# Teacher guide — How a model learns from examples

**Duration:** 120 minutes<br>
**Positioning:** Follow a complete learning loop without advanced math: example, prediction, error, small adjustment, repetition.<br>
**Expected evidence:** Established mechanisms; numerical simplifications are pedagogical.

## Observable outcomes and preparation

- Distinguish model training from model use.
- Explain the role of examples, error, and parameters.
- Run one simple numerical update.

Before class, the instructor runs the worked case and lab, prints the exercise packet, prepares a four-column board—assumption, prediction, observation, delta—and checks that every mathematical notation is paired with dimensions. The demonstration must not become slide reading.

## Diagnostic

1. Explain in one sentence: **An example and a target**. What observation would falsify your explanation?
2. Explain in one sentence: **The prediction**. What observation would falsify your explanation?
3. Explain in one sentence: **Measure the error**. What observation would falsify your explanation?

**Teaching decision:** if two of three answers remain nominal or lack a validity condition, rebuild the vocabulary with a numerical example before any formula. A fluent but unfalsifiable answer does not count as mastery.

## Timed plan

| Time | Activity | Observable evidence |
|---|---|---|
| 0–10 min | Individual diagnostic, then pair comparison | Three answers and one named uncertainty |
| 10–25 min | Situation and vocabulary | Annotated input → state → output diagram |
| 25–55 min | Develop the mechanism on the board | Shapes, assumptions, and intermediate calculation visible |
| 55–75 min | Worked case with deliberate errors | Reasoned correction, not only the right number |
| 75–95 min | Causal lab: predict, change one variable, run | Prediction / observation / delta table |
| 95–112 min | Exercises 1 and 2 with peer correction | Retained artifact and applied rubric |
| 112–120 min | Exit ticket and transfer | Mechanism, boundary, next experiment |

## Teaching notes

### 1. An example and a target

An example replaces the rule with a signal: an input plus the expected answer. For “the sky is…”, the target “blue” explains nothing — it gives a point of comparison. Our toy model keeps only the numbers: input x = 2, target y = 1.

### 2. The prediction

The model applies whatever its current setting is: p = w × x. With w = 0.4 and x = 2 it proposes p = 0.8. That 0.8 is neither good nor bad in itself: it is the raw material for the comparison with the target 1.

### 3. Measure the error

The error turns the verdict into a signed number: e = 1 − 0.8 = +0.20, so predict higher. A prediction of 1.3 would give e = −0.30: predict lower. The sign gives the direction, the magnitude gives the size of the move.

> **Working formula:** `error = target − prediction`

### 4. Adjust one parameter

The update sizes the step with the learning rate: w ← 0.4 + 0.1 × 0.20 = 0.42. The new prediction 0.84 moves toward the target 1 without reaching it: each example pulls the setting a little, none dictates it.

> **Working formula:** `new weight = old weight + learning rate × error`

### 5. Repeat across many examples

Pre-training repeats the loop over millions of varied examples, served in batches. Pulled in turn toward 0.5 and toward 0.7, w settles near a compromise (≈ 0.54) that lowers the average error — no example is perfect, all are approximated.

### 6. Training ≠ conversation

They are two distinct regimes. During training, w changes: 0.4 → 0.42. During a conversation, w is frozen; only the contents of the context window vary. Your correction lives in the context, not in the weights.

### Running the worked case

With weight 0.4, input 2, and target 1: prediction 0.8, error 0.2. With learning rate 0.1, the weight increases slightly. The lab shows the new prediction moving closer to 1.

Do not reveal the result at once. Ask learners to predict the next operation, its shape, and the expected sign. After each line ask: “What changed? What stayed fixed? Which assumption did we use?” A calculation error repaired with a causal chain is worth more than a guessed result.

### Lab protocol

1. Write a qualitative and, where possible, numerical prediction before touching a control.
2. Change one variable only; retain a capture or record initial and final values.
3. Explain the delta through the mechanism, not “the tool did that.”
4. Test one boundary value and state where the model stops representing a real system.

## Misconceptions

| # | Observable misconception | Grounded correction | Probe |
|---|---|---|---|
| 1 | “An example and a target guarantees the outcome without assumptions or measurement.” | An example replaces the rule with a signal: an input plus the expected answer. For “the sky is…”, the target “blue” explains nothing — it gives a point of comparison. Our toy model keeps only the numbers: input x = 2, target y = 1. | Ask for a counterexample, then restate the mechanism with its validity condition. |
| 2 | “The prediction guarantees the outcome without assumptions or measurement.” | The model applies whatever its current setting is: p = w × x. With w = 0.4 and x = 2 it proposes p = 0.8. That 0.8 is neither good nor bad in itself: it is the raw material for the comparison with the target 1. | Ask for a counterexample, then restate the mechanism with its validity condition. |
| 3 | “Measure the error guarantees the outcome without assumptions or measurement.” | The error turns the verdict into a signed number: e = 1 − 0.8 = +0.20, so predict higher. A prediction of 1.3 would give e = −0.30: predict lower. The sign gives the direction, the magnitude gives the size of the move. | Ask for a counterexample, then restate the mechanism with its validity condition. |

> **Boundary to maintain:** This one-weight loop is a teaching model. Real models have many parameters, gradients, and optimizers.

## Probing questions

1. If we remove or reverse **An example and a target**, which output changes first, and what observation would show it?
2. If we remove or reverse **The prediction**, which output changes first, and what observation would show it?
3. If we remove or reverse **Measure the error**, which output changes first, and what observation would show it?
4. If we remove or reverse **Adjust one parameter**, which output changes first, and what observation would show it?
5. If we remove or reverse **Repeat across many examples**, which output changes first, and what observation would show it?
6. If we remove or reverse **Training ≠ conversation**, which output changes first, and what observation would show it?

## Assessment

| Level | Criterion |
|---|---|
| 0 | Repeats terms without connecting input, transformation, and output. |
| 1 | Describes the chain but checks neither shape nor assumption. |
| 2 | Executes the case, explains the result, and names one limitation. |
| 3 | Transfers to a new case, compares an alternative, and proposes a measurement that could invalidate the choice. |

**Exit threshold:** level 2 on the worked case and at least one exercise; a memorized formula without interpretation remains level 1.

## Observation and remediation protocol

During discussion, the instructor records evidence rather than impressions. Evidence of understanding contains a named object, a justified transformation, and a checkable consequence. If a learner gives the right result without a chain, ask for the preceding line. If the chain is coherent but the result is wrong, preserve the chain and isolate the arithmetic error. If vocabulary from another concept is used, compare both mechanisms in an input, state, output, cost, and boundary table. Remediation targets the first break only: vocabulary, shapes, operation, interpretation, or claim scope. After correction, use a neighboring case with a changed value; success on the same example does not prove transfer. For pair work, assign operator and verifier roles, then swap. The verifier does not supply the answer: they request an assumption, check the shape, and ask what observation could contradict the reasoning. The instructor retains the exit ticket and classifies the dominant break. The next session opens with a three-minute problem aimed at that break instead of repeating the whole lesson.

## Differentiation

- **Support:** provide shapes and the first transformation; let the learner complete interpretation and boundary.
- **Core path:** worked case without result, lab with one assigned variable, diagnostic exercise.
- **Extension:** change one assumption, compare two mechanisms, and define the metric that would decide between them.

## Post-session follow-up

Within twenty-four hours, return each annotated exit ticket with one priority, the exact resource to reopen, and a mini-case different from the worked case. Revision requires three items: a written prediction, a retained trace, and one sentence explaining the delta. At the next session, sample two submissions: one that repaired the break and one that remains ambiguous. Discuss them anonymously, then state the criterion that separates them. Do not use completion rate as evidence of mastery. Evidence of remediation is a correct chain on a new case with a stated boundary. If the same break appears in more than one third of the group, repair the support or demonstration before blaming learners.

Follow-up closes only when the new artifact shows the causal chain, check, and boundary—not merely when a file has been submitted.

## Sources and evidence boundary

- Rumelhart, Hinton & Williams, “Learning representations by back-propagating errors”, Nature (1986).
- Course source packet supplied by the owner; named-product details remain source-reported until primary verification.

> **Scope:** Established mechanisms; numerical simplifications are pedagogical. These references support the session frame; they do not turn a reported product choice into an independently verified result.

## Exit ticket

In no more than six lines: mechanism; calculation or trace; observation; boundary; evidence level; next experiment. The instructor marks one priority causal break for revision.