When an electric current flows through a wire, the heat produced in the wire can be determined using the formula known as Joule's Law of Heating. This is mathematically expressed as:

\(H = I^2 R t\)

where:

• \( H \) is the heat produced,
• \( I \) is the current flowing through the wire,
• \( R \) is the resistance of the wire, and
• \( t \) is the time for which the current flows.

In this scenario, we are considering what factor increases the amount of heat produced for a fixed time. Let us analyze each of the given options:

1. Decreasing the length of the wire: If the length of the wire is decreased, the resistance \( R \) of the wire decreases, since resistance is directly proportional to the length of the wire. This would lead to a decrease in heat produced.
2. Reducing the current in the wire: Since the heat produced \( H \) is proportional to the square of the current \( I \), reducing the current will decrease the heat produced.
3. Making the wire thicker: Making the wire thicker decreases its resistance, as resistance is inversely proportional to the cross-sectional area of the wire. This would result in less heat being generated.
4. Increasing the resistance of the wire: According to the formula \( H = I^2 R t \), increasing the resistance \( R \) of the wire increases the heat produced, provided the current and time remain constant.

Therefore, the correct answer is to increase the resistance of the wire, which directly leads to an increase in the amount of heat produced.