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How Intelligence Works | Overjustification Effect — Expected Rewards, Intrinsic Interest and Better Motivation

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by

eduKate Asia

Did you know? A child may enjoy drawing without an incentive, then draw less during free choice after adults begin offering a promised prize simply for drawing. Under certain conditions, a reward changes not only what a person does today but how the person explains why they did it. Psychologists call this the overjustification effect: an activity that originally felt worth doing can come to feel like something done mainly to obtain an external outcome. The effect is conditional, not a rule that all prizes or praise damage motivation.

The useful question is therefore not ‘Are rewards good or bad?’ but ‘What was the learner’s initial interest, what exactly was promised, which behaviour earned it, how did the reward communicate control or competence, and what happens when the reward is removed?’ A sound intervention separates immediate participation from later voluntary interest, uses meaningful feedback and choice when possible, and evaluates real outcomes rather than guessing at a child’s private feelings.

This guide explains the classical experiment, compares genuinely different published meta-analytic interpretations, works through a fully labelled fictional classroom calculation, and transfers the reasoning into organisational incentives. It does not assert one effect size for every child or treat an illustrative pupil as a real case. The purpose is to understand how motivations and institutions interact, not to prescribe a simplistic ban on rewards.

Contents

What overjustification means—and what it does not mean

An activity can be enjoyable for its own sake. Someone sketches because exploring colour feels interesting, reads because they are curious, or solves a puzzle because discovering a rule is satisfying. Psychologists commonly call such engagement intrinsically motivated. The same person can also act for external reasons, such as receiving a certificate, money, social recognition or avoiding a penalty. Real motivations often coexist rather than forming two mutually exclusive boxes.

Overjustification is a proposed shift in how a person attributes the reason for acting. When an external reward becomes a conspicuous explanation for an activity already enjoyed, the person may give less weight to the activity’s inherent appeal. In some circumstances, this can reduce later free-choice involvement after the promised reward disappears. That later outcome is important; more activity while a prize is offered cannot alone show what happened to intrinsic interest.

The phenomenon is not equivalent to a child becoming lazy, nor does it mean external incentives are intrinsically corrupt. If an unpleasant or unfamiliar activity has little initial interest, a carefully designed reward might increase participation. If feedback signals growing competence without pressuring the recipient, it may support rather than undermine interest. Specific conditions matter more than slogans.

The canonical 1973 paper by Mark Lepper, David Greene and Richard Nisbett, Undermining Children’s Intrinsic Interest With Extrinsic Reward, tested the hypothesis using children with an observed pre-existing interest in an activity. It distinguished promised awards from unexpected awards and no award. These distinctions are central to the claim; the study cannot justify asserting that every form of acknowledgement reduces children’s motivation.

A useful daily-life question is whether the person continues to choose the activity when the immediate external prize is no longer on offer. Even then, the answer may reflect fatigue, available alternatives, social context and the quality of the task. Interpretation requires a carefully matched comparison, not an isolated observation.

Intrinsic enjoyment, extrinsic goals and mixed reasons

Consider two fictional learners who both practise writing for fifteen minutes. One started voluntarily because the story excites them. The other started to earn extra screen time. The same duration does not reveal an identical motive. A reward system that helps the second learner begin may affect the first differently if it reframes a previously enjoyable choice as a contractual exchange.

Real education rarely contains pure motives. A student can want good exam grades, enjoy understanding Science, value approval from a parent and be relieved when a difficult worksheet ends. Motivation changes across subjects, topics and days. A thoughtful teacher observes specific behaviour and asks learners about experience rather than permanently classifying a person as intrinsically or extrinsically motivated.

External structure is not automatically the opposite of autonomy. A family can require a reasonable bedtime, a school can assess progress and a workplace can pay employees without making every activity joyless. The overjustification concern becomes most relevant when a promised incentive changes the perceived reason to do an activity that already carried enjoyment or meaning.

Material rewards, verbal feedback, grades, privileges and social recognition are not identical interventions. An unexpected thank-you after genuine effort can be interpreted differently from ‘If you do this, you will get a prize’. A comment that identifies a well-chosen strategy can give information about competence, while a public comparison can feel controlling.

The intellectually serious task is to specify what psychological mechanism is proposed and what evidence would disconfirm it. One learner’s dislike of a sticker chart might reflect social embarrassment, boredom or an unrelated conflict. Calling every disappointment ‘overjustification’ would be neither precise nor helpful.

What the 1973 children’s drawing experiment actually examined

Lepper, Greene and Nisbett examined children who had shown interest in drawing with particular materials. The experiment contrasted an expected-award condition, in which children were promised recognition for participating, with an unexpected-award condition and a no-award comparison. The study then used later engagement in the target activity as an indicator of intrinsic interest.

The crucial distinction is timing. In the expected-award condition, the child could anticipate earning something before acting. In the unexpected-award condition, the reward was not advertised as the reason to participate. That difference matters because the same object can function either as an offered incentive or as acknowledgement after the activity.

The original paper’s findings supported a reduced later interest under the expected-reward condition in the studied setting. A careful summary stays close to that observation. It does not say that receiving a nice certificate always damages creativity, or that an adult teacher should never use structured incentives in any classroom.

Field experiments can provide unusually relevant behavioural evidence, but every experiment also has design boundaries. The age group, materials, baseline interest, social setting, operational definition of voluntary engagement and length of follow-up affect how the results should be applied. A modern digital classroom differs from the original 1973 context in several potentially important ways.

To apply the lesson responsibly, preserve the core comparison: what happens to voluntary engagement after an expected reward is removed, relative to groups without that promised contingency? Merely asking whether children smiled when they received a prize measures a different outcome.

How a reward can change the explanation for one’s own action

Imagine that a learner previously chose a puzzle freely. The learner may infer, reasonably, that they must enjoy puzzles because they spend time solving them without pressure. Now introduce a conspicuous promised prize for every completed puzzle. The same behaviour has an additional obvious explanation: ‘I did this to get the prize.’ An attribution-based account predicts that the learner may assign less significance to the original interest.

This is an explanatory hypothesis, not a window into a child’s private thoughts. We cannot infer a precise mental sentence merely by observing that drawing time changed. Motivation research also considers autonomy, competence, social expectations, task difficulty and the meaning of feedback. Different theoretical accounts can make overlapping predictions in some experiments.

One reason the reward’s interpretation matters is that a task can be framed as either an opportunity or an obligation. A reward used as a control signal—do this exactly my way or lose the prize—can narrow perceived choice. Feedback that explains how a learner improved may instead communicate competence. The same physical reward can play different roles in distinct social contexts.

The related psychological reactance guide concerns resistance when freedom is threatened. Overjustification is a different reader question: whether external reasons crowd out or reshape the meaning of prior enjoyment. The mechanisms may sometimes coincide but neither is a substitute for measuring the other.

A useful diagnosis separates perceived control from demonstrated competence. An adult can believe a sticker system is encouraging while a learner feels monitored. Another learner may enjoy visible progress and see stickers as information about practice. Ask what the design communicates, then examine voluntary follow-through rather than relying only on adult intent.

Why major meta-analyses disagree about reward effects

Published evidence does not support a simple universal claim that rewards always undermine motivation. Edward Deci, Richard Koestner and Richard Ryan reported a 1999 meta-analysis of 128 studies finding decreases in certain measures of intrinsic motivation for several categories of expected tangible reward, while positive feedback was associated with increased measures in their analysis.

The review reported standardised effects for reward contingencies on free-choice behaviour, including roughly d=−0.40 for engagement-contingent, −0.36 for completion-contingent and −0.28 for performance-contingent conditions in their categorisation. These are effect-size statistics across the included studies, not percentages of children harmed and not estimates of how many hours of free play any particular child will lose.

Judy Cameron, Katherine Banko and W. David Pierce reached a different broad interpretation in their 2001 meta-analysis. They argued that negative effects were more limited and dependent on reward conditions, task interest and performance contingency. Their review emphasised situations in which rewards for low-interest tasks or properly linked performance rewards could have neutral or positive effects.

These accounts disagree about how to classify studies and draw a general conclusion, but they also share an important practical theme: the kind of reward, initial task interest and contingency matter. Presenting one side as a final all-purpose law would discard those distinctions. Research summaries should say whose analysis produced an estimate and the outcome measure it used.

The best inference for a parent or team is cautious and testable. Expected tangible rewards for an already enjoyed activity deserve care, especially when the reward is experienced as controlling. But that is not evidence for removing all feedback, withholding pay or refusing incentives for initially aversive necessary tasks. A decision should follow the actual learner and the actual context.

How researchers measure interest—and how measurement can mislead

One common outcome is free-choice behaviour: after a task or reward phase, a person has an opportunity to engage voluntarily in the target activity. Researchers measure whether and how long the person returns to it. This can reveal a form of behavioural interest that immediate compliance scores miss.

Self-reported interest is another outcome, and it answers a different question. A child may say they like drawing while spending less time drawing during a particular observation period because another attractive activity is available. A learner may keep practising despite reporting lower enjoyment because mastery is personally valuable. Neither measure is inherently a complete description of motivation.

A third outcome is performance quality, such as whether a task is completed accurately. A reward might increase short-term output even if later voluntary participation falls. Conversely, interest may remain high while a child struggles with the skill. These outcomes should be reported separately rather than compressed into one vague statement about ‘motivation’.

Study design matters. Compare groups that began with similar interest, account for unequal exposure and describe exactly when the follow-up happened. If the reward group receives substantially more instruction or resources, later performance differences cannot be attributed cleanly to reward framing alone. In practical schools, changes in teacher attention, peer norms and task difficulty often accompany an incentive.

Report the denominator and observation window. Ten children who return to a task for five minutes each are not equivalent to one child who returns for fifty minutes, even though the total group time is identical. Meaningful analysis may consider the distribution of engagement and the opportunity each person was actually given.

Fictional classroom experiment: what changed after the incentives stopped?

Imagine a purely illustrative exercise with four groups of twenty students each. Everyone initially enjoys a particular voluntary word game and all groups have comparable average baseline free-choice time of twelve minutes during a standard observation. The groups then experience different incentive conditions, followed by an equivalent free-choice period without an available prize.

The expected-tangible-award group’s later average is seven minutes, a change of 7−12 = −5 minutes. The unexpected-award group’s later average is eleven minutes, a change of −1 minute. A positive-skill-feedback group’s later average is fourteen minutes, a change of +2 minutes. The no-award comparison remains at twelve minutes, a change of zero.

If an analyst uses the no-award group as the reference, the expected-award difference in changes is (7−12)−(12−12)=−5 minutes. The positive-feedback difference in changes is (14−12)−(12−12)=+2 minutes. These calculations make an illustrative contrast visible; they do not establish a population-level causal effect. They are invented group means, not reported results from the 1973 study or any eduKate classroom.

Each group contains twenty fictional students, so if they all had equal observation opportunities the expected-award group’s total later engagement is 20 × 7 = 140 person-minutes. The no-award group has 20 × 12 = 240 person-minutes. The total gap is 100 person-minutes across these two groups. It would be inaccurate to convert that into ‘100 children lost interest’ or to infer that every child reduced participation by exactly five minutes.

Actual inference would need individual-level variation, assignment method, uncertainty estimates, follow-up duration, possible spillovers and the task’s meaning. If one group included more initially enthusiastic children or a different teacher, the comparison might be confounded. The arithmetic teaches how to keep outcomes and denominators straight while recognising the limits of a fictional illustration.

Most importantly, a later reduction in voluntary word-game time does not automatically mean poorer vocabulary learning. It is evidence about that specific behavioural outcome under the invented conditions. A responsible evaluator would separately measure language performance, later retrieval, self-reported interest and whether children voluntarily choose other useful learning activities.

Why competence feedback is not identical to a promised prize

Feedback can tell a learner which strategy worked, where an error occurred or why a complex answer has improved. A phrase like ‘You tested both possible denominators and corrected the unit’ identifies a concrete action and offers information the learner can use independently. The positive-feedback outcomes in some motivation research should not be collapsed into the effects of offered tangible prizes.

Vague praise can also miss the point. Constantly saying ‘You’re brilliant’ can encourage dependence on evaluation or obscure what was actually learned. Good educational feedback is credible, specific and connected to a strategy, product or decision. It makes the next independent attempt easier instead of turning every task into a request for adult approval.

A certificate earned for demonstrating a meaningful skill differs from a sticker promised merely for sitting at a desk, but either can become controlling in context. Ask whether the recognition is informative, whether standards are transparent and whether children still have opportunities to choose meaningful tasks outside the incentive system.

One repair is to offer genuine choices within a sensible learning goal. A student might choose which short text to summarise or whether to demonstrate understanding through a diagram before writing. The teacher retains educational standards, but the learner has some control over the route. Such design may support agency; it still requires observation rather than guaranteed claims.

Another repair is to fade an initial participation incentive when the learner acquires capability and interest. The taper should not be deceptive or arbitrary. Explain the purpose of practice and help students notice their own progress. When the activity remains genuinely unpleasant or necessary, external incentives may continue to have a role.

Low-interest tasks are a different policy question

Not every useful activity begins with enjoyment. Some children resist practising multiplication tables, correcting spelling or learning a new instrument because they lack fluency and find the first attempts frustrating. Adults may introduce predictable external incentives to make initial participation possible. The question is whether the arrangement produces useful practice and whether longer-term learning or willingness to continue improves.

The 2001 Cameron, Banko and Pierce review specifically distinguishes low-interest from high-interest tasks. Their interpretation includes cases in which reward can improve free-choice involvement for initially low-interest activities. This does not mean every reward will teach the underlying skill; it means the overjustification argument should not be used as a blanket prohibition on motivation supports.

Suppose a child initially avoids practising an unfamiliar skill. A short-term goal might reward the effort to attempt a few carefully chosen items while the teacher makes the steps more accessible. As the child becomes competent, independent success and curiosity may take on more motivational weight. The teacher should track accuracy, error repair and willingness to practise without an incentive, not just the number of tokens earned.

A reward can make effort visible while still being poorly aligned with learning. If points depend only on finishing twenty questions quickly, students may rush, copy or avoid difficult examples. The design could inadvertently reward compliance rather than understanding. A better system might recognise revising an incorrect method or independently explaining a concept.

The sound response to a low-interest task starts with diagnosis. Is the task incomprehensible, frightening, poorly matched to the learner, or simply tedious but necessary? An incentive is not a substitute for first-principles teaching and practice design. It is one tool whose value depends on the cause of disengagement.

A practical parent decision: what exactly are we encouraging?

Parents often want children to read, write or explore more outside school. A well-intentioned reward chart is tempting because it makes effort visible. Before using one, ask whether the child already enjoys the activity. If so, consider whether ordinary access to books, social discussion and encouraging feedback could support continued interest without creating a new prize contract.

If a child reads voluntarily, paying per page may turn a flexible discovery activity into page-count optimisation. The child could choose shorter or easier material to accumulate rewards. That possibility does not prove the child will become less interested, but it shows why incentive design must consider what behaviour the measurement actually rewards.

For a reluctant reader, a small invitation or short-term recognition may help begin, provided it accompanies better-fitting texts and patient assistance. The aim should be real engagement and expanding skill, not permanent reliance on prizes. Observe whether the child begins selecting material without prompting and whether comprehension grows.

Discuss success in specific terms: Did the learner understand a difficult paragraph, ask an original question, return to the book or explain a new word in another context? Those behaviours can be celebrated without reducing every action to points. If the child dislikes a particular activity, avoid moralising about motivation and examine what makes the task hard.

Family constraints differ. A child with limited free time, language challenges or competing responsibilities may have fewer opportunities to show voluntary engagement. Comparing two families’ reward systems without those conditions can be misleading. Use the mechanism to ask better questions, not to judge parents or promise a universal behaviour change.

A school design problem: participation, mastery and curiosity are separate

Classroom incentives often target observable behaviours: handing in work, following a routine, contributing to a group and attempting revisions. These behaviours can create conditions for learning, but they are not the same as mastery or intrinsic enjoyment. A good programme therefore defines each outcome separately and does not call a rise in points earned a rise in curiosity.

Consider a fictional Science classroom where students earn a badge for making three predictions. If the badge is all that matters, students might write quick guesses without attending to evidence. The teacher could instead require a reasoned prediction, a relevant observation and a reflection on why the result differed. Even then, the badge’s motivational effects should be evaluated rather than assumed.

The teacher’s own tone matters. A reward framed as a bribe under threat may create a different social meaning from recognition for a student who independently tested a difficult idea. Peer competition adds further complications: public leaderboards can motivate some learners while discouraging others who repeatedly finish last. The outcome depends on classroom norms and task design.

A sensible small-scale trial could compare student choice, accuracy, persistence and later engagement before and after an intervention. Inference will be limited without a controlled design, but careful tracking can detect obvious trade-offs. Record whether a new practice helped pupils who previously could not begin the task and whether it displaced interest among those who already enjoyed it.

Parents and teachers should avoid treating reward removal as a test of moral character. If engagement drops, the activity may still be poorly supported or the incentive may have changed expectations. Diagnose the skill, environment and feedback process before blaming the learner.

Workplace incentives: pay and bonuses are not the same as a sticker experiment

Employees normally work within an explicit exchange of labour and compensation. It would be unreasonable to infer from a children’s drawing experiment that wages are harmful or that people should be expected to work for free. Organisational incentives also signal priorities, support basic needs and determine what performance is economically rewarded.

However, a workplace can unintentionally change how people approach a task by making one numerical target dominant. A team that receives a bonus solely for closing support tickets may close complex cases prematurely. This may reflect metric gaming, narrow goal substitution or weak supervision as well as any change in intrinsic motivation. Those mechanisms should not be conflated.

An organisation can protect craftsmanship by measuring quality and customer outcomes alongside throughput, recognising independent problem solving and making professional judgement visible. The goal is not to remove accountability but to prevent a crude measure from crowding out the real job. Again, the effect of any reward scheme is empirical rather than guaranteed.

A volunteer programme illustrates a closer conceptual comparison. People may join because the activity is meaningful, but a poorly framed payment-per-action can alter participation and perceived purpose. That possible change requires ethically obtained evidence, not an assumption that any payment will inevitably corrupt public service.

For team learning, it is useful to compare competency traps, where success with established routines reduces exploration. Overjustification concerns why people engage; competency traps concern which capabilities and routines an organisation continues to exploit. Both can involve incentives but they answer different reader questions.

Ten common incentive-design errors and specific repairs

Error one is rewarding an already enjoyed activity without asking what the reward might change. Repair: establish baseline voluntary interest and make the incentive’s purpose explicit. Error two is treating surprise recognition and promised prizes as identical. Repair: distinguish reward timing and whether participation was contingent on earning it.

Error three is interpreting higher task completion during a reward period as lasting motivation. Repair: measure appropriate free-choice follow-up and skill transfer after the incentive changes. Error four is measuring only pages, clicks or worksheets. Repair: include comprehension, independence and the quality of thinking where those are the true goals.

Error five is applying a universal conclusion from the 1973 experiment to all ages and contexts. Repair: read study scope and competing meta-analyses. Error six is converting a standardised effect size into a child-level percentage. Repair: report the statistical unit correctly and avoid invented personal forecasts.

Error seven is relying on vague praise. Repair: give information that helps a learner identify a method and correct errors. Error eight is using humiliating public leaderboards. Repair: evaluate social effects and consider private mastery feedback. Error nine is abruptly removing all support from a reluctant learner. Repair: build capability and adjust supports as evidence changes.

Error ten is assuming that an incentive caused every later change. Repair: compare task difficulty, opportunity, instruction, baseline interest and alternatives. A good explanatory model invites attempts to falsify its story rather than treating every response as confirmation.

Changed-context exercise: a recognition system for adult volunteers

Imagine a fictional reading club with forty adults who volunteered because they enjoy discussing books. The organiser introduces a promised voucher for every review posted online. During the voucher period, the total number of reviews increases from forty to sixty. Does this prove that motivation improved? No. It establishes that the recorded activity increased under the new incentive, not why, or whether voluntary interest will persist.

After the voucher ends, suppose only twenty-eight reviews are posted in an equivalent period. The post-voucher count is twelve below the original forty, a 12/40 = 30% reduction in total recorded reviews. Compared with the incentivised peak of sixty, the reduction is (60−28)/60 ≈ 53.33%. These percentages have different denominators and answer different questions. Both are invented observations.

Several explanations remain. Members may have exhausted the books they wanted to review, a holiday may have changed availability, or the new public posting requirement may feel less enjoyable. Overjustification is one candidate if a conspicuous external incentive changed the meaning of a previously enjoyed activity, but the counts alone cannot distinguish it from the alternatives.

To test the idea more responsibly, the organiser could compare an appropriate control group, record baseline participation, retain a choice of review formats, obtain feedback about perceived pressure and measure later engagement without rewards. Ethical and practical constraints matter; participants should not be deceived simply to create a neat comparison.

Now change one condition: new members had never enjoyed writing reviews and found the initial task intimidating. A small incentive plus supportive coaching might help them begin, while veteran members need no such inducement. The same programme could have different effects by baseline interest. That is the transferable lesson: treat reward design as conditional on people, tasks and meaning.

Limits of the research and responsible interpretation

The reward literature contains genuine disagreement about synthesis methods, effect classifications and generalisability. The two prominent meta-analyses cited here do not collapse into a universal ‘rewards are harmful’ or ‘rewards are harmless’ answer. Their shared conditional details are more useful for practical decision-making than an absolutist headline.

Free-choice activity is informative but not synonymous with learning quality, well-being or all forms of autonomous motivation. A small behavioural reduction in one experimental window cannot diagnose a child’s emotional health or explain every decline in school performance. Individual circumstances and instructional quality remain central.

Financial compensation has ethical, economic and social functions that a laboratory prize may not capture. Removing wages, access supports or reasonable accommodations in the name of intrinsic interest would be an inappropriate inference. People need fair conditions and clear expectations as well as opportunities to find meaning in work.

A short intervention may behave differently from a programme lasting months or years. The child’s age, initial preference, peer setting, reward type and available alternatives all affect what can reasonably be inferred. Long-term claims require long-term evidence; the existence of a classic experiment is not a substitute.

The fictional groups and volunteer counts in this article are transparently fabricated to teach careful interpretation. No eduKate student outcomes, educational trial or workplace implementation are claimed. Readers should distinguish illustrative arithmetic from the original published experimental evidence.

Frequently asked questions

What is the overjustification effect? It is a proposed reduction in intrinsic interest under certain conditions when an external incentive becomes a salient reason for an activity that was already enjoyed.

Do all rewards reduce motivation? No. Results depend on initial interest, type of reward, whether it is expected, its contingency and how it is perceived. Major published reviews disagree about the breadth of negative effects.

Why does expected versus unexpected reward matter? A promised prize can become the reason for engaging before the task starts; an unexpected acknowledgement cannot motivate that earlier choice in the same direct way.

Are rewards for low-interest tasks always harmful? No. Some findings suggest certain incentives can help engagement with initially low-interest tasks. That does not remove the need for good teaching, fair conditions and outcome evaluation.

Does positive feedback count as a reward? It can have rewarding qualities, but specific competence information can function differently from a promised tangible prize. Research should distinguish verbal feedback from other reward contingencies.

Can we diagnose a child’s motivation from a drop in study time? No. Changes in difficulty, opportunity, fatigue, anxiety and available alternatives can all change behaviour. Ask what was measured and consider other explanations.

Can a reward chart still be useful? Possibly. Define the real goal, compare baseline interest, observe independent learning and adjust if the chart begins to replace curiosity or quality with points.

Does this mean workers should not receive bonuses or pay? No. Wage arrangements and organisational incentives involve economic and fairness considerations far beyond the experimental child-drawing context.

What does a negative effect size mean? It is a statistical comparison under a defined study and measure; it is not a percentage of people who lost motivation or a prediction of an individual child’s change.

Reader pathways and original evidence

For a student, separate four things: interest before the reward, behaviour during the reward, later voluntary engagement and actual skill learning. Try analysing the invented four-group example with those categories. Ask which missing observations prevent a confident causal conclusion and how the group denominator affects interpretation.

For parents and teachers, start by diagnosing whether the learner already enjoys an activity or has a difficulty that blocks engagement. Where possible, preserve meaningful choice and skill-specific feedback. If an incentive is introduced, track the intended learning outcome and later independent participation, and change course without blaming the child.

For managers and community organisers, document what the reward is meant to accomplish, what metric might be gamed and how the arrangement is interpreted by participants. Give people credible feedback, fair compensation and clarity about the purpose of their work. Examine unintended effects with context rather than insisting on one psychological explanation.

Primary research and competing syntheses: Lepper, Greene and Nisbett (1973), children’s intrinsic interest and expected rewards; Deci, Koestner and Ryan (1999), meta-analysis of 128 studies; Cameron, Banko and Pierce (2001), alternative meta-analytic interpretation. Each source should be read for scope and method, not just a headline.

Continue through the How Intelligence Works collection and the education-to-life learning framework for related questions about autonomous action and evidence. The worked numbers and group descriptions here were constructed for explanation and have not been observed in any real classroom.

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