I was asked a while ago to write a blog for the National Council on Teacher Quality. I agreed and here it is. This is also posted on their website nctq.org/commentary/blog. This is re posted here with their permission.

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.

— John Benson