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How can multiplication capabilities be improved?

How do you acquire knowledge about multiplication? There are several multiplication strategies in the research literature:

  • Recalling from memory
  • Repetitive addition (2 + 2 + 2 = 3X2)
  • Writing the problem (and answering it without addition or a visible count)
  • Counting / adding object groups (3X2=||| + |||)

With the learning process, the relative frequencies of the use of strategies change. Children increase the ratio of attempts in which they recall from memory the answer and reduce their dependence on a repetitive addition and saying “I do not know.” Also, there is a retreat from saying “I do not know” in the direction of at least trying to solve any problem.

Previous studies have indicated that children make adaptive decisions regarding the use of a recall or backup strategy. The more severe the problem, the more common the use of backup strategies. These findings suggest one source of improvement in quickly and accurately solving multiplication problems: changes in “frequency of use of strategies.” Using recalling more times in each attempt increases speed and accuracy. Saying “I do not know” more and more frequently increases accuracy.

 

The Adaptive Strategic Choice Model (ASCM) was developed to predict how strategies develop. According to this model, when a problem appears a certain strategy is activated in order to answer this problem. After executing the strategy, data emerges on: the specific answer to the problem given, how quickly the answer was given and whether the answer was accurate (correct / incorrect). This data is collected and sent back to the database on the effectiveness of the strategy applied to this problem and the difficulty of the problem. The data collected on each strategy is represented at 3 levels:

  1. Effectiveness of the strategy on problems in general (average speed and overall accuracy)
  2. The effectiveness of the strategy on problems with certain characteristics (for example: odd-even)
  3. The effectiveness of the strategy on specific problems (i.e., on the specific problem that appears)

When a problem is presented, these 3 types of information are combined together to create a strong associative relationship between each strategy on that particular problem. This system also includes information on the frequency of use of the strategy in the past. Thus, the system dynamically modifies and adjusts itself so that the speed, accuracy, and frequency of use in the past are associated with each strategy as the assessment gains problem-solving experience — leading to an increase in the use of the most effective strategies and a decrease in the use of the less effective strategies.

ASCM works on each problem in two steps: 1. Choosing a strategy. 2. Implementing the actual strategy (choosing the answer)

At the strategy selection stage, strategies are selected where the associative connection between the strategy and the problem is stronger relative to the rest of the strategies. Once a strategy is selected, the model tries to implement it. According to the model, if a strategy (other than recall) is selected, this strategy will be implemented in practice. If the retrieval strategy is selected, the system selects a potential answer from a number of answers associated with the problem. If the associative connection is strong enough the answer will be given. If the connection is not strong enough the system will try to recall another answer or use a different strategy. (8X7=?: 56, 48)

In general, there is an assumption that the answers said are related to the problem; they are more connected when the answers are correct and less connected when the answers are incorrect, but in both cases the strength of the associative connection increases following the answer given. This assumption, coupled with the fact that backup strategies produce the correct answer more often than one incorrect answer, leads to the correct answer that becomes stronger relative to all other answers, and also leads to common mistakes for a particular problem to become stronger relative to less common mistakes.

Recall leads to learning just as other strategies lead to learning. When an answer is recalled (recall strategy is used), speed and accuracy information is returned to the data source existing on the strategy and the problem, in the same way that information is generated when using backup strategies.

In a long observation study conducted on 20 second-graders (the year in which multiplication is studied) in France, students were presented with single-digit multiplication exercises (1-9), and the child was required to solve them, in any way he wanted: use pages, count with fingers, count in the head. The experimenter then classified the strategies by: When the behavior was overt – using the page, fingers, etc. – it was classified as a backup strategy, when no overt behavior was observed the strategy was classified as recall. 3 tests were conducted throughout the year (January, April and June).

In this study, accuracy and speed were improved between the sessions: in the first test the children answered 45% of the problems correctly (average speed of 9.9 seconds), in the second test they answered 70% of the problems correctly (5.5 seconds), in the third test they answered 88% of the problems correctly (3.3 seconds).

Strategies used: recall 63%, repetitive addition 20%, “I do not know” 16%, object group counting and question writing 1%.

Between sessions, changes in strategies were observed: the average number of strategies that each child used increased (between sessions 1 and 2) and then decreased (between 2 and 3): First session: 3.1 strategies. Second session: 3.7 Third session: 2.4. The average increase in the number of strategies between the first test and the second test is because the children started using the two rare strategies, and the decrease between the second and third tests is because the children stopped using these strategies and also stopped saying “I do not know” as they became more skilled at multiplication.

 

If so, we see the streamlining of the application of the two strategies: speed and accuracy of the answer when using recall and repetitive addition, increased significantly between the 3 tests.

In summary, multiplication capabilities (speed and accuracy) can be improved through the following strategic changes:

  • 1. Increase in the frequency of using the best strategy (which gives a quick and accurate answer) – recall.
  • 2. Streamlining how to use strategies, for example: repetitive addition with the large number, knowing the rules of even-odd multiplication.
  • 3. An effective choice between the strategies for each question, according to the level of difficulty, and at each point in time, according to the level of knowledge and ability at this point in time.

Thus, in multiplication, it seems that the more recall is used, the better the speed and accuracy and there is no need to learn additional strategies. At the same time, it is important to know how to choose between different strategies effectively, and to learn the internal rules of multiplication (for example, when the result will be even / odd).

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