Does thinking during an action improve the quality of the action or reduce it?

I don't know if there is a Buddhist answer to this. But I would think it is more common sense.

It really depends on what you're doing.

Some acts require careful thinking before it is done. A little thought before speaking to a person, or if you're doing something like writing or programming, which are mental skills, is good.

Since you mentioned Martial Arts and Bruce Lee, I'd say martial arts has a way of repetitive action, in other words, a combination of moves being practised repeatedly, so that it just occurs naturally. Going with the flow is crucial.

Now, thinking or noting in the middle of that, may cause you to check the action unnecessarily when it is going correctly, consequentially the slowing down occurs, which may be detrimental, in that context. That's why all the thinking is done before-hand in martial arts practice, and unbroken, smooth repetition is of paramount importance.

(Commenting on movies would be inessential here. I'd advise you to take anything you hear in them with a pinch of salt. The dialogues may be used for effect rather than truth)

In conclusion, it really depends on the situation you're in. Be present in the moment and do mindfully what the situation entails.

If you're sitting in meditation practice, noting and letting go would help because that's the whole point of meditation. The mind's activity needs to be calmed, if I may say so. That's where noting and letting go helps.

In regular activities, just know where you are, be alert and aware of what you're doing, that is mindfulness. And the effect is that you're able to do things more productively (which is generally a good thing).

Before, while and after, a mental, verbal, bodily deed, in ways of skillful improvement, good householder, thinking, observing, reflecting, whether harmful for oneself, for others: Instructions to Rahula at Mango Stone, and there is actually (kammical) no deed without thought, without mind, as forerunner.

[Note that this isn't given for stacks, exchange, other world-binding trades, but for improvements toward release]

I remember in a Bruce Lee film in which he says, "Don't Think". If there is a split between action and thought, wouldn't the processing of action slow down?

In the context of practising martial art I believe that's related to the endless training involved -- after which you can "reply" to an attack effectively and automatically (i.e. immediately and without thinking, and without over-doing it).

You might see something similar if you learn to play piano or guitar -- or other "motor skill". As a beginner you must think about where to place your fingers and how to move them. To play at all well and quickly, you need to have practised so the movement is automatic (also called "muscle memory") and your mind is free (e.g. to listen to and cooperate with the other musicians you're playing with).

That's also about "not thinking" in the sense of not getting upset -- e.g. "He's attacking me! I'm frightened!" etc. -- and not "thinking" (i.e. not focusing on an "internal object of the mind") may improve your awareness of "external" objects, your mind is already "in the right place" so you shouldn't be surprised and unready.

In Buddhist terms though you might want to be thinking (especially as beginner) -- e.g. about whether something (an activity) is worth doing at all, whether it's well-intentioned, what the motive is, what the result will be, etc. (see for example the sutta referenced in Samana Johann's answer).

For example, I am writing, and I am aware that I am writing: when I repeat, "I am writing", "I am writing", in my mind, when performing the action of writing, isn't there a split between action and thought?

I think you're talking about one of the ways to practice "Vipasanna".

I'm not sure what the purpose of that practice is, maybe it's to make you aware of what you're doing and what you're feeling and what you're aware of, and maybe why and how that's so. Or its purpose is, as you said "mindfulness" or sati -- i.e. to remember or be mindful of the doctrine (dhamma).

Conversely not-thinking -- perhaps what might be called "immersion" or "absorption" in an activity -- is that a form of samadhi?

This article -- Full-Stop Mind by Bhante Boddhidhamma -- seems to say that "noting" (or labelling) is intended as a step in the right direction, i.e. to be aware of something without thinking about it too much, wandering off-topic (into other thoughts), or attaching to it in the wrong way:

Noting is the second component of the vipassana technique that Mahasi Sayadaw taught. Paradoxically, the result of noting is that it takes a meditator beyond thinking. It is not an end in itself. The Buddha taught that there are two stages of concentrated thought before full concentration is established. The first is a simple noting or naming of the object. This act of labeling, vitakka, whereby the attention is pointed at the object, is likened to a bee flying toward a flower. The label encapsulates the whole experience. In children just beginning to speak, this process is very obvious and simplistic. They rejoice at being able to name an object—“Car! Car!” At their level of linguistic development, the word “car” simply points to the object. There’s not much thought around the word, since language itself, which allows us to think about an object, is not that developed yet.

For adults, the word “car” conjures up a host of memories and desires. We are thinking about an object, which is known as proliferation (papañca). [etc]

Noting is an acknowledgement of what the body, heart, and mind are doing. [etc]

Does thinking during an action improve the quality of the action or reduce it?

What is the nature of thinking ? It is impermanent. Meaning thinking ability arises. Thinking ability changes. Thinking ability vanishes.

As a Buddhist we must realise...if we attach thinking process with any action then sometimes it will improve the action , sometimes it will reduce the quality of action as thought process will change , sometimes we will not be aware of any effects on the quality of action because thinking process will cease.

Only right thinking leads us to nirvana...wrong thinking may produce desired effects but it won’t help in spiritual progress.

When most people use the word "think" they are referring to discursive thoughts, and these are the ones you mention, but thinking is more than that.

On a neurological level, you're (obviously) capable of doing several things at once. Not all the systems rely on others to work - you don't need to think discursive thoughts to breathe, but it does rely on mental activity (you won't be doing much breathing after death;) It is quite possible that discursive thought has no effect on physical actions directly, but it is eminently possible that it can have the effect of removing attention from what is being done, and in performing actions attention is incredibly important

If you are well concentrated then discursive thought would not be a problem, but also, it will be less likely to arise of its own accord, and is probably unnecessary.

(Right) Intention will drive attention, and attention will aid the quality.

"Don't Think" -> Bruce Lee is student of Wing Chun master, Ip Man and Chi Sao can train that. It's a technique that body that react automatic instinct subconscious mind to fight. Using the conscious mind to think is too slow.

All form of Wing Chun or Tai Chi has concept of power absorption from opponent so it give some time for conscious mind to think too. If you know the one inch kinetic punch concept too then it will return enormous powerful strike.

https://medium.com/wing-chun-kung-fu/thinking-without-thinking-1ff332f46c27

one inch punch - https://www.youtube.com/watch?v=hMMS0Yx0Sa0

Well, my personal thought Bruce Lee is too over in physical strength training base on the documentary and comments by his peer movie actors. He is an atheist and free thinker base on his interview. "religious affiliation was, he replied "none whatsoever."

Bruce didnt make it to level 3 and above on Wing Chun because Ip Man refused to teach him. . Have u seen Bruce playing wooden dummy in movie action? Wing Chun Ip Man version has total of 6 levels. Certain levels of emptiness is the criteria to enter level 3. So, i personally dont recommend people to learn from his way of martial arts.

I am not sure what Bruce Lee meant by his version of emptiness too. https://www.youtube.com/watch?v=cJMwBwFj5nQ

Lets begin with our good old actor Bruce lee and ascend to Buddhism afterwards....

What you are referring to is something called "Reflexes". In martial arts the reason they practice long hours is to train their bodies to fight automatically. This is the extreme level of fighting and he who is in this level is considered an elite. Here your muscles do the thinking for you and there is no need for logical reasoning (like your hand is feeling heat from a candle and immediately taking the hand away).

The Scientific explanation is very simple on this. There is a thing called "Muscle memory". Muscle memory is your muscles learning the things that you do constantly. This allows you to do things very fast and better than a novice.

for example :- Playing a guitar without looking at the strings, Typing something without looking at the keyboard and etc.

There is a common misinterpretation of what "Mindfulness" truly is, my school teach us differently and the form you are mentioning is "Not so effective" as to my teacher. So here is what we have learned as Mindfulness.....

Lets discuss your own example,

When i'm writing,like any other time my mind starts to wander and start many discussions within. I may even get angry or bored. I might even start to focus on the debate my mind has generated within a couple of minutes and jump into it. I am no longer focused on my writing and i may even write a wrong letter or a word.

Clearly here i have lost my focus and i'm jumping back and forth on my writing and debating with my mind. this is clearly not being mindful.

So if i focus on my writing and focus my thinking on what i have written and what i am about to write that focus my mind on the task at hand. Now i'm mindful about what i am actually doing this instance.

When asked about the version of mindfulness that is mentioned on the question my teacher had a reply like this....

(The person who asked the question asked......)

Is it correct to think i'm walking now, now i'm taking the right foot, now i'm taking the left foot. as a form of mindfulness?

(My teachers answer....)

What is the reason that we as Buddhists go on to practice mindfulness?

Because we are trying to reach Nirvana. And when we are not mindful the mind start to wonder.

Where does the mind wander?

• Mind wander in the adoring nice memories of the past, Bright hopes of the future and in the moment we enjoy currently and it gets attached.
• Mind wander in the cruel worrying past, dread for what future may bring and finds a reason to start hating right now
• Mind wander in a chaos. it jumps here and there, a moment ago you were laughing and now you are feeling bad and almost about to cry. Seconds later you are angry and a half an hour later you are burning in lust.

These three phases or sides of your mind feeds the craving you have for life,to be born again,to live forever,to chase the mirage of happiness.

This is bad because it's these three phases that drives a being further into "Samsara".

These 3 has brought you here and they will drag you through the dangerous lower realms to feed on innocent little animals,to be be hunted by lions and tigers,to be eaten alive and even to be tortured and burned in "hellish" realm. This is the true danger that a being face in Samsara. Because once there in a bad realm there is virtually no hope for a poor being.

And what about the good realms.....

Yes, those 3 sides of your mind also took you to "Deva" realm,to "Brahma" realm and here to "Human" realm. Not so bad you might say,won't you?

Well lets see. you have been born so many times in this samsara. Our Lord Buddha said "Any being has been killed so many times.If someone collect all the blood that fell when he or she was decapitated,the amount of blood will be roughly equal to the amount of water in the sea."

Look at that old person next to you, after listening to the words of lord Buddha can you certify that he or she next to you has not being someone you loved?

Well lets see this in another way. We know what animals are,they are the poor beings who lost their way and ended up in a bad realm.

When you go home and eat a delicious meal like a fish curry or sausages,look at it closely. Imagine a one you had. Do you see your father or mother? How about that girl you promised to live eternity with in another life, don't you see her?

The sad fact is you do. Now you are eating them so greedily,just like they ate you. Look at this bizarre circle that we roam around from good realms to bad, from bad to good.

One day Lord Buddha smiled and there was Ven. Anandha thero nearby. As a smile from a Buddha is rare the venerable one asked why. Lord Buddha gave a strange reply, if you focus on this statement alone you can understand the true reason to practice "Mindfulness" and the true danger of Samsara. The answer was "Look Anandha, look at these ants here. There isn't a single ant in this long row of ants that hasn't been the king of earth".

Remember these words from Lord Buddha,

"Vinnana" is like a seed, attachment or craving is like water and Karma is the land.

Because your mind wander and get attached it creates Karma. Whatever you get attached to,you soon start to crave for it. There it is The land,the water and the seed, because of this you are born and will be born.

Practice to focus on the temporary nature of things and use your knowledge to see and feel the world not your body. practice to keep away from the 3 phases mentioned above.

Mindfulness Lord Buddha taught was keeping mind from those 3 phases mentioned above and maintaining your life and the very second you live in on 8 noble ways.