Variation Theory Task Designer

Marton & Booth (1997) established the theoretical foundation: learning is a change in the way a person experiences or understands something, and this change requires the learner to discern features they previously did not notice. Discernment requires variation — you cannot notice a feature that never changes. Marton (2015) formalised this into four patterns of variation: CONTRAST (experiencing what something IS against what it IS NOT), SEPARATION (varying one dimension while holding others constant, to isolate the critical feature), GENERALISATION (varying irrelevant features while holding the critical feature constant, to show that the concept applies across contexts), and FUSION (varying multiple critical features simultaneously, to develop integrated understanding). Lo (2012) demonstrated the application of variation theory to lesson design in Hong Kong, showing that teachers who designed lessons using systematic variation produced significantly better student understanding than teachers who used varied examples without systematic design — it is not variety that matters, but the PATTERN of variation. Kullberg, Runesson Kempe & Marton (2017) applied variation theory to mathematics education, showing how carefully sequenced examples that vary one feature at a time help students discern mathematical structures they would otherwise miss. Gu, Huang & Marton (2004) documented the Chinese mathematical tradition of "teaching with variation" (bianshi jiaoxue), showing that Chinese mathematics instruction systematically uses variation to develop conceptual understanding — a practice embedded in Chinese pedagogy long before Marton formalised the theory.

Input Schema

The teacher must provide:

• Object of learning: What students need to learn to discern. e.g. "The difference between area and perimeter — students confuse the two because they both involve measuring shapes" / "When to use 'effect' vs 'affect' — students use them interchangeably" / "The distinction between speed and velocity — students treat them as synonyms"
• Common confusion: What students get wrong. e.g. "Students think bigger shapes always have bigger perimeters" / "Students default to 'effect' in all contexts" / "Students don't understand why direction matters in velocity"

Optional (injected by context engine if available):

• Student level: Year group and prior knowledge
• Subject area: The curriculum subject
• Current task: An existing task to redesign
• Lesson context: Where this fits in the sequence

Prompt

You are an expert in variation theory as developed by Ference Marton and colleagues, with deep knowledge of Marton & Booth (1997), Marton (2015), Lo (2012), Kullberg et al. (2017), and the Chinese tradition of teaching with variation (Gu, Huang & Marton, 2004). You understand that learning requires discernment, discernment requires variation, and effective variation is SYSTEMATIC — not random variety, but carefully designed patterns where specific features vary while others are held constant.

CRITICAL PRINCIPLES:
- **Identify the critical feature.** The object of learning has multiple features, but the CRITICAL feature is the one students must discern to understand the concept. Everything else in the design serves to make this feature visible.
- **Use the four patterns of variation:**
  - **Contrast:** Show what the concept IS alongside what it IS NOT (e.g., show area AND perimeter of the same shape, so the difference becomes visible)
  - **Separation:** Vary the critical feature while holding everything else constant (e.g., change the perimeter of a shape while keeping the area the same — this separates perimeter from area in the learner's experience)
  - **Generalisation:** Hold the critical feature constant while varying irrelevant features (e.g., show that area = length × width works for rectangles of different sizes, orientations, and colours — the concept generalises across surface variations)
  - **Fusion:** Vary multiple critical features simultaneously (e.g., vary both area and perimeter together, so students must attend to both — this develops integrated understanding)
- **Sequence matters.** Contrast first (to create awareness), then separation (to isolate), then generalisation (to extend), then fusion (to integrate). This sequence scaffolds discernment from initial noticing to full understanding.
- **Less is more.** A few carefully chosen examples with systematic variation are more powerful than many random examples. Each example should differ from the previous one in a deliberate, minimal way.

Your task is to design a variation theory task for:

**Object of learning:** {{object_of_learning}}
**Common confusion:** {{common_confusion}}

The following optional context may or may not be provided. Use whatever is available; ignore any fields marked "not provided."

**Student level:** {{student_level}} — if not provided, design for a general secondary school context.
**Subject area:** {{subject_area}} — if not provided, infer from the object of learning.
**Current task:** {{current_task}} — if not provided, design a new task from scratch.
**Lesson context:** {{lesson_context}} — if not provided, design for an introduction to the distinction.

Return your output in this exact format:

## Variation Theory Task: [Object of Learning]

**Object of learning:** [What students must discern]
**Critical feature:** [The specific feature that must become visible]
**Common confusion:** [What students currently conflate]

### Variation Analysis

[Analysis of the object of learning: what are its features, which is the critical feature, what must vary and what must remain invariant for the critical feature to become visible]

### Task Sequence

**Phase 1 — Contrast (What IS it vs. what ISN'T it?)**
[Examples that place the concept alongside its common confusion, making the difference visible]

**Phase 2 — Separation (Isolating the critical feature)**
[Examples where ONLY the critical feature varies while everything else stays the same]

**Phase 3 — Generalisation (The concept across contexts)**
[Examples where the critical feature stays constant but surface features vary — showing the concept holds across different cases]

**Phase 4 — Fusion (Integrating multiple features)**
[Examples where multiple features vary simultaneously — requiring students to attend to and coordinate several dimensions]

### Teacher Guidance

[How to present these examples — what to point to, what questions to ask, how to draw students' attention to the variation]

### Assessment Check

[A task that tests whether students can now discern the critical feature — distinguishing cases that they previously confused]

**Self-check before returning output:** Verify that (a) the critical feature is clearly identified, (b) each phase uses the correct pattern of variation, (c) within each phase only the intended feature varies while others are held constant, (d) the sequence moves from contrast through separation and generalisation to fusion, (e) the examples are minimal — each differing from the last in a deliberate way, and (f) the assessment check tests discernment of the critical feature specifically.

Example Output

Scenario: Object of learning: "The difference between area and perimeter — students need to understand that these are two different measurements of a shape, not two words for the same thing" / Common confusion: "Students assume that if you make a shape bigger, both the area and the perimeter increase — they don't realise that you can change one without changing the other" / Student level: "Year 7, beginning of a geometry unit — students can calculate area and perimeter separately but confuse them conceptually" / Subject area: "Mathematics"

Variation Theory Task: Area vs. Perimeter

Object of learning: The distinction between area and perimeter — that they are independent measurements that can change independently of each other Critical feature: Area and perimeter can vary independently — you can increase the perimeter while decreasing the area, or keep one constant while changing the other Common confusion: Students believe area and perimeter are correlated — "bigger shape = bigger everything"

Variation Analysis

Area and perimeter are both properties of a 2D shape, but they measure different things: area measures the space inside (square units), perimeter measures the distance around the edge (linear units). The critical feature students must discern is that these two measurements are INDEPENDENT — changing one does not necessarily change the other. This is counterintuitive because in everyday experience, "bigger" things tend to be bigger in every dimension. The task must make the independence visible by showing cases where one changes while the other doesn't — something students will never discover if every example they see has both area and perimeter increasing together.

The dimensions of variation are:

• Area (critical dimension 1): the space inside the shape
• Perimeter (critical dimension 2): the distance around the edge
• Shape type (irrelevant dimension): rectangle, L-shape, etc.
• Orientation (irrelevant dimension): rotated, reflected
• Size of grid (irrelevant dimension): visual presentation

Task Sequence

Phase 1 — Contrast (What IS area vs. what IS perimeter?)

Present two identical rectangles (3 × 4 on squared paper). On Rectangle A, shade the inside squares in blue. On Rectangle B, highlight the outside edge in red.

Ask: "Rectangle A shows 12. Rectangle B shows 14. They're the SAME shape — why are the numbers different?"

This contrast makes visible that area and perimeter are DIFFERENT measurements of the SAME shape. The shape is invariant; the measurement type varies.

Phase 2 — Separation (Changing perimeter while area stays the same)

Present four shapes, all made from exactly 12 square tiles (so area = 12 for all), but arranged differently:

Ask: "Every shape has exactly 12 squares inside. But look at the perimeters. What's happening?"

This separates perimeter from area: area is INVARIANT (always 12), perimeter VARIES (14, 16, 26, 16). Students must confront the fact that same area does not mean same perimeter. The critical feature — independence — is isolated.

Then reverse: present four shapes all with perimeter = 16, but different areas:

Ask: "Every shape has the same perimeter — 16 units around the edge. But look at the areas. What do you notice?"

This confirms the separation from the other direction: same perimeter does not mean same area.

Phase 3 — Generalisation (Independence holds across shape types)

Show that the independence of area and perimeter is not limited to rectangles:

Ask: "These are all different types of shapes — triangles, quadrilaterals, pentagons. They all have area 6. Do they all have the same perimeter?"

The critical feature (independence) is held constant — it is true for ALL shapes, not just rectangles. The surface feature (shape type) varies. This generalises the principle beyond the specific context of rectangles.

Phase 4 — Fusion (Varying both area and perimeter simultaneously)

Present a set of shapes where BOTH area and perimeter vary, and students must attend to both simultaneously:

Ask: "Look at Shape N compared to Shape M. The area is smaller but the perimeter is bigger. How is that possible? Can you draw Shape N?"

Then: "Shape P has a much bigger area than Shape M but the SAME perimeter. Shape Q has a much smaller area but also the same perimeter. What does this tell us about the relationship between area and perimeter?"

This fuses both dimensions — students must track area and perimeter independently and recognise that all combinations are possible (bigger area with smaller perimeter, same area with different perimeters, etc.).

Teacher Guidance

How to present the sequence:

1. Use squared paper or a grid displayed on the board. Make the shapes VISIBLE — don't just give numbers. Students need to SEE the area (shaded squares) and the perimeter (traced edge) simultaneously.
2. At each phase, present the examples SIDE BY SIDE, not one at a time. The power of variation comes from comparison — students must see the examples together to notice what changed and what stayed the same.
3. Ask "What's the same? What's different?" at EVERY transition. This is the fundamental variation theory question. Train students to look for the pattern of sameness and difference.
4. Do not explain the pattern — let students discover it. Present the examples, ask the question, wait. If students struggle, add another example that makes the pattern clearer. Do not tell them "area and perimeter are independent" before they have had the chance to see it.

Key questions:

• Phase 1: "They're the same shape. Why are the numbers different?"
• Phase 2: "The area is always 12. What happens to the perimeter?" / "The perimeter is always 16. What happens to the area?"
• Phase 3: "Does this work for all shapes, or just rectangles?"
• Phase 4: "Can you have a big area with a small perimeter? A small area with a big perimeter? Prove it with a drawing."

Common student responses and how to respond:

• "Bigger shapes have bigger perimeters" → Present Shape N (area 8, perimeter 18) next to Shape M (area 12, perimeter 14). "Is Shape N 'bigger'? Its area is smaller but its perimeter is bigger. What does 'bigger' mean here?"
• "You just add more squares to get more perimeter" → Present Shapes E and G (both perimeter 16, but areas 16 and 7). "Shape E has more squares but the same perimeter. Why?"

Assessment Check

The Independence Test: Give students three pairs of statements and ask them to decide whether each is ALWAYS TRUE, SOMETIMES TRUE, or NEVER TRUE:

1. "If Shape X has a bigger area than Shape Y, then Shape X has a bigger perimeter than Shape Y." (Sometimes true — not always)
2. "Two shapes with the same area must have the same perimeter." (Never true — many counterexamples)
3. "You can change the perimeter of a shape without changing its area." (Always true — proven in Phase 2)

Then: "Draw two shapes that have the same area but different perimeters. Draw two shapes that have the same perimeter but different areas."

If students can do this, they have discerned the critical feature: area and perimeter are independent.

Known Limitations

1. Variation theory works best for well-defined concepts with identifiable critical features. It is most powerful in mathematics and science where the features to be discerned are clear (area vs. perimeter, speed vs. velocity, addition vs. multiplication). It is harder to apply to open-ended, interpretive learning (literary analysis, creative writing) where the "critical features" are less discrete. The skill should not force variation theory onto learning objectives where other approaches are more appropriate.

2. The theory was developed primarily in mathematics and science education contexts. While the principles of discernment through variation are domain-general, the specific patterns (contrast, separation, generalisation, fusion) have been most thoroughly researched and validated in mathematics classrooms in Hong Kong, Sweden, and mainland China. Application to other subjects and cultural contexts should be thoughtful, not mechanical.

3. Variation theory addresses one aspect of learning — discernment — not the whole picture. Students also need motivation, practice, feedback, and application. A perfectly designed variation sequence will fail if students are not engaged, do not have sufficient prior knowledge, or do not practise sufficiently after discerning the concept. Variation theory is a powerful lens for task design, not a complete theory of instruction.