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Wolfram Alpha Is Making It Extremely Easy for Students to Cheat

• res0nat0r 9y ago
  You may cheat all you want on your homework, but if calculus tests are like they were when I was in college, taking your exam with no notes or calculator, if you've only cut and pasted from Wolfram Alpha all semester is going to lead you to getting an F in the course.
  • sp332 9y ago
    But it might seem like good time management to have Wolfram Alpha do your homework sometimes. That means it's going to take longer for a teacher to notice that you're not getting a concept well, and that means a lot more painful surprise-F's on tests.
    • criddell 9y ago
      That could be painful for the teacher too. If kids don't do well on tests, then teachers can fail their evaluations. It seems like it would be an incentive to use the flipped model where exercises are worked though in class and students learn lessons at night via video and reading.
    • JadeNB 9y ago
      > That means it's going to take longer for a teacher to notice that you're not getting a concept well, and that means a lot more painful surprise-F's on tests. I'm not sure how much to read into this, so forgive me if I'm misunderstanding the tone; but calling an F after not doing any of the work oneself a "surprise F" seems a lot like calling it a "surprise loss" when one hasn't shown up for any of the team practices. (If you were just observing that students are surprised by such F's, no matter how often it's pointed out to them, and no matter how often they've met such outcomes before, then I am forced reluctantly to agree.)
  • watbe 9y ago
    Agreed. This headline makes it sound like they're cheating in exams, but this is take-home assignments we're talking about. If wolfram alpha wasn't available, they'd be asking friends or posting in maths forums. Wolfram alpha is simply more efficient at this job but all take-home assignments have this problem inherently. Students just need to be aware that they should learn from solutions provided by wolfram alpha (there's nothing wrong with that), otherwise they're going to flunk the closed-book exams anyway.
    • ntsplnkv2 9y ago
      Exactly: Why is homework considered something that can be "cheated" on anyway? The whole purpose is to get practice and learn the material. Just make the tests challenging-the ones who actually did their homework will pass. The ones who didn't won't. Wolfram Alpha helped me many times-sometimes you're simply stuck on a problem and no one is there to help. But like anything, you can use it to cheat and not do work, or you can use it to help you learn more.
  • Chardok 9y ago
    Too right; it may help being able to just provide the answers, but homework made up a small percentage compared to the actual exams, which of course were done without any notes and were simple enough not to need a calculator. It may be difficult trying to teach these kids how to perform the math itself when they can show the homework, but ultimately they are failing themselves when it comes down to needing a fundamental understanding on the process itself. This reminds me more of teachers scoffing at calculators being a crutch. It comes down to the students' willingness to learn, not how to thwart cheating on homework.
    • jandrese 9y ago
      My impression is that the article is about highschool students, where homework tends to be on the order of 75% of your grade simply because you get so much of it. If someone were really serious about abusing the service they could pull out a C in the class even with hard failures of the tests.
      • _jal 9y ago
        > where homework tends to be on the order of 75% of your grade I think I found the problem.
        • jandrese 9y ago
          It's hard to balance when you have an hour of homework assigned every day, but only get tested about once a month. So you have 20 hours of homework vs. a 30 minute test. The difference in magnitude ends up being too much. Either each homework assignment is worth almost nothing or they outweigh tests.
    • throwawayjava 9y ago
      > it may help being able to just provide the answers, but homework made up a small percentage compared to the actual exams... One of the easiest ways to inflate grades without explicitly lowering standards is to decrease the proportion of the final grade that depends on the midterm and final. Cut out the midterm for "more instruction time" and leave a 20% final. Homework is now worth an absurd 80% of the grade... that's no good. Throw in a project, participation, or auto-graded "e-labs" with infinite attempts. Suddenly it's really difficult to get anything less than a B unless you really just don't give a crap. But also kinda difficult to actually learn because So. Much. Busy. Work. And, on top of everything else, we're punishing the one student who's not cheating. Any time I see a lower division syllabus where closed-book exams are worth less than 40% of the grade, I'm instantly suspicious. > but ultimately they are failing themselves when it comes down to needing a fundamental understanding on the process itself. Yup. > It comes down to the students' willingness to learn, not how to thwart cheating on homework. In fact, you can leverage the calculator or WA to teach the material at greater depth. No longer need lots of practice with trig identities or u substitution? Great. Maybe we can write a few proofs instead, or use the time to work through a large case study.
  • makmanalp 9y ago
    This - I think part of the issue is to stop homework from becoming busywork for grades. If everyone has the understanding that this is supposed to nail down and self-test understanding of material taught in class, then there's no reason students should "cheat". To this end, I like the approach of mentioning repeatedly that homeworks are for the students' own good, then give a test early on to make students quickly understand they'll do badly without practice, and then announce after the test that you'll drop the scores, but just DO YOUR DAMN HOMEWORK! :-)
    • posterboy 9y ago
      Don't announce it, or it will work only for one or two years.
  • nerdponx 9y ago
    Can confirm. Source: did this, got a C in freshman calc.
  • 2OEH8eoCRo 9y ago
    Grumble grumble back in my day grumble grumble
    • qubex 9y ago
      Grumble grumble integration by exhaustion grumble grumble Napier bones grumble grumble
  • polpo 9y ago
    I learned this pretty quickly - I used Mathematica a bit too much for help on calculus homework and wound up paying dearly during the first few exams.
  • chongli 9y ago
    I'm well aware of this. I'm going from the Ontario high school curriculum where homework is 70% of the grade to University of Waterloo math faculty where assignments are 10% and the midterm and final combine for the other 90%. Wolfram Alpha should be used for nothing other than a sanity check afterwards.
  • nilkn 9y ago
    I was a math major in college only a few years ago, and lower-level courses tended to concentrate most of the grade in exams for this reason. The exams were always pencil-and-paper only, with no resources of any kind allowed. In upper-level classes, the trend often shifted, with most or even all of the grade coming from problem sets. Some classes didn't even have exams. But this is because the material had become entirely proof-based. Referencing books was not only allowed but encouraged.
• kiernanmcgowan 9y ago
  Its not just high school students. I had some classmates in grad school using it to breeze through feedback theory homework. The tool is as powerful as students are lazy.
• wodenokoto 9y ago
  It is not cheating. It is leveling the playing field. Some students will inevitably have access to tutors or parents who can give them the full answer. With wolfram|Alpha all students have access to this. What this really show is that homework isn't the best tool for gauging student prowess.
  • fullshark 9y ago
    Homework's largely supposed to be about reinforcing the lesson for math courses. For other courses it's largely about reading which shouldn't be done in class. Tests/essays are for assessment.
    • adamnemecek 9y ago
      Other countries don't put as much emphasis on hw and seem to do just fine. In some of them hw isn't even graded, it's for you to practice. My high school for example gave out a book with the problem solutions. For some of them, it was only the final result, for some of them it was the whole process. Hw didn't affect your grade at all and it was up to you to decide if you want to do the hw or not. The amount of hw students get in the us is astounding. It's not always productive either (more often that not it's not). And it seems like there's no pushback against it.
    • xenihn 9y ago
      Then why should homework be factored into grades at all?
      • umanwizard 9y ago
        It shouldn't be, and isn't in many countries.
      • treehau5 9y ago
        Incentive to do it.
        • Cumulonimbus 9y ago
• throwaway2016a 9y ago
  This isn't new... I had a TI-89 in college that could solve for calculus equations. As someone else pointed out... we couldn't use it on tests though.
  • 2OEH8eoCRo 9y ago
    Had a TI-92+ myself. Best money I ever spent
• jandrese 9y ago
  Or they could cheat by doing their homework, then checking the results on Wolfram Alpha to make sure they always get 100%.
  • vivekseth 9y ago
    I think we should encourage that. If students can catch their mistakes early and learn from them, I think they should. What we should discourage is students using Wolfram Alpha to just get the answers without any initial work. That robs students of the chance to learn and would definitely be cheating.
• adamnemecek 9y ago
  Playing the world's saddest song on the world's smallest violin there Denise Garcia. How about maybe updating your curriculum for the 3rd millennium. Nah, that would be to much work, complaining in wired is much more productive. The best part is that people in the school administration might do something like blacklisting Wolfram Alpha on computers in the school library and feel like they've dealt successfully with the problem.
  • yequalsx 9y ago
    Have you tried teaching calculus where everyone gets to use Wolfram Alpha and/or Mathematica? I did for three years. I discovered that the students really don't understand the concepts and more importantly don't want to. They just want to mimic problem types. They don't want to understand the why. Plug and chug is their true desire. Using the computer exposes right away if a student understands the concepts. You can't get started on a problem if you don't know what to tell the computer to do or why. I went back to the old paradigm. A few students got it but most never understood.
    • JoeAltmaier 9y ago
      Sounds like Mathematica is the way to go then? Because it turned out students who could do the problems. The 'old paradigm' leaves most behind, which serves no one.
      • yequalsx 9y ago
        Students could not do the computer based problems. Save for a few. More students get through with the old paradigm. Old paradigm problem. Here is f, using the definition of derivative find f'. New paradigm problem. Here is f. Here is a Mathematica function defining f(x+h) - f(x) divided by h. Graph this function for appropriate values of x and h to show whether or not f is differentiable at 2. The old paradigm problem you just proceed as in all the examples. In the new problem they get intimidated because it involves using a computer and not a graphing calculator. They don't understand that you keep x fixed at 2 and vary h around 0. When they start the problem they give nonsensical input to the computer, get nonsensical output and promptly blame the stupid program.
• williamstein 9y ago
  > “Stephen Wolfram, the mind behind Wolfram|Alpha, can’t do long division...” What the heck kind of article is this...? I can’t read it seriously.
  • deleted 9y ago
    [deleted]
  • Rjevski 9y ago
    Even if this was true, what's the problem? He built a successful business that many people find useful, and that's good enough for me.
    • williamstein 9y ago
      I respect the abilities and accomplishments of Wolfram. My problem with the statement I quoted from the article is that I suspect it is bullshit; I think Stephen can do long division, and much much more...
      • defective 9y ago
        "I never learned long division, and look at me." -- Stephen Wolfram http://www.stephenwolfram.com/media/meet-inventor-who-makes-complex-calculus-simple/ http://www.stephenwolfram.com/media/meet-inventor-who-makes-...
  • jtraffic 9y ago
    I agree. I got hung up on this: "Wolfram|Alpha uses natural language processing technology, part of the AI family, to provide students with an academic shortcut" WA uses NLP for some things, but not for solving equations.
• joelrunyon 9y ago
  Am I the only one here that would program their TI-83 in high school to run equations for me so I didn't have to solve them by hand? I wasn't even that dev-savvy (and still am not), but it was super easy and was a huge time-saver on timed tests. I obviously had to learn the concept in order to program the equation, but once I did - why should I have to go through everything manually every single time?
  • iddqd 9y ago
    We had the calculators force reset before every test.
    • qubex 9y ago
      A friend of mine had a graphing CASIO calculator (we sat our exams in 1999) and he figured out a way of making text files persist across resets. This was the very epitome of unfair competitive advantage, but I figured that if he put in the effort to find the work-around, he deserved to benefit from it.
  • Bedon292 9y ago
    I had a math teacher who agreed. Me and another student sat in the back and we programmed the calculator to do whatever she taught us that day, which she was reviewing it with the other students. Since we developed the tools, we obviously had an understanding of the concepts and we allowed to use them for homework, and tests and such. And it kept us from being bored and disruptive, so it worked out for everyone.
  • theonemind 9y ago
    I had a high school math teacher who actually had a policy that you could use any program on your calculator you wrote yourself, including on tests.
• mxschumacher 9y ago
  If a computer can easily answer the question, it is not worth asking it to a human (this also applies in the real sense that it won't be valued by the marketplace and translate to higher wages). How about challenging students with problems that are difficult even when modern technology is used to its fullest extend? Or teaching students how to build tools that solve their homework?
  • err4nt 9y ago
    > teaching students how to build tools that solve their homework? This would give them not only a thorough understanding of the problem they are solving, but teach them a very valuable life skill of finding ways to automate your work, and finding ways to package your expertise into software that can be run by a person without your expertise. This would amazing!
    • throwawayjava 9y ago
      Basically, in most college courses the problems are either 1) easy enough that implementing a program to solve them isn't super insightful, or 2) difficult enough that complete automation would mean "do research". I'll limit myself to Math since that's the topic of this article: Calculus sequence: CS 1 is not a pre-req. And there's not enough time to teach both CS 1 and Calc 1/2/3 in a single course. "Implement it" works well for derivatives but not integrals. You're not gonna teach Risch, and implementing integration tricks isn't particularly insightful IMO. The cost/benefit ratio explodes in Calc 3, and the physical intuitions become as important as than the calculations. Everything past that is proof-based and now you're kind of in "your homework is an open research problem in combining NLP with theorem proving" territory. Maybe with the exception of particularly bad Linear Algebra courses and a bit of the early stuff in Algebra. From a "pragmatic skills" perspective, this approach is still highly suspect. E.g. no one's going to invent their way to Risch by implementing integration tricks. Point is, every field teaches useful life skills / knowledge, and programming gets in the way as often or more often than it helps.
• thomascgalvin 9y ago
  There's a simple solution to this, which also has the advantage of creating better educational outcomes: have students do the reading at home, and the work in class. Dry lectures aren't the best way to learn something: hand-on work is. Listening to a teacher read a powerpoint is a waste of time, because you could just read (or watch a video) on your own. But having an expert on the material standing over your shoulder, helping you through tough spots while you're working through it? That's valuable.
• SteveCoast 9y ago
  When I was 18, back in 1999, I had an internship at WRI (Wolfram Research) in Illinois. I'd applied armed only with a library copy of the Mathematica book, so they sent me a CD with Mathematica on it. I made some demo things and got a slot, and flew to Champaign. I worked on polyhedra for a summer, writing code that could unroll a polyhedral model to its 2D net. Find the volume, the number of faces and all kinds of stuff. I met a bunch of interesting people and it was a blast. I also fell asleep at my keyboard more than once. It was a beautiful summer, biking to work and working with what I still think is one of, if not the, best language ever. Here's why all this is relevant - I came back to real life to study CompSci on these old Sparc machines. And it was like, here's the power button. What's an object in Java? What's a compiler? All reasonable stuff. But: Wolfram Research and Mathematica had, in a sense, ruined my undergraduate life before it started. Why were we using all these bizarre tools? Can't we do this a million times faster? Why are we learning all these bizarre integrals? It was similar to being denied graphing calculators in A-Level Mathematics (in the UK, think high school). I get it - we need to learn 'the basics' and survive without tools to some degree. But, it would have been nice to use them in some contexts and not just deny their existence. There's an anecdote I think about Milton Friedman being shown people building a dam with shovels and not digging machines, to keep people employed in some God-forsaken country. He asked, why don't you use spoons instead? Then, more people would be employed. Mathematica and Alpha are wonderful tools, and I highly recommend applying for an internship if you're of the right age or whatever the requirements are today.
  • adamnemecek 9y ago
    > we need to learn 'the basics' and survive without tools to some degree This is the wrong approach. There's no way you wouldn't understand math much better with these tools. I'm hoping that in the future, math will be less about equations and symbols and more about graphing and being able to move around in the spaces described by the equations. I would draw an analogy with a compiler. After using it for some time, your brain will take on the shape of the compiler and you'll write correct (lol is it ever tho) code without even having to compile it.
    • reikonomusha 9y ago
• hguant 9y ago
  I fail to see the problem here. In a curriculum that takes the joy out of math and attempts to reduce the student's mind to a calculation machines, students employed a calculation machine. I understand the arguement that math is important, but the way it's taught in America, even at the AP level, is criminal. It boils down to "if you see this pattern, apply these steps" without any effort at going beyond. We teach "how" but not "why", which I think is a common refrain when talking about the American education system, or any test driven education system. Math is a means, not an end.
  • aesthetics1 9y ago
    This drives me crazy in my mathematics courses. Sometimes I just do not recognize the pattern, and tools like Mathway or Wolfram Alpha can say "right here dummy" and give me the first step. It is not always about cheating, but seeing the steps for solving the problem logically outlined. I often back myself into a corner working out a difficult problem and need a lifeline.
• Kenji 9y ago
  I am deeply grateful that something like WolframAlpha existed when I was taking calculus classes. It helped me soo much. I could make up integrals, solve them and see if I was right, just like that. It even has a function to show step by step solutions of integrals and series. All extremely useful tools to deepen understanding, not to cheat.
• arnaudsm 9y ago
  Students cheat because our society values diplomas more than knowledge. It's a true problem, but the consequences are not visible yet.
• Stranger43 9y ago
  So the test is rewarding students with the skills to perform well in real life, instead of only those who have submitted blindly to the outdated traditions taught by those who cannot do. There is of cause a level where you cannot just copy solutions but those tend to be badly covered by "facts centered" written exams anyway so why not build your exams around the reality that modern tools exist and will be used, instead of testing as if the world had not changed since the teachers left school.
• innot 9y ago
  Recalling the first-year university math analysis course, our professor himself told us about Wolfram Alpha. Which followed by a whole year of equations and integrals the site could solve only numerically. The high enough level of the tasks beats any cheating attempts.
• kronos29296 9y ago
  Using a search engine is not cheating. There is nothing wrong in making use of all available resources in assignments and this is just another resource at their disposal. My $0.02
• gkop 9y ago
  I felt extremely fortunate that Wolfram Alpha was available when I took calculus in college during summer '09. Wolfram Alpha was amazingly convenient and helpful in getting through tricky problem sets when there was a queue at the tutoring center. It didn't really allow me to cheat at the problem sets because it didn't give the intermediate steps, just the final answer. And of course it's useless on exams. At least regarding calculus, it seems to me almost 100% a helpful learning tool and 0% a contributor to cheating problems.
• set92 9y ago
  Since when has wolfram alpha an AI? I always thought it was simply calls to Mathematica. Maybe when you ask general questions it uses an AI, but with functions and maths I think it only uses Mathematica.
  • err4nt 9y ago
    AI gets clicks in 2017: a story about students giving an artificial intelligence their homework to finish sells better as a story than talking about students punching formulas into a web-based calculator.
  • posterboy 9y ago
    Well, maybe it can even understand mathy questions given in natural language.
• Rjevski 9y ago
  Wolfram Alpha isn't the issue. The issue is that we're grading students on an arbitrary and meaningless metric, instead of the end result. As an employer, if I hire you to do a task, let's say build a toaster, I couldn't care less about how you achieved that - as long as I've got my toaster and it can help me grow my toast-making business I'm happy. Education should work the same.
  • vlasev 9y ago
    Plug and chug with Wolfram Alpha is very similar to plug and chug Stack Exchange answers in your code base.
    • Stranger43 9y ago
      Or passing an exam thats easy/cheap for the teachers/school to grade for that matter.
    • Chris2048 9y ago
      SE can show you what specific incantation format the API requires, but math is a bit different.
    • Rjevski 9y ago
      Math and programming are a bit different though - to my knowledge in math you're not dealing with malicious users trying to give you incorrect input in hopes of exploiting bugs in your algorithm. Edit: but to be honest if you needed to give the result Z for a set of inputs X and Y without worrying about invalid/malicious input, then a Stack Exchange copy/paste is totally fine by me.
      • JadeNB 9y ago
        > Math and programming are a bit different though - to my knowledge in math you're not dealing with malicious users trying to give you incorrect input in hopes of exploiting bugs in your algorithm. But in fact you are! Actually it's even worse; in programming, you are just up against the ingenuity of actual human (or at least humanly programmed) users, whereas, in mathematics, you are up against the whole of 'reality' (in a Platonic sense). A whole tower of consequences will be built upon one mathematician's work, and, even if no human can spot the flaw in that work, if it is there then 'reality' will find it, and somewhere in the chain of consequences there will be an error that will bring the whole thing crashing down.
• EGreg 9y ago
  My dad said to me that knowledge isn't what you remember, it's what you can remember or look up relatively quickly.
• Bedon292 9y ago
  I find it an awesome tool to advance knowledge. I recently had to take calc again after 10 years, and it helped me get back up to speed super fast. I would do the work, check the answer in the back of the book, and if I got it wrong, I would use Wolfram Alpha to do step by step to see where I got it wrong. I can see some people saying that is cheating, but it is a learning tool to me. The class was 90% tests/ quizes. HW was a participation grade. Meaning a tool like that gives no one an advantage. Unless they are using it during the test.
• carry_bit 9y ago
  The "problem" is that the students have moved to https://www.computerbasedmath.org/ https://www.computerbasedmath.org/ but the curriculum hasn't (yet). Just fix the curriculum.
• deft 9y ago
  Today I learned you can cheat on homework. So apparently giving up and having the wrong answers (or no answers at all) is what professors would prefer. Cool :)
  • throwawayjava 9y ago
    > So apparently giving up and having the wrong answers (or no answers at all) is what professors would prefer. Cool :) It's much easier to correct known unknowns than unknown unknowns. The best advice I was ever given as a student was "Pretend like you'll get an automatic A on your report card and treat your grades as a feedback mechanism for figuring out what you do and do not understand. Just focus on learning." As a simple corollary, cheating is kinda silly.
    • aphextron 9y ago
      >It's much easier to correct known unknowns than unknown unknowns. This assumes you'll get any kind of personalized feedback and instruction in a lower division maths course. Good luck with that at a public university.
• jjk166 9y ago
  Anyone who thinks using wolfram alpha for math is cheating has never tried using it for non-trivial calculations. Yeah, it will save you the time of looking up an integral from a table, but it's worthless if the integral isn't in a standard table. It will save you time on looking up the equation for the area of a dodecagon, but it won't prove that equation. It will save you time figuring out the units of the final answer, but it won't tell you if those units are logical. Wolfram alpha only saves people from the tedious busywork of basic arithmetic and algebra, the user still needs to do all the math.
• bikash_ghale 9y ago
  I think we should learn
• bikash_ghale 9y ago
  we need to learn 'the basics' and survive without tools to some degree.
• kevindong 9y ago
  The problem is that not all math problems are worth doing. For instance, when I took Calculus 2/3, some of the homework problems weren't usually hard per se, they were just arduously long. And honestly not really worth doing by hand. If I misplaced a single number and didn't realize it immediately, I would spend the next 10 minutes solving an integral that had no use, plugging it into Webassign for it to tell me I'm wrong, backtrack until I find my issue, and solve the corrected integral. I would usually do this at least once per problem (but usually more like two or three times). So by the end of the course, I would just figure out the integral (or rather, the triple integral), plug it into Wolfram, plug the outputted answer into Webassign, and if the answer was wrong I'd backtrack until eventually the answer that Wolfram outputs is correct. At which point I'd solve the integral by hand. --- There was one time when the homework problem asked me to find the intercept of this obscenely complex trigonometric equation. I tried for a solid half hour and couldn't solve it, so I went to the math help room to ask the TAs for help. I ended up stumping three of the TAs. A few days later, I went back to see if they had figured it out. Turns out they hadn't. They said we should just use Wolfram to get the answer.
• captain_clam 9y ago
  In my two semesters of college math, I've gathered that the faculty has something of a phobia to, if not wolfram in particular, students' access to help outside of the department. Homework problems were oftentimes deliberately difficult, and attending tutoring/office hours was almost certainly necessary for most students to master the material. I got my hands on an instructor's manual of the textbook, and it was a tremendous boon for my mastery of the topics. By having immediate access to the solutions of difficult problems, I was able to comprehend how to approach problems of that type, and therefore could solve more difficult but similar examples in the future. The cycle of attempt/fail/check-solution/repeat was really effective. Waiting for the instructor's office hours or the availability of tutors would have made this process, if not impossible, incredibly inefficient. Do any math educators have any insight to this? Is this math department clinging to an antiquated curriculum in which faculty is something of a gate-keeper to knowledge? Is there a good reason for their distaste for 'going around' them?
  • impendia 9y ago
    > the faculty has something of a phobia to ... students' access to help outside of the department. Math professor here. I am most certainly happy if my students get help outside the department, and I think my attitude is quite typical. We can be a little bit wary of some kinds of help. Too much math teaching consists of "If you see a problem that looks exactly like X, here are the steps you should memorize to solve it." But we don't care per se if you can solve problems of the shape X, Y, or Z. We want you to develop your skills to the point that all of these lie naturally within your skill set, that you could do them even if you've never seen one exactly like that before. As such, some kinds of tutoring can be counterproductive. But most aren't. In my opinion your professors' attitude was quite foolish. Kudos to you for seizing the initiative and figuring out for yourself how to best learn the material.
    • gregmac 9y ago
      > Too much math teaching consists of "If you see a problem that looks exactly like X, here are the steps you should memorize to solve it." A significant amount of math testing is basically checking if you've memorized some theorem (and then can solve it), so is that surprising?
• mnarayan01 9y ago
  The are many things which are difficult to measure directly, and thus people tend to use indirect measurements instead; school-learned math is full of such things. Consider a fairly simple example: "What is the factorial of a number?" Determining whether a student knows this is going to take a bit of work (particularly if they lack a formal way to specify it e.g. Pi notation or even a programming language), but we might approximate it by instead asking "What is the factorial of 5?" Now obviously this is not a perfect measure of what we're actually looking for even in the absence of calculators (e.g. someone might memorize 5! = 120), but it's easy to evaluate and is probably a decent proxy in the absence of calculators.
• DennisP 9y ago
  Sounds like Wolfram Alpha could be very helpful for anyone doing self-study, from math textbooks that don't put answers in the back.
• SimeVidas 9y ago
  “Difficult to trace”? If it generates the same output for any given input, then you can just compare the students’ submitted papers with it, no?
• NamTaf 9y ago
  Teach and test for understanding why and how the answer comes to be, not what the answer is. Maths education, in my experience, relies too much on testing of the arithmetic rather than the logic. This just results in plug and chug because it's lazy testing and students apply lazy solutions. Good examiners will ask students questions that make them think about why and how, not what the answer is. These often don't even need algebra or arithmetic because it assumes students know the equations and instead goes a level deeper to test understanding of why those equations are the way they are. I had the gamut in uni. I found that engineering more often successfully tested understanding as I describe above. Some didn't; my thermo was a case of 'learn how the equations work, then read the right graphs and go' but particularly some of my fluid and aerospace courses were great at asking questions that really tested deep understanding of the theories. One good example of this that I came across more recently is some of the edX courses that used to exist featuring Walter Lewin (before his sexual harassment came to light). He was very successfully able to question his students on the why and how, not just the what. This actually proved even more important in the MOOC environment, where you can't as tightly control the environment in which students undertake examination. It's hard, it requires good lecturers really spending a bit of time devising questions as well as supporting their tutors as they teach the students, but it's possible.
• Mathnerd314 9y ago
  SymPy is in the same ballpark as Wolfram for calculus, and not rate-limited: http://www.sympy.org/en/index.html http://www.sympy.org/en/index.html It doesn't include the databases of structured information or the NLP-like interface though.
• 6d6b73 9y ago
  I found symbolab.com much better at this.. It also has much better user interface.
• jeremymcanally 9y ago
  They may let you cheat (just like SparkNotes I guess), but these tools also let people like me who haven't had advanced calculus survive when we otherwise wouldn't. If you pay for the WA service (I don't remember what it's called), you can type complex math in and it will break down how to solve it out. This was invaluable to me in some CS classes where I knew how to implement the algorithms once the math had been detangled, but the mathematical notation was so opaque to me at the time that I would've had no hope of figuring it out. Obviously, I learned a lot from having it broken down like that so I wouldn't be as dependent on the tool now, but at the time, it was a huge learning aid.
• Chris2048 9y ago
  A few issues: 1) applications rest on the shoulders of giants, it isn't efficient to learn too much more than you need to. If you only need to "understand" at a high level, you should just use tools that "do it for you" at lower levels - It's just a shame there are not better FLOSS competitors to Mathematica. 2) It should be clear what the "understanding dependencies" are - i.e. when some knowledge of the foundations can inform the higher levels, and when they don't. If I understand what 'sorted' means, I don't need to know the details of any specific sorting implementation to use it. 3) The way mathematics is split up into a million small, set-theoretically-abstract lemmas etc makes it so much harder to understand. It makes even familiar concepts hard.
• jorgec 9y ago
  I work training and cheating professionals. I used to be a project in chief (the guy that hired, fired and evaluated new personal). I encourage to my students to do the, so called, cheat, because its what the people in the real world do, people don't need to memorize a lot of topics and people use calculator, computer and excel. Sheesh with the old and rust model of learning.
• losteverything 9y ago
  The big issue to me is things that matter take effort. So many things (like math answers) are made so available with no effort.
• vmilner 9y ago
  "while a few were still using it at their jobs as engineers or quantitative analysts" Er... which is a good thing, surely?
• pletnes 9y ago
  There's a significant point a lot of people are missing. In e.g. physics and chemistry, the math is not the point. Using e.g. Wolfram Alpha is great if it allows students to solve larger and more complex problems, without sitting there solving integrals over and over (for instance).
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• ephimetheus 9y ago
  I have a Master's degree in physics and Wolphram|Alpha would not have been out of place in the acknowledgements section of my theses.