Why is the transition from multiplication tables to algorithms flawed?

[DaveC426913]

In general, I am adcocating that we isolate the "drills" of Arithmetic/Calculus/Algebra as separate subjects from math. Math should be about thinking, and the drills reinforce the opposite. The drills teach skills that are useful:

I think that, while this may seem logical in theory, in practice, it doesn't make sense.

Centuries of education demonstrate that those who grasp arithmetic intuitively will be more likely to grasp higher mathematical concepts intuitively. And that those who don't, don't.

Regardless of how you think they should be grouped, I believe that the grouping happens naturally, and the education system follows their lead.

 

[Crosson]

Centuries of education demonstrate that those who grasp arithmetic intuitively will be more likely to grasp higher mathematical concepts intuitively. And that those who don't, don't.

Based on the personal experience of at least myself, a friend, and Albert Einstein, this is totally wrong and disgusting!

Oh yes but you said "more likely", so you feel safe in ignoring exceptions. Unfortunately this is not acceptable when it comes to such a glorious subject, participation in which needs to be more encouraged!

 

[Crosson]

Regardless of how you think they should be grouped, I believe that the grouping happens naturally, and the education system follows their lead.

The educational system is a dinosaur! Computers have changed mathematics forever and those centuries of experience matter less and less with each passing day.

 

[DaveC426913]

Based on the personal experience of at least myself, a friend, and Albert Einstein, this is totally wrong and disgusting!

Based on an anecdotal sampling of three (3), you are comfortable labelling it as "disgusting".

OK, so much for a rational discussion...

Oh yes but you said "more likely", so you feel safe in ignoring exceptions. Unfortunately this is not acceptable when it comes to such a glorious subject, participation in which needs to be more encouraged!

I'm not ignoring exceptions, I'm acknowledging the catering is to the majority. Your solution is no better, it just divides it up differently, likewise leaving its own exceptions ignored.

The only way that's going to be fixed is via an overhaul in education that accounts for the exceptions as well as the majority.

But that's a whole different ball of wax.

 

[matt grime]

Unfortunately this is not acceptable when it comes to such a glorious subject, participation in which needs to be more encouraged!

And your suggestion is to teach mathematics to begin with purely by its applications (or more accurately, by what you think are its applications), thus implying that maths is only useful when applied. If that becomes ingrained over the first 11 years of schooling what is the chance that anyone would want to do mathematics at university?

 

[matt grime]

The educational system is a dinosaur! Computers have changed mathematics forever and those centuries of experience matter less and less with each passing day.

Really? How many examples can you cite? Perhaps the four colour theorem, and some useful conjecture indicators in number theory... Computer drawing packages have certainly helped with real 3-d algebraic geometry, I suppose. Certainly computer packages are useful for checking large (numbers of) examples, but I'm not even aware of a successful proof checker for a computer.

 

[arildno]

What do you suggest, Crosson?
Increase the number of Arithmetics+Maths hours to 10 per week, rather than 5 "Maths" hours per week?

 

[Crosson]

Really? How many examples can you cite?

Take for example Euler or Gauss, lagrange, laplace etc. Among their truly mathematical accomplishments were also their feats of numerical computation No mathematicians does this in todays computer age, and I consider this to be a big difference.

Also, note that the entire field of nonlinear dynamics and fractals was created because of the invention of modern computers.

And your suggestion is to teach mathematics to begin with purely by its applications (or more accurately, by what you think are its applications), thus implying that maths is only useful when applied.

You are totally misunderstanding what I am saying. No one is saying "Math should only be applications", I am saying that we should preserve the beauty of math by teaching the subject (theorem-proving) seperately from courses that involve "drill skills" i.e. Arithmetic. In a sense, I am saying the exact opposite of what you accuse me of saying.

Based on an anecdotal sampling of three (3), you are comfortable labelling it as "disgusting".

Yes, I think that every minority reserves equal representation, in idealistic discussions such as those on 'Physics Forums'. Ideallistically, this is something that America stands for, and it is the most beautiful aspect of this country. Don't bore me with what is practical, we can all see the obvious. Still, in a discussion of the way things should be, I think it is disgusting to exclude people unnecessarily.

I'm not ignoring exceptions, I'm acknowledging the catering is to the majority. Your solution is no better, it just divides it up differently, likewise leaving its own exceptions ignored.

Please, let these minorities speak up so that we can design the system to include them. The difference between your statement and mine is that you were explicitly excluding people who are bad at arithmetic, and personally that disgusts me because arithmetic drills are so irrelevant to higher math that the exclusion of even one hopeful mathematician on this basis is at least a personal tragedy (for that person's life) but also, I think, a tragedy for the entire mathematics community.

 

[arildno]

I wonder what sort of maths you've been doing. If you've done any error estimates of some shape, you wouldn't be able to do many of these without the arithmetical skills as an elementary, hardly thought about, but extensively needed skill.

 

[Crosson]

I wonder what sort of maths you've been doing. If you've done any error estimates of some shape, you wouldn't be able to do many of these without the arithmetical skills as an elementary, hardly thought about, but extensively needed skill.

Yes, arithmetic is necessary for doing error analysis. So is writing (english), but one of these is commonly included in mathematics and the other is not, so this criterion ("gets used") is not a convincing reason to include arithmetic as part of mathematics, unless we would also admit that writing is part of mathematics.

 

[morphism]

Arithmetic is part of mathematics. End of story.

 

[symbolipoint]

Crosson,

Would you settle for Mathematics instruction concentrating
on Arithmetic including number sense, units, simple Geometry,
consumer-quality graphs, word problems, until about 6th grade;
and then push for "Pre-Algebra" and Introductory Algebra, plus
review of the earlier concepts in junior high schoo and early high
school; and then Geometry, Intermediate Algebra, Trigonometry,
and maybe Calculus or Statistics for college preparatory students
in high school? Would you also prefer each or most of these to
be their own dedicated distrinct courses? That entire arrangement
should help you to feel better.

Most basic consumers should at least understand the basics of
algebra(introductory). The average consumer still should know
how to make sense of things as found by shape, collections of data,
numeric, and logic.

 

[matt grime]

Yes, arithmetic is necessary for doing error analysis. So is writing (english), but one of these is commonly included in mathematics and the other is not, so this criterion ("gets used") is not a convincing reason to include arithmetic as part of mathematics, unless we would also admit that writing is part of mathematics.

Writing clear and precise English is far more important than an ability to do arithmetic in mathematics.

 

[matt grime]

Take for example Euler or Gauss, lagrange, laplace etc. Among their truly mathematical accomplishments were also their feats of numerical computation No mathematicians does this in todays computer age, and I consider this to be a big difference.

You don't half make far reaching, and inflexible assertions that some of us think are indefensible.

Also, note that the entire field of nonlinear dynamics and fractals was created because of the invention of modern computers.

not really true. It is perfectly possible to do non-linear dynamics without any recourse to seeing a computer.

Sure, computers make some phenomena easier to observe, to calculate, to measure, but are you claiming that they seriously advanced the cause of mathematics more than the mathematicians who worked out the theory?

You seem to think that the only mathematics there is is applied.

 

[Crosson]

Writing clear and precise English is far more important than an ability to do arithmetic in mathematics.

Wonderful!

Increase the number of Arithmetics+Maths hours to 10 per week, rather than 5 "Maths" hours per week?

Now we are thinking along the same lines, but of course we cannot simply increase the load by five hours. I suggest that all students learn arithmetic (including pre-algebra i.e. arithmetic with an unknown). Due to increased use of calculators and information technology (and corresponding decrease in the importance of pen-paper computation) this should take only part of their schooling careers.

After that, all students are required to take math, in the sense of a writing subject where students practice reading and writing theorems (about arbitrary, but interesting, topics). In addition to this they can opt to take classes like analytic geometry/precalculus/calculus if they are pursuing careers in math or science, but all students are required to take mathematical communication/reasoning/critical_reading courses, to serve counterpoint to their "English" classes of fiction/rhetoric/literature.

 

[Math Is Hard]

People in the "ivory tower" of mathematics may not have an understanding of the true needs of the general population. Those in the ivory tower of English may suffer the same misunderstanding.

What we, the unwashed masses, need from our math classes is to learn basic accounting and basic logic to understand our bank balances and the mumbo jumbo we receive back from our insurance claims. As for English studies, most people need basic reading comprehension but not the skills to analyze great works of literature in depth.

Mathematics and English are both rich and beautiful subjects for those who wish to specialize, but we can't expect everyone to feel the need to delve deeply into either one. Most people are just looking for basic toolsets to manage their everyday lives.

 

[Crosson]

Symbolipoint, your compromise sounds an awful lot like the way things currently are (at least here in America)!

I really agree with you, Math_Is_Hard. Let's give the people what they want: basic skills that are useful throughout many areas of life. The ability to decode mumbo jumbo, to solve problems rationally, to make suggestions that are consistent and well-defined etc.

not really true. It is perfectly possible to do non-linear dynamics without any recourse to seeing a computer.

Okay, here is a problem for you to solve by hand: Given the following dynamical system, describe the geometry of the underlying attractor:

dx/dt = 10(y-x)
dy/dt = 28 x - y - xz
dz/dt = xy - 2.666z

...

You may think that my question is unmathematical, as many traditionalist do, but those who work in the field (N.D. & Chaos) think that the genuinely interesting answer to this question expands what ought to be considered mathematics.

The bottom line is this: no mathematicians prior to the computer age would have assigned any imprtance to this exact set of equations. The first clue appeared when Lorenz (1964) integrated the equations numerically and got total nonsense.

After re-writing the code (with punch cards) to create a routine which emphasized accuracy, Lorenz obtained something resembling an approximation to a (long since gauranteed to exist) solution to the above system. Apparently these equations "magnify errors".

People had been integrating linear equations on machines for a while, and no one had seen anything like this. Efforts to treat this exponential error magnification using mathematics (mostly analysis) led to the notion of lyaponov exponents. Now we can prove analytic results of this nature for various dynamical systems, so a lot of mathematical understanding was spawned by this numerical experimentation.

The counter argument could be "its possible to create of concepts like lyaponov exponents or correlation dimension without ever experimenting with computers" but this is highly suspect. Even after the discovery of error magnification in Lorenz's equation's, it took nearly a decade for computer graphics to get good enough to suggest that dynamical systems such as Lorenz's have "strange attractors" with non-integer dimension.

 

[Pathway]

Is this not the difference between university and college?

In university you learn how to learn; you learn principles, academia, foundations, theory, giving you a knowledge base they will open many doors.

In college you learn a trade that will open a few doors.

(Maybe just here in Canada.)

In America you might have a university, within which are contained several colleges. You might have the College of Engineering, the College of Letters and Science, and the Law School--or you could have what Harvard does: within Harvard University is Harvard College, where all undergraduates go. College does not imply a lack of prestige or an inferiority of education in the United States.

 

[CRGreathouse]

Is this not the difference between university and college?

In university you learn how to learn; you learn principles, academia, foundations, theory, giving you a knowledge base they will open many doors.

In college you learn a trade that will open a few doors.

(Maybe just here in Canada.)

In the US the first is called "college" (or tertiary education), in which you go to a university or college (the former is large, and may have subdivisions called colleges). The latter is called "trade school" or "vocational school".

 

[Alkatran]

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



The large majority of people don't need to know how to prove theorems or why division works. We give them a little push in this direction, but that's all. There's no point in doubling the amount of time spent on math if it just doubles how much people forget.

 

[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.