Why do we still teach people to calculate?
freakonomics.com
freakonomics.com
Mental arithmetic is vital to mental self defense.
It made sense since order-of-magnitude estimations constantly come up naturally in all fields of physics, whether you are asking "is this plausibly possible", "what kind of instrument do I need for this measurement" (and later "can we even measure this in this setup") or "are the results of my experiment plausible". There's a reason we joke that to the physicist g=10 and pi=3.
It's difficult to test this kind of thinking in a written test without turning it into something entirely different, but not everything in school has to be on a test. Typically half of our marks were made up from classroom participation.
Guessing may get you a part of the way if you're very good at it, but repeatability is what really hammers it into you. It also gets you out of trouble when for any reason your guessing instinct is not at full steam (when tired or otherwise incapacitated, anxious, etc) since you can usually relatively quickly rely on the steps you have drilled for.
If you don't get the basics of arithmetic it is incredibly hard to grasp the idea of "equals" as a pivot point. Division by hand helps to give us a hint that numbers can be broken up into constituents and still be the same.
You internalise a whole set of behaviours, typically via repetition, and it unlocks understanding beyond itself imo
For example, when I taught differential equations, my non-scientific observation was that otherwise smart students struggled more with getting the algorithms down, if they struggled with algebra. Having to constantly jump down to a lower level of abstraction while performing the algorithmic steps is what caused confusion to mistakes.
The tape measure itself can be used to perform basic arithmetic, and will even return results in Freedom Fractions if that's how it is labeled.
I have seen people eyeball measurements and taking notes, and bringing in perfectly manufactured goods (blinds, cupboards, etc.) in return. Esp, in construction, nothing is precise, so being able to eyeball correctly after a quick measurement with a tape and doing some basic math on the numbers, not only accelerates tasks, but is a crucial pillar of being able to work on construction sites, and be productive.
Tell me you have never built a thing[1] in your entire life that was more complex than an IKEA bookshelf without telling me that challenge: I do hobby projects all the time and I couldn't even tell you how many mental-math problems I do per hour while doing so. It's a LOT. Depending what I'm making I might well spend more time doing math in my head and on paper than I do actually putting tools to materials to build the thing.
[1]: by this I mean actually building. With your hands.
Calculation is one of the core skills of human cognition and is correspondingly foundational for just about all of human enterprise. Not doing it means limiting your ability to participate in said enterprise.
Well, that's what tolerances are for. I don't need a perfectly accurate answer. Just one that's within 1/32" or more likely 1/4" to 1/8".
At home in my own geeky little workshop, where I don't need to satisfy anyone else's proclivities, everything is exhaustively and exclusively metric-only. My tape measures are metric, my hand tools are metric, my drills are metric, and my fasteners are metric. Working with Freedom Fractions is absolutely forbidden in my workshop (as are all screws with Phillips, JIS, Pozidrive, or Reed-Prince heads, but that's a different rant).
And, I must say: It's better this way.
Torx?
Second general choice is hex-head screws. These snap into inexpensive shallow magnetic driver bits with a satisfying click. But they're ugly and kind of rough once installed, and they're not available in flat-head versions. (Hex-head shoulder screws are my only choice for drill-point screws: They maintain positive angular alignment by default and that's crucial for drilling holes in metal.)
Torx is fine too, I suppose. I don't have an anti-Torx rule in my workshop, but I try to avoid buying them. They don't tolerate angular misalignment as well as Robertson, and they don't maintain positive angular alignment like a hexagonal shoulder screw does, and they don't stay put with friction like Robertson or snap onto a magnetic driver bit like hex. There's lots of stuff Torx is not very good at doing.
Theoretically, I can probably put more torque into Torx than any of the other options listed here, but I don't find that to be a practical advantage in this kind of application: When I can already drive a Robertson screw through a chunk of old-growth wood, I don't need to improve that part.
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For machine screws, I've standardized on stainless steel button-head socket cap screws as a first choice. I've got a deeper selection of them than a good hardware store does, and they're all sorted. (Why stainless instead of graded? Because I don't want them to rot, whether sitting in a bin for decades or used outdoors or whatever and I do not want to stock more than one kind. My fastener collection is crazy enough without also multiplying it by different grades.)
There's other stuff, though, too. For instance: Regular socket cap screws have their place -- it just isn't first place.
And at the scale of things I build, I mostly use M3, M4, and M5.
I buy regular-length Bondhus non-ball hex keys in simple bulk packaging to fit my standard M3, M4, and M5 machine screws. They're very high quality tools, and they're rather inexpensive in bulk. And thus, it is no big deal if I misplace one while I'm working -- I've got more on-hand, and I'm not afraid to get more coming if stock gets low.
Bondhus makes a decent-quality hexagonal screwdriver, too, and these are nice to keep around because the design is not like the gigantic T-handled abortions that so many other manufacturers sell: It's just screwdriver-shaped, and it works just like a familiar screwdriver does -- but for socket-cap screws! This was the discovery that allowed me to completely abolish Phillips screws forever from my workshop.
I've also got sets of hex keys -- of course I do. Long, ball-end, plain, whatever. I try to avoid cornering myself into a situations where these non-regular variations would ever begin to be useful to begin with, and the long versions are mostly only useful for disassembling stuff that someone else had overtorqued. (But overtorqued fasteners are different rant.)
The inch-foot system does a tradeoff. Easier computation when your lengths are certain combinations of simple fractions, but difficult ones when not.
- A pile without apples is a pile of apples. We call it the empty pile.
- The successor function is to add an apple to the pile. Apply the successor function to the pile of apples and you still have a pile of apples.
- A pile of apples is as large as itself
- If the left pile has as many apples as the right pile, then the right pile also has as many apples as the left pile
- If you have two piles of apples you can add them by checking if the second pile is empty. If it is empty you are done and the result is in the first pile. If not take an apple from the second pile and add it to the first pile. Then check again and repeat
And then you can teach him about the bijection between money and apples. After a little more set theory you can start constructing two piles of apples, one representing what you have one what you have to give away ...
The possibilities are endless.
I learned about commutators from a guy on the front row who was really good at Rubik's cubes. Concrete examples relatable to your hobbies are great.
I also tried to pay for my lunch by commutating, but the lunch lady wouldn't have it.
the problem with bad ideas in education is the system might be stupid enough to follow through with it, like that new math nonsense.
Primary school teachers respect core fundamentals as kids acquire "mental muscle memory" and realize they have to both create some axiomatic knowledge (axiomatic in as much as you know 9x9 is 81 from rote recall, not because of a belief in inductive reasoning) as well as try to begin an uplift to reasoned knowledge (that 2^2 x 2^2 is 2^(2+2) is 2^4) and some coding/transcoding (1/2 == 0.5) Cuisenaire rods come in and out of fashion. Crows can count. Kids are sometimes dumber than crows.
Mathematicians are very much in davis/hersh "what is mathematics anyway" -I believe Hersh noted that you can be in a field where only 3 other people worldwide can talk to you cogently about your work, and peer review is meaningless.
Statisticians are very comfortable that approximations work, but are less concered with accuracy at times, and very much concerned with methodology. I've had quite remarkable conversations with them about sample size, and how UX people can survive on 5 responses. I neve predict which side of the problem they're going to respond.
Data scientists are almost intuitive at times. sometimes the reliance on codified knowledge (numpy/pandas) and a belief in the p-jacked value or an obvious excel error is frightening. I think they divide sheep/goats into the numerate, and the highly visual.
I consider myself semi literate, mathematically speaking but in fact, I stumble over basic arithmetic all the time, and I struggle with ideas behind complex numbers, trig. I have to re-prove things which should be known, re-induce belief in things which are based on inductive reasoning, I question commutation all the time. How the hell can 2 x 3 be the same as 3 x 2 there's a fundamental left-right ordering in my brain which at times I ask myself is this inside the farsi or hebrew or thai or boudestrophon flow texts, suggesting that not all right-to-left ordered languages obey it yet alas I do.
I also still don't entirely understand why school focussed on trig so much given that very few of us are navigating by sextant. I suspect at times it was dividing us into the ones which drink from the hand, and the ones which lap from the stream.
Do they teach decimal to octal and hex and binary in primary school yet? Will the world be different when the last of the duodecimal measurement learners have died?
For children, it is fundamental that they calculate. Learning arithmetic is the first introduction into the idea of infinity, which is a vital concept all children must come to terms with. The natural numbers pop up everywhere and is a basic life skill.
Mathematicians and philosophers can pontificate later as to what the exact nature of mathematics is, and if the real numbers are actually real, but none of that means that kids shouldn't learn how to count.
Nothing.
Trig is useful for practical problems involving angles, and ubiquitous in applications in engineering and statistics. One year in school we spent a long time on logarithms and trig. Approaching 20 years later I consider it time well spent, and my one regret is that I didn't strive to understand it more deeply at the time, and had to revisit some details later (mostly about logarithms, but same category IMO).
This reminds me of getting my CS degree in the early 2000s, then slowly realizing that my comp-sci program had taught me lots of interesting things about how computers and programs work, and essentially nothing about how to write code for a living.
Also reminds me of playing Dominion (the card game) with another student, me saying "Okay, I've got 1 and 2 and 2 and 3, that's 8..." and him gasping "Wow! How did you do that so fast?"
Dude was a math grad student. I narrowly restrained myself from saying "Well, you see, I finished the 3rd grade..."
I arrived at university expecting to be writing a lot of programs and was soon struggling with a heavy load of mathematics and proofs. Our introduction to computer programming professor made it clear that that class was the only one where we would be taught to program. Computer science, he said, was mostly done with paper and pencil.
It wasn't quite true. We eventually did a lot of programming, but it was nothing like I was expecting.
My problem was that the fucking college counselors didn't know either.
When I signed up, I told the lady I wanted to be a computer programmer, and she said "You'll want computer science, then," and that wasn't true. The CS program was only interested in teaching me how to be a CS professor. A few years in I was scouring the course lists for classes on graphics, web, anything with a GUI even, and there was nothing. It was 100% command line C++, and they didn't even bother teaching us about IDEs and debuggers. Just Telnet, vi, g++.
It was like taking a pure math degree to become an architect.
Those have absolutely stood the test of time than any specific IDE or debugger.
A '57 Chevy has stood the test of time, but a modern car with an automatic transmission, power steering, power brakes, and air bags is probably a better choice for a driving school today.
Do not show restraint, and neither will I. For I am a mathematician, not a fucking calculator.
But once, early in undergrad, I was capable of multiplying 4-digit numbers rather quickly. But tallying never seemed useful except while grading exams -- during which my brain populates a lookup table over the course of an hour, and then I can tally small sums without hesitation. But after that day of grading, the lookup table is flushed for more important uses of short term memory.
I don't expect mathematicians to be lightning calculators, but expressing awe at grade-school arithmetic is a bit much.
That sounds nice in the abstract, but in practice it's slow going trying to do web dev when your experience is nothing but command line C++.
So you are typing Javascript into your browser's console to see what it does instead of a text editor and compiler. The iterative process is still very similar. Write code -> get the syntax correct -> see if output matches your mental model -> repeat.
You still want unit testing or some other system of reliably testing your code and detecting regressions. You need to use a distributed version control system to track your work over time and collaborate with others. You need to be able to figure out why your code is suddenly taking much longer to execute it than you thought it would. You need to figure out why the memory usage keeps going up and never comes down. You need to be able to gather requirements for the software you are writing.
All of those things are skills you need to learn regardless of the specific programming language.
From personal experience, yes, it is. People spend years getting really good at Java, or Ruby, or SQL, or whatever. You'll get better quicker at all of those if you start with a solid grounding in basic coding principles, but basic principles alone are not enough.
Maybe you're one of those 10x programmers I keep hearing about who can master any subfield instantly. That's lovely for you. Most of us do better with a bit of specialization, especially at the beginning of our careers.
While numeracy is certainly useful, isn't that only the surface point. The depth in deeper learning (we all learn - Edward Deming - but learn what?) is in shaping self and the internally and externally perceived worlds.
Ok, we get it. Wolfram Research has fantastic and somewhat underappreciated products (Wolfram Alpha is far more impressive than LLMs!) and you want to cash in on the AI hype. But please don't take it out on innocent students.
1. Basic math, eg long division. This is what all the commenters are focusing on, and I largely agree with folks - we need to keep teaching this.
2. HS / College math. E.g. memorizing all the rules to solve integral calculus equations. Or solving all those circuits in Intro to EE. I agree with Wolfram here. At this point, students should be mentally developed enough to start focusing on formalizing and reducing problems rather than rote memorization or hand calculations. It's good to go over all the rules and practice a few times. But to make a whole quarter of it seems a bit much.
While there is value in emphasizing things like defining/abstracting a problem rather than calculating the result of it, things like the "algorithm" typically taught to elementary schoolers to do long division is pretty much how you would naively/algorithmically program a division operator that can give a remainder. The multiplication "algorithm" given to me at a young age was what first gave me the insight into the connection between addition and multiplication. Abstracting this away in education could not possibly be doing society, math, or computation any service whatsoever, at least not in the way (I think?) he is imagining.
I do think there is value to asking whether a problem needs to be solved or redefined or abstracted in a different manner. I had a calculus teacher in college a long time ago that had a really hilarious way of emphasizing this - he'd often bury "trap" questions into his tests that looked really simple but would test your knowledge of a simple algebraic or trigonometric trick he'd only covered during lecture, and lack of knowledge of this trick would lead one down a hopeless 20+ page response to the problem, to the point where if you found yourself going off the rails with a complex solution, you knew you messed up somewhere. This forced me to look at the problem from a variety angles to see if I could solve it in a more clever, general, or simpler way, or if it could be rewritten (this was usually the correct way to solve it). I really valued that, even years and years later, but maybe that's not the kind of thing Wolfram is getting after in this interview.
You're planning to become a politician?
I would say it is intended for everyone who can not figure it out for themselves.
Definitely paid off. Can calculate numbers in my head pretty easily. Definitely used to help with tips or commissions or whatever.
1. Define the problem and why we're going to solve it (in D&D terms, I'd say this is a wisdom check)
2. Translate the problem into math language (needs imagination and experience, which is an intelligence check)
2.5. Levitt noted the importance of interleaved practice; why practice a tool intensively for the test only to never use it again? Supported by the authors of Make it Stick: the science of successful learning.
That all sounds better than "here kids, learn these tools out of context".