Succeeding in mathematics

SUMMARY: Poor mathematics performance can be attributed to a variety of factors and numerous organizations and strategies are believed to help students achieve mathematics success.

Many educational initiatives are designed to motivate U.S. students to excel in science and mathematics, with the goal of building the strong science, technology, engineering, and mathematics (STEM) workforce needed to meet twenty-first century challenges. Three overarching goals of the 2009 federal Educate to Innovate program are increasing STEM literacy for everyone; improving teaching so that American students meet or exceed those in other nations; and expanding STEM education and career opportunities for underrepresented groups. However, success in mathematics, or even literacy, can be difficult to define. Some see it as some minimum skill set, number of courses, or type of courses taken. Others conceptualize it by what sorts of problems students are able to solve or by their ability to manage real-world mathematical problems, such as budgets or loans.

Measuring Success

There are many barriers to achieving success. Broad application of standardized testing in mathematics education sometimes reduces the measure of success to a single score or change in scores over time. Modern educational approaches and programs at all levels increasingly emphasize problem solving, which the National Council of Teachers of Mathematics asserts is not well measured by standardized tests, since problem solving reaches beyond simply remembering some encapsulated set of concepts, formulas, and skills to include broader applications, novel situations, and mathematical thinking and reasoning. There are calls for innovative assessment alongside changes in educational practice in order to attempt to capture what it means to know, do, and be successful in mathematics at home, school, and work. Among professional mathematicians, measures of success vary as well, with ongoing debate about various aspects of teaching and scholarship, including how and whether to assess measures like the number of publications, the number of citations to an author, the quality of a journal, or letters from peers, and the role of other measures like student evaluations. Educational researcher Christopher Jett, who examined mathematics success among African-American men, uniquely defined mathematics success as, “being able to use mathematics as an analytical tool to educate, stimulate, and liberate the (my) people.”

Failure and Anxiety

Albert Einstein was once quoted as saying, “Do not worry about your difficulties in mathematics. I can assure you mine still are greater.” This assertion seems contrary to many people’s belief that success in mathematics is binary: people are successful or not, with no middle ground. Popular culture portrayals of mathematicians as geniuses often inadvertently support the mistaken belief that one must be gifted to succeed in mathematics. Further, while mathematicians are portrayed as wizards, mathematics itself is often shown as a sort of mysticism—a secret and arcane knowledge accessible only by a select few. In reality, mathematics encompasses a diversity of fields and professional mathematicians have varying sets of competencies, personalities, and working styles. Likewise, students at any level may have command of a wide variety of skills and concepts, and while those concepts are related and may build on one another, competence is not uniform. A student who struggles all through algebra may still be successful in geometry. A student who labors over constructing a proof may have a flair for data analysis and statistics.

Mathematics is inherently cumulative in nature. The feeling of failure at mathematics, especially given the common binary view, can seemingly be caused by a small problem that actually immediately impacts only one small area. This partial temporary failure can result in a long-lasting loss of confidence. Mathematics anxiety is an increasingly recognized phenomenon that interferes with students’ ability to learn mathematics and perform at the best of their abilities, regardless of their actual skill. Many people who are perfectly capable of learning and using mathematics feel anxiety about it and will avoid using mathematics whenever they have the option. Over time, this can lead to degradation of their abilities as they fall out of practice, which cyclically reinforces the anxiety. In addition to avoidance, mathematics anxiety can sometimes negatively affect working memory. As the anxiety grows, the student has more trouble keeping track of tasks, leading to poor performance and yet again reinforcing the anxiety. Some believe that mathematics anxiety is caused in part by poor performance on mathematics achievement tests and in part by early difficulty in mathematical skill development. The anxiety remains even after actual performance has improved and may be related to a belief that the earlier difficulties reflect some inherent character trait rather than a situational difficulty. People who later in life describe themselves as “terrible” at mathematics but who display education-appropriate competence in mathematics may not remember the original event that inculcated in them this belief that they are poor performers. Beyond performance, some cite teacher or classroom practices as contributing to mathematics anxiety. Like other mathematicians, mathematics teachers do not possess equal skills and levels of comfort in all areas of mathematics. This may be especially true of elementary school teachers who must teach a wide array of subjects on a daily basis. Classroom practices such as emphasizing the “right” answer, which also frequently occurs on standardized tests, can increase anxiety because some students attach great significance to being “wrong.” Other students may feel anxiety over being asked to “show their work,” because they are less confident in their mathematical thinking than in their ability to produce a correct answer.

The Mathematics Anxiety Rating Scale (MARS) was developed by psychologists in the early 1970s and exists in several versions, including foreign language adaptations. Researchers using this scale and other measures have identified other situations more likely to trigger anxiety. For example, tests where problems become progressively more difficult appear to trigger mathematics anxiety more often than tests in which the distribution of problems by difficulty is more random, which is true even when all the problems on the test are well within the skill level of the test-taker. Timed tests and the possibility for public embarrassment, such as working at a board in the front of the class, are also factors that can induce anxiety. Many studies suggest an association between mathematical anxiety and gender; female students are more often anxious about mathematics, perhaps because they have embraced the belief that women are not as good at mathematics as men and thus have difficulty building self-confidence in their abilities. These stereotype effects can also extend to other underrepresented groups.

Stereotype Threat

One widely studied phenomenon regarding success on standardized tests is known as the “stereotype threat” in which the stereotyping of groups in society affects an individual. Researchers found that proficient white males performed more poorly on a difficult mathematics test when researchers induced the threat of superior performance by Asians as compared to a control group. The impact of stereotype threat for many groups of students has been researched under a wide variety of conditions. For instance, Asian women performed more poorly on mathematical tests in which they were cued as women, while they performed better when cued as Asians as compared to control groups. Some researchers theorize that students must contend with a subconscious whisper of inferiority when their abilities are most taxed. Whether they consciously or unconsciously accept the stereotype or not, they may still work harder in order to avoid confirming it, because failure has a more devastating meaning. The extra burden may be enough to impact performance. The stereotype cues may be subtle, like self-identification of gender, race, or culture before an exam. Researchers have also found that removing the cues can positively impact test performance. For example, in a 2009 meta-analysis of 18,976 students from five countries who were matched by researchers using past performance, stereotyped students performed better under conditions that reduced the threat.

Research and Strategies

In the 1970s, mathematics education researchers Elizabeth Fennema and Julia Sherman developed the Fennema–Sherman mathematics attitude scales to examine eight components considered critical for success in mathematics: attitude toward success in mathematics, mathematics as a male versus female domain, parent support, teacher support, confidence in learning mathematics, mathematics anxiety, motivation for challenge in mathematics, and mathematics usefulness. Their work has been cited among the most quoted social science and educational research studies of the latter twentieth century, and many versions of their scales exist.

The cumulative body of research suggests several strategies that may help students succeed in mathematics, though one point of general agreement seems to be that the key to mathematics success is active participation, active study, and engaging the material. For example, younger students can be encouraged to ask questions in and out of class and can be given mathematical exercises that are interactive rather than requiring them to only passively listen to explanations of the material. Hands-on activities with even simple objects like buttons, dried beans, or animal counters can help children develop number, counting, and arithmetic skills. More sophisticated tools, like tangrams and algebra tiles, develop geometric concepts and thinking about functions. This method has come to include computer-based virtual manipulatives. Asking questions and engaging in hands-on learning is also valuable in the later grades and college. With regard to attitudes, educators frequently encourage students to recognize that the act of learning and doing mathematics is likely to be different than other school subjects, particularly with regard to its cumulative nature and fact that working with a variety of mathematics problems is usually the only way to learn mathematics. Some instructors include explicit problem-solving and test-taking strategies in their instruction in addition to concepts. Students may also benefit from instruction in methods of note-taking, reviewing, and reading mathematics textbooks that encourage them to think reflectively about mathematics content rather than simply summarizing. At the same time, teachers may use a variety of presentation and engagement methods, including using real-world problems, considering different learning styles, being aware of anxiety and stereotypes that may affect students, and engaging parents in an ongoing dialogue about mathematics education to gain support and make them partners in their children’s success.

Organizations

Beyond the classroom, clubs, professional organizations, and scholarship programs have been shown to contribute to success and some are particularly targeted toward groups that may be more at-risk, such as women and minorities. The Meyerhoff Scholars Program is a notable example of such a program. It was initially created in 1988 to target African-American men, though admission is no longer restricted by gender or ethnicity. A 2010 statistic noted that program participants were 5.3 times more likely to be attending or have graduated from a STEM Ph.D. or M.D./Ph.D. program than others who were invited to join but declined and attended another school. The program’s success is attributed in large part to its emphasis on mentorship, particularly since women and minorities interested in mathematics may never have met a woman or a minority mathematician or have not been exposed to research and challenges to excel rather than to imply succeed. The Hypatia Scholarship program for women at the University of South Australia, founded in 1997 and named for woman mathematician Hypatia of Alexandria, awards not only financial support but also provides women with shared office space and computer resources in close proximity to faculty to encourage interaction and build confidence. It funds summer employment to encourage the participants to use their mathematical training in industry or academia. Feedback from students indicated that the women valued the social network more than the financial support, saying it motivated them and helped reduce anxiety. Other organizations offer financial scholarships and some opportunities for networking, such as the American Statistical Association’s Gertrude Cox Scholarship. Mathematics clubs and honor societies, like Pi Mu Epsilon, provide social and academic opportunities for students with mathematics interests, and researchers have found some evidence that participation is associated with increases in retention, positive attitudes about mathematics, and higher grade point averages.

The successful Upward Bound program targets disabled, low-income, homeless, and foster care youth, as well as those who would be first-generation college students to encourage comprehensive success in secondary and higher education. It provides instruction in mathematics, laboratory sciences, composition, literature, and foreign languages, along with support like counseling, academic tutoring, and assistance with college admission and financial aid.

Technology

The twenty-first century saw the first generations of digital natives undertake math learning. Technological innovations to teach math were a natural progression for schoolchildren of this era. It fell to adult educators, schooled in traditional manners of instruction, to bridge the application of technology to these groups. Not only was math instruction possible through digital means, in many cases teaching could be enhanced. Learning technologies allowed for more visual representation of mathematic concepts for areas such as modeling and graphs. These technologies allowed for greater engagement and focus on the part of students as they included sensory applications such as audio, graphics, and text. These software tools also allowed for greater collaboration and group learning. These technologies could also assist parents, who could find the manner they were instructed in math was significantly different from how their children were learning. 

Bibliography

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