Question: "In the Old Testament we read that there is 'a time to keep silence and a time to speak' (Ecclesiastes 3:7). What is the value of silence and quiet in a time when we are always 'plugged in'?"
This passage is actually speaking about silence in terms of when we should speak. However, your question is more focused on silence in terms of removing distractions from our life, particularly for times of spiritual focus. Both aspects are important to consider.
What Ecclesiastes is saying is that there are times when we should speak, and there are times when we should not. There are times when we can betray God by speaking, and there are times when we can betray God by our silence. When to speak or when not to speak is a question of wisdom, and seeking the guidance of the Holy Spirit.
How does one acquire the wisdom needed to make the right choices? One thing we need to do is to inform ourselves by studying the Scriptures, which contain great wisdom. We should also seek wise counsel, and we should pray for divine guidance. And then, you have to make what seems to be the wisest choice, but remaining open to the correction of others and praying that God will correct you, if you have made the wrong choice.
The Fathers say quite a bit about the virtues of silence. One good source to read on this is "The Evergetinos," which is a compilation of sayings of the desert fathers, but arranged topically, by St. Nicodemus of the Holy Mountain. In volume 2, beginning on page 353, there is a section entitled "Concerning speech and silence, how and when to make use of them, and that idle talk is a sin." Here are a few excerpts from that section:
"An Elder said: "One man thinks that he is being silent, and yet his heart is judging others; such a man is always talking. Another man talks from morning until evening and yet keeps silence; that is, he says nothing that is not beneficial."
"A brother visited a certain Elder and said to him: "Abba, give me a word, so that I may be saved." The Elder replied: "If you go to someone, do not hasten to speak before you consider what you are going to say." Filled with compunction at this saying, the brother made a prostration and remarked: "Truly, I have read many books, but never have I come across such learning." Thus edified, he departed."
"Abba Isaiah said: "Wisdom does not consist in speaking; wisdom means knowing the time when you should speak and when to reply as necessary. Make it seem that you know nothing, although you have knowledge, so as to avoid great distress; for he who appears to have knowledge lays burdens on himself. Do no boast about your knowledge, for no one knows anything."
"An Elder sais: "If you acquire silence, do not pride yourself on having attained to virtue, but say: 'I am unworthy even to speak.'""
From St. Diadochos: "Just as, when the doors of the baths are left continually open, the heat inside is quickly driven out, so also the soul, when it wishes to say many things, even though everything that it says may be good, disperses its concentration through the door of the voice. Hence, the soul, deprived of suitable spiritual ideas, loses the strength to struggle against thoughts and babbles with anyone it encounters. Since in this way (through loquacity) the soul drives out the Holy Spirit, it cannot keep the intellect free from harmful fantasies; for the Good Spirit always flees from loquacity, which is the cause of every upset and fantasy. Timely silence is good, since it is nothing other than the mother of the wisest thoughts."
"Two brothers from Sketis wanted to visit Abba Anthony. They embarked on a boat, and with them embarked a certain Elder, whom the brothers did not know and who was likewise going to visit Abba Anthony. As they sat on the boat, the brothers discussed what the Fathers say about the Scriptures and also talked about their handiwork. The Elder was silent. After they had disembarked from the boat and reached their destination, Abba Anthony said to the brothers: "You found good company in in this Elder." Then he said to the Elder: "You had good brothers travelling with you, Abba." The Elder replied: "They are good, but their house has no door; anyone who wants to can enter the stable and untie the ass." He said this in order to show that they said whatever came into their mouths."Here are some thoughts on the question of distractions:
"There are a number of important things that should be observed by those seeking spiritual development. One of these is physical and mental quiet (hesychia), made possible by living in a quiet place [or turning off the TV at home, the radio in the car, etc.], away from noise, confusion, and distractions. Control of talking is another. Such control helps bring about inner silence, which strengthens a person spiritually, whereas unnecessary talking does the reverse" (Anchored in God: Life, Art, and Thought on the Holy Mountain of Athos, Dr. Constantine Cavarnos).
"Silence greatly helps in spiritual life. It is good for one to practice silence for about an hour a day: to test himself, to acknowledge his passions and to fight in order to cut them off and purify his heart. It is very good if there is a quiet room in the house which gives him the feeling of a monastic cell. There, “in secret”, he is able to do his spiritual maintenance, to study, and to pray. A little spiritual study done before prayer helps greatly. The soul warms up and the mind is transported to the spiritual realm. That’s why, when a person has many distractions during the day, he should rejoice if he has ten minutes for prayer, or even two minutes to read something, so as to drive away distractions" (Excerpts from Family Life, by St. Paisios the Athonite).We cannot avoid distractions all of the time, but we need to set aside times when we intentionally avoid them, so that we can make progress in the spiritual life. Great Lent is one tenth of the year. We should treat it as a tithe, and especially set this time apart for spiritual focus. This does mean we need to cut down the usual noise, and spend more time in prayer, more time reading the Scriptures and other spiritually edifying books, and more time in the services of the Church.