Mark H. Ashcraft defines math anxiety as "a feeling of tension, apprehension, or fear that interferes with math performance" (2002, p. 1). The academic study of math anxiety originates as early as the 1950s, where Mary Fides Gough introduced the term mathemaphobia to describe the phobia-like feelings of many towards mathematics. The first math anxiety measurement scale was developed by Richardson and Suinn in 1972. Since this development, several researchers have examined math anxiety in empirical studies. Hembree (1990) conducted a meta-analysis of 151 studies concerning math anxiety. It determined that math anxiety is related to poor math performance on math achievement tests and that math anxiety is related to negative attitudes concerning math. Hembree also suggests that math anxiety is directly connected with math avoidance.
Ashcraft (2002) suggests that highly anxious math students will avoid situations in which they have to perform mathematical calculations. Unfortunately, math avoidance results in less competency, exposure and math practice, leaving students more anxious and mathematically unprepared to achieve. In college and university, anxious math students take fewer math courses and tend to feel negative towards math. In fact, Ashcraft found that the correlation between math anxiety and variables such as confidence and motivation are strongly negative.
According to Schar, because math anxiety can cause math avoidance, an empirical dilemma arises. For instance, when a highly math-anxious student performs disappointingly on a math question, it could be due to math anxiety, or the lack of competency in math because of math avoidance. Ashcraft determined that by administering a test that becomes increasingly more mathematically challenging, he noticed that even highly math-anxious individuals do well on the first portion of the test measuring performance. However, on the latter and more difficult portion of the test, there was a stronger negative relationship between accuracy and math anxiety.
According to the research found at the University of Chicago by Sian Beilock and her group, math anxiety is not simply about being bad at math. After using brain scans, scholars confirmed that the anticipation or the thought of solving math actually causes math anxiety. The brain scans showed that the area of the brain that is triggered when someone has math anxiety overlaps the same area of the brain where bodily harm is registered. And Trezise and Reeve show that students' math anxiety can fluctuate throughout the duration of a math class.
The impact of mathematics anxiety on mathematics performance has been studied in much recent literature. An individual with math anxiety does not necessarily lack ability in mathematics, rather, they cannot perform to their full potential due to the interfering symptoms of their anxiety. Math anxiety manifests itself in a variety of ways, including physical, psychological, and behavioral symptoms, that can all disrupt a student's mathematical performance. The strong negative correlation between high math anxiety and low achievement is often thought to be due to the impact of math anxiety on working memory. Working memory has a limited capacity, and when solving mathematical problems, a large portion of this capacity is dedicated to problem solving. However, in individuals with math anxiety, much of this space is taken up by anxious thoughts, thus compromising the individual's ability to perform. In addition, a frequent reliance in schools on high-stakes and timed testing, where students tend to feel the most anxiety, can lead to lower achievement for math anxious individuals. Programme for International Student Assessment (PISA) results demonstrate that students experiencing high math anxiety demonstrate mathematics scores that are 34 points lower than students who do not have math anxiety, equivalent to one full year of school. These findings demonstrate the clear link between math anxiety and reduced levels of achievement, suggesting that alleviating math anxiety may lead to a marked improvement in student achievement.
Anxiety rating scale
A rating scale for mathematics anxiety was written about in 1972 by Richardson and Suinn. Richardson and Suinn defined mathematical anxiety as "feelings of apprehension and tension concerning manipulation of numbers and completion of mathematical problems in various contexts". Richardson and Suinn introduced the MARS (Mathematics Anxiety Rating Scale) in 1972. Elevated scores on the MARS test translate to high math anxiety. The authors presented the normative data, including a mean score of 215.38 with a standard deviation of 65.29, collected from 397 students that replied to an advertisement for behavior therapy treatment for math anxiety. For test-retest reliability, the Pearson product-moment coefficient was used and a score of 0.85 was calculated, which was favorable and comparable to scores found on other anxiety tests. Richardson and Suinn validated the construct of this test by sharing previous results from three other studies that were very similar to the results achieved in this study. They also administered the Differential Aptitude Test, a 10-minute math test including simple to complex problems.
Calculation of the Pearson product-moment correlation coefficient between the MARS test and Differential Aptitude Test scores was −0.64 (p < .01), indicating that higher MARS scores relate to lower math test scores and "since high anxiety interferes with performance, and poor performance produces anxiety, this result provides evidence that the MARS does measure mathematics anxiety". This test was intended for use in diagnosing math anxiety, testing efficacy of different math anxiety treatment approaches and possibly designing an anxiety hierarchy to be used in desensitization treatments. The MARS test is of interest to those in counseling psychology and the test is used profusely in math anxiety research. It is available in several versions of varying length and is considered psychometrically sound. Other tests are often given to measure different dimensionalities of math anxiety, such as Elizabeth Fennema and Julia Sherman's Fennema-Sherman Mathematics Attitudes Scales (FSMAS). The FSMAS evaluates nine specific domains using Likert-type scales: attitude toward success, mathematics as a male domain, mother's attitude, father's attitude, teacher's attitude, confidence in learning mathematics, mathematics anxiety, affectance motivation and mathematics usefulness. Despite the introduction of newer instrumentation, the use of the MARS test appears to be the educational standard for measuring math anxiety due to its specificity and prolific use.
Math and culture
While there are overarching similarities concerning the acquisition of math skills, researchers have shown that children's mathematical abilities differ across countries. In Canada, students score substantially lower in math problem-solving and operations than students in Korea, India and Singapore. Researchers[who?] have conducted thorough comparisons between countries, and have determined that in countries such as Taiwan and Japan, parents place more emphasis on effort rather than one's innate intellectual ability in school success. By placing a higher emphasis on effort rather than one's innate intellectual ability, they are helping their child develop a growth mindset. People who develop a growth mindset believe that everyone has the ability to grow their intellectual ability, learn from their mistakes and become more resilient learners. Moreover, parents in these countries tend to set higher expectations and standards for their children. In turn, students spend more time on homework and value homework more than American children.
Math and gender
Another difference in mathematic abilities often explored in research concerns gender disparities. There has been research examining gender difference in performance on standardized tests across various countries. Beller and Gafni's have shown that children at approximately nine years of age do not show consistent gender difference in relation to math skills. However, in 17 out of the 20 countries examined in this study, 13-year-old boys tended to score higher than girls. Moreover, mathematics is often labeled as a masculine ability; as a result, girls often have low confidence in their math capabilities. These gender stereotypes can reinforce low confidence in girls and can cause math anxiety as research has shown that performance on standardized math tests is affected by one's confidence. As a result, educators have been trying to abolish this stereotype by fostering confidence in math in all students in order to avoid math anxiety.
The principles of mathematics are generally understood at an early age; preschoolers can comprehend the majority of principles underlying counting. By kindergarten, it is common for children to use counting in a more sophisticated manner by adding and subtracting numbers. While kindergarteners tend to use their fingers to count, this habit is soon abandoned and replaced with a more refined and efficient strategy; children begin to perform addition and subtraction mentally at approximately six years of age. When children reach approximately eight years of age, they can retrieve answers to mathematical equations from memory. With proper instruction, normally functioning children acquire these basic mathematical skills and are able to solve more complex mathematical problems with more sophisticated training. (Kail & Zolner, 2005).
High risk teaching styles are often explored to gain a better understanding of math anxiety. Goulding, Rowland, and Barber (2002) suggest that there are linkages between a teacher's lack of subject knowledge and ability to plan teaching material effectively. These findings suggest that teachers that do not have a sufficient background in mathematics may struggle with the development of comprehensive lesson plans for their students. Similarly, Laturner's research (2002) shows that teachers with certification in math are more likely to be passionate and committed about teaching math than those without certification. However, those without certification vary in their commitment to the profession depending on coursework preparation.
Moreover, a study conducted by Kawakami, Steele, Cifa, Phills, and Dovidio (2008) examined attitudes towards math and behavior during math examinations. The study examined the effect of extensive training in teaching women to approach math. The results showed that women who were trained to approach rather than avoid math showed a positive implicit attitude towards math. These findings were only consistent with women low in initial identification with math. This study was replicated with women who were either encouraged to approach math or who received neutral training. Results were consistent and demonstrated that women taught to approach math had an implicit positive attitude and completed more math problems than women taught to approach math in a neutral manner.
Johns, Schmader, and Martens (2005) conducted a study in which they examined the effect of teaching stereotype threat as a means of improving women's math performance. The researchers concluded that women tended to perform worse than men when problems were described as math equations. However, women did not differ from men when the test sequence was described as problem solving or in a condition in which they learned about stereotype threats. This research has practical implications. The results suggested that teaching students about stereotype threat could offer a practical means of reducing its detrimental effects and lead to an improvement in a girl's performance and mathematical ability, leading the researchers to conclude that educating female teachers about stereotype threat can reduce its negative effects in the classroom.
According to Margaret Murray, female mathematicians in the United States have almost always been a minority. Although the exact difference fluctuates with the times, as she has explored in her book Women Becoming Mathematicians: Creating a Professional Identity in Post-World War II America, "Since 1980, women have earned over 17 percent of the mathematics doctorates.... [In The United States]". The trends in gender are by no means clear, but perhaps parity is still a way to go. Since 1995, studies have shown that the gender gap favored males in most mathematical standardized testing as boys outperformed girls in 15 out of 28 countries. However, as of 2015 the gender gap has almost been reversed, showing an increase in female presence. This is being caused by women's steadily increasing their performance on math and science testing and enrollment, but also from males' losing ground at the same time. This role reversal can largely be associated with the gender normative stereotypes that are found in the Science, technology, engineering, and mathematics (STEM) field, deeming "who math is for" and "who STEM careers are for". These stereotypes can fuel mathematical anxiety that is already present among young female populations. Thus parity will take more work to overcome mathematical anxiety and this is one reason why women in mathematics are role models for younger women.
It has been suggested that this section be checked to ensure a neutral point of view. (September 2013)
Students often develop mathematical anxiety in schools, often as a result of learning from teachers who are themselves anxious about their mathematical abilities in certain areas. Typical examples of areas where mathematics teachers are often incompetent or semi-competent include fractions, (long) division, algebra, geometry "with proofs", calculus, and topology. In many countries, would-be math teachers are required only to obtain passing grades of 51% in mathematics exams, so that a math student who has failed to understand 49% of the math syllabus throughout his or her education can, and often does, become a math teacher. His or her fears and lack of understanding then pass naturally to his or her students.
According to John Taylor Gatto, as expounded in several lengthy books,[page needed] modern Western schools were deliberately[dubious ] designed during the late 19th century to create an environment which is ideal for fostering fear and anxiety, and for preventing or delaying learning. Many who are sympathetic to Gatto's thesis regard his position as unnecessarily extreme. Diane Ravitch, former assistant secretary of education during the George H.W. Bush administration, agrees with Gatto up to a point, conceding that there is an element of social engineering (i.e. manufacture of compliant citizenry) in the construction of the American education system, which prioritizes conformance over learning.
Math is usually taught as a right and wrong subject and as if getting the right answer were paramount. In contrast to most subjects, mathematics problems almost always have a right answer. Additionally, the subject is often taught as if there were a right way to solve the problem and any other approaches would be wrong, even if students got the right answer. When learning, understanding the concepts should be paramount, but with a right/wrong approach to teaching math, students are encouraged not to try, not to experiment, not to find algorithms that work for them, and not to take risks. "Teachers benefit children most when they encourage them to share their thinking process and justify their answers out loud or in writing as they perform math operations. ... With less of an emphasis on right or wrong and more of an emphasis on process, teachers can help alleviate students' anxiety about math".
- A problem set is introduced
- A solution technique is introduced
- Practice problems are repeated until mastery is achieved
Constructivist theory says the learning and knowledge is the student's creation, yet rote learning and a right/wrong approach to teaching math ensures that it is external to the student.
There have been many studies that show parent involvement in developing a child's educational processes is essential. A student's success in school is increased if their parents are involved in their education both at home and school (Henderson & Map, 2002). As a result, one of the easiest ways to reduce math anxiety is for the parent to be more involved in their child's education. In addition, research has shown that a parent's perception on mathematics influences their child's perception and achievement in mathematics (Yee & Eccles, 1988). This means that if a parent makes it apparent that they do not enjoy mathematics or that they are not good at mathematics, this can influence the way in which their child views mathematics.
Furthermore, studies by Herbert P. Ginsburg, Columbia University, show the influence of parents' and teachers' attitudes on "'the child's expectations in that area of learning.'... It is less the actual teaching and more the attitude and expectations of the teacher or parents that count". This is further supported by a survey of Montgomery County, Maryland students who "pointed to their parents as the primary force behind the interest in mathematics".
Claudia Zaslavsky contends that math has two components. The first component, commonly focused on in many schools, is to calculate the answer. This component also has two subcomponents, namely the answer and the process or method used to determine the answer. Focusing more on the process or method enables students to make mistakes, but not 'fail at math'. The second component is to understand the mathematical concepts that underlay the problem being studied. "... and in this respect studying mathematics is much more like studying, say, music or painting than it is like studying history or biology."
Amongst others supporting this viewpoint is the work of Dr. Eugene Geist, Associate Professor at Ohio University – Athens, Ohio and an early childhood education specialist. Dr. Geist's recommendations include focusing on the concepts rather than the right answer and letting students work on their own and discuss their solutions before the answer is given. Emphasis is given that young people hate to be wrong and hate situations where they can be embarrassed by being wrong.
National Council of Teachers of Mathematics (NCTM) (1989, 1995b) suggestions for teachers seeking to prevent math anxiety include:
- Accommodating for different learning styles
- Creating a variety of testing environments
- Designing positive experiences in math classes
- Refraining from tying self-esteem to success with math
- Emphasizing that everyone makes mistakes in mathematics
- Making math relevant
- Letting students have some input into their own evaluations
- Allowing for different social approaches to learning mathematics
- Emphasizing the importance of original, quality thinking rather than rote manipulation of formulas
Hackworth (1992) suggests that the following activities can help in reducing and mitigating mathematical anxiety:
- Discuss and write about math feelings;
- Become acquainted with good math instruction, as well as study techniques;
- Recognize what type of information needs to be learned;
- Be an active learner, and create problem-solving techniques;
- Evaluate your own learning;
- Develop calming/positive ways to deal with fear of math, including visualization, positive messages, relaxation techniques, frustration breaks;
- Use gradual, repeated success to build math confidence in students
Math (and Statistics) Therapy is a combination of coaching and counseling, provided for adults by people with credentials in both counseling and math education. In Math Therapy the reasons for anxiety are addressed, as well as the mathematical skills which are lacking. New coping skills are introduced and practiced, so that fear, distaste or other negative emotions do not block math (or statistics) learning.
There are several anxiety reducing techniques that teachers can teach their children and practice periodically throughout the year. Teachers will need to learn these techniques and encourage the students to practice them at home and to use them prior to testing or when feeling anxious during math class.
Several studies have shown that relaxation techniques can be used to help alleviate anxiety related to mathematics. In her workbook Conquering Math Anxiety, Cynthia Arem offers specific strategies to reduce math avoidance and anxiety. One strategy she advocates for is relaxation exercises and indicates that by practicing relaxation techniques on a regular basis for 10–20 minutes students can significantly reduce their anxiety.
Dr. Edmundo Jacobson's Progressive Muscle Relaxation taken from the book Mental Toughness Training for Sports, Loehr (1986) can be used in a modified form to reduce anxiety as posted on the website HypnoGenesis.
Visualization has also been used effectively to help reduce math anxiety. Arem has a chapter that deals with reducing test anxiety and advocates the use of visualization. In her chapter titled Conquer Test Anxiety (Chapter 9) she has specific exercises devoted to visualization techniques to help the student feel calm and confident during testing.
The theory of multiple intelligences suggests that there is a need for addressing different learning styles. Math lessons can be tailored for visual/spatial, logical/mathematics, musical, auditory, body/kinesthetic, interpersonal and intrapersonal and verbal/linguistic learning styles. This theory of learning styles has never been demonstrated to be true in controlled trials. Studies show no evidence to support tailoring lessons to an individual students learning style to be beneficial.
New concepts can be taught through play acting, cooperative groups, visual aids, hands on activities or information technology. To help with learning statistics, there are many applets found on the Internet that help students learn about many things from probability distributions to linear regression. These applets are commonly used in introductory statistics classes, as many students benefit from using them.[original research?][who?]
Active learners ask critical questions, such as: Why do we do it this way, and not that way? Some teachers may find these questions annoying or difficult to answer, and indeed may have been trained to respond to such questions with hostility and contempt, designed to instill fear. Better teachers respond eagerly to these questions, and use them to help the students deepen their understanding by examining alternative methods so the students can choose for themselves which method they prefer. This process can result in meaningful class discussions. Talking is the way in which students increase their understanding and command of math. Teachers can emphasize the importance of original thinking rather than rote manipulation of formulas. This can be done through class conversations. Teachers can give students insight as to why they learn certain content by asking students questions such as "what purpose is served by solving this problem?" and "why are we being asked to learn this?"
Reflective journals help students develop metacognitive skills by having them think about their understanding. According to Pugalee, writing helps students organize their thinking which helps them better understand mathematics. Moreover, writing in mathematics classes helps students problem solve and improve mathematical reasoning. When students know how to use mathematical reasoning, they are less anxious about solving problems.
However, there is still a large part of school math teaching which consists of "mass-produced" memorization, repetition, and mechanically performed operations. Times tables are one example, wherein rote learning is essential to mathematics performance. When a student fails to learn the times tables at a young age, they can experience math anxiety later, when all the students' classmates can remember the tables but they cannot.
Children learn best when math is taught in a way that is relevant to their everyday lives. Children enjoy experimenting. To learn mathematics in any depth, students should be engaged in exploring, conjecturing, and thinking, as well as in rote learning of rules and procedures.
- Ashcraft, M.H. (2002), "Math anxiety: Personal, educational, and cognitive consequences", Current Directions in Psychological Science, 11 (5): 181–185, doi:10.1111/1467-8721.00196, S2CID 16387293
- Suárez-Pellicioni, Macarena; Núñez-Peña, María Isabel; Colomé, Àngels (2016). "Math anxiety: A review of its cognitive consequences, psychophysiological correlates, and brain bases". Cognitive, Affective, & Behavioral Neuroscience. 16 (1): 3–22. doi:10.3758/s13415-015-0370-7. ISSN 1530-7026. PMID 26250692.
- Hembree, R. (1990), "The nature, effects, and relief of mathematics anxiety", Journal for Research in Mathematics Education, 21 (1): 33–46, doi:10.2307/749455, JSTOR 749455
- Schar, M. H.; Kirk, E. P. (2001). "The relationships among working memory, math anxiety, and performance". Journal of Experimental Psychology: General. 130 (2): 224–237. doi:10.1037/0096-3422.214.171.124.
- Harms, William. "When People Worry about Math, the Brain Feels the Pain." UChicago News. The University of Chicago, 31 Oct. 2012. Web. 02 Mar. 2014.
- Trezise, Kelly; Reeve, Robert A. (2016). "Worry and working memory influence each other iteratively over time". Cognition and Emotion. 30 (2): 353–368. doi:10.1080/02699931.2014.1002755. PMID 25648296. S2CID 1305564.
- Trezise, Kelly; Reeve, Robert A. (2018-08-01). "Patterns of anxiety in algebraic problem solving: A three-step latent variable analysis". Learning and Individual Differences. Modelling individual differences in students' cognitions and development: Latent variable mixture model approaches. 66: 78–91. doi:10.1016/j.lindif.2018.02.007. ISSN 1041-6080.
- Beilock, S. L., & Willingham, D. T. (2014). "Math Anxiety: Can Teachers Help Students Reduce It? Ask the Cognitive Scientist". American Educator. 38 (2): 28–43. ISSN 0148-432X.
- Blazer, C. (2011). "Strategies for Reducing Math Anxiety". Information Capsule. 1102: 1–8.
- Ashcraft, M. H.; Krause, J. A. (2007). "Working memory, math performance, and math anxiety". Psychonomic Bulletin & Review. 14 (2): 243–248. doi:10.3758/BF03194059. ISSN 1069-9384. PMID 17694908.
- Scarpello, Gary (2007). "Helping Students Get Past Math Anxiety". Techniques: Connecting Education and Careers (J1). 82 (6): 34–35. ISSN 1527-1803.
- PISA 2012 results. Volume 3, Ready to learn : students' engagement, drive and self-beliefs. Programme for International Student Assessment. Paris: Organisation for Economic Co-operation and Development. 2013. ISBN 9789264201170. OCLC 865657364.CS1 maint: others (link)
- Richardson, F.C.; Suinn, R.M. (1972). "The Mathematics Anxiety Rating Scale". Journal of Counseling Psychology. 19 (6): 551–554. doi:10.1037/h0033456.
- Hopko, Derek R.; McNeil, Daniel W.; Lejuez, C.W.; Ashcraft, Mark H.; Eifert, Georg H.; Riel, Jim (2003). "The effects of anxious responding on mental arithmetic and lexical decision task performance". Journal of Anxiety Disorders. 17 (6): 647–665. doi:10.1016/s0887-6185(02)00240-2. PMID 14624816.
- Richardson, F. C.; Suinn, R. M. (1972). "The mathematics anxiety rating scale: Psychometric data". Journal of Counseling Psychology. 19 (6): 551–554. doi:10.1037/h0033456.
- Richardson, F. C.; Suinn, R. M. (1972). "The mathematics anxiety rating scale: Psychometric data". Journal of Counseling Psychology. 19 (6): 553. doi:10.1037/h0033456.
- McLeod, D.B. (1994). "Research on Affect and Mathematics Learning in the JRME: 1970 to Present". Journal for Research in Mathematics Education. 25 (6): 637–647. doi:10.2307/749576. JSTOR 749576.
- Capraro, M. M.; Capraro, R. M.; Henson, R. K. (2001). "Measurement error of scores on the Mathematics Anxiety Rating Scale across studies". Educational and Psychological Measurement. 61 (3): 373–386. doi:10.1177/00131640121971266. S2CID 144616161.
- Alexander, L., & Cobb, R. (1984). Identification of the dimensions and predictors of math anxiety among college students. Paper presented at the Annual Meeting of the Mid-South Educational Research Association (New Orleans, LA, November 16, 1984). ERIC, ED 251320.
- Fenemma, E.; Sherman, J.A. (1976). "Fennema-Sherman Mathematics Attitudes Scales: Instruments designed to measure attitudes toward the learning of mathematics by females and males". Journal for Research in Mathematics Education. 7 (5): 324–326. doi:10.2307/748467. JSTOR 748467.
- Ma, X (1999). "A meta-analysis of the relationship between anxiety toward mathematics and achievement in mathematics". Journal for Research in Mathematics Education. 30 (5): 520–540. doi:10.2307/749772. JSTOR 749772.
- Dweck, C. S. (2006). Mindset: The new psychology of success.New York: Random House.
- Stevenson, H.W.; Lee, S. (1990). "Contexts of achievement: A study of American, Chinese, and Japanese children". Monographs of the Society for Research in Child Development. 55 (1/2): 1–119. doi:10.2307/1166090. JSTOR 1166090. PMID 2342493.
- Gutbezahl, Jennifer (1995), How Negative Expectancies and Attitudes Undermine Females' Math Confidence and Performance: A Review of the Literature, Education Resources Information Center, ED380279.
- Dar-Nimrod, Ian; Heine, Steven J. (2006). "Exposure to Scientific Theories Affects Women's Math Performance" (PDF). Science. 314 (5798): 435. doi:10.1126/science.1131100. PMID 17053140. S2CID 40746692.
- Kail, R.V., & Zolner, T. (2005). Children. Toronto: Prentice Hall.
- Goulding, M., Rowland, T., Barber, T. (2002). Does it matter? Primary teachers trainees' subject knowledge in mathematics. British Educational Research Journal, 28, 689-704.
- Laturner, R.J. (2002). "Teachers' academic preparation and commitment to teach math and science". Teaching and Teacher Education. 18 (6): 653–663. doi:10.1016/s0742-051x(02)00025-2.
- Kawakami, K.; Steele, J. R.; Cifa, C.; Phills, C. E.; Dovidio, J. F. (2008). "Approaching math increases math = me, math = pleasant". Journal of Experimental Social Psychology. 44 (3): 818–825. doi:10.1016/j.jesp.2007.07.009.
- Johns, M.; Schmader, T.; Martens, A. (2005). "Knowing is half the battle: Teaching stereotype threat as a means of improving women's math performance" (PDF). Psychological Science. 16 (3): 175–179. doi:10.1111/j.0956-7976.2005.00799.x. PMID 15733195. S2CID 10010358.
- Murray, Margaret A. M. (2000). Women Becoming Mathematicians: Creating a Professional Identity in Post-World War II America. MIT Press. ISBN 9780262632461.
- Sparks, Sarah (30 November 2016). "TIMSS: A Closer Look at Gender Gaps in Math and Science". Education Week. Education Week. Retrieved 4 December 2016.
- Gatto, John Taylor. "An Underground History of American Education". Archived from the original on 2017-04-06. Retrieved 2007-04-02.
- Ruenzel, David (1 March 2001). "The World According To Gatto". edweek.org. Education Week. Retrieved 23 September 2020.
It makes no more sense to do away with them [public schools] than police departments ...
- Bosmans, Guy; De Smedt, Bert (2015-01-01). "Insecure attachment is associated with math anxiety in middle childhood". Front Psychol. 6: 1596. doi:10.3389/fpsyg.2015.01596. PMC 4606049. PMID 26528233.
- Furner, Joseph M., Berman, Barbara T., "Math anxiety: Overcoming a major obstacle to the improvement of student math performance", Childhood Education, Spring 2003
- Henderson, A. T. & Mapp, K. L. (2002), A New Wave of Evidence. The Impact of School,Family, and Community Connections on Student Achievement, Austin: Southwest Educational Development Laboratory
- Yee DK, Eccles JS. 1988. Parent perceptions and attributions for children's math achievement. Sex Roles 19:317–33.
- Zaslavsky, Claudia, Fear of Math, pages 198-199. (New Brunswick, New Jersey: Rutgers University Press, 1994)
- "Episode 54: Math Anxiety – Causes and Cures", by Michael on April 13, 2008, http://www.thepsychfiles.com/2008/04/episode-54-math-anxiety-causes-and-cures/ September 7, 2009
- Hackworth, R. D. (1992). Math anxiety reduction. Clearwater, FL: H & H.
- Arem, C. (2010). Conquering Math Anxiety (3rd ed.). Belmont, CA: Brooks/Cole. p. 43.
- HypnoGenesis.: Magazine for Hypnosis and Hypnotherapy, HypnoGenesis. "The Progressive Muscle Relaxation of Dr. Edmund Jacobson". Archived from the original on 2011-06-28. Retrieved 2011-06-30.
- Arem, C. (2010). Conquering Math Anxiety, 3rd Ed. Belmont, CA: Brooks/Cole. pp. xxi.
- Spikell, M .Teaching Mathematics With Manipulatives: A Resource of Activities for K-12 Teacher. (New York: Allyn and Bacon, 1993)
- Riener, Cedar; Willingham, Daniel (2010). "The Myth of Learning Styles". Change: The Magazine of Higher Learning. 42 (5): 32–35. doi:10.1080/00091383.2010.503139. S2CID 144349329.
- Curtain-Phillips, M. Math Attack: How to Reduce Math Anxiety in the Classroom, at Work and in Everyday Personal Use. (Atlanta: Curtain-Phillips Publishing, 1999)
- Rittenhouse (1998). Lampert, M; Blunk, M (eds.). Talking Mathematics: Studies of Teaching and Learning in School (P. ed.). New York, New York: Cambridge University Press. pp. 163–189.
- Franklin, Margaret (2006). Add-ventures for girls: building math confidence, Junior High teacher's guide. Newton, Massachusetts: WEEA Publishing Center.
- Pugalee, D. (2004). "A Comparison of Verbal and Written Descriptions of Students' Problem Solving Processes". Educational Studies in Mathematics. 55 (3): 27–47. doi:10.1023/b:educ.0000017666.11367.c7. S2CID 122937513.
- Richardson, F.C.; Suinn, R.M. "The Mathematics Anxiety Rating Scale: Psychometric data". psycnet. Journal of Counseling Psychology. Retrieved 2021-02-15.>