Views From A Teacher On Instructional Practices
When I meet my classes for the first time, I never discuss discipline policies, rules and procedures. Instead, I discuss and give examples of content that we will cover throughout the year and if it is a class where I use a textbook I make a few comments about the authors. I give them examples of content that has changed throughout the years and content that is simply “strange”. Is a “set” really an undefined term in some courses? Was “1” ever a prime number? Is there more than one level of infinity? Do the angles of every triangle sum to 180 degrees? Do all calculators perform order of operations in the same manner? (You might be surprised!) Can we add an infinite number of fractions? Always? Sometimes? Never? Can negative numbers be prime? I then mention the difference between a conjecture and a proof by giving a brief discussion of Fermat’s Last Theorem and by mentioning that Andrew Wiles after many years of hard work actually proved it. I confide to them that on many occasions, I have worked on one math problem for over three weeks. They then understand that there is no need to panic if it takes them one hour to do two problems or even if they spend two hours and don’t arrive at a solution. The one thing that they understand after my course introduction is that it sounds like their math class will consist of serious, new and exciting content. They sense that the classroom structure, procedures, supplies, and rules will seem secondary compared to the mathematics that will be learned. After a day or two I assign seats, discuss a few classroom rules, go over notebook organization and discuss grading procedures that were sent home the first day of school for student and parent review. But by this time after one or two short days, my students know that the main character is math. They never ask me when they are going to use this stuff in life because they know that our purpose in class is to study mathematics as a discipline and if we get to use it in life that day it’s simply an added bonus.
I teach high ability seventh and eighth graders at school but I also tutor students of varying abilities and I always make the mathematics the main character regardless of the student population. I mention this for the following reasons:
I vehemently disagree with the view that math has to be disguised, neutered, applied, related, justified, put in context, discovered, and presented through textbooks that resemble video games in order for average ability American students to learn it.
I also disagree with and deeply resent the view that minorities can learn math only when it is presented in a context that makes it relevant to their culture. Math is the one discipline that should be culture neutral.
Three years ago, our 1000 plus students were surveyed to select their favorite subject. The result of the survey was published in the yearbook. Math was selected as the students’ favorite subject. We even beat out PE. We teach every type of student at my school from the profoundly gifted to students with severe learning difficulties but our math teachers all have one thing in common. We make math the main character and we never make apologies when it doesn’t relate to other parts of their academic life at schools. Part of the middle school philosophy states that students’ learning takes place when their core academic courses are connected through themes, and projects. When I plan for my classes it should be in conjunction with the English, Social Studies, and Science teachers according to this philosophy. When math is presented as pure math by passionate teachers without excuses, students love it. Many people, especially administrators were surprised (and I suspect even upset) by the survey. The survey was never taken again.
I present content using various techniques. Sometimes I introduce mathematical content by presenting a problem. Not a “real world problem” but a pure math problem that can only be solved using techniques and mathematics that have not been previously learned. An example would be to add the following: 1/2 + 1/4 + 1/8 + 1/16 …… Students will eventually sense that the sum approaches 1 but they will also need knowledge of finite and infinite geometric sequences and series to actually calculate the sum. Starting with this problem, I give my algebra students a thorough review of fractions.
I sometimes introduce content through a short discussion of the mathematicians who had an important impact on it. For instance: the Euclidian Algorithm, Euclidian Geometry, the Pythagorean Theorem, Cantorean Set Theory, Cartesian Coordinates, Pascal’s Triangle, Euler’s number, the Gaussian Summation Formula, and many other topics that occur in middle and high school mathematics courses.
I also review easy content and have students practice low level procedures such as the operations of addition and subtraction by introducing new topics such as modular arithmetic and number bases that use these operations in an interesting and in some cases strange manner.
Once I have explained a sufficient amount of content, I expect my gifted students to use their knowledge of this content along with their mathematics background to solve non-routine challenging problems. I never force them to solve such problems using the newly learned content but in some cases it is much more convenient and practical for them to do so. For average students, I extend the presentations and make the problems more specific to the content. This is where reality sets in. Average students have a difficult time applying newly learned content to problem solving if the content and techniques are not sufficiently learned and practiced (see the LP report on working memory). My gifted students do a certain amount of discovering on their own but even they need a strong content background before they can make serious mathematical connections (my expert opinion).
My conclusion is that there are many different teaching styles and no one method of teaching should be force fed to teachers. Direct instruction has been shown to have very positive benefits when used with underachieving students. I use a certain amount of discovery learning techniques and problem based learning techniques with my gifted students. My teaching style may shift a bit depending on the ability (not the gender or ethnicity) of the students. I believe that all students are capable of loving math even at the lower levels but some need more academic structure and teacher guidance.
I use textbooks that are focused and content rich, but I also read math textbooks that are simply interesting such as The Heart of Mathematics. I pass on some of the more interesting content to my students. I don’t use textbooks that were written after 1990 because most authors don’t offer clear explanations, examples, diagrams, and disinfested content.
I present some lessons to my classes the same way that the content is presented in some of the best textbooks. I suspect that many teachers do this which is why it’s important to have focused well written texts in every classroom both as a student text and as a teacher resource. We tend to use textbooks to make lesson plans and to reinforce teacher content knowledge. When I graduated from college I was amazed at some of the technical content (especially vocabulary) that I had overlooked in my high school and college math courses. In the 1970s I could refer to any middle school mathematics or algebra textbook if I had a question relating to any aspect of mathematical content at that level. I would never rely on a textbook written after 1990 to answer any question that I might have relating to content or lesson presentation.
In summary, teachers use textbooks for much more than a vehicle to assign problems. They really do use them as a teaching guide, teaching assistant, lesson planer, and in some cases as their teacher.
My students set in rows and in assigned seats. The blackboard is at the front of the room and I rarely use the overhead projector to present problems. We use the computer lab for two weeks when we do a unit on analytic geometry but at no other time. I do have the students work in groups but only on occasion. I never use manipulatives and calculators are rarely used by my students or by me. I use lots of chalk but so do my students. When one enters my classroom they will see twenty and thirty year old textbooks stored on the book shelves. Most modern day administrators would describe my classroom as that of a rabid traditionalist. Yet my students own the classroom. Whenever I present lessons we have discussions, real discussions. When a student is at the blackboard discussing an idea or presenting the solution to a problem they are in charge. They call on students, I don’t. I even raise my hand for recognition if I have a question of the student. That’s why my students also use lots of chalk. The system works for me but more importantly it works for my students. I resent it when people who haven’t seen a kid in twenty years preach to me that I should use cooperative groups and become less a part of the students’ learning environment.
In summary, classroom structure and teaching style should be the teacher's call and not an administrator’s call, especially if the administrator hasn’t been in a classroom for decades.