Maths Strategies

Mathematics education leaders are often asked about strategies for teaching students to calculate both mentally and with paper and pencil. The first significant step for calculating involves learning the basic number facts associated with each operation. These facts form the basis for all future number work.

Thinking Strategies

The most effective way for students to learn the basic facts is to arrange the facts into clusters (Fuson 2003; Thornton, 1990). For example, the idea of using a double, or numbers close to a double, forms the basis of the strategy in the “use doubles” addition cluster. The other addition clusters involve “counting on” and “bridging to ten”. Between these three clusters, students can master all 100 basic addition facts. “Using addition” is the most effective thinking strategy for helping students to learn the basic subtraction facts.
The idea of using a known multiplication fact involving ten, forms the basis of the strategy in the “use tens” multiplication cluster for the fives facts. The other multiplication clusters involve “doubling” for the twos, fours, and eights facts; “using a rule” to multiply by one or zero; and “building up or down” from a known fact for the sixes and nines facts. The majority of the remaining facts are covered by the turnarounds of the above. Just as subtraction is the inverse operation of addition, division is the inverse operation of multiplication. Because of this relationship between the two operations, “using multiplication” is the most effective thinking strategy for helping students to learn the basic division facts.

Teaching Sequence and Stages

The most widely accepted sequence for teaching the addition facts is as follows:

• Count on (1, 2, 3, and 0) with their turnarounds
• Use doubles (double, double plus 1, and double plus 2) with their turnarounds
• Bridge to 10 with their turnarounds
The sequence for teaching the multiplication facts is:

• Use tens (fives facts) with their turnarounds
• Doubling (twos, fours, and eights facts) with their turnarounds
• Use a rule (ones and zeros facts) with their turnarounds
• Build up and build down (sixes and nines facts) with their turnarounds
Along with the above sequences for teaching the strategies, the activities for each strategy are also sequenced according to the following four stages of teaching and learning: introduce the strategy, reinforce the strategy, practice the facts, extend the strategy. The use tens multiplication strategy for the fives facts will be used to explain and illustrate each of these stages.

Introduce the Strategy

Hands-on materials, real-world stories, pictures, discussions, and familiar visual aids are typical of the activities that are used to model and introduce each thinking strategy in the first stage. For multiplication, a rectangular array of dots is often used to show facts and their turnarounds. For example six rows of ten dots show 6 x 10 or 10 x 6. A card such as the one below, can then be used to demonstrate the thinking required to figure out a fives fact. After determining the total number of dots, fold or cover up half the array and have the student verbalise what they see and how they can figure out the total number of dots.

Reinforce the Strategy

The activities in this stage are designed to make links between the concrete/pictorial and symbolic representations of the facts being examined. Students also reflect on how the strategy works and the numbers to which it applies. Given pictures such as the one below, students can be asked to write the answer to a tens fact they see, then colour half the array and write a number fact to match what is now coloured. Experiences such as this serve to reinforce the relationship between the two facts and the thinking strategy that is involved.

Practice the Facts

At this stage, games, flashcards, worksheets, and other activities provide students with opportunities to apply and demonstrate their knowledge of the facts. The students should use mental computation only and fast recall is stressed.
The following game for two or three players is a simple way to practice the fives facts.
Materials:
• About ten counters of one colour for each student.
• Two blank cubes with the numerals 2, 4, 3, 3, 5, and 5 written on the faces of one cube and 6, 8, 7, 7, 9, and 9 on the faces of the other. Each 6 and 9 should be underlined to avoid confusion.
• A game board. Have the students sketch the game board below onto a grid or sheet of paper.

To play the game:
• The first player rolls both cubes.
• The player then chooses one of the numbers and multiplies it by five. Encourage the player to verbalize their thinking, for example, “Four tens are forty, so four fives are twenty.”
• The player finds the product and places a counter on the game board.
• The next player has a turn.
• As the game continues, a player misses a turn if they cannot cover a number on the board (only one counter can be placed on any one space).
• The player who has the greatest number of counters on the board at the end is the winner.

Extend the Strategy

Once mastered, students are encouraged to apply the strategy to numbers beyond the range of the basic number facts. The activities in this stage are designed to further strengthen students’ number sense. To extend the Use-Tens strategy, have the students explain how they can use the first answer in each number sentence below to figure out the second answer.

10 x 18 = ____ so 5 x 18 = ____

10 x 26 = ____ so 5 x 26 = ____

10 x 22 = ____ so 5 x 22 = ____

10 x 15 = ____ so 5 x 15 = ____

Later, students will learn to vary this strategy by doubling and halving the two factors rather than doubling one factor and halving the product. For example, by doubling and halving, students can quickly calculate that 14 x 35 has the same answer as 7 x 70, which is easy to calculate mentally.

Conclusion

It has long been known that students should have instant and accurate recall of the basic number facts. But now, more so than ever before, with the increased emphasis on mental computation in contemporary curricula, it is essential that these facts be taught using strategies that serve as the foundation of mental computation.

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As co-founder of ORIGO Education, James Burnett strives to lift the profile of mathematics through dynamic professional learning and the development of quality, research-based classroom materials. James frequently presents workshops and speaks at conferences, and has authored more than 200 mathematics books for teachers and primary students.

For more information about ORIGO Education professional learning and our unique resources, such as the Box of Facts kits shown in this article (images), please contact us at: info@origo.com.au or 1300 674 461.

References

Burnett, J., Irons, C. & Turton, A. (2007). The Book of Facts. (Series title) Brisbane: ORIGO Education.

Fuson, K. C. (2003). Developing mathematical power in whole number operations. In J. Kilpatrick, W.  G. Martin & D. Schifter (Eds.), A research companion to principles and standards to school mathematics (pp. 68-94). Reston, VA: National Council of Teachers of Mathematics.

Irons, C., Burnett, J. & Irons, R. (2006). The Box of Facts (Kits) Brisbane: ORIGO Education. Thornton, C. (1990). Strategies for the basic facts. In J. N. Payne (Ed.), Mathematics for the young child (pp. 131-151). Reston, VA: National Council of Teachers of Mathematics.