Mahesh Sharma and Steve Chinn
There are many perspectives we need to consider to be able to answer that question. Mathematics anxiety is not a recent phenomenon. Despite a long history of negative influences on people learning and using mathematics we are far from a level of understanding that enables us to address the problem successfully.
It is an international problem. There are some implications and deductions we can draw from that statement.
We will try to consider some of the issues and some interventions. The essentials are obvious. They are the subject itself, the way it is taught and the culture and beliefs that underpin attitudes to mathematics. The key question is obvious, ‘Why is this such a problem for mathematics in comparison to other topics?’
Anxiety is a mental state characterized by an intense sense of tension, worry or apprehension, relative to something adverse that might happen in the future. Researchers differentiate aspects of anxiety into state and trait, respectively defined as a more transient reaction to an adverse situation, and as a more stable personality attribute in experiencing events. Trait and state anxiety are a psychological response akin to nervousness or worry. It generally brings with it a sense of unease about something with an uncertain outcome or when the outcome that is perceived or predicted to be negative. It causes an increase in heart rate and blood pressure, but also a display of symptoms of increased nervousness. Anxiety stimulates our “fight or flight” response, releasing adrenaline into the blood stream, and usually has a negative effect on performance. Both trait and state anxiety need to be controlled in learning situations. Recent neurological research shows that brain maps distinguish structural and functional features of the aspects of these two types of anxieties.
Trait anxiety refers to anxiety as a characteristic of a person, so that the person is generally anxious about unknown outcomes, it is a part of who the person is. People have various levels of trait anxiety, and an athlete or a stage performer with high levels of trait anxiety will need to develop mechanisms and strategies to control the levels of anxiety that arise with performance.
State anxiety refers to anxiety that arises in a particular situation. Everyone experiences state anxiety, but the stimulus can vary. In sports state anxiety may rise when an athlete is in a high-pressure situation and is called upon to perform. Examples of this include serving for the match in tennis, taking a penalty shot in soccer, converting a try in rugby, or shooting a free throw in basketball.
Mathematics anxiety is an example of state anxiety in most people as the person may not display any anxiety in other aspects of his life, including other academic areas. However, when a person with trait anxiety experiences mathematics anxiety, it may not be just due to mathematics factors, they may display anxiety in other areas also. The issue is not a matter of imagination or of a biased perspective. There is evidence of a neuroanatomical and functional distinction between state and trait anxiety. These neural features may be considered as additional markers evaluating early diagnosis or treatment effects of mathematics anxiety. Studies show that the area of the brain that is activated by the anxiety of facing a mathematics test is the same as that activated when anticipating physical harm.
Many people complain of suffering from mathematics anxiety. Math anxiety is a negative and potentially impairing emotional reaction to mathematics, affecting those with the condition in common day-to-day and academic situations involving anything mathematical. It is an affective, cognitive, and visceral reaction, sometimes appearing to be an irrational response to mathematics and consistent failures in mathematics tasks. This, in turn, can lead to long-term detriment to career opportunities and earning potential, and possible decreased job satisfaction. It has huge costs to both individuals and societies.
Research indicates that math anxiety is the strongest predictor of both applied and basic math performance. Nature of a person’s mathematics anxiety is reflective of a person’s cognitive and affective profile. For example, people with general (state) anxiety respond to mathematics learning differently as mathematics, in particular situations, demands an exact answer, and these individuals are fearful of being wrong and they become anxious. The state and trait anxieties, in mathematics learning are known as: (a) specific math anxiety (state anxiety) developed due to problems and difficulties with specific mathematics developmental and conceptual milestones, and (b) global math anxiety developed due to trait anxiety created by personal and socio-environmental factors. This type of mathematics anxiety could also be termed as social-cultural type mathematics anxiety. There are similarities and differences in the nature and impact on mathematics learning due to the two types of anxieties.
The emotion and cognitive areas of the brain are highly interlinked, so emotional factors like stress, anxiety, happiness and belonging need to be considered when thinking about ways to improve teaching and planning interventions. For example, Kindergarten teachers report that “a lot of children crave more assurance in their relationships with us. They want to sit by us, sit next to us, they want more encouragement from us. It seems that they are better engaged in learning when these needs are met.” These needs and demands vary, but they continue to play important role in learning as emotion uses many of the same areas of the brain that learning uses. Learning is improved when teachers consider each student’s emotional needs.
Anxiety and stress of any type place heavy demands on visual and verbal memories and may also block their optimal functioning. For example, both visual and verbal working memory are significant factors in accounting for the variance in math performance measured broadly. Math anxiety primarily impacts visual working-memory; however, some recent research indicates that anxiety is also processed in verbal working memory. Research also shows that anxiety reduces the capacity and functioning of the working memory.
The phenomenon of math anxiety—understanding it, seeking directions for diagnosing it, and to some extent finding solutions to it can be explained from three perspectives: psychological, mathematical, and pedagogical.
Mathematics learning and the related difficulties do create their own stresses and anxieties in most people. For example, people with trait anxiety show mathematics anxiety earlier and easily in demanding mathematics situations. And anxiety often shows up during the performance of any “difficult” task (especially under test, demanding contexts, and social conditions where poor performance may have consequences—personal and/or social), but there are, of course, exceptions. Difficulty in a task means: (a) the presence of many components—sub-tasks, prerequisite skills, primary concepts vs secondary concepts, etc., (b) the complexity of the sub-tasks, multi-concept and multi-step procedures, i.e., long-division, systems of equations, word-problems, etc., and (c) cognitive complexity of the tasks and sub-tasks, i.e., subtraction problem: 10,043 – 987 = ? (Involvement of re-grouping, zeros in several places, etc.
Anxiety causes worry and worry always impairs performance on tasks with high attentional or short-term memory demands such as difficult concepts and multi-step procedures in learning mathematics. Moreover, worry can also lead one to become passive and withdraw from the activity. According to the processing efficiency theory, worry has two main effects: (1) a reduction in the storage and processing capacity of the working memory system available for a concurrent task; and (2) an increment in on-task effort and activities designed to improve performance. In most other academic disciplines, the effect is on performance efficiency only. However, in the case of mathematics, because of its heavily cumulative nature and involvement of multiple concepts and procedures in new concepts, a third element comes into play; (3) effect on long-term memory functions—searching for relevant information—language (definitions, terms, formulas, theorem, results), concept (conceptual schemas, relationships, primary and pre-concepts), procedures (task analysis, procedural steps, facts, skills, etc.). Because of these factors about the type of difficulty of tasks in mathematics both performance effectiveness and processing efficiency are affected.
Each one of the developmental milestones in mathematics are progressively complex and more abstract than the previous ones and this abstraction is cumulative in nature: Number concept is abstraction of counting of discrete objects; addition and subtractions are abstraction of numbers; multiplication and division are further abstractions of addition and subtraction. And later, exponents are abstractions of multiplication. Place value—a very complex concept in scope and involves the integration of all of these concepts and spatial orientation/space organization with a spatial impact of language, particularly when the number also involves decimals. Learning these concepts is difficult as each place demands on attention, focus, and memory system—short-term, working memory, and long-term memory. At the same time the worry of learning these and demonstration of mastery begins to generate moderate and then clinical levels of anxiety for many children. Lack of understanding of the trajectory of the development of the mathematics conceptual milestones, poor instruction, poor instructional materials and strategies, improper expectations, and social factors undermine the timely development of the innate numerical concepts (i.e., subitizing, spatial orientation/space organization) and they may sometimes manifest as dyscalculia or acquired dyscalculia.
Some researchers on anxiety and performance argue that anxiety causes worry and worry beyond a threshold always impairs performance on tasks with high attentional or short-term memory demands. As mentioned, anxiety characteristically impairs efficiency more than effectiveness. Impaired proficiency on a task consumes more mental resources and as a result fewer tasks are performed in a given time resource, thereby decreasing task satisfaction and increasing fatigue. Fatigue—both physical and cognitive, cause cessation of activity and then procrastination and avoidance of the task. Students have used terms such as: ‘My brain hurts.’ ‘It takes so much out of me.’ ‘It tires me.’ And want to curtail the activity and withdraw from it.
From the perspective of their impact on a student’s mental and mathematics health the effects are similar—an aversion from and fear about mathematics, sometimes, it seems quite irrational. In terms of their diagnosis and remediation and the impact of remediation they present different situations. In many other ways math anxiety functions differently than other types of anxiety (i.e., general anxiety). In observations, we have found that specific mathematics anxiety can be handled with efficient and effective methods of teaching. We have observed that when the interest for the task is due to efficient strategies which create a taste of success, there is a change in their level and attitude toward mathematics . Some recent research is suggesting that developing a growth mind set as arousal factor has positive results.
Although effective instruction has positive affect on both kinds of math anxieties, but people with specific math anxiety respond to targeted instruction in the specific area of difficulty, the same instruction on global mathematics anxiety subjects has lower positive affect. In their case, there is a need for counseling, or a more personal level of tutoring is more effective as their trait anxiety heightens their state anxiety (mathematics anxiety) in demanding and trying situations.
The external manifestations of math anxiety—classic symptoms of general anxiety, avoidance of mathematics, and increased mathematics errors begin to appear very early in children’s schooling and, if not addressed may persist for long time, even into adulthood. The causes of these errors and reasons for a person’s failure in mathematics must be investigated and dealt with, to make success is a possibility.
The discrepancy between high and low math anxious performance appears to vary as a function of size of the problem being solved leads to a suspicion that there is a close relationship between the observed deficits and the memory system, particularly the working memory system as working memory skills are positively associated with academic performance, particularly mathematics. In contrast, high levels of trait anxiety are linked with educational underachievement, in general. Research shows that the association between trait anxiety and academic performance is significantly mediated by verbal working memory for three of the six academic performance measures (math, quantitative and non-verbal reasoning). Spatial working memory does not significantly mediate the relationship between trait anxiety and academic performance.
The Yerkes-Dodson law, in psychology about Arousal (motivation) Performance Function (APF) models the relationship between motivation, stress (anxiety) and task performance. This law defines relationship between the variables: arousal (motivation) and stress as independent variables and performance as the dependent variable. Its graphical representation is an inverted “⋂”. The left side of the curve represents low arousal, or stress. The right side represents high arousal. And at the top center is a medium level of arousal. The vertical line on the left side goes from poor performance (at the bottom) to peak performance (at the top). The optimal state of arousal and optimal performance come together in the middle of the curve. Too much arousal, in itself, causes a stronger stress reaction that can hamper performance. It means, there is an optimal level of arousal/motivation (whether instrinsic or extrinsic) that results in optimal performance. It proposes that you reach your peak level of performance with an intermediate level of stress or motivation. Hans Selye calls this narrow marker on the stress continuum as the range of eustress—a ‘good’ stress. Too little or too much motivation/arousal results in poorer performance. Eustress is known as a positive desirable stress and this level of stress acts as arousal/motivation mechanism. Too little arousal (no responsibilities, no accountability, no stress, no rewards, etc.) doesn’t provide much in the way of motivation. Having no stress at all isn’t necessarily a good thing in terms of mathematics performance for most people. In such situations, only self-motivated people perform. For example, classroom where no performance is required, no assessments, or responsibilities given and expectations are non-existent, very few will learn. Even novelty provides some motivation. When classroom teaching is all about routines and nothing ever changes, boredom sets in. There is no stress, but there is also no motivation, either. You are not being challenged and have no incentive to go above and beyond. Activities feel meaningless, so the child does the bare minimum. On the other hand, expectations for a student to perform, where the task is not accessible and difficult, can also create anxiety. Meaningless expectations from statements such as, ‘Your sister was good at maths, why aren’t you?’ to those star charts on the wall that are meant to motivate all learners are not motivators. We must remember that nothing works for everyone. I like Steve’s use of his made-up phrase, ‘unanxious expectations’.
A moderate level of stress (eustress) goes a long way. It is the moderate or normal psychological stress interpreted as being beneficial for the experiencer. For productivity and learning, this is the right amount of stress. It is manageable, motivational, and performance enhancing. One’s heart beats a bit faster. One feels a sense of clarity and alertness. The brain and body are all fired up. It is that little extra push you need when a hard deadline looms or quiz is coming up. There’s something the person wants. One has engagement with the task and some personal stake in the task and in the result. A moderate surge of stress boosts one’s performance. One needs just enough stress to provide motivation but not so much that one is overwhelmed.
That optimal level of arousal/motivation and stress differs from person to person, according to factors like the specific task, degree of skill, and confidence level. Getting to that optimal arousal zone can be difficult because some factors aren’t within a person’s control. As arousal (e.g., motivation, pep talk) surpasses that point, the performance begins to deteriorate. The learner is too stressed and anxious to do their best. So, what is the optimal amount of arousal? That depends on the task. A simpler task requires a higher amount of arousal, while a more challenging task requires a lower level of arousal.
Intense stress can lead to a fight, flight, or freeze response. It is the test that could keep one from graduating or the poor evaluation that shatters the fragile self-esteem, and the resulting disappointment and humiliation are high level stressors. In these types of situations, stress and anxiety ramp up to an unmanageable level. The heart is beating faster, but it is unsettling, distracting, even nerve-wracking. One might lose focus because of the resulting pressure in such situations and then one may not reach one’s full potential. One is all too aware of the fact that it is working against you. It is too much. That state is difficult to pinpoint in advance and will differ from person to person. A mild to moderate amount of short-term stress can result in an acute stress response that provides the motivation and energy you need, just when you need it. It lasts only long enough to help you perform your best.
Chronic stress is not likely to benefit students at all. In fact, it will negatively impact students’ physical and mental health. Some classroom and school environment are chronic stressors. Daily inefficient and poorly thought-out arithmetic fact drills, unnecessary and unproductive daily homework, unreasonable tasks, lack of and poor instructions to tasks, laborious and inefficient strategies, lack of recognition of student contributions to their learning, etc. are chronic stressors that create debilitating math anxiety. Once the math anxiety is present, it begins to have neuropsychological and neurophysiological impact on the person’s mathematics learning. General stress and stress produced by math anxiety produce the hormone “cortisol.” Cortisol has an impact on learning. It works with adrenalin to create memories of short-term emotional events; this is the proposed mechanism for storage of flash-bulb memories and may originate as a means to remember what to avoid in the future. Long-term exposure to cortisol damages cells in the hippocampus; this damage results in impaired learning and memory. Furthermore, cortisol inhibits memory retrieval of already stored information.
The presence of mathematics anxiety is not necessarily a sign of inferior intelligence. There are many persons who are highly successful in other fields of endeavors but become anxious about mathematics. The people who perceive mathematics as a difficult subject is much higher than the number of people who experience some kind of learning disability.
In many cases the mathematics anxiety is the outcome of early failures in mathematics which could have been due to a variety of factors. Early introduction to difficult abstract concepts, inefficient instruction or poorly selected or designed materials, low level of mastery expected so that the information is not internalized.
It is not true that students who have had difficulties in arithmetic are the only ones who have difficulty in learning algebra. Students facing difficulty in algebra are not just those who have done poorly in elementary school or those who have intellectual or cognitive deficits. Some students who are intellectually quite superior also face difficulty in algebra if it is presented either earlier than they are ready for it or it is presented purely abstractly. For example, in many schools, superior mathematics students from seventh grade are given the opportunity to study their first course of algebra in the middle school as the first course in an accelerated mathematics program that culminates with advanced placement calculus (AP Calculus) toward the end of high school. Unfortunately, this course in algebra proves to be a stumbling block for some students whether the course is accelerated or regular in nature. Unsuccessful students in the accelerated course either drop the algebra course and revert to regular eighth-grade mathematics or complete the course only to have to repeat it in the ninth grade. In either case, intelligent, previously high achieving students experience their first trauma of failure or underachievement in mathematics. These students experience their first bout of anxiety.
Although the difficulties, fear, and anxieties of these students are quite different from the students we mentioned earlier, a small number of them do begin to experience math anxiety.
Gender and Math Anxiety
Male and female reaction to general anxiety and mathematics anxiety are different and so is their impact on them and the ensuing behaviors are quite different. Female students also face an extra factor of social stereotypes of female performance on mathematics. Therefore, the professional interventions should take these factors in their planning and executions.
Dyscalculia and Test Taking Anxiety
Assessments can easily trigger negative thoughts and feelings, especially when students’ disabilities are related to anxiety, particularly mathematics anxiety. There are ways to help students address testing anxiety in a healthy way.
Testing anxiety shows itself in different ways for different students. It can range from refusing to do work, crying, hiding in the bathroom, and verbal aggression to physical behavior like flipping tables and desks or hitting school staff. Some students avoid school on test days, and many suffer from symptoms such as stomachaches or headaches on the days when teacher asks them to arithmetic drills (e.g., 100 arithmetic facts in 3 minutes). In special education programs, many of our students’ disabilities are closely related to anxiety, and testing can be a trigger that heightens those negative thoughts and feelings.
It’s a common belief that testing anxiety affects only older students, such as those taking high school or college placement exams. However, testing anxiety affects students of all ages. In fact, studies have shown that test anxiety is actually the worst in the middle grades. This anxiety can stem from a fear of failure, social condemnation, negative testing experiences, or feeling unprepared. Although each student’s needs are unique, there are some strategies to help students manage testing and anxiety, in general and math anxiety, in particular.
The dominance of evaluation in mathematics is frightening, intimidating, daunting, and demotivating for many learners. It can begin in very young children and last throughout a life. Hearing ‘wrong’ too often is an inflexible and harsh judgement, leaving little scope to alleviate its impact. As a result, students develop a fear of negative evaluation. Of course, we need to know when we are wrong. It can be a constructive judgement but is rarely perceived as such. The consequence is avoidance of the task that can lead to the judgement in much the same strategy of avoiding situations that may lead to physical harm. The avoidance of specific situations may well permeate to avoiding any involvement in mathematics which may include future math learning.
One of the goals of a teacher is to create a classroom culture that includes a variety of low stress formative, constructive and short assessments. There are many adjustments that can be made in a classroom that will reduce anxiety. For example, a set of questions where the first three or four are straightforward and then the difficulty increases slowly. When there is a sudden quantum leap in difficulty the learner gives up, even though there may be subsequent questions that they could tackle.
There are adjustments that can be made to those classroom behaviors of teachers that can limit the escalation of anxiety. Largely these are rooted in understanding learner profiles: How one learns? Why learning problems occur? What are efficient strategies to make difficult concepts accessible and even easier and adjusting presentations accordingly.
Here we meet a factor which is ingrained in the culture of mathematics, the speed of processing and answering questions. The normal distribution applies to many factors and that includes the speed of processing mathematical data. A classic example is the demand to answer mental arithmetic questions quickly. A natural reaction to the demand to do something more quickly than you feel you can do, is to withdraw from any involvement in the task.
Short term memory capacities are also varied. The amount of data people can retain for a short time is not the same in all individuals. Teachers, or any person interacting with another, need to be aware when they are exceeding the short-term memory load of the other person. And they need to know that when short term memory loses the data, it is irretrievable.
Working memory is now recognized as a key component in mathematics ability. Again, the capacity varies among individuals. What is additionally critical is that relatively recent research has shown that this key component in mathematics ability is negatively impacted by anxiety.
Many children, but not all love competition, but it must have a purpose and it should be healthy and possibility of growth. The incidence of failure, mistakes, errors, homework avoidance, fear of assessment most times stem from poor preparation of students, poor methods of teaching, poor task analysis, poor sequencing of tasks, and poor reinforcement. By paying attention to these elements, teachers can reduce failure, errors, and negative attitudes. Encouraging group work and problem solving, asking a lot of low stress questions, and providing answers to “how,” “why” and “when” questions, discussing students’ approach to concepts, tasks, problems, and their answers, teaching students to support classmates, asking students to make observations and then making conjectures about their observations, and letting students do error analyses to observe patterns in errors, and then fixing their mistakes not only reduces stress and anxiety but increases and creates positive attitude toward mathematics. It is also helpful to have students get used to seeing their own data and tracking their own progress, whether on their IEP goals or classroom tests and quizzes. By creating an environment where it is not only alright to make mistakes but to realize that understanding the cause and nature of these mistakes is the beginning of their being mathematicians and then helping students track their own growth. This decreases not only test and assessment anxiety but also math anxiety.
Ensuring that accommodations are customized— for example, level of task, type of questions being asked, the amount of practice and praise are based on actual student needs. Not all students with IEPs need extended time or directions read aloud, for example. Although ‘I’ in IEP stands for ‘individual’, however, good teaching is not just individualized teaching; it is teaching to meet the needs of the individual. Poor teaching performed to satisfy IEP requirements of individualization harms students rather than enhance their learning.
I think the worst thing we can do is to trivialize or dismiss students’ concerns around testing and math anxiety. For meaningful and productive engagement of students it is important to have their input into their learning. Students want to feel understood and that their input has value. With discussion and with appropriate examples, we can help them to see the role and utility of assessments and help put the assessment in perspective. We should empathize with our students, and they should realize that we understand their concerns about learning and assessment. Talking about what they believe would be the absolute worst-case scenario if they fail a particular test and how we will support them in that situation can reassure them. Often just by talking about these concerns aloud students will realize their fears are not as bad as they envisioned.
I always tell my students that while they may not have power over everything about an assessment, there are several things that are within their control. An easy way to explain this is to create a visual concept map or web with your students, with ideas of things they can control that will help them on test day. For example, they can eat a good breakfast, get enough sleep the night before, study in advance, practice self-regulation strategies such as breathing exercises, and organize notes and materials. Then, you can also work on teaching concrete test-taking strategies, such as eliminating multiple choice answers you know are not correct, systematically throughout the school year as mini lessons. These practices reinforce the idea that they are not helpless when faced with a test but can instead help themselves in many ways.
Sometimes teachers can add to student assessment stress by sharing their negative past academic experiences or current anxiety about their student’s progress. It happens quite often in math classes for special students, where many special needs teachers would sometimes project their failures and negative thoughts onto their children. Children do identify with us. We should share how we overcame our limitations and are those strategies applicable to them.
The objective is to illustrate to students that testing is not something to fear, but rather an opportunity to show what you know and discover where you might require some additional review to master the subject matter.
Finally, if you are giving out rewards, such as preferred activities or even items like stickers, make sure they are for finishing the task, not based on scores. One of the worst mishandling of rewards I have seen was when students completed their districtwide progress monitoring assessments and staff decided to give out fun necklaces only to the students who had improved scores. So many factors go into student assessment performance that this is not an equitable practice. Some of my students in special education were mortified to not earn a necklace to wear around school the rest of the day, which led to more anxiety around testing. Instead, we want assessment experiences to conjure positive memories, so that students can have less worries in the future.
Anxiety does not “look” the same for all students, so you must know when to push students and when to empathize, when to listen, and when to set limits. It is important to reiterate that the entire point of assessment is not to measure who you are by a single score on a single day for a particular area in school, but rather it is to be used as an instrument to gauge progress and direct instruction for more optimal learning. Students are a heterogeneous group.
Mathematics For All 