Mathematical anxiety

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Mathematical anxiety is anxiety about one's ability to do mathematics independent of skill.

Math anxiety[edit]

Math anxiety is a phenomenon that is often considered when examining students’ problems in mathematics. Mark H. Ashcraft, Ph.D. defines math anxiety as “a feeling of tension, apprehension, or fear that interferes with math performance” (2002, p. 1).[1] 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.[1] Hembree [2] (1990) conducted a thorough 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[1] (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 Ashcraft,[3] 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.[4]

Performance anxiety[edit]

People's fear of math can be related to test taking and performance anxiety. Some scholars have suggested a strong relation between math anxiety and math performance.[5] Current research in math anxiety concerns working memory.[6]

Anxiety rating scale[edit]

A rating scale for mathematics anxiety was written about in 1972 by Richardson and Suinn.[7] Richardson and Suinn defined mathematical anxiety as "feelings of apprehension and tension concerning manipulation of numbers and completion of mathematical problems in various contexts."[8] 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.[9] 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."[10] 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.[9] The MARS test is of interest to those in counseling psychology [11] and the test is used profusely in math anxiety research. It is available in several versions of varying length [12] and is considered psychometrically sound.[13] Other tests are often given to measure different dimensionalities of math anxiety, such as the 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.[14] 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.[15]

Math and culture[edit]

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 and Singapore. Researchers 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. 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.[16] (Stevenson & Lee, 1990).

Math and gender[edit]

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.[citation needed] 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[17] (Dar-Nimrod & Heine, 2006). 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.[18]

Mathematics and women[edit]

Related to this is gender and mathematics as younger female scholars are thought to develop anxiety towards mathematics and sciences when they become more interested in social relations in their teen years. It is thought that women experience more anxiety in mathematics as a group than men and this has also been suggested in regards computer programming. See for instance Copper, Joel, & Weaver D, Kimberlee. Gender and Computers: "Understanding the Digital Divide" who explore computing and gender and especially have done experiments relating gender and anxiety.[19] It has also been suggested that in primary elementary years, if female students have an anxious female math teacher, they are more likely to confirm the math anxiety as a gender stereotype. Girls are more likely than boys to take notice of their female teachers "negatives and fears about math", which could negatively influence their future pursuit of the subject.[20] One method to help address this issue is ensuring that teaching programs are reinforcing positive attitudes towards math, and helping teacher candidates solidify their grasp on mathematics.[20]

When men and women take math exams, there is a stereotype that women score less than men, saying they are not as good as men. The researchers explain that it is not a biological but more of a social effect. The researchers doing the experiment believe that gender stereotype threat could be a key factor in explaining women and men’s difference in performance on math exams. The gender stereotype threat would be gender references on the exams and that they could affect how a male or female answers the question and if they get it correct or not. The gender references on the exam could also be called gender labeling. The researchers did 2 experiments. In experiment 1, they created an exam consisting of 1/3 of male, female, and neutral questions. The results found that both male and female answered male-labeled questions with better proficiency than others. Even if the questions were the same, the gender label affected the result.[21] Here is a question they used on the exam, “There are 12 car pools at the plant where Mr. Holst works. Half of them contain 4 people, the other half contain 5 people. How many workers at Mr. Holst’s plant belong to car pools?”[21] The way they manipulated the questions was to change Mr. Holst to Mrs. Holst or do not include a name at all to remain neutral. The ones that performed well did best on male-labeled questions while the ones that performed poorly did best on female-labeled questions. In experiment 2, they tested university students because gender differences in test performance have been show to increase with age. Their main focus was to see if a negative stereotype towards women would affect their performance. The results showed that there were more correct answers on gender stereotype situations and men outperformed compared to women.[21] The results showed that a brief written reference to gender in the questions affected women’s performance negatively, supporting their hypothesis. Based on the results of this experiment, gender labeling on exams could cause women to underperform on math exams. Stereotype threat can interfere with performance, affecting men positively but women negatively.

Math pedagogy[edit]

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.[18] (Kail & Zolner, 2005).

High risk teaching styles are often explored to gain a better understanding of math anxiety. Goulding, Rowland, and Barber [22] (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 [23] (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 [24] (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 [25] (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; educating female teachers about stereotype threat can reduce its negative effects in the classroom.

Common beliefs[edit]

In the United States, many people believe that only a few "gifted" individuals have "what it takes" to learn math, and that hard work cannot compensate for this. Studies have shown "When asked to explain why some children do better in math than others, Asian children, their teachers, and their parents point to hard work, their American counterparts to ability."[26]

Women mathematicians in the United States have almost always been a minority according to Margaret Murray. 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].[27] "Since 1980, women have earned over 17 percent of the mathematics doctorates.... [In The United States]".[27] The trends in gender are by no means clear, but perhaps parity is still a way to go. Thus parity will take more work to overcome mathematical anxiety and this is one reason for women in mathematics being role models for younger women.

Mathematical anxiety in schools[edit]

Causes[edit]

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. As John Taylor Gatto[28] has demonstrated at length, modern Western schools were deliberately designed during the late 19th century to create an environment which is ideal for fostering fear and anxiety, and for preventing or delaying 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”.[29]

While teaching of many subjects has changed from rote memorization to the current Constructivist approach, math is frequently taught with a rote learning behaviorist approach. That is,

  • 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.

Teachers who actually understand what they are teaching tend to encourage questions from the students[original research?]. Those teachers who do not understand much about their subject, on the other hand, impose fear on the students to prevent them asking questions which might expose the teacher's ignorance[original research?].

It has long been well established[when?] that anyone (other than a tiny minority who have serious learning disabilities) can learn any area of mathematics, given a desire to learn, a coherent presentation of the information, and adequate practice. Nevertheless, many educational administrators[who?] continue to profess the belief that anything more complex than simple arithmetic is too difficult for most people.

In spite of this, a remarkably high percentage[quantify] of schoolchildren continue to find mathematics interesting, relaxing, easy, and enjoyable.

Solutions[edit]

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.".[30]

Claudia Zaslavsky[30] 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.[31] 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

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, 3rd edition, 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 regularly basis for 10–20 minutes students can significantly can reduce their anxiety.[32]

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.[33]

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 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.[34]

Studies have shown students learn best when they are active rather than passive learners.[35]

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.[36]

New concepts can be taught through play acting, cooperative groups, visual aids, hands on activities or information technology.[37] 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 understand 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.[38] 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?"[39]

Reflective journals help students develop metacognitive skills by having them think about their understanding. According to Pugalee,[40] 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 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.[37]

See also[edit]

References[edit]

  1. ^ a b c Ashcraft, M.H. (2002). Math anxiety: Personal, educational, and cognitive consequences.Directions in Psychological Science, 11, 181-185.
  2. ^ Hembree, R. (1990). The nature, effects, and relief of mathematics anxiety. Journal for Research in Mathematics Education, 21, 33-46.
  3. ^ Ashcraft, M. H., & Kirk, E. P. (2001). The relationships among working memory, math anxiety, and performance. Journal of Experimental Psychology: General, 130, 224-237
  4. ^ 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.
  5. ^ Cates, Gary L.a; Rhymer, Katrina N. "Examining the Relationship Between Mathematics Anxiety and Mathematics Performance: An Instructional Hierarchy Perspective", Journal of Behavioral Education Vol: 12, Issue: 1, March 2003 pp. 23-34
  6. ^ Ashcraft, Mark H.; Kirk, Elizabeth P., "The Relationships Among Working Memory, Math Anxiety, and Performance", Journal of Experimental Psychology: General 2001 pp. 224-237
  7. ^ Richardson, F.C., Suinn R.M., "The Mathematics Anxiety Rating Scale", Journal of Counseling Psychology, Volume: 19, (1972), pp. 551-554
  8. ^ Hopko, Derek R.; McNeil, Daniel W.; Lejuez, C.W.; Ashcraft, Mark H.; Eifert, Georg H.; Riel, Jim "The effects of anxious responding on mental arithmetic and lexical decision task performance" Journal of Anxiety Disorders Vol: 17, Issue: 6, 2003 pp. 647-665
  9. ^ a b Richardson, F. C. & Suinn, R. M. (1972). The mathematics anxiety rating scale: Psychometric data. Journal of Counseling Psychology, 19(6), 551-554.
  10. ^ Richardson, F. C. & Suinn, R. M. (1972). The mathematics anxiety rating scale: Psychometric data. Journal of Counseling Psychology, 19(6), 553.
  11. ^ 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.
  12. ^ 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, 373-386.
  13. ^ 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.
  14. ^ 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.
  15. ^ 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.
  16. ^ 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, 1-119
  17. ^ Ian Dar-Nimrod and Steven J. Heine (2006). Exposure to Scientific Theories Affects Women’s Math Performance. Science, 314, 435
  18. ^ a b Kail, R.V., & Zolner, T. (2005). Children. Toronto: Prentice Hall.
  19. ^ Copper, Joel, & Weaver D, Kimberlee. Gender and Computers: Understanding the Digital Divide (Mahwah, N.J.: Lawrence Erbaum, 2003).
  20. ^ a b Bielock, Sian, Elizabeth Gunderson, Gerardo Ramirez, and Susan Levine. "Female teachers' math anxiety affects girls' math achievement." PNAS 107, no. 5 (2010): 1860-1863.
  21. ^ a b c Walsh, Margaret, Hickey, Crystal, Duffy, Jim. (1994) Influence of item content and stereotype situation on gender differences in mathematical problem solving. Sex Roles, 219.
  22. ^ Goulding, M., Rowland, T., Barber, T. (2002). Does it matter? Primary teachers trainees’ subject knowledge in mathematics. British Educational Research Journal, 28, 689-704.
  23. ^ Laturner, R.J. (2002) Teachers’ academic preparation and commitment to teach math and science. Teaching and Teacher Education, 18, 653-663.
  24. ^ 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, 818-825.
  25. ^ Johns, M., Schmader, T., Martens, A. (2005). Knowing is half the battle: Teaching stereotype threat as a means of improving women’s math performance. Psychological Science, 16,175-179.
  26. ^ Tobias, Shiela, Overcoming Math Anxiety. (New York: W. W. Norton & Company, 1993), page 52
  27. ^ a b Murray M. A. M., Women Becoming Mathematicians: Creating a Professional Identity in Post-World War II America (Cambridge, Mass.: The MIT Press, 2000)
  28. ^ Gatto, John Taylor .""An Underground History of American Education."" http://www.johntaylorgatto.com/underground/index.htm
  29. ^ Furner, Joseph M., Berman, Barbara T., "Math anxiety: Overcoming a major obstacle to the improvement of student math performance", Childhood Education, Spring 2003
  30. ^ a b Zaslavsky, Claudia, Fear of Math, pages 198-199. (New Brunswick, New Jersey: Rutgers University Press, 1994)
  31. ^ "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
  32. ^ Arem, C. (2010). Conquering Math Anxiety. Belmont, CA: Brooks/Cole. p. 43. 
  33. ^ HypnoGenesis.: Magazine for Hypnosis and Hypnotherapy, HypnoGenesis:. "The Progressive Muscle Relaxation of Dr. Edmund Jacobson". 
  34. ^ Arem, C. (2010). Conquering Math Anxiety, 3rd Ed. Belmont, CA: Brooks/Cole. pp. xxi. 
  35. ^ Spikell,M .Teaching Mathematics With Manipulatives: A Resource of Activities for K-12 Teacher. (New York: Allyn and Bacon, 1993)
  36. ^ [1]
  37. ^ a b 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)
  38. ^ Rittenhouse (1998). Lampert, M & Blunk, M, ed. Talking Mathematics: Studies of Teaching and Learning in School (P. ed.). New York, New York: Cambridge University Press. pp. 163–189. 
  39. ^ Franklin, Margaret (2006). Add-ventures for girls: building math confidence, Junior High teacher's guide. Newton, Massachusetts: WEEA Publishing Center. 
  40. ^ Pugalee, D. (2004). "A Comparison of Verbal and Written Descriptions of Students’ Problem Solving Processes". Educational Studies in Mathematics 55 (3): 27. 

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