Does it have to be one or the other? Its hard to argue that both relationship and rigor are both necessary characteristics for a warm learning environment that breeds growth. I was talking to one of my colleagues last week who was debating how long she should hold off on doing "math" so she could start the year with better relationships. She was feeling a little stuck as far what to do on the second day of school after she had already introduced her procedures and met her students on the first day.
I told her to go ahead and begin "math" operations, but to engage the kids in a game or competition. She would still have time to get to know her students, but would also set a tone for class engagement.
I think her conundrum was part of a problem with which we sometimes see our role as teacher. We split our job into two separate roles as "instructor" and "mentor," so we view some of our work as drilling and filling kids with knowledge, and then other times as when we do "relationship." Relationship is only genuine if its an ongoing process. I think we've all had those students that during "work" time they drive us crazy, but during "relationship" time, you really enjoy. If you really value relationship building in your classroom, then everything you do should be an opportunity to build on those, not just at the "fun" times.
Here's a few posts I've written related to the subject:
“Never smile until 2nd quarter.” Ever heard this advice? You can be firm and set a good tone for the year without being cold and unemotional.
Understanding and accepting your students’ nature - http://mrcbaker.blogspot.com/2014/07/your-students-will-never-be-perfect-and.html
Goal-setting with your students allow you to get to know them better and share your own experiences.
Treat your students like “real” people by offering a mobile device charging station.
Have your students pace and lead your lesson - resist the temptation to get through a certain number of problems. Quality examples and practice over quantity.
Put your students in your word problems. Even fictional scenarios can gain a kernel of relevance if your students are part of the story.
Transparency in your grading is important for students and parents. Making your standards “kid-friendly” will help your students in goal-setting and reconceptualizing your standards for your students will help you understand them more, too.