Today we find the ecclesiastical Christian institutions assembled in such a way that provides the means for people to be seen by others. We can go into a church and show all our neighbors and virtually everyone in our community that we are religious.
Does this accomplish anything? According to Jesus, it has the opposite effect.
The word "righteousness" is being translated from the Greek word δικαιοσύνη (dikaiosynē) which means, according to the lexicon, "in a broad sense: state of him who is as he ought to be, righteousness, the condition acceptable to God."
This means that the purpose of "righteousness" is to act in a way that is pleasing to the Supreme Being.
What does this have to do with showing off to others? Nothing.
The physical world is populated by those who are self-centered. It is all about me, myself and I. We can also include our family. And our dog. And our company or school or country.
But self-centeredness is the opposite of love.
What does Jesus mean by "they have received their reward in full"?
The reward they are seeking is they want the respect of others. So they pray in public or act religiously in front of others so others will honor them and give them respect. And because this is the reward they want form that activity, they indeed receive that reward by their public activity.
This is opposed to the reward that Jesus wants his students to have as a result of their dedication to the Supreme Being.
Jesus wants each of his students to re-establish their personal loving relationship with the Supreme Being.
But to love someone means to care more about them than we care about ourselves. This is true love. And pleasing the one we love is only natural.
This mood - of wanting to please God - is the opposite of trying to impress others with our religiousness. The two are mutually exclusive. They have nothing to do with each other.
We can't have it both ways. We can't try to impress others and exchange a loving relationship with the Supreme Being. It is one or the other, as Jesus states above.
Doing what is pleasing to God means having a personal relationship with Him. It means dedicating and sharing our personal lives with the Supreme Being.
It does not mean trying to use God to impress others with how great I am.
Just consider how this might play out among friendships of this world. Let's say that we have a popular friend and we decide to do something for them in front of everyone, showing everyone how tight we are with this friend. How will the friend feel about this?
He will feel used. He will feel insulted that we tried to use our friendship with him. And he will feel that the thing we did for him was not sincere. It was just an act.
Why should the Supreme Being feel any different about us showing off our religiousness among others?
(For a translation of Jesus' statements from the Gospel of Matthew without institutional sectarian influence, see the Gospels of Jesus - translated from the original Greek texts.)