SoylentNews is people
SoylentNews
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, @07:13AM (4 children)
I think there are two separate things we are talking about here. I think, and the summary seems to agree, that while useful mathematical skills are in constant flux, the concepts are not, and are more important than ever. What distinguishes statistical predictions based on large samples from voodoo is exactly the practitioner's understanding that the expected probability of error is based on the properties of the normal CDF, which implies the understanding of the laws of probability, the real number line, a function, an inverse function, a curve on the xy-plane, and an antiderivative, though the latter may be not by that name. This is also the the bare minimum a skeptical non-statistician needs to know in order to research the validity of a published result. They do not need enough statistical background to detect bullshit, but these are the skills required to understand white papers which report whatever bullshit detected.
So I would go on a limb and suggest that we should expand and build up the entire pure math, and try to give everyone a really good understanding of at least statistics and logic, as well as computer-assisted computational skills, all at the expense of more traditional computational skills.
(Score: 1, Insightful) by Anonymous Coward on Monday February 05 2018, @07:30AM
Meanwhile, back at the farm, the University I teach at has decided that deductive logic is not a foundational skill. So students now have to take some class that involves more "calculaton". Blatant move by the Math dept to steal FTEs from the Philosophy Department. And do you know who else had a PhD in Math?
(Score: 1) by khallow on Monday February 05 2018, @07:01PM (2 children)
I disagree on two points. First, you can't separate the skills from the concepts in practice. While it is possible to treat math as a bunch of black box algorithms (and some students manage that well enough to pass tests), humans won't remember a pile of those things for the rest of their lives without context.
Second, I disagree that mathematical skills change much. While being able to calculate a cube root by hand has devolved from useful in very rare circumstances to a curiosity, the skill itself didn't change any. We haven't come up with new ways to calculate cube roots in living memory (it's pretty stable since the 18th century). But due to calculators and computers, the relative usefulness of the known approaches to calculating cube roots has changed. For example, it's better to calculate cube roots via calculator than by hand. On that calculator, it's better to calculate cube roots by Newton's method [mathforum.org] than trying to implement some of the known [wikihow.com] by hand algorithms or using logarithm tables [wikipedia.org].
There are new algorithms developed every day, but most of these are far from any relevance to the layman. For most of the math in normal use, they've been around for a while.
(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.