Department policy possibly affecting inclusion and diversity I am a last year PhD math student in an university in USA-Canada. 

The university I work in offers graduate students the option to teach first year calculus courses as a special kind of TA. The demand for this is high and not everyone gets a position for a given term. However, the department's policy is that everyone who passes a one-term course on teaching and wishes to teach can do it at least once before graduating.

So far so good. 

The problem arises when it comes to wanting to teach a second time. It seems like whatever metric is being used to decide who gets the job heavily favours a specific ethnic group. This is, the vast majority of students who get to teach a second time are from a specific ethnic group, while  people from other less represented groups seldom seem to get a second chance.

Clearly the department's argument for this is that these people get better teaching evaluations, but in part this is an echo chamber resulting in the exclusion of members of other less represented ethnic groups in the student body.

The two immediate problems that come to mind are: 


Reducing the diversity of teachers for a very diverse base of students. 
When it comes to job applications teaching experience can be a valuable asset. Access to it shouldn't depend on our ethnic background.


My (perhaps naive/biased/egotistical) conclusion is that the department's criteria is flawed and it should be modified in some way to reflect this in the future. My questions are:


Who should I talk about this first? I think the best idea is to talk with the person in charge of assigning the positions, who inherited the criteria from the previous person in the position.
How should I approach this person? I think this can be treated as a potential opportunity to improve the department, but I feel like I need to be careful on my approach so that no one feels threatened or accused of anything.
What should I expect? Let me be clear: I don't expect to get a teaching position since I'll be graduating soon, but I'd like this issue to stop and I'm willing to make some noise and bother some people if necessary.
It's an interesting problem, and wanting a fairer representation is a noble goal. But the goal of fair representation does need to be balanced against another goal: giving students a good education. The first goal of having TAs is to help educate students, not to provide people a career stepping stone to the TAs.


  The problem arises when it comes to wanting to teach a second time. It seems like whatever metric is being used to decide who gets the job heavily favours a specific ethnic group. This is, the vast majority of students who get to teach a second time are from a specific ethnic group, while people from other less represented groups seldom seem to get a second chance.


There's a lot of uncertainty here. If you want to successfully challenge your institution on the facts, you need more facts. 

If you go to them and say "you seem to hire a lot of X", they can just say "well it may seem like that to you but we don't recognize ourselves in that feeling". If you come and say "group X is represented as 20% of first term TAs and 70% of second term TAs", it's a lot harder to turn you away at the doorstep.

But make sure you get your ethnic classifications right. For example, a pale ginger guy from Argentina with Spanish as a first language, would you classify him as Caucasian or Latino?

The other thing you need to know is what metric they actually use. If you try to argue that their decision-making process is flawed without knowing how they actually make their decisions, you're not going to achieve much.

How are TAs evaluated? Professors are typically evaluated at least partially on feedback forms. But I don't often see specific TA feedback forms. Do other departments/universities do it differently (and do they have a different/fairer ethnic spread)? Can you point to a best practice that your institition could adopt?

The article linked by @PatriciaShanahan points out the value of blind auditions. Once feedback from students has been acquired, how is it processed? To find that out you're going to have to talk to some insiders in your institution. I would recommend finding some progressive-minded professors and asking them if they can explain to you how the internal process actually works. Not accusing them: just asking to explain.


  Clearly the department's argument for this is that these people get better teaching evaluations, 


Sure.


  but in part this is an echo chamber resulting in the exclusion of members of other less represented ethnic groups in the student body.


Well, maybe. What if this ethnic group really has a property that gives them an edge in teaching? For example, they speak the local language better than other ethnic groups (such as foreign exchange students). If, as a student, I have to rate two TAs and one is easier to understand, and therefore I learn more from that TA than from the other one, is it really unfair that the first one gets a better evaluation?


  My (perhaps naive/biased/egotistical) conclusion is that the department's criteria is flawed and it should be modified in some way to reflect this in the future. 


I think your conclusion is logically flawed. You don't like the outcome of the process, therefore you assume the process must be bad. But what if the process is okay, and the favored ethnic group really did have some advantage that makes them better TAs? 

Are students better served by having the best TAs, or the fairest ethnic spread of TAs?

So yeah, your reasoning is noble, but naive. That said, a system which seems to produce ethnically biased outcomes should be viewed with suspicion. So you need to dig deeper into the causes, before you try to convince people to change it.


  Who should I talk about this first? I think the best idea is to talk with the person in charge of assigning the positions, who inherited the criteria from the previous person in the position.


Asking them is a possibility. But I think I'd start with a progressive-minded professor who'd been there for a couple of years already. The professor would have some idea of what happens behind the scenes, but not be as directly challenged, and might become interested themselves. Could result in you gaining an ally for your cause.


  How should I approach this person? I think this can be treated as a potential opportunity to improve the department, but I feel like I need to be careful on my approach so that no one feels threatened or accused of anything.


In the beginning I would focus on gaining information, not on insisting on change. You don't have strong enough evidence (yet). Ask how evaluations are done. Is there any broader analysis of evaluation results, that give you a picture of the ideal TA? For example, if students mention "good English" as an important criterion, you can look at your non-favored TAs and ask if there's a noticeable difference in their English. 

Another question to ask is whether TAs receive any kind of training to prepare them for the job. Does the training match the evaluation criteria? Maybe the training doesn't contain something that already comes naturally to the favored ethnic group. So by evaluation time, they do fine on that criterion, but others struggle because nobody taught them.


  What should I expect? Let me be clear: I don't expect to get a teaching position since I'll be graduating soon, but I'd like this issue to stop and I'm willing to make some noise and bother some people if necessary.


That's going to depend on a number of factors, such as how progressive your institution is/wants to be, and what the reasons are that this ethnic group seems to be doing better than the rest, and on how diplomatic you are/whether you manage to find the right allies.



TL;DR - 


You need more facts about ethnic representation. Numbers.
You need to know more about how the evaluation is done, what is it testing and how.
You need to find out why other ethnic groups are being evaluated worse. Look for causes, don't just complain about bad people with biases.
You need allies on the inside.
Remember that the institution can always say "our primary goal is to give our students the best education, so we prioritize having the best TAs over having the most diverse TAs". And they're not entirely wrong. You need to find out why more ethnically diverse TAs are being rated as "not the best".