MATHS
In Maths, our kids seem to have really benefitted from a concretepictorialabstract approach. That means every topic is first learned with real, physical objects, before being represented with a picture and only then with abstract symbols, such as written numbers or a + sign.
With this in mind, I've mostly drawn on Montessori and Maths  No Problem!. I don't think it's worth buying all the fancy Montessori materials, but the Montessori approach has some nice physical examples of maths play for young children which are easy to recreate using simple household materials and objects.
Maths! No Problem is currently offering free parent resources if you buy their books. I like Maths  No Problem! because it has something physical to demonstrate most new concepts learnt. It is really well designed, engaging and really helps the child fully grasp each new topic in baby steps. One downside is that by year 3, there are slightly too many lessons per year, so we sometimes end up doing 2 of their lessons in one day so we can give ourselves big chunks of time for nontextbook, explorative Maths during the year.
This is my scheme of work for my own kids, starting age 0. I move onto the next step when the child is ready regardless of age.

From birth, count everything you can see. Poppers, stairs, stacking plastic bowls, plastic lions  everything.

Name shapes with physical objects, shape toys and books.

Read counting books in our chosen languages.

Recognise numbers out and about.

Stacking and length ordering games. E.g. stacking cups, towers with the biggest at the bottom and smallest at the top, putting sticks in length order.

Count to 10.

Count to 10 alternating with a sibling.

Count backwards from 10. At zero, lift up the child like a rocket  Meg on the Moon is a useful book.

Read number flashcards. Trace the numbers with child's finger.

Read dot cards  flashcards with dots in 2 rows of 10, from 010.

Try and get the child to count objects with a onetoone correlation  one point = one number. "One point one count!"

Counting Caterpillars game, Orchard Toys.

Collect 10 sticks/carrots/children and put them in height order.

Make number rods of unifix cubes  a tower of one blocks, a tower of 2 blocks, a tower of 3 blocks all the way to 10. Put them in the wrong order and get the child to order them from 110.

Put unifix cube towers in the right order from 110 with the right number card on top of each one.

Eat 10 grapes. Every time you eat one, shout "take away one!" with arms out like a minus sign. Then say, "101=9. 9 grapes" all the way down to zero.

Give a number and ask for one more and one less.

Use unifix cube towers (110) to work out number bonds to 10. Say a number, the child has to run to the other end of the room to collect its number bond to 10. Check the two number towers together are the same length as the 10 tower. So if I say 2, the child has to collect 8.

Write numbers with a stick in mud or in rice grains in a baking tray.

Write numbers in a wipeclean book.

Play Frog Party, Orchard Toys.

Slowly start Maths! No Problem 1A (textbook and workbook)  even though 1A is a half year's worth of work, I start this at age 3 and take a whole year doing it slowly with a long time on all the manipulatives.

Use Montessori layered numbers to 1000 and a base 10 set to teach numbers 120.

Magic Cauldron Game, Pop to the Shops, Tell the Time, Orchard Toys

I spend another whole year on Maths  No Problem! 1A until the child is ready to go faster. With our kids, we spent a the whole prereception year on book 1A (Year 1, first half), the whole of Reception Year slowly ambling through 1B content (Year 1, second half) and by their actual age Year 1, they were ready to do the Year 2 book at a normal speed, i.e. books A and B within one year. My first two are a year ahead without any feeling of rush.

Discuss real Maths, political Maths, cooking etc. whenever possible!

Try and use Maths lots in other subjects.

Try and use lots of maths in real life, like cooking or shopping.

By year 4, you can get a Maths Teaser set from Maths  No Problem! We do one a day and have found it really helps with confidence and calm while solving puzzles. Many of the questions have the type of answer that somebody with a geeky mind would find pretty or satisfying  e.g. the answer being 12:34.

Don't rush. Only move on to the next topic when the current one has been conquered in depth. Frequently go back and review old topics.
We've just started combining Maths with the computer programme Python. We're using this website for Python activity ideas.