In this English grammar sentence improvement question, choose the best word so that the idiomatic expression becomes correct. Original sentence: We were not the wiser after hearing the explanation.

Introduction / Context:
This question focuses on correct usage of a fixed English idiom. The expression "none the wiser" is commonly used to say that after some effort, explanation, or action, a person still does not understand anything more than before. The original sentence slightly distorts this idiom, and you must select the option that restores the correct form.

Given Data / Assumptions:
- Original sentence: We were not the wiser after hearing the explanation.
- The intended meaning is that the explanation did not increase understanding.
- The idiom should follow the standard pattern used in English.

Concept / Approach:
The proper idiomatic expression is "none the wiser", not "not the wiser". It is a fixed phrase meaning "no better informed" or "not having gained any understanding". Therefore, the correct improvement is to change "not" to "none" while keeping the rest of the sentence the same. Recognising such set expressions is important for scoring well in idiom based questions.

Step-by-Step Solution:
Step 1: Identify that the sentence uses an idiom relating to understanding: "the wiser". Step 2: Recall the fixed phrase "none the wiser" from common usage and reading. Step 3: Compare the given options to see which one exactly matches the standard idiom. Step 4: Option A, "We were none the wiser after hearing the explanation", uses the idiom correctly while preserving the intended meaning. Step 5: Confirm that the other options either change the meaning or create unnatural expressions.

Verification / Alternative check:
You can confirm the idiom by checking any reliable dictionary or phrase dictionary, which will list "none the wiser" as meaning "as ignorant or uninformed as before". Additionally, example sentences typically read "After all that discussion, I was still none the wiser". This exactly matches the situation described in the question. Hence Option A is verified as correct.

Why Other Options Are Wrong:
Option B, "neither the wiser", is not a recognised idiom and sounds incorrect. Option C, "nevertheless the wiser", changes the meaning to "in spite of everything, we were wiser", which is opposite of what is intended. Option D, "No improvement", keeps the faulty phrase "not the wiser". Option E, "no wise", is grammatically wrong and does not form a meaningful expression in this context.

Common Pitfalls:
Students sometimes overlook that certain expressions are fixed and cannot be modified freely. Small changes like replacing "none" with "not" may appear minor but actually break the idiom. Regular reading of quality English material helps you become familiar with fixed phrases, which makes such questions much easier. Always ask yourself if you have seen the phrase exactly in that form before choosing it in an exam.

Final Answer:
The correct idiomatic sentence is We were none the wiser after hearing the explanation.