was described by Ebbinghaus in 1885 and has been replicated many times. Established Retrieval practice, which means trying to recall or solve without looking, strengthens later memory more than rereading does (Roediger and Karpicke, 2006). In a broad review of learning techniques, Dunlosky and colleagues (2013) rated practice testing and distributed practice as high utility and rereading as low. Cepeda and colleagues (2006, 2008) found that spacing practice out helps retention and that the best gap depends on how long you need to remember. Most of this evidence comes from verbal and classroom materials, with fewer long-term studies in higher mathematics, so treat it as strong in principle and approximate in detail.

Expanding intervals and rotating skills

After learning or repairing a skill, retrieve it on a new problem after roughly 1, 3, 7, 14, 30 and 60 days. These are practical intervals, not uniquely optimal ones. If a retrieval fails, repair it and drop back to the shorter intervals. Rotate skill families so you are not retrieving the same one every day. A simple rotation:

Mixing problem types (interleaving) also helps; in a randomized classroom study, Rohrer, Dedrick and Stershic (2015) found interleaved practice improved later mathematics test scores. Established for that setting and age group.

After a long absence

Do not restart the course. Start with your error log and a mixed check across what you had learned. A few weeks away usually calls for a short mixed check; longer gaps may justify a broader assessment if gaps appear widespread. Repair the named failures only. One forgotten method does not cancel a legitimately completed course.

A disrupted week

When life interrupts, shrink the plan instead of abandoning it: three short sessions of about 15 minutes, keeping the current prerequisite and one retrieval item alive. Do not add catch-up sessions later. Debt-based plans fail because they turn one missed day into a reason to quit. How to structure the learning itself is in Learning Mathematics: Procedures and the Six Abilities.

Related routines in other fields

The same logic applies where skill decays without use. Musicians keep scales and repertoire alive by short daily retrieval rather than occasional long sessions, and strength training schedules sessions by recovery and progresses only when targets are met; see Strength Training for Busy Lives: The Recovery-Based Cycle. In each case the principles are the same: observable targets (Deliberate Practice: Choose Targets You Can Observe), fresh attempts, honest records, and a regular review such as The Weekly Loop: Read, Reconstruct, Make, Defend, Log.

Try this

1. Take 20 mixed problems from skills you believe you know. Do them without notes, then label every error with one of the four types.
2. Pick one skill you repaired recently and schedule fresh retrieval problems at 1, 3, 7 and 14 days. Record whether each succeeds.
3. Write a "disrupted week" plan of 60 minutes total that you could follow on your worst day.

Further reading

• Roediger and Karpicke, "Test-enhanced learning", Psychological Science, 2006.
• Dunlosky et al., "Improving Students' Learning With Effective Learning Techniques", 2013.
• Cepeda et al., "Distributed practice in verbal recall tasks", Psychological Bulletin, 2006.
• Rohrer, Dedrick and Stershic, "Interleaved practice improves mathematics learning", 2015.
• Peter Brown, Henry Roediger and Mark McDaniel, Make It Stick (Belknap Press of Harvard University Press, 2014), an accessible summary.