I recently read a series of very interesting posts, discussing doing “silent prayer” in a Jewish service. Some people were very much in favor of it; some less so.
When synagogue-goers use the phrase “silent prayer,” they most often mean the “Amidah” — the prayer that’s said silently. But I got the feeling that the above discussion was, rather, about allowing time in a service when people could pray their own prayers silently.
There can be many “pros” to this, but a major “con” is the problem that many people have with prayer in general: distracting thoughts. They’re difficult enough with prayers that are actually recited; the “Shema,” for example. Given a period of “silent” prayer with a head full of racing, random thoughts, the experience can be distinctly unpleasant.
Rabbi Morris Lichtenstein taught his students (members of “The Society of Jewish Science”) to pray silently. But there was a structure to how they did it. Rather than simply a period set aside during the service for personal things expressed silently, the Rabbi taught his students to focus on positive statements (“affirmations”) or positive images (“visualization”). This kind of prayer can be done individually, outside of services, too. He goes into this in detail in his text, “Jewish Science and Health“  and elsewhere.
His wife did too. Some of her essays/sermons were published in the 1980’s under the title “Applied Judaism,” but she always acknowledged the Rabbi as her teacher. Her ideas are most clearly understood with reference to his writings, which preceded hers. Still, one of my all-time favorite spiritual quotes is hers:
“…G-d cannot be perceived through the mind alone. If you would know G-d, do not seek merely to prove His existence, but turn to Him with your heart; affirm your union with Him, affirm His responsiveness to prayer, pray to Him; if you actually turn to G-d … speak to Him in your heart, you will be astonished to find how close He is to you, you will feel His nearness, you will have found G-d.” 
When I attended meetings in the 1980’s, many of the members, who had known both Rabbi and Mrs. Lichtenstein personally, had been practicing this kind of “affirmative” or “visualized” prayer for decades. When the time came for it in the service, a sweet silence filled the room. The word “Shechinah” comes readily to mind.
So, perhaps the question is not just whether to pray silently, but how to do so. _____________________________________________________________
 Lichtenstein, Rabbi Morris; Jewish Science and Health; “Prayer”; p. 43-56
 Lichtenstein, Tehillah; Applied Judaism; “Can We Prove That G-d Exists?”; p. 96