Cognitive load, chunking and dual coding: why a worked example teaches and a wall of text does not

Working memory holds about four things at once. A new procedure asks for more than that, so when you try to solve a new kind of problem cold, every slot is taken by the search for the next step and nothing is left for learning the pattern. That is cognitive load, and it is why a worked example teaches a beginner faster than ten problems, why a formula list chunked by "what is missing" is easier to use than five formulas in a row, and why a labelled sketch beside the words sticks. Here are the theory and its three tools, with one physics problem done both ways.

What cognitive load is

John Sweller (1988) put it in one sentence: working memory is small, learning happens there, and a task can demand more of it than it has. When the demand exceeds the space, you may still get the answer, but nothing of the method is left behind, because the space that would have held the pattern was holding the search.

How small is working memory? Miller (1956) gave the famous figure of seven items, plus or minus two, and said in the same paper that he half meant it as a joke. Cowan (2001) went back over the evidence and argued for about four chunks when rehearsal and other tricks are stripped away. The honest number is "a few", and either way it is fewer than a new procedure needs.

Cognitive load theory names three kinds of load, and a student can feel each one:

• Intrinsic load is how many ideas interact at once in the task itself. A related-rates problem is intrinsically heavy: a geometric relation, a derivative, a chain rule and a substitution all have to be held together.
• Extraneous load is what the presentation adds. A diagram on the next page, a formula sheet in a different notation from the textbook, a phone face up on the desk.
• Germane load is the effort that builds the pattern: noticing what changed between two lines, naming the method, asking why this step and not another. This is the one you want.

Learning fails when the first two fill working memory and leave no room for the third. Every tool below cuts the first two or feeds the third. Where this sits among the study methods is on the study-methods hub: it is the theory under several of them.

The worked-example effect

Sweller and Cooper (1985) gave algebra students either a set of problems to solve or the same problems paired with worked examples to study. The students who had studied the examples solved similar problems afterwards faster and with fewer errors than the students who had practised solving from the start. Cooper and Sweller (1987) followed it up and found that with more practice the advantage spread to problems in a new shape, not only the ones that looked like the examples. That is the worked-example effect: for a beginner, studying a solved problem teaches the procedure better than struggling through an unsolved one.

Why: the example carries the search. You do not have to find the next step, so working memory can hold the step and the reason for it, which is the pattern you came to learn. The way to read one is the way the maths page puts it, read it like a mechanic: line by line, saying what changed and why, and predicting the next line before you uncover it.

Two more results shape how to use examples.

Expertise reversal. Kalyuga, Ayres, Chandler and Sweller (2003) reviewed studies in which the same instruction helped beginners and hurt students who already knew the procedure. Once you can do it, an example you no longer need is extraneous load, and the problem teaches more. So the examples must stop, and the question is when. Renkl and Atkinson (2003) proposed fading: a full worked example first, then the same kind of problem with the last step missing for you to supply, then two steps missing, then a bare problem. The load rises one step at a time as your pattern grows.

Split attention. Chandler and Sweller (1991) ran six experiments on material where the text and the diagram had to be read together. When the two were physically separate, students had to hold one in mind while finding the other, and learning suffered; integrating the words onto the diagram helped, as long as the words were ones the diagram needed. Extra commentary beside a diagram that already said it all made things worse. For you: labels on the diagram, not in a paragraph on the next page.

Chunking

Chase and Simon (1973) showed chess positions to a master, a strong club player and a beginner for a few seconds and asked them to set the pieces up again from memory. On positions from real games the master was far better. On boards with the same pieces scattered at random the master's advantage disappeared. The master was not remembering more pieces; the master was seeing groups of pieces as single familiar patterns, and a random board had none.

A chunk is a pattern you recognise as one thing. Working memory holds a few chunks, so the more of a problem you can see as chunks, the more of it fits. You build them two ways:

• Name the pattern on every worked example. "Difference of squares." "The inner derivative." "A strong acid, so no equilibrium table." A step with a name is a chunk; a step without one is a line to be rediscovered.
• Group formulas by the question they answer, not by the chapter. The Grade 12 physics (SPH4U) kinematics equations for constant acceleration are the standard case. Written in a row they are five formulas to search through. Chunked by what each one leaves out, they are one rule: pick the equation that does not contain the quantity you neither know nor need.

Leaves out	Equation
Displacement d	v = v₀ + at
Final velocity v	d = v₀t + ½at²
Time t	v² = v₀² + 2ad
Acceleration a	d = ½(v₀ + v)t
Initial velocity v₀	d = vt − ½at²

Five lines, one decision. That is a chunk.

Dual coding

Paivio (1986) argued that words and images are handled by two connected systems, so a fact you have both read and drawn is encoded twice and has two routes back. Mayer's work on multimedia learning (2009) finds the same from the other direction: people learn better from words with relevant pictures than from words alone. Words and pictures stored twice is the paragraph on the memory-techniques page; here is the rule that follows for problems.

Draw a sketch beside every problem, before the equations, and label it with the given values. The sketch is not decoration. It takes the situation out of working memory and puts it on the page, which frees the slots for the method. And a diagram you draw is dual coding; a diagram you copy from the book without thinking is a copy. Redraw the diagram from memory is the test of whether the picture went in.

Worked example: one physics problem, two ways

The first idea in SPH4U, two-dimensional kinematics and projectile motion, on the SPH4U course map. The problem: a ball rolls off a cliff 20.0 m high at 12.0 m/s horizontally. How far from the base of the cliff does it land? Take g = 9.8 m/s² and ignore air resistance. Three significant figures throughout.

As a wall of text. The ball has a horizontal velocity of 12.0 m/s which does not change because there is no horizontal force and a vertical velocity that starts at zero and increases because of gravity so we need the time it takes to fall 20.0 m which we find from the vertical motion using the equation for displacement with zero initial velocity and then we multiply that time by the horizontal velocity to get the horizontal distance which comes out to about 24 m.

Every fact in that paragraph is true. Try to use it to solve the next problem and you will find you cannot, because nothing in it is where you can see it: no sketch, no labels, no line you can point at and ask "why that?"

As a worked example. First the sketch, described in words because this page has no images: the cliff as a vertical line 20.0 m tall, the ball at the top, an arrow to the right labelled vₓ = 12.0 m/s, an arrow down labelled g = 9.8 m/s², and vᵧ₀ = 0 written at the top. Then the lines:

1. Split the motion. Horizontal: constant velocity, vₓ = 12.0 m/s, no acceleration. Vertical: starts from rest, vᵧ₀ = 0, accelerates down at 9.8 m/s². The two share only the time.
2. Vertical: find the time. With vᵧ₀ = 0, d = ½gt², so t = √(2d/g) = √(2 × 20.0 / 9.8) = √(40.0 / 9.8) = √4.08 = 2.02 s.
3. Horizontal: find the distance. d = vₓ · t = 12.0 × 2.02 = 24.2 m.

The ball lands 24.2 m from the base. Each line has a name, the reason is beside it, and the sketch carries the given values so the lines do not have to.

The common wrong line is putting 12.0 m/s into the vertical equation, as if the ball's speed were its downward speed. The sketch is what stops it: the 12.0 is drawn sideways, and the vertical arrow at the top says 0.

The fade. The same problem with line 3 removed: you have t = 2.02 s and must supply the horizontal distance yourself. Then a problem with lines 2 and 3 removed. Then this sibling, solved cold:

A ball rolls off a 25.0 m cliff at 9.0 m/s horizontally. How far from the base does it land? (g = 9.8 m/s².)

1. Split: vₓ = 9.0 m/s constant; vertical from rest.
2. Vertical: t = √(2 × 25.0 / 9.8) = √(50.0 / 9.8) = √5.10 = 2.26 s.
3. Horizontal: d = 9.0 × 2.26 = 20.3 m.

A higher cliff gives more time in the air; a slower ball covers less ground in it. Both effects are in the answer, and the check against the first problem is that the time went up and the distance went down. Chapter 4 of the free playbook has the example-to-problem ladder as a one-page template.

What this means for how you study

If the procedure is new to you: one example, read line by line with the pattern named, then a faded version, then problems. Solving cold on day one is not grit; it is load with nothing left for learning. The order from worked example to past exam is the full sequence.

If you can already do it: skip the examples. Do the problem, and read an example only when you are stuck. More examples now are extraneous load, which is what expertise reversal means.

Everyone: cut the extraneous load before you start. One page, one source, the labels on the diagram, the phone in another room; how to focus when you cannot is the page on the phone. And never confuse "it felt easy" with "I learned it". A well-made example makes the reading easy, and that ease is the thing the judgement-of-learning trap warns about. The test of an example is the problem done cold.

The mistakes

• Reading ten examples and solving none. The fade exists because examples stop teaching once the pattern is in.
• Solving cold on day one and calling the struggle learning. Struggle helps once you have the pattern to struggle with, not before.
• A diagram copied instead of drawn. Copying is not dual coding.
• Notes as a wall of text with no sketch. They will read well and solve nothing.
• Formulas grouped by chapter. Group them by the question each answers.

The theory on one page

• Working memory holds a few things. Three kinds of load: intrinsic (the task), extraneous (the presentation), germane (the useful effort).
• Three tools: worked examples for beginners, faded to problems as the pattern grows; chunking, by naming patterns and grouping formulas by what they answer; dual coding, a labelled sketch before the equations.
• The fade: full example, last step missing, two steps missing, bare problem.
• Expertise reversal: once you can do it, examples slow you down.
• Cut the extraneous load first, and test with the book closed.

Where YSR Studies does the bookkeeping

On paper, you choose when the examples stop and you draw your own fade. In YSR Studies a tutor coaches before it reveals the course's verified answer, so the worked example arrives after your attempt, not before it; a card's answer stays on the server until you have tried it, and the answers are checked by code. The course map shows each idea as Learned, Shaky or Not shown yet, which is where each idea stands on the ladder from examples to problems. Five free hours every month.