Stereotype and its Effect: Math Anxiety and Math Achievement Part One

We all see the world through the frame of meaning. We engage in activities we believe have meaning, value, and importance to us. Our desire for meaning is our desire for discovery of the world around us and our own strengths and limitations. The emotional act of discovering meaning is at the heart of everything we do everyday. Meaning is the greatest factor/driver of fulfillment. More than happiness, more than even achievements and profits, people want their lives to have value, and people who report higher levels of meaning are less anxious, healthier, and more satisfied of their lives. They are also less worried about others’ judgment of their values. In other words, they have the antidote to the effects of stereotype.

Ordinarily when others do not see value in our work, we may begin to doubt whether that activity is worth pursuing, particularly, in the stage before we have formed an autonomous self and acquired a healthy self-worth. However, when we see value in our work, we are not affected by the fear of value judgment by others and we persevere in the venture. For example, we learn because we see value and meaning in that learning.

We, as teachers and other adults in students’ lives, help provide value and meaning to our students’ learning. This becomes evident in the tasks we assign, the language we use in our teaching and the quality of our interactions with them, the type and number of questions we ask during teaching. It is also evident in the value we assign to their work through our assessments, the feedback we provide on their work (achievements and failures), and the type and nature of encouragement we give. Their learning is their work and through that they derive meaning. When they do not find the work in mathematics classes meaningful, neither on short-term basis (e.g., that particular class or test) or the long-term (e.g., the course or degree), they lose interest in that endeavor.

Therefore, teachers should have deep concern about the implicit and sometimes explicit bias in their teaching of mathematics and their classrooms. This bias is seen in the number of questions asked of different groups during teaching. When these questions are probing yet supportive and scaffolded, then they promote learning. The bias is also evident when there are low expectations. Setting high expectations is the mark of an effective teacher. They set tasks for them to de that are moderately challenging, but accessible. They assign projects that have meaning and purpose. They constantly monitor their students’ progress—their cognitive, affective, and psycho-motoric growth, in their classroom and their courses and program. They form groups that are welcoming, nurturing and collaborative, yet competitive in healthy ways. Their assessments are realistic with constructive, supportive suggestions.

A. The Problem: Math Stereotype and Its Impact
People’s fear and anxiety about math—over and above actual math ability—are impediments to their math achievement. Social conditions such as gender, class, race, and/or ethnic stereotypes about mathematics further compromise their achievements. The most prevalent are gender and gender stereotype that undermine female and minority participation in mathematics related activities.

Of particular concern are the low enrollment of females and minorities in higher mathematics (e.g., calculus, etc.) classes in high school and high attrition rates of undergraduates at colleges and universities from science, technology, engineering and mathematics (STEM) majors. They drop out of STEM fields or fail to complete a degree in a STEM field.

The proportion of college freshmen intending to major in STEM fields has remained around 25 percent over the past 15 years. STEM degrees, as a proportion of total bachelor’s degrees have remained relatively constant at about 15-17 percent. The gap between the percent of freshmen intending to major in STEM fields and the percent of awarded bachelor’s degrees in these fields is a persistent and unwanted trend. Women, for example, earned about 18% of all computer science degrees and make up less than 25% of workers in engineering and computer-related fields. The number of degrees earned by African- and Hispanic American students is even lower. This is in stark contrast to the gains women have made in law, medicine, and other areas of the workforce.

While dearth of women and minorities in STEM fields is often attributed to lack of innate ability or lack of desire on their part, in most cases these are not the factors. Many attribute their decisions in part to the poor quality of instruction or lack of faculty interest in them, but the biggest factor is: gender and race stereotype that female and minority students do not do well on mathematics. Math stereotype and its impact is widespread.

Math Stereotype
The performance of high-achieving female math students on challenging math tests can be impaired by a social-psychological experience of stereotype threat. This “threat” occurs when a female student is taking a difficult math test, and the challenges she experiences with it bring to her mind negative stereotype about female math ability. Once female students identify with the stereotype, they feel they will be judged accordingly and their performance may confirm it.

Numerous experiments have found that the experience of stereotype threat is sufficiently distracting and upsetting to cause women to score lower on difficult math tests than equally skilled men. This threat works in the case of minority students as well. When individuals are confronted with a test or situation in which they are in danger of confirming a stereotype about their group, their performance plummets. For example, if one tells women that women generally score lower on particular math and spatial tests than men, they actually score lower on those tests than they would have had the stereotype not been made salient. However, when subjects are told that woman and men have the same ability on the particular test, the disparities in performance disappear and there are no gender differences in ratings of aptitude, assessments of competence, or interest in fields requiring that ability.

Research and experience of many math and science teachers and students alike show that stereotype has impact on learning. To understand and counteract the impact of this stereotype, it is important to understand this social phenomenon:

  • When and how gender, race, ethnicity, and class stereotype about mathematics are formed?
  • What is the effect of these stereotypes on mathematics learning, achievement and math anxiety?
  • What can math educators do to minimize the effect of stereotype and provide math education that does not allow these stereotypes to happen?

The development of math stereotype is gradual and insidious. The psychological constructs behind this phenomenon are:

  • formation of implicit self-concept—being aware of the presence of and personally experiencing stereotype in math learning situations,
  • forming attitudes toward learning mathematics and its role in life as a result of these, and
  • How and how much does the individual identify with math, as in “math is for me?”

This means to develop an antidote to this stereotype, our concern should be: Are we helping students to form healthy math self-concept?

Answers to these questions lie in a complex combination of social, cultural, and intellectual environmental factors. For example:

  • Many students still consider studying math by some people as a “geeky” activity—not feminine,
  • Many students have poor perception of the math major. They may not know how to study math. They may not know what is involved in learning math and related STEM fields. Why should one learn these subjects? What kinds of job do they lead to?
  • Many students have negative reaction to feedback on math test and assessments—they believe that it is a difficult subject—difficult to get good grades—it is either right or not—so precise.

Such perceptions and beliefs drive many people away from mathematics and come in the way of attracting and keeping females and minorities.

For girls, lack of interest in mathematics may come both from overt and covert culturally communicated messages at home, classrooms and schools, about math being more appropriate for boys than for girls. In many research studies, almost half of participants report believing that men are “better at math” than women—whereas less than 1% report that women are better!

The “math is for boys” stereotype has been used as part of the explanation for why so few women pursue STEM careers. The cultural stereotype may nudge girls, albeit initially quite subtly, to think, “math is not for me,” which can affect what activities (toys, games, hobbies, readings, projects, etc.) they engage in and the career aspirations they may develop.

The ethnic, race and class stereotype are also socio-cultural that African-American and Hispanic-American children are perceived to have lack of interest and pre-requisite skills in mathematics and the Asian-American children are expected to do well on mathematics is also socially transmitted in our schools. Children from higher socio-economic backgrounds, because of their better communication skills, are given the benefit of doubt that they may be better in mathematics as they take lead in asking and answering questions.

Stereotype Construct
One can observe or capture a stereotype behavior by measuring, for example, how strongly a person associates various academic subjects with either masculine or feminine connotations. The stronger the stereotype is, the faster the response to such questions. Researchers have examined three key concepts: Gender identity, or the association of “me” with male or female; math-gender stereotype, or the association of math with male or female; and math self-concept, or the association of “me” with math or reading.

When children are asked to sort four kinds of words: boy names, girl names, math words and reading words, children expressing the math-gender stereotype should be faster to sort words when boy names are paired with math words and girl names are paired with reading words. Similarly, they should be slower to respond when math words are paired with girl names and reading words are paired with boy names. As early as second grade, children (particularly, American children) demonstrate the stereotype for math: boys associate math with their own gender while girls associate math with boys.

On self-concept formation, boys identify themselves with math more than girls do. Even on self-report tests on all three concepts children give similar responses. Cultural stereotype about math is absorbed strikingly early in development, prior to ages at which there are gender differences in math achievement.

The discrepancy, in part, could be due to socio-cultural factors: home, classroom and school environmental influences, geographical variables—urban, rural, and suburban (e.g., the quality of teacher preparation in urban and rural areas is inferior compared to suburban schools), historical, teacher and pedagogical biases, assessment and recognition systems in schools.

The differences in math and science achievements of minorities and females have serious implications for the future careers of these groups and the size of the pool of innovative, research scientists. It has been a source of concern for educators everywhere.

Much of human progress depends on innovation. It depends on people coming up with breakthrough ideas to improve life. We have seen the impact of the invention of wheel, pulley, steam, penicillin or cancer treatments, electricity or the silicon chip. For this reason, societies have a big interest in making sure that as many people as possible have the opportunity to become scientists, inventors, innovators, and entrepreneurs. It is not just a matter of fairness. Denying opportunities to talented people can hurt everyone.

If we do not do something about it, women, African-Americans, Latinos, and low- and middle-income children are far less likely to grow up to become innovators and inventors.[1] Our society appears to be missing out on potential inventors from these groups. We do a very good job at identifying and retaining children who are good at throwing a football or playing a trumpet. But we do not do even a satisfactory job of identifying children who have the potential of creating a phenomenal new product, service, invention, or a discovery. As a result, we all suffer—the whole society loses.

Stereotype and Self-assessment and Self-value
The negative stereotype about women and minorities can hinder their performance, depress their self-assessments of their ability, and bias the evaluations made by key decision makers. Combination of these effects can subtly influence the aspirations and career decisions, keeping them away from degrees and careers in STEM subjects.

If a person is exposed to a negative stereotype about a group to which she belongs (e.g. women, Asians, African-Americans, etc.), she will then perform worse on tasks related to the stereotype.  This is particularly problematic for women in the STEM fields, as there are many societal beliefs about how women do not have strong mathematical ability and about how men make better engineers and scientists.

This has significant implications for real-world situations; for instance, when women are asked to indicate their gender before taking the AP Calculus exam, it is enough to trigger stereotype threat and significantly suppress their scores  whereas it does not have the same effect when this condition is not there. Thus, more women would receive AP Calculus credit in the first year of their college program if the stereotype was not present. This demonstrates the powerful effect of negative stereotype on performance.

Negative stereotype can lower self-assessments of ability and leads individuals to judge their performance by a harsher standard. Beyond diminishing performance and self-assessments of ability, stereotype lowers their goals and ambitions. These effects make women less likely to enter STEM fields because they are less likely to believe they have the skills necessary for a particular career and less likely to develop preferences for that career.

Disconnect with Ability and Achievement
Cognitive, neurological, and educational research indicate that there is no reason why women, for example, cannot succeed in mathematically demanding fields, including advanced research, serious and useful applications, and innovation. Despite these conclusions, women and several groups of minorities still are underrepresented in advanced levels in STEM related fields.

On most international assessments, females outscore males on language: reading, literacy, and verbal skills usage in every country and continue to exhibit higher verbal ability throughout high school. On the other hand, although there are no significant differences between the performance of boys and girls up to fourth grade on mathematics, boys begin to perform better than girls on science and math ability tests beyond fourth grade. The reading gender-gaps narrow during the upper elementary grades, but gender gaps in math achievement and development of negative attitudes towards math grow. The possible explanations for these phenomena are:

  • there is no stereotype in reading abilities,
  • although small stereotype exists in mathematics abilities as early as first and second grade, children may not observe them overtly and may not have yet internalized their role and impact, however,
  • children begin to discern and act on these behaviors by age ten and above.

The results of the stereotype —low achievements and avoidance of math related activities, increase with age.

Similarly, this disconnect with ability and achievement is evident in the case of Hispanic children. For example, Hispanic bilingual children have higher level of executive function skills than their monolingual counterparts as preschoolers and continue to have this advantage even later. Whereas, executive functions (EF) are an important aspect of school readiness that have been shown to predict higher achievement in language, math, and science starting in the early years. But, by third grade many of them are doing poorly in arithmetic. This association has been found among children of different ages, languages, and socioeconomic statuses. These findings can help inform teachers and policy-makers that these children are capable of doing well, the only reasons one can give are factors outside of them. These social factors may involve low expectations, poor instruction, lack of support, and stereotype they experience.

Teachers’ Math Anxiety and Stereotype
Research has found that girls’ math achievement is lower if they have a female teacher who is anxious about math. This may be because the girls in such classrooms pick up on gender stereotype earlier. A large number of early elementary school teachers in the United States are female (>90%) and the percentage of math anxious individuals in that group is larger than average. Many of them may not understand the true nature of mathematics learning and the developmental trajectories of important mathematics concepts and procedures.

Arithmetic has been in use for such a long time that many concepts and procedures are taken for granted by most people and math anxious teachers may find it difficult to explain the reasons and concepts behind them. Therefore, many of them only emphasize the computational (e.g., procedures, recipes, short-cuts, mnemonic devices, and “tricks”, etc.) aspect of arithmetic rather than developing the language, conceptual schemas, and then computational procedures. They may teach only simplistic methods rather than focus on deep structures, patterns, concepts, and the true nature of mathematics. They may not realize that math is the study of patterns in quantity and space. For example, arithmetic is the study of patterns in quantity—number concept, numbersense, and numeracy; geometry is the study of patterns in shapes and their relationships; probability is the study of patterns in chance and possibilities; algebra is the study of patterns and relationships in variability; and calculus is the study of patterns in rate of change, etc.

A math anxious female teacher’s math anxiety has impact on girls’ math achievement through the process of girls’ forming perceptions, attitudes, and beliefs about mathematics (e.g., who is good at math early on through these identifications). Many a times, female teachers fail to observe girls’ novel strategies in math and do not encourage them. This may happen because they may not see these strategies as novel. However, it happens in the case of boys also, but boys generally persist in advocating for their strategies. Teachers with math anxieties may also have lower expectations from girls. They may justify and rationalize a girl’s poor mathematics performance with the belief that “girls are supposed to be poor in mathematics.” In addition, observations suggest that when boys and girls have the same math performance and behaviors in math classes, teachers perceive and express sometimes overtly and other times covertly that the boys are better at math. This “differential rating and mental evaluation” of boys and girls contributes to gender-gaps in math performance.

This is not to suggest that teachers are to blame for gender differences in math performance. Teachers’ views simply reflect those of society as a whole.

When math achievement of the students in math anxious teachers’ classrooms is assessed a very important phenomenon is observed. Although there is no relation between a teacher’s math anxiety and her students’ math achievement and attitude about mathematics at the beginning of the school year, by the school year’s end, however, the more anxious the teacher is about math, the more likely the girls (but not boys) are to endorse the commonly held stereotype that “boys are good at math, and girls are good at reading.” This is particularly true about the girls with lower math achievement. Indeed, by the end of the school year, girls who endorse this stereotype have significantly worse math achievement than girls who do not and than boys overall. Thus, in early elementary school, where the teachers are almost all female, teachers’ math anxiety carries serious consequences for girls’ math achievement by influencing girls’ beliefs about who is good at math.

Research has been mixed about whether today’s children hold gender stereotypes about math at the same level as in the past. Children often report being aware of gender stereotype about mathematics, but they less often indicate that they believe the stereotype. Attitudes towards math may have changed as many parents and educators are beginning to take proactive actions about these matters.

Classroom Environments and Stereotype
Beyond math anxiety, other characteristics about teachers and classroom environments also have been identified as contributors to this gender gap. Students from middle to high school on surveys and interviews identify a math or science teacher as a person who made math, science, or engineering interesting to them. At the same time, many female students report the classroom environments not conducive to their becoming as successful math and science students. For example, many report being passed over in classroom discussions, not being encouraged to have high expectations of themselves and by the teacher, and made to feel inadequate and incompetent.

Classroom environments can be made to feel more female-friendly by:

  • helping students to develop prerequisite skills (sequencing, spatial/orientation space organization, pattern recognition and extensions, visualization, estimation, deductive and inductive reasoning) for mathematics learning in order to neutralize gender and socio-economic differences,
  • incorporating cooperative competitions, public discussion and sharing of their strategies, thinking and practices in solving problems,
  • teacher paying focused attention to achievement for all, and, in particular, on female and minority math achievement,
  • having high expectations for all and to encourage students to have high expectations of themselves,
  • using appropriate, efficient and universal concrete/visual models for mathematics teaching and learning and retiring them when the students have generated the language, developed the conceptual schemas, and have arrived at efficient and effective procedures,
  • exposing students to female and minority role models,
  • using non-threatening, non-discriminatory, and constructive assessment methods, and
  • using nonsexist, non-racist books and materials.

Stereotype threat may emerge even during everyday experiences. The performance of female students decreases with the stereotype threat, for example, when a woman takes a math test in a room with two male test-takers rather than two other women. Other researchers have also found similar reductions in test performance among women who, before taking a difficult math test, are asked to watch TV commercials that depict women in a trivializing ways, that is, in ways that are inconsistent with the stereotype about being good at math. This suggests that the experience of stereotype threat contributes to women’s lower standardized math test scores and to their decreased persistence in quantitative fields in multiple settings. Therefore, to stay in STEM fields and to be successful, they have to exercise a higher level of effort, energy—both psychic and cognitive. Classroom environment can play an important role in this.

B. Other Social Factors for Lower Math Achievement
Stereotype causes some of the math anxiety, particularly, in the case of many women and minorities. However, there are also other factors for the low participation by women and minorities.

Quality of Interaction and Stereotype
Some of this bias is also evident in the applications of mathematics to other disciplines—the kinds of applications we expose our students to engage. Minority students are not shown meaningful scientific applications, fewer girls are in robotic exhibitions and applications. Our choices of applications show class and gender bias in   exposure to extracurricular situations (i.e., for female and minority students the membership in clubs, math competitions, math Olympiads, etc.) are low. This minimizes and marginalizing the voices of female and minority students in mathematics education, higher education and later in the work place.

Children assimilate the stereotype exhibited by parents, educators, peers, toys and games, and the media and in turn, they may exhibit the same later in their dealings with others—classmates, younger students, and later as subordinates. To fight this, we should depict math as being equally accessible to boys and girls of all backgrounds by treating all students in the classrooms the same way—to helping them to reach their potential. We should help broaden the interests and aspirations of all our children.

To counter stereotype that students may bring to the classroom, teachers should always look for opportunities to promote positive gender and minority representations and give their students a broader perspective on their options and capabilities in mathematics and related subject areas. For instance, gender representation should not mean only showing what women can do but also what men can do in an activity that is seen as stereo-typically female activity.

Emphasizing that participation for females in STEM related fields is important may be the key to increasing the number of participants, but what is even more important is to demonstrate what are the skills needed to succeed in these fields and then how the skills gained in these programs will help them to find and get jobs in careers that they may want to pursue and then to succeed.

There is need for all teachers and specially the mathematics teachers, to educate our youth, particularly female and minority students, to understand what is involved in STEM disciplines and what set of skills can make them successful. In the world of information technology (IT), for example, that means that students should be shown directly how STEM educational tracks will help them pursue jobs in IT administration, software development, systems integration, product development, communication, and related fields.

Many middle and high school students are not even aware of what is involved in STEM related jobs. For example, surveys show that almost half of them do not even know anyone who has a job in STEM fields and 1 in 4 has never spoken to anyone about jobs in STEM fields. Almost half of them do not know what kinds of math jobs exist and almost 3 out of 4 do not even know what engineers do in their work. Almost 9 out of 10 of these young people think that people who study STEM subjects work at organizations like NASA, and almost half think that people with such backgrounds work for computer and Internet related companies only.

Only about 1 in 4 believe that people with STEM backgrounds might work for consumer companies like super-markets, non-profit organizations, insurance companies, banks, and entertainment fields, such as gaming.

To change the trend of women abandoning the STEM fields, is to re-brand and redefine the kinds of job tracks and careers that STEM education can lead to for students across the nation. Programs offering expanded educational opportunities for students in science, technology, engineering and mathematics should be seen as the incubators where young people can delve into these subjects more deeply, while showing them and opening wide options for their careers.

Because of these misunderstandings and serious shortages of qualified candidates to fill STEM jobs in a wide swath of industries, employers, educators and human resource professionals should clarify what STEM entails and how widespread the job possibilities in these fields are. Companies seeking STEM workers, schools, teachers and parents can all contribute to making these shifts happen. They should visit schools to create interest and encourage students to put more effort in and develop positive attitudes toward these fields. Industry and STEM researchers need to explain these opportunities to students in depth, including clear information about what kinds of jobs and careers can be pursued with skills and degrees in science, technology, engineering and math.  They should share with them their education and their career trajectories.

Mathematics classes should be more creative so that students can get more engaged and interested in exploring a variety of STEM fields and careers. Teachers should invite business representatives involved with STEM related careers and decision making to their classrooms to discuss STEM careers and they should share the worker needs in STEM fields.

Stereotype and Course and Degree Choices
Gender and race differences are evident in course choices, degree options, and projects selected for internships and research. These differences specifically affect certain groups at the higher levels of math and science. This is evident in enrollments in majors in mathematics and computer sciences, in areas of research and innovation, and in fields of mathematics applications—modeling and problem solving. Females enter STEM areas at the undergraduate level, but many of them leave at the graduate and research levels. For example, men are much more likely than women to pursue graduate degrees, post-graduate research, and careers in STEM fields and economics.

Complex cultural, social, political, and economic factors contribute in the selection of majors of study and career choices for individuals. Reasons and questions behind career choices, wage gaps and the disparities in representations of minorities and women in some STEM require multifaceted answers. For example, professions based on biological and medical sciences have mostly achieved equitable representations of male and female, but mathematics, engineering, and physics have significantly lower female representations. This is also true in history, finance and accounting.

Despite higher percentages of females attempting to enter STEM fields, issues related to stereotypes in schools and higher education and discrimination in the work place continue to exist. These can create hostile environments and gaps in offering challenging and interesting opportunities, meaningful advancements, and adequate compensation increases in comparison to their male counterparts.

The inhospitable climate is partly a result of STEM field’s imbalanced gender and race ratios. For example, women make up only a quarter of employees and about a tenth of executives in the tech industry. There have, of course, been other male-dominated fields notorious for similar environments, including Wall Street and Madison Avenue. But, part of what differentiates tech is the industry’s self-regard, as a realm of visionary futurists and tireless innovators who are making many aspects of the world better. In many ways, the tech world does represent the future—it is defining the future and its contours from social interaction to how do we learn, live and work. It has attracted the most promising and creative mathematicians, engineers, and scientists. That should assure that they would have influence over the nation’s ideas and values for years to come. It’s deeply troubling, then, that many of these companies have created an internal culture that, at least when it comes to gender and race inequality, resembles the past rather than the future.[2]

Tech companies have been promising to improve their hiring of women and underrepresented minorities for nearly two decades. Hiring women, blacks and Latinos is apparently so challenging for these companies that the vice-president of diversity and inclusion for a leading tech company lowered the bar by saying that it is difficult find qualified individuals. Despite challenges, they need to redouble their efforts. They need to collaborate with schools, colleges, and universities.[3]

Investigations on sexism, racism, and ethnic stereotypes in STEM fields demonstrate the bias and stereotyping, and later it turns into the multi-faceted repression experienced by women and minorities. That forces many to quit these fields. For example, men in academia act as gatekeepers. People, by virtue of their positions, have the ability to keep members of certain groups from achieving their full potential. This gatekeeper bias has real consequences. Many female and minority scholars leave the field or they get so disappointed that they do not actively encourage other female or minority graduate students or younger scholars to stay in the field. Studies show that academic colloquium speakers are more likely to be men than women, even when controlling for rank and representation of men and women in the disciplines that sponsor the events. Men give more than twice as many colloquium talks over all (69 percent) as do women (31 percent).

The question is what measures can we take to minimize, root out, and create environments that do not foster such situations in future and then create the learning environments that anyone can be part of innovation, if they desire.

Mathematics as a Gateway to STEM Fields
In the spring of 1986, during a professional development workshop for the faculty of the newly established Illinois Mathematics and Science Academy, Noble laureate, physicist Leon Lederman introduced my workshop on mathematics learning by emphasizing that at the base of all sciences is mathematics. Mathematics is the key to the study of physics, physics explains chemistry and chemistry, in turn, is the basis of understanding biology. Physics and mathematics lead to engineering, whereas, biology and chemistry with the help of mathematics tools lead to understanding microbiology, neurology, and medicine. The base of the pyramid of STEM is mathematics.

Today, even academic fields such as anthropology, psychology, and history rely on mathematical modeling, for example, two to three decades ago, one would see a rare equation, graph or table in an undergraduate textbook in these subjects. Today mathematical modeling—quantitative and qualitative representations using variables, patterns analysis, graphs, tables, charts, equations, and inequalities are common in most academic fields. Of course, the organization and serious study of economics and business is impossible without high level of competence in mathematics, particularly application of mathematical modeling and processes.

STEM fields are the gateway to equity, equality, and major means of participating fully in society. Throughout history, in every century, the latest technologies have generated the maximum number of jobs, most wealth, and higher living standards. Technologies change and reorganize the social structures in a country and now internationally. With appropriate skills in the latest technologies one can be on the forefront of this change. With mathematics competence all sciences are within the reach of a person. Mathematics, thus, is the gateway to STEM and related fields. Mathematics teachers, at all levels, need to make access and equity as their goal in their interactions with students and mathematics.

Many math teachers and math departments in schools and colleges, in a very long tradition, have made mathematics the gatekeeper to many interesting fields. It started when Socrates’ Academy displayed a motto: “No one not well-versed in geometry enter this academy.” We, as math teachers, place hurdles in the path of students’ entry into mathematics particularly for those who do not have the appropriate preparation, attitude, or achievement. Teachers in STEM fields also traditionally view themselves as gatekeepers, choosing the elite students who deserve to be scientists, engineers or doctors — and discarding everybody else. It begins very early in school where rather than challenging all students to interesting problems we assign children to various groups according to ill-defined criteria and abilities.

We need to transform mathematics teaching (e.g., courses, programs, pedagogy, etc.) from a gatekeeper function to a gateway to more mathematics and STEM fields. The aim of math education should be to foster student empowerment—developing critical constructs of mathematics identity, agency, and teaching mathematics as a major instrument for social justice—for access. Our students should learn math as a tool for solving problems—both personal and social. For example, algebra in eighth grade, for many should be a civil rights issue.  Without mathematics competence, we create “third world” economies, living standards, and opportunities in the midst of the “first world.” On the other hand, fortunately, mathematics is creating “first worlds” in the midst of “third worlds” in many places, in several countries.

Current mathematics education often reinforces, rather than moderates, inequalities in education and preparation for life and careers. Effective mathematics education not only should moderate inequalities but should also seek to remove the structural obstacles that stand in the way of achieving equitable outcomes.

[1] Lost Einsteins: The Innovations We’re Missing by David Leonhardt, New York Times, December 3, 2017.

[2] New York Times, December 3, 2017

[3] Wong, Julia Carrie (December 31, 2017). New Year’s resolutions for big tech: how Silicon Valley can be better in 2018, The Guardian.

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Stereotype and its Effect: Math Anxiety and Math Achievement Part One

Enriching Mathematics Experiences for Kindergarten through Second Grade Students: Calendar Activity

The focus of the first three years of formal schooling for children Kindergarten through second grade is to provide experiences that help them develop:

  • Neuro-psycho-physiological maturation
  • Socio-linguistic maturation
  • Quantitative Reasoning
  • Spatial orientation/space organization.

All experiences from psychomotoric/physical (games, toys, activities on water and sand tables, kitchen, playing with pieces of wood, bead-work, etc.) to social (games, toys, story time, sharing, etc.) to emotional (games,toys, community building, making friends, sharing, etc.), and cognitive (reading, writing, language acquisition, number work, building/taking apart, organizing, classifying, making and observing patterns, designing, etc.) develop the above areas.  These experiences should support each other and in order to have maximum impact, they should be integrative. The objective of these experiences is to move children from their egocentric, centered, and perception bound perspective to observe and appreciate others’ perspectives, focus on more than one idea, and take initiative.

Kindergarten through second grade is the most important period in children’s lives. Children make more neural connections, acquire a large number of brand new words, begin to understand and use the structure of language in communication and socialization. They learn—recognize, extend, create and apply patterns, gather and use information, and begin to form and test social relations in and out of school.  The learning habits—personal and social, they form in this period are the bedrock of their future studentship.  One can predict what their achievements will look like later in life based on what happens to them in these years.  For example, the most important skills that can predict achievement, with a high degree of predictability, at high school and beyond are:

  • phonemic awareness, a “good” vocabulary in the native language, and the ability to read and willingness to apply basic reading skills,
  • decomposition/recomposition of numbers up to ten and the related sight facts to show the foundations of quantitative reasoning, and mastery of additive reasoning—the concept of addition, addition and subtraction facts, procedures of addition and subtraction, and, most importantly, the understanding that addition and subtraction are inverse operations, and
  • spatial awareness of objects around him/her (to my right, left, above me, below me, next to me, near me, far away from me, etc.) to understand space organization/spatial orientation (by the end of second grade children should be able to identify objects not only from their perspective but also from the opposite perspective).

Quantitative reasoning and spatial orientation/space organization form the basis of mathematics and mathematical way of thinking.  In the next few posts I focus on how we can transform classroom routines to enrich mathematics activities, mathematical thinking, and mathematics content—language, concepts, and procedures during these years. One of those routines during the Kindergarten through second grade is the calendar activity.  There are definite goals to be realized from this activity.  We want to focus on the mathematics component of this activity.

A.  Calendar Activity—Introduction to the School Day
All over the United States, teachers from Kindergarten through second grade open their day by gathering children around in a circle.  Circle time is a social activity—a content rich process of community building.It has the potential of providing an opportunity for every child to become a contributing member of this learning community. Effective teachers are able to set the tone for the day through this activity.  Here the rules and responsibilities of the membership to classroom learning community are acquired and are the harbinger of being a productive member of the future world they will inhabit. Circle time also serves as a venue not only for social learning but also for exploration and testing of one’s potential.

This circle activity is based on the principle that all learning is socially constructed while we individualize it for personal competence.  During this socialization period several things happen:  New children are welcomed to the class, special events in individual lives (e.g., birthdays) are acknowledged, children share their accomplishments, and they learn about the day—the day of theweek, the date, the temperature, the weather, the number of school days passed and remaining, important historical events, etc.  It is also an opportunity for the development of socio-linguistic, emotional, and quantitative reasoning. It is planned to integrate cognitive, affective, and psychomotoric development. Effective teachers make use of this time for important learning in all of the domain related to children’s development.  In this post, I want to focus on the quantitative reasoning component.

1. The Setting and Activities
It is another day in Mrs. Hills’ first-grade classroom. Nineteen children are sitting around her in a circle. Each one occupies one letter on the rug. The rug has all the letters of the alphabet woven into it.

Mrs. Hills has begun her class just like each day. The routine is predictable, and the children know it well. They knowtheir place on the rug. She takes the same seat.

On the surface, the day appears just like another day—things appear to go almost the same way: she takes attendance, the lunch count, assigns jobs to children and reminds them of the old and regular assignments and selects one of the children as the person of the day.

She looks out the window. As she looks out, children’s eyes follow her eyes. They begin their comments about the weather—the physical aspects, their feelings about it, and wishes. They talk about the leaves turning color. They mention their mothers talking about the weather and winter clothes. One of the students, David, almost as if reacting to a pat on his back goes to the window to observe the weather outside and tries to read the temperature. David is having difficulty reading the number/numeral.  There is a little line before the numeral. Mrs. Hills asks him to tell her what is creating the difficulty in reading the temperature. “There is a line just before the number,” David announces.  “Yes, this time of the year, we will see this line quite often.  Does anyone know about this line?” Mrs Hills asks the class. Several children raise their hand to help him read the temperature. Jonathan is always there to help, but Mrs. Hills sees the raised hands and asks Roland to help David read. Roland helps David read: “− 2 degrees.” Mrs. Hills now asks Jonathan to explain to the class what the line before the number means. Jonathan is pleased to explain the reason. Mrs. Hills talks about the relationship of the weather and the temperature—she talks about different seasons, temperatures, and surroundings outside the classroom. After several questions and comments from the children, she steers the class discussion to their daily opening activity—the calendar activity.

Mrs. Hills is a veteran teacher of thirteen years. She used to teach Kindergarten before she was moved to first grade five years ago. She also used to begin her teaching day in the Kindergarten class by the calendar activity with her children.

Mrs. Hills: Susan go to the calendar and point to today on the calendar.

Susan stands near the calendar on the easel and touches the square of the day and moves her finger above the day and points to Thursday.

Susan: Today is Thursday andit is the 29th of October.

Mrs. Hills: Look at the number line.  Can someone point to the number that tells us today’s date?

Several students raise their hands. Two children try to point to the date.  Finally, Mrs. Hills asks one of the students who is looking for 29 in the nineties. He points to the number 92.  Mrs. Hills asks him to point to 20 and then asks him to count sequentially till he reaches 29.  She asks children to look at the number the child is pointing to.

Mrs. Hills: Read the number.

The child reads the number. Mrs. Hills asks another child to read the number on the calendar.

Mrs. Hills: Michal you go to the calendar and put your finger on today’s date.

Michael points to the location where 29 is written on the calendar. Mrs. Hills asks the whole class to give Michael a hand.

Mrs. Hills: Does any one knowhow to make 29 using Cuisenaire rods?

Only a few years ago, Mrs. Hills used to use Unifix cubes, blocks, and other counting objects (coffee stirrers, straws, buttons, etc.) to make the number representing the date and the number of school days.  It used to take a long time, and as a result the only mathematical skill the children would learn was one-to-one counting. For example, using straws, children will make bundles of ten straws to represent ten or they will fasten ten unifix cubes to makes groups of ten. Then, she started using Base Ten blocks.  That cut down the time as the “longs” in the Base Ten blocks represented 10s and the “flats” represents 100s.  Even with these materials, children counted the units when the one’s place was a number bigger than 5 and some even counts the ten marks on the 10-rod. Now she uses Base Ten blocks for hundreds and tens and Cuisenaire rods for ten’s and the one’s places. Now children, in her class, routinely make numbers using Cuisenaire rods and Base Ten blocks together.  For example for displaying the number 124, they would use a one hundred block, two orange rods, and the purple Cuisenaire rod. They have become quick and fluent in making numbers, place value, and number relationships. Their numbersense is so much better.  She is able to cover the curriculum in allotted time with almost all children demonstrating mastery.  Even her children know the definition of mastery (efficient strategies, fluency, and applicability).

Early in the academic year Mrs. Hills defines what she considers mastery of a math idea: One understands the mathematical idea, can derive the answer using efficient strategy, can do it in more than one way, has fluency (where fluency is needed, e.g. arithmetic facts, key formulas, etc.), and can apply the idea in solving problems. Every time she introduces a new language, concept, or a procedure she reiterates the definition of mastery.

Contrary to her earlier fears that children will take long to learn how to use Cuisenaire rods, she found that it took only a few days for them to learn their Cuisenaire rods—the relationship between numbers and colored rods.  First, she helped them discover the number names of each rod and then memorize them by using them and discovering patterns and number relationships. She kept a graphic of the Cuisenaire rods (stair case of rods from smallest to largest) for a few weeks and then removed it when the children knew the rods well.

Mrs. Hills has realized that the earlier her students know the rods, the sooner they will learn, master, and apply number relationships—facts and place value.  Today also, before the children make today’s numbers, she does a brief exercise:  She says a number and children in turn show the corresponding rod (if the number is less than or equal to 10) or make their number (if the number is larger than 10) using Cuisenaire rods. The children have already mastered the number names of Cuisenaire rods that match the colors (Sharma, 1988). They have been using these rods to make numbers and add and subtract numbers.

Similarly, it took Mrs. Hills some time to accept the definition of mastery.  She always believed that children can have either conceptual understanding or fluency.  She thought if children could arrive at answers by counting objects, on fingers, on number line, or hash marks on paper, they knew the fact. She thought fluency of facts was not necessary and it was counter productive to mathematical thinking. But she now realizes that the language, conceptual understanding, procedural fluency, and applications are complementary and support each other. She is sure of the idea that it is better to achieve mastery in the current concept before going on to the next concept.  In the beginning of the year, it takes longer to master concepts, facts, or procedures, but later because of the mastery of earlier concepts the new concepts become easier to master and applications are much easier. In fact, she and her classes are able to do more meaningful mathematics, efficiently andeffectively in less time. She really understands what effective teaching is all about.  Now, she routinely practices the following concepts almost every day:

  1. Counting forward, backward, from a given number beginning the academic year by 1 and then progressing to 2, 5, and 10. Towards the  end of the year, her children are able to count by 100 from any given number.
  2. Number names of the rods till children are fluent.
  3. One more and one less than a given number as a preparation for introducing strategies for developing arithmetic facts.
  4. She picks up a rod and asks what number will make it ten?
  5. What two numbers make a particular teen’s number?
  6. She practices Sight Facts of a particular number using Visual Cluster cards till they master all of the 45 sight facts.
  7. She uses Cuisenaire rods and Base Ten blocks for making the numbers during calendar time (the date, number of school days, number of the day).

For example, Mrs. Hills asks children to make 29 (today’s date) using Cuisenaire rods. Children make 29 using the Cuisenaire rods.  They display 29 as 2 ten-rods and a nine rods.

She displays the number by using the magnet

ic cuisenaire rods on the board and writes 29 below the rods (2 below the orange rods and 9 under the blue-rod.)

Mrs. Hills: What two numbers make 29?

Screen Shot 2017-09-10 at 3.00.52 PM

Children: 20 and 9.  29 = 20 + 9 or 9 + 20!

Mrs. Hills:  Great!  What two digits make the number 29?

Children: 2 and 9.

Mrs. Hills: Very good!  What is the value of digit 2?

Children: Two tens or 20!

Mrs. Hills: Very Good! That is true! Yes, 2 is in the tens’ place. What is the value of digit 9?

Children:  9 ones or 9!

Mrs. Hills: Great! Can anyone tell me what two other numbers make 29 as you saw in the Cuisenaire arrangement of 29? What two numbers, other than 20 and 9, make 29?

Child 1: 10 and 19, 10 + 19 = 29.

Child 2: Or, 19 + 10 = 29.

Mrs. Hills: That is very good.  Can you show me this by the rods?

Child 1: Yes!  See.
The child shows 10 and 19 (as seen in the figure below).

Decomposition of 292017-09-07 at 2.53.31 PM

Mrs. Hills:  That is very good! Please give her a big hand.

Child 2:  What about 0 and 29?

Mrs. Hills:  That is also right.  Give him a big hand too!

Mrs. Hills, then children to write the combination of two numbers that make 29 as seen earlier on their white-boards.  Children write:

0 + 29 = 29; 29 + 0 = 29;

10 + 19 = 29; 19 + 10 = 29.

Every child holds their white-boards and she checks them from her seat. If there any corrections to be made, she solicits children’s input.  In case of a child having difficulty, she asks the child to make the number using the Cuisenaire rods or points to the board, where these number combinations are displayed using the Cuisenaire rods. After this she continues the calendar activity.

Mrs.  Hills:  Could someone tell me what will be the date tomorrow?

Child 3:  That will be 1 more than today.  Just add one to 29. It will be 30.

Mrs. Hills:  That is right!  Can you show us?

Child 3: Yes, I will take 30 Unifix Cubes.

The child first counts 29 Unifix Cubes and then adds one and declares:  “Here are 30 cubes.  These show tomorrow’s date.”

Mrs. Hills: That is correct. Can someone show us another way?

Children show 30 using several counting materials. Some children make 30 using Cuisenaire rods.  Mrs. Hills observes their progress.

Mrs. Hills: Can someone show how to make tomorrow’s date more efficiently?

Child 4: I can do it more efficiently. Let me show it.

The child shows 29 and 1.  He places 1 above the 9 in the number 29. (as seen in the diagram below). And then replaces 9 + 1 by 10 (an orange rod). He also writes the equation for the operation,

Screen Shot 2017-09-07 at 2.54.32 PM

Mrs. Hills:  That is great!  He deserves a long hand.

Children applaud the child with several claps.

(The rule in Mrs. Hills’ class is “big hand” means two claps and “long hand” is several claps or till Mrs. Hills stops clapping.  Children yearn for Mrs. Hills’ “big hand” and “long hand.” When they get the long hand, that is a big day for the child. Children keep score of the big and long hands earned. Generally, they only get “great” or a “great job.”)

Mrs. Hills: Great! What two digits make 29?

Children: 2 and 9.

Mrs. Hills: Great! What is the value of digit 2?

Children: 20.

Mrs. Hills: Great! What is the value of digit 9?

Children: Nine ones.

She asks children to write number 29 in the expanded form on their white-boards. 29 = 20 + 9.

She writes the expressions:  Number in standard form (29) and the number in expanded form (29 = 20 + 9)

Mrs. Hills writes few more numbers on the board to assess that all of her students have understood the number concept and its decomposition/recomposition and asks questions from each of her students and makes sure that each one of them had a chance to answer few questions.

B.  Number of School Days
Another important and interesting activity related to calendar time is the number of the school day.

Today is the 67th day of school. The number pouch next to Mrs. Hills displays the school day from yesterday in symbols 66 and the six orange rods in the pocket marked tens and one dark green rod in the pocket marked ones. The pocket marked with 100s is empty. Children are eagerly waiting for the hundred pocket to have something in it.

She takes the number 66 from the pouch and displays the number on the white board next to her. She also displays the number with the help of rods as they are magnetic. Then she asks children to read the number displayed on the board.

Mrs. Hills:  What number day was yesterday?

Almost all children have their hands up. Mrs. Hills picks David—a shy little blonde whose hand is half way up.

David:  Sixty-six.

Mrs. Hills: Who is going to tell me how many tens are in sixty-six?

Mrs. Hills picks Marina.

Marina:  I think six.

Mrs. Hills: Touch the six tens.

Marina touches the six orange rods.

Mrs. Hills:  What is the number of the school days today?

Children shout out 1 more than 66.

Mrs. Hills: Cameron, what is 1 more than 66?

Cameron:  67.

Mrs. Hills:  Great!  How will you make 67 from 66?

Cameron puts the 1-rod on top of the 6-rod (dark green) and then replaces the 6-rod by the 7-rod (black Cuisenaire rod). Mrs. Hills all children to make their own 67.

Each child makes 67 using 6 Orange Cuisenaire rods and a black rod. Each child has a small white board to write the numbers on.  They place their rods making the number 67 in front of them in the same way as they will write the number on paper.  Below this arrangement they write the numbers ‘67’—6 below the six orange rods and 7 below the black rod.

Mrs. Hills, then, asks all the same questions she asked in the case of “29” to make sure that children knew how to make the number and decompose it as:

67 = 60 + 7 = 50 + 17 = 40 + 27 = 30 + 37 = 20 + 47 = 10 + 47 = 0 + 67, concretely, orally, and then in writing.

By the end of the activity, each child has been asked questions related to these numbers.  The time devoted on these activities varies, depending on what other pressing demands of the day are.  However, this period is used for “tool building” for her main concept lesson little later in the day.  She generally teaches reading and mathematics in the morning.  The formal mathematics period involves a three-part lesson:  (a) Tool Building, (b) Main Concept, and (c) Supervised individual, small-group, and large-group practice to achieve mastery. She conducts formative assessment during all three segments.  The formative assessment is to collect information about her teaching and children’s learning.  It informs her immediate teaching activity and her work with children. The formative assessment information also helps children to assess themselves as learners.  The information from the first segment informs her how to shape the main concept teaching, and the formative assessment information from the main concept teaching how to design/redesign children’s practice activity (the quantity and quality).

C.  The Hundredth School Day Necklace
Many Kindergarten and first grade teachers celebrate the hundredth school day by children making a necklace of hundred fruit-loops. The completed necklace for their mothers.  Children count colorful fruit-loops and then string them.  It is a very interesting and engaging activity for children. However, there are two problems: (a) first, it is a very time consuming activity, the mathematics payoff is very little. Of course, there are payoffs—social, fine motor coordination exercise, and emotional satisfaction; (b) by the time the child presents this necklace to his/her mother many of the fruit-loops have fallen down and the child still thinks there are 100 fruit loops on the necklace.  There are better alternatives for making these necklaces.

1.  Necklace One
Children can make a 100-necklace by taping (or stapling) ten strips (cut from orange oak-tag paper/heavy stock and each the size of orange colored Cuisenaire rod) and writing 10 on each strip. The following shows the partial necklace (with 4 tens).

Screen Shot 2017-09-07 at 2.55.04 PM

This will help children to learn, very easily, that ten 10-rods (or 10 groups of 10) make 100 (without counting).

2.  Necklace Two
Children can also make a hundred necklace by taping (or stapling) ten strips (cut from oak-tag paper/heavy-stock and each the size of an orange colored Cuisenaire rod) and writing 10 on each strip. Each strip is equal to ten. The following shows the partial necklace.

Screen Shot 2017-09-07 at 2.55.23 PM

Whereas making the first necklace teaches children that 10-tens make 100, the second teaches children all the sight facts of 10 and that 10 can be made in several ways. Making ten is fundamental to learning addition and subtraction facts as most efficient strategies for deriving addition facts are dependent on making ten.  When children know the sight facts of 10, they can easily arrive at all the other arithmetic facts. With the mastery of arithmetic facts and place value prepares them for arithmetic operations.

She knows children have mastered place value when they can answer the following questions, correctly, consistently, and fluently.

  • What digits make this number?
  • What is this place (pointing on a digit in the number)?
  • What is the value of this digit?
  • What digit is in the ___ place?
  • What place is the digit __in?
  • Can you write this number in the expanded form?
  • Can you write this expanded form in the standard form?
  • What numbers make this number?

Making the number representing the date and the number of school days so far help children to learn to answer the questions posed above. Mrs. Hills knows that if children can answer these questions correctly and fluently for three-digit numbers, they can easily extend this knowledge to any digit whole number.

 

Enriching Mathematics Experiences for Kindergarten through Second Grade Students: Calendar Activity

Mathematics Anxiety and Mathematics Achievement (Part V)

How to Overcome Math Anxiety
Mathematics anxiety and poor mathematics performance do not have a single cause but are, in fact, the result of many factors. On one hand, these may include: teacher attitudes about mathematics learning, implementation of pedagogy (e.g., undue emphasis on learning mathematics through drill and rote memorization and without understanding), and opaque assessment methods. On the other hand, the factors may relate to the personal learner differences such as cognition, executive function skills, language and reading levels, mathematics learning personality, as well as attitude toward learning, poor self-image, poor coping skills, and truancy.

Math anxiety is, thus, a complex problem and not amenable to simplistic approaches and short interventions. To begin to address math anxiety, teachers, parents, and students benefit from more information about the nature of mathematics learning, learning problems and math anxiety, guidance in minimizing the impact of mathematics anxiety, and understanding the nature of interventions in learning mathematics that are more efficient. Fortunately, there is help available for students with math anxiety and also for minimizing the incidence of math anxiety. One can learn to prevent and even overcome math anxiety and be successful in learning and using mathematics skills. I believe that all people are capable of meaningful mathematics engagement; however, instruction, infrastructure, and skills depend on the context and the skills of the instructor. Of course, I acknowledge the challenge of meeting the diverse needs of students who have already given up on mathematics and mathematics lessons. Here are some suggestions.

A. Components of a Mathematics Lesson
Since the relationship between math anxiety and mathematics achievement is reciprocal, all interventions must address both issues simultaneously: (a) improving mathematics achievement and (b) alleviating math anxiety. Each mathematics lesson (or tutorial/intervention session) should have three components:

1. Tool Building
The teacher develops the tools for the concept/procedure to be taught in the lesson. This requires a fine task analysis of the new concept/procedure and seeing its developmental trajectory—what prerequisite (mathematical and non-mathematical) is needed for its successful delivery and learning by students, how this concept relates to earlier mathematics language, concepts, and procedures, what models are most effective in getting access to this new concept, and what behaviors of a skilled learner appear in this concept. Tool building is akin to pre-teaching and bringing the related information from long-term memory to the working memory in order to make connections. For example, if one wants to teach multiplication of fractions, one reviews the four models of multiplying whole numbers (e.g., repeated addition; groups of; array; and the area of a rectangle) and shows which ones are not applicable to the multiplication of fractions (repeated addition can be extended to multiplication of fractions only in limited cases: ¼×3=¼+¼+¼, but not in the case of ½×⅓; array can be extended to any fraction multiplication) and why. And then one derives the multiplication of fraction procedure using the area model.[1] The area model applies and can be extended to multiplication of whole numbers, fractions, decimals, integers, and algebraic expressions.

2. Teaching the Main Concept/Procedure
This requires that students see the development of the concept/procedure from intuitive to concrete to pictorial to abstract to applications to communication rather than just jumping into the abstract aspect of it. They should understand it at each level and then integrate the levels. For example, in the case of multiplication of fractions, the teacher introduces the meaning of multiplication of fractions using “groups of” and “area model” and actually derives the multiplication of fractions.[2] Many teachers simply give the formula for multiplying of fractions (e.g., “numerator × numerator/denominator × denominator”), but this creates misconceptions in students’ minds and they never feel confident about the estimates of products of fractions.

3. Practice: Achieving Accuracy and Fluency
Once students have arrived at the concept/procedure at the concrete and pictorial levels, they should practice it at the abstract level and the concrete and pictorial models should be given a “sunset.” Many teachers stay at these levels too long, even after their utility has been reached. This is particularly true in the case of special education students. Overuse of concrete and pictorial models makes them dependent on these models.

First, students should work towards achieving accuracy under supervised conditions, so that the teacher can help them use the strategies correctly, efficiently, and nuances and subtleties in the procedures are brought to students’ attention. After accuracy, they should work on efficiency, fluency, and automatization. When fluency has been achieved or is being achieved, they should apply this new concept/procedure. Choice of exercise problems, language, questions, models, strategies, and “scaffolds” for achieving accuracy, fluency, and applicability are the marks of an “effective teacher.” She also knows when to remove scaffolds from student performance, i.e., when the strategies have been learned. Accuracy and fluency are achieved only when scaffolds are removed. For example, when the procedure for multiplying binomials has been derived and understood by students, the Cuisenaire rods, Base-ten blocks, Algebra-tiles, and the “arrows” showing the partial products should be removed and generalizations should be made by invoking the patterns in the product. Only after understanding and accuracy, should one provide mnemonic devices, graphic organizers, scripts, lists, etc. When mnemonic devices are provided before conceptual understanding, students do not move to higher mathematics, do not develop positive attitudes towards mathematics, and do not appreciate the power and beauty of mathematics.

B. Strategies for Reducing Mathematics Anxiety and Increasing Math Achievement
With proper methods, one can alleviate the negative impact of math anxiety on math achievement. When anxiety is regulated or reframed, students often see a marked increase in their math performance. The beginning of any approach to reduce or prevent math anxiety is a positive learning environment, free from tension and possible causes of embarrassment or humiliation. The following suggestions are for teachers who want to avoid students’ mathematics anxiety and reduce its impact. The goal of this work is understanding students and their learning needs, giving them agency, and letting them do engaging and exciting work. That means:

1. Accommodating for Multi-Sensory Input and Learning Styles
Multi-sensory input invokes and encourages the development of executive function. That in turn, as one makes decisions, sees connections, and uses working memory, develops metacognition (understanding one’s own learning processes, connecting the success with the causes of success). Metacognition, in turn, enhances cognitive and perceptual skills. Improved cognitive and perceptual skills make students better learners. In other words, they process more from the learning situations – the input to the learning system (short-, working, and long-term memory complex).

2. Creating a Variety of Assessment Instruments and Environments
Mathematics anxiety is less linked to mathematics performance when mathematics tests are not timed, indicating that anxiety resulting from time-pressure reduces test performance. During the development of accuracy, achievement process timed-tests (flash-cards, mad minutes, etc.) should be avoided. Only when accuracy has been achieved, then one can use one’s own time to “beat.” After one has reached a level of fluency, then one can use timed assessments. Short—a few minutes long, but frequent assessments with immediate feedback are better than long and infrequent assessments with delayed feedback. Initially, it is better to give only a few problems as assessment.

Many studies provide evidence for the cognitive interference proposed by the Debilitating Anxiety Model by highlighting the negative effects math anxiety can have on mathematics test performance. This does not mean children should not do anything timed. Actually, when children have acquired accuracy and have mastered the concept, they should practice fluency and timed activities. Further, teachers should let students have some input into their own assessments and selection of work to practice and demonstrate (as long as it meets the standards). However, the teacher should refrain from tying self-esteem to success with math tasks. Praise should be not just for the successful outcome; it should also be for the causes of the outcome – the planning, the hard work, perseverance, proper and efficient use of strategies and ideas, the keen interest in monitoring the effort and success, and finally the emergence of new skills and perspectives. This kind of praise develops metacognition, the awareness of one’s own learning processes, and it is the beginning of higher self-esteem. 

3. Designing Positive Experiences in Mathematics Classes
One of my students, a nine-year old girl, always uttered “I hate math” every time she was asked to solve a mathematics problem or asked for a computation. Once I asked her to leave “I hate math” on the bench outside of my office before she came in and to pick it up when she left my office. When she came to the next appointment, she exclaimed: “I did not use the words ‘I hate math’ this whole week in school.” I asked her, “Why?” She said: “I forgot to pick it up from the bench. This week we were working on the table of 4 as we worked here. I knew all the answers. Would you believe I could even find 12 × 4, 4 × 15, even 4×20 and 4×24.” Another student said the same thing: “We mastered the table of 1, 2, 5, 10, and 4 here. I was so happy I was able to reduce the many of the fractions into lowest terms easily. I guess it helps to know the multiplication tables.” Young children are very suggestible. Once they feel successful, they begin to lose their negative feelings about their past failures and limitations. And this applies to older students too!

4. Nature of the Tutorial
The tutorial/intervention sessions for students suffering from math anxiety should have a three-pronged approach: (a) making up the gaps using efficient strategies (e.g., arithmetic fact mastery with decomposition/recomposition), (b) connecting the current work with the grade level work using vertical acceleration (e.g., master a multiplication table and then connect it to fractions and solving equations), and (c) ending with a successful experience (solving a problem on the current topic). At the same time, the tutor/teacher must connect each of the student’s successes with the cause of the success (e.g., you mastered multiplication tables and now you are able to convert fractions into their simplest forms). This helps develop a student’s metacognition, making him/her a better learner, thereby increasing his/her cognitive and learning potential. 

5. Role of Mistakes in Mathematics Learning
Emphasizing that everyone makes mistakes in mathematics and making mistakes is a means of learning and improving mathematics thinking. To do this, the problems, tasks, exercises should be moderately challenging, yet accessible to the child. Through mistakes and in the process of alleviating them one acquires stamina.

6. Making Connections
Make the current mathematics relevant to other mathematics concepts and procedures (intra-mathematical), other disciplines (inter-disciplinary), and problem solving situations (extra-curricular). 

7. Role of Social Setting in Learning Mathematics
The teacher should allow for different social approaches to learning mathematics. The social conditions set the opportunity for learning. While all learning is thus socially constructed, we individualize it for personal competence. When a concept or procedure is introduced to children, they should work in pairs and have opportunities to talk and convince each other of their approaches and outcomes. These discussions are invaluable for reducing mathematics anxiety. Different approaches used by students should be displayed to the class and opportunities given to explain (as a pair—one writing on the board and the other communicating the reasons and the strategies) to the class. Then class discussion should ensue to discuss the exactness and efficiency of different approaches.

8. Stress and its Management
Children should be taught and learn stress management and relaxation techniques. Techniques such as deep breathing and meditation that help them to relax in any stressful situation can also be help deal with the nervousness and tension that affect students with math anxiety. They should be taught how to free up their minds by relieving some of their physical responses to stress, for example, by asking them to get up and move around (in the hall or classroom) for a minute before the test or squeeze a stress ball during the test.

9. Role of Technology in Mathematics Learning
Students should use multi-sensory learning models including technology (when appropriate and when they have understood the language, concept, and the procedure) for solving problems. However, when middle and high school students cannot do mathematics at grade level and have a high degree of math anxiety, giving technology with minimal instruction to do math is not a solution.

Every child should have access to grade level material via technology along with effective instruction with proper language and relationship to concept. At the same time, we should still teach them to do mathematical thinking, not just press buttons/keys. Life requires more than touching keys. While technology is useful, children deserve quality mathematics instruction with and without technology. Having worked with children in all grades, I have made significant gains using multi-sensory teaching intensively, and students are amazed that they can do mathematics. The growth of esteem and confidence, in such situations, is remarkable.

10. Role of Meaningful Mathematics in Remediation
Respect for students and the mathematics they learn is critical. The foci of a mathematics teacher’s respect are: (a) students and (b) mathematics. Respect for students means that the teacher does not judge them from past performance but for their potential. This also means each student should be exposed to meaningful mathematics at a meaningful level with effective and efficient strategies, not simplistic approaches to mathematics—addition is counting objects and counting up on number line, subtraction is counting down, multiplication and division are skip counting (forward and backward, respectively) activities on the number line. If we offer students small bits of mathematics and these too procedurally in order to make it simpler and if we do not develop their mathematical way of thinking, we are not respecting them. They may learn that procedure, but they will become anxious with the next mathematics concept or procedure. Focus on mathematics means: students are exposed to and taught meaningful mathematics (with an emphasis on all its components—linguistic, conceptual, procedural, and problem solving). We should make space for students to practice asking and exploring mathematical questions so that they feel that they belong there. When given the opportunity, students with significant math anxiety and even learning disabilities offer impressive questions and deep insights.

11. Building Confidence
One of the impacts of math anxiety is a loss of confidence, which can be a major impediment for students learning new mathematics. Teachers and counselors should replace negative thoughts (“I can’t do this”, “I’ve never been good at math”, “I won’t finish in time”) with confidence-building affirmations (“I know this”, “I’m prepared”, “I can do this”). This is only possible if students taste success in mathematics and then use that for encouraging them in learning more mathematics. Athletes use the technique of “visualization” to prepare for major competitions. Similarly, students can imagine themselves being relaxed doing math and confidently solving problems during a test.

Because successful experiences produce memory-binding neuro-transmitters, students should be taught to begin with solving “easiest” problems to experience success. Students build their confidence by first doing those problems in an assignment or on a test that they “know” best. This will help them relax when they tackle the “harder” stuff. Moreover, remembering and experiencing success will give them confidence.

12. Role of Challenge in Learning Mathematics
Some level of engagement with challenging tasks is essential. Challenges call upon potential reserves to be actualized thereby entering the zone of proximal development. Challenges also create cognitive dissonance and that creates disequilibrium. Because it is every organism’s nature to resolve disequilibria, the resolutions result in new learning from a high vantage point.

Many students have a negative attitude toward automatizing facts. For example, when we ask students to memorize multiplication times tables, they respond with statements such as: “I cannot memorize facts.” Students should be helped to understand the “why” of math concepts rather than just memorize. With understanding and efficient and effective strategies, it is possible to automatize not only arithmetic but also important concepts and procedures in algebra. I have seen many high achieving students become overwhelmed when they have no algorithm to follow. Mathematics is hard and inherently difficult but brain research tells us that all of us have the capacity to learn math. We all may not become mathematicians, but with efficient strategies almost all can learn basic mathematics to be productive members of society.

When one is under stress, the first thing to be affected is short-term memory and difficulty with retrieval of information from long-term memory. This is one reason it is so important to understand that math is not just a set of rules to memorize but that each concept builds on what came before and that the mathematics language must be appropriate. Language results in conceptual schemas for mathematics ideas and helps retain the information. If one understands the reason behind the rules, one will remember the concepts better and apply them to many different types of problems (not just ones seen before). If one tries to “cram” the material quickly without understanding, one is likely to forget it quickly too. But if one practices the material over a period of time, one will have a better understanding of it and is less likely to forget it when under stress.

High and low mathematics anxiety adults show a significant performance difference in their attitudes and mathematics achievements. However, some high mathematics anxiety individuals are able to use their higher cognitive functions to mitigate the effects of mathematics anxiety on performance. This partially explains and reveals why correlations between mathematics anxiety and performance tend to be relatively low, albeit significant. It appears that individuals who are better able to suppress their negative emotional response to mathematics have less of a performance deficit, and therefore it suggests that the original performance deficit was caused by negative and intrusive thoughts produced by the effect of math anxiety. Some studies explain this phenomenon by concluding that mathematics anxiety does not affect activation in brain areas known to be involved in numerical processing (cognitive areas). Mathematics anxiety is linked with a preoccupation with the emotional value of numerical stimuli. This suggests that performance deficits in high mathematics anxiety individuals are more related to emotional interference than cognitive deficits. This also suggests that math anxiety affects adults more profoundly than children.

C. Executive Function Skill Levels Can Improve
Mathematics anxiety affects executive functioning negatively, and math anxiety affects performance more in cases where executive function skills are poor. However, the most important point is that executive function skill levels are not fixed. Everyone has the ability to improve executive function skills with practice while improving proficiency in math at the same time.

Exercises should be with a focus on mathematics related pre-requisite skills and executive functions—ability to follow sequential directions, spatial orientation/spatial organization, pattern analysis, visualization (working memory), exploring flexible strategies (assessing competing strategies for efficiency and generalizations), etc.

Exercises can strengthen executive function. For example, setting goals that include sub-goals improves prioritization. Fixed daily routines inhibit distractions (physical, emotional and cognitive) and strengthen impulse control. Exercises that emphasize time management, efficient concrete and pictorial models, graphic organizers, mnemonic devices (only after language, concepts, and procedures have been derived) and apps can also help with staying focused. All these improve organizational skills and flexible thinking in moving from one task to the next. Training in those areas can accompany mathematics lessons for better performance overall.

Complex mathematics problems require prioritization because operations must be solved in a specific order. Impulse control is required to stick with these problems long enough to completely solve them. Many children lose points in math not because they got the answer wrong but simply because they gave up too soon. Limited storage space in their working memory prevents them from connecting the logic strings that many math problems require; organization skills are required to know which formula to apply and where to look to find the right ones; flexible thinking is necessary to help the math student forget about the previous problem and cleanly move on to the next. By focusing efforts on building up these executive function skills, math proficiency is sure to improve.

D. Mindfulness, Tests, and Math Anxiety
A newly popular method for shifting a student’s focus from task to task and from fear to attention is “mindfulness.” It is a form of attention training in which students—and sometimes teachers—engage in breathing exercises and visualizations to improve focus and relieve stress, thus indirectly trying to improve the executive function skill of inhibition. The method shows promise in reducing anxiety about tests and math and related behavior problems in children and adolescents. Most anti-stress programs involve at least some aspects of mindfulness such as breathing exercises and students learning to identify their emotions and managing them.

The mindfulness approach trains teachers and students to recognize their physical and emotional symptoms of stress and understand how they could affect their thoughts in the lead-up to a test or a math activity. For example, if students learn to just watch their anxiety and see that it gets stronger and weaker—not to push the emotion away but just to notice it—they can surf the waves of anxiety. Similarly, a teacher may be able to detect the onset of stress or anxiety and may suggest actions to mitigate it. For the student, the suggestion may be as simple as just taking three deep breaths before a math test. With young children, it may be to time their inhalations and exhalations by tracing the fingers of one hand with the other, both to help them count and to give tactile feedback. If a teacher is practicing mindfulness with their class consistently, it is a seamless transition. As a result, mindfulness may be something that the students will naturally do when they shift from one problem to another during a math test or math activity.

Studies have attempted to solve the problem of math anxiety and mathematics deficits by manipulating mathematics anxiety and its impact on performance. For example, it has been observed that freewriting about emotions prior to a mathematics test in order to alleviate math anxiety related intrusive thoughts increases performance. Thus, one way to reframe anxiety is to have students write about their worries regarding math ahead of time of taking a test. This type of “expressive writing” helps students to download worries and minimizes anxiety’s effects on working memory. For younger students, expressive picture drawing, rather than writing, may also help lessen the burden of math anxiety. Teachers can also help students reframe their approach by helping them to see exams as a challenge rather than as a threat.

E. Role of Pedagogy and Math Anxiety
Numerous causes develop students’ mathematics anxiety. More specifically, rote-memorized rules and the manipulation of symbols with little or no meaning are harder to learn than an integrated conceptual structure, and this can result in a stumbling block for the child. The principle cause of mathematics anxiety has been teaching methodologies. Our math classes do not encourage reasoning and understanding. Teachers can create anxiety by placing too much emphasis on memorizing formulae, learning mathematics through drill and practice, applying rote-memorized rules, and setting out work in the ‘traditional’ way.

Mathematics anxiety may therefore be a function of teaching methodologies used to convey basic mathematical skills, which involve the mechanical, ‘explain-practice-memorize’ teaching paradigm and emphasize memorization rather than understanding and reasoning. I strongly believe that a lack of understanding is the cause of anxiety and avoidance and that understanding based learning is more effective than drill and practice. This does not develop confidence in students. A lack of confidence when working in mathematical situations then may become a cause of mathematics anxiety. When students have mastery of numeracy (related concepts, skills, and procedures) they develop confidence and a will to engage in new learning. The role of the mastery of fundamental basic skills (arithmetic facts and place value) is invaluable for competence in numeracy. In the mastery of basic skills, the fundamental steps are: Numberness, sight facts (automatized addition facts with sums up to 10; teen’s number), decomposition/ recomposition, and flexibility of strategies (8 + 6 = 8 + 2 + 4 = 10 + 4 = 14; = 4 + 4 + 6 = 4 + 10 = 14; = 2 + 6 + 6 = 2 + 12 = 14; = 7 + 1 + 6 = 7 + 7 = 14; = 8 + 8 – 2 = 16 − 2 = 14).

Explorations and efficient strategies develop understanding, fluency, and flexibility and a growth mindset. Procedural teaching results in a fixed mindset. When students’ basic skills are not well developed with understanding, fluency, and flexibility, they experience difficulty and failure and then possibly math anxiety. I suspect these children also do not have the right mindset of what math is. They think that math is memorizing or math is following procedures. If we can show that math is thinking and figuring things out instead of trying to recall things, then the cycle of failure and anxiety and fear and failure can be curtailed.

To improve mathematics achievement so that not only students’ mathematics anxiety is reduced but also students do not develop it in the first place, students should have opportunities to explore and reason, see patterns and develop conjectures, reason deductively and inductively and communicate mathematics by engaging in stimulating discussions and activities.

An antidote to math anxiety is engagement in learning experiences – exploration, structured learning, practice, reflection, and communicating learning (journal and discussion). Students must explore and discover mathematical ideas before structured, procedural learning. Very often a student will be given structured learning far too early, but exploration is essential so they must explore before structured learning. Then they must practice under supervision. They must also practice documenting and communicating their thinking both in groups and individually, such as using a journal where they write ideas that are discussed in class and outcomes of their reflections. The journal is not for taking notes; instead, it is for making notes, documenting ideas, and reflecting.

The mathematics lessons that are planned around levels of knowing: Intuitive, concrete, representation, abstract/symbolic, applications, and communications, not only reflect the development of an idea but also provide an entry for learning for every student with an individual learning personality. At the concrete level, students engage in hands-on learning experiences using concrete objects. This is followed by drawing pictorial representations of the mathematical concepts that help them to generalize an idea at the abstract level. Students then solve mathematical problems in abstract ways by using numbers and symbols.

Because learning is a social activity, another important antidote to math anxiety is group work. Learning of any subject should always commence in a group; that is why school was created in the first place: to bring students together. Often in traditional teaching situations and environments, teachers isolate student learning and performance. We are social creatures and we learn best by interacting. Yet, many classrooms do not allow students to interact or work with concrete materials, so math is hard because of the way it is taught. If students learn math from the concrete before pictorial, pictorial before abstract, then all students can handle the abstraction and symbolism.

[1] How to Teaching Fractions Effectively and Easily: A Vertical Acceleration Model (Sharma, 2008).

[2] Same as above (See chapter on multiplication of fractions.)

Mathematics Anxiety and Mathematics Achievement (Part V)

Mathematics Anxiety and Mathematics Achievement (Part IV)

Educators, researchers, and the general public are not sure whether math anxiety is the result of poor performance or poor performance is the result of math anxiety. It is therefore important to explore the causal relationship between the two for planning instruction and remediation.

The anxiety-performance link has two possible causal directions. They have been extended into the specific field of mathematics anxiety. The first direction is explained by the Deficit Theory. Mathematics performance deficits, for example on mathematics tests, generate mild to extreme mathematics anxiety, which may lead to higher anxiety in similar situations. For example, students who have not mastered non-negotiable skills with efficient strategies at their grade level (number concept—Kindergarten, additive reasoning by the end of second-grade, multiplicative reasoning by the end of fourth-grade, etc.)[1] attempt fewer problems on tests, thereby lowering their score. For example, if a student does not have the mastery of (a) multiplication tables, (b) divisibility rules, (c) short-division, and (d) prime factorization before they do operations on fractions, they will have difficulty in mastering them; they will, therefore, be afraid of proportional reasoning (fractions, decimals, percents, ratio, proportion, scale factor, etc.) and then algebraic operations. To turn lower performance into high-level of math anxiety requires time. But, in the case of a vulnerable child (e.g., learning disability, lower cognitive and executive functions, etc.), anxiety may take less time to manifest and may escalate quickly. Thus, children with mathematical learning disabilities are often found to have disproportionately higher levels of mathematics anxiety than typically developing children, supporting the Deficit Theory.

The second causal direction is that anxiety, particularly math anxiety, reduces mathematics performance by affecting any or all of these processes:

  • the pre-processing (initiating or responding to mathematics tasks—attitudinally and cognitively, negative predisposition for mathematics in general and particular mathematics),
  • processing (making sense of the problem—linguistically, conceptually, and/or procedurally, connecting multiple presentations of the problem—data into table, graph, or diagram, etc.),
  • retrieval of information (relevant prior knowledge—formulae, definition, equations, concept, or skills),
  • comprehending (understanding the problem, making connections between the incoming information and prior information and knowledge, translating the words and expressions into mathematical expressions and equations, etc.), and
  • perseverance (engaging and staying with the problem and showing interest in the outcome of the problem),

thereby reducing the level of performance. This is referred to as the Debilitating Anxiety Model.

The Deficit Theory Model of Anxiety
In at least some cases, having especially poor mathematics performance in early childhood could elicit mathematics anxiety. This poor performance could be the result of environmental factors such as poor math teaching, lack of resources and experiences.

Studies of developmental dyscalculia and mathematical learning disabilities indicate that specific cases of mathematics anxiety are related to poor performance, but that poor performance could be attributed to these deficits and then the resultant math anxiety. However, only 6-8% of the population suffers from developmental dyscalculia and such findings cannot be generalized to the typically developing child. It should also be noted that cognitive resources are not the only possible deficits that could cause poor mathematics performance and math anxiety. For example, self-regulation (one of the components of executive function) deficits have been associated both with mathematics anxiety and decreased mathematics performance. The condition of acquired dyscalculia (e.g., children without learning disabilities who show gaps in their mathematics learning will fall in this category) is a clear example of poor mathematics performance.

It has been found that significant correlations exist between a student’s mathematics performance, both at elementary and adolescent age, in one year and their mathematics anxiety in the following year. These correlations are stronger than those found between a student’s mathematics anxiety in one year and their academic performance in the following year, indicating that mathematics performance may cause mathematics anxiety, thus providing support for the Deficit Theory.

The Debilitating Anxiety Model
Mathematics anxiety can impact performance at the stages of pre-processing, processing and retrieval of mathematics knowledge. Recent research suggests that anticipation of mathematics tasks causes activation of the neural ‘pain network’ in high math anxiety individuals, which may help to explain why high math anxiety individuals are inclined to avoid mathematics. In young children, task-avoidant behaviors have been found to reduce mathematics performance. Similarly, many adolescents with mathematics anxiety avoid math-related situations, suggesting that mathematics anxiety influences performance by reducing learning opportunities.

Adults with high mathematics anxiety answer mathematics questions less accurately but more quickly than those with lower levels showing that mathematics anxiety is associated with decreased cognitive reflection during mathematics word problems. Because of poor numerical skills, adults do not have resources to check their answers for correctness. Such data suggest that adults with mathematics anxiety may avoid processing mathematical problems altogether, which could lead both to reduced mathematics learning and to lower mathematics performance due to rushing, lack of engagement, and lack of comprehension. Adults with mathematics anxiety are less likely to enroll in college or university courses involving mathematics.

The worry induced by mathematics anxiety impairs mathematics performance during mathematics processing by taxing processing resources and minimizing their impact. Worry reduces working memory’s processing and storage capacity, thus reducing performance. For instance, research shows a negative correlation between college students’ math anxiety levels and their working memory span. Further, there is an interaction between adults’ mathematics anxiety and their performance on high and low working-memory load mathematics problems, with high working-memory load questions being more affected by mathematics anxiety. Thus, mathematics anxiety appears to affect performance by compromising the working-memory functions of those with high math anxiety.

Mathematics anxiety also affects strategy selection, leading individuals to choose simpler and less effective problem-solving strategies and thus impairing their performance on questions with a high working-memory load. This is supported by evidence suggesting that those with high working-memory, who usually use working-memory intensive strategies, are more impaired under pressure than those who tend to use simpler strategies.

Mathematics anxiety may manifest as (a) lack of willingness to engage in the activity because of previous negative impressions of mathematics, (b) poor reception and information processing, therefore disposing individuals to avoid mathematics related situations, (c) poor comprehension of mathematics information in mathematics learning tasks, thereby abandoning the tasks prematurely and giving up too easily, and (d) later, at the stages of processing and recall, mathematics anxiety may influence performance by cognitive interference. Math anxiety, thus, may negatively tax executive function resources, such as working memory, which are vital for the processing and retrieval of mathematical facts and methods. All of these affected behaviors impact the Standards of Mathematics Practice (SMP)[2] identified and recommended by the framers of Common Core State Standards in Mathematics (CCSS-M)[3].

On the other hand, positive emotions enhance learning by increasing the willingness to initiate tasks, develop persistence, use effective strategies and recruit cognitive resources. The idea that emotions have an effect on general achievement and particularly on math achievement is strongly supported by studies across all ages that manipulate anxiety to reveal either a decrement or improvement in performance. This effect of mathematics anxiety on performance is likely through executive function skills. This is particularly so in the case of working memory. The working memory functioning is impaired by the intrusive negative thoughts, negative talk, and poor self-esteem generated by math anxiety.

The mechanisms of influence of math anxiety, particularly cognitive interference, may be more immediate than from one academic year to the next. Since the effect of anxiety on recall would cause a fairly immediate performance decrement in those with high mathematics anxiety, this supports the debilitating anxiety model—the impact of math anxiety on performance.

To conclude, the evidence for the relationship between math anxiety and mathematics performance is mixed. Neither theory can fully explain the relationship observed between mathematics anxiety and mathematics performance. While some studies provide data, which fit the Deficit Theory, others provide more support for the Debilitating Anxiety Model. The mixture of evidence suggests a bidirectional relationship between mathematics anxiety and mathematics performance, in which poor performance can trigger mathematics anxiety in some individuals and mathematics anxiety can further reduce performance in a vicious cycle.

The belief about a causal relationship should prompt articulating educational policy, program planning in mathematics education, developing initial and then remedial mathematics instruction, assessment, particularly for those who suffer from math anxiety.

For example, if policy-makers share the belief that math anxiety is just another name for ‘bad at math,’ to reduce students’ math anxiety, effort and money will be targeted at courses to improve their mathematics performance. It will involve searching alternative teaching methods to mitigate this situation. In some cases, this may be (a) the development of computer-adaptive programs that may offer a way to ensure that students do not experience excessive failure in their math learning, (b) adjusting the difficulty level of mathematics tasks to an individual student’s ability, or (c) adapting remediation to student’s mathematics level and his/her mathematics learning personality[4].

If the relationship is in fact in the other direction, such efforts are likely to be ineffective and it would be better to focus on alleviating mathematics anxiety in order to improve mathematics performance. Then, it is important to understand the nature of classroom teaching that may produce math anxiety and focus on remediation of math anxiety. This will focus, particularly on methods, which may be undertaken in the mathematics classroom and during interventions. For example, writing about emotions prior to a math test has been seen to increase performance in those with high math anxiety. Because low mathematics self-concept is related to mathematics anxiety, when teaching, teachers should strengthen students’ academic self-concept, which has been identified as a factor related to academic performance.

The mechanisms proposed by the Deficit Theory are long-term, with the detrimental effect of poor performance on anxiety levels occurring over years. This may be why the Deficit Theory is often supported by

longitudinal studies. On the other hand, the Debilitating Anxiety Model, particularly cognitive interference, proposes some immediate mechanisms for anxiety’s interference with performance (e.g., taxing working memory resources). This could explain why the Debilitating Anxiety Model is best supported by experimental studies such as those investigating stereotype threats.

The Reciprocal Theory
The tension between the deficit and debilitating anxiety theories is indicative of the very nature of the mathematics anxiety-mathematics performance relationship. Whilst poor performance may trigger mathematics anxiety in certain individuals, mathematics anxiety lowers or further reduces the mathematics performance in others.

This relationship suggests a model in which mathematics anxiety can develop either from non-performance factors such as social, emotional, biological predisposition or from performance deficits. Mathematics anxiety may then cause further performance deficits, via avoidance and working-memory disruption, suggesting the bidirectional relationship of the Reciprocal Theory. The question of whether the mathematics anxiety-mathematics performance relationship is in fact reciprocal is likely to be best answered by longitudinal studies across childhood and adolescence since only longitudinal data can determine whether mathematics anxiety or weak performance is first to develop.

Some data suggest that previous achievement may affect a student’s mathematics levels of performance and that mathematics anxiety in turn affects future performance, and further proposes indirect feedback loops from performance to appraisals and emotions.

Mathematics anxiety in adults may result from a deficit in basic numerical processing (poor number concept, poor numbersense, and lack of mastery in numeracy skills), which would be more in line with the Deficit Theory. For instance, adults with high mathematics anxiety have numerical processing deficits compared to adults with low mathematics anxiety. Mathematics anxiety may result from a basic low-level deficit in numerical processing that compromises the development of higher-level mathematical skills. Highly mathematics anxious adults’ basic numerical abilities are impaired because they have avoided mathematical tasks throughout their education and in adulthood due to their high levels of mathematics anxiety, supporting the Debilitating Anxiety Model. 

Genetics, Environment and Mathematics Performance
Genetic studies may help to elucidate whether mathematics performance deficits do in fact emerge first and cause math anxiety to develop. One such study suggests that some (9%) of the total variance in mathematics performance stems from genes related to general anxiety, and 12% from genes related to mathematics cognition. This may indicate that for some, mathematics anxiety is caused by a genetic predisposition to deficits in mathematics cognition. However, this does not preclude the possibility that the relationship between mathematics anxiety and performance is reciprocal.

Parental (and other authority figures in a child’s life) math anxiety could be transmitted to children; in other words, parents likely play an important role, either positive or negative. In that case, it is more of social transmission of attitudes towards mathematics rather than genetic.

Sometimes, some of the genetic factors are translated into or affected by stereotypical reactions. Stereotype threats also elevate anxiety levels, thereby affecting participation in and processing of math activities. Stereotype threat is the situation in which members of a group are, or feel themselves to be, at risk of confirming a negative stereotype about their group. Under stereotype threat, individuals are seen to perform more poorly in a task than they do when not under this threat. It is posited that this is due to anxiety elicited by the potential to confirm or disconfirm a negative stereotype about one’s group. This particularly applies to some minority and women’s groups.

The effect of increasing anxiety by stereotype threat can be seen in adults as well as in children. For example, research shows that 6–7 year-old girls showed a performance decrement on a mathematics task after they completed a task that elicited stereotype threat. Similarly, it has been observed that presenting women with a female role model who doubted her own mathematics ability reduced their performance in mathematics problems compared with a control group who were presented with a confident female role model.

Deficits in mathematics performance in women under mathematics stereotype threat appear because math anxiety coupled with the stereotype affect the working memory. This phenomenon supports the idea that mathematics anxiety taxes the working-memory resources and that reduces mathematics performance. The same phenomenon is active when mathematics anxiety affects mathematics performance as the compounding of stereotypes based on race, income level, and gender.

[1] See Non-negotiable skills in mathematics learning in previous posts of this blog.

[2] Visit earlier posts on SMP on this blog.

[3] Visit earlier posts on CCSS-M.

[4] See The Math Notebook on Mathematics Learning Personality by Sharma (1989).

Mathematics Anxiety and Mathematics Achievement (Part IV)

Mathematics Anxiety and Mathematics Achievement (PART III)

Students’ prior negative mathematics experiences, their mindsets about the content of mathematics and learning it, and their view of their mathematics problems determine the type of anxiety they exhibit. Remediation and interventions have a better chance of succeeding if the interventionist determines the nature of students’ math anxiety and possible causative or related factors.

Math Anxiety of the Math-Type (or Specific Math Anxiety)
Specific math anxiety is triggered by certain language, concepts, or procedures: for example, difficulty in memorizing multiplication tables when understanding is not there; long-division procedure; estimation when place value and facts have not been mastered; operations on fractions (why multiplications of two fractions may result in smaller numbers than the fractions being multiplied); understanding place value—particularly decimal places (where there is no one’s place); understanding and operations on negative numbers (how addition of two numbers is smaller than the numbers being added); algebraic symbols—the radical sign (one student declared how can a letter be a number, you cannot count with this); certain mathematical terms (how can a number have a value less than 4, e.g., p = 3.14159265358…, if it is going on for ever and it is not exact; x ≤ 4, how can any thing be equal to and smaller than something at the same time, etc.

These students are not able to come to terms with what their intuitive thinking tells them and what the new concept calls for. The conflict between their intuitive understanding of the mathematics ideas and the new mathematics concept creates a dilemma in their minds—a situation of cognitive dissonance. They may not have a strong conceptual framework and/or this particular concept to resolve the cognitive dissonance. The trigger for the resulting anxiety may be a symbol, a certain procedure, a concept, or a mathematical term. For some reason, that specific mathematics experience creates a mental block in the process of learning the new mathematics concept. Then, they doubt their competence in mathematics and, therefore, distrust mathematics. They find it difficult to go any further, give up or develop an antipathy towards the concept or procedure. Moreover, they declare incompetence in specific aspects of mathematics (self-diagnosis—I am terrible in fractions, equations, etc.). At this juncture of their math experience, fear of mathematics is the result and not the cause of their negative experiences with mathematics.

However, in some cases, since students remember the times they were successful and felt that they were good at mathematics; they do not fear all of mathematics. They have tried to understand that particular part of mathematics but now, as a result of unsuccessful and frustrating experiences, have developed anxiety about a specific aspect of mathematics. A particular concept becomes the locus of their math anxiety.

The reaction of persons with the specific math anxiety is also specific. When they seek help they have specific goals about mathematics and have specific need and their reactions about mathematics are also very specific. For example, they are apt to say:

“My teacher started doing geometry in class and I have always had difficulty in geometry. Can you help me go over this part of the course?”

“I have to take this exam. I always do poorly on exams, can you help me in passing this exam?”

“I used to be good in math up to sixth grade, but now with algebra I am lost.”

“I like geometry but I get lost in algebra, particularly the radical numbers and expressions.”

“I like arithmetic and algebra, but geometry is something else.”

“I had a really bad math teacher in eighth grade, it was all downhill after that.”

“I understand what you are saying, but I don’t see the meaning, I am sorry.”

“Calculus is so abstract. Can you show me this concretely?”

“Why can’t you explain the way my sixth grade teacher used to do?”

“If my sixth grade teacher had explained this material this way, I would have learned this material better.”

“I always got into arguments with the geometry teacher. I could not see the meaning of invisible points and lines.”

Key phrases by such students are: “Sorry!” and “I tried my best.” Their reactions are mild and of disappointment rather than of fear and inadequacy.

Many of these students are willing to try. They believe that if proper methods, materials, and examples are given, they can learn mathematics. These students complain about the teacher, the textbook, the class size, the composition of the class, anything outside of them. It could be anything related to their mathematics experience. As soon as that particular thing is changed, they feel they will be able to learn mathematics.

Whereas people with global math anxiety generally avoid taking mathematics courses, students with specific math anxiety will register for math courses, but if one of these conditions are not met, they may use that as an excuse for dropping the course. In that sense, they are easier to teach. They are looking for somebody to break the cycle of failure in that specific aspect of mathematics. They are eager to talk to math teachers willing to listen. They are not particularly afraid of math or math teachers, but they do not want to repeat the same experience of failure. They need help, and an effective math teacher can usually help them.

In the previous post we mentioned that social myths have created conditions for the prevalence of people with global math anxiety. It would seem that there are more people with global math anxiety, and that used to be the case only a few decades ago. Today however, specific math anxiety is much more prevalent than global math anxiety. There are several reasons for this phenomenon:

  • A student may understand the concept on the surface level but may not truly understand the concept or procedure;
  • A student may not practice the concept or procedure enough to the level of mastery so easily forgets the material. In the long run, the lack of mastery of nitty-gritty aspects of math is the source of the problem. Practicing problems of different types relating to the same concept helps students see the subtleties in the concept, and applications of the concept becomes easier. That builds stamina for mathematics learning.
  • When important developmental concepts are not taught properly, students may not connect concepts properly, which means every new concept looks novel and unrelated, thereby creating mental blocks in the process of learning.
  • When transitions of concepts are not handled properly, students may have difficulty learning concepts. For example, the transition from addition and subtraction (one dimensional—linear) to multiplication and division, is not just the extension of repeated addition/subtraction to skip counting for multiplication/division but is abstracting repeated addition to groups and developing it to a two-dimensional model of multiplication (as an array and area of a rectangle). It is a cognitive jump that requires effective and efficient concrete and pictorial models, language, and conceptual framework.
  • Specific math anxiety can occur if mathematics is taught procedurally, without the proper base of language and conceptual development. Language serves as the container for concepts and concepts are the structure of a procedure. Without the integration of the three, students have to make extra effort to understand and master a concept. This takes a toll on their enthusiasm and motivation for mathematics learning.

These habits and inclinations do not help students learn mathematics easily and sufficiently well. They do poorly on examinations and tests and feel anxious about math because they lack the practice in integrating the language, concept and procedures. Timely help from a sympathetic mathematics teacher who uses efficient and effective methods of teaching that motivate these students to practice is key for improving math achievement and lowering math anxiety and thus breaking the cycle.

The first step, in addressing specific mathematics anxiety is to identify the specific area of mathematics deficiency or where the students faced the first hurdle in mathematics.

The second step is placement in an appropriate math class, instructional group, or matching with the right tutor with an individual educational plan. Then the teacher should develop a plan to attack first the student’s perceived and real incompetence/difficulty in mathematics. The perceived incompetence is often the result of negative experiences. Then the remedial help that they receive should begin with the focus on one’s deficient areas of mathematics and create success using vertical acceleration.

Vertical acceleration is taking a student from a lower level concept (where the student is functioning) to a grade level concept (where the student should be) by developing a vertical relationship (a direct path) from the lower concept to the higher concept. An example is when a student is having problems in fractions or solving algebraic equations because she does not have the mastery of multiplication tables. The teacher should focus on one multiplication table, say the table of 4, and, she should help the student to derive the entries on the multiplication table and learn the commutative, associative and distributive properties of multiplication using effective and generalizable model (area model) and efficient materials (e.g., Cuisenaire rods). Then she should help the student to practice the table of 4 using Multiplication Ladder[1] for 4 and then master the extended facts (×40, ×400, .4, etc.). Then the teacher should practice (a) multiplication of a multi-digit number by 4 (e.g., 12345×4, etc.) and division of a multi-digit by 4 (78695 ÷ 4, etc.), (b) form equivalent fractions and simplify them where the numerators and denominators are multiples of 4, and (c) solving one-step equations (e.g., 4x = 36; 40x = 4800, etc.). When this skill/concept is mastered, it should be connected to the current mathematics. In the next session, the focus should be another table. Supplying students with multiplication tables and using calculators is not a solution.

When a student feels successful in one small area, then related metacognition helps manage learning and then math anxiety. Soon, it begins to disappear. When one provides successful experiences in mathematics at some level (even at a lower level than the chronologically expected mathematics complexity) to this type of student, he/she may lose the anxiety and feel better about mathematics and him/herself.

For this reason, I begin work with these students (say a ninth grader with gaps and anxiety) with simple algebraic concepts, integrating the corresponding arithmetic concepts or taking a simple arithmetic concept and relating it to algebraic concepts with the help of concrete materials and patterns. This process develops in students the feeling that they are capable of learning mathematics and begins to remove their fear. It is not uncommon to hear: “Is that all there is to algebra?” We then build on this newly acquired confidence by taking digressions to make-up for the arithmetic deficiencies by providing successful mathematics experiences using vertical acceleration techniques that result in further building of confidence and reduction in mathematics anxiety. Vertical acceleration is applicable in both global and specific math anxiety situations and in the case of all developmental mathematics concepts.

Math Anxiety and Working Memory
Working memory[2] is a kind of ‘mental scratchpad’ that allows us to ‘work’ with whatever information is temporarily flowing through our consciousness. It is of special importance when we have to do math problems where we have to juggle numbers, apply strategies, execute operations in computations, or conceptualizing mathematical ideas. For example, during computations (e.g., long-division, solving simultaneous linear equations, etc.), we have to keep some of the outcomes of these operations in our mind. These processes take place in different components of the working memory. Increased math anxiety with it demands on working memory reduces working memory’s functions that in turn affects performance. The cycle of poor performance and math anxiety ensues. However, the effect of math anxiety on working memory is limited to math intensive tasks. Thus, the role of working memory and its related component parts is a significant factor in accounting for the variance in math performance.

Just like general anxiety, math anxiety affects both aspects of working memory—visual and verbal, but there is no relationship between math anxiety and processing speed, memory span, or selective attention. However, in the case of mathematics, the effect on visual component of the working memory is more pronounced. Worries and self-talk associated with math anxiety disrupts and consumes a person’s working memory resources, which students could otherwise use for task execution.

Although there are similarities in the effect of general and math anxieties, math anxiety functions differently than general anxiety and other types of specific behavioral anxieties. Whereas general anxiety affects all aspects of human functions to differing degrees, there is no or only a limited relationship between math anxiety and performance on a non-math task.

There is an inverse relationship between math anxiety and performance on the math portion of working memory intensive math tasks. One reason for this is that math anxiety is directly related to the belief that mathematics seeks perfection (e.g., there is only one answer to a problem and there is one way of arriving at it) and there is a fear associated with the perceived negative evaluation when one gets a wrong answer.

It is true that people who are anxious in general often get test anxiety, but a lot of people who are not particularly anxious can still develop stress around tests in subjects like mathematics. What is actually going on when a student stresses out over a test? The moment an anxious student begins a test, the mind becomes flooded with concerns about the possibility of failure. Between the worry and the need to solve the problems on the test, a competition ensues for attention and working memory resources. That divided attention leads to a stalemate—called “choking.” The impact of this is the shutting down of the brain to that task.

This choking can be particularly visible in younger students. High school students may respond more like adults; they may find and use excuses for this shutting down—lack of preparation, poor teaching, irritability, lack of sleep, too early in the morning, too late in the day, etc. Young children just shut down—may start crying, won’t write much, withdraw from the activity, get angry, etc. They just get overwhelmed and don’t know how to deal with it.

Interestingly, due to anxiety, the fear response appears in both low- and high-performing students. However, the impact on students is different. It doesn’t matter how much the student actually knows, but rather how well he or she feels they have the resources to meet the demands of the test and how tightly the performance on the test is tied to the child’s sense of identity. Students who see themselves as “math people” but perform poorly on a math test actually repress their memories of the content of the class, similar to the “motivated forgetting” seen around traumatic events like death. The effort to block out a source of anxiety can actually make it harder to remember events and content around the event. So the student may feel, “I’m supposed to be a math person, but I’m really stressed out, so maybe I’m not as big a math person as I thought I was.” That stress becomes a major threat to the student. So, most surprisingly, math anxiety harms more the higher-achieving students who typically have the most working memory resources.

Changing a student’s mindset about the anxiety itself could boost test performance. For instance, students can be trained to reinterpret physical symptoms—a racing pulse or sweaty palms, say—as signs of excitement, not fear. Those students have better test performance and lower stress than students who interpret their symptoms as fear. Experiencing a sense of threat and a sense of challenge actually are not that different from each other. Ultimately, by changing one’s interpretation, one is not going from high anxiety to low anxiety but from high anxiety to optimal anxiety.

On mathematics tests and examinations, however, it is difficult to separate the effect of test taking anxiety from the mathematics anxiety; thus there is a compound impact. Specifically, for example, there is an effect of math anxiety on the SAT’s total score and individual SAT English, Math, and Science scores. In this case, the impact of test taking anxiety is a factor. A moderate amount of anxiety (irrespective of focus) has a positive impact on performance. For example, low math anxious individuals have higher SAT total and Math scores than both moderately and highly math anxious individuals. High math anxious individuals have low mathematics scores.

Although math anxiety begins to manifest more during the upper elementary school grades, studies show that younger children are beginning to demonstrate math anxiety. Some students report worry and fear about doing math as early as first grade. Research shows that some high-achieving students experience math anxiety at a very young age — a problem, if not treated, that can follow them throughout their lives, and they become underachieving gifted and talented students.

Studies have also found that among the highest-achieving students, about half have medium to high math anxiety. Still, math anxiety is more common among low-achieving students, but it does not impact their performance to the same levels, particularly on less demanding, simpler numeracy problems. Their performance is more affected by math anxiety on higher mathematics—multiplicative reasoning, proportional reasoning (fractions, decimals, percents, etc.), algebraic thinking (integers, algebra, etc.), and geometry.

A high degree of math anxiety undermines performance of otherwise successful students, placing them almost half a school year behind their less anxious peers, in terms of math achievement. High achieving students want to utilize efficient and multiple strategies that place higher demands on working memory and if these strategies are not properly taught, high achieving students begin to do poorly.

Less talented younger students with lower working memory are not impacted by math anxiety in the same way as it affects the students with high working memory. This is because less talented students develop (or taught, particularly, in remedial special education situations) simpler and inefficient ways of dealing with mathematics problems, such as counting on their fingers, on number line, or concrete materials. For example, they are taught that addition is counting up and subtraction is counting down, multiplication is skip-counting forward and division is skip-counting backward.

Counting is a less demanding mathematics activity on working memory when the counting objects are present—counting blocks, fingers, number-line, etc. However, when these students do not use these materials and want to do it without them in their head, then the same task is a heavily demanding working memory activity. However, counting mentally occupies the working memory completely and does not leave any space for higher order thinking or strategy learning. For example, to find the sum of 8 + 7 requires a student, whose only strategy, to hold two sets of numbers: 9, 10, 11, 12, 13, 14, and 15 and the matching numbers 1, 2, 3, 4, 5, 6, and 7. These 14 numbers fill the working memory space completely. Thus, these students have difficulty learning efficient strategies as they place more demands on working memory. In the absence of efficient strategies, they hardly achieve fluency without paying a heavy price on rote memorization.

Ironically, because these lower-performing students do not use working memory resources to solve math problems, their performance does not suffer when they are worried. However, their performance on demanding, complex and longer performance goes down as they demand the involvement of working memory and math anxiety undermines it. Because if these limitations, these students do not progress very high on the mathematics skill/concept continuum.

Academic abilities, size of working memory, and fear of mathematics interact with each other. Sometimes, due to mathematics anxiety even the higher cognitive ability and working memory are undermined. Such interaction affects the high achieving students more than low achieving students. Higher achieving students apt to apply higher order strategies in mathematics and these strategies demand more from working memory and math anxiety may undermine it.

Teachers who give choices in their classrooms lower the anxiety of students. Mathematics classrooms where students have the flexibility to choose some “must do” each day, as well as some “may dos” offer opportunities for them to succeed and make mistakes. Tasting success at the same time as learning to make mistakes is a sure way to improve learning skills. They should also have the opportunity to work with a group or alone. A more open-ended approach allows students to play to their strengths – choosing the problems that they are most comfortable with. This encourages them to stretch themselves a little, try out new things, and worry less.

[1] Improving Fluency Using Multiplication Ladders (Sharma, 2008).

[2] See previous posts on Working Memory and Mathematics Learning.

Mathematics Anxiety and Mathematics Achievement (PART III)

Mathematics Anxiety and Mathematics Achievement (Part II)

Stories and legends told by human beings through the ages to explain our abilities and the acquisition of abilities shape our relationships with learning. Whereas entering into the learning process of language, art, or music seems natural (almost unconscious and involuntary), formal teaching and learning is human and has to be organized. The message and meaning of myths about learning evolve. Over the last few centuries, as quantification has entered ordinary life, myths about its learning have become part of life too.

The myths that we inherit about a subject shape our thoughts and our journey in acquiring the knowledge and competence in it. Many times, individuals need courage to get out of those mythical ideas that we have formed to be truly open to learning.

In mathematics learning, as children we come to the subject matter without preconceived ideas, but very quickly we are shaped by the ideas and myths about mathematics learning our caregivers consciously and/or unconsciously share with us. Then we struggle between meaning making from our real experiences about learning math or forming ideas about it and reading these experiences in the shadow of imparted myths. Early on, many students, when they do not have positive experiences or do not have skills to make meaning from their own experiences, succumb to the prevailing myths about math learning.

In spite of many efforts by mathematics educators, psychologists and social reformers, the myths about mathematics learning and achievement persist. These myths color students’ mindsets about mathematics and its learning. Even administrators who are well-meaning and able but not well-versed in mathematics perpetuate these myths by emphasizing gimmicks and easy solutions to the problem of mathematics achievement. Moreover, even the experts in learning assumed that ability to learn (particularly mathematics) was a matter of intelligence and dedicated smarts and therefore did not study the issue. They assumed, it seems, that either people had the skill of learning or they did not. For them, intelligence –and thus the ability to gain mastery—was an immutable trait.

Yes, for learning mathematics, one needs some cognitive abilities, but one also needs to engage in the process of learning. The field of learning is rife with vague terms: studying, practice, know, mastery, etc. For example, does studying mean reading the mathematics textbook? Does it mean doing sample problems? Does it mean memorizing? Does practice mean repeating the same skill over and over again (like memorizing flash-cards, doing mad minutes)? Does practice require detailed feedback? What kind of feedback? Should practice be solving hard problems or easy problems? Should practice be intense or small chunks? There are so many imprecise terms, which feed into the myths a person selects. Effective teachers help students to achieve freedom from these myths.

Myth 1
Mathematics ability is inherent. You have to be born with a mathematical brain.
People, children as well as adults, who are successful in mathematics are not usually born that way. Many individuals with mathematics anxiety tend to believe that you either have the ability or you don’t, rather than assuming that your skills and abilities are the result of study and practice. When students’ mentors—teachers, parents, sports coaches, successful and intelligent people brag about not being good at math, not being numbers people, they reinforce disinterest in mathematics.

Learning math, like learning in general, takes knowledgeable teachers with high expectations, willing students, and, most importantly, a great deal of time and practice that result in success during each session. A growing number of studies shows that learning is a process, a method, a system of understanding. It is an activity that requires focus, planning, and reflection, and when people know how to learn, they acquire mastery in much more effective ways.

Learning math is much like learning a language—both need a great deal of exposure, “gestation” time, and with usage learners get better. Learning mathematics takes time, effective practice, and help in making connections. The symbols and notations make up the rules of grammar and the terminology is the vocabulary. Doing math homework is like practicing the conversation of math. Becoming fluent (and staying fluent) in math requires years of practice and continuous use. That is true about any field. To be good at anything we need to practice. Learning mathematics is a dedicated, engaged process; it is not a spectator sport. It is not about memorizing facts (static data); it is more about what we do with that data (look for patterns) and how we think better (convert patterns into strategies) by the help of that data (learning).

Myth 2
Mathematics is a very difficult subject to learn.
Many believe that only the very few can do mathematics, that it is difficult to learn mathematics. It is not that mathematics learning is difficult. Many times, it is the method of teaching, efficient models, and effective language usage by the teacher and the students that are the key to learning mathematics. Learning and teaching methods affect the outcomes in every field of learning. For example, to make sense of a concept and to make connections with other concepts, the teacher should use efficient language and effective models to conceptualize and should ask a great deal of enabling questions to help internalize the learning.

Effective strategies and practice boost performance from baseball and tennis to balancing equations and proving theorems. On the other hand, when mathematics teaching is approached with an emphasis on procedures and memorization and when concepts and topics are taught in a fragmented manner, students see mathematics as a difficult subject.

Apart from effective teaching, the mathematics curriculum has to be well orchestrated at each grade level. To engage all students and take many more to higher levels, it is important that school systems emphasize (a) articulation and mastery of non-negotiable skills at each grade level, (b) common definition of knowing a concept or procedure by everyone concerned with mathematics education, (c) knowing the trajectory of developmental milestones in mathematics learning, and (d) the most effective and efficient pedagogy that respects the diverse needs of all children.

Many teachers feed into the myth about the difficulty of mathematics when they begin a topic with statements such as: “Fractions are difficult.” “Algebra is not for everyone.” “Irrational numbers are truly irrational, they generally do not make sense.” These kinds of statements make mathematics look difficult and then it becomes truly difficult.

Mathematics is the integration of language, concepts, and procedures. It is the study of patterns. As Godfrey Harold Hardy (mathematician) said: A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. Statements such as Hardy’s can serve as counter narratives to the myth of the difficulty of mathematics.

Another way to counter the myth is when teachers show students how it took many mathematicians a significant amount of time to develop the formulas and equations that they are attributed with creating and that they are studying now in middle and high schools. Teachers need to show that, as in any other field of endeavor, most mathematicians, even though they made mistakes, persisted in the process.

Students should be encouraged to learn that mistakes are part of learning and creating mathematics. Mistakes and efforts literally improve the cognitive abilities that are needed for any learning and, therefore, mathematics. Students struggle to find an immediate solution when solving problems and they give up so easily, but many mathematicians took many years to solve a single problem. This shows students that math is not about speed but rather devotion and perseverance. A teacher’s selection of a problem and the methods of attacking the problem should be well-thought out so that they begin to take interest.

Myth 3
You cannot be creative and be good at math.
When people asked what I did for living, I would hesitate to admit that I teach or do mathematics. As soon as I announced this to a person at a party, for example, most times the response was: “Math was never my cup of tea. I am right hemispheric person.” “I am a creative person. I like creativity.” “I am humanities and people person.” “Math is not interesting. I wanted to be in an interesting field.”

Can you be an artist, writer, or musician and be good at math too? Yes! Math is found throughout literature, art, music, film, philosophy, and it is essential to many “creative” fields. Although the structure of mathematics is created by man, every aspect of life can be modeled by mathematics principles. It is pervasive in nature, society, and all edifices. Mathematics is constantly making new tools that help all aspects of human endeavor. Its collection of tools shaped the imaginations of Leonardo DaVinci, Mozart, M.C. Escher, and Lewis Carroll. These are just a few of the artists who used math extensively in their works.

Geometry is the right foundation of all painting. – Albrecht Dürer (artist)

I am interested in mathematics only as a creative art. …Mathematics is the study of patterns. – GH Hardy (mathematician)

The mathematician’s best work is art, a high perfect art, as daring as the most secret dreams of imagination, clear and limpid. Mathematical genius and artistic genius touch one another. – G Mittag-Leffler (mathematician)

In addition to increasing student interest in learning math, a deeper understanding of how concepts were developed and which mathematicians were responsible for them adds content knowledge. Explaining and showing how the Babylonians and Greeks worked to get more precise values of pi only gives teachers additional credibility as masters of content.

Incorporating the history of math also allows for an interdisciplinary approach to teaching. As we move from a STEM philosophy to a STEAM philosophy (including the arts), the history of math shows us a relationship between music and mathematics, in particular with Pythagoras, as well as between art and mathematics, such as in geometry. There are many ways to incorporate the history of math into your classroom.

Myth 4
Mathematical insight comes instantly if it comes at all.
Most people have an incorrect conception of creativity and insight because they think of them as spontaneous and automatic. But ideas, insights, and creativity in all areas most of the time come after a person devoted time thinking about them. The second wind in running comes only after one has run for sometime. When one has basic skills, thought about an idea, and then continues to think about it, one begins to make connections, see relationships, and get insights. For this reason, we need to move from thinking about why acquire new skills and knowledge to practicing that is more dynamic, productive and efficient.

The learning cycle in mathematics, as in other subjects, is predictable: it begins with information gathering, and we need some basic skills for this. Then, with discussion and exchange of ideas, we convert this information into knowledge. With application and usage in multiple settings (intra-mathematical, interdisciplinary, and extra-curricular applications) we convert our knowledge into insights. And over a period of time, with deeper engagement with the subject, we might even become wise (we may have expert thinking skills or ability to solve unstructured problems) in that area. For tough unstructured problems one begins to look for analogous thinking in other fields. Imagination and creativity become prominent as soon as we cross the threshold of information gathering to conversion into knowledge. 

Myth 5
Mathematics is a male domain—women are not as good at math as men.
During the last twenty-five years, a great deal of effort and progress have been made in recruiting a large number of girls and women into mathematics courses and the field of mathematics. However, the myth of mathematics as a male domain still keeps many women out of the field. Even today, young girls may not be encouraged to investigate the world in the same way that boys are. The type of games and toys we give boys and girls to play and the type of language we use with the children is different. Boys are given blocks, science kits, and construction tools and are encouraged to explore, represent, and express their world in more mathematical ways than are girls. If more girls were given the same support and opportunities that boys have to excel at mathematics, there would be many more high-achieving girls and women in mathematics.

Myth 6
I am not going to use mathematics, so I don’t need to learn it.
Most teachers want to answer this question by giving “pie in the sky” kinds of applications of mathematics. Which is not difficult to do. Application of mathematics is the real story of human civilization. However, the message behind this question is: “I have never succeeded in mathematics. I do not know the previous content, so how can I learn this new content.” When teachers are not able to provide success in simple mathematics, they create a population who is going to ask the question: “Where am I going to use it?” every time they introduce a new concept. Interestingly, few successful students ever ask this question, and definitely not in the tone that this question is generally asked. My own approach is to take a simple concept that students find difficult and then provide success in that concept. Then repeat this many times. It is impossible to learn if one does not want to learn, so to gain expertise, we have to see the skills and knowledge as valuable. What is more, we have to create meaning. Learning is a matter of making sense of something.

 

 

 

Mathematics Anxiety and Mathematics Achievement (Part II)

Mathematics Anxiety and Mathematics Achievement (Part I)

Do not worry about your difficulties in mathematics; I assure you that mine are greater. Albert Einstein

I hated every minute of training, but I said, don’t quit. Suffer now and live the rest of your life as a champion. Muhammad Ali

Learning mathematics takes a special kind of courage and enthusiasm for the subject, and to be successful in it, one needs effective teachers. Effective teachers prevent failures while poor teaching may create them. Mathematics is a major hurdle for many elementary school students, particularly when teachers rely on drilling and memorization.  Failures in early mathematics can develop into full-fledged anxiety and may result in avoidance of mathematics and more failure.

For a significant number people, young and adult, the failure to feel successful and consistent lowered achievement in mathematics turn into mathematics anxiety, ranging from mild to severe. Math anxiety is the strongest predictor of both applied and basic math performance. Math anxiety may continue at the work place and in higher education.

In addition to socio-cultural (e.g., educational) factors, differences in students’ mathematics achievement are also due to attitude, motivation, language and intellectual abilities (e.g., executive function skills). Learning difficulties, disorders, and mathematics disabilities, like dyscalculia, often arise as a result of issues with cognitive and executive functioning. However, some students experience failure in mathematics due to environmental/social factors only (e.g., inefficient teaching, lowered expectations).

A pertinent question related to mathematics anxiety is whether mathematics anxiety causes poor mathematics performance, or whether poor mathematics performance elicits mathematics anxiety.

Executive Function
Numerical skills are important for success with meaningful problem solving, higher order thinking and higher mathematics, but other cognitive factors and executive skills (domain-specific and domain-general) also play an important role. Some of these critical skills include: (a) Working memory—holding and manipulating information in one’s mind, (b) Inhibition—ability to suppress distracting information and unwanted responses, selecting effective strategies/procedures, and using overlearned strategies wisely, and (c) Shifting—the ability to flexibly switch attention between different tasks—language, multiple concepts, strategies and multiple procedures involved have also been implicated in mathematics achievement.

Executive function skills are critical at every stage of mathematics learning: from language processing for decision-making, monitoring, and controlling emotions, thought, and action—and then these abilities are required for planning and assessing one’s actions (What mathematical operation to choose in this problem? e.g., long division or prime factorization, etc.), attention (What is the role of these symbols and words here? What is the relationship between them?), and self-control and perseverance (What do I know here? What else can I do here?). These processes[1] allow us to respond flexibly to a mathematics problem or concept and to engage in deliberate, goal-directed, thought and action.

Executive functions also affect processing skills. The term processing disorder is an umbrella term covering a continuum from auditory processing to language processing—from acoustic/central auditory processing, to linguistic-phonemic, to decoding, to meaning within language.

An auditory processing disorder will impact language processing and therefore language related aspects of mathematics such as: conceptualization (e.g., 2 × 3 can be read as (i) 2 groups of 3, (ii) 2 repeatedly added 3 times, (iii) a 2 by 3 array, and (iv) area of a 2 by 3 rectangle; ⅔×¾ is read as ⅔ of ¾, and the fraction, ¾ can be read as: (i) three-fourth/three-fourths, (ii) 3 out of 4 equal parts, (iii) 3 divided by 4, (iv) the ratio of 3 is to 4, (v) 3 groups of ¼, or (vi) ¼ repeated 4 times), of and problem solving. A child may have language processing difficulty in varying degrees of severity. Such a demand of language in mathematics creates difficulties, and with math anxiety the language processing is further compromised.

Math anxiety compromises executive functions, contributing to difficulty in learning mathematics, thereby lowered achievement, and more mathematics anxiety. Math anxiety affects students’ intellectual factors such as learning styles, persistence, working memory, and organization.

Brain research shows that anxiety makes learning harder because it activates the amygdala and the limbic system and pulls processing away from the pre-frontal cortex.  If a student who makes a mistake stays calm, his brain will stay in a mode to learn, whereas if a student freaks out with a stress response at each mistake or unfamiliar answer, his brain will be less able to process the incoming information or make connections between different pieces of information both new and old.  This can create a spiral: General anxiety —> making errors or lack of understanding —> math anxiety —> more mistakes or less understanding—> greater anxiety —> poor performance —> math anxiety.

Math Anxiety
Mathematics anxiety is a state of discomfort and helplessness around the performance of mathematical tasks, and is generally measured using self-report trait anxiety questionnaires or can be observed by mathematics educators. On one hand, there is broad consensus that presence of mathematics anxiety is linked to poor math performance. On the other, poor mathematics performance is the result of math anxiety. Thus, debilitating emotional reaction to mathematics activity is either the cause of or an outcome of underachievement in mathematics.

Mathematics anxiety is a child’s emotional reaction to negative mathematics experiences, an intense feeling of helplessness about quantity. This is a global phenomenon. Even in countries where students on the average do better than most countries, there are students who are anxious about mathematics. Math anxiety is present in many students poorly performing in mathematics, but even many high performing students exhibit math anxiety. In cases of high-performing students, math anxiety makes a major impact on their performance. High achieving students are more fearful of poor performance, and this may translate into math anxiety. But, it is the feeling of helplessness about mathematics that grips a large percent of students from elementary school to adulthood. Math anxiety is a serious and pervasive problem, especially in the middle and high school and college setting.

In the earlier stage, math anxiety may indicate something is not right in a child’s mathematics learning. At this stage a child may not have internalized the mathematics anxiety. However, once math anxiety begins to be internalized, it undermines child’s self-esteem, and then it is a causative factor for poor mathematics achievement. Early math anxiety, if not treated, may lead to a cycle in which fear of math interferes with learning math that exerts an increasing cost to math achievement by changed attitude and motivation towards math, curtailing aspirations, increasing math avoidance, and ultimately reducing math competence. This may lead students to delay or stop taking math courses, limiting their educational opportunities.

Students experience math anxiety in many forms and degrees, from a feeling of mild tension and anxiety that interferes with the manipulation of numbers and solving mathematical problems in ordinary life and academic situations to freezing up during a math exam/test and avoiding anything having to do with quantity. Symptoms may be physical or psychological and may include:

  • Physical: Nausea, shortness-of-breath, sweating, heart palpitations, increased blood pressure, fidgeting, lack of attention, avoiding direct eye contact, etc.
  • Psychological: Short-term memory loss, feeling of panic, paralysis of thought, mental disorganization, loss of self-confidence, negative self-talk, helplessness, math task avoidance, isolation (thinking you are the only one who feels this way), etc.

In many cases, math anxiety is a unique kind of anxiety; for example, there is increased heart rate when people are tested on math but not on other academic areas. However, math anxiety is not restricted to tests or classroom settings, people develop severe avoidance of situations involving quantitative facts and/or reasoning and formal mathematics. They may not choose careers involving application of mathematics even if cognitively they are capable. There is overlap between math anxiety and other general types of anxiety, especially related to test taking.

Math anxiety has a variety of sources. Its development is tied to social factors such as a teachers’ and parents’ anxiety about their own math ability and cognitive skills or individual factors such as students’ own quantitative and spatial competencies. All those factors that have a negative influence on mathematics achievement are potential factors for the development of math anxiety.

Students can develop mathematics anxiety by the presence of teacher anxiety, societal, educational or environmental factors, innate characteristics of mathematics, failure and the influence of early-school experiences of mathematics.

Types Of Math Anxiety
Three types of math anxiety are identified by the factors that may cause them or by the nature of their manifestations. They are:

  • Math type (Specific Math Anxiety) is caused by mental blocks in the process of learning math and related to specific mathematics language, concept, or procedure. Generally, it relates to a difficulty or negative experience with one of the key developmental milestones—number concept, number relationships (arithmetic facts), place value, fractions, integers, algebraic or spatial thinking.
  • Socio-cultural type (Global Math Anxiety) is the result of socio-cultural factors. It relates to socio-cultural conditions that may influence a learner in forming negative attitudes about and aversion towards mathematics once they experience difficulties in executing mathematics tasks.
  • Handicap type is caused by some physical or mental handicaps.

These types are generalizations of cases. Sometimes one finds the presence of each, and other times the math anxiety may be because of the integration of more than one. To illustrate the three types of math anxieties, below one example of each reflects general principles of a type of math anxiety.

Example of Math Type (Specific Math Anxiety). Ms. Gamble had a master’s degree in humanities from a university and had an excellent academic record. After graduating she worked for an insurance company, taught at a school and finally decided to become a lawyer, so she wanted to take the law board examinations. She asked a friend, a professor of mathematics at the university[2] to help her with the math/quantitative reasoning portion of the law board exams. They had one or two sessions a week for tutorials. Most of these sessions were successful, but at times, she became irrational about math tasks. She would throw down her pencil and say things like, “You just don’t understand me. I can’t stand these things any more. It does not make any sense.” She would become visibly angry and upset. She would become quite anxious.

The professor started exploring the reasons behind her unpredictable behavior about mathematics. The professor ruled out any socio-cultural factor because she was very independent, assertive and knew what she was doing. She had not accepted female conditioning to the point of dependence, non-assertion and helplessness. She was intelligent, well educated and handled most of the math she was learning quite well. Finally, it became clear to him that anytime he tried to make her do problems involving the radical sign “√” and exponents, she had fits of math anxiety. Otherwise, she was quite comfortable with math and learned well. Somehow, she was uncomfortable with square roots and the radical sign. The radical sign and related concepts had become a mental block in her efforts to learn math. She had difficulty understanding how one could have an exponent less than a whole number, and she had difficulty conceptualizing the concept.

Example of Socio-Cultural Type (Global Math Anxiety). Ms. Cook needed to pass a required math course to graduate with a liberal arts degree. She put the math course off till her last semester at the university. She finally decided on taking an elementary course called Topics in Mathematics, designed especially for liberal arts majors. Soon she had problems in the course. She claimed: “Nothing made sense, I did not understand anything.” She dreaded coming to class and always felt inadequate—“stupid.” She did not take the first quiz, avoided the class and avoided the teacher. One day, the teacher saw her in the university cafeteria and asked her what was wrong, why did she not take the quiz. Rather hesitantly, she said, “Math makes me sick, I don’t need math, people are right math is not for girls. I cannot do math.”

Then, with tears in her eyes, she told the professor how her elementary school teacher used to tell her “You need to work hard. You should get help from your father every night to review the material and do the homework.” Anytime she had difficulty in arithmetic, the teacher would say: “Did you ask your dad?” She did not have a dad around. She lived with her mother—a single parent. When she asked her mother for help, her mother got angry with her and the teacher, and her mother told the teacher, “Why are you doing this to my daughter and me? I never learned this stuff in school. You know as well as I do that math is not for girls. She will be fine without it.” Since then Ms. Cook avoided math with the belief that math was not for girls.

Example of Handicap Type. John, a bright young man, was crippled in a car accident. He registered in a linear algebra course at the university. John read the first test and thought, “This looks like a fair and easy test. I am going to do well.” But after about half an hour, his face became red, he began to perspire profusely as he tried one problem after another, and he looked sick. The professor asked if he was well and informed him that he could arrange a make-up test. John said he was all right and just made some mistakes somewhere and the answers he was getting were not checking out. The professor knew that even though John was confined to the wheelchair, he had enjoyed being in the class and worked hard.

When the professor corrected the test, he noticed that John had poor handwriting—a clear case of dysgraphia. He had difficulty rounding up letters and numbers like O, 0, 9, and 6. His writing showed that words, sentences, and expressions were a mixture of upper and lower case letters. Numbers were mixed and illegible; it was not possible to distinguish whether John was writing 6 or 0. Later in his office, the professor pointed this out to John and suggested that he should do his test on a computer. With this arrangement John did well in the course.

Handicaps may be of two types: physical handicaps and cognitive handicaps. Not every handicapping condition results in poor math performance or math anxiety. But, in some situations they may cause learning disabilities/difficulties in learning mathematics. Learning disabilities and some physical handicaps, generally, may affect some of the prerequisite skills necessary for mathematics learning. These prerequisite skills are: (a) the ability to follow sequential directions, (b) pattern analysis—pattern recognition, extension, creation and usage, (c) spatial orientation/space organization, (d) visualization, (e) estimation, (e) deductive reasoning, and (f) inductive reasoning. The absence of any or more of these skills affects mathematics achievement, and the lowered achievement may create math anxiety.

Today, the law requires that we do everything possible to facilitate the physically handicapped and learning disabled to provide access to a quality learning experience by providing appropriate accommodations and a least restrictive learning environment without compromising the standards of mastery and competence. To comply with the law, it is therefore important to recognize and address math anxiety.

Global Math Anxiety Caused by Socio-Cultural Factors
Math anxiety of the type experienced by Ms. Cook is caused by socio-cultural factors. This type of math anxiety is not about any specific or particular concept or procedure in mathematics. There is a global negative emotional reaction to any thing mathematical. Such a student may have general anxiety to start with—may have an overall anxious personality, worrying about things in general. But, the anxiety is exacerbated and manifested more profoundly in mathematics classrooms if the teaching is not appropriate and other sociological factors come into play.

Some of these individuals may experience failure in mathematics and also in other academic areas, but they admit to having anxiety only in mathematics because in mathematics students feel they are required (student perception) to give exact answers. This perceptual demand brings out the underlying anxiety to the forefront and converts into math anxiety. The anxiety is triggered whenever a student is asked to respond to specific information—facts, formulas. The subject could be grammar, a foreign language, etc. But, it happens more often in mathematics classes, particularly because mathematics teaching is driven by executing procedures and based on memorization. It happens when the teaching is compartmentalized and no connections are made.

Some students have the belief that mathematics problems always have exact answers; therefore, they may feel especially pressured (e.g., many students get upset when they are solving an equation and get a fraction answer). To these students mathematics is “solving.” The only instruction they know is “solve.”

When he/she makes mistakes, a person with general anxiety is much more susceptible to socio-cultural factors related to mathematics learning and performance. They search for a rationale for their negative reactions and socio-cultural factors come into play. Common causes of this type of anxiety are:

  • distrust of one’s intuition, especially as applied to math
  • illusion that math practices are an exact science
  • common myths about mathematical ability
  • the myth that boys naturally do better in math than girls
  • the female fear of competing on man’s turf
  • negative experiences of people one trusts and respects, in school, at home or society in general

When exposed to any one or more of these negative experiences consistently, individuals may develop a negative attitude towards mathematics and perform poorly, reinforcing the anxiety.

People with this type of anxiety have difficulty in most aspects of mathematics, so their response to mathematics is more global in nature and more intense. Their self-diagnosis is: “I am just very bad in mathematics. Mathematics is beyond me.” Therefore, we call this kind of anxiety global mathematics anxiety.

When referred to my Center for evaluation or remediation, even before we begin the evaluation process, persons with global math anxiety volunteer comments about their incompetence in mathematics and their feelings about it. They are apt to say:

  • “I’m not smart enough to learn mathematics.”
  • “I can’t do math.”
  • “I just don’t understand math.”
  • “Math makes me sick.”
  • “Complete blackout when I see a math problem.”
  • “So and so is super bright, he is really good in math.”
  • “You would not believe it, but math makes me throw up.”
  • “I will always stay away from math. I hate math.”

Key phrases used by these people are “I Hate Math”; “Math Makes Me Sick.” Such a person is likely to show irrational behavior towards math or mathematicians.

Much of the anxious, blocking, fear-stricken behavior, and helplessness that many students experience and exhibit in mathematics often is not primary to mathematics as a subject but is caused by other factors. People experiencing this type of anxiety will usually talk about their moment of disinterest in math, about friends and relatives who are good in math and about tricks they played on math teachers whom they hated. There is usually a moment of “sudden death syndrome” when they felt that as far as math was concerned, they were through. These math anxious people stay away from mathematicians and anybody who is good at math. For his reason, it is almost impossible to get them to talk to a math teacher or to take them near a math building.

Persons with global math anxiety are less willing to deal with their math deficiencies because they are not aware of what they are. They spend a great deal of time and energy in making decisions about how to avoid math.

Global math anxiety is on a decline due to the efforts of organizations involved in improving math education for all and making special efforts to recruit women and minorities into mathematics courses and programs. But, due to the new efforts of education leaders to improve skill levels by making the math curriculum more rigorous and because many teachers are not yet prepared to teach effectively, we are seeing more students experiencing specific math anxiety.

[1] See previous three posts on Executive Function and previous three posts on Working Memory.

[2] Most mathematics teachers in high schools and colleges report cases of this type; however, these particular cases were reported by Dr. Dilip Datta, Professor of Mathematics, University of Rhode Island.

Mathematics Anxiety and Mathematics Achievement (Part I)