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.

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Mathematics Anxiety and Mathematics Achievement (PART III)

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