What should teachers know in order to teach? (Mathematics)

02 Aug

Do you ever wonder what training your child’s teacher gets to become an educator?  What do colleges require for teachers to earn their teaching credentials?  What do states require to hire a teacher into their school system?  Why do we seem to have such a small number of truly exceptional teachers?  Have you ever gone to or ?  What do these sites say about our teachers?  Some report, “avoid this teacher at all costs.”  Sometimes might say, “you can get an easy A but you won’t learn anything about the subject matter.”  What do we want in a teacher?  Parents may want different things than students.  Some students may not actually care about learning the content, they only care if they get a good grade.  Some parents feel that way too.  There are some private schools where you basically pay for this option, give the school your money, they will make sure the classes are easy enough for students to get A’s and B’s.  Don’t get me wrong, there are plenty of private schools that do not work this way but I have seen a few (and I am sure there are more) that this seems to be the approach when I compare their class expectations and work to public or charter schools.  The bottom line is that “good grades” becomes a desire of parents and students and this sometimes can trump good teaching, which is unfortunate.

Taking that out of the equation for the sake of this blog, let’s assume all grades are equal.  What skill set should a teacher have?  The first answer is usually that they should know the knowledge area that they are teaching.  That seems obvious but the problem with this requirement is that it can be weighed so heavily that many states and schools will allow that to be the only qualification.  This is called “Lateral Entry” and I see it all the time in job postings.  If a person hasn’t earned a degree in mathematics or science education but they have a higher degree (masters or Ph.D.) in the field, they are allowed to join the work force as a teacher despite not having any formal education as an educator.  They may or may not have to take a state mandated test.  These tests can be in both content and pedagogy.  However, the graders of the pedagogical tests seem to accept a wide range of answers for a “pass” since it doesn’t seem to be a stumbling block for lateral entry candidates to get hired.  Besides, in general, the pedagogical tests don’t test that you know alternative approaches to teaching your subject such or effective ways to reach non-auditory learners or students with learning disabilities.  To be honest, you jot down some B.S. and your done.

Most of our teachers, however, do go through the traditional route of getting a true certification in education.  Let’s look at my own experience.  To become certified as a 9-12 math teacher, I had to meet all the requirements of a Bachelors of Science in Mathematics (which were quite rigorous at my particular college compared to others – I needed to take courses such as Calc 1, 2, 3, Differential Equations, Abstract Algebra 1 and 2, Real Analysis 1 and 2, Topology, Statistics 1, 2, plus computer programming classes, Discrete Math, and probably some others I can’t even remember.  Most schools do not require that much mathematical “theory;” for those who don’t know what that means think about your proofs from high school geometry but much more difficult.)  Additionally, I had to take a methods class.  This methods class was to prepare me for the actual teaching!  This should have been THE SINGLE MOST IMPORTANT class I took.  Why was there only ONE class that would show me HOW to teach math when clearly I can do math and all my teaching was going to be at a level far lower than anything I took for my Bachelors except maybe Calculus?  So, what did I do in that one methods class?  Well, not much.  We learned a format for how to write lesson plans.  Learning how to “format” lesson plans doesn’t really teach someone how to teach.  We were given assignments where we wrote 2 lesson plans.  That’s it, 2 lesson plans!  We explored some hands-on manipulatives such as base 10 blocks and Algebra tiles and we did some content work at the high school level to practice our lower level high school skills (we practiced from the NY Regents exams).  This is what was supposed to prepare me for teaching.  I also had to take a course on the history of mathematics, somehow that also would make me a better math teacher.  Finally, you must do student teaching.  I ended up waiting until grad school to do my student teaching.  This is your “on the job training.”  You are given the content material of the lesson you are supposed to teach, then you teach it and your cooperating teacher and college supervisor watch you and provide feedback.  In general, you can’t fail at this.  You may discover that teaching is not for you if things don’t go well.  In my case, I was lucky, since it was through grad school, we taught summer school and were the ONLY teachers (a group of us).  We were videotaped and then watched our videos and we all critiqued them together.  As a doctoral student, however, I worked as supervisor of the more traditional student teachers who took over for a classroom teacher for a semester and I provided the feedback along with the cooperating teacher so I got to experience student teaching in this fashion as well.

If you got lost or your eyes glazed over in the last paragraph, let me summarize:  there was very little done to prepare a teacher for teaching.  It wasn’t done for me and it continues to not be done for new rising teachers.  It seems that you either have the natural ability to figure out how to effectively reach students or you follow the same traditional path of how many of you were taught: students sit and listen while teacher lectures, teacher provides a lot of  basic procedural steps or possibly conceptual theory at a level that is too far above a student’s level and students get lost and teacher doesn’t even realize it.

So, what are the qualities of a good teacher?  What information should be imparted to all students learning to be teachers and all teachers taking Continuing Education Credits?  Here is my list for mathematics.  I would have different lists for other subjects although some of this can be generalized:

1.  Know your content (although sometimes those who were the strongest students in school can make the worst teachers because they don’t understand how students can’t just “see” the solutions like they can.)

2.  Know and read your students (you should know how each student is doing in your class at all times, during lectures you should be able to “read” when you have lost the class and make adjustments, this isn’t about YOU, it is about the students.)

3.  Be able to represent the material in multiple ways for visual and auditory learners (don’t make the assumption that everyone learns the way YOU do, present the same material in different ways and be sure to draw pictures and explain the pictures for visual learners.)

4.  Write down the steps for the procedures for the students (don’t assume that the student is going to be able to remember that when you complete the square that you have to take half the x term, square it, add it both sides, then create a factor of (x- 1/2term)^2 = total, take the square root of both sides, remember the plus/minus, and solve.  You can show a problem with this all worked out (sorry for those that I lost with this example but the point is to include a list of the steps so you don’t get lost!) but the students will forget how those numbers magically showed up without you helping them remember.

5.  Learn TRICKS to make things easier:  for Geometry and proofs, did you know you should start backwards with what you are trying to prove?  With factoring something like 6x^2-5x-4, learn the BOTTOMS UP approach, it is the quickest and easiest way to solve quadratics when the leading coefficient is not equal to one, there are many more TRICKS, learn them and teach them.

6.  Teach developmentally.  Start with easier problems and then build on that while adding slightly harder pieces, this is the easiest way to learn.

7.  Assign homework.  Students can’t master material without practice – don’t assign 4 problems it isn’t enough for students to learn but also don’t assign 50 problems, pick and choose the BEST problems for the student to do, don’t just day 1-50 odd, are all those GOOD problems and worth the time of the student.  Do your homework before you give them homework.

8.  Watch your tests, if students are failing, then YOU are not doing your job right and need to make adjustments.

In a perfect world, I would have students take a MINIMUM of 3 methods classes to become certified to teach.  During these classes, I would make sure all of the above were taught to students and that students had practice implementing these things.  How can we make STUDENTS better in mathematics?  The answer is simple, make our TEACHERS better.


Written by:  Lynne Gregorio, Ph.D.

Owner:  Apex Learning Center in Apex, NC



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