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.)

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Mathematics Anxiety and Mathematics Achievement (Part V)