How to Improve Numbersense Part One

As we expect every child to read fluently with comprehension by the end of third grade, we should expect every child to have mastery of numeracy with understanding by the end of fourth grade so that they can access and learn mathematics easily, effectively, and efficiently. Then, they can appreciate the reach of mathematics, its utility, power, and beauty. To do so is to understand number, acquire numbersense and build the brain for fluency in numeracy and beyond. Through learning, practicing, and applying knowledge of the number concept, numbersense, numeracy, and mathematical way of thinking children will have access to higher mathematics.

When I ask teachers in my workshops, from elementary through high school and college, what their major concern is in teaching students mathematics, comments about numbersense are the most frequent:

  • Many of my students do not have “good” numbersense. How do I develop “good” numbersense in my students so that I can teach my curriculum at grade level?
  • I wish my students knew their facts. No numbersense!
  • How can I teach my grade level material, when they do not even have a sense of the place value of whole numbers? No numbersense!

The concept of numbersense is important for learning higher mathematics and also for day-to-day living. When I evaluate children from elementary to high school and even college students and adults, for learning problems in mathematics, most times it is the lack of mastery of numbersense that is at the base of many students’ difficulties in mathematics.

When I ask teachers: What do you mean by numbersense? Everyone gives his/her own definition. Numbersense is a key concept, but the meaning of this term is only vaguely understood. A term that is not well-defined cannot be effectively taught and assessed. Therefore to teach and to assess numbersense, it is important to know:

  • What is the meaning of numbersense? How do we define it?
  • How does it develop in children?
  • How do we develop and teach it?
  • What student behaviors must be evident when it is present?
  • What are the component skills necessary to learn and master it?
  • How do we know the child has acquired it? How do we assess it?
  • What are the levels of its achievement? How should it manifest at each grade level?
  • What is its role in learning other mathematics concepts?
  • What are the implications if it is not acquired?

Till these questions are answered in the teacher’s mind, the concept cannot be developed in children and assessed. A teacher’s instructional methodology for this concept depends on how it is understood. In the next several blogs, I plan to answer the questions posed above.

What is Numbersense?
From Kindergarten to upper elementary school, three major concepts form the foundation of arithmetic. They are also essential elements and building blocks in learning higher mathematics concepts, skills, and procedures. These are number concept, numbersense, and numeracy. Numbersense depends on the mastery of number concept, and its mastery is essential for the development of numeracy. The mastery of the concept and numbersense skills is the integration of three major components and skills.

  • Number Concept
  • Arithmetic Facts
  • Place Value

When students have mastery of these individual skills, they develop competence in numbersense and numeracy by integrating these skills.

Numbersense is a developmental and hierarchical concept. The type and level of mastery of arithmetic facts and place value varies from grade to grade; therefore, the concept and skills related to numbersense change and become complex and more demanding from grade to grade. The following are the non-negotiable component skills for the development of numbersense at each grade level. This does not mean nothing else other than these needs to be learned at these grade levels. Non-negotiable skills[1] at any grade level mean that other concepts, procedures, and skills can be mastered easily if these non-negotiable skills are mastered.

1. Numbersense at the end of Kindergarten

  • Mastery of number concept
  • Mastery of 45 sight facts
  • Place value (two digits)

2. Numbersense at the end of First Grade

  • Mastery of number concept
  • Mastery of 100 Addition facts
  • Place value (three digits)

3. Numbersense at the end of Second Grade

  • Mastery of number concept
  • Mastery of 100 subtraction facts (assuming the 100 addition facts have been mastered)
  • Place value (four digits)

The major goal of the first three years of mathematics curriculum (K through second grade) is to master additive reasoning: understanding the concepts of addition and subtraction, fluency of addition and subtraction facts, mastery of addition and subtraction procedures, applying these skills into solving problems, and knowing that they are inverse operations of each other. It means that: (a) given an addition problem, one can transform it into a subtraction problem and vice-versa, 23 + 12 = 35, 12 + 23 = 35 and 35 – 12 = 23, and 35 – 23 = 12, (b) when two numbers (10 and 9 are subtracted) from a given number (27), then the (27 –10 = ?, 27 – 9 = ?), then are being subtracted from a given number, then the remainder is larger from the given number in the case of the smaller number being subtracted from it (27 –10 17, 27 –9 = 18), (c) the difference of two number (51—29 = ?) will remain the same when the problem is translated by a number ((both numbers are translated by 2 units: 52 – 30 = 22 and 51 –29 = 22), etc.

4. Numbersense at the end of Third Grade

  • Mastery of number concept
  • Mastery of 100 multiplication facts (multiplication tables from 1 through 10)
  • Place value (at least 5 to 7 digits and ultimately any digit whole number)

5. Numbersense at the end of Fourth Grade

  • Mastery of number concept
  • Mastery of 100 division facts
  • Place value (any number of digit whole number) and to hundredth place

The major goal of the third and fourth grades mathematics curriculum is to master multiplicative reasoning: understanding the concepts of multiplication and division, fluency in multiplication and division facts, mastery of multiplication and division procedures, applying these skills into solving problems, and knowing that they are inverse operations of each other (given a multiplication problem, one can transform it into division problem and vice-versa).

Numeracy
Numeracy is both dependent on numbersense and aids in the development of numbersense; in this sense it is the culmination of numbersense. Numeracy is the ability and facility of a student to execute four whole number arithmetic operations correctly, consistently, and fluently in the standard form with understanding. By the end of fourth grade, every child should have mastered numeracy.

[1] See an earlier post on this blog on non-negotiable skills at elementary school level.

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How to Improve Numbersense Part One

Stereotype and Its Effect: Math Anxiety and Math Achievement Part Two

 

What is Needed to Fight Stereotype?
In order to effectively minimize the effects of stereotype, eradicate the conditions that foster stereotype in institutions, and create environments where our children do not encounter these conditions will take time, will, and effort. For the moment, we need to focus on a few key factors. Here, our focus is only on the mathematics education related factors:

  • Classroom environment (physical, affective, cognitive, mathematical),
  • Teacher characteristics (training as a teacher, subject matter mastery, attitude—toward the discipline and learner differences, usage of language in communicating mathematics, questioning and assessment techniques, mastery of teaching and learning tools and their effective and flexible use, collaboration with colleagues and students, interest in learning, etc.),
  • Instructional strategies (knowledge of pedagogy, choice of instructional materials/models, selection and sequencing of introductory and practice exercises, amount of time devoted on mathematics instruction—tool/skill building, main concept, collaboration, practice, problem solving, etc.).

As educators and policy planners we react to situations where stereotypes are manifested. Then we seek easy solutions: we increase the number of female and minority faculty, provide mentors, and actively recruit students in STEM programs. These changes provide only opportunity for positive impact, but to have long lasting effect, they need to accompany significant changes in pedagogy, understanding of learning issues and the aspirations, assets—strengths and weaknesses of these students, the nature of classroom interactions, type of assessments, and the nature of feedback.

For example, even when the number of females and minorities increases in STEM programs, not enough students remain in the programs. It is because they may not have the information about what kinds of pre-requisite skills they need to succeed in these programs. How to acquire these pre-skills? What efforts should they make to succeed? They may not know how to be effective and successful learners. They may not know what types of jobs they can get if they succeed in STEM programs. When they perceive that they are not succeeding, they change course, programs, and aspirations for careers.

Many students change majors during their undergraduate years. The rate at which students change their major varies by field of study. Whereas 35 percent of students who originally declare a STEM major change their field of study within 3 years, 29 percent of those who originally declare a non-STEM major do so. However, about half (52 percent) of students who originally chose math major switch major within 3 years. This change of major is much higher than that of students in all other fields, both STEM and non-STEM, except the natural sciences. [1]

The challenge of keeping students—especially women and underrepresented minorities—is on the agenda of every policy decision at every level of government—local to federal and education—from early childhood to graduate school. According to studies, among the culprits of attrition in STEM programs are uninspiring introductory courses, a culture that can be unwelcoming, and, and lack of adequate preparation of students, specifically in mathematics. [2]

Learning Strategies and Stereotype
Differences in students’ familiarity with mathematics concepts explain a substantial share of performance disparities between socio-economically advantaged and disadvantaged students and males and females. Many children, particularly girls and minorities, do not get exposure to quality mathematics content and effective pedagogy. Access to proper and rich mathematics language, transparent and effective conceptual schemas, and efficient and generalizable procedures is the answer to higher mathematics achievement for all students and fewer inequalities in mathematics education and in society.

When gender differences in math confidence, interest, performance and relations among these variables are studied over time, results indicate that gender differences in math confidence are larger than disparities in interest and achievement in elementary school. Research shows that confidence in math has become a major problem for girls and many minority children. It is one of the reasons women are vastly outnumbered by men in STEM professions later in life. Differences in math confidence between boys and girls show up as young as grade 2 and 3, despite girls and boys scoring similar marks. That trend continues through high school.

According to several studies, about half of third grade girls agree with the statement that they are good at math compared to two-thirds of boys. The difference widens in grade 6, where about 45 per cent of girls say that they are good at math compared to about 60 per cent of boys. This information is important for teachers as these attitudes are significant predictors of math-related career choices.

Early gender differences in math interest drive disparities in later math outcomes. At the same time, math performance in elementary school is a consistent predictor of later confidence and interest. There is a reciprocal relation between confidence and performance in middle school. Thus, math interventions for girls should begin as early as first grade and should include attention to developing math confidence, in addition to achievement. Even in preschools, the kind of games and toys children play with can determine the development of prerequisite skills for mathematics learning.[3] Confidence in learning is a function of metacognition and that in turn develops executive function and cognitive flexibility.

In elementary school, boys often utilize rote memory when learning math facts whereas girls rely on concrete manipulatives such as counting on fingers, number line, etc. And they use them longer than they should. Their arithmetic fact mastery puts boys in a better preparation for related arithmetic concepts (e.g., multiplication, division, etc.). These differences in strategies result in girls demonstrating slower math fluency (i.e. the ability to solve math problems related to arithmetic facts quickly) than boys. Both these methods (mastery by rote memorization and prolonged use of counting materials) are inefficient for arithmetic fact mastery for anyone. But, these inefficient strategies reinforce the gender stereotype for girls. Girls may blame their slower mastery of facts on being girls rather than the inefficient methods and strategies.

These inefficient models, methods, and strategies might bring higher achievement in elementary mathematics up to grade 3 and 4 (one can answer addition, subtraction, multiplication, and division problems by sheer counting), but they do not develop skills that are important for later concepts (e.g., difficulty dealing with fractions, ratio and proportion, and algebra as they are not amenable to counting). As a result, the average mathematics achievement of an average American is fifth-to-sixth grade level.

Clear understanding of concepts, fluency in procedures, and application of proportional reasoning (e.g., fraction concepts and procedures, ratio, proportion, etc.) are the gateway to algebra. Algebra is the main door to STEM fields. Students who do not opt for STEM related topics are the ones who have experienced difficulty understanding and operating on fractions. To become competent in understanding and fluently operating on fractions, students need:

  • Mastery of multiple models of multiplication (e.g., repeated addition, groups of, an array, and the area of a rectangle),
  • Mastery of multiplication tables (tables of 1 to 10),
  • Divisibility rules (for 2, 3, 4, 5, 6, 8, 9, and 10),
  • Short-division (computing division with one digit divisor and multi-digit dividend without long-division procedure),
  • Prime factorization, and
  • Knowing that a/b = 1,for any a and b and b ≠0, and that multiplying by a/a (by a fraction whose value is 1) gives an equivalent fraction.

Skills listed above cannot be achieved by being proficient only in counting, memorizing and using manipulatives. These skills are acquired with having strong numbersense (e.g., mastery of number concept, arithmetic facts, and place value). Students have mastery of an arithmetic fact if they demonstrate:

  • Solid understanding of number concept and sight facts,[4]
  • Efficient strategies for arriving at the fact (e.g., using decomposition/ recomposition of number),
  • Fluency (giving a fact in 2 seconds or less orally, 3 seconds or less in writing), and
  • Applying it to other facts, mathematics concepts, and/or to problems.

Developmental Trajectory of the Competence in Mathematics
Strong numbersense (Additive and multiplicative reasoning and facts, place value, decomposition/recomposition of number)
to Numeracy (Ability and facility in executing four whole number operations, correctly, consistently, flexibly, and fluently in the standard form with understanding
to Proportional reasoning (e.g., Fractions, and all of its incarnations—decimals, percent, ratio, proportion, scale factor, rate, slope, etc.)
to Algebra [Seeing it as generalization of arithmetic ideas and procedures; arithmetic of functions and expressions; modeling of problems by algebraic systems (e.g., linear, exponential and logarithmic, absolute and piece-wise); concept of and operations on polynomials—with specific emphasis on quadratic and trigonometric; concept of transformations and their applications; systems of equations, inequalities, etc.
to Continuous modeling (Calculus) and discrete modeling (probability and statistics
to Higher mathematics
to STEM fields.

To be prepared for higher mathematics, all children should learn, master and apply strategies that are based on

  • decomposition/ recomposition of numbers (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14; 8 + 6 = 4 + 4 + 6 = 4 + 10 = 14; 8 + 6 = 2 + 6 + 6 = 2 + 12 = 14; 8 + 6 = 8 + 8 – 2 = 16 – 2 = 14; 8 + 6 = 7 + 1 + 6 = 7 + 7 = 14),
  • flexibility of thought (able to arrive an answer to a problem in more than one way),
  • generalizable skills (moving from strategies that give the exact answer to efficient strategies that give the accurate answer easier and quicker with less effort and then move to strategies that are elegant—that can be abstracted into formal systems, and can be extrapolated), etc.

Inefficient strategies and simplistic definitions and models such as addition is counting up/forward, subtraction is counting down/ backwards, multiplication is skip counting forward, and division is skip counting backwards (e.g., counting objects, fingers, number positions on the number line, etc.) are inefficient formulations of these concepts. They are counter-productive to creating interest, flexibility of thought and confidence in mathematics because they do not expose children to the beauty and power of mathematics. Mathematics is the study of patterns, it has deep underlying structures, and it is based on the regularity of principles in its concepts. Its power lies in the collection of concepts and tools for the modeling of problems from diverse fields from anthropology to space and methods of solving them.

Teachers and parents need to emphasize that mastery of math facts and concepts is not just memorizing or arriving at the answer by counting. It is:

  • Deriving facts with efficient strategies, strong conceptual schemas, precise language, and elegant procedures,
  • Accuracy and fluency, and
  • Ability to apply this information in diverse situations (e.g., intra-mathematical, interdisciplinary, and extra-curricular applications).

When girls are encouraged to continue counting to find answers, they become self-conscious of their strategies and give up easily. This happens to boys too. But, in most cultures boys are given more support and encouragement. In addition, the stereotype that “boys are good in math and girls are good in reading” gives boys the benefit of doubt—they will ultimately outgrow inefficient strategies.

All students must understand that mastery of certain math concepts is important for any quantitative problem solving in most professional fields. To acquire efficient and effective strategies for number relationships and gain confidence in their usage is even more important. For example, students in early grades show high interest in STEM. But in later grades, without fluency in basic skills, lack of flexibility of thought and poor/or no conceptual schemas for key mathematics concepts, they tend to lose that interest. They also have difficulty connecting the diverse strategies and experiences in problem settings to related disciplines. For example, they have difficulty in applying conversion of units and dimensional analysis algebraic manipulations from mathematical setting to physics and chemistry. As a result, the sheer size of numbers and complexity of concepts and procedures they encounter in the STEM fields overwhelms them. As another example, a calculus course requires students to have an in-depth understanding of rates of change (e.g. proportional reasoning and its applications). The foundation of the concept should be introduced to students early in their mathematics education, and their understanding of it should evolve from middle school up to and including calculus. They should explore rates of change using numbers, tables, graphs and equations:

  • Investigate and model applications of rates of change, and
  • Explore how integrating concepts and technology appropriately enhances student understanding across grades.

When students are exposed to interesting and challenging problems from early grades and are shown clear developmental trajectory of each concept and procedure, they see connections between concepts. When students see the relationships between mathematics tools—strategies, skills and procedures and problems and where do these problems come from they remain engaged. This is particularly true about many female students as they are not sure of their competence. When problems are selected and their relationship with the STEM fields is made transparent, students get interested in these fields. Many students do not know what types of problems are solved in different fields. For example, in surveys 34 percent more female students than male students say that STEM jobs are hard to understand, and only 22 percent of the female respondents name technology as one of their favorite subjects in school, compared to 46 percent of boys.

Increasing STEM Participation is a Whole School Activity
Turning students’ interest toward mathematics and then STEM has to be a school-wide effort. Every educator (teachers—regular and special education, administrators, guidance counselors, coaches, para-professionals, etc.) should be aware of their own beliefs about math, gender/minorities, and their biases. For example, I have observed interactions between many adults (including principals) openly admitting their incompetence in mathematics to students. Here is a sample of interaction between a guidance counselor and two ninth grade students.

Female Student: Mr. Wilson, I am having very difficult time in my algebra I class. It looks like I do not have what it takes to be successful in Algebra I. I guess, I need to be taking the simpler algebra course or pre-algebra again. Could you please sign this paper for change of course?
Guidance Counselor: Let me see! Do you have a note from the teacher or your parents? Yes, algebra is kind of difficult. I have seen, over the years, more girls changing from this Algebra I class to easier courses. Have you tried getting some help from your algebra I teacher?
Female Student: I tried. I went to her a couple of times. It did not work. I will get a note from my father. He did warn me that algebra might be difficult. I will see you tomorrow.
GC: OK!

Another day:
Male Student: Mr. Wilson, I am having great deal of difficulty in my algebra I class. It looks like I do not have what it takes to be successful in Algebra I. I guess, I need to be taking the simpler algebra course or pre-algebra again. Could you please sign this paper for change of course?
Guidance Counselor: Let me see! Did you do poorly on the first test? You know the first test in a course is not really an indication of poor preparation for a course. One has to get used to the new material and the teacher—her style of teaching and her expectations. Now do you know what the teacher wants? Have you tried getting some help from your algebra teacher? You know she is one of the best teachers in our school. I know she is a little demanding, but she is an excellent teacher.
Male Student: Yes, she is demanding. Not a little, but a lot.
GC: You should join a study group. David, your friend on your soccer team, he is very good at math. Have you asked him for help? He even lives near you. Why don’t you try the course for few more weeks, maybe till the next test and then you still have difficulty come see me. Meanwhile, I will talk to your teacher. By the way, before you come see me next time, get a note from the teacher explaining that you did try. And, I also need a note from your parents so that they know about your changing the course? I know, algebra is kind of difficult, but trying is even more important.
Male Student: I guess, I will give it another try. If it doesn’t work, I will come to you, again. Yes, I will get a note from my mother. My father wants me to have algebra on my transcript. He says: “It looks good for college applications to have algebra in eighth grade or latest in ninth grade. I will see you later.
GC: OK!

Quality of Concepts and Quality of Instruction
Quality instruction has the greatest impact on student achievement and the development of a positive attitude about a subject matter. The major changes in student outcomes are obtained by teachers’ instructional actions. Generally, the premise is that teachers who implement effective instructional strategies will, in turn, help students use mental processes that enhance their learning. However, it is not enough to merely use an instructional strategy; it is more important is to ensure that it has the desired effect on student learning.

The opportunity to learn and the time students spend learning quality mathematics content and practicing meaningful and rigorous mathematics tasks assure higher mathematics achievement. Differences in students’ familiarity with mathematics concepts explain many performance disparities between socio-economically advantaged and disadvantaged students and between females and males. Widening access to meaningful mathematics content—proper mathematics language, efficient, effective, and generalizable conceptual schemas, and efficient and elegant procedures—are the answers raising levels of mathematics achievement and, at the same time, reduce inequalities in mathematics education.

Poor learning environments and poor mathematics teaching create gender, race, and class disparities in quantitative fields, and the gaps begin to develop as early elementary school. Initially small and subtle, they grow into causative factors for low achievement in and avoidance of mathematics in high school, college, and even graduate school. They become most pronounced in quantitative professions such as university-based mathematics research and STEM fields. It is worth noting that women who drop out of quantitative majors do not tend to have lower scores on college entrance exams or lower freshman grades than their male peers. Females are leaving math fields when they are performing just fine; it is therefore worth considering that the reasons hardly lie in them, but in our educational environments that might induce them to leave.

Attitudes and Values
Various explanations exist for gender differences, beginning with small differences in elementary school to consequential differences in high school, undergraduate mathematics classes, advanced mathematics, and in math-related career choices. Below we summarize some of the factors that contribute to gender, race, and ethnic differences in mathematics and math-related courses and career choices.

Attitudes toward numeracy
Even from a young age, girls are less confident and more anxious about math than boys. These differences in confidence and anxiety are larger than actual gender differences in math achievement. The differences are shaped by the social attitudes of adults toward work, careers, education, and achievement. These attitudes are important predictors of later math performance and math-related career choices.

Early perceptions and attitudes formed at home and in early childhood classrooms form the basis of future attitudes toward learning. For example, many parents read to their children regularly and with interest. This instills the love for literacy and learning. During this reading, some parents do discuss the quantitative and spatial relations—ideas about number, number relationships, and numeracy, in the reading material. Many families play board and number games and with toys that develop prerequisite skills for mathematics learning [5].

From early childhood, traditionally, boys and girls play with toys and engage in games that are responsible for the development of different types of skills. Boys engage in activities that develop spatial skills and girls participate in games and toys that develop sequencing skills. For example, boys tend to be stronger in the ability to mentally represent and manipulate objects in space, and these skills (ability to follow sequential directions to manipulate objects mentally, spatial orientation/space organization, visuo-spatial representation, rotations, transformations, pattern recognition and extensions, visualization, inductive reasoning, etc.) predict better math performance and STEM career choices. In teaching mathematics, it is important to use those models—concrete materials, visual representations (diagrams, figures, tables, graphs, etc.) that develop these skills and to make up earlier deficits in these skills and aid in the development of numeracy skills effectively.

Attitudes thus formed about quantitative and spatial relations become the basis of later interest and competence in numeracy and its applications. By high school, these skills and related attitudes are well established. Personal aptitudes and motivational beliefs in the middle and high school have profound impact on individuals’ interest in science, technology, engineering, and mathematics in college and later in choice of occupations and professions.

Attitudes toward work and professions
Occupational and lifestyle values, math ability, self-concept, family demographics[6] (particularly, financial and educational status of family), and high school course-taking more strongly predict both individual and gender differences in STEM careers than math courses and test scores in undergraduate years.

People’s life styles, values, attitudes, and interests and that of those around them influence gender and class differences about occupational and career choices and the role of work in their lives. For example, women tend to care more about working with people, and men tend to be more interested in working with things. This difference, in turn, relates to gender-gaps in selection of math-related careers and even within STEM disciplines—health, biological, environmental and medical sciences (HBEMS) versus mathematics, physical, engineering, economics, accounting, and computer sciences (MPEEACS).

Women’s preferences for work that is people oriented and altruistic predict their entrance into HBEMS instead of MPEEACS careers. For example, for the first time ever, women make up a majority (50.7 percent) of those enrolling in medical school, according to the Association of American Medical Colleges. In fall 2017, the number of new female medical students increased by 3.2 percent, while the number of new male students declined by 0.3 percent.[7]

Women prefer biological sciences, where they represent 40% of the workforce, with smaller percentages found in mathematics or computer science (33%), the physical sciences (22%), and engineering (9%). To change this phenomenon active intervention and education are needed.

Role of Problem-solving Strategies
Mathematics is the study of patterns in quantity, space, and their integration. This means mathematics is thinking quantitatively and spatially. For example, in elementary school, understanding the concept of place value in representing large numbers is the integration of quantity and space. We are interested in a digit’s quantitative value in the number by its location in relation to other digits in the number. Similarly, coordinate geometry is a good example of this integration: each algebraic equation and inequality represents a curve in space and every curve can be represented by a system of equations/inequalities. Thus, to do better in mathematics and in subjects dependent on mathematics, one needs to have strong visual/spatial integrative skills: the ability to visualize and see spatial organization and spatial orientation relationships. Students who are poor in these skills, generally, have difficulty in mathematics. Those students who can do well in arithmetic up to fourth grade by just sequential counting begin to have difficulty later when concepts become complex (e.g., fractions, ratio, proportion, algebraic thinking, geometry, etc.). Then they blame themselves for their failures in mathematics – “I cannot learn mathematics.” “Mathematics is so difficult.” However, to a great extent, the reality lies in lack of these prerequisite skills and inefficient strategies.

The prerequisite skills for mathematics learning can be improved through training and intervention. Gender differences in spatial abilities and visual-spatial skills can be reduced and/or stronger compensatory strategies can be developed with effective interventions. The pattern of differences in the prerequisite skills for mathematics learning can be broken through these intervention programs. The types of games and toys children play, in early childhood, determine the fluency in these skills[8]. This means that one way forward is to ensure that all students spend more “engaged” time learning core mathematics concepts, solving challenging mathematics tasks, acquire prerequisite skills for learning mathematics.

Success in STEM fields depends on a person’s ability to apply efficient math strategies and exposure to diversity of problem solving strategies. For that, students need to engage in thinking both quantitatively—analyze ideas and strategies (deductive thinking), and qualitatively—synthesize ideas (inductive thinking). Thinking quantitatively means developing a strong numbersense and its applications. Thinking spatially/qualitatively means seeing patterns in numbers, shapes, objects, and seeing connections amongst ideas.

Research and observations show that in our schools boys tend to and are encouraged to use novel problem-solving strategies whereas girls are likely to follow school-taught procedures. In general, girls more often follow teacher-given rules in the classroom. It could be that girls are trying to fit in the class and they learn that these behaviors are rewarded. This tendency inhibits their math explorations, innovations, and the development of bold, efficient, and effective problem-solving skills. They need to explore and acquire strategies that can be generalized to multiple situations than just solving specific problems.

Such differences in learning approach and types of experiences contribute to gender related achievement gaps in mathematics as content becomes more complex and problem-solving situations call for novel approaches rather than just learned procedures. The rigorous use of the Standards of Mathematics Practices (CCSS-SMP)[9] in instruction at K-12 level and student engagement in collaborative and interdisciplinary research and internships at the undergraduate level can better prepare our students for higher mathematics and problem solving. Then, they will stay longer in STEM fields.

To be attracted to and stay in mathematics, students need to engage from a very early age with appropriate and challenging mathematical concepts. That means to experiment more and experience widely. This happens when they collect, classify, organize, and display information (quantitative and spatial); analyze, see patterns and relationships, arrive at and make conjectures; and communicate these observations using mathematics language, symbols, and models. These skills are central to a person’s preparedness to tackle problems that arise at work and in life beyond the classroom. Unfortunately, many students do not have a rigorous understanding of basic mathematics concepts (integration of language, concepts, procedures, and skills) and are not required to master these skills. In school, they practice only routine tasks procedurally that do not improve their ability to think quantitatively and qualitatively and solve real-life, complex problems—involving multiple concepts, operations, and meaningful ideas.

Mathematics also means communicating mathematical thinking using mathematics language, symbols, diagrams, models, mathematical systems: expressions, systems of equations, inequalities, etc. All these skills are central to a person’s preparedness to tackle problems that arise at work and in life beyond the classroom. The best approach to keeping students in the STEM fields is not only to give them skills but also to give them the “taste” of success in applications of mathematics in problem solving.

Collective Course Design
In 2011 the Association of American Universities started a project to improve the quality of STEM teaching at the undergraduate level. Among the conclusions of this project are:

Success is more likely when interdisciplinary departments take collective responsibility for introductory course curricula in STEM fields. For example, mathematics teachers should select application problems from the STEM disciplines so that students see connections between use of mathematics tools and concepts and the nature of the problems they can solve from other disciplines.

Along with interdisciplinary integration, there should be active collaboration between K-12 and undergraduate curricula, pedagogy, instructional strategies, and teachers. To improve the situation, colleges and universities should collaborate more, with K-12 schools, industry, and one another.

Applications of mathematics should involve its language, concepts, skills and procedures, not just formulas and procedures using calculators, computer packages, and apps. The objective of applications, therefore, is to see the utility, the power, and the beauty of mathematics. The application of mathematics fall in three categories:

  • Intra-mathematical applications: Applying a concept, method, procedure, or strategy from one part of mathematics to solve a problem in another part of mathematics. For example, solving a geometry problem using algebraic equations; seeing the study of coordinate geometry as the integration of algebra and geometry; understanding statistics as the integration of algebraic concepts (e.g., permutation/combination, binomial theorem), geometry (e.g., representation and presentation, graphing, and displaying of data, etc.), and calculus (differentiation and integration of probability functions), etc.
  • Interdisciplinary applications: Applying mathematics modeling to problems in other disciplines (e.g., understanding and explaining concepts and principles in physics, chemistry, economics, psychology, etc. using mathematical models and systems). Students should understand and realize that mathematics is used first for understanding and explaining physical phenomenon and then mathematics modeling for solving problems in natural, physical, biological, and social sciences. For example, understanding the airline routing problem using mathematics (linear and non-linear programming, and permutation/combination, etc.) and then extend the approach to modeling similar problems.
  • Extra-curricular applications: Applying mathematics in solving problems in real life through group projects, independent and small group research, etc. This involves applying combination of skills from different branches of mathematics in solving real life problems. The skills involved are: identifying a problem; asking right questions about the problem and solution requirements and constraints on solutions; defining knowns and unknowns; identifying unknowns as variables; identifying and articulating relationships between knowns and unknowns—functions, equations, inequalities, etc.; identifying already known facts relating to the problem and the variables involved, postulates, assumptions, results that apply in this situation; developing strategies for solving the problem; collecting data, classifying, organizing, displaying the data; analyzing data; observing patterns in the data; developing conjectures/hypotheses, and results; solving the problem; relating the solution to the original problem; conclusion(s); if needed rethinking/redefining the problem with modified conditions and restraints; etc.

The course planning, design and course delivery are, therefore, more than a sole faculty member’s task. Colleges and schools showing the most improvements in attracting and retaining students in STEM use teams of faculty, instructional-design experts, data analytics on student learning, administrative supports like teaching-and-learning centers, creative-learning spaces, mentoring and tutoring, and multiple means of delivery, and meaningful and timely feedback.

An approach that is attracting more students into mathematics is undergraduate research, where students engage in independent individual or small group research projects for a sustained period of time under the supervision of a faculty member.

To keep many more students in STEM fields cannot be the activity of an isolated individual or an office. To address the issue of female and racial achievement gaps university and school reforms must be campus-wide and embraced by all faculty members in order for women, black and Latino students to truly thrive.

Schools must also move away from forcing students of color into remedial programs before their participation into proper programs. Those students need to learn how to navigate the boundaries of the different social worlds that make up higher education. They have to learn how to “try on” the identities of the professions, to feel that they own them and have the right to play in them. This approach is in opposition to remedial programs, which lead people to think of themselves as outsiders, inferior, and not worthy of achievements. With remedial programs, they learn math as a compliance activity.

Achieving change takes total campus/school commitment, with the most powerful and knowledgeable people involved. Diversity offices can be supportive, but the power of the academy is in the hands of faculty. They can either motivate or demotivate a student from a course, program or degree. When everybody – faculty, staff, and administration, makes the success of students from different backgrounds a top priority on campus, only then can we make a difference.

Minimizing the Effect of Stereotype
Our main goal should be to create conditions so that stereotype does not exist in our schools, but that will take time and a great deal of effort. Concurrently, we also need to minimize the effect of current conditions that have been affected by stereotype. Here are some strategies for improving mathematics instruction for all and for making sure that children do not adopt the cultural stereotype that math is for a select few.

Tracking and Its Role
Placing students in ability groups, particularly minority and low performing female students, is the beginning of closing doors for meaningful mathematics and STEM. Some instruction grouping may be considered at sixth grade and beyond. But these groupings should be to accommodate interventions—both for gifted and talented and those who struggle, not in place of regular classes. From seventh grade on there may be two levels: honors and regular. However, each group should be provided challenging and accessible instruction that is grade appropriate and rigorous in content. The difference should be on time on task rather than in the quality of content or nature of instruction.

Teachers should ensure that each and every student has access to meaningful curriculum and effective instruction that is balanced with respect to rich language of mathematics, strong conceptual understanding using multiple models and representations, efficient procedures and fluency, diverse and flexible problem solving, and the development of a productive disposition for mathematics.

Each teacher should provide every student the opportunity to learn grade-level or above mathematics using efficient strategies and provide the differentiated and targeted instructional support necessary for every student to successfully attain this goal. However, some may need more and others may need less practice to reach proficiency. For example, to differentiate, all students should be asked questions appropriate to their level but on the same concept or procedure being taught in the class.

Differentiation does not mean making groups and teaching them lower or higher level mathematics. Differentiation should offer children exercises and problems at different levels but on the same grade level concept or procedure. Small groups and individualization can be organized for brief but frequent periods of practice, reinforcement, and deepening their understanding, but not for initial teaching and for long intervals. Students learn more from other students than from most teachers. Teachers can make this happen.

Teachers should affirm and help students develop their mathematical identities by respecting their mathematics learning personalities.[10] For example, each student falls on the mathematics learning personality continuum of learning mathematics processes. On one end of this continuum are students who process mathematics information parts-to-whole. They process information sequentially, deductively, and procedurally. They are known as quantitative mathematics learning personality students. They are very strong on procedural parts of mathematics. They need more work on language and concepts of mathematics. On the other extreme are students who process information from whole-to-parts. They look for patterns, relationships, and commonalities in concepts, ideas and procedures. They use inductive reasoning to process mathematics ideas. This is called qualitative mathematics learning personality. They are strong in concepts, making connections and applications. However, they need support and reinforcement in mastering standard procedures. Instruction should be to make sure that the needs of all students are met and complement their mathematics learning personalities.

To engage all students, it is important to be cognizant of different ways people learn mathematics. Teachers should view students as individuals with strengths, not deficits. This manifests when they value multiple contributions and student participation and recognize and build upon students’ realities and strengths.

Multiple Models
Teachers should provide students multiple opportunities to grow mathematically by providing:

  • multiple entry points and multiple models for the same concept (e.g., for multiplication—repeated addition, groups of, an array, area of a rectangle; for division—repeated subtraction, groups of, an array, area of a rectangle),
  • multiple procedures for the same problem (e.g., the quadratic equation: 2x2 – 5x – 7 = 0 can be solved by using algebra tiles, by graphing, by factoring, by completing the square, or by quadratic formula;
  • multiple strategies for deriving a result (i.e., the sum 8 + 6 can be derived as 8 + 2+ 4 = 10 + 4 = 14, 2 + 6 + 6 = 2 + 12 =14, 8 + 6 = 4 + 4 + 6 = 4 + 10 = 14, 8 + 6 = 8 + 8 – 2 = 16 – 2 = 14, 8 + 6 = 7 + 1 + 6 = 7 + 7 = 14, etc.), and
  • multiple expressions for and demonstrate their knowledge in multiple ways—models (in words, symbols, tables, graphical, equations), forms, and levels (concrete, pictorial, abstract/symbolic, etc.).

Intervention and Remedial Instruction
We should provide additional targeted instructional time as necessary and based on the results of common formative assessments—make instructional time variable, not student learning. A teacher has four opportunities for remedial instruction:

  • Tool building—identifying the tools necessary for students to be successful in the main lesson and quickly reviewing them (orally using the Socratic method) before the main lesson (e.g., commutative property of addition, N+ 1, making tens, and what two numbers make a teens’ numbers before beginning addition strategies; rules of combining integers before solving equations; prime factorization before reducing fractions to lowest terms; differential coefficients of important functions before starting integration of functions, etc.).
  • During the main lesson—if during the lesson a concept is found to depend on a previous concept, briefly review that concept in summary form and write important formulas to be used in the new concept (e.g., divisibility rules during fraction operations; laws of exponents during combining polynomials; addition strategies during teaching subtraction strategies; important algebraic expressions before factoring: (a + b)2 = a2 + 2ab + b2, (a – b)2 = a2 – 2ab + b2, (a + b) (a – b) = a2 – b2, etc.).
  • Individual and small group practice—at the end of the group lesson on a major concept, making small groups and helping each group of students to practice skills, concepts at different levels and helping them to practice previous concepts.
  • Intensive intervention—organizing and providing intensive intervention for select students (e.g., those with dyscalculia, learning problems in mathematics, gaps in previous concepts and procedures, etc.). This type of intervention should be provided by specialists after or before class and should be in addition to regular math education instruction (math specialists with mastery of mathematics concepts and understanding of learning problems in mathematics using efficient and effective methods, not by a special educator who is weak in mathematics).

Instructors and Pedagogy
Quality instruction goes a long way toward keeping students — especially underrepresented minorities and women in the STEM fields. But measuring educational quality is not easy. Assessing the quality and impact in STEM at the national level will require the collection of new data on changing student demographics, instructors’ use of evidence-based teaching approaches, deeper and meaningful student engagement, student transfer patterns and more.

Most experienced, effective teachers (who have a clear understanding of the trajectory of the development of mathematics concepts and procedures and how children learn) should provide instruction to students who need more support rather than the least trained and least effective teachers and paraprofessionals. Highly effective teachers have the skills to support students who may not have previously been successful in mathematics.  Effective teachers can make up almost three years of the result of poor teaching. Similarly, a poor teacher can nullify the gains of three years of effective teaching and even turn students off from mathematics.

Keeping Diversity in Math to Fight Stereotype Threat
An important way to address underrepresentation of minorities and women in mathematical pursuits is to create environments without stereotype threat (gender, race, ethnicity)—environments in which these groups are not concerned about being judged according to negative stereotypes. This can be done by mentoring where students are assured that:

  • They are respected. Assigning simplistic work just to keep students in a program is not respecting learners of any kind and is insulting to their intelligence. A mathematics teacher should have fidelity only to (a) students and (b) to mathematics; teach meaningful math to all children in meaningful ways.
  • The work should be challenging. Students should realize that although the work they are asked to do is challenging, it is accessible to them, they do indeed have the ability to succeed at it, and they believe that the teacher is there to help them succeed. The role of the teacher is to help them develop self-advocacy of their abilities, their strengths, and their usage.
  • They trust the mentor and her intent and capacity to help achieve the mentee’s success. This means that they should feel secure that someone is there to help them to reach their goals; they believe in the process that they will have a higher level of skill set as the outcome and they will increase their potential to learn.
  • Importance of learning skills. Student work should focus not just on the content of mathematics but on how to learn—planning, goal setting, organizing, gaining learning skills—developing executive functions, marshaling resources, self-assessment, self-advocacy, self-regulation, self-reflection, and seeking and using feedback properly.

The emphasis in this kind of mentoring is on stressing the importance of the expandability of one’s learning potential and realization of one’s goals—in a sense intelligence itself—students should grasp and internalize the idea of the role of the plasticity of the brain in learning potential and learning.

The intent of this mentorship is that at the end of these experiences (math courses, STEM program, etc.), mentees realize the idea that intellectual ability is not something that one has a finite amount of, and that it can be increased with genuine effort, experience and training. The role of the mentor here is more than cheerleading; it is helping the mentees to reach new personal heights in success and the attitude that they are capable of learning mathematics.

The increase in female representation in faculty and mentor roles, for example, has positive effects as long as they do not emphasize the uniqueness of their achievements and do not place undue pressure and importance on their mentees to view themselves as female mathematicians or scientists.

The more our female math students are exposed to women role models who can show them that not only can women “do math” but also that their feminine identities need not be viewed as a liability, the more they are likely to view math environments as places where they can belong and succeed. The same applies to other underserved groups.

Identity and Mathematics
The effects of stereotypes are far reaching. Research shows that stereotype threats induce undue pressures on women in the quantitative fields who are in the process of shaping their identities. Do women in mathematical arenas bifurcate their identities in response to prolonged exposure to threatening stereotype in these environments, or are women with bifurcated identities simply more likely to study mathematics? Research shows that because of the threat of negative stereotypes about female math ability many female math students (but not their peers in other fields) bifurcate their feminine identities. For example, in responding to stereotype threats, women in mathematics related fields often bifurcate their feminine identities, cutting off feminine traits they view as related to negative stereotype about female math ability and potential, and continuing to identify with feminine traits that they view as relatively unrelated to the stereotypes.

They report keeping fewer feminine traits (e.g., sensitivity, nurturance, and even fashion-consciousness) in order to avoid negative judgments in math environments. They foster fewer feminine traits even though these may not lead to negative judgment. Thus, the stress of engaging in such adaptation could constitute yet another deterrent to women’s persistence in quantitative fields. Although sacrificing fashion-consciousness as an aspect of one’s identity may seem trivial, sacrificing an interest in having children does not.

Similar pressures in identity formation are present on students from minority groups. Many minority students try to adapt to situations by sacrificing some of their positive traits. We need to understand that stereotypes affect not only others’ judgments but also people’s own judgments of their own competence. For example, new immigrants to the country respond to discrimination and stereotype by blaming themselves. They blame their lack of knowledge, skills—language, experience, and knowhow, and as a result harshly judge themselves. Some of them are negatively affected and others may succeed by paying a price in both cases. For example, first- and second-generation children of immigrants respond to these situations with skills and knowhow (e.g., Chinese, Indian, and East European immigrants children flock to STEM majors and want to succeed to fight discrimination); they do not fight against stereotype and place higher expectations on themselves. Many of these children feel a great deal of pressure to succeed.

To reduce and nullify the effect of stereotype organized response is needed on several fronts. There is need for both systemic and tactical changes; change in systems and people that inhabit these organizations as these effects are situational.  For this reason, the focus should be on both the people who are vulnerable to stereotypes and the organizations where the stereotypes exist (which applies to almost all organizations).

First, the change has to come in these organizations to reduce and then to eradicate these conditions.

Second, the affected groups need self-advocacy skills in responding to these situations. They need skills to achieve. They need skills in organizing and taking advantage of the positive situations. A good example of this is the METCO program in the Metropolitan Boston area. In this program, volunteer minority students (both male and female) from Boston public schools are bused to suburban schools instead of Boston Public Schools. These students get the opportunity to get a first-rate education in their suburban host schools. Moreover, the suburban students who realize the importance of this program and the schools who treat METCO students as their own also derive benefits. Many METCO students have gained skills from this program and many of them have gone on to become leaders in STEM fields.

As the METCO example shows, along with systemic change, there is need to work at the individual level. There is need for a mindset change on the part of individuals to actively gain skills to minimize the negative effects of stereotypes. For example, we can a priory determine a group’s needs and aspirations. Women’s ideas about themselves, their academic and career needs, and aspirations cannot remain fixed. Therefore, just focusing on organizations may not be enough – we also need to focus on the individuals.

Organizations—schools/colleges, social organizations, work places, and parents can do three key things to effect change.

First, they can control the messages they are sending by making sure there are no negative beliefs about any group in the organization.  For instance, an experimental study on the evaluation of engineering internship applicants found that the same resume was judged by a harsher standard if it had a female versus a male name. Applications should be judged by the same standard.

Second, they can make performance standards unambiguous and communicate them clearly because when people don’t know what the standards are, stereotypes fill in the gaps.

Last, organizations can hold gatekeepers in senior management accountable for reporting on gender, race, and ethnic disparities in hiring, retention and promotion of employees.

[1] National Center for Education Statistics (NCES). (2017). Percentage of 2011 – 12 First Time Postsecondary Students Who Had Ever Declared a Major in an Associate’s or Bachelor’s Degree Program Within 3 Years of Enrollment, by Type of Degree Program and Control of First Institution: 2014. Institute of Education Sciences, U.S. Department of Education. Washington, DC. https://nces.ed.gov/datalab/tableslibrary/viewtable.aspx?tableid=11764

[2] President’s Council of Advisors (2012).

[3] Pre-requisite Skills and Mathematics Learning (Sharma, 2008, 2016).

[4] Sharma (2015). Numbersense a Window to Understanding to Dyscalculia in An International Handbook of Dyscalculia (Steve Chinn, Editor).

[5] For the role of prerequisite skills in mathematics learning see Games and Their Uses in Mathematics Learning by Sharma. A list of games that develop prerequisite skills can be requested (Mahesh@mathematicsforall.org) from the Center free of cost.

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

[7] Scott Jaschik (December 19, 2017), Women are Majority of New Medical Students. Inside Higher Ed.

[8] See Games and Their Uses by Sharma (2008). A shorter version of this book, Pre-requisite Skills and Mathematics Learning, in electronic form, is available free of cost from the Center. This document includes a list of games to develop these pre-requisite skills.

[9] See www.CCSS.org/math for the eight Standards of Mathematics Practices.

[10] See The Math Notebook on Mathematics Learning Personalities (Sharma, 198?)

Stereotype and Its Effect: Math Anxiety and Math Achievement Part Two

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.

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)