Should Quizzes Count?

As I passed back our most recent chapter assessment, I struck up a conversation with one of my Algebra 1 students.  When she looked at her scores (I give each problem a score which is tied to a standard rather than one overall score), she noted that she had never received such high scores in math.  This student has said that after every chapter test usually in amazement at her success.  This last time I decided to sit down and have a chat with her about two things: my confusion of her lack of confidence and her continued disbelief in her math ability.  She has passed every essential standard on every test with a proficient or advanced score, usually an advanced score (based on our department expectations). 
I took a seat next to her and we looked through her assessment.  I asked her some questions about how she solved some of the problems, how she chose strategies to use, why she solved problems the way she did.  These  questions proved what I already knew, this is a strong math student who knows what she is doing and why she is doing it.  In my eyes math makes sense to her. 
So then I asked the burning question I was really wondering: "How is math different for you this year?"  She sat and thought for a minute or two.  While she contemplated the question, I considered why I asked this and what I was expecting to gain from her response.  Since I am in my second year of implementing standards based grading, I am conducting action research in my classroom constantly.  I was hoping to gain some insight on how/if standards based grading had anything to do with the improvement of her math grade.  As it turns out, the implementation has had an indirect effect on her improved grade.
She told me that the biggest thing for her was our weekly formative assessments (quizzes) and the immediate feedback on classwork that had enabled her to practice, learn, make mistakes, identify her misconceptions, and then allow her to re-learn the things she struggled with without being punished by a failing grade.
"Mrs. Beck, you record our standard scores in the practice phase so that we can track our progess, understanding, and misunderstanding but those scores do not count against our grade AT ALL.  When it is time for the chapter test, the scores I receive on that are the scores I have in the class.  So, this year I am not stressed if I do not know how to do something, I just know that I need to get help and practice more.  I am able to focus on learning rather than focusing on not making mistakes.  The mistakes I make while practicing allow me to adjust and build a lasting understanding of algebra 1.  Last year I was so stressed about not making mistakes that I couldn't focus on learning the math standards."
When my winter break ends in a week, I know that I need to bring this issue up with my department to discuss if/how quizzes should be counted.  The most common comment I hear from my colleagues is that if they do not "count" the quiz, then the students will not take it seriously.  So, when I hear that comment my first question will be: what is the purpose of your quiz.  For me, the purpose of a quiz is to see where my students are with their understanding.  There has been more than one occaision that I thought my students had a concept down up until I gave them a formative assessment on the standard and was proven wrong.  Thankfully, my students will write me notes and let me know they have no idea what/how to do something because there is not the punishment of a grade looming over them.  They readily admit their lack of understanding knowing that there will be help and support to get them where they need to be.
I only hope I can convince the math departement that fear of a failing grade will not motivate students to learn and be successful math students.  It will be a conversation that will continue for many months with many uncomfortable moments.  But it is a conversation that I will start.  There will be many challenging questions asked from both sides and the parody will enable all involved to grow. 
I apprreciate any suggestions and/or advice you have to share!