The Expansion Sequence

A key to deep, flexible, knowledge and understanding

Podcast is AI generated, and will make mistakes. Interactive transcript available in the podcast post.

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We’ve mentioned the expansion sequence in passing a few times.

Lee Wheeler

mentioned it as one of four techniques he attributed to the outstanding success of his students:

I bet you've had a student say this to you

As a quick recap, at the point of initial instruction, there are three possible sequences of instruction:

1. Initial Instructional Sequence

2. Initial Testing Sequence

3. Expansion Sequence

(1) teaches them the concept.

(2) makes sure your teaching worked.

(3) is optional.

Say you wanted to teach your children to expand a single bracket. You say ‘they can all expand a single bracket now.’ Great. Does that mean they can only expand this:

\(3(x + 2)\)

Or does it mean they can expand this:

\(3x \left(2 + 4b - 3\sqrt{x} + y\right)\)

Both fit the description of ‘expanding a single bracket,’ but there is a clear distinction between the two.

You’ve taught them to add fractions. Does that mean they can add these:

\(\frac{3}{4}+\frac{10}{3}\)

Or that they can add these:

\(\frac{3a}{4x}+\frac{10y}{3z}\)

If they can do the first then it’s not a huge leap to learn how to do the second (albeit learning the meaning of the second takes a little additional work.)

You’ve taught them to share in a ratio. Does that mean they can share 10 in the ratio 2:3, or that that they can answer the following:

Given that a ratio is 2a:3a:3x:4p and that one part of the ratio is worth 5Q, what would calculate each share?

You’ve taught them to find the area of a trapezium. Can they tell you what would calculate the area of this trapezium?

The expansion sequence makes all of these possible.

It makes them possible in minutes.

It makes it all so possible that children don’t even realise they have achieved something far outside the bounds of what we typically expect of them. ‘That was easy,’ is what they’ll tend to say if you set it up as a challenge.

When I shared an expansion sequence for integration, a few weeks ago,

John

reached out to let me know he’d tried it with his 8 year old who hasn’t yet been taught algebra. Not only did he get them all right, he did something very interesting with the final task:

His response was this:

\(\frac{ax^{O}}{O} + \frac{bx^{2}}{2} + \frac{cx^1}{1} + C\)

He had understood the transformation, but since he hadn’t been taught algebraic conventions, he took his best guess as to how to write ‘n + 1.’ John’s son wasn’t ‘wrong’ at all in his thinking. He had fully grasped the transformation, and this response demonstrates that he was thinking.

If the initial instructional sequence communicates the concept, then the expansion sequence ‘expands’ the range of contexts to which is can be applied. It teaches students to generalise. It teaches them to apply the same idea to increasingly unfamiliar examples.

It does all this without any further instruction from you, setting up one ‘micro-problem’ after another for students to apply their newfound knowledge towards.

Here is an example of an expansion sequence for adding equations:

Each task asks students to do something new, to apply the idea that ‘I add the like terms in each equation’ to fourteen different contexts.

It works, without additional instruction, because of the sequence. The sequence of prompts (what Engelmann calls their ‘juxtaposition,’) is what does the teaching. If students ‘take their best guess’ each time they will very likely get it right.

Whereas, pluck a task from the sequence at random and assign it after initial testing and most students will fail.

If you have just taught students to expand a single bracket:

\(7(3x+10)\)

And then you ask them to expand this:

\(5(-6t+10+3p-2a)\)

Very likely to fail.

But put that at the end of this sequence:

Very likely to succeed.

Same with the ‘impossible’ ratio question above. Ask even maths teachers to solve that one and most can’t. Ask it as a part of this sequence, however:

All of a sudden, dead easy. ‘Is that all, sir?!’

Designing expansion sequences is tremendous fun.

Helping kids realise that seemingly intractable problems are ‘dead easy,’ well within their capabilities, is immensely rewarding.

Knitting together the atoms into coherent chains of deep relational understanding over time is profoundly satisfying.

Below are similar examples of expansion sequences for:

• The area of the trapezium above

• Calculating the mean of a, b, c, d, e

• Proportional reasoning: ‘There are 8n squarks for every 2x - 3 squirks. How many squarks are there if there are 17 squirks?’