This week, I read the article "To Know and to Teach: Mathematical Pedagogy from a Historical Context" by Frank Swetz. In the article, the author notes that particular techniques for teaching mathematics, such as the use of instructional discourse, a logical sequencing of problems and exercises from simple to complex, and the use of manipulatives is much the same as it was thousands of years ago. He goes on to expand on each of these by pulling examples from different times in history, including some geometry problems from the Old Babylonian period approximately 3000 years ago! I actually worked through some of the problems myself and found them to be quite fun.
The authors' section on mathematical exercises was particularly intriguing to me, as I was constantly thinking of current school mathematics texts and the structure of exercises in those book. What struck me the most was the total amount of exercises. Within these old texts, the number of exercises seems to be capped around 20. In regards to current calculus texts, one section could have at least 80 problems!! Most of the problems are essentially the same, with some slight variations here and there. In regards to some Chinese problems from 100 BC, the author states that in order to solve them "thinking is required." This made me stop to think. In the current school system, how much "thinking" do our students actually do? Even if problems in textbooks go from simple to complex, how much thinking are students doing on these problems?
Moreover, do current mathematical textbooks encourage students to think critically? I am tempted to say no for the most part. Of course, I'm running off of my own personal experience at this point. I remember the majority of my homework in math class being the same problem over and over, tests being variations of those problems, and this evolved to the point where I "didn't have to think about it." If I am remembering correctly, there was a Schoenfeld article I read recently about students' beliefs about mathematics. He was observing a high school geometry class and when the teacher was referring to studying geometric construction problems for the state exam, he told the class "they shouldn't even need to think" (or something along those lines). Unfortunately, I think I'm also guilty of this as a teacher, in telling my students that certain types of problems should be essentially automatic. This is certainly not what I want for my students, but it is what expected in the system.
Does performing 40 variations of the same problem enhance student learning and understanding of particular mathematics concepts? Why did texts evolve in such a way? I am left with many questions as to why textbook exercises are the way they currently are. I look forward to the discussion on Wednesday.
Vanessa, I find what your saying so appropriate and it brings me back to the video we watched, "how old is the shepherd." If we are constantly giving our students variations on the same problem, I believe we are setting our students up for using repetition rather than critical thinking. I, also, find myself getting caught in this trap as I teach my grade 4's their math curriculum unit by unit. I try to throw in questions from past units in order to give them some variation but definitely find it challenging. Where is the fine balance of practicing skills that they have just learned and also creating critical thinkers? Often textbooks will merely show questions in a different format thinking it is mixing up the process, however, I do not think this is the case.
ReplyDeleteI also wonder if repetition results in the automaticity that is strived for? Is automaticity what we are aiming for or are we striving for more? I guess it brings me back to the old quote "practice makes permanent." In sports often skills are practiced over and over again in order to get muscle memory, however, skills are varied during a given practice time, is this something we should be looking at with math practice? I too am left with many questions and the more I think about it the more torn I become!
Thanks for the blog post Vanessa.
ReplyDeleteInteresting what you mention about historical textbooks. It would be very interesting to do a comparison of textbooks on a particular subject across the ages and see how they have evolved. Have textbooks moved towards more repetitive exercises? I think the answer is yes. Anecdotally, I used Stewart's Early Transcendentals as my first year calculus book, which is probably the most popular book at the moment. Every chapter has a ridiculous amount of boring repetitive exercises at the end.
Another interesting project would be to create a course around a historical textbook. This would be a real challenge as many ancient textbooks used a different notation. The Babylonian's used base 60! This would make the course interesting and result in flexible thinking, not to mention incorporate a history lesson into a math class.
In terms of why we have gotten to the point where textbooks are filled with repetitive exercises, I think the answer has a lot to do with the very top: The so-called prescribed learning outcomes (PLO). These are chosen so that they can be measured easily. This is a long debate and a can of worms that I probably shouldn't open. Suffice it to say, one of the reasons I'm reluctant to go into education research is my utter discouragement with the way our education system is set up: Students study based on how they will be assessed, and what is assessed is based on what is easy to assess. This is true even at the highest levels of education at the most enlightened universities.