Will it rot my students' brains if they use Mathematica? (2002)
theodoregray.com
theodoregray.com
Something that occurs to me is that most of us have a poor grasp of why we even teach school math: It's treated as some kind of tournament to get into college, or a general IQ building exercise, or even a form of obedience training. The math that is "real" to them is just what they happened to learn, the way they learned it. But most people admit that they forgot all of their math immediately upon finishing school.
And school math isn't even how "math people" do math. I still enjoy doing a pencil and paper derivation for fun sometimes, but mostly I do math at a computer. The world does math in Excel.
I'm not too worried about Mathematica being proprietary. Like with programming, if you learn one language, you can usually learn another in a jiffy.
> It's treated as some kind of tournament to get into college
and
> But most people admit that they forgot all of their math immediately upon finishing school.
in those 2 quotes, you summarize the current issues with school mathematics. There is no real benefit in learning school math as most people will just use up to 6th/7th grade math in their daily work. Even professions that "do math" daily (like carpenters) have, most of the time, figured out ways to do it "quickly" without really doing math (see, for example, the diamonds on a measuring tape and how they are used to solve the Pythagorean theorem).
That leaves school math as "the great sorter" of students. Students that "can" take advanced math classes and use those as leverage in college admissions. To see this process in motion, look up parents' concerns with California's new math sequence and how those usually involve college readiness (as in, college admission advantage) over career readiness (as in, calculus II is needed for whatever profession my kid will do after high school).
He might be talking about the 3-4-5 trick (pythagorean triples). But the only place I can think of where that is really useful (roofing) has other, better, tricks for figuring out lengths of hypotenuses.
I am not sure what this similarity with mathematics means for either mandatory art or math classes, but perhaps there is something to consider in our arguments.
[0] https://www.maa.org/external_archive/devlin/LockhartsLament....
I had this experience myself. Math during grade school was delivered kind of like a set of unquestionable facts. Much of which was valuable but I wish it didn't take me until college and some random tangential exercise to realize i didnt really understand how real numbers work.
This continues well into 200-level and even 300-level college mathematics. Gallian's Contemporary Abstract Algebra spends a lot of time exploring groups of the form a + b√n without explaining why.
In general, it seems like math education should step back and read John Dewey and stop teaching mathematics as inert knowledge.
Doing mathematics on a computer is pretty darn tricky. It's not impossible but tools like Mathematica tend to lack the flexibility that pen and paper grant you.
Of course computers are better at computation, but I don't think that counts as 'doing math'. That would be like saying a loudspeaker composes music.
Sounds like a case of:
"The math that is "real" to them is just what they happened to learn, the way they learned it"
Programming is doing mathematics, and computers do a great job of enabling one to program.
At the places where I've worked, general math or quantitative problems that go beyond the capabilities of Excel are brought to the "math person."
One of my favorite math courses was a college class that focused on concepts - technically not an education class but intended for those in education majors. It covered 5th-8th grade math and showed me the why of so many things I had just taken for granted previously, and I'm sure if math was taught that way it would be much better remembered.
Of course, it could still be argued over whether math should be intended for practical use or not, but I think it's important that math be taught from (at least one of) a conceptual basis, or a historical basis (eg. [0] on the creation/discovery of imaginary numbers - it explains why they were needed, as well as interesting historical background).
(This was typed in a hurry, sorry if it's not entirely cohesive.)
https://www.psychologytoday.com/us/blog/iq-boot-camp/201606/...
Once you understand the why, the how becomes so much easier.
The interesting thing is how each student has a different path to understand the "how". It sounds like you have particularly vivid mental imagery, and gravitate towards visualizations for solving problems. This isn't true for everyone; it's not necessarily aphantasia but a unit circle or slope/area curve just isn't as useful for some as it is for you, others might prefer to think in terms of limits or series or patterns or a myriad of other techniques. You can repeat "The derivative is the slope of the curve" until you're blue in the face but some people just won't 'get it', finding the means by which they can get it is the fun part.
Then you have to explain why the slope of a line is significant. For example if you have the position of something in the form of an equation you can take the derivative and get the velocity. Anybody who understands why the slope of the line is interesting will fair far better.beyond that the area under the curve of a velocity graph will give you the distance traveled.
I actually agree with you in general but, this isn't a guarantee.
I've known plenty of people who "only" know how to "program" in Visual Studio. They don't understand how to use git outside of a helper GUI, they didn't understand the distinction between a compiler and a linker, or build options, or pathing issues--at least not down to the bare wires of what's actually going on.
I'm not saying "everyone" is like that, but when you give exhaustive encapsulating tools, oftentimes people will only learn what they have to, to get running and never delve deeper. Of course, never delving deeper is the key issue but some frameworks/systems make delving deeper much easier than others, and better yet, encourage it.
For an example, it was way easier for me to learn deep aspects of "what is really going on" when I started learning Linux--even immediately when I had to decide how to partition my hard drive correctly just to install it. I "could" just hit auto format but... what are all these buttons? Why is there different types of partitions? Is there a benefit? Not hiding me the inherent complexity allowed me information to better my understanding.
Perhaps this applies more to programming than math, as, the core is to do math... right? The rest is just implementation details of "how do I tell this [math thing] to do [thing I want]". So as long as you know what you're asking "I need a line integral" that's a general enough term. You're learning the _techniques_ not only the framework itself.
Also, the “video games make people violent” argument rears its ugly head.
There is an argument to be made that gangsta rap made a generation of impressionable kids much more rough around the edges, as it defined the new “cool” for them. Even though it wasn’t “supposed to be for them”, they knew all the words, which are loaded with expletives and misogynistic lyrics, and sowing distrust of the police etc.
Maybe I am just a social conservative but I agree with this lady who tried, in vain, to fight against the capitalist machine that is ths music industry: https://m.youtube.com/watch?v=Pr6gb1w72xA
Kids are impressionable — the Merchants of Cool documentary clearly shows how capitalist industries exploit that to sell more stuff: https://www.pbs.org/wgbh/frontline/film/showscool/
It also helped shape a rift, in my opinion, between the Black and the White youth that grew up now, which had been closing thanks to Liberalism and efforts to move past it.
Why don’t we set aside the rap battle and Mathematica concerns and focus on well known human society features failing people.
Cut the crap with the pearl clutching “there’s an argument to made other peoples behavior did…” and consider yours as a member of an implicitly caste based society.
Treat yourself like the subject of study instead of abstractly focusing on the flock, dad.
Western psychotherapy role play is the worst personality trait.
Music is a reflection of culture, and when there is violence in culture, artists make art out of it to cope. If black music seems more violent to you than white music, please meditate on how much more violence black people experience than white people, and why someone would presume one genre is reactive and another genre is causative.
If you want to hear the arguments against gangsta rap and the exploitative music industry (exploiting externalities just like any capitalist industry)
Listen to me do it in rap form, from 2015 … and please comment
Why has Blaxploitation became not ok but gangsta rap is still considered “just art” with misogynistic and violent lyrics portraying Blacks self describing themselves as materialistic thugs, they were role models “poppin bottles”? Are women not 50% of the population, why is it OK? Why is it OK to have lyrics about mistreatinf Black women etc. as “just art” and Ice Cube who uses B’s and H words about women in his music comes on to Bill Maher’s show to lecture him about how using the N word even about yourself in a joking way is “not cool”? Yet it is cool to be a non-woman and using slurs against women? I think both are uncool.
[1]https://www.flavorwire.com/489278/25-of-musics-most-obnoxiou...
Edit: Thanks for your Soundcloud link. I like your flow. You make fair points, but I see no evidence that any of the issues you raise are worse in hiphop than any other genre. (Or that hiphop's issues are internally generated, and not amplified by an exploitative industry that makes bank off stereotypes.)
Ok, then make the argument. There must be a mountain of evidence if this is true.
Or maybe I’m wrong, maybe NWA caused the LAPD to brutalize Rodney King.
I think Monster Hunter World was the first time I worked out a dynamic programming algorithm in anger.
Care to elaborate?
If there is, then it's a genuine grievance.
But if there's not — what's wrong with paying for a good product? I have yet to see an equally good alternative to other software products, like Excel, Photoshop or Logic Pro. There are alternatives that have the same functions, but none that have the same level of quality.
After all, most of us here on HN are being paid to write code. It's really weird that we would expect to use someone else's software with thousands of years of developers, designers, testers and other professionals without having to pay for it.
Between SageMath, Numpy, scikit-learn, Pandas, Folium etc. Python can probably match Mathematica feature for feature and even beat it in some cutting edge research areas.
However Mathematica makes it a lot easier to get started if you don't know how to program. Just start Mathematica, enter the equation like it looks on the page in your math book using the equation editor GUI, click solve and you're done. Also since everything is built in you don't have to worry about working out which package implements Moving Median or Facial Recognition.
Mathematica also comes with a bunch of real time data sources which are really fun to play with. Want to plot Germany's GDP vs Natual Gas prices or Gold prices vs global average temperatures? The datasets are there and ready to be explored.
The only place where Python is better if you need to share your code - not everyone has Mathematica license.
For instance in SageMath, to get the eigenvalues of a matrix M, you have to do A.eigenvalues(). What is that supposed to mean? To a programmer, that means A is an object (in the sense of OOP), and you are calling an internal method on it.
But that's not how you think about mathematics. Mathematics is functional. A mathematician knows that eigenvalues(A) means, to choose an appropriate eigenvalue-finding algorithm and apply it to the matrix A.
The mathematics program should conform to how one should think about mathematics, rather than forcing mathematics students to think like programmers. Mathematica does that.
Haven't tried Folium, but from people who are familiar with both: The number of ways Mathematica has an edge over Sage (in terms of math capabilities, not interface), significantly outnumbers the reverse comparison.
Sage is better in a few ways. Mathematica is better in many ways.
Still, for the typical undergrad curriculum, Sage is probably good enough.
You can pay to get access to Mathematica, but you can’t pay to get the freedom you would get from open source. No matter how much you pay, Mathematica is still proprietary. There is, indeed, no price for freedom.
Here is an editorial from an American Math Society journal that expresses very well one aspect of Mathematica that lots of us find troubling: http://www.ams.org/notices/200710/tx071001279p.pdf
Nothing in principle.
But if we teach people to do math with Mathematica, and they graduate and don't have access to Mathematica any more, how are they going to do math?
Most of the practical lab skills you're taught for doing "wet" biology require quite an expensive lab, and doing computational biology often requires a much more expensive supercomputers.
But the MATLAB skills I honed in university have seen little direct use after graduating - because it's faster to teach myself to do whatever it is in Python than to get an employer to adopt MATLAB.
You can use it for free if you use a Raspberry Pi (non commercially)
But sure! Take the limited motivation and effort the student's got and make them go through 16th century tech to learn it all.
It's such a nice, accessible environment for learning. My daughter is still too young to begin exploring SageMath, NumPy, SciPy, etc., but we've had fun together plotting things in Mathematica.
Can't stand this kind of rhetoric. Arguments aren't scary monsters.
For anyone not familiar with SymPy, check out this short tutorial I wrote: https://minireference.com/static/tutorials/sympy_tutorial.pd... ; also available as notebooks https://github.com/minireference/sympytut_notebooks#play .
Obviously you might want to get a nice recent version of the Pi and splurge on getting the max memory, but...it's the cheapest way into Mathematica (and the Wolfram language) other than winning a copy.
The first time I had a problem and received a technical fix via support email, I was sold on being a subscriber. Whether I could justify the commercial license to an employer is another matter…
Also, while even some of the best educational software (like Duolingo) have problems, they’re generally not the problems that the authors expected.
The discourse around Mathematica, educational software, and video games has changed a lot in 20 years. I've the whole thing, not much is particularly relevent. Wolfram asks simply too much per license to be widely used in primary and secondary education.
In addition to the military training the book also talks about dis-association of killing through different methods and how that increased the "shoot to kill". The basic argument is that it is easier to kill someone when seeing them at a distance through the scope of a rifle than being right next to them and stabbing them. The other part of the dis-association is campaigns to dehumanize the enemy in war and not refer to them or acknowledge that they are individual humans.
The video games argument in "On Killing" isn't part of the main thesis of the book, it is more tacked on the end. Whereas the rest of the book is looking at the past and evidence of what has happened. The video game part of it was (at the time) more focused on current feelings and attitudes towards it and was not substantiated.
With programming/mathematical software, it finally clicked and it gave me a powerful, practical tool to make mathematics work. Finally it was not so much about not messing up, but about actually building something. I enjoyed it a lot and in the end I obtained a PhD in theoretical particle physics and became a software developer.
I kind of wonder if this was part of how the child of two math professors¹ was taking upper-division college level math classes at Harvey Mudd as a high school student. It's not so much that he learned K5 mathematics quickly and was on to pre-algebra and algebra as that he was likely introduced to algebraic concepts much sooner than his peers.
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1. I never met him, his time was a little before mine, but I recall someone who had met him saying that his parents taught him a lot of mathematics but not so much by way of social skills.
I wonder what things we consider "lazy" at the moment, will become the standard in 20 years
wow
I'd rather call it "information". Knowledge is one level above that, knowledge is information integrated in a coherent whole. Nowadays it's very easy to find information about anything, but most people still know very little about most things. Confusing information with knowledge and knowing nothing at the same time is why misinformation spreads so easily.
People have access to limitless information but have little interest in actually looking for and using it.
(Granted, that was in physics where the graders usually use the final answer as a check and rarely deduct many marks if all of the work leading up to an incorrect numerical answer is correct.)
When I taught high school in the 00s, I had to impose rules about what sort of calculators were permitted to avoid cheating, especially as devices which could be used for communication like network-connected palm pilots were available.¹ I also wanted to have the kids avoid buying a $99 TI-89 when a much cheaper calculator was sufficient for the work they were doing.
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1. I remember one cheating technique that my friends in high school AP Chem used where a group of us were sharing a calculator. You would type in the question number you needed an answer for and someone else would type in the letter of the answer (we generally had multiple choice tests).
I recall wondering out loud in hs geometry if one infinity could be bigger than another in typical early high school "big brain" thinking, and at the time my high school geometry teacher basically shrugged their shoulders.
Years later when I discovered cardinalities of infinite sets, which comes up pretty quickly when studying math in college, I was fairly disappointed that such a great potential teaching moment was passed over. I'm pretty sure early high school me would have been dramatically more engaged with math if a teacher has said "funny you should mention that, do you know about the difference between integers and real numbers..."
The tragedy of this is that most Americans, even the ones teaching math, believe mathematics is calculation, and even at the advanced levels will only rarely be exposed to real mathematical thinking.
He had a policy of no calculators/computer use, but he relaxed it for this assignment as he didn't want students to needlessly suffer.
All the kids that used software (Mathematica and similar ones) got it wrong. All of them. Each and every one made a typo somewhere.
Thing is, had any of these students done a very simple dimensional analysis, they would have known very quickly that their answer was incorrect. I'm sure they would have done that analysis had they attempted solving it by hand. However, people tend to forget sanity checks when working on a computer.
Personally, I think that kids should learn simple mental arithmetic. It's not that I expect them to use it when they become engineers or cashiers, but it is useful for making quick approximations. It is useful to know when they have to pull out a calculator to check the price they are being charged at the store or to know when to check what they fed the computer (since they have the tools to figure out when an answer is wrong).
While I agree with being able to check the results with many modern calculators, my argument is that it is useful to know when the results have to be checked. For example: if you know that all of the points are less than 5 and you are adding 20 of them together, a result greater than 100 means you have to go back and check what was entered.
(I agree with you, btw).
It is interesting that you can learn to calculate a lot of useful stuff in memory and there is a lot of tricks and you can get quite proficient at it after one semester of classes and some training.
But once I left this school it was completely unusable skill. It sort of helps me estimate things, a little bit? I guess? But I also carry a phone with me at all times and I have a powerful calculator on my desk just within reach.
A skill that is not being used deteriorates, and now I can no longer multiply 5 digit numbers in memory and I use calculator for anything that involves more than 3 digits. And I don't feel like I lost anything.
There are so many useful skills to learn. If you want to spend time, learn something else that is more useful. And if you don't want to learn something useful learn something that gives you joy or impresses people.
I am not sure I agree. The argument only seems to work if you consider a very simplistic teleological model of skill learning: that the only benefit in learning a skill is mastering a very specific task, e.g. multiplication, or derivation. But you learn a lot of other skills on the way that may be useful for completely different tasks - or even tasks which aren't known yet and thus cannot yet be trained specifically.
As my university teacher in Latin once said: sure, you don't need Latin anymore, but learning it will unlock areas in your brain previously unknown to you.
Now we're going to have people who know algebra programming apps to help people who don't know algebra solve extremely specific problems with no ability to generalize.
Web apps for visualizing math are nothing new, profs were doing it with Java 20 years ago. The manifesto on this page is the new and bad element being added.
Contrary to the manifesto, symbolic manipulation is not a freakish knack possessed only by an elite minority of humans. Barring severe and rare developmental issues, everyone can understand and manipulate written symbols. And in countries with effective educational cultures, people learn it.
Our problem is a damaged education culture, not a general human inability to do basic algebra.
Yes, but it doesn’t mean the symbols can’t evolve. Like Feynman Diagrams—they made the concepts way more accessible.
Couldn't you just learn some live language, that has those same grammar features as Latin (declensions for nouns, conjugation for verbs, whatever) that are new to English speakers?
Now of course that's vastly simplified and I'm sure there are other mental affects (and numerous benefits to learning live languages), but those are the ones I've noticed, and the effects would definitely be different.
In some sense - yes. Every cognitive prosthetics or enhancer means that we do not need to develop a particular set of skills. At the same time - it allows us to move further!
Thanks to writing, we don't need to remember everything. Thanks to the Internet, we can retrieve this information in seconds (not days or months, depending on if a book was in a nearby library).
Integrals are the art of the past. Sure, it is still worth doing some, but rather than memorizing hundreds of tricks, let's focus on something different than reinventing the wheel.
There is also a pretty nice introductory book that is aimed at students: https://www.wolfram.com/language/elementary-introduction/2nd...
My professors liked it so much they standardized on it and licensed the student edition for the entire department.
Now to be honest, the skills I learn over Gurobi, at the level I am, are super transferable to other software, so I'm not trapped into that proprietary stuff.
If you need a free solver with reasonable performance, you should consider the COIN-OR project [0].
This argument seems to be from that techno-utopian era, when we thought that just having access to computers would elevate civilization, and that we'd always have access to computers, so things would keep getting better and better as long as we embraced technology.
What happened instead was a bit more complicated than that: when people got access to ubiquitous computing, it did teach them a new set of skills, and they did lose some old ones. But, we don't live in the world many people expected would come about as a result.
We live in a different world, with many tradeoffs we hadn't expected. Some good things, some bad things. But one thing I think it's safe to say is that the average person is not using computers as tools to think in more abstract and sophisticated ways than they were capable of in 2002.
So, the argument he's making is phrased as pragmatic, but I wonder to what extent that argument just assumes a certain, positive outcome. Would the author make the same argument, with the same level of confidence today?
IMHO, a big problem with math education in general is that it focuses on formalisms and equations first while leaving the big picture as just an addendum. That's not so great for those having a hard time keeping motivated without seeing a real life application that is relevant to them.
The other day I watched "Cargo" on Netflix, in which after a civilisational collapse, many indigenous Australians are depicted as going "back to the old ways". I found this quite problematic: it suggests that the pre-European way of life was so easy that a bunch of adults can just _start doing it again_. As if it's some natural animalistic habit you can just fall into, and not a set of skills and knowledge that evolved over millennia.
I guess it's possible we were supposed to read that these people's communities had passed down that knowledge and experience across the intervening centuries. Maybe they have.
For teaching id broadly day either set it as a prerequisite to examine something it can't do or leave it until the last lecture to show the course material can be applied more efficiently.
Learning is as much getting used to using tools correctly as understanding what the tools do and how they do it.
I've worked with those who yes, have had their brain rot due to blindly trusting Mathematica or equivalent and they never spot first step errors from their blind spot. That being said I've worked with one person who used it really well and I think they were genuinely more efficient for it existing.
A big wide open "environment" to get things done in the same way that people in the real world do. Incidentally, I'm also thinking that this might also apply to "programming?"
I actually wrote a whole python/mathematica interface that was very elegant, but most of what I learned was cool Python tricks.
*By software I mean both programming languages (Python) and software suites (MatLab, Mathematica, Sage, etc.),
TL;DR: Teachers, please encourage computer usage for problem solving, but make students understand they need to grasp the fundamentals first!
Also, why must pages be without styles and so w i d e? What is this, a manpage? Long form text should not need to be so wide.
Basic CSS can be applied to a site to make it readable, don't even need to add classes to the HTML to do it.
Here is the page with readability turned on: https://imgur.com/a/0yzRY61
> Converted by Mathematica May 30, 2002
It's sad that we've forgotten why web rendering has fluid layout: the whole point was to enable the user to make a web document work for them. In contrast, the modern websites are essentially PDFs emulated with CSS and JS.
1) open devtools, 2) resize the page, 3) make devtools a popup, 4) then minimize the popup.
It works wonders on amazon cloud reader, read.amazon.com, although I am thinking about writing an dev-tools extension to make it even prettier. I believe text should be human scale, which means minimizing the amount of left-right muscle movement of the eyes, resembling as much as possible the format of paperback books.
- I see you know about Water.css: I've used the 'waterize' bookmarklet on pages like this. Similarly for Sakura. - Most browsers have a reader mode, and it works perfectly with this page, as you point out. - Pages like this work really well when printing, either to paper or PDF. Most websites just don't. - The total size of the page is...44k.
Overall, I actually prefer pages like this. Ironically, I wouldn't publish my own writing this way because so many people have the reaction you have; I guess I realize I'm in the minority on this one.