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!

TYPES OF LEARNERS

Learning Style

learning style as “the composite of characteristic cognitive, affective, and physiological factors that serve as relatively stable indicators of how a learner perceives, interacts with, and responds to the learning environment.” .

Active Learners

Active or kinesthetic learners retain and/or understand information by doing something active, such as discussing, explaining, or experiencing. These learners prefer group work to sitting through lectures.

Reflective Learners

Reflective learners prefer to think quietly about information. They tend to be loners and do not like group work.

Sensing Learners

Sensors like learning facts and performing well-defined tasks. They are good at memorizing facts and are practical and do not like courses of study that are not relevant.

Intuitive Learners

One who is an intuitive learner is content to discover possibilities and relationships and likes innovation. They are quite comfortable with abstractions and mathematical formulations.

Visual Learners

A person who is a visual learner remembers best when he/she can see a picture, diagram, flowchart, film, or demonstration.

Verbal Learners

Verbal learners obtain their information through spoken or written explanations.

Multiple Learning Styles of Students

Much research has been done on the rich and complex ways in which students learn at the pre-school and elementary instructional levels. Teachers now understand that most young children learn holistically by creating webs of association as they integrate new information and master new skills. Early childhood education some decades ago was often structured as if children were tiny adults. Classrooms were set up with 25 desks all in neat rows, and children were expected to sit still and simply absorb information, interacting with each other and their environment only in carefully controlled ways. Rote learning was heavily emphasized in some settings, and punishment for inattention or for “coloring outside the lines” in various ways was routine. Thankfully, early childhood educators these days have developed much more effective ways to support children’s learning.

• Good teachers create a “safe” learning space in which varying skill levels are respected, children are directed away from framing their learning in terms of competition, and each child is affirmed enough that s/he can dare to take the risks that learning always involves.

• Good teachers engage all of children’s senses, not just their cognitive abilities.

• Good teachers support children’s learning in social groups, sometimes using peer teaching relationships across different grade levels.

• Good teachers respect and affirm what children already know, while inviting them to expand and deepen their knowledge and learn new things.

• Good teachers invite children into decision-making and the development of judgment.

• Good teachers encourage imaginative play in classroom centers that provide a variety of related kinds of learning opportunities.

Many of the above insights apply as well to adult learners. Consider the following points too:

1. All learners have a need for freedom of cognitive range, affirmation, and a sense of safety so that they can take the risks associated with learning.

2. Applying new skills in concrete contexts is a crucial part of learning for children and adults alike.

3. Learners learn best when they experience respect for their judgment, latent knowledge, and emerging abilities.

4. Imaginative work and collaboration in groups can reinforce and deepen learning. 

Frustration, a heightened sense of risk, and confusion can result for adult learners when teachers do not engage the variety of learning temperaments and kinds of expertise that exist in the classroom. It is essential that teachers learn to recognize and, where possible, affirm the different kinds of learning strengths that adults bring to the classroom. Sometimes a single course cannot structure assignments that bring out all of students’ different learning strengths. For example, it is not normally feasible in a large lecture course to assign artistic projects, field trips, or community-based experiential learning. Still, teachers can do a lot to engage different learning temperaments in the classroom in small creative ways. Think about creative possibilities for your sections, in line with your own temperament, of course, but responsive to other types as well.

Below is a summary of research done on learning styles by David A. Kolb, creator of the Kolb Learning Style Inventory, who based his work on a number of “theories of thinking and creativity,” including the theories of Jean Piaget and J. P. Guilford. There are other kinds of assessment tools and research on learning styles as well. One resource potentially of interest is the Keirsey Temperament Sorter, based on the Myers-Briggs Personality Type Indicator. Kolb plots learners’ strengths and affinities on a grid with four components, below. He calls these the ‘four phases of the learning cycle.’

Concrete Experience (Experiencing)

learning from specific experiences

relating to people

being sensitive to feelings and people

Reflective Observation (Reflecting)

carefully observing before making judgments

viewing issues from different perspectives

looking for the meaning of things

Abstract Conceptualization (Thinking)

logically analyzing ideas

planning systematically

acting on an intellectual understanding of a situation

Active Experimentation (Doing)

showing ability to get things done

taking risks

influencing people and events through action

Different learners privilege those four areas of learning differently, depending on temperament and contextual variables. There is no one “right” or “best” way to learn, although there are learning styles that are better suited to one learning task or another. Well-balanced learners will learn in many of the ways listed above, drawing on one or another aspect as needed. Other students may have significant strengths in one area but find learning daunting or uninteresting in one of the other areas, or they may wish to become stronger in a new skill but be uncertain how to proceed. Facilitating skills for learning itself is part of your job! The Kolb Learning Style Types are based on various combinations of two of the four phases of the learning cycle, as in the following material quoted from the Learning Styles Inventory.

DIVERGING

Combines Concrete Experience and Reflective Observation

People with this learning style are best at viewing concrete situations from many different points of view. Their approach to situations is to observe rather than to take action. If this is your style, you may enjoy situations that call for generating a wide range of ideas, such as brainstorming sessions. You probably have broad cultural interests and like to gather information. This imaginative ability and sensitivity to feelings is needed for effectiveness in arts, entertainment, and service careers. In formal learning situations, you may prefer working in groups to gather information, listening with an open mind, and receiving personalized feedback.

ASSIMILATING

Combines Reflective Observation and Abstract Conceptualization

People with this learning style are best at understanding a wide range of information and putting it into concise, logical form. If this is your learning style, you probably are less focused on people and more interested in abstract ideas and concepts. Generally, people with this learning style find it more important that a theory have logical soundness than practical value. This learning style is important for effectiveness in information and science careers. In formal learning situations, you may prefer lectures, readings, exploring analytical models, and having time to think things through.

CONVERGING

Combines Abstract Conceptualization and Active Experimentation

People with this learning style are best at finding practical uses for ideas and theories. If this is your preferred learning style, you have the ability to solve problems and make decisions based on finding solutions to questions or problems. You would rather deal with technical tasks and problems than with social and interpersonal issues. These learning skills are important for effectiveness in specialist and technology careers. In formal learning situations, you may prefer to experiment with new ideas, simulations, laboratory assignments, and practical applications.

ACCOMMODATING

Combines Active Experimentation and Concrete Experience

People with this learning style have the ability to learn primarily from “hands-on” experience. If this is your style, you probably enjoy carrying out plans and involving yourself in new and challenging experiences. Your tendency may be to act on “gut” feelings rather than on logical analysis. In solving problems, you may rely more heavily on people for information than on your own technical analysis. This learning style is important for effectiveness in action-oriented careers such as marketing or sales. In formal learning situations, you may prefer to work with others to get assignments done, to set goals, to do field work, and to test out different approaches to completing a project.

While the pedagogical objectives of the course cannot be tailored to each student’s learning strengths, it is important for the teacher to be as responsive as possible to the ways in which students learn and the diverse motivations students bring to the classwork. You can shape class exercises to call on various skill sets among your students. You can have them work on the relevant texts in small groups. You can challenge them to move outside of the “comfort zone” of their most familiar learning style. If you are flexible and responsive as a teacher, your students will see how they can be flexible and responsive as learners.

Differing Roles and Temperaments of Teachers

There are many ways in which a teacher can teach creatively and effectively. You will draw on a variety of skills and various kinds of knowledge throughout your career as a teacher. You may find that your teaching temperament changes as you move from one job to another or from one classroom context to another. Your teaching style may change as you go through professional and personal life changes. Attending to how you understand your role, your strengths, and your vulnerabilities is an important part of preparing to teach effectively.

The following two elements are crucial for effective pedagogy with adult learners, whatever your teaching style may be. You have your own ideas about what is essential for good pedagogy as well. Share them with your teaching fraternity colleagues!

1) Flexibility regarding the ways in which you judge students competently to have addressed themselves to the learning in the course. Some students with strong training in literary criticism may be able to write papers extremely well but have little sense of the issues at stake in a particular argument. Other students, perhaps second-career types or pastors who are only now coming back for their academic credentials, may express their ideas more haltingly in writing but may have a wonderfully seasoned, mature view of issues related to its proper interpretation. Some students may feel the pressures of their own interpretive communities (ecclesial or academic) so acutely that they cannot risk opening themselves to some of the ways of thinking that we emphasize in the course. They may seem not to have mastered the finer points of a method when what is (also) going on is that they are resisting the world view that the method implies. Yet other students may be solid or even gifted students who do not apply themselves to the writing tasks or to studying for the exam, so they may present the beginnings of good ideas without much follow-through. It is the task of the good teacher to be as flexible as is reasonable in judging how students have fulfilled the requirements of the course, without being too “soft”—low expectations encourage poor performance—or too rigid.

2) Ability to communicate a passion for the subject matter and an interest in how people learn it. What are you convinced is most important about the subject matter you are teaching? Why should someone else care about it? Communicate your enthusiasm, your curiosity, your contagious excitement in whatever ways are natural to your temperament. The old-school model of the “jaded expert” who sits sphinx-like with awed disciples at his feet does not motivate most adult learners, unless the “jaded expert” has managed to create a cult of personality around himself. Communicate why this material is exciting! Help your students see why learning it should be a rewarding experience for your students. Of course, in order to do that, you will need to know what matters to your students as well as what is intrinsically fascinating about the subject matter.

Which of these roles fit your view of your own teaching? More than one may apply:

credentialed expert

midwife

leader of an expedition

learned companion

guardian of the discipline

guide

shepherd

boot-camp sergeant

motivational speaker

advocate for my students

facilitator

sage

Sometimes you can discern that your motivation may not be entirely about maximizing student learning. For example,

• You may find yourself thinking, “Look, I had to spend 5 hours a day studying maths. . .” or, “I stayed up all night to keep up with the reading in X class. . .” or “I had to learn to deal with Mr.Z’s withering comments in front of my peers . . . so darn it, these students should have to do that too!” This is a variation of the “I had to walk ten miles to school in the snow, so quit yer whining” kind of thinking. It’s not healthy. While holding students to rigorous expectations can be an important motivating factor for them, you are crossing the line if you think they should have to suffer as you suffered! Give students tips about effective study habits instead. Share the wisdom you’ve gained from your own student experiences.

• As a way of overcompensating for past experiences of demanding or inaccessible teachers in your own life, you may find yourself trying to be all things to all students, endlessly accessible and affirming even if they test the boundaries of appropriate behavior or ignore course expectations. Reflect on what roles are appropriate for you as teacher: while “advocate” is appropriate, “pastoral counselor,” “mother,” and “best friend” are not.

Why are questions important for learning? ‘Questioning and Understanding Improves Learning and Thinking’ (QUILT)

Why are questions important? Questions play an important role in the processes of teaching and learning because children’s achievement, and their level of engagement, depend on the types of questions teachers formulate and use in a classroom . Recent models of teaching and learning view learning as a social activity in which children construct knowledge with the teacher and other children. In this context, learning is seen as a situated social practice where children are developing identities as a member of a particular community and it is seen as a socially negotiated and arbitrated process. This view of teachers and children acknowledge questions as a core function for both learning and teaching. As Hunkin notes, ‘We are shifting from viewing questions as devices by which one evaluates specifics of learning to conceptualizing questions as a means of actively processing, thinking about, and using information productively’.

An education research consultant in some where, sat in on a kindergarten class where the teacher said the day’s lesson was the colour green. After pointing out green on the colour wheel she asked the children to find green items among their classmate’s clothing. The children quickly found green stripes on a shirt, green socks, a green hair ribbon, and green stitching on a little girl’s jumper. Then, for the next ten minutes the teacher held up green object after green object (e.g., a stuffed frog, a fern, and ivy growing in a

plastic container) that she pulled from boxes near her chair. For each item, the teacher asked, ‘What colour is this?’ and the children chimed in unison, ‘Green!’ But in no time, the kids were sprawled across the floor, bored with the activity and indifferent to the teacher’s desperate attempts to hold their attention.

‘Green! Green! Green!’ a little boy shouted in exasperation. ‘Lime, Lime, Lime!’ another one yelled and the whole class disintegrated into howls of laughter and relief (Black, 2001).

This scenario demonstrates one of the teaching strategies that is common in classrooms. The knowledge and skills used in asking different types of questions in a classroom is one important, but critical, aspect of the teaching and learning process. The classroom above demonstrated that the children became bored because of the type of questions that the teacher had asked even though the teacher was desperate to get and keep the children’s attention. While the national and international literature has mainly focused on the importance of questioning as a teaching technique and as a strategy in promoting interactive classrooms, teachers are not necessarily taught the essential knowledge and skills to conduct effective questioning episodes which facilitate higher-order thinking.

Teachers’ questions are imperative to children’s learning because they mediate the interactive processes in the learning environment in a number of important ways. Firstly, the questions that teachers formulate and ask children are considered to be cues and clues which focus their attention on what needs to be learned. Secondly, teachers’ questioning patterns affect which students learn and how much . Thirdly, the tendency of teachers to wait (or not) for students’ responses has been found to vary from high achievers to low achievers. Teachers tend to call upon high achievers more frequently because these children usually sit in the teachers’ line of vision area (action zone) in a classroom.

International evidence suggests that children engage differentially in interactions in classrooms and this is partly due to their proximity to the teacher. Sadker and Sadker’s study of 100 different classrooms found that a few salient students received more than three times the number of teacher interactions than their classmates. In other words, their research suggested that where a student sits in a classroom determines how much interactions the student will have with the teacher. Those students that received the most verbal interactions were seated in the front rows and the centre seats of the other rows.

Contemporary researchers support different seating patterns to facilitate more effective questioning by teachers. For example, Kerry (2002) proposes an ‘arc of vision’ in which children are positioned in rows of six where the teacher is at the front of the group. Dantonio and Beisenherz (2003) suggest a U-shaped design as being useful for sound teacher-student questioning. Researchers such as Cazden (2001) have found that teachers who extend the wait times to three to five seconds between the initial question and the student response gain a number of benefits, such as: (1) students give longer responses, (2) students give more evidence for their ideas and conclusion, (3) students speculate and hypothesize more, (4) students ask more questions and talk more to other children, and (5) more children participated in responding. These changes in pacing facilitate more social interactions and higher-level thinking in children.

Questions that are inconsistent, ambiguous and imprecise can confuse children and they are less likely to be able to engage and be involved in the discussions. Questions that are formulated and conceptualised at low levels of Bloom’s taxonomy are also likely to limit the level of challenge children experience in the learning environment. The implicit message given to the children through such low-level questions is that this level of learning is more important while they are unlikely to motivate them to engage in higher-level learning.

These low-level questions initially formulated by the teachers required only one correct answer and these answers were already determined by the teachers. An important implication of asking these types of question are that co-construction of learning is limited. The learning process is determined by the teacher. Such questions also have implications for scaffolding children’s learning. Experiencing questions at repetitively low levels limits children’s opportunities to further develop their ideas and to be supported to reach higher cognitive levels. Teachers need to be aware of these errors in conceptualising and formulating questions because if questions are unclear, ambiguous and imprecise, students’ understandings can be hindered and there is a possibility that little learning and thinking occurs.

However, a substantial improvement in post-training data was shown. Teachers’ questions, and the way the questioning episodes were structured, improved as a result of acquiring new skills and knowledge through the research and training process.

In sum, teachers’ questions and their specific approaches towards and during questioning are imperative for the development of children’s learning and thinking.

Experience--The Means and Goal of Education

IN SHORT, the point I am making is that rejection of the philosophy and practice of traditional education sets a new type of difficult educational problem for those who believe in the new type of education. We shall operate blindly and in confusion until we recognize this fact; until we thoroughly appreciate that departure from the old solves no problems. What is said in the following pages is, accordingly, intended to indicate some of the main problems with which the newer education is confronted and to suggest the main lines along which their solution is to be sought. I assume that amid all uncertainties there is one permanent frame of reference: namely, the organic connection between education and personal experience; or, that the new philosophy of education is committed to some kind of empirical and experimental philosophy. But experience and experiment are not self-explanatory ideas. Rather, their meaning is part of the problem to be explored.

To know the meaning of empiricism we need to understand what experience is. The belief that a genuine education comes about through experience does not mean that all experiences are genuinely or equally educative. Experience and education cannot be directly equated to each other. For some experiences are miseducative. Any experience is miseducative that has the effect of arresting or distorting the growth of further experience. An experience may be such as to engender callousness; it may produce lack of sensitivity and of responsiveness. Then the possibilities of having richer experience in the future are restricted. Again, a given experience may increase a person's automatic skill in a particular direction and yet tend to land him in a groove or rut; the effect again is to narrow the field of further experience. An experience may be immediately enjoyable and yet promote the formation of a slack and careless attitude; this attitude then operates to modify the quality of subsequent experiences so as to prevent a person from getting out of them what they have to give. Again, experiences may be so disconnected from one another that, while each is agreeable or even exciting in itself, they are not linked cumulatively to one another. Energy is then dissipated and a person

becomes scatter- brained. Each experience may be lively, vivid, and "interesting," and yet their disconnectedness may artificially generate dispersive, disintegrated, centrifugal habits. The consequence of formation of such habits is inability to control future experiences. They are then taken, either by way of enjoyment or of discontent and revolt, just as they come. Under such circumstances, it is idle to talk of self-control.

Traditional education offers a plethora of examples of experiences of the kinds just mentioned. It is a great mistake to suppose, even tacitly, that the traditional schoolroom was not a place in which pupils had experiences. Yet this is tacitly assumed when progressive education as a plan of learning by experience is placed in sharp opposition to the old. The proper line of attack is that the experiences, which were had, by pupils and teachers alike, were largely of a wrong kind. How many students, for example, were rendered callous to ideas, and how many lost the impetus to learn because of the Way in which learning was experienced by them? How many acquired special skills by means of automatic drill so that their power of judgment and capacity to act intelligently in new situations was limited? How many came to associate the learning process with ennui and boredom? How many found what they did learn so foreign to the situations of life outside the school as to give them no power of control over the latter? How many came to associate books with dull drudgery, so that they were "conditioned" to all but flashy reading matter?

If I ask these questions, it is not for the sake of whole sale condemnation of the old education. It is for quite another purpose. It is to emphasize the fact, first, that young people in traditional schools do have experiences; and, secondly, that the trouble is not the absence of experiences, but their defective and wrong character-- wrong and defective from the standpoint of connection with further experience. The positive side of this point is even more important in connection with progressive education. It is not enough to insist upon the necessity of experience, nor even of activity in experience Every- thing depends upon the quality of the experience, which is had. The quality of any experience has two aspects. There is an immediate aspect of agreeableness or disagreeableness, and there is its influence upon later experiences. The first is obvious and easy to judge. The effect of an experience is not borne on its face. It sets a problem to the educator. It is his business to arrange for the kind of experiences which, while they do not repel the student, but rather engage his activities are, nevertheless, more than immediately enjoyable since they promote having desirable future experiences Just as no man lives or dies to himself, so no experience lives and dies to itself. Wholly independent of desire or intent every experience lives on in further experiences. Hence the central problem of an education based upon experience is to select the kind of present experiences that live fruitfully and creatively in subsequent experiences.

Here I wish simply to emphasize the importance of this principle for the philosophy of educative experience. A philosophy of education, like any theory, has to be stated in words, in symbols. But so far as it is more than verbal it is a plan for conducting education. Like any plan, it must be framed with reference to what is to be done and how it is to be done. The more definitely and sincerely it is held that education is a development within, by, and for experience, the more important it is that there shall be clear conceptions of what experience is. Unless experience is so conceived that the result is a plan for deciding upon subject-matter, upon methods of instruction and discipline, and upon material equipment and social organization of the school, it is wholly in the air. It is reduced to a form of words which may be emotionally stirring but for which any other set of words might equally well be substituted unless they indicate operations to be initiated and executed. Just because traditional education was a matter of routine in which the plans and programs were handed down from the past, it does not follow that progressive education is a matter of planless improvisation.

The traditional school could get along without any consistently developed philosophy of education. About all it required in that line was a set of abstract words like culture, discipline, our great cultural heritage, etc., actual guidance being derived not from them but from custom and established routines. Just because progressive schools cannot rely upon established traditions and institutional habits, they must either proceed more or less haphazardly or be directed by ideas which, when they are made articulate and coherent, form a philosophy of education. Revolt against the kind of organization characteristic of the traditional school constitutes a demand for a kind of organization based upon ideas. I think that only slight acquaintance with the history of education is needed to prove that educational reformers and innovators alone have felt the need for a philosophy of education. Those who adhered to the established system needed merely a few finesounding words to justify existing practices. The real work was done by habits, which were so fixed as to be institutional. The lesson for progressive education is that it requires in an urgent degree, a degree more pressing than was incumbent upon former innovators, a philosophy of education based upon a philosophy of experience.

I remarked incidentally that the philosophy in question is, to paraphrase the saying of Lincoln about democracy, one of education of, by, and for experience. No one of these words, of, by, or for, names anything which is self- evident. Each of them is a challenge to discover and put into operation a principle of order and organization, which follows from understanding what educative experience, signifies. It is, accordingly, a much more difficult task to work out the kinds of materials, of methods, and of social relationships that are appropriate to the new education than is the case with traditional education. I think many of the difficulties experienced in the conduct of progressive schools and many of the criticisms leveled against them arise from this source. The difficulties are aggravated and the criticisms are increased when it is supposed that the new education is somehow easier than the old. This belief is, I imagine, more or less current. Perhaps it illustrates again the Either-Or philosophy, springing from the idea that about all which is required is not to do what is done in traditional schools.

I admit gladly that the new education is simpler in principle than the old. It is in harmony with principles of growth, while there is very much which is artificial in the old selection and arrangement of subjects and methods, and artificiality always leads to unnecessary complexity. But the easy and the simple are not identical. To discover what is really simple and to act upon the discovery is an exceedingly difficult task. After the artificial and complex is once institutionally established and ingrained in custom and routine, it is easier to walk in the paths that have been beaten than it is, after taking a new point of view, to work out what is practically involved in the new point of view. The old Ptolemaic astronomical system was more complicated with its cycles and epicycles than the Copernican system. But until organization of actual astronomical phenomena on the ground of the latter principle had been effected the easiest course was to follow the line of least resistance provided by the old intellectual habit. So we come back to the idea that a coherent theory of experience, affording positive direction to selection and organization of appropriate educational methods and materials, is required by the attempt to give new direction to the work of the schools. The process is a slow and arduous one. It is a matter of growth and there are many obstacles, which tend to obstruct growth and to deflect it into wrong lines.

I shall have something to say later about organization. All that is needed, perhaps, at this point is to say that we must escape from the tendency to think of organization in terms of the kind of organization, whether of content (or subject-matter), or of methods and social relations, that mark traditional education. I think that a good deal of the current opposition to the idea of organization is due to the fact that it is so hard to get away from the picture of the studies of the old school. The moment "organization" is mentioned imagination goes almost automatically to the kind of organization that is familiar, and in revolting against that we are led to shrink from the very idea of any organization. On the other hand, educational reactionaries, who are now gathering force, use the absence of adequate intellectual and moral organization in the newer type of school as proof not only of the need of organization, but to identify any and every kind of organization with that instituted before the rise of experimental science. Failure to develop a conception of organization upon the empirical and experimental basis gives reactionaries a too easy victory. But the fact that the empirical sciences now offer the best type of intellectual organization which can be found in any field shows that there is no reason why we, who call ourselves empiricists, should be "pushovers" in the matter of order and organization.

The sound- ness of the principle that education in order to accomplish its ends both for the individual learner and for society must be based upon experience--which is always the actual life-experience of some individual. I have not argued for the acceptance of this principle nor attempted to justify it. Conservatives as well as radicals in education are profoundly discontented with the present educational situation taken as a whole. There is at least this much agreement among intelligent persons of both schools of educational thought. The educational system must move one way or another, either backward to the intellectual and moral standards of a pre-scientific age or forward to ever greater utilization of scientific method in the development of the possibilities of growing, expanding experience. I have but endeavored to point out some of the conditions, which must be satisfactorily fulfilled if education takes the latter course.

For I am so confident of the potentialities of education when it is treated as intelligently directed development of the possibilities inherent in ordinary experience that I do not feel it necessary to criticize here the other route nor to advance arguments in favor of taking the route of experience. The only ground for anticipating failure in taking this path resides to my mind in the danger that experience and the experimental method will not be adequately conceived. There is no discipline in the world so severe as the discipline of experience subjected to the tests of intelligent development and direction.  Hence the only ground I can see for even a temporary reaction against the standards, aims, and methods of the newer education is the failure of educators who professedly adopt them to be faithful to them in practice. 

As I have emphasized more than once, the road of the new education is not an easier one to follow than the old road but n more strenuous and difficult one. It will remain so until it has attained its majority and that attainment will require many years of serious co-operative work on the part of its adherents. The greatest danger that attends its future is, I believe, the idea that it is an easy way to follow, so easy that its course may be improvised, if not in an impromptu fashion, at least almost from day to day or from week to week. It is for this reason that instead of extolling its principles, I have confined myself to showing certain conditions which must be fulfilled if it is to have the successful career which by right belongs to it. I have used frequently in what precedes the words "progressive" and '"new" education. I do not wish to close, however, without recording my firm belief that the fundamental issue is not of new versus old education nor of progressive against traditional education but a question of what anything whatever must be to be worthy of the name education. I am not, I hope and believe, in favor of any ends or any methods simply because the name progressive may be applied to them. The basic question concerns the nature of education with no qualifying adjectives prefixed. What we want and need is education pure and simple, and we shall make surer and faster progress when we devote ourselves to finding out just what education is and what conditions have to be satisfied in order that education may be a reality and not a name or a slogan. It is for this reason alone that I have emphasized the need for a sound philosophy of experience.

Why predict the future?- In Education

More than an attempt at being Nostradamus, the value is in providing targets against which others may compare their thoughts and to stimulate efforts to either facilitate or inhibit possible futures implied by the predictions. As technology plays a larger role in education, any predictions concerning the future of education must include an analysis of technological trends. The purpose of this paper is to do just that: Analyze the trends in technology and how they relate to education, and then to extrapolate these trends in an attempt to predict the future of technology and education. Much of what is predicted in this paper might offend ardent supporters of our traditional educational system and a large portion of it will probably miss the target substantially. However, it will be clear that as technology is adopted into education, the end result will be change.

For over a century, education has remained largely unchanged. Classrooms full of students deferring to the wisdom of an all-knowing professor has, is, and many believe, will continue to be the accepted mode of instruction. Despite many technological advances and the introduction of new pedagogical concepts, the majority of today's classrooms continue to utilize this traditional mode. Educators have thrived in a bubble immune from advancements in technology, but the increasing rate of change of these advances now look to be threatening to burst this bubble.

The world is changing -- it is getting both smaller and bigger at the same time. Our world shrinks as technologies now allow us to communicate both synchronously and asynchronously with peers around the world. Conversely, the explosion of information now available to us expands our view of the world. As a result of the ability to communicate globally and the information explosion, education must change. Most educators might not want to change, but the change is coming -- it is a matter of when not if. The challenge is to prepare the children of today for a world that has yet to be created, for jobs yet to be invented, and for technologies yet undreamed.

The current teaching paradigm of the teacher as the possessor and transferor of information is shifting to a new paradigm of the teacher as a facilitator or coach. This new teacher will provide contextual learning environments that engage students in collaborative activities that will require communications and access to information that only technology can provide.

It is no secret that education is slow to change, especially in incorporating new technologies. This is described by Jukes and McCain (1997) as paradigm paralysis, the delay or limit in our ability to understand and use new technology due to previous experiences. It takes new experiences to replace the old ones, and this simply takes time. Unfortunately, education can no longer take the time it wants. The trends in technology are creating a future that is arriving faster than education is preparing for it. We must therefore ask what are these trends and how will education adapt to them? To answer these questions, the techniques of H.G. Wells will be used. Wells, the father of futures studies, "had a gift for seeing how all the activities of humankind -- social, cultural, technological, economic, political -- fit together to produce a single past, and by extension a single future" .

"Using technology can change the way teachers teach. Some teachers use technology in 'teacher-centered' ways...On the other hand, some teachers use technology to support more student-centered approaches to instruction, so that students can conduct their own scientific inquiries and engage in collaborative activities while the teacher assumes the role of facilitator or coach."

Right now, education is moving along at a snail's pace, while the world outside is speeding by at a supersonic rate. According to Fulton , "Classrooms of today resemble their ancestors of 50 and 100 years ago much more closely than do today's hospital operating rooms, business offices, manufacturing plants, or scientific labs." If you put a doctor of 100 years ago in today's operating room, she would be lost, yet if you placed a teacher of 100 years ago into one of today's classrooms she wouldn't skip a beat. Does this mean that the end is in sight for education? The answer is YES, if your asking if it means the end of education as we know it today. Let us take a peek at what the future might look like.

I could continue with some loftier predictions, but to do so would only trivialize what I will predict. So, knowing exactly what happens in our future is not important. It is important that educators have a sense of where the world is headed. Only then will they be able to adequately prepare current and future students to thrive in this ever-changing world. We must always keep in mind that a good driver doesn't watch the car's hood while they are motoring down the road. Instead, a good driver carefully watches the road ahead, looking for the obstacles and challenges that lie before them. It is time that education quit watching its hood and start looking at the road ahead.

Keep in mind always

When the rate of change inside an institution is less than the rate of change outside, the end is in site...

                                                                                                        Jack Welch, CEO of General Electric