What research has modelled the difficulty of mental mathematical calculation?

What research has modelled the difficulty of mental mathematical calculation?

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I posted this also on mathoverflow.

What research has modelled the difficulty of evaluating a formula mentally (for your average, numerate, person, not a trained mental calculator)?

For instance, evaluating an arithmetic expression requires in general requires three registers, but some only require two and are considerably easier to deal with. Additions is easier than subtraction which is easier than multiplication which is easier than division. Multiplication by small integer number is easy, as are multiplications and divisions by 2,5,10…

Many rule of thumbs are intuitive, but I'm looking for a more comprehensive work on the topic.

There is a huge amount of research on this topic, particularly in developmental psychology.

Siegler: As @Joel has noted, Robert S. Siegler provides a great entry point into this literature (see his list of selected publications with PDFs). He has done a lot of research on the strategies that children use to solve mathematical problems. His research includes both empirical and computational modelling approaches. He also has a number of studies that compare performance across features of arithmetic problems. For example, you could look at Siegler and Lemaire (1997) which shows in an adult sample the differences in speed and accuracy for different kinds of multiplication problems (e.g., multiples of 10 or not, single versus double digits). As another example, Siegler and Shrager (1984) show empirical error rates and strategy use in children for all possible combinations of single digit addition.

Act-R: There are also a number of publications using the ACT-R framework on mathematical problem solving (see this list from the ACT-R website). Perhaps check out Ritter et al (2007) for a review.

Defining difficulty: In general, there are questions about how you would define difficulty. Typical indicators would include task completion time and accuracy. Also, when talking about the difficulty of a mathematical operation, you need to consider the strategy that an individual is using. In particular, there is general shift with practice from using an algorithmic strategy to using a retrieval strategy (for a review, see Delaney et al, 1998). Thus, at first children use strategies like repeated addition to solve multiplication problems whereas with practice, they typically learn to retrieve the answer from memory. The ability to retrieve is broadly related to the number of times the specific problem has been encountered with the correct answer available, along with forgetting effects related to periods where there is no exposure.


  • Delaney, P.F., Reder, L.M., Staszewski, J.J. & Ritter, F.E. (1998). The strategy-specific nature of improvement: The power law applies by strategy within task. Psychological Science, 9, 1-7. PDF
  • Siegler, R. S., & Lemaire, P. (1997). Older and younger adults' strategy choices in multiplication: Testing predictions of ASCM via the choice/no choice method. Journal of Experimental Psychology: General, 126, 71-92. PDF
  • Siegler, R. S., & Shrager, J. (1984). Strategy choices in addition and subtraction: How do children know what to do? In C. Sophian (Ed.), The origins of cognitive skills (pp. 229-293). Hillsdale, NJ: Erlbaum. PDF1 PDF2
  • Ritter, S., Anderson, J.R., Koedinger, K.R. & Corbett, A. (2007). Cognitive Tutor: Applied research in mathematics education. Psychonomic bulletin & review, 14, 249-255. PDF

Frailties of Human Reasoning

The processing capacity of human working memory is limited (26). Our intuitive system of reasoning, which is often known as “system 1,” makes no use of it to hold intermediate conclusions. Hence, system 1 can construct a single explicit mental model of premises but can neither amend that model recursively nor search for alternatives to it (22). To illustrate the limitation, consider my definition of an optimist: an optimist =def a person who believes that optimists exist. It is plausible, because you are not much of an optimist if you do not believe that optimists exist. Yet, the definition has an interesting property. Both Presidents Obama and Bush have declared on television that they are optimists. Granted that they were telling the truth, you now know that optimists exist, and so according to my recursive definition you too are an optimist. The definition spreads optimism like a virus. However, you do not immediately grasp this consequence, or even perhaps that my definition is hopeless, because pessimists too can believe that optimists exist. When you deliberate about the definition using “system 2,” as it is known, you can use recursion and you grasp these consequences.

Experiments have demonstrated analogous limitations in reasoning (44, 45), including the difficulty of holding in mind alternative models of disjunctions (46). Consider these premises about a particular group of individuals: Anne loves Beth. Everyone loves anyone who loves someone.

Does it follow that everyone loves Anne? Most people realize that it does (45). Does it follow that Charles loves Diana? Most people say, “No.” They have not been told anything about them, and so they are not part of their model of the situation. In fact, it does follow. Everyone loves Anne, and so, using the second premise again, it follows that everyone loves everyone, and that entails that Charles loves Diana—assuming, as the question presupposes, that they are both in the group. The difficulty of the inference is that it calls for a recursive use of the second premise, first to establish that everyone loves Anne, and then to establish that everyone loves everyone. Not surprisingly, it is even harder to infer that Charles does not love Diana in case the first premise is changed to “Anne does not love Beth” (45).

Mathematics Anxiety and Attitudes to Mathematics

Attitudes to mathematics, even negative attitudes, cannot be equated with mathematics anxiety, as the former are based on motivational and cognitive factors, while anxiety is a specifically emotional factor. Nevertheless, attitude measures tend to correlate quite closely with mathematics anxiety. For example, Hembree (1990) found that in school pupils, mathematics anxiety showed a mean correlation of 𢄠.73 with enjoyment of mathematics and 𢄠.82 with confidence in mathematics. In college students, the equivalent mean correlations were a little lower than in schoolchildren, but still very high: 𢄠.47 between mathematics anxiety and enjoyment of mathematics, and 𢄠.65 between mathematics anxiety and confidence in mathematics.

Mathematics anxiety seems to be particularly related to self-rating with regard to mathematics. People who think that they are bad at mathematics are more likely to be anxious. Most studies indicate a negative relationship between mathematics self-concept and mathematics anxiety (Hembree, 1990 Pajares and Miller, 1994 Jain and Dowson, 2009 Goetz et al., 2010 Hoffman, 2010).

However, as most of these studies are correlational rather than longitudinal, it is hard once again to establish the direction of causation: does anxiety lead to a lack of confidence in one's own mathematical ability, or does a lack of confidence in one's mathematical ability make one more anxious? Ahmed et al. (2012) carried out a longitudinal study of 495 seventh-grade pupils, who completed self-report measures of both anxiety and self-concept three times over a school year. Structural equation modeling suggested that each characteristic influenced the other over time, but that the effect of self-concept on subsequent anxiety was significantly greater than the effect of anxiety on subsequent self-concept. The details of the results should be taken with some caution, because although the study was longitudinal, it was over a relatively short period (one school year) and also a different pattern might be seen among younger or older children. However, it provides evidence that the relationship between mathematics anxiety and mathematics self-concept is reciprocal: each influences the other.

A closely related construct is self-efficacy. Ashcraft and Rudig (2012) adapted Bandura's (1977) definition of self-efficacy to the topic of mathematics, stating that “self-efficacy is an individual's confidence in his or her ability to perform mathematics and is thought to directly impact the choice to engage in, expend effort on, and persist in pursuing mathematics” (p. 249). It is not precisely the same construct as self-rating, as it includes beliefs about the ability to improve in mathematics, and to take control of one's learning, rather than just about one's current performance but there is of course significant overlap between the constructs. Studies have demonstrated an inverse relationship between self-efficacy and math anxiety (Cooper and Robinson, 1991 Lee, 2009).

Attitudes to mathematics also involve conceptualization of what mathematics is, and it is possible that this is relevant to mathematics anxiety. Many people seem to regard mathematics only as school-taught arithmetic, and may not consider other cultural practices involving numbers as mathematics (Harris, 1997). Also, people may not recognize that arithmetical ability (even without considering other aspects of mathematics) is made up of many components, not just a single unitary ability (Dowker, 2005). This can risk their assumption that if they have difficulty with one component, they must be globally � at maths,” thus increasing the risk of mathematics anxiety.

Most studies of mathematics anxiety have not differentiated between different components of mathematics, and it is likely that some components would elicit more anxiety than others and that the correlations between anxiety about different components might not always be very high. Indeed, studies which have looked separately at statistics anxiety and (general) mathematics anxiety in undergraduates have suggested that the two should be seen as separate constructs, and differ in important ways. For example, as will be discussed in the Section Gender and Mathematics Anxiety, most studies suggest that females show more mathematics anxiety than males, but there are no gender differences in statistics anxiety (Baloğlu, 2004).

The Big Issue

Before discussing the significance of the possible behavioral underpinnings of retirement, it is important to disentangle the different meanings of the term "retirement." That is, "retiring" may mean different things to different people. First, retiring can mean exiting the workforce when individuals no longer want to or are no longer able to work, they may decide that it is time to leave the workforce. Second, retiring may refer to claiming Social Security benefits. For many retirees, those two events likely are one and the same, but those events do not always temporally coincide&mdashindividuals may claim benefits while continuing to work or they may stop working without claiming benefits.

When individuals decide to stop working, they must have a way to support themselves financially, as their income from work will no longer be available. Thus, the question of how to support oneself in retirement should be an important consideration in the retirement decision. Traditionally, income during retirement is thought to come from three main sources, or what is generally referred to as a "three-legged financial stool": Social Security benefits, pensions, and personal savings. Unfortunately, many individuals fail to consider the issue of financial well-being in retirement until retiring becomes imminent (EBRI 2008), which can mean that the "personal savings" leg of the stool is weaker than it should be. In addition, the number of workers who participate in an employer-sponsored defined benefit pension plan has decreased over the past two-to-three decades (Buessing and Soto 2006). 5 Individuals consequently may be left financially unprepared for retirement, leading them to rely heavily on Social Security benefits.

Coile and others (2002) highlighted a number of additional factors that may affect the relationship between retiring and benefit claiming, including life expectancy, age at retirement, and marital status. Importantly, however, the authors noted that many people may simply claim benefits immediately at age 62, without taking into account the far-reaching financial effects of this uptake decision. As such, the authors suggested that "claiming behavior should be better understood by those interested in Social Security" (384).

The claiming decision for individuals who must leave the workforce early citing poor health or a layoff very likely depends entirely on their financial condition. For those retirees, choosing the option to delay claiming may not be possible if they do not have sufficient savings or an employee pension. In addition to those needing or forced to leave the workforce, a substantial number of retirees choose to stop working before reaching their FRA . According to EBRI 's (2006) report, 38 percent of individuals reported retiring early although 39 percent of early retirees surveyed said they did so because they could afford to, 24 percent reported that they wanted to do something else and 22 percent indicated that they retired early for family reasons. If individuals in those latter two groups have little personal retirement savings and no pension, they will quite likely claim Social Security benefits upon retiring.

Regardless of the specific financial needs of a potential retiree, if individuals work longer, they are less likely to claim benefits whether they have sources of retirement funding outside of Social Security or not (Gustman and Steinmeier 2002). That is, individuals who continue to earn wages through working are less likely to claim benefits, regardless of their personal savings or pensions. 7 Therefore, when encouraging individuals to delay claiming Social Security so that they receive a higher monthly benefit for the rest of their lives, it may behoove policymakers to shift their focus from delaying claiming to encouraging prolonged labor force participation. 8 With this in mind, many of the issues raised later focus on behavioral and psychological impediments to working longer, and many of the suggested interventions focus on working longer and claiming later.

New political psychology research models the epidemic-like spread of hate speech

Even though a precise definition of hate speech is hard to pin down, it is a well-known phenomenon, pervasive online and, studies show, increasingly so. It is particularly insidious in that, as demonstrated by a review of the literature, exposure to hate speech primes individuals to experience feelings of contempt and behave in antisocial ways towards victims, both individually and collectively. This, of course, reinforces the use of hate speech.

A desire to understand the epidemic-like spread of hate speech prompted two Polish researchers to devise an agent-based mathematical model, wherein a simple set of rules played out across hundreds of independent, interacting digital agents in an iterative fashion can reveal important patterns and provides a model against which researchers can measure real-world observations. Their study, Hate Speech Epidemic. The Dynamic Effects of Derogatory Language on Intergroup Relations and Political Radicalization, appeared in Advances in Political Psychology.

“The problem of hate speech is increasingly visible both in people’s online environments, as well as in political discourses,” explained study author Michał Bilewicz, an associate professor of psychology at the University of Warsaw.

“Twenty years ago you would not find a single prominent politician using such language in his or her political speeches or statements. Many survey studies show that day by day our exposure to hate speech increases, both online and offline. This motivated our desire to explain dynamics of hate speech proliferation in social media and psychological consequences of being exposed to hate speech.”

“We used the term ‘hate speech epidemic’ because the pattern of hate speech proliferation resembles the spread of epidemic diseases,” Bilewicz said.

Following an extensive and highly insightful literature review, the authors set to defining their model. The literature reveals three important properties that the authors use as agent parameters.


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First, all agents are given a level of contemptuous prejudice (CP), varied randomly from 0 to 1, but which changes over time as a result of exposure to hate speech. Second, individuals respond to levels of hate speech in their peers: adjusting up or down to match it. Thus, an agent’s neighbors determine in part its trajectory in use of hate speech. Third, individuals vary in their emotional reaction to hate speech, which becomes blunted with exposure.

According to their model, agents don’t engage in hate speech if: they already have a low level of CP or they are pressured by anti-discriminatory norms (neighbors don’t use hate speech). Agents use hate speech if: they have a high level of CP and their neighbors use hate speech (radicalization) they have a high level of CP and their neighbors don’t use hate speech, but they fail to recognize it (“faux pas”).

In each iteration, an agent who is exposed to hate speech will increase in their level of contemptuous prejudice and decrease in their sensitivity to hate speech (and thus their ability to identify it or respond to social pressure).

The results of the model demonstrate just how easily a population can slide into common and prevalent use of hate speech, and helps examine why and how this might occur.

It’s important to remember that this is only a model. Nonetheless, it does provide some important insights. For example, in almost all cases where agents did not use hate speech despite high levels of contempt, it’s because their neighbors were “low-prejudiced nonhaters.” Likewise, individuals with relatively low levels of CP may still engage in hate speech based on their neighbors’ propensity to do so. Additionally, agents tended to form many, small clusters of hate, rather than large, centralized clumps of hate speech.

“The main message from this study is quite fatalistic,” Bilewicz told PsyPost. “People immersed in social media become less sensitive to hate speech, they start treating offensive language as something normal. This, in turn, alters their perception of minorities, immigrants, or gay people — as well as emotions felt toward these groups. Finally, they start using hate speech themselves. We observed that this process is a consequence of moving from traditional media as key sources of information to social media and online journalism.”

The authors also provide some real-world evidence to corroborate their model. Unregulated social network Gab, absent moderators and where hate-speech policies have been deliberately forgone, was witness to steady increases in hate speech from 2016 to 2018. Furthermore, new Gab members became more rapidly hateful as hate speech increased, and language used in the Gab community became more homogeneously hateful. Finally, hateful users were more likely to become popular in their network and become influencers.

The model, as the authors state, is limited by its simplicity and the fact that there is a stepwise movement towards ever more hate speech, curbed only by social norms. In the real world, these are clearly not the only influences. Individuals are capable of becoming less prone to use hate speech, even less contemptuously prejudice, with education, time, and experiences. However, there is an undeniable growth in hate speech in many parts of society.

“The emotional nature of how hate speech affects its recipients is still unclear. The term ‘hate speech’ suggests that hate is the key emotion elicited by offensive language. Our studies suggest that it is contempt, and sometimes even enjoyment, rather than hate. The fact that offensive language can generate positive emotions might be responsible for its very rapid spread in the society,” Bilewicz told PsyPost.

These simple but powerful observations about hate speech require experimental and observational evidence to better understand why and how hate speech spreads, and what we can do to combat it. If mere exposure to hate speech dulls individuals to its harmful nature, as has been shown time and again, what avenues are there for reversing these effects?

The present research suggests that, on a personal level, speaking up against hate speech in one’s own social circles can greatly limit its spread. Doing so at the levels of social media as well could powerfully curb hate speech tendencies. The model is based on agent interaction, rather than spontaneous self-education, so its implications are necessarily social in nature, but no less powerful for this limitation.

Scientific literature shows that hate speech paves the way for more violent and aggressive anti-social, widespread behaviors, all the way up to genocide. The Nazis made studied and cruelly reasoned use of these observations to convince entire populations of the necessity to exterminate the Jews, while Hutu propaganda before the Rwandan genocide used similar tactics.

Most frighteningly, even social media platforms with strict anti-hate speech policies seem to have great difficulty curbing the growth of hate speech.

Globally, hate speech is on the rise. Simple countermeasures, like speaking out against hate speech when one encounters it, can be highly effective, but understanding the sociopsychological mediators of its spread are paramount, be they modelled mathematically or observed empirically.

“The key question to be addressed is how to translate our knowledge about the hate speech epidemic into possible effective strategies counteracting the proliferation of hate speech ,” Bilewicz said. “Currently, working with companies applying artificial intelligence in social media environments, we are experimenting with real life interventions against hate speech that are based on our theoretical models that were developed in laboratory experiments and correlational studies.”

Mental Model

Mental models developed from experience can be resistant to instruction. In the case of curvilinear momentum cited above, even students who had learned Newton's laws in physics classes often maintained their belief in curvilinear momentum. One technique that has been used to induce model revision is that of bridging analogies (Clement 1991 ). Learners are given a series of analogs. The first analog is a close match to the learner's existing model (and therefore easy to map). The final step exemplifies the desired new model. The progression of analogs in small steps helps the learner to move gradually to another way of conceptualizing the domain.

Mental models have been used in intelligent learning environments (see Intelligent Tutoring Systems ). For example, White and Frederiksen's ( 1990 ) system for teaching physical reasoning begins with a simple mental model and gradually builds up a more complex causal model. Early in learning, they suggest, learners may have only rudimentary knowledge, such as whether a particular quantity is present or absent at a particular location. By adding knowledge of how changes in one quantity affect others, and then progressing to more complex relationships among quantities, learners can acquire a robust model.

Another implication of mental models research is that the pervasiveness and persistence of mental models needs to be taken into account in designing systems for human use. Norman ( 1988 ) argues that designers' ignorance of human mental models leads to design errors that plague their intended users. Sometimes these are merely annoying—e.g., a door that looks as though it should be pulled, but that needs to be pushed instead. However, failure to take mental models into account can lead to serious costs.

An example of such a failure of mental models occurred in the Three-mile Island nuclear disaster. Early in the events that led to the melt-down, operators noted that the reactor's coolant water was registering at a high pressure level. They interpreted this to mean that there was too much coolant and accordingly they pumped off large amounts of coolant. In fact, the level was dangerously low, so much so that the coolant was turning into steam—which, of course, led to a sharp increase in pressure. Had this alternate model been at hand, the operators might have taken different action.

General conclusions

In sum, we have presented an overview of the impacts of anxiety on cognition. Both threat of shock𠅊 translational anxiety induction𠅊nd pathological anxiety disorders promote the detection of potentially harmful stimuli at multiple levels of cognition from perception to attention to memory and executive function. At the most basic level this tends to be associated with improved perception of environmental changes irrespective of valence, but at more complex levels of cognition, leads to promotion of cognitive processes relevant to harm avoidance at a cost to certain functions such as working memory, while leaving still further processes (such as planning) unperturbed. However, we also draw attention to a number of processes, such as spatial learning, PPI and non-emotional Stroop which are discrepant across threat of shock and anxiety disorders. We argue that this discrepancy, largely in cold cognitive functions, may reveal the differences between adaptive and maladaptive anxiety. Future work should attempt to delineate the causes of these differences, as well as explore the possible use of (1) cognitive interventions for the treatment of anxiety and (2) the use of threat of shock as an analog screen for candidate anxiolytics. The precise neural mechanisms underlying these effects are far from clear this review, which is the first to collate the growing number of studies using the translational threat of shock paradigm, aims to highlight the value of this paradigm as a means to clarify these neural mechanisms. Given the large burden represented by anxiety disorders, such research is of pressing concern.

Conflict of interest statement

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Problem Solving as an Instructional Goal

If our answer to this question uses words like exploration, inquiry, discovery, plausible reasoning, or problem solving, then we are attending to the processes of mathematics. Most of us would also make a content list like algebra, geometry, number, probability, statistics, or calculus. Deep down, our answers to questions such as What is mathematics? What do mathematicians do? What do mathematics students do? Should the activities for mathematics students model what mathematicians do? can affect how we approach mathematics problems and how we teach mathematics.

The National Council of Teachers of Mathematics (NCTM) (23,24) recommendations to make problem solving the focus of school mathematics posed fundamental questions about the nature of school mathematics. The art of problem solving is the heart of mathematics. Thus, mathematics instruction should be designed so that students experience mathematics as problem solving.

The National Council of Teachers of Mathematics recommends that problem solving be the focus of school mathematics in the 1980s. An Agenda for Action (23)

We strongly endorse the first recommendation of An Agenda for Action. The initial standard of each of the three levels addresses this goal.
Curriculum and Evaluation Standards (24)

Limitations and future research

While multiple databases were used in this scoping review, some articles may be missed due to using specific terms in the search strategy. The disciplines covered in this scoping review were psychology, engineering, mathematics and some of the health disciplines such as nursing. Future research might focus on numerical ability and maths anxiety in university students who need maths and calculation in their future careers as engineers and health care professionals.

For example, the relationship between medication and drug calculation errors and maths anxiety in the health care field can be researched. Moreover, the relationship between self-awareness and numerical ability and maths anxiety and their impact on the performance and ability of the university students can be a future research topic. Finally, developing a new teaching package or strategy that reduces maths anxiety can be tested on university students.


We gratefully acknowledge the ongoing contribution of the participants in the Twins Early Development Study (TEDS) and their families. TEDS is supported by a programme grant [G0901245 and previously G0500079] from the UK Medical Research Council our work on environments and academic achievement is also supported by grants from the US National Institutes of Health [HD044454, HD046167 and HD059215]. CMAH is supported by a British Academy Research Fellowship OSPD is supported by a Sir Henry Wellcome Fellowship [WT088984] RP is supported by a European Research Council Advanced Investigator Award [295366] YK's and SM's research is supported by a grant from the Government of the Russian Federation [11.G34.31.0043].

Figure S1. Cholesky decomposition of the genetic (1a), shared environmental (1b) and non shared environmental (1c) influences on the 3 mathematical measures and the Jigsaws Test with 95% confidence intervals in brackets, below the estimates. The direct paths (vertical arrows) represent specific genetic and environmental influences, the oblique paths indicate genetic and environmental influences shared among the measures. The comparison between the fit statistics of the multivariate saturated model (-2LL = 158139.81, df = 38211, BIC = -179612.25, ep = 88) and the multivariate ACE model (-2LL = 91102.13, df = 38265, BIC = -247127.25, ep = 34) indicates a good fit of the model to the observed data.

Figure S2. Correlated Factor Solution of the model including the Jigsaws Test and the 3 mathematical sub-tests. The curved arrows represent the genetic (2a), shared environmental (2b) and non-shared environmental (2c) correlations among the 4 measures. 95% confidence intervals are in brackets, below the estimates. The vertical paths represent the estimates of the heritability (2a), shared (2b) and non-shared (2c) environmental influences.

Figure S3. Cholesky decomposition of the genetic (3a), shared environmental (3b) and non shared environmental (3c) influences on the 3 mathematical measures and the Hidden Shapes Test with 95% confidence intervals in brackets, below the estimates. The direct paths (vertical arrows) represent specific genetic and environmental influences, the oblique paths indicate genetic and environmental influences shared among the measures. The comparison between the fit statistics of the multivariate saturated model (-2LL = 121703.89, df = 38469, BIC = -218328.67, ep = 88) and the multivariate ACE model (-2LL = 91296.14, df = 38523, BIC = -249213.73, ep = 34) indicates a good fit of the model to the observed data.

Figure S4. Correlated Factor Solution of the model including the Hidden Shapes Test and the 3 mathematical sub-tests. The curved arrows represent the genetic (4a), shared environmental (4b) and non-shared environmental (4c) correlations among the 4 measures. 95% confidence intervals are in brackets, below the estimates. The vertical paths represent the estimates of the heritability (4a), shared (4b) and non-shared (4c) environmental influences. Small discrepancies between the above estimates the estimates reported in Fig 1S and 2S are due to rounding up the decimal places in the two different model fitting.

Figure S5. Cholesky decomposition of the genetic (5a), shared environmental (5b) and non shared environmental (5c) influences on the 3 mathematical measures and the spatial composite. Confidence intervals are in brackets, below the estimates. The direct paths (vertical arrows) represent specific genetic and environmental influences, the oblique paths indicate genetic and environmental influences shared among the measures.

Table S1. Intra-class correlations.

Table S2. Univariate sex-limitation.

Table S3. Correlated factor solution.

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