How to update MongoDB documents with arrays of sub

I am working with a MongoDB database whose collections model classes, students, subjects, and [academic] performance. Below are the Mongoose based schema and models:

var mongoose = require('mongoose');
var Schema = mongoose.Schema;
var ObjectId = Schema.Types.ObjectId;

// SUBJECT collection
var subjectSchema = new Schema({
    name        : { type: String, required: true },
    category    : { type: String, required: true }
});
var Subject = mongoose.model('Subject', subjectSchema);

// STUDENT collection
var studentSchema = new Schema({
    name        : { type: String, required: true },
    grade       : { type: Number, required: true }
});
var Student = mongoose.model('Student', studentSchema);

// PERFORMANCE collection
var performanceSchema = new Schema({
    subjectId   : { type: ObjectId, ref: Subject.modelName, required: true },
    score       : { type: Number, required: true },
    maximum     : { type: Number, required: true },
    grade       : { type: String, required: true }
});
var Performance = mongoose.model('Performance', performanceSchema);

// *Note*: This is just to use as a sub-document schema, and not as a collection
var classStudentSchema = new Schema({
    studentId   : { type: ObjectId, ref: Student.modelName, required: true },
    performance : [performanceSchema]
}, { _id: false });

// CLASS collection
var classSchema = new Schema({
    name        : { type: String, required: true },
    scores      : [classStudentSchema]
});
var Class = mongoose.model('Class', classSchema);

The class collection's documents are the most complex of the lot; an example document would be:

{
  "_id" : ObjectId("57758f15f68da08c254ebee1"),
  "name" : "Grade 5 - Section A",
  "scores" : [{
    "studentId" : ObjectId("5776bd36ffc8227405d364d2"),
    "performance": [{
      "subjectId" : ObjectId("577694ecbf6f3a781759c54a"),
      "score" : 86,
      "maximum" : 100,
      "grade" : "B+"
    }, {
      "subjectId" : ObjectId("5776ffe1804540e29c602a62"),
      "score" : 74,
      "maximum" : 100,
      "grade" : "A-"
    }]
  }, {
    "studentId" : ObjectId("5776bd36ffc8227405d364d5"),
    "performance": [{
      "subjectId" : ObjectId("577694ecbf6f3a781759c54a"),
      "score" : 94,
      "maximum" : 100,
      "grade" : "A"
    }, {
      "subjectId" : ObjectId("5776ffe1804540e29c602a62"),
      "score" : 81,
      "maximum" : 100,
      "grade" : "A"
    }]
  }]
}

I was able to retrieve an existing class document and add a student to it's scores using the following code:

Class.findOne({ name: 'Grade 5 - Section A' }, function(err, class) {
  if (err) throw err;
  Student.findOne({ name: 'John Doe' }, function(err, student) {
    if (err) throw err;
    class.scores.push({
      studentId: student.id
    });
  };
});

But how do I add/update/delete that particular student's performance? I need to be able to interact with the class collection in the following ways:

Retrieve the scores for all students, or for a specific student (retrieve specific element in the scores array)
Add/ update/ delete a specific student's score, for a specific subject (in case of update or delete, retrieve a specific element in the scores[n].performance array; for add, append to the same array.

There are a few way to do this I'll answer point by point

Retrieve the scores for all students, or for a specific student (retrieve specific element in the scores array)

Class.findOne({ name: 'Grade 5 - Section A'})
     .populate('scores.studentId')
     .exec(function(err, class) {
       if (err) throw err;
       //now class.scores.studentId becomes ObjectStudent
       //hence you have all scores for all students
});

Add/ update/ delete a specific student's score, for a specific subject (in case of update or delete, retrieve a specific element in the scores[n].performance array; for add, append to the same array.

Class.findOneAndUpdate({name: 'Grade 5 - Section A'
                        ,'scores.studentId': ObjectId('5776bd36ffc8227405d364d2')
                        , 'scores.performance.subjectId' : ObjectId('577694ecbf6f3a781759c54a')}
                        , {$set: {scores.performance. score: 50}}
                        , function(err, data) {
           if (err) throw err
    });

I hope that helps