Does the following sentence contain a gerund?

Walking to the store, Allison saw her teacher.

No, this sentence does not contain a gerund.

Explanation:

Gerund and participles look the same, although their function is different. Gerunds function as nouns in a sentence, whereas participles function as adverbs.

If the sentence said - "Allison likes walking", then the word walking would be a gerund (you can ask the question, what does she like?)

In this sentence, the participle walking can be rewritten as "while she was walking to the store," which has the adverbial function of time, and therefore can only be a participle, but never gerund.

Gerunds can be replaced by a noun, whereas participles only with an adverb/clause.

absolutism the political belief that one ruler should hold all of the power within the boundaries of a country.