My initial internal reaction was to try to convince her that my pedagogy was sound, that it would indeed be better for her long term to struggle and make sense of novel situations, apply and stretch herself, learn how to tinker and problem solve rather than regurgitate algorithms repeatedly, but I felt that this would be minimizing her experience and negating her sense of her learning and mathematical identity. She had clearly stated that things make sense to her after she is given a method and does a lot of similar problems - only then does she believe that she is able to generalize and form an underlying concept. This isn't how our program is designed and I absolutely believe that it is better for most students to experiment and play first, forming conjectures and identifying patterns before coming to or seeing more formal methods (if needed), but maybe it's not better for her. At the very least, if she is convinced that this is the wrong way for her to learn, then it will be very difficult for her to interpret her experience otherwise, thus creating a self-perpetuating cycle.
So I'm trying something new, and I'm not sure how well it's going to work. Every week, I'm going to email her a list of concepts that we will be working on next week, along with resources either in the textbook or online for her to see these concepts explained and practice problems for her to work on. A preview, if you will. Class will then not be a time for her to explore and invent, like it is for other students, but a time for her to generalize and prove the patterns that have already been revealed and practiced. In exchange, she has agreed that in a few weeks, she will again try exploring a new topic and be open to coaching by me in order to also get better at this way of learning.
I'm hoping that by engaging in good faith, I am able to bridge the divide in expectations and meet this student at her current level of need and that she is able to grow over time in the mathematical habits of mind that I believe are just as important as, if not more than, content knowledge. It is certainly possible that she will continue preferring doing math in predictable and routine ways, following a pattern shown to her by someone else, on mathematical autopilot. I really hope that I can convince her that she can be successful and that it's worthwhile to engage in math in a different way than she has in the past. But it's okay if that's not where she is right now. I have a whole semester to build a relationship of trust and forment and celebrate moments of mathematical success for her.
Have you had students who actively and eloquently resisted your view of math or ways of teaching? What are some ways that you've made progress over time in their willingness to go there with you? Are there students who never changed their minds? Any and all advice welcome, as always :)