It sounds like you have three independent variables and one dependent variable. As such, it would make sense to use regression. Depending on how "teacher qualification" is coded, you may need to use dummy variables for that predictor.
For the class size it clear that I can select three (3) classes (Small, Medium and Large) and their mean score, But What If I assign different teachers with different qualifications to different classes and measure whether if there is significant difference between the students? will it be meaningful? regarding the teacher adequacy(Availability in numbers), can I purposively select those schools that have adequate number of mathematics teachers and those that do not have and give them a test and compare whether if there is significant difference between the two groups?
With regard to your last question, "can I purposively select those schools that have adequate number of mathematics teachers and those that do not have and give them a test and compare whether if there is significant difference between the two groups?" the answer is yes, because this is basically just a t-Test.
For your first question, "What If I assign different teachers with different qualifications to different classes and measure whether if there is significant difference between the students?" I am not exactly sure what you mean, but it sounds like this would be a quasi-experimental design.
David L Morgan thank you sir. the topic that I want to work on is " Effect of Class-size, Teacher Qualification and Adequacy on Students' Performance in Mensuration in Bichi Educational Zone, Kano State, Nigeria."
I will use 15 senior secondary schools,6 out of 15 schools will have three (3) classes, small class (15 students) medium (30 students) and large (45 students). i will assigned a teacher with a certain qualification to teach all the three classes(low, medium and large) in a particular school. Example, a teacher that have a bachelor degree will teach school A, a teacher that have diploma will teach school B and so on. Is it appropriate?
The remaining 9 schools will be selected purposively to measure the teacher adequacy. 3 schools that have inadequate mathematics teachers, another 3 schools that have moderately adequate and the other 3 that have adequate mathematics teachers.
How many teachers will you have? Is it 6*3 + 9*3? The issue is one of statistical power. If you have a pre -test score to model progress the full interaction model will have 2 * 3 (for class size) * 3 (teacher quality) parameters which looks seriously underpowered to me. If class size was continuous, there would be fewer. You need to do a proper power analysis but with software that respects the clustered nature of you data, eg. Mlpowsim.
If teacher adequacy is different from teacher qualifications you have two separate studies and your effective sample size is halved. This is a large and ambitious project and you need experienced help on the design and the analysis of this complex design.
The choice of design may depend on the research approach you intend to use (i.e. qualitativ, quantitative or mixed methods approach).
I will suggest you use the mixed methods research approach. Thus, by using mixed methods approach, you can consider the convergent parallel design which will help you take qualitative and qualitative data so that you can compare or relate the two and then interpret them accordingly.