Let's analyze the given statements to determine the correct answer.

Assertion $\mathbf{A}$: Houses made of concrete roofs overlaid with foam keep the room hotter during summer.

Reason $\mathbf{R}$: The layer of foam insulation prohibits heat transfer, as it contains air pockets.

Explanation:

Foam insulation generally acts as an effective thermal barrier because it contains numerous air pockets that impede heat transfer. The primary function of foam insulation is to reduce the amount of heat that penetrates into or escapes from a building, thereby maintaining a more constant internal temperature. If a house has a concrete roof overlaid with foam, the foam will work to keep the interior cooler in summer by prohibiting the transfer of heat from the outside into the room. Hence, the foam insulation should actually help in keeping rooms cooler during summer, not hotter.

Therefore, the Assertion $\mathbf{A}$ is false because foam insulation typically helps in keeping rooms cooler during summer. Meanwhile, the Reason $\mathbf{R}$ is true because air pockets within the foam insulation impede heat transfer.

Based on this analysis, the correct option is:

Option B: A is false but R is true.