Math Is Fun Forum

Given that for a vector

,

and for a matrix

,

Show that:
1.

2.

3.

Thankes

I have always felt that the potential of educational software has been vastly under-appreciated and as a result, vastly under-funded. I'm trying to brainstorm different ways to fix this, and this is one of my ideas. I just dished this up quickly so it's not supposed to look professional, it's just a brainstorm idea. Please tell me what you think about this and if it's in the right direction. Also any thoughts on learning in general would be greatly appreciated.

Edit: I don't think my pdf is attaching ??

Anyway here it is, copied from the pdf, with the image attached:

Introduction:
There are a few reasons why I am choosing math as a prototype. One, math doesn't require all the fancy animations and interactive media that physics etc require - math is simple in its abstractness, so it would be a good prototype choice. Of course we would like to use graphics and audio as much as possible even in mathematics, but perhaps for now at least, they won't be needed as much. Two, there may disagreement over this, but math is widely perceived as one of the drier subjects in the sciences. This may be because of its abstractness and obscurity for people trying to pierce its cold, hard exterior. If we can spice up mathematics, we can spice up anything. Three, math proofs are can vary a lot, and so in a sense they can be one of the hardest to tackle when writing educational software. However, they could serve as an insightful proof-of-concept.

The Philosophy:
When reading math books, especially the rigorous ones that act more like encyclopedias rather than textbooks, it can be very easy to gloss over crucial information in definitions, theorems and proofs. Mathematics is a subject in which a lot of care must be taken when reading. Notes must be taken and questions must be asked. Normally, students engage in problem solving in the exercises at the end of a long chapter, or in the often insufficient class assignments. If you are self-studying, you may not even have solutions on hand to check with, which can throw you way off course, or reduce confidence and motivation. For true mastery of a subject, I think more interactivity is required. Interactivity stimulates learning and lets you practice what you have learnt.

Ultimately, I believe that learning mathematics should be a fully interactive activity, where progress is monitored nearly all the time, examples are provided, and tests are given to ensure that the student fully understands and appreciates what they have learned. In addition, everything, even the most vital proofs, should be proven by the student (guided by the software) when they are ready. For example, highschool students would be given the highschool version of the implicit function theorem to prove, while college students would be asked to prove the implicit function theorem in full generality. The software can be `` played" on various difficulty settings. More or less guidance will be provided to the student in the proof, based on the difficulty selected. By proving the crucial and important theorems themselves, students will gain an appreciation for how the theorems work from the inside, and gain insight into the proof techniques.

I believe that the greatest dilemma in writing educational software is that student responses can vary a lot. Unlike computational problems where the answer is simply a number, in a proof, for example, there may be many steps to solving a proof. How is a program supposed to keep track of these many steps, evaluate them, and give feedback? If we define one way of solving the proof as being the definitive way, then other, possibly more elegant ways of proving may be wrongfully rejected.

Until we have AI as advanced as humans in proof-reading, I fear we have no other choice but to implement only the methods of proof that we are aware of and reject all proofs that have not yet come to our attention or that are ridiculous. For each proof we would like to offer several avenues for the student to solve the proof, and the number of avenues available will depend directly on the amount of time we have to implement these avenues. If a student believes they have found an alternate method of proof not covered in the software, they can give us feedback about this and we can try to implement it, or at the very least note it as an alternative.

Right now, it seems that using math input to write proofs in the software as you would on paper is too complicated to implement. You would need advanced systems for reading and evaluating proofs (for example, teachers). This probably is possible at the moment, but only with a lot of people working at how to implement it. In the proof of concept, we will be using an easier-to-implement approach.

We prove proofs by applying ``proof techniques", otherwise known as ``abilities" (to get in touch with the RPG element of the software - I hope in the end the software develops into a fully-fledged Math RPG, combining the purposefulness of learning with the incentives of the most addictive RPG games), which are collected throughout the course of the game by solving problems, taking part in examples, etc. Eventually, the student will have amassed a library of techniques. When given a proof to solve, the student would reach into the library of proof techniques and pick one to start with. The student will then be asked how to apply the technique - this will have to be tailored to suit each proof. In this way, it is a little like a detective game, using clues to solve a crime. We expect the student to play the game with pen and paper in hand and to do scrap work while playing through the software. The software is there to provide guidance, incentive, and interactivity. We may need to implement certain tests to make sure the student is testing their ideas on paper and not blindly guessing.

As the student gains experience using proof techniques, they will gain the ability to automate certain proof techniques, such as basic algebra techniques like expanding brackets, or, even more advanced techniques when the time comes. After a technique is learnt, it will have its own page with examples and tutorials on how to use it. I would like this game to be {\em very} comprehensive in this respect. We must never let the student feel lacking in learning resources.

I believe there is even a potential advantage of this ``proof technique" method over traditional ``written down" proofs. By emphasising {\em techniques}, we are training the student's intuition more than their mechanical computation. After all, when you attempt a proof, you do not start with crystal clear ideas of where to go. You have to feel around in the dark until you can latch on to something. Here, we can teach students to draw from a library of techniques they have learnt and to apply them, following leads like a detective. Not every lead will succeed, but soon students will gain an intuition of which might.

Example
In my following example, we are asked to prove the Cauchy-Schwarz inequality for real numbers given the axioms for inner product spaces. You may have to zoom in to see the small writing:

*IMAGE*

At the end, we would ask several questions, like, why is it ok to take the square root of both sides. The student would then use axiom 1 in the inner product axioms to show that norms are positive, and hence the RHS is greater than the LHS, regardless if the sign of

. We would also ask when equality occurs, which is another proof of its own. We finally conclude with a recap of why the proof method worked, any techniques we took from the proof, a grading based on how many mistakes we made (in arithmetic or blatantly wrong guesses - and only blatantly wrong guesses), and we then continue on to applications etc.

Evolving into a game... and my more distant daydreams
I optimistically believe it is possible for this to evolve into a full RPG game with characters and a story to captivate. The quests would be new discoveries in science, and the powers would be the abilities you have gained. It would be challenging and rewarding. Perhaps it would be hard for a math game to go RPG, but I believe it would be very easy to turn a physics or chemistry, or biology game (Come to think of it, a biology game would work REALLY WELL) into an RPG of some sort. One can dream of a mega-RPG where you can learn anything. Perhaps that would end up being a virtual Earth. Anyhow, even one tiny successful piece of software may start a revolution in education, and make textbooks redundant.  It is a shame that the huge budgets and production values of today's best games are not also used for educational software, since I believe that the budgets used for those games could easily make the next World of ScienceCraft possible today.

Let

be a constant integer.

For what values of

is the fraction

in simplest form?

I've been trying to use Euclid's algorithm to find

, but since they aren't numbers I'm having a lot of trouble setting up the division algorithm.

I'm familiar with how to solve systems of DEs of the form

where

and

is a constant matrix, but how do you solve DEs of the form

where

is a constant translation?

Thanks

I must be getting really rusty from not having done geometry in ages, but I'm stuck on how to prove that

in the following diagram:

(Note that the two angles x are equal)

Thanks

If

, where

, then

Can I get some pointers on proving this? Thanks

NEVERMIND, GOT IT WITH THE CHAIN RUlE

If you write:

,

then

,

Then if you have a function

for

,

I don't really see how these come about, can someone please show me? Thanks