Acquiring a Working Knowledge of the Multiplication Tables

The young learner first learns to relate the concrete 3D presentation of number value with the related four operations i.e. add, subtract, multiplication and division. A working knowledge and automatic recall of Number Bonds to ten facilitate mathematical operations involving addition and subtraction. A working knowledge and automatic recall of the multiplication tables facilitate mathematical operations involving multiplication and division.
The earlier these skills are established the easier it is to embrace the practical application of further mathematical skills and related problem solving techniques. However, learning the multiplication tables remains an awesome task for many learners who are unable to secure the automatic accuracy required to aid everyday and higher mathematical operations. The author has therefore given below some carefully described descriptions of some of the multi-sensory learning techniques she has invented alongside some of the more recognised supportive scaffolding.
Traditionally the multiplication tables were taught through the dominantly auditory mode of verbal repetition. However the ability to automatically chant a multiplication table, e.g. the two times tables, does not necessarily mean that the learner can automatically recall random elements such as 7×2 or 2×3. For some the answer to 7×2 can only be found by chanting the two times tables form the beginning through to the required seventh element. 1×2=2, 2×2=4, 3×2=6, 4×2=8, 5×2=10, 6×2=12, and 7×2=14! This extended process of recall is time consuming and likely to distract the learner from the mathematical understanding required for successful completion of the task in hand.
Steiner schools place a focus on rote learning of the multiplication tables once the children have entered main school class one at seven plus years. In Steiner schools repetitive rote learning is accompanied by rhythmic movements such as stamping or clapping, catching and throwing together with random participation. This encourages individual children to recall a specific answer without recalling the whole of the multiplication table from the beginning. This combination of repetitive chanting and random recall within a rhythmic structure of movement is considered more beneficial than standard rote learning through repetitive recall of the tables from one times through to ten or twelve times.
The importance of learning is superseded by the nature of enjoyable participation, the development of understanding and later application within a wide range of associated intellectual activity. Multisensory activities are designed to meet the widest possible range of learning and future application. Ideally they may need to present both auditory information and visual information within structural and rhythmic movements and actions with associated kinaesthetic sensory perception. The wide range of examples presented by the author in this chapter are designed to support enjoyable learning and thereby help every learner gain an instant recall of the multiplication tables and a wide spectrum of understanding. Later development of mathematical skills involves recording this information as 2D drawn numerical symbols and/or written numbers that illustrate sums, mathematical formulas and algebra systems of investigation and resolution.

Using Coloured Bricks

A similar theme related to building the multiplication tables is illustrated in the following pictures,. This activity starts with a box of coloured bricks. The bricks used in the pictures are 18 bricks of each of the six different colours. In this set the bricks that are placed as the tenth brick in the positions of 10, 20, 30 etc are marked with a dot sticker (on the top and on the side) to mark the special significance of creating groups of ten within our base ten number system. The bricks marking ten’s are also placed in a special forward position. A basket is used to collect the correct number of bricks for each added multiple for example if constructing the six times tables, six bricks of the same colour would be selected from the main box of bricks and placed in basket. Then the ten’s bricks can be marked with stickers as required or if the bricks are already marked, a marked brick can be chosen or taken in exchange for a plain brick as required. Ideally there would be ten different coloured bricks and enough of each colour to successfully construct all the multiplication tables using permanently marked bricks for the ten’s places.
The following illustrates a line of bricks laid out as a presentation of the three times tables. The brick that completes a set of ten can be marked with stickers and/or pulled out of line. This gives a clear visual presentation of the number answers presented in the three times tables, as illustrated below:-

A number-line and a set of multiplication tables cards can also be provided as an added extra option for this activity.

Reversibility

The reversible nature of a simple multiplication e.g. that 2 x 8 is the same as 8 x 2 can be presented as a concrete example using blocks.

Thus the reversibility of all tables can similarly be seen and believed through practice with blocks until no more visual proof is needed for a genuine understanding this area of mathematical knowledge.
This mathematical phenomenon also halves the number of tables to be learnt because the automatic answer for 2×8 is the same as 8×2 and so on.
Once the reversibility of tables has been established those who can use a Tables Square can see that the multiplication for the answers in the blue squares is the reverse of the multiplication for the same answers in the pink squares. Therefore, once one set has been learnt the other answers are equally easy because the multiples are the same
i.e. 5 x 3 = 3 x 5. The squared numbers shown in green will need to be learnt as well as answers to either all the pink or all the blue squares

The bricks above represent the numbers 1 to 10 in a twined pattern that illustrates the concept of odd and even numbers. These blocks can also be organized as a collection that offers a plentiful collection of unmarked blocks together with a selection of bricks marked up to support the recognition of how these numbers may be used in relation to building number bonds to ten. Thus when one brick is added on as a unit within an unfinished block of ten – this brick is chosen
When a single unit is added that completes a block of ten then the unit block which is marked up to signal ‘a completed block of ten’ is used
Similarly when a two unit block is used as units within an incomplete block of ten this plain block is used If a one brick of a two unit block is needed to complete a ten block then the block with one brick marked up is used e.g. 9 + 2 = 11 and if both the bricks are needed to complete a ten block then the block with two units marked up is chosen e.g. 18 + 2 = 20.

The bricks above represent the numbers 1 to 10 in a twined pattern that illustrates the concept of odd and even numbers. These blocks can also be organized as a collection that offers a plentiful collection of unmarked blocks together with a selection of bricks marked up to support the recognition of how these numbers may be used in relation to building number bonds to ten. Thus when one brick is added on as a unit within an unfinished block of ten – this brick is chosen

When a single unit is added that completes a block of ten then the unit block which is marked up to signal ‘a completed block of ten’ is used

Similarly when a two unit block is used as units within an incomplete block of ten this plain block is used If a one brick of a two unit block is needed to complete a ten block then the block with one brick marked up is used e.g. 9 + 2 = 11 and if both the bricks are needed to complete a ten block then the block with two units marked up is chosen e.g. 18 + 2 = 20. The following set of nine bricks illustrate how the blocks can be used to build the nine times tables and give visual proof of how the tens number and the units number presented in the answers always add up to nine.