Four Principles of Sequencing

Together they describe the climb from simple introduction to abstract understanding.

Principle 1: Start with simple steps

The first principle: a teacher must start the lesson or unit with simple steps. Not the most basic possible (that would be too elementary), but a starting point that students can reach without prior learning.

There is a specific requirement: the simple step must come with at least one example. A bare definition without an example is too thin. With an example, students have something concrete to anchor to.

For a lesson on parts of speech: start with the simplest part (the bigger picture - what a part of speech is) along with an example. “Words are grouped by their function in a sentence. For example, in ‘The boy runs’, ‘boy’ and ‘runs’ do different jobs.”

For a lesson on fractions: start with the simplest fraction along with an example. “A fraction is part of a whole. If a pizza is cut into two equal pieces, each piece is one-half.”

The simple start gives students a foothold. From there, the lesson can climb.

A common error: teachers start with too much complexity, assuming students can keep up. They lose the simpler students immediately. The fix is to make the start simple enough that every student can engage.

Principle 2: Use concrete examples

The second principle: build the early part of the lesson on concrete examples. Students learn abstract ideas by first connecting them to concrete cases they can see, touch, and recognize.

’s distinction: a concrete example is something students can see and relate to physically. The chair in the classroom is a concrete example of “chair”. The river running near the school is a concrete example of “river”. A real plant from the garden is a concrete example of “plant”.

Compare with abstract examples. A “department chair” is an abstract use of the word “chair” (it refers to a position, not an object). For a young child learning the word “chair”, introducing department chair too early would confuse them. The teacher starts with the chair they sit on.

For each new concept, the teacher offers concrete examples first. Multiple concrete examples help students see what stays the same across cases. The chair in the classroom, the chair in the staff room, the chair at home all have arms and a back and a seat. Across these, students extract what makes a chair a chair.

One example is enough to start, but multiple examples deepen the concept. Each new example is a chance for students to refine their understanding.

Principle 3: Add complexity

The third principle: a lesson cannot stay at the simple-and-concrete level. It must add complexity over time.

A lesson that only uses the simplest examples never develops higher thinking. Students learn the basic case but cannot apply it more broadly. A lesson that only stays concrete never develops the conceptual understanding behind the examples.

Adding complexity means introducing harder cases, edge cases, and connections to other concepts. For chair: after the classroom chair (concrete), the teacher introduces a folding chair (still concrete but with a complication). Then a high chair for a baby (concrete, special purpose). Then maybe a swivel chair (concrete, additional feature).

For fractions: after one-half of a pizza (concrete, simple), the teacher introduces one-third of a pizza, then one-quarter, then improper fractions, then mixed numbers. Each step adds complexity without losing the concrete grounding.

Complexity must be added within the lesson and the unit. A whole unit at the simple level does not build understanding. The teacher who plans for complexity from the start produces students who can handle hard cases.

Principle 4: Introduce abstractions

The fourth principle: at some point in the sequence, the teacher must introduce abstractions. These are ideas that students cannot directly see or touch but can understand through reasoning.

Abstract ideas matter because real understanding lives in abstractions. A student who only knows concrete cases understands instances but not principles. A student who reaches abstraction understands principles that apply to many instances.

For chair: an abstract use is “department chair” - a leadership role named for a piece of furniture. A student who has reached abstraction can extend “chair” to leadership in this way without confusion.

For circulatory system: the abstractions are heartbeat, blood pressure, oxygen transport efficiency, cardiovascular health. A student who only knows the concrete heart and blood vessels does not yet understand these abstract ideas. A student who has reached abstraction connects them to the concrete system.

A lesson that never reaches abstraction stops at the concrete level. Students leave with examples but not principles. Real understanding requires abstraction. The teacher’s job is to take students from concrete to abstract over the sequence.

Flashcard

What is the difference between concrete examples and abstractions in a lesson?

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Answer

What students can see vs what they must reason about

Concrete examples: things students can see, touch, and recognize directly (the chair in the classroom, the river near the school, a real plant).

Abstractions: ideas students cannot directly see but can understand through reasoning (department chair as a role, blood pressure, heartbeat efficiency).

A lesson that stops at concrete never develops principles. A lesson that jumps to abstract too fast loses students.

Worked example: human circulatory system

Principle 1: Simple step. Begin with the simple idea that the body has a system to move blood around. Students can grasp this without complex anatomy.

Principle 2: Concrete example. Compare the circulatory system to a river system. Pakistan’s rivers (which students may have seen or know about) supply water across the country: the main rivers split into canals, then smaller rivers and streams. The teacher draws this on the board. Students recognize the concrete river system.

The teacher then connects: “The human body has a similar system. The heart is like the main source. Big blood vessels are like rivers. Smaller vessels are like canals. The smallest vessels are like the streams that bring water to remote places.”

Students who have not seen blood vessels can now picture them through the river analogy. The concrete is a bridge to the unfamiliar.

Principle 3: Add complexity. Introduce more concrete examples. Electricity is another network: power lines, transformers, household wiring. A telephone network is another. Each new network shows the same pattern (central source, branching distribution, capillary endpoints) in a different domain.

By multiple examples, students see the pattern itself. The pattern is becoming concept.

Principle 4: Introduce abstractions. Now the teacher introduces abstractions about the circulatory system. Heartbeat. Blood pressure. The efficiency of oxygen transport. The risks of high or low blood pressure. The relationship between heart health and exercise.

These abstractions connect to the concrete system students already understand. A student with low blood pressure, high blood pressure, or arrhythmia can now see what is happening at an abstract level.

By the end of the lesson, students have moved from a simple statement (the body has a system to move blood) through concrete examples (rivers, electricity, telephones) to abstractions (blood pressure, cardiovascular health). The full sequence builds understanding.

The recycling mode

There is an important detail: the four principles are not strictly one-way. Sometimes students fail to reach abstraction and need to recycle back.

A graphical representation: arrows pointing up from simple to concrete to complexity to abstraction. But also a recycling arrow: if students fail at abstraction, return to complexity. If they fail at complexity, return to concrete.

For example, a teacher reaches the abstraction phase of the circulatory system lesson (blood pressure). They notice that several students do not understand. The teacher does not push forward. They go back to the concrete examples. They reinforce the network analogy. They add another concrete example (perhaps a city water distribution system). Then they try abstraction again.

The recycling can happen multiple times. A teacher who recycles when needed eventually gets all students to abstraction. A teacher who refuses to recycle leaves struggling students behind permanently.

This is also why the time span of sequencing varies. Recycling adds time. A complex concept that needs significant recycling takes longer than a simple one. The teacher must allow enough time for the full climb, including recycles.

Flashcard

What is the recycling mode in instructional sequencing?

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Answer

Returning to earlier steps when students struggle at later ones

The four principles (simple → concrete → complex → abstract) are not strictly one-way.

If students fail at abstraction, return to complex examples.

If they fail at complex, return to concrete examples.

A teacher who recycles eventually gets all students to abstraction. A teacher who refuses leaves struggling students behind.