DigiMat starts in pre-school to explore the World by counting using several symbolic systems including Abacus of different designs:
Counting is the basic form of mathematics formed by repetition of +1 carrying important information of age, size and value. Even birds are able to count the number of eggs in the nest and tell if an egg is missing. But it is not known what system birds use.
There are many things which children can find pleasure in counting when exploring the World, such as fingers, age, amount, length, distance, duration, area, volume, money, eggs, candy, points…you name it.
In particular, children could be encouraged to create their own symbolic systems for binary numbers, and discover rules for computation within the system.
Here is a list of sessions introducing counting using binary numbers to children of age 4 – 6 years, which does not require any reading skill or familiarity with numbers in decimal form including the decimal digits 1, 2, 3,…, 9.
Session 0. Start counting using identical cards, sticks, markers et cet. Count number of objects in a chosen set of objects, such as the number of fingers on one hand, or the number of people in the group, by pointing at each object and then putting a card/marker on the table, arranging the cards/markers in a row, or putting marks on a paper in a row (compare picture below). See that putting a card on the table (a mark on paper) corresponds to doing +1 as adding one. Compare the number of objects in differents sets by comparing the lengths of corresponding rows of cards/markers. Add numbers by continued operation with cards/markers. After some time ask if there may be a way of counting using fewer cards/markers, maybe with cards of different type like color, shape or symbol. Check Symbols Check Spiders.
Session 1. Introduce binary counting according to Counting with Binary Numbers in Symbolic Form using several different symbolic forms such as cards with dots (1, 2, 4, 8,..), colors (green = 1 dot, yellow = 2 dots,…), or symbols (mouse = 1 dot, rat = 2 dots, …), as well as systems freely invented by the children. Use Ada’s World for inspiration.
Session 2. Use a standard Abacus with rows of coloured beans naming first row “one-beans”, second row “two-beans”, third row “four-beans”, following the principle of replacing two beans (e g shifting to the right) in one row with one bean in the next row.
Session 3. Use the different systems for counting of anything that can be counted, number of fingers, number of children, number of teachers, age, length, distance, height, weight, size, money,….Display the principle of counting by repetition of +1.
Session 4. Practice addition of binary numbers in the different systems as continued counting/repetition of +1. Practice subtraction of binary numbers as reversed addition. Play selling-and-buying using a commonly agreed system. Transfer one system to another.
Session 5. Let the children construct an Abacus for binary numbers such as this one-bead abacus (see picture below) or some other design invented by the children, using material like flirting ball, straw, egg carton et cet. Compare with this one. Compare using the standard Abacus from Session2.
Session 6. Complement chosen representation (green = 1 dot, yellow = 2 dots,…) with a positional representation using the digits 0 and 1 (green = 1, yellow = 10, red = 100,…). Display the addition table: 0+0=1, 0+1=1, 1+0=1, 1+1=11. Practice addition in digital form using the addition table, see here and here. Get inspiration form these binary addition machines. Suggest children to invent binary addition machines using available material.
Session 7. Practice subtraction in digital form supported by symbolic form. Invent subtraction machines.
Session 8. Connect to numbers in binary form to decimal form, to the extent the children are familiar with decimal form, including addition and subtraction. See how simple addition and subtraction are in binary form.
Session 9. Extend to multiplication in binary form, starting with multiplication by 2, 4, 8, et cet, see Binary Multiplication.
Session 10 -… There are endless possibilities of using the acquired skills of counting to explore the World by measuring the World…
Music is the pleasure the human mind experiences from counting without being aware that it is counting. (Leibniz)
Read: Dawn of Mathematics