ࡱ> y|vwx bjbj eeت  cccccwww8Lwuwwwwww$Bbc"cc$$$Rccu$u$$ wC߃Paƺ0/!r&cى0$[#r ): EXPLORING THE RELATIONSHIP BETWEEN TEACHERS IMAGES OF MATHEMATICS AND THEIR MATHEMATICS HISTORY KNOWLEDGE Danielle Goodwin. Institute for Mathematics and Computer Science Mathphilosopher@Gmail.Com Ryan Bowman, Kristopher Wease, Jeffrey Keys Vincennes University John Fullwood, Kelly Mowery Penn State Erie: The Behrend College ABSTRACT This nationwide survey (n = 4,663) was conducted to explore the relationship between teachers images of mathematics and their mathematics history knowledge. Most respondents believed mathematics is connected to the real world, makes a unique contribution to human knowledge, can be done by everyone, and is fun and thought-provoking. The median score on the mathematics history test was 37.5%. Mathematics history knowledge scores were related to teachers views on mathematics. Teachers with high history scores were more likely to believe that investigating is more important than knowing facts and that mathematics is ongoing and shows cultural differences. On the other hand, teachers with low history scores were more likely to believe that mathematics is a disjointed collection of facts, rules and skills. What teachers believe about the nature and role of mathematics affects the development of mathematics curricula in schools, as well as the way mathematics is taught (Dossey, 1992; Lerman, 1986; Thompson, 1992). Thompson (1992) goes so far as to say that teachers approaches to mathematics teaching depend fundamentally on their system of beliefs, in particular on their conception of the nature and meaning of mathematics (p. 131). Hersh (1997), Barr (1988), Lubinski (1994) and others have indicated that the different beliefs teachers have about the nature of mathematics create a dominant force that shapes their teaching behaviors. They concur that teachers conceptions about the nature and structure of mathematics affect planning and instructional choices, the curriculum in general and what research is conducted by action researchers and mathematics education researchers. In turn, teachers behaviors affect student learning. Teachers who have rule-oriented images of mathematics can weaken student learning by representing mathematics in misleading ways. Balls (1990a) research suggests that teachers who see mathematics as nothing more than a collection of rules think that giving a rule is equivalent to settling a mathematical problem. Such teachers value memorization more than conceptual understanding. When teachers view mathematics in these very narrow ways, they teach mathematics as a set of unconnected fragments, definitions and tricks that foster algorithmic learning in classrooms (Ball, 1990b). Mathematics is the product of human inventfulness (Romberg, 1992, p. 433). The idea that mathematics is a set of rules, handed down by geniuses, which everyone else is to memorize and use to get the right answer, must be changed. If teachers do not believe that mathematics involves creativity, this may deter them from assisting their students in exploring possible approaches to problems. If a teachers view of mathematics is that it is a set of disjointed rules to be followed, s/he may fail to help students understand the processes of making connections and problem-solving (Ball, 1990b). Examining Teachers Images of Mathematics Attitudes, beliefs and views of the nature of mathematics, mathematical ability and mathematics education are all aspects of what Sam and Ernest (1998) call the individuals image of mathematics. Alba Thompson (1984), one of the strongest proponents of the importance of studying teachers conceptions of the nature of mathematics, stresses that imprudent and erroneous efforts to improve mathematics education will likely be the result of not properly considering the role that teachers conceptions of mathematics play in shaping their teaching; however, the relationship between teachers views of the nature of mathematics and their teaching practices is not a direct or simple relationship. Barbin (1996) proposes that studying the history of mathematics allows teachers to form a broader view of the nature of mathematics and positively transforms their teaching practices. Further, mathematics teachers need to learn the history of mathematics, because that history is a part of mathematics itself (Kline, 1980). Shulman (1987) contends that through study of the history and philosophy of a discipline, teachers can come to understand its structure. The Relationship between Teachers Images of Mathematics and Classroom Practice Clark (1988) and Shulman (1987) assert that teachers construct and maintain implicit ideas about the discipline that they teach and that these personal theories come from their personal experiences, beliefs and studying the history and philosophy of their discipline. Teachers beliefs about the nature of the subject they teach guide their actions in the classroom (Shulman, 1987). Lampert (1988), in her case study of secondary teachers, showed that conveying the nature of mathematics to future teachers affects how they teach. Future teachers can learn to have a more complex view of mathematics and what it means to learn mathematics. Lerman (1986) and Ernest (1998) contend that teachers beliefs about mathematics shape their image of what teaching and learning mathematics should be like. Lerman (1983) asserts that teachers who believe that mathematics is a cumulative and value-free body of knowledge convey to their students that one must first learn mathematical processes and understand usefulness or relevance afterwards, sometime in their future, perhaps after they are finished schooling (during employment or even later). Lerman (1983) also reports that holding the alternative view that mathematics is a human process (and therefore possibly fallible) leads teachers to portray mathematics as growing and changing, encouraging students to think mathematically, proposing ideas and suggesting methods. Lerman warns that the fundamental issue from which mathematics teachers cannot escape is that a commitment to a theory of mathematical knowledge logically implies a particular choice of syllabus content and teaching style (p. 65). If teachers images of mathematics are to be consistent with the views of mathematics advocated in literature, then teachers must create opportunities for students to experience the construction of mathematics (Dossey, 1992). Teachers should provide students with ideas that illustrate the evolution of the solution process and that supply the historical and cultural insights behind the problem (Swetz, 2000). Learning should not just be an accumulation of facts. For meaningful learning to take place, ideas about why the concepts arose, the historical conditions surrounding the development of the concepts and the development of the concepts themselves must be addressed (Grugnetti, 2000). Understanding Mathematics through Its History Mathematics is a cumulative discipline, and the past, present and future of mathematics are all closely connected. The historical development of mathematical ideas serves as a background to mathematics, so mathematics must not be dissociated from its history (Giacardi, 2000; Man-Keung, 2000). Heine (2000) claims that without an understanding of the history of mathematics, one cannot understand the motivations for studying mathematics because todays motivations may not be the same motivations as of those who have studied mathematics before us. The development of mathematics is intimately related to religion, society and politics (Gellert, 2000) and in turn these have influenced past and contemporary perspectives on the philosophy of mathematics (Ernest, 1998). Further, mathematicians throughout history -- influenced by more than a pure pursuit of knowledge -- have decided what problems to study, what mathematical objects to create and what axiomatic systems to adopt. Mathematics is a living, exciting discipline that has taken many twists and turns during its long history (Fauvel, 1991; Heiede, 1996; Kleiner, 1996; Liu, 2003). The soundness of mathematics can be shown only by understanding its historical development (Davis & Hersh, 1986). Lakatos (1976/1999) asserts that regarding mathematics as a polished set of deductive proofs hides the struggle, hides the adventure....The whole story vanishes (p. 142). Mathematics teachers are the carriers of mathematical culture (Rickey, 1996, p. 252). History shows us that people from all cultures and all levels of education have contributed to the development of mathematics (Fauvel, 1991; Liu, 2003). Housewives, high-schoolers, and a host of other amateurs have changed the course of mathematics as we know it and many studies have shown that teachers can benefit from knowing this history (Barbin, 1996; Bruckheimer & Arcavi, 2000). Exploring the history of mathematics allows teachers to see its importance and encourages their enthusiasm for the subject (Kleiner, 1996). Heiede (1996) goes so far as to say that mathematics without its history...is mathematics as if it were dead (p. 232). Many mathematics philosophers have used mathematics history to determine the nature of mathematics. For instance, Lakatos (1976/1999) used a historical case study to attempt to show that mathematics is a process rather than a product, and that it is indeed a fallible process. Some mathematics philosophers define math as the study of certain social-historic-cultural objects, thereby explicitly weaving history into their philosophy (Fauvel, 1991; Hersh, 1997). Hersh (1997) goes further, remarking that an adequate view of the nature of mathematics must be cognizant of and compatible with the history of mathematics. Teachers form beliefs about mathematics and mathematics teaching based upon their own schooling experiences that are not easily changed during teacher education programs (Cooney, Shealy & Arvold, 1998). Textbooks, for the most part, present the formal, polished mathematics long after all of the details have been worked out (Liu, 2003). In this sense, the textbooks that most teachers learn from suggest that mathematicians are infallible and that doing mathematics is completely predictable (Hersh, 1997). History shows us this is not true. In the development of calculus, the details of computing limits were developed some two hundred years after differentiation. Todays calculus textbooks present limits first, then derivatives, as if all of the details were worked out in perfect order. Sometimes, studying the historical development of concepts is the only way to examine how mathematical knowledge really comes about (Lakatos, 1976/1999; Liu, 2003). Ernest (1998) asserts that to delve into many aspects of the nature of mathematics, historical inquiry is a necessity. Studying mathematics history shows that mathematics is situated within the larger context of human history (Barbin, 1991; Brown, 1991; Ernest, 1998; Fauvel, 1991; Lerman, 1986; Liu, 2003). The history of mathematics also shows that mathematics is not a linear process it takes many twists and turns during development (Ernest, 1998; Fauvel, 1991; Liu, 2003; Russ, 1991). Mathematics history reveals that mathematics is intimately connected within itself, to other disciplines and with the real world. Mathematics history exposes the fact that mathematics has been done by people of all ages, from all walks of life, from all cultures (Fauvel, 1991; Liu, 2003). Research Questions The questions that guided this research are: What images do teachers have of mathematics? What do teachers know about the history of mathematics? What is the relationship between teachers images of mathematics and their mathematics history knowledge? Methodology To explore these questions, a non-experimental, survey research design was employed. The combined survey instrument (consisting of the Mathematics Images Survey, a Mathematics History Test, and demographic items) was developed by the researcher to collect the primary data. The chosen survey method was a questionnaire e-mailed to the study participants. The teacher sampling consisted of approximately 28,395 randomly-selected teachers. Roughly 10% of school districts with teacher email addresses listed online were selected randomly, with a random sampling of elementary teachers and secondary mathematics teachers selected from the chosen districts. There were no incentives to participate and a high proportion of the emails were unfortunately relegated to junk email. It was determined before the e-mail distributions began that an acceptable response rate for an incentive-free email survey from someone unknown to the recipients would be 10% (Survey Response Rates, 2014). This minimum was met and exceeded, as 4,663 surveys were returned for a 16.4% response rate. Mathematics Images Survey Items from many (Andrews & Hatch, 1999; Benbow, 1996; Brendefur, 1999; Carson, 1997; Coffey, 2000; Mitchell, 1998; Mura, 1995; Ruthven & Coe, 1994; Schoenfeld, 1989) studies about the various dimensions of images of mathematics were combined and modified to form the Mathematics Images Survey. Mathematics History Test No mathematics history tests relevant to mathematics teachers were found during an exhaustive literature search. A Mathematics History Test that contains mathematics history items relevant to K-12 instruction was created. To assure reliability and eliminate subjectivity in coding of the responses to the history test, closed response questions were chosen. The history questions were written and formatted to reflect the most important elements from the historical development of K-12 mathematics. The items were constructed so that knowledge of the precise dates that historical events occurred was not necessary. For example, on questions that require the respondent to identify the time period of an important development, the answer choices have very broad ranges of years (no less than a 400 year time span per answer choice). Also, the chronological ordering items were chosen very specifically, so that if the respondent understands the development of these concepts, then they must know which event came first. Many clues and a picture are given on items that require the respondent to identify a famous mathematician. Each item was specifically crafted to be closed response and yet test the understanding of a significant portion of the historical development of K-12 mathematics. The Combined Survey Instrument In developing the combined survey instrument, the items went through a series of refinement steps as they were field-tested for format and clarity. The combined survey instrument underwent several revisions before asking for feedback from two small focus groups and one large (N = 38) focus group of central Massachusetts masters and doctoral level mathematics and science education students. This helped begin establishing the validity and reliability of the instrument. The combined survey instrument, containing the Mathematics Images Survey, the Mathematics History Test, and demographic items was pilot-tested. Pencil-and-paper copies of the survey were sent to 300 randomly-selected public high school teachers in California. The pencil-and-paper format was chosen so that the teacher respondents could write comments and concerns on the combined survey instrument and then return it anonymously. Of those, 193 completed surveys were returned. A Kuder-Richardson reliability of > 0.60 on the Mathematics History Test was sought and met, with a reliability of 0.78. The images and demographics items were checked to be sure that no major concerns had been written in and no questions had been left blank or answered in an invalid way on more than 5% of the responses. Having met the requirements set for instrument reliability, the researcher continued with the full study. Findings Demographics A frequency analysis of the demographic items for the study respondents is presented in Table 1. The frequency analysis of the demographic data shows that over 60% percent of respondents had a Masters as their highest degree. About one-quarter of the respondents teach at the elementary school level, one-quarter at the middle school level, and almost half teach at the high school level. There were at least 24 respondents from each state in the nation. When the states were grouped into the geographic regions designated by the U.S. Census Bureau (Census Regions and Divisions of the United States, 2014), the respondents were almost evenly split with about one-quarter of the respondents teaching in each of the four geographic regions. Almost 50% of the respondents had been teaching for 12 or more years. Table 1 Demographic Characteristics of Respondents Characteristic N % Highest Degree Completed 4,381Bachelor Master Doctorate35.0 63.5 1.5Highest Grade Level Taught 4,366Elementary Middle High28.8 23.7 47.5Geographic Region Currently Teaching In Northeast Midwest South West 4,424 24.4 25.8 26.5 23.3Geographic Region Prepared In Northeast Midwest South West 4,378 26.5 28.7 24.4 20.4Years of Teaching Experience 0-3 years 4-7 years 8-11 years 12 or more years 4,426 14.6 19.8 15.9 49.7 Images of Mathematics Two items were included on the survey for a gross measure of the overall image of mathematics. Table 2 shows a frequency analysis of the two overall images items. Table 2 Frequencies of Responses to Overall Images of Mathematics Items Ideally, doing mathematics is like: (N = 4,639) %Cooking a meal Playing a game Conducting an experiment Doing a puzzle Doing a dance Climbing a mountain10.7 14.1 7.7 60.9 3.6 3.0Mathematics is(N = 4,427) %Creating and studying abstract structures, objects Logic, rigor, accuracy, reasoning and problem-solving A language, a set of notations and symbols Inductive thinking, exploration, observation, An art, a creative activity, the product of the A science; the mother, the queen, the core, a tool A tool for use in everyday life2.9 32.5 3.3 16.9 2.6 13.4 28.4 The frequency analysis shows that the majority of respondents believe that mathematics is like doing a puzzle. Approximately one-third of respondents believe that mathematics overall is logic, rigor, accuracy, reasoning and problem solving, while almost 30% believe that mathematics is most accurately characterized as a tool for use in everyday life. The rest of the images items were Likert-Type items formatted to a scale with a score of 1 corresponding to Strongly Disagree, 2 corresponding to Disagree, 3 corresponding to Slightly Disagree, 4 corresponding to Slightly Agree, 5 corresponding to Agree, and 6 corresponding to Strongly Agree. Table 3 shows the means and standard deviations for each of the Likert-Type images items. Table 3 Means and Standard Deviations for Likert-Type Images of Mathematics Items ItemNMeanSDMathematics is fun.4,6415.300.802Math is thought provoking.4,6485.580.658Mathematics is a disjointed collection of facts, ...4,6362.011.268Everything important is already known 4,6262.091.090Some people are good at math and some people are not.4,6374.131.196Math is intricately connected to the real world.4,6395.500.707The ability to investigate is more important than facts.4,6344.611.150The process a mathematician uses is predictable.4,6303.231.214Mathematics makes a unique contribution to knowledge.4,6335.460.661Mathematical objects exist only in the human mind.4,4102.281.141Mathematics shows cultural differences.4,3993.091.335Mathematics supports different ways of solving 4,4605.280.727In mathematics, you can be creative.4,4415.140.881Mathematical knowledge never changes.4,4452.291.068Math can be separated into many different areas 4,4052.751.247[Doing mathematics] can change your mind about it.4,4214.890.799 Table 3 indicates that the average respondent agreed with the statements mathematics is fun, math is thought provoking, math is intricately connected to the real world, the ability to investigate a new problem is more important than knowing facts, mathematics makes a unique contribution to human knowledge, mathematics supports many different ways of looking at and solving the same problems, in mathematics, you can be creative, and the process of trying to prove a mathematical relationship can change your mind about it. The average respondent disagreed with the ideas mathematics is a disjointed collection of facts, rules and skills, everything important about math is already known, mathematical objects and formulas exist only in the human mind, and mathematical knowledge never changes. The average respondent was on the fence about the ideas some people are naturally good at math and some people are not, the process a mathematician uses when solving a problem is predictable, mathematics shows cultural differences, and mathematics can be separated into different areas with unrelated rules. History of Mathematics The Mathematics History Test portion of the survey instrument was found to be reliable, with a Kuder-Richardson reliability of 0.7. The Mathematics History Test has 16 items. The mean was approximately 6.0 correct with a standard deviation of roughly 3.9. The median score on the mathematics history test was 6 correct out of 16. Over 26% of respondents knew at least half of the correct answers on the mathematics history test. Figure 1 shows the number of respondents with 0-4, 5-8, 9-12 and 13-16 out of 16 correct. No single question was answered correctly by less than 17% of respondents or more than 65% of respondents.  SHAPE \* MERGEFORMAT  Figure 1. Bar Chart of History Scores. Relationships between History Score and Images ANOVA with Tukey post hoc testing revealed that the lower the history score, the more likely it was for a respondent to select cooking a meal on the first overall images of mathematics item Ideally, doing mathematics is like: (F(5, 4,633) = 6.9, p = 0.000). For example, the teachers who selected doing a dance scored over 7% higher on the history test on average than the teachers who selected cooking a meal. ANOVA with Tukey post hoc testing revealed that the respondents with lower history scores were more likely to select a tool for use in everyday life on the second overall images of mathematics item, Mathematics is (F(6, 4,420) = 69.9, p = 0.000). For example, the teachers who selected an art, a creative activity, the product of the imagination scored over 25% higher on average on the history test than the teachers who selected a tool for use in everyday life. A Pearson product-moment correlation coefficient was computed to assess the relationship between history score and the Likert-type images item scores, revealing that every Likert-type image item except one (mathematical objects and formulas exist only in the human mind) were significantly correlated to history score (see Table 4). Teachers who scored better on the mathematics history test more strongly agreed with the statements mathematics is fun, math is thought provoking, math is intricately connected to the real world, the ability to investigate a new problem is more important than knowing facts, mathematics makes a unique contribution to human knowledge, mathematics shows cultural differences, mathematics supports many different ways of looking at and solving the same problems, in mathematics, you can be creative, and the process of trying to prove a mathematical relationship can change your mind about it than teachers who scored lower on the history test. As history score increased, respondents disagreed more strongly with the statements mathematics is a disjointed collection of facts, rules and skills, everything important about math is already known, some people are naturally good at math and some people are not, the process a mathematician uses when solving a problem is predictable, mathematical knowledge never changes, and mathematics can be separated into different areas with unrelated rules. Table 4 Correlations between History Score and Likert-Type Images of Mathematics Items Likert-Type Images of Mathematics ItemN r  p Mathematics is fun.4,6410.1170.000Math is thought provoking.4,6480.1380.000Mathematics is a disjointed collection of facts, ...4,636-0.1980.000Everything important is already known 4,626-0.1600.000Some people are good at math and some people are not.4,637-0.0370.012Math is intricately connected to the real world.4,6390.0630.000The ability to investigate is more important than facts.4,6340.0510.001The process a mathematician uses is predictable.4,630-0.0610.000Mathematics makes a unique contribution to knowledge.4,6330.1210.000Mathematics shows cultural differences.4,3990.0320.032Mathematics supports different ways of solving 4,4600.0700.000In mathematics, you can be creative.4,4410.1600.000Mathematical knowledge never changes.4,445-0.1240.000Math can be separated into many different areas 4,405-0.1690.000[Doing mathematics] can change your mind about it.4,4210.1200.000 Discussion of Findings Images of Mathematics Overall, the views expressed by the respondents about mathematics seem to be mostly positive. Respondents were split in their characterizations of mathematics, with the highest percentage (32.5%) indicating that mathematics overall is a logical, rigorous process involving reasoning and problem solving. Most respondents agreed that mathematics is connected to the real world and makes a unique contribution to human knowledge. Many believed that the ability to investigate a new problem is more important than knowing facts and that mathematics is fun and thought-provoking. Most respondents did not see mathematics as unchanging or as a disjointed collection of facts, rules and skills. History of Mathematics While the median history score was only a 37.5%, over one-quarter of respondents knew at least half of the correct answers on the mathematics history test. So, it seems that a sizeable minority of the respondents valued mathematics history and were somewhat proficient in it. Relationships between Images and History Knowledge Respondents with more history knowledge exhibited more favorable views of mathematics. Respondents with low history scores were more likely to indicate that they believed mathematics overall was like cooking a meal or a tool for use in everyday life. Respondents with high history scores were more likely to indicate that they believed mathematics overall is like doing a dance or an art, a creative activity, the product of the imagination. Respondents with a low history score were less likely to agree that mathematics can be done by everyone and shows cultural differences than respondents with high history scores. Respondents with low mathematics history scores were more likely to believe that mathematics is a disjointed collection of facts, rules and skills than respondents with high history scores. Respondents with high history scores disagreed more often with the statement everything important about mathematics is already known than did their low-scoring counterparts. Respondents with lower history scores appeared to be more likely to agree with the statement that the process of doing mathematics is predictable than those with higher history scores. By and large, the teachers with low history scores in this study were the teachers who exhibited narrow, negative views of mathematics. Implications for Practice Implications for Practice Involving Images of Mathematics Like the teachers in this study, future teachers need to come away from their mathematics and mathematics education courses with positive images of mathematics. What teachers believe about the nature and role of mathematics affects their actions in the classroom, planning, the development of mathematics curricula in schools, as well as the research that is done in classrooms. Research on teacher education tells us that teachers can learn to use and choose behaviors (Lampert, 1988). Future teachers can learn from teacher training programs to have a more complex view of mathematics and what it means to learn mathematics. Shulman (1987) also remarks that teacher education must work with the beliefs that guide teacher actions, with the principles and evidence that underlie the choices teachers make (p. 13). This is important, as teachers images of mathematics are largely shaped by their own experiences of mathematics long before they enter teacher training programs (Cooney, Shealy & Arvold, 1998; Lampert, 1988). So, efforts to widen nave images of mathematics in future teachers during mathematics education courses must be dramatic to effect change. Efforts to be sure that teachers hold favorable images of mathematics are extremely important as teachers of mathematics do more than just teach content, they are a students chief source of information about the nature of mathematics (Rickey, 1996). Implications for Practice Involving Mathematics History The teachers in this study who had command of mathematics history held more positive, informed views about the nature of mathematics. So, it seems that teacher education, both in mathematics and mathematics education courses, should involve exposure to mathematics history. Because many states require a substantial number of mathematics content courses prior to, or concurrent with, mathematics education courses, it is important that both mathematics and mathematics education courses involve the historical development of concepts, as teachers should develop an appreciation of the contributions made by various cultures and individuals to the development of mathematics. This study highlights the importance of knowing mathematics history. Mathematics history supplies the context of mathematics. History shows teachers and students the nature of mathematics. Future teachers of mathematics should know the history of mathematics. What a preservice teacher is taught is a major influence on what they learn (Barr, 1988). If they are not taught the history of mathematics, they often times do not learn it. Including mathematics history courses in teacher preparatory programs deepens understanding, changes teachers views about the nature of mathematics and provides teachers with information for direct use in their classrooms. Avital (1995) indicates that future teachers must be exposed to approaches for the inclusion of mathematics history that are directly applicable in their own practice. Seeking out important problems in the teaching and learning of mathematics and highlighting related historical developments can help teachers to cope with these issues. Using history in teacher training programs is a way to help future teachers experience the process of creating mathematics and when the historical development of concepts are used in an integrated way, they can show future teachers what it can be like to participate in a class that values questioning and investigation (Avital, 1995). Final Conclusions Asserted relationships between knowing mathematics history and images of mathematics were supported by this research. Knowing that mathematics history is important because it is related to teachers images of mathematics, researchers must now move toward studies that examine how best to incorporate mathematics history into our K-12 mathematics classrooms and mathematics and mathematics education courses for future teachers. The causal relationships between mathematics history knowledge and images of mathematics in teachers and students must be further studied, so that it can be determined if mathematics history can be used to positively influence images of mathematics. Mathematics history is part of mathematics itself (Giacardi, 2000; Man-Keung, 2000). Mathematics, as revealed through studying its history, is an artful discipline that is very much alive and ongoing (Barbin, 1991; Benbow, 1996; Brown, 1991; Carson, 1997; Ruthven & Coe, 1994). Teachers benefit from knowing the history of mathematics because it shows them the whys of mathematics as well as the hows (Barbin, 1996; Bruckheimer & Arcavi, 2000; Kleiner, 1996). When mathematics teachers know historical stories that they can bring to their classes, they can cajole, exasperate, stimulate, motivate, seduce, amuse - all welcome didactic traits (Kleiner, 1996, p. 261). But, a historical approach does more than that, it allows teachers to select good pedagogical sequences and teaching methods, as well as contributing to a deeper understanding of key concepts. Studying mathematics history changes what teachers believe about mathematics and mathematical knowledge and transforms their teaching practices (Barbin, 1996). If mathematics history courses are integrated into teacher education programs, perhaps teachers will have deepened understanding of concepts, more information to take to their classrooms and changed views about the nature of mathematics. Presenting the history of mathematics is valuable pedagogically. There are many historical ideas and problems from the field of mathematics that are readily accessible to students today (Bruckheimer & Arcavi, 2000; Swetz, 2000). Man-Keung (2000) also asserts that mathematics history motivates the students and gives them a sense of the nature of mathematics while a concept is being explored. There are many benefits of bringing mathematics history to the classroom. Using history of mathematics in the classroom...allows us to talk about more advanced and more recent results in mathematics where it would have been impossible to present the mathematical details (Rickey, 1996, p. 255). Historical stories are a device for talking about mathematics in laymans terms, which makes these stories about mathematics accessible to students even if the mathematics itself is not (Rickey, 1996). By referring to the motives and stories of mathematics, properly used historical materials lead to a deep understanding of and appreciation for mathematics. History shows that there are often unforeseen outcomes and unanticipated applications of mathematics. History is a not only a source of attention-grabbing problems to consider, but it also suggests devices and aids for assisting in students construction of mathematical concepts. 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