tkosan

“Fundamental Mathematics” fourth edition by Wade and Taylor is the best book I am aware of for learning mathematics from scratch.

tkosan

8 anni fadiscuss

I am the main developer of the MathPiper CAS (http://mathpiper.org), and I forked it from the Yacas CAS in 2008. MathPiper is a rewriting system that is similar in design to Mathematica. Most CASs don't solve elementary algebra equations the way humans solve them by hand, so these systems can't show the steps they took in a form that is easily understandable by a human.

However, a few CASs do solve elementary algebra equations the way humans do, and the most famous of these is probably PRESS (Prolog Equation Solving System). The story of PRESS's development is an interesting one.

In the 1970s and 1980s, a group of artificial intelligence researchers led by Dr. Alan Bundy at Edinburgh University conducted research on how computers can be "taught" to solve elementary algebra equations the way humans do. The first thing they did was to try and figure out exactly how mathematicians do mathematics. They were surprised to learn that a significant number of the techniques that mathematicians used to perform mathematics were not written down anywhere. They were not in any textbooks, nor were they in any journals or research papers. As the researchers dug deeper, they discovered that the techniques did not have names, and they were not taught explicitly. The researchers concluded that mathematicians were using these techniques unconsciously. (Alan Bundy. The Computer Modelling of Mathematical Reasoning. Academic Press, 1983, p.164.)

Why were these researchers the first people in history to discover this information? I think it’s because computers were the first "students" in history that absolutely refused to learn any mathematics that was not taught explicitly. The researchers then devoted years of effort to discovering and naming the unwritten techniques that mathematicians used to perform mathematics. When they "taught" PRESS these techniques, it was able to perform mathematics similar to the way humans typically would.

The step-by-step solver I am building is based on PRESS, and here is an example of what I have working so far:

https://www.youtube.com/watch?v=cy6bwNBkAK0

tkosan

8 anni fadiscuss

I am in the process of writing a step-by-step elementary algebra equation solver that solves these problems like a human does. One thing I have learned from this experience is a good way for a person to discover how much they don't know about even the simpler parts of mathematics is to try teaching them to a computer. This leads me to suspect that Meai is correct about most mathematicians not having a complete understanding of the material they teach.

tkosan

8 anni fadiscuss

Doug Lenat (who started the Cyc project) says Cyc is the "last man standing" of the large logic-based AI projects. So I don't think any open-source projects that are similar to it exist.

If aigrant.org was able to provide accounts on a Cyc system to community members, would you be interested in having one?

tkosan

8 anni fadiscuss

Cyc is used in the MathCraft application, which is currently in beta:

http://www.cyc.com/mathcraft/

I don't think Google, Bing, Wolfram Alpha, or Siri can be used for applications like this.

tkosan

8 anni fadiscuss

I spend 1-2 hours each day learning Cyc, and each day I see it do something new. For example, I just asked Cyc "Where did the 2012 Summer Olympics take place?" and this was its response:

  %cycl,microtheory="InferencePSC"
  
  (eventOccursAt 
      (SummerOlympicsFn 
        (YearFn 2012))
      ?VAR)
  
  %/cycl
  
      %output,mpversion=".257",preserve="false"
         #  ?VAR                          
        --- ---------------------------------------------------------------------
          1 UnitedKingdomOfGreatBritainAndNorthernIreland
          2 England                       
          3 EasternContinent              
          4 ContinentOfEurasia            
          5 Occident                      
          6 MilkyWay-Galaxy               
          7 PlanetEarth                   
          8 NorthernHemisphere-Region     
          9 ContinentOfEurope             
         10 CityOfLondonEngland           
  .   %/output

tkosan

8 anni fadiscuss

I have had access to the research version of Cyc (http://www.cyc.com/) for over a year, and its capabilities are breathtaking. If aigrant.org had a subcommunity of people who were working on Cyc-related side-projects, I would definitely be interested in contributing to it.

What would be helpful is if aigrant.org could provide an account on a Cyc instance to community members.

tkosan

8 anni fadiscuss

Use the Teensyduino software:

https://www.pjrc.com/teensy/teensyduino.html

tkosan

9 anni fadiscuss

I am the main developer of the MathPiper computer algebra system (http://mathpiper.org). I found that a great way to improve one's understanding of math is to learn how a computer algebra system works. I would be happy to teach the fundamentals of a CAS to anyone who is interested.

tkosan

10 anni fadiscuss

How well do you understand the kind of elementary algebra equation solving that is taught in high school? If you don't understand it well enough to write a program that can solve easier elementary algebra equations, then learning how to write a program like this is a great way to start learning how math works.

If you are interested in discussing this topic, feel free to contact me using the email address that is in my profile.

tkosan

10 anni fadiscuss

I am the main developer of the MathPiper open source computer algebra system (http://mathpiper.org). An effective kind of programming language to learn that will help you learn math is a programming language that is part of a computer algebra system. The following video shows MathPiper solving an equation step-by-step:

https://www.youtube.com/watch?v=cy6bwNBkAK0

If you are interested in learning the MathPiper programming language and how this solver works, feel free to contact me using the email address that is in my profile.

tkosan

10 anni fadiscuss

I am the main developer of the MathPiper computer algebra system. When students ask "when will I use algebra?" it is a symptom of a deeper problem, which is what they are being taught is not algebra. Almost no math teachers know how the "game" of algebra actually works, and therefore they are incapable of teaching others how to play it. What are they teaching that is being called algebra? That's a good question. What would the game be called that operated exactly like chess, except the rules for how the pieces are moved were never written down and never taught explicitly? Lets call it c_es_. One critical ability that is needed to be good at c_es_ that is not needed to be good at chess is figuring out how the pieces are moved by only observing how other people move them.

Most people don't like playing games they can't win because they have not been taught how to play. Most people would not like playing c_es_ for this reason, so they would only play it if they were forced to. They would also start asking questions like "why am I being forced to play this illogical game?". The way to help your son is not by providing him with reasons for why he is being forced to play a game he has not been properly taught how to play. The way to help him is to simply teach him how algebra actually works. I can show you how to do this (my contact information is in my profile).

tkosan

10 anni fadiscuss

I think new education systems that are based on software will eventually replace most teacher-based education systems. Not only do teachers not scale, most of them don't understand the subjects they are teaching at a deep enough level to teach these subjects effectively.

I am convinced that software tutors that are based on AI research which was primarily done in the 1970s and 1980s will form the core of these new software education systems. For the past few years I have been working on an AI tutor for solving elementary algebra equations that can already teach parts of this subject better than a human can. If you are interested in discussing how AI tutors like this can be used to help realize the vision you have for Bloom, contact me using the email address that is in my profile.

tkosan

10 anni fadiscuss

The book “A Logical Approach to Discrete Math” by David Gries is excellent for learning math on a theoretical level, and for understanding proofs.

I have found that writing software that automatically solves elementary algebra equations using the techniques humans use is also a very good method for understand how math works at a deep level. Here is a demo of a step-by-step equation solver that I am currently writing:

https://youtu.be/cy6bwNBkAK0

Contact me by email if you are interested in learning more about either of these options (my email address is in my profile).

tkosan

10 anni fadiscuss

No currently used step-by-step equation software I am aware of shows all of the steps (inferences) that are needed to solve an equation, and why it took those steps. For example, I just had cymath solve 8x - 2 = -9 + 7x. The solution it provides is missing steps, and there is no explanation of why it took the steps it did. The solver I am writing shows all of the steps needed to solve an equation, and it is capable of explaining why it took these steps (although the "why" part is not shown in the video).

tkosan

10 anni fadiscuss

Is Maple able to show the steps a human would typically take to solve a given elementary algebra equation?

tkosan

10 anni fadiscuss

Lesson 6 at the following website explains the fundamentals of how the solver works:

http://patternmatics.org/temp_1/mathfuture/

The other lessons explain some of the concepts used in lesson 6.

tkosan

10 anni fadiscuss

This video shows what I have working so far:

https://www.youtube.com/watch?v=cy6bwNBkAK0

tkosan

10 anni fadiscuss

I think teaching fundamental mathematics to programmers using ITS (Intelligent Tutoring System) technology is a very promising idea. For example, an ITS that is capable of doing step-by-step elementary algebra equation solving would be able to teach this subject, and it would also be able to provide instant feedback which is much more detailed than a human would typically provide.

I have been working on a step-by-step elementary algebra equation solver for a few years now, and here is an example of what I have working so far:

http://p1.ssucet.org/tkosan/misc/mathfuture/steps/solve_equa...

If you are interested in discussing the possibility of teaching programmers fundamental mathematics using technology like this, feel free to contact me (my email address is in my profile).

tkosan

10 anni fadiscuss

Douglas Lenat recently gave the following talk at CMU about how Cyc works and its current capabilities: https://www.youtube.com/watch?v=4mv0nCS2mik

Submissions

Ask HN: I am building a step-by-step equation solver to learn how math works

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