As committed as he is to improving the quality of education in urban school,
I don't understand why the word used is improving, not improve (root verb)
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To understand what is going on here, first let's undo the "As X as Y is" subordinating construction and recover a standalone sentence:
He is committed to improving the quality of education in urban school
"Is committed to" is a verb phrase, acting as a single unit. "Improving [...]" is its direct object. It is a rule of English that in the construction commit to [action], the [action] must be described using gerunds and not infinitives. This rule doesn't have a rationale; when you have the choice between a gerund and an infinitive as a direct object, it's pretty much arbitrary which is correct (depending on the main verb: sometimes one, sometimes the other, sometimes both can be used and it doesn't matter, sometimes both can be used but the meaning is different).
This article discusses gerunds versus infinitives at some length and goes into more detail about when you should use which.
Incidentally, "is" and "to" in this context are acting as pseudo-tense markers:
is X-ed ≡ state X was established in the past and still pertains now
commit to X-ing ≡ make a firm plan to X in the future and/or continually from now
Without them, you would have the very different
He committed improvements to the quality of education ...
which indicates that he's done doing whatever he did to improve the quality of education, and connotes that whatever he did was criminal (! but seriously, commit [act], without the to, is basically only used when act is a crime).
Improving in this sentence is the present participle of improve. It is a non-finite form, which means it is not inflected for person, number or tense.
The choice between the ing form and the to-infinitive can be a difficult one for foreign learners, and to understand the different uses fully, you really need to consult a qualified teacher of English as a foreign language, or a grammar book written for foreign learners.
The first time I read this question, I didn't see any problem with either As committed as he is to improving ... or As committed as he is to improve .... Both sound fine to me.
I don't know which grammar rule I could use to argue, but since this is very specific, Google is useful.
I searched for BBC "committed to improving", and CNN "committed to improving, and found about 6 million and 4 million results. Here are some examples,
Gulf Air committed to improving all aspects of products, services.
Sammy Wilson says executive remains committed to improving A5.
Speaking to BBC London's Tim Donovan Mr Johnson said he is committed to improving safety for cyclists ...
But he said President Pena Nieto “is very committed to improving the economic lot in Mexico.”
... we continue to be committed to improving the health and well-being of a diverse Michigan community.
On the other hand, we are committed to improving the team ...
I searched for BBC "committed to improve", and CNN "committed to improve, and found about 140 and 72 thousand results. Here are some examples,
The BBC has committed to improve understanding in the following ways: ...
You have not committed to improve the website access information ...
... the Media Action is committed to improve the quality of ...
Canadian Solar is committed to improve the environment ...
In 2010, Landesa and its partners committed to improve economic ...
“We are committed to improve the state of the human rights ..."
Although the number of "committed to improving" is overwhelming, I don't believe that we should rule out the use of "committed to improve". Besides, my searches are very restrictive (to either BBC or CNN). When I simply searched for "committed to improving" and "committed to improve", both in quotes, I will get about 39 million and 2 million hits, in that order.
It appears to me that even though the use of committed to improving is recommended, but committed to improve has become at least acceptable.