LocusBlog

Redo the ones you missed, later

A student misses this one on Tuesday.

2x + 5y = -1
3x - 2y = 27

She goes after x. She multiplies the first equation by 3. She multiplies the second by 2 and writes 6x - 2y = 54. The -2y never got multiplied. Every line after that is wrong.

Mark it, hand it back, move on. That is the standard loop, and it fails in a specific way. She sees the red mark, reads the worked solution, agrees with it, and forgets it by Thursday. Reading a solution is a different act from producing one.

So bring the problem back. The interesting questions are what to bring back and when.

Same skill, different numbers

Her mistake was not x = 7. Her mistake was a habit: she does not distribute a multiplier across every term, and a negative coefficient is where it shows.

Handing her the identical problem later does not test that habit. She can pass it on recall of the answer, or on recall of the shape of the worked solution she read. Both are cheaper than doing the work, and students take the cheap route because it produces a green check.

Give her this on Thursday instead.

4x + 3y = 5
5x - 2y = 12

New coefficients. New answer. Same trap: to eliminate y she multiplies the first by 2 and the second by 3, and the -2y has to get multiplied. If she has the habit, this takes ninety seconds. If she memorized x = 7, y = -3, she has nothing to work with.

This is why Locus generates problems from templates. Every retry is a fresh variant with new numbers, so the retrieval is forced. Roediger and Butler reviewed the retrieval literature and describe retrieval practice as “a powerful mnemonic enhancer, often producing large gains in long-term retention relative to repeated studying.” They also take on the obvious objection, that testing only drills a fixed response. It does not. Repeated testing produced better transfer than repeated studying, including on new questions in domains the student had not been tested on.

What the gap should be

Rohrer and Taylor ran the cleanest version of this in math. 216 college students learned a procedure for counting permutations. Everyone got ten practice problems. One group massed all ten in a single session. The other did five, then five more a week later.

Tested one week after practice, the two groups were level: 75 percent for the spaced group, 70 percent for the massed. Tested four weeks after, spaced scored 64 percent and massed scored 32 percent. Same problems, same total effort, twice the score.

The second experiment in that paper is the one I would put in front of a department. Students did either nine practice problems or three, all in one sitting. The extra six changed nothing at one week or at four. Volume in a single session bought no retention. Scheduling did all the work.

The meta-analysis backs the general effect. Cepeda, Pashler, Vul, Wixted and Rohrer pooled 839 assessments from 317 experiments across 184 articles. Averaged over all retention intervals, spaced study produced 47.3 percent correct against 36.7 percent for massed. Only 12 of 271 comparisons failed to show a benefit. Their practical line is that separating learning episodes by at least one day, rather than packing everything into one session, is what maximizes long-term retention.

How far apart, in days? Cepeda, Vul, Rohrer, Wixted and Pashler taught over 1350 people a set of facts, varied the gap before review out to 3.5 months, and varied the delay before the final test out to a year. For every test delay, performance climbed as the gap grew, peaked, and then fell off. The optimum gap was about 20 percent of the test delay for delays of a few weeks, dropping to about 5 percent at a year. In days, for recall: a test 7 days out peaked at a 1-day gap, 35 days out at an 11-day gap, and 70 days out at a 21-day gap.

Read that against a unit test five weeks away. The review that helps most lands somewhere around a week and a half after the miss. Not that evening. Not the night before the test. Those are the two moments a normal class uses.

Locus follows the same shape. A missed problem enters a review queue, comes back after a short gap first, and gets pushed further out each time the student gets it right. What comes back is a variant, never the original.

A queue like that interleaves by construction, because what surfaces on a given day is a mix of whatever the student has missed recently. That mix matters on its own. Rohrer, Dedrick and Stershic gave 126 seventh graders the same practice problems over three months and changed only the arrangement, blocked or interleaved. On an unannounced test one day after review, interleaved scored 80 percent against 64. Thirty days after review, 74 percent against 42. Nothing was added. The problems were reordered.

The correction is where the learning happens

Retrieval alone helps. Feedback on top of it helps more, and the timing is not what most people expect.

Butler, Karpicke and Roediger tested students on prose passages and then varied the feedback. On the final recall test, previously tested material scored .42 against .26 for material never tested. Adding feedback moved that to .65 against .42 with none. Then the surprise: delayed feedback scored .70 against .60 for immediate feedback. Handing over the correct answer at the instant of the mistake was worse than handing it over later.

That is awkward advice for a teacher standing at a desk. It is a clean argument for making the correction its own scheduled event.

Metcalfe’s review of learning from errors goes further. Error avoidance is the default in American classrooms and it is counterproductive. Errorful learning followed by corrective feedback beats a clean run. The effect is strongest for errors committed with high confidence, which get corrected more readily than low-confidence ones.

The student who was sure about 6x - 2y = 54 is the one most likely to fix it permanently once she watches it fail. A confident wrong answer is the best material a test produces. Throwing it away with a score and a red pen is the waste.

So assign corrections, and make them count for something. On Locus a teacher assigns test corrections and each student gets fresh variants of the problems that student missed. There is nothing to copy from a neighbor, because no two students get the same set. The original answer key is worthless, because the numbers moved. The only route through is the procedure.

A student who can reproduce the answer she saw has learned an answer. A student who can produce it from different coefficients has learned the method. The test only ever asks for the second one.

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