Thanks! It's a year now........

I want to thank you for your patronage. I love seeing you and your friends on a regular basis and guiding them to become readers, lifelong learners and blog lovers!

Over the past year,  I have added many new articles to accommodate different aspects of maths learning, 

e-learning and education as a whole increased interest. I am proud to write in a community where the learning is central to people’s lives. There is always an effort from my side to make these articles more interesting and to be as fair as possible. 

“The hunger and thirst for knowledge, the keen delight in the chase, the good humored willingness to admit that the scent was false, the eager desire to get on with the work, the cheerful resolution to go back and begin again, the broad good sense, the unaffected modesty, the imperturbable temper, the gratitude for any little help that was given - all these will remain in my memory though I cannot paint them for others” Quotes Frederic William Maitland .

This is what ‘MATHAMIT’ looks forward to.‘MATHAMIT’ aims at pruning and grooming minds to bloom into flowering personalities and spread fragrance wherever they go.Things do change fast. We are in the second year of ‘MATHAMIT’. From being ‘mentored’ to becoming a mentor themselves. The fact that many of our readers have attained greater heights in their success has held our heads high. The research articles on various topics will give you a glimpse of the various activities that our minds have indulged in, which in turn has helped them to unveil their talents.

Now, I have come up with the second year of ‘MATHAMIT’ and every year we will choose a theme for the cover page, which is either related to recent issues in the field of science and technology or  mathematics or something that is the need of the hour. I will try to focus upon the need of merging our creative scientific ideas, thoughts and research skills with the business forums, which would thereby strengthen the economic status of our country. Knowing that the development of society as a whole solely lies in the hands of the future generation, it is imperative to train the youngsters for the required skill sets. I hope that we attain success in achieving what we have always aimed for i.e. nurturing and bringing out research from Lab to Life and now we could take a step further in the form of service to humanity.

Last but not the least, we take this opportunity to thank our readers and Google+ community, for their cooperation and guidance in releasing this year long journey of ‘MATHAMIT’. My profound thanks to all the students, teaching and the non-teaching people and the ‘MATHAMIT’ committee for willingly coming forward and helping in making the dream of this blog a reality. 

Wishing you a scintillating and happy reading.

Trends from recent thinking on effective learning

• E-learning should not be a mass of online material for individual access without guidance on how to learn from it effectively.
• Courses involving e-learning need to be planned for, and grounded in an understanding of the roles of teachers and learners, of learning, and of how students learn.
• The role of prior knowledge in learning is critical and must be taken into account in e-learning design. Ongoing formative assessment is part of this.
• The brain is a dynamic organ shaped by experiences. Conceptual links are reorganised through active engagement with information in various contexts.
• Learning is an active process. It is the result of carrying out particular activities in a scaffolded environment where one activity provides the step up to the next level of development.
• Learning needs to be meaningful to learners and they should be supported in developing the skill of relating new material to what is meaningful to them.
• Learners should be enabled to become adaptable and flexible experts in their own current and future learning.
• Learning takes time and effective learning practices enable learners to work with materials from a variety of perspectives while they become fully conversant with it.
• Weaving e-learning into existing teaching and learning practices adds more ways for students to be actively and deeply involved with subject area materials.

Most of the evidence related to e-learning is available in case studies of individual courses. There is more evidence related to university than to polytechnic courses. Case studies cover a wide range of subjects, including both skill-based and conceptual. We have focused on blended learning, where e-learning complements some class-based interaction, since that appears more popular with students and teachers, is easier to introduce, and appears to offer some advantages over fully online learning.

Studies comparing student outcomes for e-learning and conventional courses show comparable results in terms of achievement, with indications that student outcomes can be broader if elearning is used well. Student retention shows mixed patterns, and is dependent on a number of factors. Students value the flexibility of e-learning, but it is different from classroom learning, and can demand more.

The main messages from a survey of the available evidence are consistent with the messages from recent thinking on effective teaching and learning.They are:

1.E-learning can improve understanding and encourage deeper learning, if there is careful course design and choice of technology in relation to learning objectives that aim to encourage deeper learning.

2.It can free up face-to-face teacher:student time for discussion, rather than using it to cover information or provide skill practice, depending on the use made of technology.

3.It can improve and sustain motivation by offering interesting tasks and material. .. Students need formative feedback throughout the course. This requires careful structuring and the development of channels and projects encouraging student-student interaction as well as strategic use of teacher time to provide feedback; online tasks, tests, and quizzes are also useful in giving students a picture of their learning progress.

4.Student-student interaction can also be enhanced through careful structuring, creating additional support for learning, and even a “learning community”. Participation in discussion groups, etc. is supported by linking it to assessment or tasks and measures that “matter”.

5.It is important that students have a clear picture of the learning objectives for the course, and that assessment methods reflect and support the learning objectives.

6.Students need very clear course information, and if accessing the course externally, initial face-to-face sessions are valuable to ensure understanding and skills needed to access the web material, to lay the ground for student-student interaction, particularly if some collaborative work is to be done, and for teacher-student web interaction.

7.While asynchronous formats offer students more flexibility, they may also spend more time on a course using the web.

8.The technology used has to be reliable, simple, and easily accessed by students.

9.E-learning is easier for students who are self-managing, which may mean it is easier for mature students.

10.Barriers to making the most of e-learning can arise from students’ familiarity with classroom based methods and assumptions that this is how learning occurs, and from a greater interest in superficial learning to pass a course, than in increasing understanding.

However, increasingly attention is focusing on the creation of tasks, material, and feedback mechanisms and channels that will increase motivation and hence encourage self management, and on course structures, processes, and requirements that provide some additional frameworks for those who need them.

Proportionality,Computataion and Equality - Prerequisite Knowledge for the Learning of Algebra

Proportionality

Fractions commonly appear in beginning algebra in the form of proportions, which provide wonderful examples of naturally occurring linear functions. Because of this, Post, Behr, and Lesh feel that proportionality has the ability to connect common numerical experiences and patterns, with which students are familiar, to more abstract relationships in algebra. Proportions can also be used to introduce students to algebraic representation and variable manipulation in a way that parallels their knowledge of arithmetic.

In fact, proportions are useful in a multitude of algebraic processes, including problem solving, graphing, translating and using tables, along with other modes of algebraic representation. Due to its vast utility, Post et al. (1988) consider proportionality to be an important contributor to students’ development of pre-algebraic understanding. Similarly, Readiness Indicator number 4 focuses on the importance of ratios, rates and proportions in the  study of algebra (Bottoms, 2003).

Proportional reasoning requires a solid understanding of several rational number concepts including order and equivalence, the relationship between a unit and its parts, the meaning and interpretation of ratio, and various division issues (Post et al., 1988). Therefore, these concepts could be considered, along with proportional reasoning, prerequisite knowledge for the learning of algebra.

Computations

In addition to understanding the properties of numbers, algebra students need to understand the rules behind numerical computations, as stated in Readiness Indicator number. Computational errors cause many mistakes for algebra students, especially when simplifying algebraic expressions. Booth (1984) claims elementary algebra students’ difficulties are caused by confusion surrounding computational ideas, including inverse operations, associativity, commutativity, distributivity, and the order of operations convention. These misconstrued ideas are among basic number rules essential for algebraic manipulation and equation solving (Watson, 1990). The misuse of the order of operations also surfaced within an example of an error made by collegiate algebra students that Pinchback (1991) categorized as result of lack of prerequisite knowledge. Other errors deemed prerequisite occurred while adding expressions with radical terms and within the structure of long division (while dividing a polynomial by a binomial) (Pinchback, 1991).

Mentioned by Rotman (1991) as a prerequisite arithmetic skill, the order of operations is also included in Readiness Indicator number 10 (Bottoms, 2003). In fact, this convention has been found to be commonly misunderstood among algebra students in junior high, high school, and even college (Kieran, 1979, 1988; Pinchback, 1991). The order of operations relies on bracket usage; however, algebra requires students to have a more flexible understanding of brackets than in arithmetic. Therefore, according to Linchevski (1995), prealgebra should be used as a time to expand students’ conceptions of brackets.

Kieran (1979) investigated reasons accounting for the common misconception of the order of operations and alarmingly concluded that students’ issues stem from a much deeper problem than forgetting or not learning the material properly in class. The junior high school students, with which Kieran worked, did not see a need for the rules presented within the order of operations. Kieran argues that students must develop an intuitive need for bracket application within the order of operations, before they can learn the surrounding rules. This could be accomplished by having students work with arithmetic identities, instead of open-ended expressions.

Although teachers see ambiguity in solving an open-ended string of arithmetic operations, such as 2 + 4 x 5, students do not. Students tend to solve expressions based on how the items are listed, in a left-to-right fashion, consistent with their cultural tradition of reading and writing English. Therefore, the rules underlying operation order actually contradict students’ natural way of thinking. However, Kieran suggests that if an equation such as 3 x 5 =15 were replaced by 3 x 3+ 2 =15, students would realize that bracket usage is necessary to keep the equation balanced (Kieran, 1979).

Equality

Kieran’s (1979) theory assumes that students have a solid understanding of equations and the notion of equality. Readiness Indicator number 10 suggests that students are familiar with the properties of equality before entering Algebra I (Bottoms, 2003). However, equality is commonly misunderstood by beginning algebra students (Falkner, Levi, & Carpenter, 1999; Herscovics & Kieran, 1980; Kieran, 1981, 1989). Beginning algebra students tend to see the equal sign as a procedural marking that tells them “to do something,” or as a symbol that separates a problem from its answer, rather than a symbol of equivalence (Behr, Erlwanger, & Nichols, 1976, 1980). Even college calculus students have misconceptions about the true meaning of the equal sign (Clement, Narode, & Rosnick, 1981).

Kieran (1981) reviewed research addressing how students interpret the equal sign and uncovered that students, at all levels of education, lack awareness of its equivalence role. Students in high school and college tend to be more accepting of the equal sign’s symbolism for equivalence, however they still described the sign in terms of an operator symbol, with an operation on the left side and a result on the right. Carpenter, Levi, and Farnsworth (2000) further support Kieran’s conclusions by noting that elementary students believe the number immediately to the right of an equal sign needs be the answer to the calculation on the left hand side. For example, students filled in the number sentence 8 + 4 =  __ +5 with 12 or 17.

According to Carpenter et al. (2000), correct interpretation of the equal sign is essential to the learning of algebra, because algebraic reasoning is based on students’ ability to fully understand equality and appropriately use the equal sign for expressing generalizations. For example, the ability to manipulate and solve equations requires students to understand that the two sides of an equation are equivalent expressions and that every equation can be replaced by an equivalent equation (Kieran, 1981). However, Steinberg, Sleeman, and Ktorza (1990) showed that eighth- and ninth-grade algebra students have a weak understanding of equivalent equations.

Kieran (1981) believes that in order to construct meaning while learning algebra, the notion of the equal sign needs to be expanded while working with arithmetic equalities prior to the introduction of algebra. If this notion were built from students’ arithmetic knowledge, the students could acquire an intuitive understanding of the meaning of an equation and gradually transform their understanding into that required for algebra. Similarly, Booth (1986) notes that in arithmetic the equal sign should not be read as “makes”, as in “2 plus 3 makes 5” (Booth, 1986), but instead as “2 plus 3 is equivalent to 5”, addressing set cardinality.

Mathematics a Gatekeeper: A Historical Perspective

Discourse regarding the “gatekeeper” concept in mathematics can be traced back over 2300 years ago to Plato’s (trans. 1996) dialogue, The Republic. In the fictitious dialogue between Socrates and Glaucon regarding education, Plato argued that mathematics was “virtually the first thing everyone has to learn…common to all arts, science, and forms of thought” . Although Plato believed that all students needed to learn arithmetic—”the trivial business of being able to identify one, two, and three” —he reserved advanced mathematics for those that would serve as philosopher guardians2 of the city. He wrote: We shall persuade those who are to perform high functions in the city to undertake calculation, but not as amateurs. They should persist in their studies until they reach the level of pure thought, where they will be able to contemplate the very nature of number. The objects of study ought not to be buying and selling, as if they were preparing to be merchants or brokers. Instead, it should serve the purposes of war and lead the soul away from the world of appearances toward essence and reality. 

Although Plato believed that mathematics was of value for all people in everyday transactions, the study of mathematics that would lead some men from “Hades to the halls of the gods”  should be reserved for those that were “naturally skilled in calculation” ; hence, the birth of mathematics as the privileged discipline or gatekeeper. This view of mathematics as a gatekeeper has persisted through time and manifested itself in early research in the field of mathematics education in the United States. In Stanic’s review of mathematics education of the late 19th and early 20th centuries, he identified the 1890s as establishing “mathematics education as a separate and distinct professional area” , and the 1930s as developing the “crisis” in mathematics education. This crisis—a crisis for mathematics educators—was the projected extinction of mathematics as a required subject in the secondary school curriculum. Drawing on the work of Kliebard , Stanic provided a summary of curriculum interest groups that influenced the position of mathematics in the school curriculum:

 (a) the humanists, who emphasized the traditional disciplines of study found in Western philosophy; 

(b) the developmentalists, who emphasized the “natural” development of the child; 

(c) the social efficiency educators, who emphasized a “scientific” approach that led to the natural development of social stratification; 

and (d) the social meliorists, who emphasized education as a means of working toward social justice.

Stanic noted that mathematics educators, in general, sided with the humanists, claiming: “mathematics should be an important part of the school curriculum” . He also argued that the development of the National Council of Teachers of Mathematics (NCTM) in 1920 was partly in response to the debate that surrounded the position of mathematics within the school curriculum. The founders of the Council wrote: Mathematics courses have been assailed on every hand. So-called educational reformers have tinkered with the courses, and they, not knowing the subject and its values, in many cases have thrown out mathematics altogether or made it entirely elective. …To help remedy the existing situation the National Council of Teachers of Mathematics was organized. 

The question of who should be taught mathematics initially appeared in the debates of the 1920s and centered on “ascertaining who was prepared for the study of algebra” . These debates led to an increase in grouping students according to their presumed mathematics ability. This “ability” grouping often resulted in excluding female students, poor students, and students of color from the opportunity to enroll in advanced mathematics courses . Sixty years after the beginning of the debates, the recognition of this unjust exclusion from advanced mathematics courses spurred the NCTM to publish the Curriculum and Evaluation Standards for School Mathematics (Standards, 1989) that included statements similar to the following:

The social injustices of past schooling practices can no longer be tolerated. Current statistics indicate that those who study advanced mathematics are most often white males. …Creating a just society in which women and various ethnic groups enjoy equal opportunities and equitable treatment is no longer an issue. Mathematics has become a critical filter for employment and full participation in our society. We cannot afford to have the majority of our population mathematically illiterate: Equity has become an economic necessity. 

In the Standards the NCTM contrasted societal needs of the industrial age with those of the information age, concluding that the educational goals of the industrial age no longer met the needs of the information age. They characterized the information age as a dramatic shift in the use of technology which had “changed the nature of the physical, life, and social sciences; business; industry; and government” . The Council contended, “The impact of this technological shift is no longer an intellectual abstraction. It has become an economic reality” . The NCTM believed this shift demanded new societal goals for mathematics education: 

(a) mathematically literate workers, 

(b) lifelong learning,

(c) opportunity for all, 

and (d) an informed electorate.

They argued, “Implicit in these goals is a school system organized to serve as an important resource for all citizens throughout their lives” . These goals required those responsible for mathematics education to strip mathematics from its traditional notions of exclusion and basic computation and develop it into a dynamic form of an inclusive literacy, particularly given that mathematics had become a critical filter for full employment and participation within a democratic society. Countless other education scholars have made similar arguments as they recognize the need for all students to be provided the opportunity to enroll in advanced mathematics courses, arguing that a dynamic mathematics literacy is a gatekeeper for economic access, full citizenship, and higher education. In the paragraphs that follow, I highlight quantitative and qualitative studies that substantiate mathematics as a gatekeeper.

The claims that mathematics is a “critical filter” or gatekeeper to economic access, full citizenship, and higher education.In the today context, mastering mathematics has become more important than ever. Students with a strong grasp of mathematics have an advantage in academics and in the job market. The 8th grade is a critical point in mathematics education. Achievement at that stage clears the way for students to take rigorous high school mathematics and science courses—keys to college entrance and success in the labor force.

Students who take rigorous mathematics and science courses are much more likely to go to college than those who do not. Algebra is the “gateway” to advanced mathematics and science in high school, yet most students do not take it in middle school. Taking rigorous mathematics and science courses in high school appears to be especially important for low-income students. Despite the importance of low-income students taking rigorous mathematics and science courses, these students are less likely to take them. The report, based on statistical analyses, explicitly stated that algebra was the “gateway” or gatekeeper to advanced (i.e., rigorous) mathematics courses and that advanced mathematics provided an advantage in academics and in the job market—the same argument provided by the NCTM and education scholars. The statistical analyses in the report entitled, Do Gatekeeper Courses Expand Educational Options? presented the following findings:

Students who enrolled in algebra as eighth-graders were more likely to reach advanced math courses (e.g., algebra 3, trigonometry, or calculus, etc.) in high school than students who did not enroll in algebra as eighth-graders. Students who enrolled in algebra as eighth-graders, and completed an advanced math course during

high school, were more likely to apply to a fouryear college than those eighth-grade students who did not enroll in algebra as eighth-graders, but who also completed an advanced math course during high school. The summary concluded that not all students who took advanced mathematics courses in high school enrolled in a four-year postsecondary school, although they were more likely to do so—again confirming mathematics as a gatekeeper.

The concept of mathematics as providing the key for passing through the gates to economic access, full citizenship, and higher education is located in the core of Western philosophy. The school mathematics evolved from a discipline in “crisis” into one that would provide the means of “sorting” students. As student enrollment in public schools increased, the opportunity to enroll in advanced mathematics courses (the key) was limited because some students were characterized as “incapable.” Female students, poor students, and students of color were offered a limited access to quality advanced mathematics education. This limited access was a motivating factor behind the Standards, and the subsequent NCTM documents. NCTM and education scholars’ argument that mathematics had and continues to have a gatekeeping status has been confirmed both quantitatively and qualitatively. Given this status, I pose two questions:

(a) Why does our education system not provide all students access to a quality, advanced (mathematics) education that would empower them with economic access and full citizenship? 

and (b) How can we as mathematics educators transform the status quo in the mathematics classroom?

To fully engage in the first question demands a deconstruction of the concepts of democratic public schooling and an analysis of the morals and ethics of capitalism. To provide such a deconstruction and analysis is beyond the scope of this article. Nonetheless, I believe that Bowles’s argument provides a comprehensive, yet condensed response to the question of why our education remains unequal without oversimplifying the complexities of the question. Through a historical analysis of schooling he revealed four components of our education: 

(a) schools evolved not in pursuit of equality, but in response to the developing needs of capitalism (e.g., a skilled and educated work force); 

(b) as the importance of a skilled and educated work force grew within capitalism so did the importance of

maintaining educational inequality in order to reproduce the class structure;

(c) from the 1920s to 1970s the class structure in schools showed no signs of diminishment (the same argument can be made for the 1970s to 2000s); 

and (d) the inequality in education had “its root in the very class structures which it serves to legitimize and reproduce” . 

He concluded by writing: “Inequalities in education are thus seen as part of the web of capitalist society, and likely to persist as long as capitalism survives” . Although Bowles’s statements imply that only the overthrow of capitalism will emancipate education from its inequalities, I believe that developing mathematics classrooms that are empowering to all students might contribute to educational experiences that are more equitable and just. This development may also assist in the deconstruction of capitalism so that it might be reconstructed to be more equitable and just. The following discussion presents three theoretical perspectives that I have identified as empowering students. These perspectives aim to assist in more equitable and just educative experiences for all students: the situated perspective, the culturally relevant perspective, and the critical perspective. I believe these perspectives provide a plausible answer to the second question asked above: How do we as mathematics educators transform the status quo in the mathematics classroom?

Numbers - Prerequisite Knowledge for the Learning of Algebra

Researchers have focused on the value of understanding numbers prior to algebra introduction (Booth, 1984, 1986; Gallardo, 2002; Kieran, 1988; Rotman, 1991; Wu, 2001). According to Watson (1990), a better understanding of number basics would give students a stronger ability to handle algebraic operations and manipulations. What types of numbers need to be studied prior to learning algebra? The Readiness Indicator for algebra focuses on students’ ability to read, write, compare, order, and represent a variety of numbers, including integers, fractions, decimals, percents, and numbers in scientific notation and exponential form (Bottoms, 2003). Some of these forms have also been mentioned in research addressing prerequisite number knowledge for the learning of algebra.

Gallardo (2002) focused on the fact that the transition from arithmetic to algebra is where students are first presented with problems and equations that have negative numbers as coefficients, constants and/or solutions. Therefore, she believes that students must have a solid understanding of integers in order to comprehend algebra. Lack of this understanding will affect students’ abilities to solve algebraic word problems and equations. However, Gallardo’s research showed that 12- and 13-year-old students do not usually understand negative numbers to the fullest extent.

Misconceptions of negative numbers were identified in earlier research done by Gallardo and Rojano (1988; cited in Gallardo, 2002) while investigating how 12- and 13-year-old beginning algebra students acquire arithmetic and algebraic language. One major area of difficulty involved the nature of numbers. Specifically, students had troubles conceptualizing and operating with negative numbers in the context of prealgebra and algebra. Therefore, Gallardo (2002) argues that while students are learning the language of algebra, it is imperative that they understand how the numerical domain can be extended from the natural numbers to the integers.

Kieran (1988) also found misunderstandings regarding integers to affect the success of algebra students in grades 8-11. During interviews with Kieran, students who had taken at least one year of algebra made computational equation-solving errors involving the misuse of positive and negative numbers. Furthermore, when these students were required to use division as an inverse operation, they tended to divide the larger number by the smaller, regardless of the division that was actually required within the operation. Therefore, students’ errors extended into the division of integers, which implies a lack of understanding of fractions.

An opinion article regarding how to prepare students for algebra further supports the inclusion of fractions as prerequisite knowledge for the learning of algebra. According to Wu (2001), fraction understanding is vital to a student’s transformation from computing arithmetic calculations to comprehending algebra. Wu believes that K-12 teachers are not currently teaching fractions at a deep enough level to prepare students for algebra. In fact, she believes that the study of fractions could and should be used as a way of preparing students for studying generality and abstraction in algebra.

Fractions were also stressed when Rotman (1991) chose number knowledge as a prerequisite arithmetic skill for learning algebra. During a research project that mounted evidence against the assumption that arithmetic knowledge is prerequisite for successful algebra learning, Rotman constructed a list of arithmetic skills he considers as prerequisite to algebra. Based on his experiences as a teacher, Rotman argues that algebra students need to understand the structure behind solving applications, the meaning of symbols used in arithmetic, the order of operations and basic properties of numbers (especially fractions). Of course, in order to operate with fractions students are required to know basic number theory ideas including least common multiple. Therefore, the necessity of fraction knowledge partially supports Readiness Indicator, which states that students need to be able to determine the greatest common factor, least common multiple, and prime factorization of numbers (Bottoms, 2003).

What is algebra?

Colin Maclaurin in his 1748 algebra text defined it like this:

“Algebra is a general method of computation by certain signs and symbols which have been contrived for this purpose, and found convenient. It is called a universal arithmetic, and proceeds by operations and rules similar to those in common arithmetic, founded upon the same principles.” (Katz & Barton, 2007, p. 185).

Leonhard Euler, in his own algebra text in 1770, defined algebra as:

“the science which teaches how to determine unknowns’ quantities by means of those that are known.” (ibid., p. 185).

Katz and Barton (2007) categorize the historical development of algebra in four stages: the rhetorical stage, the syncopated stage, the symbolic stage and the purely abstract stage, but they also name another four conceptual stages:

“the geometric stage, where most of the concepts of algebra are geometric; the static equation-solving stage, where the goal is to find numbers satisfying certain relationship; the dynamic function stage, where motion seems to be an underlying idea; and finally the abstract stage, where structure is the goal” (p. 186).

As an old science, algebra has a complicated historical background according to Katz and Barton (2007). Algebraic procedures have developed slowly. There are different opinions about where the evolution of the term “algebra” started. It is commonly believed that algebra first appeared among the Egyptians, the Babylonians, the Greeks or the Arabs. The geometrical influence on algebraic reasoning was strong in ancient Greece. However, the word algebra originated in Baghdad, where the Arabic scientist al-Khwarizmi (A.D. 780-850) published a short book about calculating with the help of al-jabr (restoration) and al-muqabala (reduction). Today’s algebra has its root in Arabic algebra. Western mathematics tended to turn algebraic operations into symbols and later developed abstract algebra. The process of algebra development was slow and the whole history lasted 4000 years (Katz & Barton, 2007).

Mathematics Self-Efficacy

Mathematics self-efficacy is defined to be the confidence a person has in his/her ability to learn and do mathematics.

Self-efficacy theory was originally reported by Bandura (1977) and referred to a person’s beliefs concerning his/her ability to successfully perform a given task or behavior. The factors that influence this measure are

• Performance accomplishments,

• Vicarious learning or modeling,

• Verbal persuasion, and

• Emotional arousal or anxiety.

Lent (1996) stated that these four sources interact dynamically to affect self-efficacy judgments. Bandura (1977, 1982) stated that performance accomplishments were hypothesized to be the most powerful source of self-efficacy, and that self-efficacy expectations could be learned and/or altered. In fact it was shown that task performance significantly and strongly influenced ratings of task selfefficacy, task interest, and global ability ratings. Success experiences produced elevations in self-efficacy, task interest, and ability ratings over time, while failure experiences depressed these same ratings (Campbell & Hackett, 1986).

Utilizing this cognitive theory, Betz and Hackett (1983) developed the Mathematics Self-efficacy Scale (MSES). The MSES is currently utilized both for research and counseling intervention and is intended to measure a person’s perception of his/her ability to perform various mathematics related tasks. There have been several iterations of the original MSES and the present scale contains a 34-item questionnaire which yields three scores in the following areas:

1. perceptions of ability to utilize mathematics in everyday tasks and activities,

2. perceptions of ability to complete mathematics and science related college

courses with a final grade of “A” or “B”, and

3. overall mathematics self-efficacy.

Mathematics self-efficacy is an important factor for the prediction of success of students in mathematics classes (Lent et al., 1993; Matsui, Matsui, & Ornish, 1990; Pedro, et al., 1981; Sherman & Fennema, 1977). College counselors reported that students who believe that they cannot succeed no matter what measure they take will avoid special tutoring sessions or avoid arranging special one-on-one help sessions with their instructors. They will not ask questions for clarification in class nor seek help from instructors during office hours. A student who does not believe that anything he/she does will affect the grade in a positive manner will not take advantage of outside activities specifically designed to help improve understanding of the mathematics. With regard to this avoidance, Pajares (1995) stated that

Self-efficacy beliefs … strongly influence the choices people make, the effort they expend, the strength of their perseverance in the face of adversity, and the degree of anxiety they experience. In part, these self-perceptions can be better predictors of behavior than actual capability because such self-beliefs are instrumental in determining what individuals do with the knowledge and skills they have.

Additionally, mathematics self-efficacy has been reported to significantly contribute to career choices (Post-Kammer & Smith, 1986). According to Betz (1978), mathematics anxiety may be a critical factor in a student’s educational and vocational decision and, in addition, may influence the student’s achievement of his/her educational and career goals. Self-efficacy theory (Bandura, 1977, 1982; Hackett & Betz, 1981) and research investigating the role of mathematics self-efficacy in the career choice process (Betz & Hackett, 1983; Hackett & Betz, 1984) provide support for the view that mathematics-related self-efficacy, as influenced by gender, socialization, and math level and background, is more strongly predictive of math-related major and career choices than ability, math background, or gender alone or in combination…

In fact…, at least with college-aged women and men, self-efficacy expectations with regard to occupations and career-related domains are much more important than measured abilities (Hackett, 1985). The relationship of mathematics anxiety to performance and career choice is undeniable (Fennema, 1980). In studies completed in 1978, Betz (1978) found the following:

• students avoided college majors and careers if an extensive mathematics background was required

• older women reported higher levels of anxiety than did the younger women, and

• high school mathematics preparation strongly influenced a college student’s attitude about mathematics.

Tobias and Weissbrod (1980) found that students would stop studying mathematics to avoid having anxiety. Meece et al. (1990) found that “a large percentage of students stop taking mathematics courses by the 10th grade.” This action could severely limit the students’ educational and career aspirations. They saw this decision as affecting career options for women, and they reported that fewer women than men elect to take advanced mathematics courses in high school, which causes women to continue to be underrepresented in mathematics intensive career fields. With respect to race, Post et al. (1991) found that self-efficacy and confidence played a greater role in selection of career for African-American males than African-American females; however, the African- American males considered a broader choice of careers regardless of whether the field was mathematics or science related. Other noteworthy researchers pursuing an understanding concerning attitude towards mathematics, reported the following:

• Sternberg (1986) stated that there are many reasons other than intelligence which affect the level of a person’s performance,

• Dessart (1989) reported that some educators believe that attitude is more important than ability in predicting success,

• Seigel and Shaughnessy (1992) found that women were more insecure and anxious than men in calculus classes,

• Goleman (1994) stated that at best IQ accounts for only 20% of the life factors that determine life successes,

• Shaughnessy et al. (1994) found that significant predictors of success in calculus were “exacting in character, persevering, responsible, and conscientious” individuals, and

• Shaughnessy, et al. (1995) found that the personality factors of “privateness, intelligence, and emotional stability” contributed to the prediction of college calculus grades.

There is much to be learned from the cognitive theorists regarding the influence of self-efficacy on both high school students and college students with respect to level of mathematics courses taken and to career choices made. The researchers reviewed here emphatically urged educators of mathematics to take heed and understand that mathematics ability is secondary to the student’s perceptions (self-efficacy) of how well he/she can perform. Mathematics educators must understand that the student’s perception is his/her reality. This body of evidence should be enlightening to educators who have been unable to understand why students fail to meet their expectations with regard to asking questions for clarification and/or participating in extra help sessions. This would explain why the educators holding office hours rarely see the students who need the most help. In the students’ minds, nothing will help, and they are doomed to fail!