Spiral problems

Recasting single-answer problems so that working them runs the Learning Spiral

The book states the principle in Part II, in the subsection Writing the problem to start the spiral, and shows one recast. This page carries what one example cannot: the recast as a method any course can run, shown across a writing course, a data course, and a lab. The middle of a recast disciplines the work. The closing move is where it asks for creative thinking: what the answer means, and what question it opens. The same move also refreshes a course, because a recast is new work for the student even when the instructor changed no numbers and no readings.

First, a pair from a writing course

Single-answer form. In 800 words, argue whether the printing press caused the Reformation. Support your argument with evidence from the assigned readings.

Spiral form. In 800 words, argue whether the printing press caused the Reformation, as follows.

Before rereading the sources, write your one-sentence answer and name the two pieces of evidence you expect to carry it.
Draft the essay.
Give an AI tutor the assigned reading list, or your own summary of each reading, then ask for the strongest objection your thesis faces from those readings. Ask for the objection, not the rebuttal. Answer it in your own words in your revision.
Check the objection against the sources: find the passage it rests on, read it in full, and quote that passage where you answer it.
In your closing paragraph, state the best case for the opposite answer in two sentences, and name the evidence that would change your mind.

What each item forces, in the Spiral’s own stage names:

Item 1 forces Try: the thesis is committed before the sources or the tutor can supply one.
Item 2 keeps Work with the student: the draft and the revision are theirs.
Item 3 forces Ask and keeps it help: the student may request an idea, never prose.
Item 4 forces Verify without the tutor: the check runs against the readings themselves.
Item 5 forces Reflect and opens the next turn: the opposite case, argued briefly, is the deeper question.

The stages are all forced, in the order this task allows: commit, draft, ask, check, open.

The method

Keep the task. The original work stays; nothing is removed. Item 2 above is the untouched assignment.

Open with a commitment. Before the main work, the student writes a guess, a stance, or a prediction, with one sentence on what it rests on, and optionally how sure they are. The commitment is what powers the ask: a miss gives the student something specific to bring to the tutor. Item 1.

Write the ask in. Say when the student may turn to the tutor and what they may ask for: a hint, a diagnosis, an objection. Never the answer, never the steps, never wording for the submission. Two dials are available. A trigger makes the ask conditional; the data pair below uses one. An escalation gives a stuck student somewhere to go: one hint, then try again; still stuck after another attempt, ask what the problem turns on and where the attempt goes wrong. Item 3.

Add a second road to the answer. One check that does not pass through the tutor: back to the sources, the chart, the equipment, or a second computation. Item 4.

Close by opening. The last item asks what the answer means and what it opens. Pick one move from the repertoire at the end of this page and write it as a commitment the student makes before checking. This is where the recast asks for creative thinking. Item 5.

Fit the order to the problem. Every stage must be forced, or set behind a trigger, somewhere in the wording. The diagram’s order is not required; the pair above runs commit, draft, ask, check, open.

The test

Read the recast as your weakest student would, and put six questions to it. A stage the wording merely permits is a stage most students will skip.

Try: must the student produce something before the main work begins?
Ask: does the wording say what may be asked for, and is all of it help rather than product?
Work: does the student build the deliverable themselves after any help?
Verify: is there a check that does not pass through the tutor?
Reflect: does the last item ask what the answer means, or what it opens?
Category: does every ask the wording licenses stay on the tutor side of the AI-as-Tutor (AIT) line? One licensed request for wording, steps, or the answer moves the problem out of AIT.

Two more pairs

A data course.

Single-answer form. A handout chart shows the share of American teenagers holding summer jobs from 1990 to 2020; the line falls by roughly half. In one paragraph, say what the chart shows.

Spiral form.

Before looking at the chart, predict whether the share rose or fell since 1990, say why in one sentence, and say how sure you are.
Read the chart and write the paragraph.
If the chart did not match your prediction, tell an AI tutor what you predicted and how sure you were, then ask what your reasoning missed. Ask about your reasoning, not for a summary of the chart.
Say what the chart counts, what it leaves out, and whether its scale starts at zero, using only what is printed on it.
Describe one honest way the same numbers could be drawn to tell a different story, and say what in the numbers resists it.

The trigger in item 3 is the first dial from the method, and the confidence clause in item 1 is what it consumes: a sure miss and an unsure miss are different conversations. Item 5 is the repertoire’s reinterpret.

A lab course.

Single-answer form. Hang a weight on a string. With a stopwatch, time ten full swings, repeat three times, and report the average time for one swing.

Spiral form.

Before starting, write your guess for the time of one swing and one sentence on what the guess rests on.
Time ten swings, three times, and report the average for one swing.
If your three runs disagree more than you expected and you cannot say why, ask an AI tutor for one hint and look at your setup again. Still stuck, ask what the measurement turns on and where your explanation goes wrong. Do not ask it to explain your numbers.
Check your average a second way: time thirty swings once and divide. Say whether the two agree, which you trust more, and why.
Predict before testing: with double the weight on the string, does one swing take more time, less, or the same? Test it, and say what the result tells you about what the swing time depends on.

Item 3 is the method’s escalation, capped at diagnosis. Item 5 is the repertoire’s perturb, and its answer surprises most students, which is the point.

Where this sits

Plan-Use-Verify, in the book’s Part II, wraps a submission structure around any problem; recasting rewrites the problem itself, and the two compose. The Choose and justify before solving template in assignment templates shares the opening commitment; a recast goes further and writes in the ask, the second check, and the next question. Everything on this page assumes AIT; for a No AI (NAI) assignment, drop the ask, reword any later item that leaned on it, and keep the rest.

A repertoire of next questions

A recast’s closing item needs a question, and the question is a choice. Twelve moves follow. Not every move fits every course; two or three will fit yours. They belong to a long tradition of asking what a solved problem opens, and Polya’s How to Solve It is the classic source.

Reverse it. If your answer follows from the evidence, would the evidence follow from your answer? In a writing course: argue the opposite conclusion briefly, and name what would make you switch.
Weaken it. If the full claim fails, what smaller claim survives the same evidence?
Strengthen it. Does the work show more than was asked? A measurement that settles one question sometimes settles a second.
Generalize it. Does it hold beyond this case: another decade, another text, another material?
Perturb it. Change one condition and predict the direction before checking. The lab pair closes on this move.
Iterate it. Do it again, or feed the answer back in: ten more trials, or this conclusion as the next argument’s starting point.
Quantify it. Replace “it changed” with “by roughly how much,” and say whether a change of that size matters.
Reinterpret it. Tell the same result another way: another picture, another audience, another course’s eyes. What would this make an engineer, a historian, or a nurse ask?
Test it further. Try the smallest case, the next case, an odd case. Does the pattern hold?
Try to break it. Hunt for the input, the passage, or the setup where it fails. The failure is the lesson.
Ask why it works. Name the one thing the result depends on. If you cannot, that is the next question.
Find the boundary. Where exactly does it stop being true, and what decides that? Often the most productive question on this list.

Add your course

The pairs above cover a writing course, a data course, and a lab. Yours may be a fourth kind. Recast one problem from your next assignment, put the six questions to it, and send the pair back: CONTRIBUTING.md explains how.