## fredag 28 augusti 2020

### DigiMat and the Multiplication Table

DigiMat is new reformed mathematics education. DigiMat starts by constructing the natural numbers (1, 2, 3, 4,.., in decimal notation) by repetition of the basic operation +1, starting from 0:

• 1 = 0 + 1
• 1+1 (= 2 )
• 1+1+1 (= 3 = 2+1)
• 1+1+1+ 1 (= 4 = 3+1)
• 1+1+1+1+1 (= 5 = 4+1)
• and so on
DigiMat starts with a binary representation instead of the usual decimal representation, because it is more basic and simpler (and therefore preferred by the computer): Instead of the ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 of the decimal system, the binary system has only two digits 0 and 1, which when used in a position system gives the following representation
• 1 = 1 (decimal)
• 10 = 2 (decimal)
• 11 = 3
• 100 = 4
• 101 = 5
• 110 = 6
• 111 = 7
• 1000 = 8
• 1001 = 9
• ...
In DigiMat for preschool different binary systems can quickly be mastered such as for example
• one red (apple) is 1
• two red (apples) is one yellow (apple)
• two yellow is one green
• two green is one blue
• two blue is one violet
• ...
with the following representation
• 1 =  red (apple)
• 2 = yellow (apple)
• 3 = yellow and red (apples)
• 4 = green (apple)
• 5 = green and red (apples)
• 6 = green + yellow (apples)
• 7 =  green + yellow + red (apples)
• 8 = blue (apple)
• 9 = blue and red (apples)
• ...
Alternatively, a one-bead abacus (column of beads with bead left = 0 and bead right = 1) is quickly mastered.  See Basics on DigiMat School.

With such binary representations any preschool child quickly learns to (without effort) perform the basic operations of addition, multiplication (as repeated addition) and subtraction of natural numbers, followed by division.  This is carefully described on the DigiMat School site.

This is to be compared with standard school mathematics, where the decimal multiplication table is both corner stone/top catch and greatest hurdle. The discussion on the necessity of learning the multiplication table by heart is filling endless texts on teaching school mathematics, with no consensus after 100s of years. Some experts claim that knowing by heart that 7 times 8 is 56, is immensely useful, while others claim this blocks true understanding, which is better expressed in a computation of the form

• 7 x 8 = (10 - 3) x (10 -2) = 10 x 10 - 3 x 10 - 10 x 2 + 2 x 3 = 100 - 30 - 20 + 2 x 3 =  50 + 2 x (2 + 1) = 50 + 2 x 2 + 2 x 1 = 50 + 2 x (1+1) + 1 + 1 = 50 + 2 x 1 + 2 x 1 + 1 +1 = 50 + 1 + 1 + 1 + 1 + 1 +1 = 56
requiring familiarity and understanding of in particular the distributive law. Here is a recent article presenting this conundrum with the shocking title:
with references to authorities such as Arthur Baroody, Jo Boadler and Gina Kling.

DigiMat lifts school mathematics out of this paralysis. Recall that the binary multiplication table is so simple, that it is not necessary to memorise: 0 x 0 = 0, 0 x 1 = 0, 1 x 0 = 0, 1 x 1 = 1.

Nor the (slightly more complex) addition table: 0 + 0 = 0, 0 + 1 = 1, 1 + 0 = 1, 1 + 1 = 10.