It would be good to see if there is a significant difference in learning outcome of students using programs that offer answer input versus those that offer procedural input (or both).

We have access to several programs that offer questions and allow students to input answers. The programs immediately provide correct/incorrect feedback. This is great for one-step problems. Most students get "lost" in the multi-step problems. It would be great to have a program that (easily) allows students to input several steps, or have "checkpoints," along the way and really utilize it. The students would be able to determine if they are "on the right track" before finishing half a page of work. It would also build confidence. My experience is that these programs take much time to set up and are not easily edited (or tuned to a specific course). 

It would be nice to show research to support the funding for such programs (if deamed more effective), or at least justify the time and effort.

The school where I teach has smartboards, clickers and CBRs with graphing calculators.  In addition to using the blackboard, I have used the smartboards and the CBRs in my lessons, but have not used the clickers yet.  The CBRs allow very quick feedback for the students in a qualitative way for parabolas, slope and calculus.  I've found though that the smartboards are not quite as useful because although they are interactive, it is really only with one student at a time.

Many of my students also have iPads, and there are a number of apps that they will use, or access to Khan academy to reinforce their learning.

You might look into classroom response systems (clickers). These allow instant feedback for a given problem/set of problems including number right/wrong, which distractor(s) are getting answered the most, etc...

You might also look into CBL/CBR units for graphing calculators. These allow dynamic input.

I have noticed that there has been a huge growth in apps, both for smart phones and tablets.  I often wonder how many of these educational apps are effective and how they might be used by students and in classrooms.  It would be a fascinating study to look at the current app marketplace and determine where the apps are being created and where there is a need, and how effective they are.

It would be cool to test how the use of iPad apps affects math skill acquisition and retention. I've been reading some interesting research about how young people think of iPads and other mobile technologies as friends. Such technologies provide motivation, feedback, and fun. For young people, they are more than just "tools." I wonder if that makes the technologies useful in motivating students to learn and practice. 

• The speed and success in maths problem solving in lower grade mathematics and higher grade between girls and boys.

It is said that girls are good in lower grade and boys are good at higher grade.

So in lower grades boys need much attention and at higher grades girls need more attention from teachers. The method of teaching may also be different for girls and boys in respective grades.