Mathematics is done by the computer, problem solving not yet

I’ve been doing a lot of maths recently and whenever I come across a problem which I don’t know how to evaluate and solve at the end I ask WolframAlpha (https://www.wolframalpha.com) or Symbolab (https://www.symbolab.com). As you can see if you type any high school – college problem you will almost certainly get a positive result, a solution to the problem. Not only will you get the solution but you will also get the steps towards the solution which is insane.

Let’s go back to year 2000, no let’s got back to year 2010. You had none of this things alive. You were really great-full if you could get a script or a book with solved problems including the steps.

Now let’s see some of the stuff that you can easily do with WolframAlpha or Symbolab

  1. WolframAlpha quadratic equation solverhttp://www.wolframalpha.com/widgets/view.jsp?id=bf424cb7b0dea050a42b9739eb261a3a  We are all familiar with this one, it can be easily programmed and you can find thousands of such example on the Internet and of-course Casio Calculators are the best (the fastest and the easiest) for solving quadratic equations. You get the result instantly and see if an equation has 0, 1 or 2 results.
  2.  WolframAlpha  intersection points of two curves/lineshttp://www.wolframalpha.com/widgets/view.jsp?id=7b9037eb9f5f7493a73df97a38bc58e6  You can use this app to draw any desired function-graph no matter the order. You can draw a parabola, third order equations, of-course, lines (first order equations) etc. You instantly get the x coordinates and if you subscribe and pay (6 dollars per month for students you can see the steps towards the x solution). This can be useful if you don’t want to draw things on paper. As we all know computers will always be much more precise and accurate than humans. Although someone may think that is an easy app to program it certainly is not and takes much effort to achieve such thing.
  3.  WolframAlpha derivatives: http://www.wolframalpha.com/widget/widgetPopup.jsp?p=v&id=f4ccdf86d504bc91f8652fa6d7b76db6&title=+Derivative+Calculator&theme  Put the function from which you want to take the derivative from and you’ll get the result instantly. You can take the n-th derivative up to number 10 if you wish (maybe for Taylor series I cannot think why would you do it for anything else at this time). If you’ve got a function with a lot of x-is and a function which is really big and you know that the probability of making a mistake is high then I suggest you to use this app. Again, computers are “smarter” and cannot make a mistake. Well they can, but they don’t. Although this app is nice I think that this is time when Symbolab kicks in. Symbolab is magnificent at doing this job, also the output looks nice. Here is the link: https://www.symbolab.com/solver/derivative-calculator The greatest thing of all is that you get the steps for free at Symbolab. You don’t have to pay anything. The steps are much more nicer than at WolframAlpha. I think we really have to appreciate such things because the engineers have put great effort into it, I really don’t know how they earn their money to support such platform.
  4. Wolfram alpha Integralshttp://www.wolframalpha.com/calculators/integral-calculator/. Put inside whatever function you want and you will most certainly get the result. Again, I prefer Symbolab over Wolfram generally for all the calculus problems, but for the complex calculus problems WolframAlpha still beats Symbolab. Symbolab integrals link: https://www.symbolab.com/solver/definite-integral-calculator
  5.  WolframAlpha Surface & Integrals: http://www.wolframalpha.com/widgets/view.jsp?id=8ab70731b1553f17c11a3bbc87e0b605  Write in the functions (subtract functions of-course, see Newton-Leibniz rule) that you want to integrate over it and write down the limits for integration for the surface. Now if you are unsure about the limits of the two function then the best thing to do is to go to number 2 on the list and find the intersection points and insert them in the two boxes which represent the limits. Direct solutions (similar to this one) is not offered by Symbolab, however you can always use the definite integration if you know the limits of your integration, after you know them go to number 4 on the list
  6. WolframAlpha Function domain: http://www.wolframalpha.com/widgets/view.jsp?id=b0160688b805d84769cebe1afb71895   We all know how important that is and how many times we’ve searched for the domain whether it’s a function with multiple or just a single variable. There isn’t an equivalent at Symbolab but you don’t really need it. I believe this is one of the easiest problems in mathematics no matter which function is in the game…
  7. Symbolab Limits: https://www.symbolab.com/solver/limit-calculator Limits just kick ass at Symbolabs they can’t be compared to WolframAlpha. The steps are all shown and even the hardest problems are solvable here. The alternative WolframAlpha still deals with the most difficult limits to solve (I believe that the engineers at Symbolab will soon have solution for everything).

The counting goes on and on, almost all calculus types of differential equations can be solved, sequences, Laplace transformations, Fourier series and many others. In 2016 the computer can do a lot of things, consider that another lot of things is not revealed to the public. The reason why I posted so much mathematical stuff is because I’ve realized that we are taught mathematics in a wrong way. Through elementary and high school we are always doing some kind of equations and when we have to apply them on a problem the majority and when I say majority I mean 90% of us do not know how to apply the learned stuff (logarithms, roots, exponential equations, derivatives, limits, differential equations etc.). As we can see, all of this stuff that we are doing in school is solved in a couple of seconds by the computer. The point is, we should not be doing this stuff, well, we should but in low amounts. What we should do is concentrate on PROBLEM SOLVING stuff (word problems). Let’s be honest, word problems are nightmares for most of us. As soon as we see an example, task with a lot of words written people become frightened.

Word problems are indeed hard at the beginning but once our brains gets used to them, we begin to understand them. The most important thing is that we are becoming aware of the tools that we’ve learned in maths and can solve the problem sometimes in many ways. It’s better for us to spend a whole day on solving two word problems then to solve 50 long equations pulling the x out of them. There are multiple reasons why this word problem approach is better than the others but let’s focus on year 2016-2017.

As I’ve said, computers are here to help us and whenever we can use them we should do so to save us time and to put this time into something that computers are unable to do at this moment, but what we are doing is rejecting or partially rejecting the technology and following the education path of our parents/grandparents. Of-course, we should learn how to solve “equation problems” so that we can solve the word problem but the education is exaggerating with these type of problems because it assumes that the word problems will take too much of a time for a pupil to learn/grasp and therefore won’t learn anything or will learn very little but in contrary the pupil/student will benefit hugely out of it because it will try many ways to get to the result and therefore learn how not to solve the problems and how problems should be solved (depends on the type of problem being solved). We can see this kind of problem solving in programming. It takes a lot of time to develop something valuable, there are times when a computer programmer just sits and looks at three lines of code trying to work things out. The same should be applied on children early in school so that they develop a logical thinking habit.

The time of AI will certainly come at some point in the future and God knows what will happen then. I used to have an AI subject at my faculty and was skeptical with this field. I did not believe that something of that kind could ever be build in the future. As my exam was approaching and as I gained more knowledge I found out that I was wrong at the beginning. AI is indeed plausible, however, it will always be rejected from a person who has never done maths, programming and studying the AI (neural networks, deep learning etc.).

AI is basically mathematics trying to simulate a brain. With such concept, researchers are trying to make a software (and already have made quite few) which learns and applies the knowledge afterwards. Every once in a while we can see how a computer (software) outsmarted a human. For instance, the latest achievement  presents a software beating the world champion in “Go” game https://www.youtube.com/watch?v=EOWnzHAgPgE.  Experts could not believe that it happened so soon. This is one of many examples (but one of the most discussed examples recently) where AI algorithms surpass the human even though AI hasn’t yet come near its full potential.

AI will continue to expand, algorithms will appear, new concepts will emerge. We cannot stop it neither should we. Why is that so, why do we want to harm ourselves perhaps? It’s quite easy to answer the question because all the discoveries in the past as well the discoveries which haven’t happened yet are due to people’s CURIOSITY. We are curious on how things work and what things could do once build. Even if a government program against AI is to be introduced there will always be a good amount people digging and trying to build the forbidden (in this case AI).

As soon as we realize and accept the upcoming AI events, the better for all of us.

The summarize this article:

  • Learn how to think and solve problems don’t spend too much time on solving problems which a computer can easily solve such as solving numerous equations as mentioned above. Use On-line tools to speed up your work.
  • Adapt to current technologies
  • Practice daily on your field, no matter what type of field you are involved into.