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In a world where Wolfram Alpha can do all the algebraic calculations for you, isn't learning to solve problems more relevant? I'd say that problem solving is th

In a world where Wolfram Alpha can do all the algebraic calculations for you, isn't learning to solve problems more relevant? I'd say that problem solving is the core of mathematics. Am I wrong?

flatline 17y ago
The conceptual understanding is certainly essential, but if you want to use a computer to do symbolic/algebraic manipulation, how could you be sure to understand the results or even properly phrase the question without a thorough understanding of the symbols and the rules for their manipulation? The best I think that something like Alpha could do is to help elucidate the underlying principles or give you a quick answer for guesswork or estimation, like a calculator.
- jey 16y ago
  You contradict yourself. As you said, one does not need to be skilled in combinatorial search, er I mean, algebraic manipulations to be able to solve problems; one only needs a solid conceptual grounding and understand the symbols and the rules for their manipulation. I wouldn't be surprised if most students who pass college calculus courses with good grades are entirely unable to recognize the situations where calculus is applicable.
yequalsx 17y ago
Yeah, Wolframalpha is a disruptive technology. Is it still relevant to learn algebraic manipulations? I don't know. There are compelling arguments for both points of view. When you say problem solving do you really mean so called practical applications? If so then I disagree. If you mean solving problems to understand the essence of mathematical ideas then I agree. For instance, linear equations are the easiest equations to solve. All of them can be manipulated to yield the solution (or be reduced to showing all numbers are solutions or have no solutions). Linear equations are really 1st degree polynomials. What about second degree polynomials? After a clever bit of insight and some work we get that all 2nd degree polynomials can be solved. The quadratic equation is a solution to a problem in mathematics. It's useful as it leads to a greater understanding of mathematics. I believe that this is what mathematics courses ought to focus on. Not on problems that involve throwing a ball in the air and predicting how long it takes to hit the ground.
- jimbokun 17y ago
  "Yeah, Wolframalpha is a disruptive technology. Is it still relevant to learn algebraic manipulations?" We've had calculators for decades now, and my kids are still learning how to do arithmetic.
  - yequalsx 17y ago
    But is it worthwhile or important?
    - jimbokun 17y ago
      I think it is. If you can't perform the algorithms yourself, I don't think you really understand what the symbols on the machine really mean. When I worked at a bank, I remember complaints about an employee whose job involved running certain calculations in a spreadsheet. Sometimes the numbers would be off orders of magnitude due to some error somewhere, but the employee would just carry on as normal. If you have some clue as to what the calculations are supposed to be doing, there should be some alarms going off if the numbers are not at least reasonable. Or what if you just wrote a program that includes some arithmetic calculations, and need to verify that you are getting the right answers for your test data? I remember another time at that same bank, where a developer did not know that multiplication had precedence over addition, which led to errors in a financial calculation. I'm sure you could easily think up many more examples like this.
      - yequalsx 17y ago
        There was a time when it was important to know how to use interpolating polynomials. Rationalizing the denominator was only useful before calculator became ubiquitous. Being able to make fire was without use of matches or flint used to be essential. There are plenty of ways to teach algorithms without using arithmetic. Maybe a knowledge of doing basic arithmetic by hand is essential to understanding higher math. I don't know. It's worth exploring though.
deleted 17y ago
[deleted]
zackattack 17y ago
Let me share a frustrating experience I recently had. I hired some dude off Craigslist to install my TV wall mount because I don't have the proper tools. I tried to explain where I wanted the TV: I want it to be centered in the wall, but then moved a few inches to the right. He simply couldn't understand what I was saying. It blew my mind, I had to just measure it out myself and mark the spot. If he had a solid of understanding of basic high school algebra, he would have been able to formulate the problem in his head. Done.

In a world where Wolfram Alpha can do all the algebraic calculations for you, isn't learning to solve problems more relevant? I'd say that problem solving is the core of mathematics. Am I wrong?

flatline 17y ago

The conceptual understanding is certainly essential, but if you want to use a computer to do symbolic/algebraic manipulation, how could you be sure to understand the results or even properly phrase the question without a thorough understanding of the symbols and the rules for their manipulation? The best I think that something like Alpha could do is to help elucidate the underlying principles or give you a quick answer for guesswork or estimation, like a calculator.

jey 16y ago
You contradict yourself. As you said, one does not need to be skilled in combinatorial search, er I mean, algebraic manipulations to be able to solve problems; one only needs a solid conceptual grounding and understand the symbols and the rules for their manipulation. I wouldn't be surprised if most students who pass college calculus courses with good grades are entirely unable to recognize the situations where calculus is applicable.

yequalsx 17y ago

Yeah, Wolframalpha is a disruptive technology. Is it still relevant to learn algebraic manipulations? I don't know. There are compelling arguments for both points of view. When you say problem solving do you really mean so called practical applications? If so then I disagree. If you mean solving problems to understand the essence of mathematical ideas then I agree. For instance, linear equations are the easiest equations to solve. All of them can be manipulated to yield the solution (or be reduced to showing all numbers are solutions or have no solutions). Linear equations are really 1st degree polynomials. What about second degree polynomials? After a clever bit of insight and some work we get that all 2nd degree polynomials can be solved. The quadratic equation is a solution to a problem in mathematics. It's useful as it leads to a greater understanding of mathematics. I believe that this is what mathematics courses ought to focus on. Not on problems that involve throwing a ball in the air and predicting how long it takes to hit the ground.

jimbokun 17y ago
"Yeah, Wolframalpha is a disruptive technology. Is it still relevant to learn algebraic manipulations?" We've had calculators for decades now, and my kids are still learning how to do arithmetic.
- yequalsx 17y ago
  But is it worthwhile or important?
  - jimbokun 17y ago
    I think it is. If you can't perform the algorithms yourself, I don't think you really understand what the symbols on the machine really mean. When I worked at a bank, I remember complaints about an employee whose job involved running certain calculations in a spreadsheet. Sometimes the numbers would be off orders of magnitude due to some error somewhere, but the employee would just carry on as normal. If you have some clue as to what the calculations are supposed to be doing, there should be some alarms going off if the numbers are not at least reasonable. Or what if you just wrote a program that includes some arithmetic calculations, and need to verify that you are getting the right answers for your test data? I remember another time at that same bank, where a developer did not know that multiplication had precedence over addition, which led to errors in a financial calculation. I'm sure you could easily think up many more examples like this.
    - yequalsx 17y ago
      There was a time when it was important to know how to use interpolating polynomials. Rationalizing the denominator was only useful before calculator became ubiquitous. Being able to make fire was without use of matches or flint used to be essential. There are plenty of ways to teach algorithms without using arithmetic. Maybe a knowledge of doing basic arithmetic by hand is essential to understanding higher math. I don't know. It's worth exploring though.

deleted 17y ago

[deleted]

zackattack 17y ago

Let me share a frustrating experience I recently had. I hired some dude off Craigslist to install my TV wall mount because I don't have the proper tools. I tried to explain where I wanted the TV: I want it to be centered in the wall, but then moved a few inches to the right. He simply couldn't understand what I was saying. It blew my mind, I had to just measure it out myself and mark the spot. If he had a solid of understanding of basic high school algebra, he would have been able to formulate the problem in his head. Done.