The Math Square - A Fresh Way Forward

Allan Lawson has devised a system of dynamic dot pattern maths that he believes improves teaching methods for addition and subtraction

Why are so many children still failing in numeracy skills? I believe one of the causes lies with the use of rote memory in the early years of maths education. Childrenʼs natural ability to visualise and mentally manipulate patterns can change things.

Banish rote memory!

The use of rote memory is both an inadequate and an inappropriate teaching method in maths. Yet rote memory is still often relied upon to learn the 55 addition facts up to 10 + 10, and the further 55 multiplication facts in the same range. Thatʼs a total of 110 facts -quite a task for young children - enough to put anyone off maths for good!

I believe rote memory should be left to language teachers, and for learning poetry.

But removing rote memory from early arithmetic is not easy: for what other methods are there enabling pupils to recall the 110 addition and multiplication facts?

And is instant recall even necessary?

Domino inspired solution

Around 40 years ago, when teaching arithmetic to children, I noticed how easily quite young children (4 or 5 years of age) could identify the dot pattern groupings on domino pieces. This lead to a method enabling children to carry out a few additions and subtractions by identifying the numerical value of combined or remaining dominoes.

But using dominoes had severe limitations. Some combinations of two dominoes did have an equivalent domino pattern to identify. For example:-

However many combinations had no equivalent domino pattern. The merged pattern below does not exist as a single domino. In this case, a pupil would have to learn two patterns for the number 4 - the one below and the pattern in the example above. With 55 combination of 2 dominoes in the range 1 to 10 for additions up to 20, this would often mean learning to identify many different patterns for the same resultant number. Readers may be relieved to learn I did not experiment to see if children could cope with this!

Attempting to use dominoes to teach addition and subtraction proved quite unsatisfactory.

The Math Square

The limitations that dominoes presented acted as an incentive for me to create new dot patterns which could be more useful in carrying out arithmetic. I decided upon a systematic set of dot patterns to represent numbers to 20 - rather than a set of random dots that werenʼt in some form of pattern - to cover all additions to 20.

The dot patterns can be displayed using plastic counters or be printed on square cards. I arranged up to 9 such dots on each card in a 3 by 3 square arrangement and named the series of dot patterns the “Math Square” numeral system.

But with cards containing 9 dots in a square pattern, where could the tenth dot be placed? The answer came from children themselves playing with printed “9 - cards” and loose counters: the tenth counter was neatly placed on top of the centre dot of the card!

Bearing in mind more advanced maths lessons later, when cards and loose counters would no longer be used, and diagrams alone would be employed, it seemed sensible to represent the ninth and tenth dots at the centre of a ten pattern card with a single dot of a different colour, for example, purple:

The pattern above shows ten represented by a dot pattern in a Math Square. Later, we shall see alternatives, where the tenth purple dot can be positioned elsewhere in the ten dot pattern.

Numbers between 10 and 20 were arranged on cards double in size, with a 10-pattern on the left-hand half of the card and patterns from 1 to 10 arranged on the right-hand side.

Before using the new cards with children, I used only plastic counters to arrange the dot patterns in order to provide plenty of hands-on activity, with pupils adding or taking away counters from various pattern arrangements in doing sums.

As sums of 5 + 5 and higher were approached, it became useful to use a combination ofMath Square cards and plastic counters. The dots printed on the cards were made in the same colour and size as the loose plastic counters. Counters can then be placed on the cards to create new dot patterns in addition or dots covered with white card in subtraction. Variations of patterns for some numbers

For some of the numbers, such as 4 or 5, I found that two or more variations of patterns were found to be useful in carrying out additions. Below are patterns that teachers may wish to use: they are not exclusive and teachers may ﬁnd other patterns more useful.

The essential point I found is this: in order for children to re-assemble and re-arrange dot patterns into otheridentiﬁabledot patterns, a dot pattern numeral system must already be established in their minds, no matter what dot patterns are used. Once a set of dot pattern numerals has been chosen, it must be memorized by pupils. The use of ash cards will help. Here are my suggestions :

The patterns should be identiﬁable from any orientation, whether as mirror images or revolved images as this will help pupils in performing dot pattern arithmetic.

Performing additions

In performing additions of two numbers, the aim is to merge or re-arrange and then re-assemble the 2 dot patterns to form a single identiﬁable new pattern. Where it seems possible to create a 10 pattern, this must be the ﬁrst step. Making a 10 pattern is essential knowledge, as it is when using, say, Dienes blocks, an abacus or any other decimal system.

I suggest below some strategies to assist in creating a 10 pattern where one of the numbers in the addition sum is a 6, 7, 8 or 9. These numbers can be “built up” to form 10 by removing dots from the other number “n” in the addition.

So, where we have (6 + n), n being in the range 1 to 4, the aim is to build up the 6 dot pattern by attaching the n pattern to the 6 pattern, or where n≥5, by removing 4 dots from the n pattern, attaching them to the 6 pattern, thus forming a new 10 pattern. The resulting pattern formation will then be in identiﬁable form.

Below is displayed a single strategy for the solution of each example addition: the many equally good alternatives for re-arranging and re-assembling the dot patterns are left for you and your pupils to explore.

i) 6 + 3 : the 6 and 3 patterns merge to make the 9 pattern:

ii) 6 + 4 : the 6 and 4 patterns merge to make the 10 pattern: in the process of merging, two dots overlap at the center of the square - and is represented by a purple dot. Only a single overlap is allowed in any ten dot square. Overlapping dots are not allowed in smaller numerals.

iii) 6 + 5 : in dot patterns :

Before merging, the 5 pattern can be re-arranged as a 4 pattern, plus one:

Then the 4 pattern can be merged with the 6 pattern to make a 10 pattern, with 1 dot remaining – resulting in the 11 pattern:

iv) 6 + 6 : in dot patterns :

Before merging, the right-hand 6 pattern can be re-arranged as a 4 pattern plus a two pattern:

Now the 4 pattern can be merged with the 6 pattern, the re-assembled dots making a 10 pattern, with 2 dots remaining; resulting in the 12 pattern:

In additions of (7 + n) the rule to follow is to remove 3 dots from n, the other number, and merge these with the 7 pattern, so forming a 10 pattern. Eg:-

i) 7 + 3 : the 7 and 3 patterns merge to make another form of the 10 pattern:

ii) 7 + 8 : by removing 3 dots from the 8 pattern and merging these with the 7 pattern, a 10 pattern is formed, with a 5 pattern remaining, the two new patterns forming a 15 dot pattern:

In additions of (8 + n), remove the 2 dots from the other number n, and merge these with the 8 pattern, to make a 10 pattern. Eg:

i) 8 + 2 : the 8 and 2 patterns merge to form a 10 pattern:

ii) 8 + 8 : the suggested steps needed to form a 16 pattern are:

In additions of (9 + n), we simply remove one dot from n and merge it with the 9 pattern, making a 10 pattern e.g:

9 + 9 :

10, 20 and beyond

The Math Square dot pattern numeral system does not stop at 20. However, to reduce the number of dots required when dealing with larger numbers, we could dispense with some of the red unit dots for all numbers larger than 9. So instead of 10 consisting of 9 red dots and a central purple dot, it could be represented by the purple dot alone. And 20 would be represented by two purple dots, thirty by 3 purple dots, and so on. One hundred could be represented by a third coloured dot, say yellow, with each successive power of ten being represented by yet another colour. For example 1,342 might be represented as:

The Math Square numeral system could in theory represent enormous numbers, limited only to the extent that colours might be exhausted in representing higher and higher powers of ten.

In a future article, I will describe how the Math Square was developed into an ordinal place-value numeral system capable of representing all positive whole numbers to inﬁnity - and beyond!

Subtraction

I have not provided examples of how subtraction can be performed. But teachers can experiment with the dot pattern cards, in a similar way as some additions are carried out, by the removal of counters from a dot pattern or by covering up dots on cards to show the resulting dot pattern in an identiﬁable form.

Mental arithmetic

One remarkable development that came to light from children who had played with the dot pattern cards and counters for some while was their ability to visualize the dot patterns in their mindʼs eye, then to re-arrange and re-assemble them into identiﬁable patterns, without the use of any physical materials.

Before I conclude: Q - Why have an overlapping tenth dot?

A criticism that could be leveled at this numeral system is this: wonʼt an overlapping tenth dot in the 10-dot pattern cause confusion with the 9-dot pattern? Indeed, why introduce a dot numeral system with an overlapping tenth dot? After all, there are other regular 10-dot patterns without overlapping dots that can form the basis of a numeral system. For example: -

However, my justiﬁcation in choosing the Math Square format lies with itʼs development into a similar square with a tenth position lying at itʼs centre, for use in solvingthe multiplication facts. The tenth overlapping central position becomes vital in this development.

Conclusions

1. Childrenʼs visual mental ability:

Children can easily identify dot pattern numbers; they can also visualize dot patterns in their mindʼs eye and re-arrange and re-assemble these patterns to form previously known patterns. And they have no difﬁculty in distinguishing the 9-dot pattern from the 10-dot pattern.

The Math Square system provides children with good opportunities to exercise their visual mental ability to manipulate dot patterns, and a fresh way to explore the possibilities to be discovered in dot pattern maths.

In my opinion psychologists should re-examine the role that childrenʼs visual mental ability could play in the teaching of mathematics generally.

2. Memorizing facts unnecessary:

Employing these methods to carry out additions and subtractions has shown me that committing the facts to memory for instant recall is unnecessary. Children can quickly arrive at the results of the 55 additions mentally using this system of dynamic dot pattern maths.

Next

The following article shows how the Math Square format was developed into a positional framework with vectors enabling children to ﬁnd the facts of the multiplication tables mentally in a systematic way in place of rote memory. Visualized spatial abilities are here too employed.

For those who wish to try some Math Square cards, the on-line version of this article has a printable page of cards, which can be cut or guillotined to provide a set of 20 cards.

[Note about the author: Allan Lawson, 62, is a retired English lawyer now living in Canada, who has had a lifelong interest in numeral theory and the teaching of mathematics]