Essentially, Bartra should be 3rd choice centerback ahead of Mathieu because, unlike Mathieu, he doesn't freak out when he has the ball at his feet.
The trick is that former coaches haven't trusted to Bartra too much (and he did make some horrible mistakes in key matches). I expect now:
-- but he was young
-- he hasn't played and similar answers
Ok, but the thing with Bartra is:
1. he played some bigger matches in the past and he did some mistakes in those matches (for example, when he asked Tito to take him out of Bayern game because he was scared, even though we hadn't any other available options. Plus, mistakes like against Bale. Or when he is sometimes easily caught out of a position against bigger teams)
-- ok, people will again say: he need to play more to improve in that area.
-- true. Maybe he will improve.
The point is that till now he made mistakes against bigger teams, and that since then, he almost ALWAYS plays only against a very crappy teams
-- and he is more or less good versus crappy teams, because:
1. those teams are crappy
2. or we don't need to defend that much
3. if he makes mistakes, crappy teams can't punish us and score goals
He don't get too many opportunities against bigger teams.
And when he does, it is either on Camp nou, or in cup matches (Atletico), or when all other options are injured, like currently.
For example, a list of Bartra's matches this Season:
http://www.soccerbase.com/players/player.sd?player_id=53395
-- based on the current La liga table, this is the rankings of the teams, against whom Bartra played:
Home matches:
-- 15. Elche
-- 18. Levante
-- 19. Granada
Away:
-- 7. Malaga
-- 11. Rayo
-- 14. Eibar
-- 15. Elche
-- 16. Almeria
-- 17. Deportivo
-- 19. Granada
-- he did play in a few tougher matches like:
-- Champions league (Ajax, Ajax, Apoel, Apoel, Psg home)
-- and all cup matches, plus cup match against Atletico at home
Now, let's check matches where Mathieu played:
http://www.soccerbase.com/players/player.sd?player_id=32238
Home:
5. Villareal
6. Sevilla
7. Malaga
8. Bilbao
9. Espanyol
10. Celta
11. Rayo
14. Eibar
15. Elche
19. Granada
Away:
2. Real Madrid
3. Valencia
5. Villareal
8. Bilbao
11. Rayo
12. Sociedad
13. Getafe
18. Levante
19. Granada
Champions league:
-- Psg, Psg, Man. City
Cup:
-- Atletico away
-- Villareal home
-- Villareal away
So, let's do some maths and stats:
Bartra's matches against teams from Top10 of La Liga:
-- Home: -
-- Away: Malaga (7)
-- Ajax, Ajax, Psg (Champions league)
-- Atletico (4) cup
= 5 matches in total
Mathieu's matches against strong teams and Top10 La Liga teams:
-- Home: Villareal (5), Sevilla (6), Malaga (7), Bilbao (8), Espanyol (9), Celta (10)
-- Away: Real (2), Valencia (3), Villareal (5), Bilbao (8)
-- Champions League: Psg, Psg, Man. City
-- Cup: Atletico (4), Villareal (5), Villareal (5)
= total 16 matches
So:
1. Mathieu hasn't done any horrible mistakes, right?
2. bigger matches with Mathieu:
-- Villareal 1:0
-- Bilbao 2:0
-- Psg 2:3
-- Real 1:3
-- Celta 0:1
-- Sevilla 5:1
-- Valencia 1:0
-- Espanyol 5:1
-- Psg 3:1
-- Atletico 3:2
-- Villareal 3:2
-- Bilbao 5:2
-- Villareal 3:1
-- Malaga 0:1
-- Man. City 2:1
-- Villareal 3:1
= total, 12 wins, 4 defeats
-- so, it's not like that we play bad with Mathieu, right?
3. Mathieu has more experience and he is more or less=a proven La liga defender
4. More or less, he has proven even this Season in Barca that he can play in matches against Top10 teams and that we will win most of those matches
5. he played more, so he has a better chemistry with other teammates, right?
Bartra:
1. he is young, La Masia product, loves Barca
2. in the previous Seasons, hasn't convinced till now that he is exactly ready for the biggest matches
3. This Season he played mostly against the weakest teams
4. he played only 5 stronger matches
-- in La Liga, against Top 10 sides, Lucho played Barta only once (!!) till now against any Top10 team (Malaga away 0:0)
So, honestly, how EXACTLY is now a moment to put faith in Bartra in key moments of the Season, when he didn't play bigger matches nor in previous Seasons, nor in this Season (only 1 bigger La Liga Match in 27 rounds)
All numbers and stats indicates that Mathieu is the 3rd option and the safest, the most logical option in the final weeks of the Season.