According to Diane Boehm, Instructional Support Programs, Saginaw Valley State University, the following are benefits of writing to learn: (easily adaptable to homeschooling)
1. WTL promotes active learning.
2. Students use their own language to understand course concepts; they "own" rather than "rent" the language and ideas.
3. WTL stimulates participation and discussion (/every/ student has a response to the question).
4. Teachers discover what students are thinking and learning, what's clear and what isn't.
5. WTL creates opportunity for teacher/student and student/student dialogue.
6. Students can "rehearse" ideas and strategies before tackling formal writing assignments; they can "practice" before the "big game."
7. WTL creates a way for students to reflect upon what they are learning, to think meta-cognitively and personally about their learning processes in the course.
8. WTL assists students in discovering what they know and what they don't know.
9. WTL gives everyone a stake in the class.
10. WTL can be adapted for whole group or small group activities.
11. WTL creates opportunities to write for audiences other than the teacher.
12. WTL allows for formative (assisting in the process) rather than only summative (evaluating a product) assessments.
Maybe helping your student see how writing to learn will benefit them, learning to write will not be as big a chore for them.