During my forty-two years of teaching high school mathematics in Evanston, Illinois, I concluded that an essential ingredient for providing quality learning is that the teacher be well versed in the subject that the student is learning as well as the content that comes before and after the subject being learned. This may sound obvious, but it often happens that teachers have mastered what is in the textbook they are using without having knowledge far beyond. I believe this greatly inhibits their ability to help students make connections and often such teachers make poor choices about instruction because they fail to see the entire picture.
I taught a two semester Algebra 1 class, Empirical Geometry, Mathematics--A Human Endeavor, as well as Traditional Euclidean Geometry, Trig, Calculus, Multivariable Calculus and Linear Algebra. I found deep knowledge to be useful at all levels, all the time.
A teacher who has mastered the material well beyond the course being taught will understand why certain topics are presented the way they are and will anticipate what's next. A less prepared teacher will emphasize tricks and shortcuts that will get the students through Friday's test, but leave them ill-prepared for future courses. For example, a student who learns to multiply binomials using FOIL (First, Outer, Inner, Last) instead of the distributive property of multiplication over addition may do well on the problems involving multiplication of two binomials, but will be hopelessly confused when multiplying more than two, or when one of the factors is a trinomial.
Part of good teaching involves understanding the importance of what is being taught and how it can be applied. Sometimes application of the content does not come until the student studies physics, or calculus, but a teacher who is not well versed in those subjects will not understand their importance. For instance, a teacher who is not familiar with Linear Algebra will not understand the importance of row-reduction of matrices and probably will not present it as the tool of choice for solving systems. In fact, many of those teachers will never ask their students to solve two equations with three unknowns because they do not see the big picture, limiting their students.
I have also observed that students can have remarkable insights into the subject at hand, but those insights may not be well formed. A teacher with deep content knowledge will be able to see the gem the student has noticed and clarify it for the rest of the class. A less prepared teacher will not. I found that by giving students a problem and walking around observing their work, I could find the teachable moment for the concept I was trying to teach, and I could make intelligent use of student work in bringing that moment to life in the class. This would have been very difficult if I was not confident in recognizing good and bad mathematical work.
To ensure that our students receive a rich math education rather than a string of rules, I think we should move forward by insisting that certified math teachers know a lot more mathematics than what they will be expected to teach and that they know it well.