How do you describe your teaching style? Do you show students examples and then assign homework doing multiple similar problems that mirror those examples, or do you prefer a more constructivist/reform approach?
I used to be a very traditional teacher, but over the years I have made a pretty significant reversal in my thinking about good mathematics teaching.
A typical lesson in my class now centers around only a few very rich problems that employ multiple tasks and concepts. I focus my assessments on students understanding of the concepts and less on skill acquisition.
One thing I have started doing recently that has helped my students develop fluency is giving my students a problem and four sample-student-solutions with work shown using multiple representations (one solution using a graph, one an equation, one a diagram, etc), then tell them that two of the solutions are correct and two are incorrect. They have to identify the correct solutions and add additional work and explanations to all the answers to justify their choices. Just doing a few of these per lesson has made a tremendous difference in student performance on my assessments. I just created a problem last night using this same strategy, I'll try to get it posted on here so you can take a look if you are interested.
I think it was Parker Palmer who said, "We teach who we are." Some of us might be inherently conservative and consequently more traditional, and others of us might be inherently non-traditional and enjoy embracing the next new thing. But I do think that any teacher must be able to connect with the student in a way that reflects self-respect and respect for the student, no matter what.
I also think that no matter who we are, we need to understand that some kinds of learning are constructivistand other kinds of learning might be more traditional. For example, I cannot imagine anyone being able to have the multiplication tables mastered without the involvement of some rote learning, and the quadratic formula really needs to be memorized. But having said that, I must also say that neither is of any use whatsoever if not used in a hands-on way. I agree with akannen that we need to understand both approaches and anything along the continuum that will work.
I am becoming increasingly disenchanted with the phrase "best practices" because each interaction between teacher and student is unique and ineffable, a connection between who we are and who they are.
I think that this type of question goes at the heart of modern teaching. Having said that, I think there is a way to bring both reform based and traditional notions of teaching into the same pantheon. It would be using different tools for different jobs. There are times when innovative approaches work extremely well and moments when the traditionalist methodology would be more effective. The teacher has to gauge what works for the students, the pulse of the classroom and what approach works best with which groups of students. A group of higher end students might favor one approach, while a group of lower end students could respond better with another. The middle of the road students, who always seem to fall through the cracks, might respond better to one or the other, or even a hybrid of both. Teachers need to be able to be fluent in as many approaches as possible and examining their classroom dynamics, determine which one works for which student at which given moment.
In answering the original question, I'd say I'm traditional. Coming to teaching high school from the manufacturing sector, I jumped "into the fire" so to speak. What I found was that students lost the basics before they arrived. As a consequence, I am constantly stressing "fundamentals" in my Algebra 2 classes. Weekly, I give quizzes and include on tests problems such as add three numbers, add, subtract, multiply, and divide fractions, multiply and divide numbers and expressions with exponents, etc. Then as we progress through the textbook, I add to the fundamentals radicals and logarithms.
For me, when trying to teach students a new subject, I like to start first with the problem. I intrdocude problem for students then discuss with students how to fing the answer. This method jas proven to be very effective because it allow students to obtain the method themselves so they would never forget the formula because they helped obtained it.
This way also helps you make sense of math for students always asking "what's the point" or "why are we learning this". So introducing the problem makes sense of the subject even before introduce it.
I agree with the preceding posts. There is a lot of flexibility in the field of teaching. If you are not considering multiple learning styles in your lesson plans, you aren't reaching as many of the students as you could if your lessons varied to reach visual, auditory, and kinesthetic learners.
That having been said, I think I am more traditional--I have set routines, etc., but I also adapt my lessons to appeal to all kinds of learners.
I once read that an ideal classroom is one where, learning continues even when the teacher is not there. I structured my classroom around this idea...trying to have the students really enjoy the learning activities, encouraging them to talk with each other about the topics, asking them to work together, helping each other. When I needed to write lesson plans for a substitute I could just write "have them work on their independent work" and know that their time was not being wasted. I gave them many choices of things that could be worked on, so that they felt in charge of their learning...discipline problems were very minimal, as a result.
I teach management subject to students at post-graduate level. The method of teaching that I adopt certainly need to be different from what may be appropriate for younger students. Yet I find that I need to vary my teaching style from situation to situation based on following three consideration.
- Nature of the subject content. Some topics are quite mechanistic - like Forecasting Methods largely consist of mathematical procedure. These methods can be best taught by describing the methods in classroom using simple example. Other subjects like Strategic Management require students to develop ability to think innovatively based on some general principles. IN cases like these it is better to rely on independent study and research by students followed up by discussions in class.
- Level of students: When students are able to understand a subject easily, I would rely more on directed self study. However when I find students have difficulty in understanding a subject, I give more stress on classroom lectures with simple examples. I also try to give students some simple assignments to help them reinforce learning in the class.
- Time available to teach the subject: In most of the institutions, where I teach, I have the flexibility to vary the width and depth of course coverage to some extent, but still there are some limitation. When I find that I am short of time, I find the classroom lectures is the faster method.
Thanks, Phil.. the reason I giving sample student solutions is that it takes a great deal of critical thinking to figure out what the student was thinking. Even when my students are able to figure out the correct solution strategy (or strategies), they find it much more challenging to explain why the strategy works.
#1: I love your idea of presenting actual student work and having them discover the 2 correct and 2 incorrect answers. This encourages students in finding multiple ways of arriving at the correct answer and shows them that there isn't just one way of doing it. I also like that they get to see what it means when we say "SHOW YOUR WORK"