I think in another thread you are seeking advice on moving your DC and I do understand you are not happy.
To try and answer your points:
I’m not sure what you mean by under notice from the LA. Has the Improvement Officer looked at your progress and outcomes and given you a poor review? We have a system where a school can be given a red warning. This is, in effect, a notice that, if Ofsted came they would find the school RI or worse. So I’m assuming your school is in this position.
Schools with worrying progress/attainment must have a school improvement plan that addresses this. Schools should check with their Improvement Officer that this is fit for purpose and Ofsted will need it to be the case. Your role as a governor should be within this framework. When you visit under the remit of Maths Governor, you must check for evidence that the Improvement Plan is having an effect.
You cannot sack individual governors so they can remain in position and oversee school improvement. If the whole GB is removed after an Ofsted inspection, that’s a different matter. It could be that the LA seek to remove this GB snd start again but that’s up to them, not the Governors themselves. The LA should be a lot more proactive in training the GB to do their job effectively.
The Head should be drawing up an Improvement Plan as discussed above and the Governors should draw up their own plan to improve. This should be lead by the Chair. If you have an ineffective one, the process will be muddled and not fit for purpose. This is another reason for the LA to step in. However as you appear to be a C of E (Aided or Controlled?) it might have to be the Diocesan Education Board that also gets involved. In a good school, you would get whole GB training on how to become effective. This could be a way forward in giving you ideas for a GB improvement plan. Roles and responsibilities should be clearly identified and also what evidence the GB want to see regarding school improvement. Setting the Heads targets is also tied into this.
I have rarely seen Governors go round to good schools. You really would not learn much. They are also busy and I don’t think it’s a great plan. It’s far far better to get trained so you can ask pertinent questions. You cannot go around criticising class teaching, because you think you have seen better elsewhere, is operational and you would be very foolish to do this.
What you must do is ensure your Head gives you robust data in terms of assessment and progress in the form of a detailed report for each year group at every meeting of FGB and any Teaching and Learning Committee or equivalent you have. You must be able to trust this data. If you don’t have robust assessment procedures then it’s very difficult for GBs. You have to ask those difficult questions! When everything is being done correctly and teaching is good, the progress improves and the attainment will come. It’s not a quick fix and it’s virtually impossible with the wrong people in post. You must have s detailed and fit for purpose improvement plan and this should spell out what success looks like, how you are going to get there and how you are going to monitor it along the way. At every meeting, the Head should update the governors on progress on every item. The items should mirror the areas where Improvement is needed and there could be a lot of them. Governors should be attached to each item. To some extent this links into subjects, SEN and PP but Govs doing fun things and not important things are wasting their time and not helping with improvement. They simply don’t understand their role.
As for the governor teaching: it’s the least of your worries, but yes they can. Vicars often do! As long as it’s in the curriculum and they are Safeguarding checked! It would be better if they concentrated on being a governor though! Strategy and challenge! Hope it gets better!