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416

HE MERITS OF INCORPORATING HISTORY INTO

mathematics education have received consid-

erable attention and have been discussed for

decades. Still, before taking as dogma that history

must be incorporated in mathematics, an obvious

question is, Why should the history of mathematics

have a place in school mathematics? Answering

this question is difﬁcult, since the answer is subject

to one's personal deﬁnition of teaching and is also

bound up with one's view of mathematics. Fauvel's

(1991) list of ﬁfteen reasons for including the histo-

ry of mathematics in the mathematics curriculum

includes cognitive, affective, and sociocultural

aspects. My purpose in this article is not to provide

complete and satisfactory answers but rather, on

the basis of theoretical arguments and empirical

evidence, to attempt to pinpoint worthwhile consid-

erations to help high school teachers think about

what history really can do for the curriculum and

for their teaching. On the basis of Fauvel's list and

other scholars' arguments, I propose ﬁve reasons for

using the history of mathematics in school curricula:

•

History can help increase motivation and helps

develop a positive attitude toward learning.

•

Past obstacles in the development of mathemat-

ics can help explain what today's students ﬁnd

difﬁcult.

•

Historical problems can help develop students'

mathematical thinking.

•

History reveals the humanistic facets of mathe-

matical knowledge.

•

History gives teachers a guide for teaching.

History can help increase motivation and helps

develop a positive attitude toward learning

As sometimes taught, mathematics has a reputation

as a "dull drill" subject, and relevant studies report

a steady decline in students' attitudes toward the

subject through high school. The idea of eliciting

students' interest and developing positive attitudes

toward learning mathematics by using history has

drawn considerable attention. Many mathematics

education researchers and mathematics teachers

believe that mathematics can be made more inter-

esting by revealing mathematicians' personalities

and that historical problems may awaken and main-

tain interest in the subject. By comparing two college

algebra classes, McBride and Rollins (1977) probed

the effects of including the history of mathematics

and found a signiﬁcant improvement in the stu-

dents' attitudes toward mathematics when history

was included. Philippou and Christou (1998) also

reported that prospective teachers' attitudes and

views of mathematics showed radical change after

they took two history-based mathematics courses in

a preparatory program. One teacher responded,

History of mathematics provided me with a vari-

ety of interesting new experiences. . . . Through

the journey I realize that mathematics has

always been and continues to be a very useful

subject. . . . The course showed me that mathe-

matics is, at least sometimes, a human activity. I

felt more conﬁdent when I realized that even

great mathematicians did mistakes as I fre-

quently do. (Philippou and Christou 1998, p. 202)

Contrary to the previously cited studies, Stander

(1989) conducted two short-term experiments along

this line but found that studying the history of

mathematics had no signiﬁcant effect on improving

Edited by Barbara Edwards

edwards@math.orst.edu

Oregon State University

Corvallis, OR 97331-4605

Margaret Kinzel

kinzel@math.boisestate.edu

Boise State University

Boise, ID 83725-1555

Po-Hung Liu, pohung66@yahoo.com.tw, teaches at the

National Chin-Yi Institute of Technology in Taiwan. He is

interested in incorporating the history of mathematics in

the mathematics curriculum and studying the relation-

ship between students' mathematical beliefs and learning

behavior.

MATHEMATICS TEACHER

History

increases

motivation

and develops

a positive

attitude

toward

learning

Po-Hung Liu

Connecting Research to Teaching

Do Teachers Need to Incorporate

the History of Mathematics in

Their Teaching?

This material may not be copied or distributed electronically or in any other format without written permission from NCTM.

V ol. 96, No. 6

•

September 2003 417

students' attitudes toward mathematics. The out-

come indicates that using history only for sake of

using history is superﬁcial, as well as impractical.

Past obstacles in the development of

mathematics can help explain what today's

students ﬁnd difﬁcult

During the development of mathematical ideas, cer-

tain concepts were slowly recognized by mathe-

maticians. It is reasonable to assume that today's

students would also encounter difﬁculties when

they begin to learn these concepts. For instance,

the concept of function is taught to students as

early as middle school, yet many students in high

school (and even college students) hold incomplete

and inappropriate ideas about this concept (Carlson

1998; Williams 1991).

Generally speaking, the beginnings of an implicit

use of functions can be traced back to the ancient

Babylonians. The earliest explicit recognition of the

concept of function did not appear until the time of

Nicole Oresme in the fourteenth century. James

Gregory gave the ﬁrst explicit, although incomplete,

deﬁnition of function in 1667. Johann Bernoulli and

Leonhard Euler systematically investigated the

theory of function, yet both failed to distinguish

between function and value of function. Their state-

ments did not indicate that they recognized the

uniqueness of function value. The concepts of

domain and range , terms commonplace in modern

textbooks, did not come into play until the late

nineteenth century. We must be aware that the

present deﬁnition of function is a result of long-

term historical evolution. Students' negative out-

look toward the formal deﬁnition of function is

therefore not difﬁcult to understand.

A moderate or convenient mathematical nota-

tion can assist our thinking in understanding

mathematical concepts (Pólya 1945), whereas one

of the major obstacles in learning algebra is the

difﬁculty in using and understanding the meaning

of mathematical symbols. History may also explain

students' troubles in this respect (Avital 1995). As

A History of Mathematical Notations (Cajori 1928)

demonstrates, the evolution of mathematical nota-

tion was sluggish and played a signiﬁcant role in

developing mathematical ideas. Ancient Greek

mathematics did not go beyond geometry, partially

because the Greeks did not recognize the enormous

contribution that using the alphabet could make to

increase the effectiveness and generality of alge-

braic methodology (Kline 1972). The decline of

ancient Chinese mathematics was also partly

caused by the absence of a simple and effective

symbolic system. Knowing the historical struggle

to pick suitable notations can increase teachers'

comprehension of students' barriers to symbolic

understanding.

Many people view mathematics as a rigid, dry

subject, particularly because of its rigorous and

abstract features. Failure to appreciate the rigor

and abstraction of mathematics may be based on an

individual's mathematical maturity. Students prob-

ably do not appreciate the necessity of rigor unless

they have accumulated enough appropriate experi-

ence. In this regard, a knowledge of history can

give teachers and students a feeling for how stan-

dards of rigor evolved through the centuries (Arcavi

1991). What today is regarded as a nonrigorous

mathematical argument was widely accepted with-

out doubt centuries ago. Many calculus students

become frustrated in grasping the formal

deﬁni-

tion of limit. The modern concept of limit eluded

several outstanding mathematicians in history;

thus, expecting students to comprehend and freely

use the formal

deﬁnition of limit within a short

period of time is probably naive. Cornu (1991) indi-

cates that students' cognitive obstacles may reﬂect

the historical difﬁculty in the development of the

concept of limit.

Historical problems can help develop students'

mathematical thinking

The idea of using historical mathematics problems

in teaching has recently received considerable

attention among scholars. In contrast to telling sto-

ries to attract students' interest and improve their

attitudes, using historical problems in class has the

advantage of improving students' attitudes about

mathematics, as well as improving their under-

standing of mathematics. Many mathematical con-

cepts have evolved and have been revised through

the ages. The wisdom behind these great endeavors

may provide insight into the essence of mathemati-

cal thinking. As Ernest says, "Mathematicians in

history struggled to create mathematical processes

and strategies which are still valuable in learning

and doing mathematics" (1998, p. 25). Mathemati-

cal thinking is a combination of complicated

processes: guessing, induction, deduction, speciﬁca-

tion, generalization, analogy, formal and informal

reasoning, veriﬁcation, and so on. Yet modern text-

books usually present mathematical concepts in a

neat and polished format that "hides the struggle,

hides the adventure. The whole story vanishes"

(Lakatos, 1976, p. 142). By posing historical prob-

lems and analyzing the approaches by mathemati-

cians of previous eras, students can better under-

stand mathematical thinking and appreciate its

dynamic nature. Siu (1995a, 1995b) discusses

numerous examples of Euler's approaches to solv-

ing problems to explain how Euler's mind worked.

For instance, in solving the problem of the seven

bridges of Königsberg, Euler illustrated how gener-

alization and specialization complement each other,

introduced good notation, broke the problem into

Historical

problems

can help

develop

students'

mathematical

thinking

418 MATHEMATICS TEACHER

subproblems, and reassembled them to obtain a

solution to the problem. These typical traits in the

work of mathematicians are certainly worth point-

ing out to students.

In addition to presenting single typical solutions,

demonstrating multiple methods for a particular

problem provides another effective way to teach

problem solving and develop mathematical insights

(Swetz 1995). Alternative solutions for particular

historic problems from different persons, time peri-

ods, and cultures can be assembled and assigned as

exercises for students to contrast and compare. Stu-

dents thus can be advanced from knowing to under-

standing, even to appreciating, these approaches.

Before introducing the concept of integration, I

prefer to ask students to propose a method for

deriving the area of a circle without using the for-

mula. I ask them to imagine themselves as middle

schoolers and solve the problem merely by employ-

ing basic geometric and algebraic knowledge. They

usually begin with complaining about the restric-

tion, yet they come up with diverse approaches.

After they present their solutions, I show solutions

proposed by Archimedes and Liu Hui and ask stu-

dents to compare the different methods of the two

ancient ﬁgures. Most students are impressed by

Archimedes' inconceivable idea for converting a cir-

cle into a right triangle. As shown in ﬁgure 1,

Archimedes seemingly regarded the circle as a com-

bination of inﬁnitely many concentric circles and

then straightened the circumferences of all concen-

tric circles to form a right triangle by stacking

them. Students are also impressed by Liu Hui's

manner of partitioning a circle into inﬁnitely many

regular polygons and rearranging them to form a

parallelogram, as shown in ﬁgure 2. Some stu-

dents are surprised to ﬁnd that their ideas are close

to the ideas of these mathematics masters.

Calculating the sum of the harmonic series

1 + 1 +1 +1 + …

234

may also serve as a good example. The fact that the

sum is inﬁnite frequently astonishes students.

Before uncovering the secret, I encourage students

to explore and discover facts on their own; then I

delineate the approaches employed by such mathe-

maticians as Johann Bernoulli, Nicole Oresme, and

Pietro Mengoli (as described in Dunham [1990]).

After learning about three different methods, stu-

dents can better appreciate the intrinsic nature of

each approach. As Dunham indicates, Bernoulli's

approach is trickier, Oresme's idea is clear and con-

cise, and the beauty of Mengoli's method is its self-

replicating nature. In this fashion, students learn

to reject the stereotyped thinking that problems

always have only one rigid and strict method of

solution. As Siu (1993) indicates, multiple

approaches collected from history do not merely

convince students but can enlighten them.

History reveals the humanistic facets of

mathematical knowledge

Considerable research suggests that many students

believe that mathematics is ﬁxed, rather than ﬂexi-

ble, relative, and humanistic. Mathematicians' pol-

ished style in published mathematics usually elimi-

nates the human side of grappling, of perseverance,

of the ups and downs experienced on the road to

ﬁnal achievement (Avital 1995); and mathematics

teachers pass on neatly deductive formats to stu-

dents without modiﬁcation. The National Council of

T eachers of Mathematics proposes that helping stu-

dents learn the value of mathematics is an a priori

goal of school mathematics (NCTM 1989) and that

all students should develop an appreciation of

mathematics as being one of the greatest cultural

and intellectual achievements of humankind

(NCTM 2000). Yet not much has been done to

achieve these objectives. By virtue of its logical and

deductive traits, mathematics is typically deemed

History

reveals the

humanistic

facets of

mathe-

matical

knowledge

Fig. 1

Archimedes' method for deriving the area of a circle

Fig. 2

Liu Hui's method for deriving the area of a circle

V ol. 96, No. 6

•

September 2003 419

the most reliable and certain body of knowledge

among all school subjects. Nevertheless, history

reveals that this widely accepted impression is

questionable. The history of mathematics consis-

tently highlights the fact that the initial driving

forces of mathematical knowledge are plausible

conjectures and heuristic thinking; logical argu-

ments and deductive reasoning later come into

play. Acceptance or rejection of a concept is mainly

tied to mathematicians' beliefs about what mathe-

matics should be. These beliefs can be illogical,

even metaphysical. Examples like the Pythagore-

ans' rejection of irrational numbers, Kronecker's

objection to an inﬁnite number of real numbers,

and Cauchy's denial of complex numbers indicate

illogical and irrational aspects of mathematical

progress. Actually, in the early 1800s, no branch of

mathematics was logically secure (Kline 1980). The

history of mathematics notes human intellectual

adventure in mathematical ideas, thus manifesting

limitations of the human mind.

In addition to augmenting students' grasp of

mathematical thinking, using historical problems

humanizes mathematics by illustrating mathemati-

cians' struggles in attacking problems and estab-

lishing concepts. Students are pedagogically

enlightened when they realize that such problems

are not created in a vacuum and more important,

that mathematicians make mistakes too. These

recognitions have not only cognitive merit but also

affective merit. The importance of introducing

humanistic aspects of such knowledge in education

can be best summarized by Tymoczko's argument:

It took human beings thousands of years to

progress to the mathematical level of today's

high school students, and perhaps teachers

should mention this to students. . . . Educators

ignore humanistic mathematics at their peril.

Without it, educators may teach students to com-

pute and to solve, just as they can teach students

to read and write. But without it, educators can't

teach students to love or even like, to appreciate

or even understand, mathematics. (Tymoczko

1993, pp. 12–14)

History gives teachers a guide for teaching

The teacher always needs to determine the best

approach of assisting students in grappling with

and understanding ideas. History is one valid

approach (Katz 1997). In responding to the ques-

tion of whether history is important in mathemat-

ics teaching, Morris Kline indicates,

I deﬁnitely believe that the historical sequence is

an excellent guide to pedagogy. . . . Every teacher

of secondary and college mathematics should

know the history of mathematics. There are

many reasons, but perhaps the most important is

that it is a guide to pedagogy. [italics added]

(Albers and Alexanderson 1985, p. 171)

Kline's argument explicitly and clearly delineates

the chief rationale for using history in mathematics

teaching.

Integrating history into school mathematics cur-

ricula not only helps improve students' attitudes

and enhance higher-level thinking, but it also helps

expand teachers' understanding of the nature of

mathematical knowledge. Along with the growth in

their understanding of "real mathematics," that is,

the dialectical nature of mathematics in addition to

its deductive nature, teachers are expected to

restructure their beliefs about mathematics. This

restructuring may in turn affect their thinking

about curriculum design and instructional behav-

ior. Planning curriculum involves far more than

choosing the content to be taught. Teachers must

decide the instructional sequence and the methods

to use in teaching the content.

In this respect, Pólya was convinced that the

"genetic principle" offers an important guide.

By "genetic principle," Pólya means retracing the

great steps of the mental evolution of the human

race. Pólya (1965) indicates that understanding

how the human race has acquired knowledge of cer-

tain facts or concepts puts us in a better position to

judge how a human child should acquire such

knowledge. The German mathematician Otto

T oeplitz (1963) also proposed that a genetic

approach is best suited to bridge the gap between

high school and college mathematics:

Follow the genetic course, which is the way man

has gone in his understanding of mathematics,

and you will see that humanity did ascend grad-

ually from the simple to the complex. . . . Didac-

tic methods can thus beneﬁt immeasurably from

the study of history. (Toeplitz 1963, p. vi)

For instance, the approaches that ancient mathema-

ticians used in deriving the area of a circle can de-

monstrate a wide variety of mathematical thinking.

Speaking of the idea of incorporating the history

of mathematics in mathematics teaching, we

should not neglect a question that many teachers

would ask: How can a teacher incorporate the evo-

lution of mathematics concepts and cover all the

required curriculum in the short time that we have

with students? My answer is that teaching the his-

tory of mathematics is teaching mathematics itself,

too. The history of mathematics is better treated as

part of the lesson plans, not as an "extra" activity.

After participating in a workshop, one teacher's

reaction to incorporating the history of mathemat-

ics in teaching was as follows:

When a colleague asks me if and how to use his-

tory I answer: Do not talk about the history of

mathematics in your classroom, but do it, use it!

Use historical problems in your teaching for rea-

sons of variety and to give your pupils something

extra! The extras that historical problems bring

History

gives

teachers

a guide for

teaching

420 MATHEMATICS TEACHER

to your pupils are historical insights and mathe-

matical insights. (Furinghetti, 1997, p. 56)

I am convinced that, as Furinghetti indicates, a

good knowledge of the history of mathematics may

foster pedagogical creativity for integrating history

into mathematical activities.

IS INCLUDING THE HISTORY OF

MATHEMATICS IN MATHEMATICS

TEACHING EFFECTIVE?

A panel discussion, "On the Role of the History of

Mathematics in Mathematics Education," at the

second International Conference on the Teaching of

Mathematics (ICTM-2), held on Crete in July 2002

addressed the role of the history of mathematics in

education. Following the panel's reports, an Ameri-

can mathematics educator raised a critical ques-

tion: "Is there any evidence showing that including

the history of mathematics is effective in the teach-

ing of mathematics?" Answering this question is

difﬁcult for anyone who advocates the importance

of including history in the mathematics curriculum.

We have to clarify one critical conception before

answering this question. Namely, what is meant by

"effective in the teaching of mathematics"? If it

means improving students' performance on stan-

dardized examinations, my attitude would be

reserved. To my best knowledge, no empirical study

indicates that learning the history of mathematics

helps students perform better on traditional tests.

Although studying the history of mathematics may

improve students' attitudes toward mathematics,

the linkage between attitude and achievement is

neither linear nor straightforward.

Y et if effectiveness means developing students'

views of thinking and further improving their

learning behavior, I am convinced that including the

history of mathematics in the curriculum can help.

After experiencing a problem-based course that

used a historical approach, many Taiwanese students

were likely to hold active views about mathematical

thinking and were able to demonstrate multiple

approaches to problems (Liu 2002). Particularly,

when learning about "peculiar" methods used by

ancient mathematicians, those students better

appreciated the role of imagination in problem solv-

ing, and some students were more willing to think

and try the problems. After seeing Archimedes'

derivation of the area of a circle, one of my students,

who had initially emphasized the deductive nature

of mathematics, reconsidered his view:

I consider imagination more important [than log-

ical thinking] because of Archimedes. I feel he is

so strange. He derived the volume of a sphere by

means of a lever. . . . How did he think of it?

Plus, he transformed a circle into a triangle. I

feel his imagination is quite strange.

That response is typical of those of students in

my class. Nevertheless, the empirical evidence

accumulated thus far is insufﬁcient for us to con-

clude what history can or cannot do for teachers

and students. The International Study Group on

the Relations between History and Pedagogy of

Mathematics (HPM) is attempting to delineate a

role for the history of mathematics to play in school

teaching. With cooperation between researchers

and teachers, we hope that a clear picture of that

role can be drawn in the near future.

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ematical People: Proﬁles and Interviews. Boston,

Mass.: Birkhäuser, 1985.

Arcavi, Abraham. "Two Beneﬁts of Using History." For

the Learning of Mathematics 11 ( June 1991): 11.

Avital, Shmuel. "History of Mathematics Can Help

Improve Instruction and Learning." In Learn from

the Masters, edited by Frank Swetz, John Fauvel,

Otto Bekken, Bengt Johansson, and Victor Katz, pp.

3–12. Washington, D.C.: Mathematical Association

of America, 1995.

Cajori, Florian. A History of Mathematical Notations .

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of the Development of the Function Concept." In

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Ernest, Paul. "The History of Mathematics in the

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1991): 3–6.

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Linking Different Domains." For the Learning of

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Mathematics in Mathematics Education." Panel dis-

cussion at ICTM-2, Crete, July 2002.

Katz, Victor. "Some Ideas on the Use of History in the

T eaching of Mathematics. Fo r the Learning of Math-

ematics 17 (February 1997): 62–63.

Kline, Morris. Mathematical Thought from Ancient to

Modern Times. New York: Oxford University Press,

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———. Mathematics: The Loss of Certainty . New York:

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Mathematical Discovery. New York. Cambridge Uni-

versity Press, 1976.

Students

better

appreciated

the role of

imagination

in problem

solving

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for Research in Mathematics Education 8 (January

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ADDITIONAL RESOURCES FOR

TEACHERS

Calinger, Ronald, ed. V ita Mathematica: Historical

Research and Integration with Teaching. Washington,

D.C.: Mathematical Association of America, 1996.

Katz, Victor, ed. Using History to Teach Mathematics .

Washington, D.C.: Mathematical Association of

America, 2000.

National Council of Teachers of Mathematics (NCTM).

Historical Topics for the Mathematics Classroom.

Reston, Va.: NCTM, 1989.

Rogers, Leo. "History of Mathematics: Resources for

T eachers." Fo r the Learning of Mathematics 11 (June

1991): pp. 48–51.

Swetz, Frank, John Fauvel, Otto Bekken, Bengt

Johansson, and Victor Katz, eds. Learn from the

Masters. W ashington, D.C.: Mathematical Associa-

tion of America, 1995.

‰

... The question of whether the history of mathematics should be taught during regular mathematics lessons is not new (Liu, 2003;Clark, 2012;Fried, 2001;Jankvist, 2009;cf. Arcavi et al., 1982;Fauvel, 1991). ...

... Addressing the questionof whether to teach the history of mathematics and why it should have a place in school mathematics is not easy as it seems (Liu, 2003). Heiede (1992) argued that there can only be personal answers to the question of whether to teach the history of mathematicsand that history enters into mathematics just as it does into every other field because humans have by nature a sense of history. ...

... Fauvel (1991) offered more detail, listing fifteen reasons to teach the history of mathematics during regular classes. Liu (2003), building on Fauvel's research, focused on five elements that made the teaching of the history of mathematics necessary at the school level: ...

Daniel Doz

Several studies have explored the importance and benefits of teaching the history of mathematics as part of regular math classes. Some of these studies addressed the question of using the history of mathematics as a motivational factor. For instance, some found that teaching or using the history of mathematics boosted students' interest in the topics, lowered mathematical anxiety, and increased motivation, as well as supporting student learning and increasing the understanding of mathematical concepts. In the present paper, we analyze the positive effects that integrating elements of the history of mathematics into regular math classes could have on student motivation. We argue that students could greatly benefit from the inclusion of topics from the history of mathematics in regular classes.

... The history of mathematics has thus got rid of the situation … and it has gradually been valued …" (p.783) The history of mathematics is a field of study that put forwards past obstacles and difficulties which mathematicians have overcome in the development of mathematics; reveals mathematics' dynamic nature (Liu, 2003) and "shows the evolutionary and progress of mathematical knowledge through civilizations" (Baki, 2014, p.3). In other words, history of mathematics is a comprehensive area that deals with the growth processes of mathematics, the lives, works, achievements or failures of leading figures who have contributed to mathematics, the social and cultural dimension of mathematics, and development and progression of mathematical knowledge (Bidwell, 1993;Burton, 2003;Eves, 1990;Katz, 1993;Otte, 2007;Pepe, & Guerraggio, 2017;Yee and Chapman, 2011). ...

... The findings of various research studies pointed out that the use of history of mathematics in learning and teaching process bears potential contributions to both students and teachers. Specifically, it is stated that the history of mathematics helps students to comprehend the formation of mathematical thinking, improve problem-solving skills, assess mathematical topics in a comparative way between the past and present, establish relationships between mathematical topics and other disciplines, and appreciate that mathematics is a constantly evolving discipline (Alpaslan, 2011;Ho, 2008;Jankvist, 2009;Lim & Chapman, 2015;Liu, 2003;Sullivan, 1985;Wilson and Chauvot, 2000). Besides, the history of mathematics has a supporting role for teachers to gain different perspectives, to comprehend mathematical facts unnoticed before, and to move from product-oriented instruction to process-oriented (Radford, 2014). ...

... Teachers, while blending their qualified knowledge about the history of mathematics with in-class activities, can develop their creativity and also acknowledge the reason for teaching each specific topic. As a result, their teaching skills might improve (Furinghetti, 1997;Guillemette, 2017;Haile, 2008;Kjeldsen, 2011;Liu, 2003;Nataraj & Thomas, 2009;Pengelley;. Bidwell (1993, p.461) notes that students think mathematics "as a closed, dead, and emotionless island; where teachers can rescue them for replacing them on an alive, open, full of emotion, and always interesting mainland" when they integrate history of mathematics in the learning and teaching of mathematics. ...

The aim of this study was to investigate the faculty members' and the middle school mathematics teachers' perspectives regarding the use of the history of mathematics in the learning-teaching process of mathematics. As a phenomenological study, the qualitative data were collected through semi-structured interviews from 27 middle school mathematics teachers and seven faculty members and then, subjected to the content analysis. The findings revealed that both teachers and faculty members believed that using the history of mathematics is a worthwhile effort, with the potential to not only provide meaningful learning opportunities for students but also enrich teachers' professional development. However, it was also found that lack of historical perspective in the curriculum, teachers' inadequate knowledge, time constraint, no room for the history of mathematics in the textbooks and exams, overloaded curriculum and students' inadequate desire to learn were some of the reasons for rarely-use of the history of mathematics. Based on the overall findings of the study, it is concluded that teacher education (both pre-service and in-service), the structure of mathematics curriculum, teachers' and students' characteristics were the most important dynamics to integrate the history of mathematics into teaching effectively.

... Son zamanlarda matematik tarihiyle matematiği öğretme faaliyetlerinin ne şekilde bütünleştirileceği, matematik tarihinin öğretim sürecinde hangi gereksinimi karşılayacağı, matematik tarihiyle zenginleşen derslerin öğrencilere ne kazandıracağına dair sorular, kısacası matematik tarihinin öğretimdeki rolü araştırmalara konu olmuştur (Baki, 2018;Özcan, 2014). Bu bağlamda ilgili literatürde matematik tarihinin öğretimde kullanılması gerektiği dile getirilmiştir (Baki, 2018;Başıbüyük, 2018;Bütüner, 2011;Clark, 2012;Dündar & Çakıroğlu, 2014;Fauvel, 1991;Fried, 2001;Georgiou, 2010;Liu, 2003;MEB, 2018;NCTM, 2000;Özdemir & Yıldız, 2015;Tzanakis & Arcavi, 2002). Araştırmacılar matematik tarihinin öğrencilerin matematiğin gelişim aşamalarını görmelerine (Baki, 2008;Fauvel, 1991;Liu, 2003;Marshall, 2000;Özdemir & Yıldız, 2015;Tözluyurt, 2008), öğrencilerin motivasyonunun artmasına (Awosanya, 2001;Baki & Bütüner, 2013;Fauvel, 1991;Gulikers & Blom, 2001;Lawrence, 2006;Liu, 2003;Percival, 2004), matematiğe yönelik olumlu tutum geliştirmelerine (Awosanya, 2001;Baki, 2008;Bütüner, 2014;Ersoy & Öksüz, 2016;Karakuş, 2009;Lim, 2011;Liu, 2003), akademik başarılarının artmasına (Bayam, 2012;İdikut, 2007;Karaduman, 2010;Lawrence, 2006;Lim, 2011), problem çözme becerileriyle birlikte (Bell, 1992;Dündar & Çakıroğlu, 2014;İdikut, 2007;Karaduman, 2010;Liu, 2009;Wilson & Chauvot, 2000) muhakeme becerilerinin gelişimine (Bell, 1992;Dündar & Çakıroğlu, 2014;İdikut, 2007;Özdemir vd., 2012) katkı sağladığını belirtmişlerdir. ...

... Bu bağlamda ilgili literatürde matematik tarihinin öğretimde kullanılması gerektiği dile getirilmiştir (Baki, 2018;Başıbüyük, 2018;Bütüner, 2011;Clark, 2012;Dündar & Çakıroğlu, 2014;Fauvel, 1991;Fried, 2001;Georgiou, 2010;Liu, 2003;MEB, 2018;NCTM, 2000;Özdemir & Yıldız, 2015;Tzanakis & Arcavi, 2002). Araştırmacılar matematik tarihinin öğrencilerin matematiğin gelişim aşamalarını görmelerine (Baki, 2008;Fauvel, 1991;Liu, 2003;Marshall, 2000;Özdemir & Yıldız, 2015;Tözluyurt, 2008), öğrencilerin motivasyonunun artmasına (Awosanya, 2001;Baki & Bütüner, 2013;Fauvel, 1991;Gulikers & Blom, 2001;Lawrence, 2006;Liu, 2003;Percival, 2004), matematiğe yönelik olumlu tutum geliştirmelerine (Awosanya, 2001;Baki, 2008;Bütüner, 2014;Ersoy & Öksüz, 2016;Karakuş, 2009;Lim, 2011;Liu, 2003), akademik başarılarının artmasına (Bayam, 2012;İdikut, 2007;Karaduman, 2010;Lawrence, 2006;Lim, 2011), problem çözme becerileriyle birlikte (Bell, 1992;Dündar & Çakıroğlu, 2014;İdikut, 2007;Karaduman, 2010;Liu, 2009;Wilson & Chauvot, 2000) muhakeme becerilerinin gelişimine (Bell, 1992;Dündar & Çakıroğlu, 2014;İdikut, 2007;Özdemir vd., 2012) katkı sağladığını belirtmişlerdir. Ancak yapılan çalışmaların bazılarında ise matematik tarihinin öğrencilerin akademik başarılarında (Başıbüyük, 2018;Bütüner, 2014;Lit vd., 2001) ve tutumlarında (Bayam, 2012;İdikut, 2007) etkisinin olmadığı görülmüştür. ...

Dönsel DANACI
Ömer Şahin

This study aims to determine the effect of history of mathematics on the development of quantitative reasoning skills of 7 th grade students. In the study, quasi-experimental research design, one of the quantitative research design, was used. The sample of the study consists of 41 7 th grade students studying at an elementary school located in the Central Anatolia Region of Turkey. In Mathematical Reasoning Test (MRT)", developed by Erdem (2015), was used as "Quantitative Reasoning Test (QRT)". The Mann Whitney U test and Wilcoxon signed-rank test were used to analyze quantitative data. After the analysis, it was seen that the experimental group students' quantitative reasoning skills improved more than the control group students although there was no statistically significant difference between the quantitative reasoning post-test scores of the experimental and control group students.

... Therefore, the usage of HoM in teaching can be considered a brilliant step to help students in learning Mathematics, enrich teaching content and create an exciting learning environment. Moreover, the integration of HoM in the classroom has been widely supported by many researchers (Astin et al., 2016;Bidwell, 1993;Fauvel, 1991;Fried, 2001;Gulikers & Blom, 2001;Jankvist, 2009;Liu, 2003;Tzanakis et al., 2002). Several groups such as the National Council of Teachers of Mathematics (NCTM) and The Mathematical Association of America (MAA) have also proposed and recommended the integration of HoM in the classroom (Baki & Guven, 2009). ...

... In addition, most studies have found that the usage of HoM in teaching Mathematics has many positive effects on teachers, students and the process of teaching and learning Mathematics itself (Hickman & Kapadia, 1983;Philippou & Christou, 1998;Marshall & Rich, 2000;Liu, 2003;Charalambous, Panaoura, & Philippou, 2009). HoM is also seen as a medium that can serve as a guide for developing mathematical instruction designs that include a variety of aspects such as teaching strategies, problem solving and interesting topics that can be used for classroom discussion (Galante, 2014). ...

... Matematiksel muhakeme ve kanıt; matematiksel kavramların anlam kazanmasını, kavramlar arasındaki ilişkilerin görünür kılınmasını ve bu ilişkiler sayesinde yeni bilgilerin öncekilerle bütünleşmesini sağladığı için matematiksel düşünmenin gerçekleşmesi bakımından son derece önemlidir (Flores, 2002;Liu, 2003;Mubark, 2011). Bu nedenle kanıtın öğrencilerin algılama ve bilişsel düzeylerine uygun olacak şekilde her sınıf düzeyinde ele alınmasına ve muhakeme ile birlikte matematik öğretiminin bir parçası olmasına vurgu yapılmaktadır (De Villiers, 1990;NCTM, 2000). ...

... The mathematical proof that enables mathematics to be a coherent whole is a universal concept that fits all contexts of mathematics and at the center of mathematics, as it reveals the relationships between subjects in the mathematics curriculum (Balacheff, 1991;Blanton et al., 2009;Knuth & Elliot, 1998). Mathematical reasoning and proof; it is extremely important to provide for the realization of mathematical thinking as it enables mathematical concepts to gain meaning, to make the relationships between concepts visible, and to integrate new information with the previous ones thanks to these relationships (Flores, 2002;Liu, 2003;Mubark, 2011). When the literature is examined, there are many studies in which different proof categories and proof schemes are determined in order to evaluate the processes of students understanding and producing evidence (Balacheff, 1988, Sowder & Harel, 1998Yang & Lin, 2008;Waring, 2001;Weber, 2005). ...

In order to support the development of reasoning skills, it is very necessary and important for students to meet mathematical proof from an early age and to carry out actions in this process, and this is why "can young students make proof?" brings to mind the question. In line with this question, the purpose of this study is to examine the reasoning of middle school students in the process of proving the given problems and propositions. For this purpose, in the 2019-2020 academic year, seventh-grade students studying at a state middle school were selected as participants. In this study, in which qualitative research approach was adopted, the data were collected through open-ended tasks. The study findings revealed that students who did not understand the tasks left the task unanswered, rewrote it, used the data in task incompletely, incorrectly or went outside the data and sometimes did not understand the premise of the proposition in the task. When the reasoning of the students who understood the tasks was examined, it was determined that they generally made empirical verification in number tasks that require direct proof. In geometry tasks that require direct proof, it was determined that they mostly made incorrect justification by giving an example as opposed to another quadrilateral covered by a quadrilateral, justification with symbols on the prototype drawing, or justification based on incorrect drawing, and some students make deductive reasoning. In tasks in which they need to give counterexamples, was observed that some students made non-logical justifications caused by a lack of prior knowledge in the tasks, while many students made deductive reasoning. In this study, it can be said that students in general have difficulty making arguments in the proving process, often tend to generalize using special situations and use their previous learnings without transformation.

... En este contexto, Area et al. (2014) argumentan que los profesores, formadores de profesores, deben encaminar su instrucción hacia enseñar a aprender historia, es decir, es indispensable que el formador cuente con los instrumentos didácticos necesarios para ese fin, para que luego sea capaz de enseñar a los profesores en formación. En este contexto, Tzanaski et al. (2002) presentan un estudio en el que concluyen la importancia de la incorporación de la historia en la enseñanza de las matemáticas porque permite el enriquecimiento y perfeccionamiento del aprendizaje de la ciencia y puede generar una predisposición afectiva hacia las matemáticas al considerarla una actividad cultural y humana, en la diferenciación entre la naturaleza y la práctica matemática; así mismo, Liu (2003) precisa que la historia en la enseñanza de las matemáticas ayuda a desarrollar el pensamiento matemático en los estudiantes y se convierte en una guía metodológica para los profesores, en tanto que permite afianzar una actitud positiva hacia el aprendizaje de esta ciencia. ...

Se presenta una revisión sobre los rasgos de personalidad, las habilidades y las competencias que las personas poseen y ejecutan en los procesos de negociación en los que participan. Se focaliza en los elementos individuales involucrados en estos procesos, en particular de las competencias y habilidades de un negociador efectivo. El método empleado comenzó por establecer los temas a consultar, para luego hacer una investigación documental de ellos en las bases de datos Scopus, JSTOR, Science Direct, Scielo y Dialnet. La búsqueda se basó en palabras clave del tema de la negociación y se recuperaron y analizaron en su contexto las diferentes definiciones de estrategias, estilos y competencias asociadas. Los resultados se clasificaron por tema y, finalmente, se elaboró el marco teórico que los integra. No se encontró un modelo unificado que integre la personalidad, los estilos y las competencias en un solo perfil personal de negociación, por lo que se propone una agrupación y delimitación en dimensiones y sub-dimensiones de los rasgos y competencias personales positivas para la negociación, como parte de un constructo conceptual que pueda, posteriormente, ser medible por medio de pruebas psicotécnicas. Los resultados contribuyen al mejoramiento de los procesos de calidad y planeación en los campos de la administración y la gestión humana.

... Liu (2003, p. 417) defined mathematical thinking as "a combination of guessing, induction, deduction, specification, generalization, analogy, formal and informal reasoning, verification, and similar complex processes". Based on Liu's (2003) definition, reasoning can be said to be an important process for mathematical thinking. Artz and Yaloz-Femia (1999), who consider reasoning as a part of mathematical thinking, stated that reasoning involves attaining valid conclusions and generalizations. ...

Mehmet Ertürk Geçici
Elif Türnüklü

Reasoning is handled as a basic process skill in mathematics teaching. When the literature was examined, it was seen that many types of reasoning related to mathematics education were mentioned. In the present study, it was focused on visual reasoning, which is one of the types of reasoning and also used in different research areas. The purpose of the study was to propose a conceptual framework for what visual reasoning is and what its components are. The conceptual framework constructed consists of three components as visual representation using, visualization, and transition to mathematical thinking. In this framework, a clear distinction was made between the concepts of visual reasoning and visualization, which are thought to be intertwined with each other in the literature. At the same time, we tried to explain where visualization will take place in visual reasoning. Additionally, how visual reasoning will relate to mathematical thinking also distinguishes the framework from other frameworks.

... Mathematical thinking is the process of finding the unknown from the known that includes assuming, gathering evidence, and generalization (Baki, 2008;Breen & O'Shea, 2010). Liu (1996) also defined mathematical thinking as the union of the prediction, induction, deduction, representation, generalization, formal and informal reasoning, and verification. It is seen that the definitions of mathematical thinking highlight higher-order thinking, reasoning, and problem-solving. ...

This study is correlational research and aims to investigate the relationship between preservice mathematics teachers' mathematical thinking levels and attitudes for courses in mathematics. We also examined whether gender, reasons for career choice, and academic achievement lead to significant differences in pre-service teachers' attitudes and mathematical thinking levels. Participants are 109 senior pre-service mathematics teachers from three different state universities that have similar conditions. Participants are selected via convenience sampling. Seventy-nine of the participants are female, and 30 are male. "Attitude scale for courses in mathematics" and "Mathematical Thinking Scale" are used to collect data. Data were analyzed by using SPSS package program. Pre-service teachers are found to have moderate attitudes while their mathematical thinking levels are at a high-level in the sub-domains of higher-order thinking tendency, reasoning, and problem-solving and at a moderate level in the subdomain of mathematical thinking skill. Pre-service teachers' attitudes for courses in mathematics have a significant moderate relationship with higher order thinking tendency, and reasoning and have a significant and weak relationship with problem-solving.

... Given the importance of teacher's knowledge about the HoM, various opinions have been put forward and studies have been made on which component of this information should be in the component of MKT (Smestad 2015;Huntley and Flores 2010;Jankvist, Clark and Mosvold 2019;Molvold et al. 2014;Smestad, Jankvist and Clark 2014). Some of these views are that teachers can better understand subject rankings in teaching programs by learning how mathematics has evolved in the past (Schubring et al. 2002), that teachers can enrich teaching strategies with their knowledge of the HoM, and that it can be used as a tool to reveal the nature of mathematical activities (Liu 2003;Tzanakis et al. 2002). ...

This study aimed to determine how the primary school mathematics teacher candidates' subject matter knowledge about the history of mathematics and their attitudes and beliefs about the use of the history of mathematics in mathematics education changed after the history of mathematics lesson and to explain to what extent their subject matter knowledge about the history of mathematics predicted their attitudes and beliefs towards the use of the history of mathematics in mathematics education. Furthermore, it is aimed to better explain quantitative data by referring to the opinions of the teacher candidates. In this respect, the research was designed according to the exploratory sequential mixed methods in which quantitative and qualitative methods were used together. In the quantitative part, pretest/posttest experimental design without a control group was used, while in the qualitative part, teachers' opinions were considered. A total of 40 primary mathematics teacher candidates participated in the study. As a data collection tool, the knowledge test of history of mathematics (KTHM), Attitudes and Beliefs towards the Use of History of Mathematics in Mathematics Education (ABHME) Questionnaire, and interview form were used. Descriptive analysis, paired samples t test, correlation and regression analysis were used for the analysis of quantitative data. Also, In the analysis of qualitative data, content analysis was employed. After the experimental procedure, KTHM and ABHME scores of teacher candidates were significantly increased. However, it was concluded that the candidates' KTHM scores significantly predicted their ABHME scores.

Frank J. Swetz

Many teachers believe that the history of mathematics, if incorporated into school lessons, can do much to enrich its teaching. If this enrichment is just the inclusion of more factual knowledge in an already crowded curriculum, the utility and appeal of historical materials for the classroom teacher is limited. Thus to include a historical note in a student's text on the life or work of a particular mathematician may shed a historical perspective on the content, but does it actually encourage learning or illuminate the concept being taught? The benefits of this practice can be debated.

Cecil C. McBride
James H. Rollins

To determine the effects of the history of mathematics on the attitudes toward mathematics of college algebra students, an experiment was conducted in which a test group was exposed to historical items in the classroom and a control group received no such exposure. Analysis of a measure of attitude change indicated that there was a difference in attitude change for the two groups.