> It isn't "length of the school year" or "length of the school day" that is > the critical factor. It is how much time students actually spend "doing > mathematics" that counts (and the kind of math activities in which they > engage).
I agree! With the human-mind compatible approach ( confidence building, discovery learning to get general concept and simple algorithm, followed by introduction of all available algorithms and adequate, consistent practice on the daily basis to get to the deep understanding and complete picture of concept, strong math intuition and skills then built ) and timely, accurate guidance, students only need to spend less than one hour each day, they can achieve the kind of math ability that a lot of math teachers now a day do not have ( as some of you telling us).
I am seriously thinking, should there be a reform, we need to have a panel consist of the real mathematicians ( from different field of math ) to tell us what mathematics really is and how were they developed ( so we understand that each development of the concept or algorithm has its base and reasons, if we can let students see that, they won't be struggling ); the scientists to tell us what kind of math discipline the future scientists need; the engineers to tell us what kind of math skills or abilities of using technology students need to become a good engineer and how technology was developed upon mathematics; the psychologist tell us what kind of approach can most easily break the mind barrier to the math concept then the educators tell us what they can deliver. Then, we can have a genuine reform. Otherwise, I am afraid that it will only become a joke or if it really comes true, it will at least sacrify one generation of students. I know I will become very unpopular by saying this, but, I don't feel people here alone are capable nor have the right to make this kind of reform decision.
Honestly, how many of you people here really understand what mathematics is all about? seems many teachers' teacher still struggling? How many of you people here know exactly how the technology, which you are so eager to use in the classroom to escape your own challenge, is developed? and what kind of math algorithm behind today's technology? I am afraid that with what you are pushing toward, very soon, you will have no technology to use anymore, no need to mention the power breakerage. The real world is much more complex than what is happening in the classroom. I know it might sound to harsh to you people, but, after all, mathematics is the foundation of all the science and technology, it is a concern of all, not only yours.
Let's all work together to settle down what should be the goal or goals, and the content. Then, you guys can work hard at how to deliver. My personal feeling of the traditional approach is not that there is anything seriously wrong, but there are a lot things missing because of limitted resources. With the help of new technology, and new approach, we can do much better. But don't just abandon the content just because it is difficult to deliver. Retraining program for the teachers in math ability and teaching methods seems more urgent than reform.
Also, the reform has to be realistic in a sense that the teacher retraining program can catch up, otherwise, the victims are both students and teachers. If the retraining program and resources are not available, I suggest we all go back to the old teaching method which has been proven to work to some degree. One reason is in the old day the teachers are mostly well trained on the curriculum they need to deliver, job was much easier for them. It seems to me that part of the problem of today's education is that we too frequently urge to get reform and come up with new idea and new approach ( how much it came from the real urgent need? how much it came from the personal eagle to prove the value of existence? ) in a very unstructural unsystematic way, and most importantly, without a good retraining program, and it put all the burden on the teachers who don't know what to follow anymore.
There is another point, I think, too much concentrate on "teaching student to think" so that learning algorithm becomes less important is a rather phony arguement. Are we too much self-expanding? Do all subjects should teach students how to think?
Different from human language, the math concept and algorithm are usually developed in a spiralling way. No math concept can be fully realized without a logically meaningful algorithm. Very often, new concepts were triggered by algorithm, of course, vise versa. Having students do exercise on proper algorithm is a natural way to help student learn how to think. If a student only memorize "one" algorithm ( discovered by himself or taught to him by teacher) without understanding and alternative approach, a simple test can easily find it out. ( so who said "how do you know if they really understand?" very easy!) And this is one of the reasons that students should do many practices on algorithm itself to help them strengthen their real "mathematics intuition" and deeper understanding of the concept behind.
On the other hand, a student can be good at solving real world problems with limitted math skills, and still a mathematically illiterate. I think many people here still think "solving the real world problem is mathematics", that is totally wrong.
Of course I won't let student memorize any algorithm before they understand the "basic" concept and reasoning behind. But as soon as they understand the basic concept and the reasons for the develpoment of the algorithm, showing them all alailable approaches and having them practice on all different kind of problems should follow to further reinforce the understanding and skills. Most deep realization of the concept and intuition won't come without going through all variaties of practices on algorithms. Using technology to serve to help student get through these processes is definitely a plus. But, never a replacement for human mind development. I have not seen any person who is good in math and later having trouble using simple thing like calculator. Be cause they can picture exactly how the calculator is functioning inside. So the aguement about teaching them how to use calculator is also seems to be a disguise of escaping responsibility.
Of course, I think the concept and some simple algorithm of numerical analysis ( which often time requires a lot of manipulation in head ) can and should be included in the school curriculum, it is a must. But this is very different from having students using calculator to replace so called "useless alogrithms". This is exactly helping student not to think. Just take what ever available to get the answer without understand what is going on. The temptation and habbit to just jump to get solution is the worst way of learning. Students learn to urge for fast easy solution, whenever it is not available in an obvious way, they tempt to give up. No discipline on deep thinking ability. Doing all different kind of algebraic manipulations, like playing puzzle or playing chess, is a very good training on the parallel, logical, vigorous, deep thinking. Not only for the mathematics sense. After all, all mathematic algorithm follow rogorous reasoning.
( I don't think I understand what is "linear thinking". Then what is "nonlinear thinking"? To me, it seems to be "sequential" opposes to to "parallel" thinking, "vigorous" or "logical" verses "random" or "chaotic" thinking. Also, often time, we refer exponential growth as "linear" growth for a very good fundamental reason. )
I also disagree with what you said about mental calculation, I will not encourage any student do mental calculation until they understand the concept and the proper algorithm. Taking the example of time table, I usually have student do addition problems like 7x4 = 4+4+4+4+4+4+4 for about a week or two depending on each students' ability, then I happily show them the time table and have them use the addition to check each of them to make sure all are correct on the time table, then, I have them go home and memorize them one line each night untill they remember all of them. At this stage, they will be more than happy to accept and willing to absorb the time table like sponge to water. Averagely, they can all master the time table in 2 weeks. I also show them some trick for memorizing 5 times and 9 times and show them the reason behind. They all like to learn a few "trick" and learn some deep concept behind. Here is the kind of psychology we can play. Not just help them avoiding the difficulty.
Generally speaking, the memorization of those simple operation should come naturally with the adequate practice, as long as they "think" when they do the problem, no matter if it is algorithm or concept related. Calculator just let them not to "think". Praticing the proper algorithms is actually one of the best discipline of "thinking" because all algorithms follow logical reasoning.
Your wish that your kids can get a good intuition for algebra without going through the painful arithmatic manipulation is also a big wishful thinking. All the development of the math concept has its human mind related natural course --> that is, existing bases and/or practical reasons. Except for a few genius, most people can not appreciate algebra without good arithmatics intuition. You got to break the psychological barrier by having them appreciate the power of the new concept and new algorithm. For so many years of teaching since my high school years, I have not seen a single student who can master arithmatics while having hard time understanding algebra. NOT ONE!
Finally, I like to share with you the following, as I always tell my students of all ages, mathematics is not to make your life more difficult, it is created to make the complicated world look so pretty and so simple to deal with.
It should not be difficult, if you know what it is and how to show it to the students.