I have recently completed a first reading of the second edition of Dave Moursund's book "Computational Thinking and Math Maturity: Improving Math Education in K-8 Schools." Dave Moursund is a Professor Emeritus of Education at the University of Oregon with a PhD in Mathematics and has worked as a professor of mathematics and as a professor and chair of computer science and information science at the University of Oregon. This book contains many excellent ideas about the problems of mathematics education in America. Though the focus is on K-8 math education, many of the ideas in his book carry over to high school and early college math education. The book is available at http://uoregon.edu/~moursund/Books/ElMath/K8-Math.pdf.
One significant idea in his book is that students need teaching methods to get them to move to the next cognitive level of thinking, the thinking about the abstract. Most students struggle with math because they haven't gotten beyond concrete thinking. Moursund mentions that only about a third of students graduate from high school having reached Piaget's Formal Operations level of cognitive thinking in mathematics and that many math teachers have not either. That is, these students and teachers lack mathematical maturity. Math education neither today nor in the past does not and has not helped most students reach the mathematical maturity they need to succeed in algebra and other math courses with symbolic reasoning. Chapters 5-7 discuss these ideas in depth. This is something that I knew all along, but Dave Moursund states it in ways more precisely than I have done.
It is interesting to note that Moursund has divided what Piaget calls the Formal Operations level of cognitive thinking into several additional levels: one level for reasoning in algebra and early college math courses, another level for advanced undergraduate and graduate math courses, and a final level for mathematicians. I do believe that Piaget's Formal Operations level does consist of several additional levels of cognitive thinking, at least in mathematics. But the fact that only about a third or so of high school graduates have reached the Formal Operations level says that only about a third of them have the minimum cognitive ability to have any chance of learning algebra and other mathematics with symbolic reasoning (symbols besides those in arithmetic). No wonder so many students struggle immensely with college mathematics.
Another significant idea he explores is that neither current nor traditional teaching methods teach students to read mathematics and learn from it and do not teach them how to become independent learners of mathematics. I would not doubt this claim in the least bit because none of my K-12 math teachers did anything to teach us how to read a math book and to learn from it and never taught us anything about how to teach ourselves mathematics. None of the current courses I teach provide students any materials on how to do this; that is, these course developers assume that students already have these skills. And few of them actually do. I try to provide such resources, but I know I need to try to improve those resources because they don't seem to be helping much (I say "don't seem" because I have no way to verify if these students are or are not reading the information I give them; I doubt most of them are because in many other cases I've seen students act as if they haven't read something, but I can't say for sure). And it seems that hardly anyone else addresses this issue in math education except those who support reform teaching methods. On top of that, the weak reading skills of students today make such learning extremely difficult for them.
These two significant ideas give students the necessary general skills they need to learn mathematics. Without these skills, they cannot learn mathematics successfully. And, if they forget any necessary math they need later, they cannot relearn the math for themselves. Finally, all these specific skills they try to learn in math make no sense to them because they haven't learned what it means to learn mathematics, to learn to think mathematically.
Finally, another significant idea he explores is the integration of computers into teaching. He notes that little has changed in the teaching methods and curricula of mathematics despite the significant recent changes in technology because mathematicians and mathematics educators are still unsure about the best ways to integrate such technology into mathematics education. I believe such integration that is sane and intelligent is possible and is necessary because much of today's mathematical work is done via computers. One concern is that the integration of computers will cause students not to learn mathematics successfully and become too dependent on technology to do their thinking for them; this is certainly a valid concern, but I believe there are ways to overcome such challenges. One possible way is to assign computer work that cannot be done successfully without thinking about the mathematics behind it; that is, if the student tries to complete the work blindly, the computer gives nothing but elaborate nonsense. And we should stress this idea repeatedly in class. Moore and McCabe's "Introduction to the Practice of Statistics" does a good job of stressing that computers are an aid, but not a substitute, to thinking and understanding and that using a computer blindly produces nothing but elaborate nonsense. Another possible way is to give quizzes and exams where at least some of the questions cannot be answered by a computer and must be answered using the necessary understanding of mathematics.
In short, here are the Big Ideas (as he calls them) that Moursund explores in his book (as quoted from the preface):
1. Thinking of learning math as a process of both learning math content and a process of gaining in math maturity. Our current math education system does a poor job of building math maturity.
2. Thinking of a student?s math cognitive development in terms of the roles of both nature and nurture. Research in cognitive acceleration in mathematics and other disciplines indicates we can do much better in fostering math cognitive development.
3. Understanding the power of computer systems and computational thinking as an aid to representing and solving math problems and as an aid to effectively using math in all other disciplines.
4. Placing increased emphasis on learning to learn math, making effective use of use computer-based aids to learning, and information retrieval.
I may write more comments later about Moursund's book.