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Topic: Re: length of school day or year
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Chi-Tien Hsu

Posts: 144
Registered: 12/6/04
Re: length of school day or year
Posted: Jun 23, 1995 6:26 AM
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> 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

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

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

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

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.

Good luck to you all!

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