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Mathematician Keith Devlin writes about how the capabilities to work with maths have changed since the late 1960s. He summarizes what he considers to be the essential skills and knowledge that people can focus on as more and more is turned over to software.
The shift began with the introduction of the digital arithmetic calculator in the 1960s, which rendered obsolete the need for humans to master the ancient art of mental arithmetical calculation. Over the succeeding decades, the scope of algorithms developed to perform mathematical procedures steadily expanded, culminating in the creation of desktop and cloud-based mathematical computation systems that can execute pretty well any mathematical procedure, solving—accurately and in a fraction of a second—any mathematical problem formulated with sufficient precision (a bar that allows in all the exam questions I and any other math student faced throughout our entire school and university careers).
So what, then, remains in mathematics that people need to master? The answer is, the set of skills required to make effective use of those powerful new (procedural) mathematical tools we can access from our smartphone. Whereas it used to be the case that humans had to master the computational skills required to carry out various mathematical procedures (adding and multiplying numbers, inverting matrices, solving polynomial equations, differentiating analytic functions, solving differential equations, etc.), what is required today is a sufficiently deep understanding of all those procedures, and the underlying concepts they are built on, in order to know when, and how, to use those digitally-implemented tools effectively, productively, and safely.
Source : What Scientific Term or Concept Ought to be More Widely Known?
(Score: 2) by melikamp on Monday February 05 2018, @10:10PM (1 child)
I can. For example, the notion of a cubic root is a concept, which is a special case of a more general concept known as rational exponent. Using something like R to compute exact and approximate cubic roots is a skill. One can clearly understand and manipulate the rational exponent in a pure setting without having a clue about what to do in order to compute the cubic root of 2 down to 5 digits after the dot. On the other hand, one can easily learn to type 2^(1/3) into R without knowing anything about the rational exponent, and copy down the answer. In my GP post I offered a view that understanding the rational exponent, for example, is crucial to an intelligent application of relevant skills, but not the other way around.
Well you are not disagreeing with me. I said, useful mathematical skills are in constant flux, and your example of calculator skills plays right into my goal. That skill used to be useful, now it's not. For a netizen, It would be far more effective to have a conceptual understanding of underpinnings of algebra and number theory. The only skill they need is basic programming, so that they can translate all of their computational questions into something an app can understand, and then use their broad understanding of concepts again to interpret the results.
I would be first to suggest that even the most foundational concepts in mathematics will eventually need a review, which is kind of what happened 100 or so years ago when everyone switched to the axiomatic approach, and several times before that (paradigm shift they call it). But the rate of change of useful concepts is at least an order of magnitude slower than the rate of change of useful skills, it seems like. And even when professional mathematicians rearrange the cornerstones of their cathedral, it rarely trickles up to the users. Defining every branch of math as a subfield of the set theory, for example, was a tremendously successful endeavor which failed to effect any change in the actual mathematical work being done, or 95% of it anyway. Analysts stopped banging their heads against the wall and spawned the field of topology, while everyone else safely ignored everything Cantor ever did. Feels like something truly amazing would have to happen before we change the definition of the derivative, for example, or stop explaining what a prime number is.
(Score: 1) by khallow on Monday February 05 2018, @11:57PM
Then compute the cube root without knowing what a cube root is.
Yes, I agree. They can learn this. But why expect them to remember this lesson ten minutes later?
Review for what? What's the justification?
Note that you aren't actually disagreeing with me. Useful skills is a proper subset of all skills.