Создание таблиц

class Predmet < ActiveRecord::Base
  has_many :pudatas
  belongs_to :specialnost
end

class Uchitel < ActiveRecord::Base
  has_many :pudatas
  validates :imya,:familiya, :otchestvo, :presence  => true
end

class Pudata < ActiveRecord::Base
  belongs_to :predmet
  belongs_to :uchitel
end

class Specialnost < ActiveRecord::Base
  has_many :pudatas
end

specialnost.pudatas.each # ...

class Predmet < ActiveRecord::Base
  has_many :pudatas
  has_many :uchitels, :through => :ppdatas
end

class Uchitel < ActiveRecord::Base
  has_many :pudatas
  has_many :predmets, :through => :ppdatas
end

class Pudata < ActiveRecord::Base
  belongs_to :predmet
  belongs_to :uchitel
end

teacher = Teacher.first
subject  = Subject.where(name: :mathematic).first
teacher.subscription << subject
teacher.save

class Subject < ActiveRecord::Base
  has_many :subscriptions
  has_many :teachers, :through => :subscriptions
end
 
 
class Teacher < ActiveRecord::Base
  has_many :subscriptions
  has_many :subjects, :through => :subscriptions
end
    
 
class Subscription < ActiveRecord::Base
  belongs_to :subject
  belongs_to :teacher
end

# SubscriptionController#create
# ...
@subject.bind(@teacher)
 
# Subject model
 
def bind(other_teacher)
  subscriptions.create!(teacher_id: other_teacher.id)
end

<%= render @subscriptions %>