#worked examples

What it is:

Worked examples are models of how experts would find the solution to a problem

Step by step examples of how something is done

What it does:

Better encoding

Better procedural knowledge

Greater accuracy

Greater Efficiency

Greater conceptual knowledge

Provides a starting point for beginners / acts as scaffolding

Leads to procedural flexibility if done correctly

Examples + Applications:

Teacher shows how to do a problem on the board and explains steps

A very descriptive cooking/baking recipe

Watching a video of someone showing how to do something

Why/How it works:

Decreases cognitive load, so person is able to focus on doing the skill rather than remembering what comes next

By using deliberate practice and slowly taking parts of the worked example away, person is able to not use the

Risks:

Can be wrong skill level

Not shown or explained well due to expert blindspot

Worked examples tend to need to be for specific procedures

If used for too long, scaffolding can become a crutch / unable to do it without the worked example

May become too rigid in problem solving and unable to ‘think outside the box’

difficult to use for extremely flexible procedures

Implicit differentiation of equations

Cont’d from “Worked examples: Example 2”

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A process is followed to implicitly differentiate a function. We’ll start by considering the equation:

a + f(y(x)) = g(x)

Since we have a function of y it is difficult (or impossible depending on the function) to rearrange to be solely in terms of x on the right hand side.

Now, let’s recall what we learnt about the differential operator (See “The differential operator”). To first differentiate any equation we apply the differential operator to both sides of the equation. So, if we do this the equation becomes

d/dx [a + f(y(x))] = d/dx [g(x)]

which represents the first derivative of our original equation. Now, using the summation rule of differentiation, which states that d/dx (f(x) + g(x)) = d/dx f(x) + d/dx g(x), the left hand side (LHS) of the equation becomes

d/dx [a + f(y(x))] = d/dx a + d/dx f(y(x))

and since we know that the derivative of any constant (i.e. something that is independent of x) is zero; d/dx a = 0, Therefore this equation becomes

d/dx f(y(x)) = d/dx g(x).

We can now apply the chain rule to the LHS in the same way we did in example 2, to get

d/dx f(y(x)) = d/dy f(y(x)) d/dx y(x)

which we can substitute back in to the equation to get

dy/dx d/dy f(y) = d/dx g(x)

⇒ dy/dx = d/dx g(x) [d/dy f(y)]−1

Worked examples for radical substitution Cont'd from "Radical substitution mechanism"

↓ If I got anything wrong in this section, please let me know here.

Reaction of bromine and heptane: Br2 and C7H16

(i) Write an equation for the formation of bromine radicals.

    UV Br2 → 2Br•

(ii) Bromine radicals react with heptane to form bromoheptane (C7H15Br). Write equations for the propagation steps involved.

Br• + C7H16 → C7H15• + HBr C7H15• + Br2 → C7H15Br+ Br•

(iii) Using skeletal formulae, show three possible structural isomers that can be formed when bromine reacts with heptane to form bromoheptane.

John Seely Brown on: 'Learning in the Digital Age'

Een start in de zoektocht naar Worked Examples...

"Digital Media and Learning as an Emerging Field, Part II A Proposal for How to Use 'Worked Examples' to Move Forward" – James Paul Gee

‘Play exemplars’ (JG, p.3)    > “proposal about what an exemplar might look like.” (JG, p.4)                 > “In a sense, exemplars, as they historically engendered the discussions and debates that led to their acceptance as exemplars, served in the process as proposed worked examples for field that did not yet exist.” (ibid.)                 >> Public debate as “a sort of communal public working of the example.” (JG, p.5)

“I am, then, proposing a new sense of and a new use for worked examples: namely, as attempts to imagine exemplars for a new field and ways to create collaboration and debate around such proposed exemplars, in service of hastening actual exemplars and the growth of the field.” (ibid.)