Matthew 6: 3...let not your left hand know what your right hand doeth.

What is the meaning?

Don't be a pharisee and do good to be seen.

Don’t give cognitive thought to helping just do it and move on, God will reward that which was done in secret.

Mat 6:1 Take heed that ye do not your alms before men, to be seen of them: otherwise ye have no reward of your Father which is in heaven.

Mat 6:5 And when thou prayest, thou shalt not be as the hypocrites are: for they love to pray standing in the synagogues and in the corners of the streets, that they may be seen of men. Verily I say unto you, They have their reward.

Mat 6:16 Moreover when ye fast, be not, as the hypocrites, of a sad countenance: for they disfigure their faces, that they may appear unto men to fast. Verily I say unto you, They have their reward.

Mat 23:3 All therefore whatsoever they bid you observe, that observe and do; but do not ye after their works: for they say, and do not.
Mat 23:5 But all their works they do for to be seen of men: they make broad their phylacteries, and enlarge the borders of their garments,
Mat 23:12 And whosoever shall exalt himself shall be abased; and he that shall humble himself shall be exalted.

Joh 12:42 Nevertheless among the chief rulers also many believed on him; but because of the Pharisees they did not confess him, lest they should be put out of the synagogue:
Joh 12:43 For they loved the praise of men more than the praise of God.

Do not give glory to yourself, exalting yourself, and wanting people to think you are a great person of God for following Him, when all glory goes to God.

God said He does not give His glory to another.

And if you think you are something when you are nothing you deceive yourself.

And we can do nothing without Christ.

And who makes you to differ from another, and what do you have that you did not receive, and if you did receive it why do you glory as if you did not receive it.

Paul was given a thorn in the flesh to keep him humble so he would not exalt himself, for God resists the proud, but gives grace unto the humble.

If we want people to think we are someone special, especially over other people, and exalt ourselves, then we have our reward for that is behavior of the world that likes to brag and boast about themselves in arrogance.

All glory goes to God, and it is only because of Him that we receive anything spiritual, and not of ourselves, for the natural man does not understand the things of God, neither can they know them, but God revealed the kingdom to the saints by the Spirit.

So how can we brag it if is all comes from God, and think we are something special to exalt ourselves.

Paul said he is the least of the least of the saints, and knew not to exalt himself, and he labored great for the kingdom of God, and endured much hardship, which God said he shall suffer many things for the kingdom's sake, so who are we to brag, and think we are special.

We are special but not in a bragging way on our part, for glory to God.

When you tithe or give, do it discretely so as to not be notices by men and gain their praise. Rather seek the approval of God who sees what you do in secret and will reward you for it. If you drop your money into the offering box with your left hand, do it so discretely that if your right hand had eyes, not even it would see your offering.

This can be applied to integrity in the work place also:
Eph 6:5-6 “Servants, be obedient to your masters, with respect, in the sincerity of your heart as to Christ; not by way of eyeservice, as men-pleasers, but as servants of Christ doing the will of God from the heart.”

Really it applies to all service

The right hand is a symbol used in biblical language when someone does the right thing.

The left hand.

Jesus was kinda pointing out that those making a show of doing the right thing were doing it with their left hand.

The left hand, is works that do not have the right motive, purpose or intent. Therefore ,Jesus is saying, do not DO your good things to seek praise, ( that would be a left hand motive) and do not publish or tell of your good deeds, so as to have the praise of men, another left hand motive, but rather in secret, and your Father who sees in secret will reward you openly.

The meaning_ Did you read before and after this teaching_ Do not do your good works for man to see you for in so doing you have your reward. Instead do your works in secret so that only the Father sees you, and you will be given reward for all to see come th Kingdom.. Meanwhile give all glory to God always, amen.

It has also a touch of spontaneity (rather than planning) in our giving

This'll only work if you are in your right mind.

That's difficult for the left-brained folks.

The second sentence is biblically sound, but the first is eisegesis. Absolutely nothing in the text supports that interpretation.

the right half of the brain generally controls the left half of the body, and vice versa.
why do you suppose God made us this way?

((yes you could technically say only left-handed people are 'in their right mind' and no i am not left handed))

Your right about that, motive is in the mind not in the hands, rather using a left or right hand both are able to do the wrong thing if the mind chooses it.