Hi mummy123456
I'm a long-term expat - DH is English.
I think you have to start by contacting Local Education Authority once you know where you will be living. If you're renting first, once that's arranged ring the LEA.
You will be a late entry (both for YR and Y2) now - so that does present rather a problem as you don't have priority over applicants to Year R. Britain is experiencing something of a baby boom - so it may mean that initially you don't get your first choice/ nearest school.
Don't panic. You can wait list for the schools nearest you (approach them directly) and may well get a place a few months later. Once one child is in - this improves the position of the second child (as children with siblings already in the school are usually given priority).
Going the private route means you can organise schooling right now and have everything set up before arriving. However, it is a question of whether you can afford it - and I don't want to presume that is or isn't an option for you.
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In terms of Y2 curriculum:
Have a look at the new national curriculum by subject area here: www.gov.uk/government/collections/national-curriculum - just scroll down to the bottom and chose area of curriculum. Look at Y1 to see what notionally your DS should know by Y2.
Year R (Reception) = kindergarten is under the Early Years Foundation Skills curriculum and is in fact an extension of nursery.
For starting Y2 the essential thing is that they are starting to read (able to sound out simple words) and can make plausible attempts at writing (so may have larger vocabulary than they can actually spell).
Joined up (cursive) writing starts to feature Y2/ Y3 - You may find doing a bit in a workbook helpful - Try Collins Easy Learning Handwriting Workbooks 1 - 3 (available amazon & most bookstores/ large newsagents). These were a great help to my DD2 and also teach a bit of grammar/ spelling rules along the way.
Children will only be tested here in standardised testing on English/ Maths/ Science - so don't worry about history (which in primary is big general themes like The Romans, The Greeks, The Egyptians, etc...).
With Math (or Maths as they say here) - Y2 is about securing addition/ subtraction skills to at least 100 and maybe introducing the first times tables (x0, x1, x2, x5 and x10). In Y1 you might already by counting by 2s to 20, and 5s and 10s to 100 and should be able to add/ subtract numbers to 20 in preparation for this.
UK schools are different from US State schools (no experience of private sector myself) in that you don't get letter grades. At present the English system is in flux. They are dropping National curriculum levels (hard to explain best read MN guidance on how assessment works here: www.mumsnet.com/learning/assessment/introduction but there are a lot of changes going through at the moment and one of them is that national curriculum levels are being dropped.
Be prepared for religion to be taught in schools and daily worship. There is no separation of church and state as in the US. Church of England schools are seen as better schools and often are very popular - but attending one can mean a lot of hymns, church services, etc... There are many CofE/ faith schools because these were the first institutions to provide education to children in the 19th century and then were later incorporated into the state education system (ca. 1870 - en.wikipedia.org/wiki/History_of_education_in_England)
However - they won't pledge allegiance to the UK Flag and won't sing God Save our Gracious Queen daily, although they will learn it and perhaps sing it occasionally.
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Expect a small novel of a report card if you end up in a state school. Ours runs to 5 pages including full attendance records with little statements under each area of the curriculum. So for RE in Y2 DD2 had something along the lines of: DD2 especially enjoyed the unit on the story of Noah's ark and is developing a wider understanding of different religions in the world. It's all very 'fluffy' - each child develops at their own pace - and schools can be very loath to openly and honestly discuss how your child is performing (as in avoiding at all costs warning a parent their child is failing and really are struggling to keep up).
I don't have personal experience but from what I understand from MN private schools are often highly geared to prepping children for entrance exams at prestigious private schools or for grammar schools. Grammar schools can be either private or state funded. Here they are state funded and free but entrance is by examination on English, maths, verbal and non-verbal reasoning. Some areas the exam is straightforward and roughly 25% of students pass and go to grammar school. Other areas, like here, it's highly competitive and maybe 7 - 8% go to a grammar school.
Most areas do not offer middle schools - so your 11 year old will immediately go to a high school setting (11-18 year olds) - called senior school here. I'm not a big fan of that - I'd prefer middle schools - but they don't exist in this area. So you also need to be thinking a bit ahead (if you're moving to UK for > a few years) about what options you want for senior school. Unfortunately, my experience has been that primaries are relatively o.k. but quality options for senior school become very scarce - so when choosing where to live think through what the educational options are all the way through.
HTH