I love that diagram. I've always thought that maths should be thought of that way. But the real diagram can be infinitely branched and detailed. You might be able to make a simplified one for all maths but here's another suggestion.

Can you devise a way to record the **success fraction** for each topic? ie. how many questions attempted are answered correctly over the total number of attempts. That way you've got a crude measure of what people are finding difficult, so you know where to concentrate your efforts. Make intelligent links for those topics first (and add them to an initally blank diagram so you know what topics fall into this help category ... maybe publish the diagram too so others can contribute ideas and those who find this sort of help useful can find more of it)

What this means is you've got the framework for a demand driven development plan; the topics that give the most trouble are the ones you provide extra help on.

anonimnystefy: Oh thanks. I'm thinking any answer can be justified if you spend enough time on it. If any of my candidates had put down a formula like that .......

But I don't have your intensely analytic brain. Why n squared ?

Bob

]]>Yikes, that tree is enormous! What do we really know about the multiplication of two decimals? It seem easy enough and everyone thinks they know how. But what about if the two decimals were millions of digits long? Then we learn about Strassen multiplication, Karatsuba and FF methods and find out we still do not know the fastest way to multiply two decimals, or even if there is one.

What if the decimals are reasonably sized and we multiply them as part of an algorithm that is run millions of times? We learn about smearing and numerical analysis and again find out very little is known about multiplication.

Take Bob's sequence up there, a computer can not get his answer because it can not really count by 1 / 10's.

The mathopolis question database is already huge. Filled with interesting questions. It is a valuable self teaching resource. What will it be like when it has 30 000 or 300 000 questions? Maybe you can continue what you started there and it will do by sheer numbers what you are thinking of right now.

]]>for bob's sequence question,the first two answers are correct,and the third can also be the answer if the sequence's general term is:

So the process of multiplying decimals requires many concepts, and we would have to put them into a map and design questions that determine if the user has a good or poor grasp of that concept.

And then continue that process through calculus and beyond.

To make it interesting for the user we could show them all the concepts they have mastered.

A bit like this, but at a "concept" level: http://curtiee.files.wordpress.com/2010/05/a-level-formula.jpg

]]>bobby: so we have subjects and scores, and give them more questions in subjects they seem bad at.

Templates can be made for many types of questions. Then the program could generate as many as are needed, all with different numbers. This would be easy for some type of problems and difficult for others.

]]>Go to the zoomable number line ?? Or the start page on decimals. Or to a page which says (kindly of course), your answer suggests that you might need some help with this topic. Here are some links to pages that may help .......

Bob

]]>bob: excellent question. And if they get it wrong ... what would the question database do next?

]]>Many years ago I had to set, and mark, an exam, with grading G, F, E, D, C, B, A (best).

One question I set was:

**Give the next number in this sequence:**

**0.6, 0.7, 0.8, 0.9, ....**

There were just three answers given by candidates:

**11.00.10**

When the totals were determined and converted into grades this question turned out to be a 'perfect discriminator' at grade E. That is to say, every candidate who got it right got an E or better; every candidate who got it wrong got an F or lower.

So I have used this question ever since to diagnose understanding of how decimals work.

Bob

]]>That is a tough one! I do not know whether this is being implemented but the first thing that maybe can be done is repetitive questions.

If someone gets that example wrong then a bias should be put on questions of a similar type and more of them should be asked to him. Like trial and error game programs the questioning program should have that feature.

Even just that much is a huge accomplishment.

]]>I want to take it one step further ... I want the answers to learn your weaknesses and provide more learning and more questions to help you. In short "Intelligent Questions"

An example: What is 12 times 0.1? If the answer is 0.12 it shows trouble with decimal place.

This is no trivial task, requiring a map of how people do and understand math. And tailor-made questions. This could be a wonderful research program for some student. Or several research projects.

But I am not sure how to get this started.

]]>