How are calculation errors related to working memory?
Read the full dissertation (in Hebrew)…
One of the main mathematical competencies is the ability to perform mental calculations that involve several steps. This ability entails a variety of difficulties and is supported by specific cognitive mechanisms such as counting and number processing, as well as general mechanisms such as working memory. However, only a few studies examined in detail the precise role of these cognitive mechanisms that support mental multi-digit calculation, and the specific cognitive impairments that disrupt this calculation process.
We examined 4 participants with difficulty in multi-digit calculation. They solved orally addition exercises with pairs of 3-digit numbers, while saying aloud the calculation stages they performed. For example, as response to the exercise 23+45 they could say “20+40=60 –– 3+5=8 –– 60+8=68”. We examined their errors throughout the calculation process, and distinguished between incorrect repetition on an operand word (e.g., in the example above, saying 20+30=50) and incorrect repetition on an interim result they said themselves in an earlier calculation stage (e.g., in the example above, saying 60+9 in the third stage). In line with previous studies, for each participant the error rate in repeating operands was lower than in repeating interim results.
No participant had difficulty in single-digit arithmetic facts, but all showed a difficulty in multi-digit mental addition. Thus, they did not have a general difficulty in mathematics or even in arithmetic, but a more specific difficulty, in mental multi-digit calculation. Moreover, the origin of the difficulty was not the “arithmetic building blocks” – single-digit facts – but appeared to be a more-general mechanism, e.g., working memory.
We examined 3 hypotheses for the origin of the operand-result dissociation. The different representations hypothesis postulates separate representations for the operands and the interim results, and explains the dissociation by assuming a deficit in the operand representation, with spared representation of the interim results. Contrary to the prediction of this hypothesis, the dissociation was not observed in any calculation task, but was reversed when the task instructions were slightly changed.
The maintenance-over-time hypothesis assumes that working-memory representation decay with time, so a word’s degree of memory declines as a function of the time since that word was said last. The high rate of errors in result words occurs because consecutive utterances of a result word are presumably spaced apart more than consecutive utterances of an operand word. Indeed, this time factor affected the error rate, but contrary to the hypothesis, it failed to account for the operand-result dissociation.
The results support the information-transfer hypothesis. This hypothesis builds on cognitive models that assume that working memory entails activating specific items in long-term memory with different levels of activation. The dissociation arises from a selective deficit in “transferring” an item from one activation level to another, specifically in deactivating items. The participants’ deficit causes difficulty in any word that was deactivated in a previous calculation stage, such as the words of the interim results, but does not cause difficulty in words that were not yet deactivated, such as the operand words.
The study emphasizes that mathematical difficulties are not a homogeneous phenomenon; rather, there are different types of highly-specific difficulties that disrupt mathematical performance. The origin of these difficulties is not always mathematical, domain-specific cognitive processes, but may be – as it is for our participants – domain-general processes such as memory. Finally, the study supports the idea that deactivation, or removal of information, is a critical and challenging aspect of working memory.