However, that's not all there is to it! The "-ing" form doesn't always turn into the "to" form!

Consider these two sentences:
1) I hear birds singing.
2) I hear the singing nearby.
They seem to have the same "singing" word, but not really.

With the first, let's play the same substitution trick as before - "I hear birds singing" - can we say it differently? Yes, more or less - "I hear that\how the birds sing." So let's use it! it's just the present tense here - אֲנִי שׁוֹמֵעַ שֶׁהַצִּפּוֹרִים שָׁרִים.

More examples:
I see them jumping. אֲנִי רוֹאֶה שֶׁהֵם קוֹפְצִים.
I remember him telling... אֲנִי זוֹכֶרֶת שֶׁהוּא אָמַר...
I feel the earth shaking. אֲנִי מַרְגִּישׁ שֶׁהָאֲדָמָה רוֹעֶדֶת.

The second one, though, makes us think. "I hear the singing nearby" - it cannot be said differently. This is how it is. What do we do now? Let's pay attention to one detail - it's "the singing." it has the definite article. So it's like an object, like a noun (called "gerund" in grammar). Let's search for a noun in Hebrew to express it, then!
"I hear the singing nearby" - אֲנִי שׁוֹמֵעַ אֶת הַשִּׁירָה לְיָדִי.
(Doesn't sound very natural, but let it be, it's just an example.)

A better example would be:
And then the crying stopped. וְאָז הַבֶּכִי נִפְסָק.
He never showed any understanding of the subject. הוּא אַף פַּעַם לֹא גִּלָּה שׁוּם הֲבָנָה שֶׁל הָעִנְיָן.
My reading\listening\writing (skills) improved very fast. מְיֻמָּנֻיּוֹת הִקְרִיאָה\ שְׁמִיעָה\ כְּתִיבָה שֶׁלִּי הִשְׁתַּפְּרוּ מַהֵר מְאֹד.