SPLIT: Clearing Fractions (in the future?)

[mmm4444bot]

When/where did I say [that clearing fractions is called the Denis Rule]?

If vBulletin's search facilities were more than PsOS, I could probably answer this question by quoting. I've seen more than one Denis Rule, over the years. :wink:

[Getting rid of fractions is] just common sense, isn't it

Often, yes. Not always in classes where the lesson of the day concerns working with rational functions, however. If they ain't givin' specific instructions, then to each their own!

Maybe I am in the minority, but when looking at something like \(\displaystyle \frac{1}{3}-\frac{1}{x}\) I see \(\displaystyle \frac{x-3}{3x}\). That's not really a step, for me. I just see it.

And, when I see a compound fraction, my general inclination is to move toward applying (a/b)/(c/d)=(a/b)(d/c). I must really love that rule more than the Denis Rule. :-D

 

[mmm4444bot]

Now Sir Lookagain can take both our names in vain!

He wouldn't be the first! That's for sure.

Many ways to skin a cat, too. Each of them is a valid way.

Points posted in this (soon-to-be split?) thread will become moot, when schools begin viewing algebraic-manipulation skills the same as they viewed slide rules (after calculators were widely available). Some schools have already stopped requiring memorization of the multiplication table. What's next?

In other words, I think that we're already moving toward a future where schools no longer teach algebra. Eventually, nobody will have need of these rules, as people will solve math using handheld devices or by speaking to the air around them...

Schools will be reduced to teaching people how to explain problems to machines; there will be little need to understand what's happening under the hood.

 

[Deleted member 4993]

Schools will be reduced to teaching people how to explain problems to machines;

We do that now - we teach programming...

there will be little need to understand what's happening under the hood.

I don't think so. There will be need for different understanding. Our ancestors in the agricultural society understood cows and horses better - we got away from that to understand different things.

.

 

[mmm4444bot]

We do that now - we teach programming...

I'm not talking about programming languages; I'm thinking way beyond that. For example, a calculus project that today requires lots of knowledge will not require nearly as much work, in the future. It will only require communicating with machines sufficient for them to produce the finished project. No need to understand any rules or techniques; machines will do ALL the solving. Machines will probably program machines, so most people won't even need to understand that -- only the "elites" (who may be viewed in their time as "priests", heh, heh, heh).

These changes won't be in my lifetime.

there will be little need to understand what's happening under the hood.

I don't think so. There will be need for different understanding. Our ancestors in the agricultural society understood cows and horses better - we got away from that to understand different things.

I'm speaking of the general public. Schools as we know them today are on the way out.

Schools of the future will not be for the general public; the average person will get their education and training "on-line" -- whatever form that takes. I believe the only thing resembling what we think of as schools today will be extremely specialized institutions -- only for the few...

Schools today are already teaching less "hands-on" math.

Here's a guess: 25 years from now, most secondary schools will not teach how to factor polynomials (for example). There won't be any need for factoring by hand.

I make that guess because I'm confident that I won't be around in 25 years.

 

[JeffM]

Mark

I agree that much of the current mechanics of math will not be taught, or at least not taught intensively, in the future because technology will do it for us. I see this already: tutors here give answers that take advantage of graphing calculators. Fifty years ago, in order to make a graph, we would have had to solve problems that today we use a graphing calculator to solve. Slide rules, abacuses, and books of mathematical tables are now curiosities. Nevertheless, I suspect that it will still be advantageous to know how to do some of the mechanics: for example, I suspect just about all adults will find it advantageous to be able to do simple arithmetic in their heads. Mechanics will just be judged as far less important and worth much less time.

What I do not see being eliminated is teaching how to apply math. That is, kids will still be given word problems. The task will be to translate a problem into mathematical concepts and then into the notation that machines expect without worrying much about how the machine then produces answers. In other words, math education will involve mostly concepts and applications, with little emphasis on mechanics. The machine can't give an answer to a problem until it is syntactically well formulated, and it won't give the right answer unless what is syntactically well formulated is conceptually appropriate. I may be wrong of course; machines may become so attuned to the human mind that they can parse problems stated in natural languages without the need for human intervention.

 

[Bob Brown MSEE]

Subtraction and Division are unnecessary

when looking at something like [FONT=MathJax_Main]1[FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Math]x[/FONT][/FONT] I see [FONT=MathJax_Math]x[FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Math]x[/FONT][/FONT].

This has been an interesting discussion.

When my son was learning basic arithmetic, he had my HP-35 with tape over the [-] and [/] keys.
I told him that they were unnecessary and confusing. I introduced him to the elements that define the field of rational numbers by listing the rules for [+] and
[*]. The [-] and [/] keys did not follow those associativity and commutativity rules. That is why these are confusing.

He used [chs] and [1/x] to perform all of the workbook problems (he loved workbooks)
I told him that "invert and {+ or *}" happens so often that teachers invented the [-] and [/] as a kind of short-hand.
We ALWAYS worked w/o shortcuts -- and could avoid confusion.

This problem would be worked by first getting rid of [-] and [/] by applying reverse "invert and {+ or *}" then multiply both sides by multiplicitive inverses.


Let A=answer then...

A = (1/3)-(1/x)	Given
A = (1/3)+(-1)(1/x)	Def of Subtract
3A=3((1/3)+(-1)(1/x))	Mult ='s
3A=3(1/3)+(-3)(1/x)	Distribution
3A=1+(-3)(1/x)	Def Mult ID
x3A=x(1+(-3)(1/x))	Mult ='s
3xA=x+(-3)x(1/x)	Distribution & Assoc
3xA=x+(-3)	Def Mult ID
A=(x+(-3))(1/(3x))	Mult ='s & Def Mult ID
A=(x-3)/(3x)	Def of Subtract & Divide

Soon this approach became so familiar that this is what he sees when presented with a ratio.

 

[mmm4444bot]

What I do not see being eliminated is teaching how to apply math. That is, kids will still be given word problems. The task will be to translate a problem into mathematical concepts and then into the notation that machines expect without worrying much about how the machine then produces answers.

Word problems may be solved not by applying math, but simply by knowing what to tell a machine.

EG: Person speaks the following.

Partial-volume request

Spherical tank

Radius five meters

Depth six inches

Machine then displays the volume, in various units cubed. (If it's Microsoft software, the machine also takes several minutes to compile and display a bunch of multimedia diagrams and animations. )


I wonder whether schools would be teaching how to add 1/2+1/3 today, if electronic calculators had displayed the answer as 5/6=0.83333333 from the beginning.

In other words, if ubiquitous machines (eg: handheld calculators, personal computers, cell phones, etc.) had been available AND doing exact arithmetic with fractions over the past 40 years, would there be a need to teach how to add 1/2+1/3 by hand today?

Already, it seems like some schools are "skipping" the multiplication table, and drills like 2345.4792 times 32.6582 times 653.3463 are a thing of the past. :cool:

 

[JeffM]

Word problems may be solved not by applying math, but simply by knowing what to tell a machine.

EG: Person speaks the following.

Partial-volume request

Spherical tank

Radius five meters

Depth six inches

Machine then displays the volume, in various units cubed. (If it's Microsoft software, the machine also takes several minutes to compile and display a bunch of multimedia diagrams and animations. )


I wonder whether schools would be teaching how to add 1/2+1/3 today, if electronic calculators had displayed the answer as 5/6=0.83333333 from the beginning.

In other words, if ubiquitous machines (eg: handheld calculators, personal computers, cell phones, etc.) had been available AND doing exact arithmetic with fractions over the past 40 years, would there be a need to teach how to add 1/2+1/3 by hand today?

Already, it seems like some schools are "skipping" the multiplication table, and drills like 2345.4792 times 32.6582 times 653.3463 are a thing of the past. :cool:

Mark

I am sure that you are right about the Microsoft hardware.

And of course you may well be right about the parsing problem. But how many commands would a kid have to learn: a partial volume command and its associated syntax, plus how many hundreds or thousands more? Further, even if the machines are powerful enough that very few commands would be necessary and the machine then requested the information it required to implement the command, the kid would still have to know how to identify the required command and the necessary information from the given information. This might require making word problems more opaque (that is, more like real world problems). That would be an interesting type of math problem to create and a useful one to learn how to solve.

And I fully agree with you that a whole lot of mechanics will be eliminated from the curriculum. In fact, a lot should be dropped now. I answered a question last night using logs and anti-logs because the technology of my student days made that approach the only one feasible, and so the approach is familiar to me. The poster, however, had a much more efficient solution that takes advantage of modern calculators. For probably over 95% of today's student population, there is no need for them to know anything about logs.

My opinion only. Cheers

 

[mmm4444bot]

how many commands would a kid have to learn: a partial volume command and its associated syntax, plus how many hundreds or thousands more?

I don't think that people will be memorizing much of anything. Machines will guide people, instead.

Maybe there will be machines to help people communicate with machines. That is, one machine will ask yes-or-no type questions until there's enough info to tell other machines what it is that the human actually wants to accomplish. (Transfers from one machine to others will be seamless.)

Those are my thoughts about the distant future. What I see happening today is a slow shift away from certain types of hands-on math. I'm thinking that arithmetic with ratios and some algebra rules and techniques will be on the chopping block within 10 years...

 

[JeffM]

I don't think that people will be memorizing much of anything. Machines will guide people, instead.

Maybe there will be machines to help people communicate with machines. That is, one machine will ask yes-or-no type questions until there's enough info to tell other machines what it is that the human actually wants to accomplish. (Transfers from one machine to others will be seamless.)

Those are my thoughts about the distant future. What I see happening today is a slow shift away from certain types of hands-on math. I'm thinking that arithmetic with ratios and some algebra rules and techniques will be on the chopping block within 10 years...

I don't try to worry about the distant future. I have enough trouble worrying about tomorrow.

In any case, I think I m going to reread Forster's "The Machine Stops."

 

[mmm4444bot]

I'm with you; I don't try to worry.

If you don't want to worry on behalf of mankind's future, though, maybe you ought to look for a different book to read, lol.