Series route: This guide belongs to the complete PSLE Learner’s Guide volume directory. The original PSLE Learning Guide supplies the wider English, Mathematics and Science route, while PSLE English, PSLE Mathematics, PSLE Science provide verified subject-specific continuation.
Find your route through this guide
This volume solves one precise learner problem: the learner accepts “if A, then B” and silently reverses it to “if B, then A”, even though B may have several causes, a larger mathematical category or alternative explanations that make the converse false. The goal is durable control, not a remembered slogan. A Primary 6 learner should be able to identify the trigger, explain the mechanism, perform the decision under time pressure and reject an attractive answer that exceeds the evidence.
- Start with the exact mistake and the evidence boundary.
- Build the mechanism before memorising a shortcut.
- Study worked contrasts and name why the near-miss fails.
- Repair misconceptions with one targeted check each.
- Attempt graded independent practice before reading the explanations.
- Return after a delay and prove that the method transfers.
- Use the parent and tutor notes after the learner has attempted the work.
The problem beneath the mistake
“If a shape is a square, then it is a rectangle” is true. Reversing it gives “If a shape is a rectangle, then it is a square”, which is false because a non-square rectangle is a counterexample.
The original statement moves from a condition to a consequence. Its converse swaps the two parts. A true original does not automatically certify the swapped statement.
In Science, “If rain falls on an uncovered path, the path may become wet” does not justify “If the path is wet, rain fell”. A hose, spill or cleaning could also produce wetness.
In English comprehension, a character who is anxious may speak quickly. Hearing quick speech does not prove anxiety unless the passage rules out excitement, urgency, habit and other explanations.
The central question is whether the observed consequence belongs only to the original condition. If several conditions can lead to the same consequence, the reversal is unsafe.
A counterexample is decisive. To disprove “If B, then A”, find one valid case where B is true but A is false. The case must satisfy the reversed condition exactly, not merely resemble it.
Some converses are true, but they need their own proof. “A whole number is even if it is divisible by 2” and “if a whole number is even, it is divisible by 2” are both true because even is defined by divisibility by 2.
Necessary and sufficient conditions help organise the logic. If A implies B, then B is necessary for A; A is sufficient for B. This does not say B is sufficient for A.
Exam questions often hide reversal inside words such as therefore, must, proves, caused by or can only be. Slow down whenever the conclusion travels backward from an effect, property or result to one possible source.
The reliable routine is write A→B, build B→A separately, search for an alternative cause or broader category, test one counterexample and accept the converse only when independent evidence proves it.
Core decision: From “if A, then B”, do not automatically conclude “if B, then A”. The converse is a separate claim. Test it with definitions, alternative causes, category relationships and counterexamples.
Cross-subject reasoning improves when learners make direction visible. Arrows, cause paths and category diagrams prevent a true statement from lending borrowed authority to a different reversed statement.
Mechanism: why the rule works
1. Write the original direction
Represent the statement as A→B, naming A and B in words.
A visible direction prevents an unnoticed swap during reading or explanation.
Diagnostic check. What is the condition and what is the consequence?
2. Construct the converse
Swap the parts deliberately to form B→A.
The converse should be treated as a new sentence, not as a restatement of the original.
Diagnostic check. What exactly does the reversed claim assert?
3. Check category size
Ask whether B describes a broader group that contains cases beyond A.
A square belongs to the rectangle category, but the rectangle category contains additional shapes.
Diagnostic check. Can B occur outside A?
4. Check alternative causes
List other mechanisms that could produce B.
An effect with multiple possible causes cannot identify one cause by itself.
Diagnostic check. What else could make the observation happen?
5. Build a counterexample
Find one case satisfying B while violating A.
A universal converse fails when even one valid exception is found.
Diagnostic check. Does the example meet every stated condition?
6. Distinguish clues from proof
Treat B as evidence that may raise a possibility without making A certain.
Comprehension and Science often require combining several observations before choosing the best explanation.
Diagnostic check. Is the clue diagnostic or merely consistent?
7. Recognise true converses
Use a definition or two independent proofs when both directions genuinely hold.
Some relationships are equivalent, but equivalence must be established rather than assumed.
Diagnostic check. Can each arrow be justified on its own?
8. State the bounded conclusion
Say that A is possible, supported or one explanation when certainty is not earned.
Precise modal language preserves useful evidence without overstating it.
Diagnostic check. Should the answer say may, suggests, must or if and only if?
Worked examples: make the decision visible
Pause before each answer. State the controlling evidence in one line, then complete the operation. The explanations deliberately contrast a valid route with a plausible but invalid one, because transfer grows when the learner can say why a tempting route does not apply.
Worked example 1: Square and rectangle
Task. Original: if a shape is a square, then it is a rectangle. Test the converse.
Reasoning. A 3 cm by 5 cm rectangle satisfies rectangle but not square.
Answer. The converse is false.
Independent check. The counterexample has four right angles yet unequal adjacent sides.
Contrast-and-transfer pass. Keep the governing principle from “Square and rectangle”, but change one surface detail in the task: Original: if a shape is a square, then it is a rectangle. Test the converse. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—The converse is false.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
Worked example 2: Multiple of four
Task. Original: if a number is divisible by 4, then it is even. Test the converse.
Reasoning. Six is even but not divisible by 4.
Answer. “If even, then divisible by 4” is false.
Independent check. The original remains true; only the reversed claim fails.
Contrast-and-transfer pass. Keep the governing principle from “Multiple of four”, but change one surface detail in the task: Original: if a number is divisible by 4, then it is even. Test the converse. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—“If even, then divisible by 4” is false.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
Worked example 3: Wet path
Task. The path is wet. Must it have rained?
Reasoning. No. Sprinklers, washing or a spill are alternative causes.
Answer. Rain is one possible explanation, not a certain conclusion.
Independent check. Seek weather evidence or observe the water source.
Contrast-and-transfer pass. Keep the governing principle from “Wet path”, but change one surface detail in the task: The path is wet. Must it have rained? Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—Rain is one possible explanation, not a certain conclusion.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
Worked example 4: Bulb off
Task. A bulb is not lit. Must the bulb be broken?
Reasoning. No. The switch may be open, the cell depleted or a connection incomplete.
Answer. The observation does not identify one cause.
Independent check. Test the circuit components separately.
Contrast-and-transfer pass. Keep the governing principle from “Bulb off”, but change one surface detail in the task: A bulb is not lit. Must the bulb be broken? Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—The observation does not identify one cause.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
Worked example 5: Character inference
Task. A character avoids eye contact. Must the character be dishonest?
Reasoning. No. Shyness, sadness, cultural habit or distraction could fit the same behaviour.
Answer. Dishonesty requires additional textual evidence.
Independent check. Use speech, actions and context together.
Contrast-and-transfer pass. Keep the governing principle from “Character inference”, but change one surface detail in the task: A character avoids eye contact. Must the character be dishonest? Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—Dishonesty requires additional textual evidence.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
Worked example 6: Definition case
Task. If a whole number is even, it is divisible by 2. Does the converse hold?
Reasoning. Yes. A whole number divisible by 2 meets the definition of even.
Answer. Both directions are true for whole numbers.
Independent check. The domain and definition are explicit.
Contrast-and-transfer pass. Keep the governing principle from “Definition case”, but change one surface detail in the task: If a whole number is even, it is divisible by 2. Does the converse hold? Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—Both directions are true for whole numbers.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
Worked example 7: Equal fractions
Task. If two simplified fractions are equal, their cross-products are equal. Does equal cross-product imply equal fractions for non-zero denominators?
Reasoning. Yes, because ad=bc allows both sides to be divided by bd.
Answer. Within the stated domain, both directions hold.
Independent check. Non-zero denominators are essential.
Contrast-and-transfer pass. Keep the governing principle from “Equal fractions”, but change one surface detail in the task: If two simplified fractions are equal, their cross-products are equal. Does equal cross-product imply equal fractions for non-zero denominators? Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—Within the stated domain, both directions hold.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
Worked example 8: Plant growth
Task. A plant grows poorly in darkness. Does poor growth prove darkness?
Reasoning. No. Water, minerals, disease, temperature or root damage may also limit growth.
Answer. Darkness is a candidate cause, not proof.
Independent check. Use controlled comparisons to isolate light.
Contrast-and-transfer pass. Keep the governing principle from “Plant growth”, but change one surface detail in the task: A plant grows poorly in darkness. Does poor growth prove darkness? Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—Darkness is a candidate cause, not proof.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
Worked example 9: Language trigger
Task. A passage says “Whenever the bell rings, pupils line up.” Pupils are lining up. Did the bell definitely ring?
Reasoning. Not necessarily; a teacher may have instructed them directly or another signal may exist.
Answer. The converse needs evidence that the bell is the only trigger.
Independent check. Look for words such as only if or no other signal.
Contrast-and-transfer pass. Keep the governing principle from “Language trigger”, but change one surface detail in the task: A passage says “Whenever the bell rings, pupils line up.” Pupils are lining up. Did the bell definitely ring? Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—The converse needs evidence that the bell is the only trigger.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
Worked example 10: Repairing a conclusion
Task. “The floor is muddy, so the window must have been left open during rain.”
Reasoning. Mud is consistent with that story but could be brought in on shoes or spilled from a pot.
Answer. Write “An open window is one possible explanation; more evidence is needed.”
Independent check. The repair keeps possibility without claiming certainty.
Contrast-and-transfer pass. Keep the governing principle from “Repairing a conclusion”, but change one surface detail in the task: “The floor is muddy, so the window must have been left open during rain.” Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—Write “An open window is one possible explanation; more evidence is needed.”—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
The method lab: change one condition at a time
Create two columns headed original and converse. For each of ten statements, write both directions before judging either. This prevents grammar from hiding the logical swap.
Build category diagrams for square/rectangle, multiple of 4/even number and mammal/animal. Mark which arrows follow containment and which reversals fail.
For five observed effects, list at least three possible causes. Then add a discriminating test that would separate the leading explanations.
Design counterexamples that satisfy every part of the converse condition. Reject examples that break the domain or quietly alter a definition.
After a delay, find one true converse and one false converse in each of English, Mathematics and Science, explaining the proof or counterexample.
Common misconceptions and targeted repairs
Misconception 1: Reversal keeps truth
What the learner may think. The learner thinks swapping clauses preserves meaning.
Why it fails. The converse is a different claim with a different set of possible cases.
Targeted repair. Write both arrows separately.
Misconception 2: One matching example proves the converse
What the learner may think. The learner finds a case where B and A are both true.
Why it fails. A universal converse must work for every B case.
Targeted repair. Search deliberately for B with not-A.
Misconception 3: Any alternative is enough
What the learner may think. The learner invents an impossible or out-of-domain counterexample.
Why it fails. A counterexample must satisfy the reversed condition and all stated constraints.
Targeted repair. Verify each condition explicitly.
Misconception 4: An effect identifies its cause
What the learner may think. The learner ignores multiple causal routes.
Why it fails. Many effects are non-unique and need discriminating evidence.
Targeted repair. List alternatives before choosing.
Misconception 5: A clue has no value unless conclusive
What the learner may think. The learner overcorrects and discards evidence.
Why it fails. B may support A without proving it.
Targeted repair. Use may, suggests or is consistent with.
Misconception 6: No converse can be true
What the learner may think. The learner turns caution into another absolute.
Why it fails. Definitions and equivalences can justify both directions.
Targeted repair. Prove the reverse independently.
Graded independent practice with explained answers
Attempt each item without looking ahead. Write the evidence or relationship before the final response. Early items check recognition; middle items require explanation; later items change the surface form so that copying a worked example is insufficient.
Practice 1: Geometry
Question. If a shape is a square, it has four equal sides. Does four equal sides prove square?
Explained answer. No. A rhombus may lack right angles.
Why this earns trust. The converse needs angle conditions.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “No. A rhombus may lack right angles.” from memory, but to recreate a justified answer from the new evidence.
Practice 2: Number
Question. If a number is divisible by 10, it ends in 0. Does ending in 0 imply divisibility by 10 for whole numbers?
Explained answer. Yes.
Why this earns trust. Place value supplies the reverse proof.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “Yes.” from memory, but to recreate a justified answer from the new evidence.
Practice 3: Parity
Question. If divisible by 6, a whole number is even. Is every even whole number divisible by 6?
Explained answer. No.
Why this earns trust. Eight is an even counterexample.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “No.” from memory, but to recreate a justified answer from the new evidence.
Practice 4: Weather
Question. If strong sun heats the pavement, it becomes warm. Does warm pavement prove strong sun?
Explained answer. No.
Why this earns trust. Stored heat or another heat source may exist.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “No.” from memory, but to recreate a justified answer from the new evidence.
Practice 5: Circuit
Question. If a complete working circuit is closed, the bulb lights. Does a lit bulb support a complete circuit?
Explained answer. Yes under the stated simple setup, but the reverse still depends on component and domain assumptions.
Why this earns trust. Conditions must remain visible.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “Yes under the stated simple setup, but the reverse still depends on component and domain assumptions.” from memory, but to recreate a justified answer from the new evidence.
Practice 6: Character
Question. If a character is frightened, she may tremble. Does trembling prove fear?
Explained answer. No.
Why this earns trust. Cold or illness are alternatives.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “No.” from memory, but to recreate a justified answer from the new evidence.
Practice 7: Counterexample
Question. Disprove “if a number is odd, it is prime”.
Explained answer. Use 9.
Why this earns trust. Nine is odd but composite.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “Use 9.” from memory, but to recreate a justified answer from the new evidence.
Practice 8: Category
Question. Why does animal not imply mammal?
Explained answer. Animal is the broader category and includes birds, fish and others.
Why this earns trust. Containment has direction.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “Animal is the broader category and includes birds, fish and others.” from memory, but to recreate a justified answer from the new evidence.
Practice 9: Evidence wording
Question. Repair “The leaves are yellow, so the plant definitely lacks water.”
Explained answer. Yellow leaves may be consistent with water stress, but other causes must be tested.
Why this earns trust. The certainty is reduced to the evidence level.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “Yellow leaves may be consistent with water stress, but other causes must be tested.” from memory, but to recreate a justified answer from the new evidence.
Practice 10: True converse
Question. Give a relationship true in both directions.
Explained answer. For whole numbers, a number is even if and only if it is divisible by 2.
Why this earns trust. Each direction follows the definition.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “For whole numbers, a number is even if and only if it is divisible by 2.” from memory, but to recreate a justified answer from the new evidence.
Practice 11: Arrow
Question. Write the converse of “If A, then B”.
Explained answer. If B, then A.
Why this earns trust. The swap is explicit before evaluation.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “If B, then A.” from memory, but to recreate a justified answer from the new evidence.
Practice 12: Routine
Question. State the sequence.
Explained answer. Write A→B, form B→A, inspect categories or causes, find B with not-A, prove separately and bound the conclusion.
Why this earns trust. Direction and evidence are controlled.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “Write A→B, form B→A, inspect categories or causes, find B with not-A, prove separately and bound the conclusion.” from memory, but to recreate a justified answer from the new evidence.
A four-session mastery cycle
Session 1: construct. Read the mechanism once, cover it and write a three-line model containing the trigger, the operation and the check. Add one example and one near-miss. Compare only after the model is complete. If the operation is right but the trigger is missing, the knowledge is unsafe because it may be applied in the wrong place.
Session 2: discriminate. Mix six items: two that clearly require this method, two that tempt the method but violate a condition, and two that require another skill. For every item, write “use” or “do not use” before answering. Accuracy in method selection matters more than speed at this stage.
Session 3: explain. Give a complete answer to an imagined classmate. Include the smallest sufficient evidence, the mechanism and the independent check. Remove any sentence that merely repeats the question or adds an unsupported claim. A concise causal explanation is stronger than a longer collection of familiar words.
Session 4: perform. Attempt a timed mixed set without notes. Mark uncertain decisions, but continue. During review, classify each error as trigger, evidence, operation, scope, unit or check. Reattempt one changed version after a delay. Mastery is shown when the learner chooses and adapts the method without needing the original example beside it.
Delayed transfer: prove that the learning lasts
Transfer test 1: New categories
Test class relationships from shapes, animals and number sets.
Success criterion. Every reversal is checked with containment or a counterexample.
Transfer test 2: Cause diagnosis
Use one observation with three possible causes.
Success criterion. A discriminating test is proposed.
Transfer test 3: Comprehension
Separate behaviour consistent with a trait from behaviour proving it.
Success criterion. Certainty matches the textual evidence.
Transfer test 4: Delayed logic
Rebuild the converse routine after one week.
Success criterion. The learner uses arrows and an original counterexample.
Examination control
1. Name the target. State exactly what the question asks before selecting a rule or explanation.
2. Expose the relationship. Write the grammatical, numerical, causal or representational relationship that controls the answer.
3. Use decisive evidence. Choose the smallest fact, calculation or observation that settles the decision.
4. Check the boundary. Reject the nearest tempting claim that the evidence does not support.
5. Present precisely. Keep conditions, units, certainty and scope visible in the final response.
Under time pressure, compress the method to A→B → write B→A → broader category or alternative cause? → find B and not-A → separate clue from proof → bounded claim. The compressed routine is useful only after the full reasoning is understood. If it produces uncertainty, expand the step instead of guessing.
A compact error log
Use labels such as direction reversed, converse borrowed truth, counterexample invalid, alternative cause ignored, clue upgraded to proof, domain omitted or true converse rejected without testing.
Reattempt with one category and one causal example because the same logical error can hide under different subject language.
Notes for parents and tutors
Use everyday statements and ask the child to say them backward before judging them.
Praise a valid counterexample that meets every condition; this is disciplined proof, not mere disagreement.
Ask “What else could cause that?” when an effect is used to name one source.
Model careful language such as possible, supported and proven so uncertainty is not confused with ignorance.
Frequently asked questions
What is the converse?
The statement formed by swapping condition and consequence.
Does a true statement guarantee a true converse?
No.
Fastest disproof?
Find B true while A is false.
Can a converse be true?
Yes, but it needs its own proof or definition.
Why do causes create traps?
Several causes can produce the same effect.
Is a clue useless if not proof?
No; it may support a possibility.
What wording protects the boundary?
May, suggests, is consistent with, or one possible explanation.
Fastest check?
Draw A→B and ask whether B can occur without A.
Next routes
Return to the complete volume directory to locate the series, use the original PSLE Learning Guide for the foundation route, and continue through PSLE English, PSLE Mathematics, PSLE Science for verified subject guidance. The next useful task is a mixed set in which this idea is sometimes relevant and sometimes irrelevant; that is where the learner must choose a method rather than imitate one.