Why is the transition from multiplication tables to algorithms flawed?

[complexPHILOSOPHY]

Universities are distinguished as such because of their ability to distribute accredited doctoral degrees, while colleges are denoted as colleges, because they don't offer doctorates (I believe).

This is purely anecdotyl but my observations concerning people's interaction with mathematics and arithmetic, honestly leads me to think most people really don't want to do maths. I have a lot of intelligent friends who can in fact, do a lot of the math I do but lack the same passion to do it. I can explain my reasons for loving math and physics and why I think it's the most important subject but to them, it's useless. They are much more content with Political science and Law school, then mathematics and phd research.

I can sit down and derive what I consider, interesting mathematics and physics and demonstrate the usefulness and subtle cleverness that the mathematicians invoked when approaching interesting problems but they are still completely uninterested. I work at a law firm part-time and even here, they don't respect the work of math and physics students -- they value business and political science majors in a much higher regard than science or math majors.

 

[Crosson]

So... this post is all about one guy wanting to radically changing how math is taught and expecting it to go well?

So...your post was all about belittling a thread you didn't read?

The large majority of people don't need to know how to prove theorems or why division works.

I agree. They need to know how to think, and how to communicate. Could you imagine using math do teach them that?

There's no point in doubling the amount of time spent on math if it just doubles how much people forget.

Rather than doubling the time spent on math, I advocate making things like long division part of a different subject not related to critical thinking/communication.

This is purely anecdotyl but my observations concerning people's interaction with mathematics and arithmetic, honestly leads me to think most people really don't want to do maths.

I totally agree with you, and this thread is about improving peoples attitude towards math by seperating good math (teaches critical thinking/ communication) from drudgery like arithmetic (teaches useless, outdated "skills").

 

[jing]

Crosson you will have to help me as I am struggling to understand what you are actually advocating. Perhaps my understanding of your words and your understanding of your words are different. Until I do understand what you are advocating it is difficult to agree or disagree with you.

What I think I understand so far is that for you Mathematics is

The science of theorems and teaches critical thinking/communication.

Now any taught course has at least two parts

Course Content -'What is taught'
Course Process - 'How it is taught'

Thinking about content

I imagine your term 'Science of Theorems' to mean teaching about the structure of theorems, how they are constructed, what are the processes in proving a statement true or false, what constitutes proof.

Is this correct? If not please elucidate on what you mean by the 'Science of Theorems'

Would you give some examples for the content of a course on communication that would help me see why it would be the province of Maths rather than English.

 

[HallsofIvy]

Universities are distinguished as such because of their ability to distribute accredited doctoral degrees, while colleges are denoted as colleges, because they don't offer doctorates (I believe).

This is not true. As Pathway said, the technical definition of "college" is that it gives degrees in only one area, Liberal Arts, say, or Engineering, or Economics, rather than having a number of different "colleges" under its own roof, offering degrees in many such areas.

Today, because even relatively small "colleges" offer both Liberal Arts and Education, they can legitimately use the name "university". On the other hand, some universities, such as The College of William and Mary, in Virginia, U.S.A, retain the "College" title for historical reasons.

 

[Alkatran]

So...your post was all about belittling a thread you didn't read?

I agree. They need to know how to think, and how to communicate. Could you imagine using math do teach them that?

Rather than doubling the time spent on math, I advocate making things like long division part of a different subject not related to critical thinking/communication.

I totally agree with you, and this thread is about improving peoples attitude towards math by seperating good math (teaches critical thinking/ communication) from drudgery like arithmetic (teaches useless, outdated "skills").

I did read the whole thread, arithmetic is a necessary part of math, the majority people are never going to like math, and you sound like a troll.

 

[Crosson]

arithmetic is a necessary part of math

Again, so are writing skills. I debunked this line of thinking by saying that, if we include arithmetic under the heading math because it is a necessary part of math, then we should do the same thing with writing skills (which Matt Grime admits in this thread are a much more important part of doing math than is arithmetic).

the majority people are never going to like math

You may think I am a forum troll, but you sound like an old cynical troll who is encouraging me to give up. Pure mathematicians shouldn't have to pretend that there research might relate to applications someday, they should be given funding for producing intellectual art, and that can't happen until the people with the money start to appreciate more of what is really going on.

Jing - your understanding of my views is indeed correct. Here is an example of how math can teach communication: Have the students read a newspaper article, and find sections that are:

1) Vague

2) Internally-inconsistent

3) Rhetorically cumbersome (too many words saying too little)

One of the biggest problems is that most non-mathematical writing contains so much crap in the 3 categories above, that typical readers are trained to skim through this writing with very little effort or concentration. We need to help them write pieces that are worth reading carefully, and by doing so teach them how to respect other peoples writing by reading it carefully.


*******


One hope of mine revolves around streaming audio technology. It is my hope that all of the crappy written english will eventually become streaming audio, and that only the things which will be truly better off written will remain written, and hence everything written will be worth reading.

 

[Hootenanny]

Again, so are writing skills. I debunked this line of thinking by saying that, if we include arithmetic under the heading math because it is a necessary part of math, then we should do the same thing with writing skills (which Matt Grime admits in this thread are a much more important part of doing math than is arithmetic)

Are you suggesting that schools start timetabling separate 'arithmetic classes', universities start offering degrees such as 'Mathematics & Arithmetic', students major in math and minor in arithmetic?

 

[arildno]

One of the biggest problems is that most non-mathematical writing contains so much crap in the 3 categories above, that typical readers are trained to skim through this writing with very little effort or concentration. We need to help them write pieces that are worth reading carefully, and by doing so teach them how to respect other peoples writing by reading it carefully.

This is indeed correct. At the very least, people should from a very early age learn to handle MATHEMATICAL texts in the following way:
1. Excise redundant text
2. Itemize crucial information.
3. Furnish&itemize information known elsewhere thought necessary in problem solving
4. THEN proceed to "solve" the problem (with possible re-doing of the previous points as part of the problem solving process).

To learn maths in this way, therefore, will have a cross-over value for non-mathematical persons in developing a reading skill that enhances their ability to cut through vagueness and, say, superficially benevolent "power talk".

 

[arildno]

I'd like to raise one issue (of many!) where I believe that the approach in elementary school teaching of maths is seriously flawed:

The too-fast transition from use of multiplication tables to the multiplication algorithms.

I think that many elementary school teachers fallaciously believe that a multiplication table is merely a list of ready-made "answers" and that the true art of multiplication lies in performing the multiplication algorithm.
That is, the multiplication table is regarded as something trivial, a necessary evil, and to find a particular answer through table extension would be regarded as a sort of "cheating".
Furthermore, it is believed that not many insights of mathematical and pedagogical value can be gained from the use of the multiplication table, that is mathematical "insight" is to be judged by rating your ability to perform the multiplication algorithm.

That the tabular form lies much closer to the proper mathematical perspective on multiplication as a binary operation than the particular algorithm taught in schools, is wholly lost on these teachers.

A lot of fundamental mathematical insights can be gained a lot more easily by the use of multiplication tables (or sections of them) than honing the pupil's skill on performing an incomprehensible algorithm (incomprehensible to the pupil, that is, and sadly enough for many teachers as well):

1. Just because the commutative property of multiplication is readily seen from the multiplication table does not of course imply that commutativity of multiplication is a trivial, unimportant insight to be gained!

2. Furthermore, the easily seen connection between the values in adjoining boxes is a very nice illustration of the distributive property of multiplication over a sum, and this insight should become firmly lodged into the pupil's mind by exercises designed to do so.

For example, the following types of exercises will seem easy to most pupils, and teach them important insights besides:

1. Extension of multiplication tables, or sections of them
2. Detect particular products by walking table-wise from a known product, using sum and subtraction to fill out the intervening boxes.
(Here, clever choices of paths might involve use of the commutative property to reduce the number of steps necessary)
3. Detect flaws in a multiplication table

4. Break up factors in smaller numbers so that the law of distributivity+use of multiplication table+summation skills can yield the answer.*



Since the logic of what occurs is transparent, the heads of the tiny ones won't hurt so much, and they will also, as an added bonus, enhance their ability to see a lot of symbols on a piece of paper without getting lost before starting. To develop mathematical stamina happens to be very important if you are to get any better, and to do so gently from the beginning is a good, pedagogical approach, in my opinion.


Once kids are in possession of such table skills, they will be in a much better position to appreciate the sole advantage the multiplication algorithm has:
It is a way more efficient method to find a product** than to laboriously extend the multiplication table.
The latter method is logically transparent and should therefore be mastered first, the multiplication algorithm is more logically dense, and should therefore be taught later on.


*As an illustrating example, the pupil might have been given the multiplication table square section going from 16 to 20 both ways, and is asked to find 35*16.
Problem solving would then be to either use the partition (16+19)*16, or the partition (17+18)*16, and then sum the two numbers read off the given table.


**More precisely: Find the product's denary representation.