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110-398: The unnamed narrator and his schoolmates Miruka and Tetra are Japanese high school students with an interest in mathematics. Together they explore the world of mathematics by helping each other solve problems spanning a wide range of difficulty, from extensions of high school mathematics to extremely difficult problems previously addressed by famous mathematicians. While the book is presented as

220-444: A Weierstrass factorization representation as an infinite product: sin ⁡ ( π x ) π x = ∏ n = 1 ∞ ( 1 − x 2 n 2 ) . {\displaystyle {\frac {\sin(\pi x)}{\pi x}}=\prod _{n=1}^{\infty }\left(1-{\frac {x^{2}}{n^{2}}}\right).} The infinite product

330-1046: A complete orthonormal basis in the space L per 2 ( 0 , 1 ) {\displaystyle L_{\operatorname {per} }^{2}(0,1)} of L2 periodic functions over ( 0 , 1 ) {\displaystyle (0,1)} (i.e., the subspace of square-integrable functions which are also periodic ), denoted by { e i } i = − ∞ ∞ {\displaystyle \{e_{i}\}_{i=-\infty }^{\infty }} , Parseval's identity tells us that ‖ x ‖ 2 = ∑ i = − ∞ ∞ | ⟨ e i , x ⟩ | 2 , {\displaystyle \|x\|^{2}=\sum _{i=-\infty }^{\infty }|\langle e_{i},x\rangle |^{2},} where ‖ x ‖ := ⟨ x , x ⟩ {\displaystyle \|x\|:={\sqrt {\langle x,x\rangle }}}

440-621: A better understanding of math anxiety. Goulding, Rowland, and Barber suggest that there are linkages between a teacher's lack of subject knowledge and the ability to plan teaching material effectively. These findings suggest that teachers who do not have a sufficient background in mathematics may struggle with the development of comprehensive lesson plans for their students. Similarly, Laturner's research shows that teachers with certification in math are more likely to be passionate and committed to teaching math than those without certification. However, those without certification vary in their commitment to

550-1236: A change of variables ( x = − i t {\displaystyle x=-it} ): − 1 2 t 2 + π cot ⁡ ( − π i t ) 2 i t = ∑ n = 1 ∞ 1 n 2 + t 2 . {\displaystyle -{\frac {1}{2t^{2}}}+{\frac {\pi \cot(-\pi it)}{2it}}=\sum _{n=1}^{\infty }{\frac {1}{n^{2}+t^{2}}}.} Euler's formula can be used to deduce that π cot ⁡ ( − π i t ) 2 i t = π 2 i t i ( e 2 π t + 1 ) e 2 π t − 1 = π 2 t + π t ( e 2 π t − 1 ) . {\displaystyle {\frac {\pi \cot(-\pi it)}{2it}}={\frac {\pi }{2it}}{\frac {i\left(e^{2\pi t}+1\right)}{e^{2\pi t}-1}}={\frac {\pi }{2t}}+{\frac {\pi }{t\left(e^{2\pi t}-1\right)}}.} or using

660-456: A cherry tree on the day of their entrance ceremony to high school. She is a tall, beautiful girl with long black hair and a dignified demeanor. She wears metal-frame glasses. She has the top grades for math in her class. She tends to act without consideration of others. The narrator interprets her habit of helping herself to his notebook and starting conversations with others at inopportune times as signs of her lack of inhibitions. At one point she

770-507: A child's educational processes is essential. A student's success in school is increased if their parents are involved in their education both at home and school. As a result, one of the easiest ways to reduce math anxiety is for the parent to be more involved in their child's education. In addition, research has shown that a parent's perception on mathematics influences their child's perception and achievement in mathematics. Furthermore, studies by Herbert P. Ginsburg , Columbia University, show

880-459: A friend, but she rarely reveals her emotions, so it is difficult to read her true intent towards him. She thinks of Tetra as being "cute". At one point, she mentions to the narrator that she could never be as cute as Tetra. She has a great love for mathematics. She is normally quiet and subdued, but immediately becomes loquacious when she starts to talk about math. She rarely talks about anything else, for example starting her first conversation with

990-420: A higher emphasis on effort rather than one's innate intellectual ability, they are helping their child develop a growth mindset . People who develop a growth mindset believe that everyone has the ability to grow their intellectual ability, learn from their mistakes, and become more resilient learners. Rather than getting stuck on a problem and giving up, students with a growth mindset try other strategies to solve

1100-450: A means of improving women's math performance. The researchers concluded that women tended to perform worse than men when problems were described as math equations. However, women did not differ from men when the test sequence was described as problem-solving or in a condition in which they learned about stereotype threats. This research has practical implications. The results suggested that teaching students about stereotype threat could offer

1210-991: A more direct route to expressing non-recursive formulas for ζ ( 2 k ) {\displaystyle \zeta (2k)} using the method of elementary symmetric polynomials . Namely, we have a recurrence relation between the elementary symmetric polynomials and the power sum polynomials given as on this page by ( − 1 ) k k e k ( x 1 , … , x n ) = ∑ j = 1 k ( − 1 ) k − j − 1 p j ( x 1 , … , x n ) e k − j ( x 1 , … , x n ) , {\displaystyle (-1)^{k}ke_{k}(x_{1},\ldots ,x_{n})=\sum _{j=1}^{k}(-1)^{k-j-1}p_{j}(x_{1},\ldots ,x_{n})e_{k-j}(x_{1},\ldots ,x_{n}),} which in our situation equates to

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1320-399: A novel, the bulk of its content is related to finding the solution to complex math problems, so could also be considered a form of textbook. At the start of his first year of high school, the narrator meets a new classmate, a girl named Miruka. Without introducing herself, she gives him the beginning to number sequences, to which he answers with their continuation. One year later, the narrator

1430-485: A practical means of reducing its detrimental effects and lead to an improvement in a girl's performance and mathematical ability, leading the researchers to conclude that educating female teachers about stereotype threat can reduce its negative effects in the classroom. According to Margaret Murray, female mathematicians in the United States have almost always been a minority. Although the exact difference fluctuates with

1540-624: A probability that approaches 6 / π 2 {\displaystyle 6/\pi ^{2}} , the reciprocal of the solution to the Basel problem. Euler's original derivation of the value π 2 / 6 {\displaystyle \pi ^{2}/6} essentially extended observations about finite polynomials and assumed that these same properties hold true for infinite series. Of course, Euler's original reasoning requires justification (100 years later, Karl Weierstrass proved that Euler's representation of

1650-411: A result, educators have been trying to abolish this stereotype by fostering confidence in math in all students in order to avoid math anxiety. While on the other hand, results obtained by Monika Szczygiel show that girls have a higher level of anxiety on testing and in total, although there is no gender difference in general learning math anxiety. Therefore, the gender gap in math anxiety may result from

1760-402: A teacher and a student, at times working on problems given to him by Miruka or Mr. Muraki, at other times teaching Tetra. He does not like to speak before large groups of people, but he is not hesitant to speak up when teaching Tetra and Miruka. Miruka is a second year high school student, in the same homeroom as the narrator. She studies mathematics with him following their first encounter under

1870-494: A variety of ways, including physical, psychological, and behavioral symptoms, that can all disrupt a student's mathematical performance. The strong negative correlation between high math anxiety and low achievement is often thought to be due to the impact of math anxiety on working memory. Working memory has a limited capacity. A large portion of this capacity is dedicated to problem-solving when solving mathematical tasks. However, in individuals with math anxiety, much of this space

1980-506: Is always a rational multiple of π 2 k {\displaystyle \pi ^{2k}} . In particular, since π {\displaystyle \pi } and integer powers of it are transcendental , we can conclude at this point that ζ ( 2 k ) {\displaystyle \zeta (2k)} is irrational , and more precisely, transcendental for all k ≥ 1 {\displaystyle k\geq 1} . By contrast,

2090-560: Is analytic , so taking the natural logarithm of both sides and differentiating yields π cos ⁡ ( π x ) sin ⁡ ( π x ) − 1 x = − ∑ n = 1 ∞ 2 x n 2 − x 2 {\displaystyle {\frac {\pi \cos(\pi x)}{\sin(\pi x)}}-{\frac {1}{x}}=-\sum _{n=1}^{\infty }{\frac {2x}{n^{2}-x^{2}}}} (by uniform convergence ,

2200-518: Is a classmate of the narrator's. According to the narrator, he has the best grades in his grade, and is also good at sports. At one point Miruka corners him and gives him a lecture on mathematics, but he moves away from her at the first opportunity. Ms. Mizutani is the school librarian. When it becomes time for school to close, she moves quietly to the middle of the library and announces that everyone must leave. She wears dark glasses that obscure any expression. See other language versions of Misplaced Pages for

2310-640: Is a need for addressing different learning styles. Math lessons can be tailored for visual / spatial , logical/mathematics, musical, auditory , body/kinesthetic , interpersonal and intrapersonal and verbal/linguistic learning styles. This theory of learning styles has never been demonstrated to be true in controlled trials. Studies show no evidence to support tailoring lessons to an individual students learning style to be beneficial. New concepts can be taught through play acting, cooperative groups, visual aids, hands on activities or information technology. To help with learning statistics, there are many applets found on

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2420-480: Is a phenomenon that is often considered when examining students' problems in mathematics. According to the American Psychological Association, mathematical anxiety is often linked to testing anxiety. This anxiety can cause distress and likely causes a dislike and avoidance of all math-related tasks. The academic study of math anxiety originates as early as the 1950s, when Mary Fides Gough introduced

2530-496: Is a problem in mathematical analysis with relevance to number theory , concerning an infinite sum of inverse squares. It was first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, and read on 5 December 1735 in The Saint Petersburg Academy of Sciences . Since the problem had withstood the attacks of the leading mathematicians of the day, Euler's solution brought him immediate fame when he

2640-534: Is apparently because he wants the students to think about every step of the problem, including its creation. In fact, when he once gives Miruka a problem for which she already knows the answer, he tells her that it is not necessarily the answer that he's after. He tells her that if she already knows the answer, to use the problem to find something interesting. He shows a strong interest in his students, for example by giving them problems tailored to their abilities. He eventually begins giving Tetra cards, as well. Tsunomiya

2750-523: Is considered psychometrically sound. Other tests are often given to measure different dimensionalities of math anxiety, such as Elizabeth Fennema and Julia Sherman's Fennema-Sherman Mathematics Attitudes Scales (FSMAS). The FSMAS evaluates nine specific domains using Likert-type scales: attitude toward success, mathematics as a male domain, mother's attitude, father's attitude, teacher's attitude, confidence in learning mathematics, mathematics anxiety, affectance motivation and mathematics usefulness. Despite

2860-554: Is defined in terms of the inner product on this Hilbert space given by ⟨ f , g ⟩ = ∫ 0 1 f ( x ) g ( x ) ¯ d x ,   f , g ∈ L per 2 ( 0 , 1 ) . {\displaystyle \langle f,g\rangle =\int _{0}^{1}f(x){\overline {g(x)}}\,dx,\ f,g\in L_{\operatorname {per} }^{2}(0,1).} We can consider

2970-460: Is depicted as lecturing about math to another student who had no particular interest because the narrator was not there to listen to her. She tends to sulk when she feels that she is being ignored. This is shown in scenes where the narrator ignores what she is saying, or when he finds himself daydreaming. Her actions become even more extreme when those she considers an "outsider" try to insinuate themselves into her social sphere. In one scene she finds

3080-498: Is detailed in expository fashion most notably in Havil's Gamma book which details many zeta function and logarithm -related series and integrals, as well as a historical perspective, related to the Euler gamma constant . Using formulae obtained from elementary symmetric polynomials , this same approach can be used to enumerate formulae for the even-indexed even zeta constants which have

3190-434: Is equally attractive with Miruka. When not in class, she spends most of her time in front of a piano. Mr. Muraki is a math teacher at the narrator's high school. The narrator refers to him as being "strange", but knows that Mr. Muraki likes him and the others. He gives the narrator and Miruka math problems. The problems are always written on index cards, and many are just equations with no explanation. His reason for doing so

3300-427: Is for" and "who STEM careers are for". These stereotypes can fuel mathematical anxiety that is already present among young female populations. Thus parity will take more work to overcome mathematical anxiety and this is one reason why women in mathematics are role models for younger women. According to John Taylor Gatto , as expounded in several lengthy books, modern Western schools were deliberately designed during

3410-473: Is handed a letter by another girl, a new student named Tetra. The letter she wrote is a request for the narrator to tutor her in math. He begins teaching her, making Miruka jealous. The narrator balances his friendship with Tetra and his romantic interest in Miruka until Miruka and Tetra become friends after Tetra demonstrates her dedication to learning mathematics. The protagonist of the books. The story in each book

Math Girls - Misplaced Pages Continue

3520-514: Is much more like studying, say, music or painting than it is like studying history or biology." Amongst others supporting this viewpoint is the work of Eugene Geist . Geist's recommendations include focusing on the concepts rather than the right answer and letting students work on their own and discuss their solutions before the answer is given. National Council of Teachers of Mathematics (NCTM) (1989, 1995b) suggestions for teachers seeking to prevent math anxiety include: Hackworth suggests that

3630-453: Is registered. And Trezise and Reeve show that students' math anxiety can fluctuate throughout the duration of a math class. The impact of mathematics anxiety on mathematics performance has been studied in more recent literature. An individual with math anxiety does not necessarily lack ability in mathematics, rather, they cannot perform to their full potential due to the interfering symptoms of their anxiety. Math anxiety manifests itself in

3740-585: Is related to poor math performance on math achievement tests and to negative attitudes concerning math. Hembree also suggests that math anxiety is directly connected with math avoidance. Ashcraft (2002) suggests that highly anxious math students will avoid situations in which they have to perform mathematical tasks. Unfortunately, math avoidance results in less competency, exposure and math practice, leaving students more anxious and mathematically unprepared to achieve. In college and university, anxious math students take fewer math courses and tend to feel negative toward

3850-479: Is relaxation exercises and indicates that by practicing relaxation techniques on a regular basis for 10–20 minutes students can significantly reduce their anxiety. Dr. Edmundo Jacobson's Progressive Muscle Relaxation taken from the book Mental Toughness Training for Sports, Loehr (1986) can be used in a modified form to reduce anxiety as posted on the website HypnoGenesis. According to Mina Bazargan and Mehdi Amiri, Modular Cognitive Behavior Therapy (MCBT) can reduce

3960-549: Is served by solving this problem?" and "why are we being asked to learn this?" Reflective journals help students develop metacognitive skills by having them think about their understanding. According to Pugalee, writing helps students organize their thinking which helps them better understand mathematics. Moreover, writing in mathematics classes helps students problem solve and improve mathematical reasoning. When students know how to use mathematical reasoning, they are less anxious about solving problems. Children learn best when math

4070-419: Is somewhat careless, though, which frequently leads to her forgetting to take mathematical conditions into consideration when trying to solve math problems. She occasionally shows flashes of deep mathematical insight that surprise even Miruka, however. Ay-Ay is a friend of Miruka's. She is in the same grade as the narrator and Miruka, but in a different class. She is the leader of the piano club "Fortissimo", and

4180-509: Is soon abandoned and replaced with a more refined and efficient strategy; children begin to perform addition and subtraction mentally at approximately six years of age. When children reach approximately eight years of age, they can retrieve answers to mathematical equations from memory. With proper instruction, most children acquire these basic mathematical skills and are able to solve more complex mathematical problems with sophisticated training. High-risk teaching styles are often explored to gain

4290-590: Is taken up by anxious thoughts, thus compromising the individual's ability to perform. In addition, a frequent reliance in schools on high-stakes and timed testing, where students tend to feel the most anxiety, can lead to lower achievement for math-anxious individuals. Programme for International Student Assessment (PISA) results demonstrate that students experiencing high math anxiety demonstrate mathematics scores that are 34 points lower than students who do not have math anxiety, equivalent to one full year of school. Besides, researchers Elisa Cargnelutti et al show that

4400-400: Is to calculate the answer. This component also has two subcomponents, namely the answer and the process or method used to determine the answer. Focusing more on the process or method enables students to make mistakes, but not "fail at math". The second component is to understand the mathematical concepts that underlay the problem being studied. "...   and in this respect studying mathematics

4510-494: Is told from his perspective. He is a second year student in a Japanese high school (equivalent to 11th grade in the US school system). His name is not given throughout the series. During middle school, he spent his time after school working on mathematics in the library. He begins to repeat this pattern in high school, but as his friendship with Miruka and Tetra develop he spends most of his time working on math problems with them. Other than

Math Girls - Misplaced Pages Continue

4620-710: The x terms (we are allowed to do so because of Newton's identities ), we see by induction that the x coefficient of ⁠ sin x / x ⁠ is − ( 1 π 2 + 1 4 π 2 + 1 9 π 2 + ⋯ ) = − 1 π 2 ∑ n = 1 ∞ 1 n 2 . {\displaystyle -\left({\frac {1}{\pi ^{2}}}+{\frac {1}{4\pi ^{2}}}+{\frac {1}{9\pi ^{2}}}+\cdots \right)=-{\frac {1}{\pi ^{2}}}\sum _{n=1}^{\infty }{\frac {1}{n^{2}}}.} But from

4730-542: The Basel problem , they become close friends. She has a deep knowledge of mathematics, and in most cases is seen leading the narrator and Tetra through discussions. She can play the piano, and sometimes plays arrangements for four hands with Ay-Ay during lunch. Tetra graduated from the same middle school as the narrator, and is a first year high school student, one year behind the narrator and Miruka. She suffers from mathematical anxiety , and after entering high school approaches

4840-1214: The arctangent addition formula and integrated with respect to α {\displaystyle \alpha } by means of trigonometric substitution , resulting in I ( α ) = − 1 2 arccos ⁡ ( α 2 ) 2 + c . {\displaystyle I(\alpha )=-{\frac {1}{2}}\arccos \left({\frac {\alpha }{2}}\right)^{2}+c.} The integration constant c {\displaystyle c} can be determined by noticing that two distinct values of I ( α ) {\displaystyle I(\alpha )} are related by I ( 2 ) = 4 I ( 0 ) , {\displaystyle I(2)=4I(0),} because when calculating I ( 2 ) {\displaystyle I(2)} we can factor 1 + 2 e − x + e − 2 x = ( 1 + e − x ) 2 {\displaystyle 1+2e^{-x}+e^{-2x}=(1+e^{-x})^{2}} and express it in terms of I ( 0 ) {\displaystyle I(0)} using

4950-880: The integral test , or by the following inequality: ∑ n = 1 N 1 n 2 < 1 + ∑ n = 2 N 1 n ( n − 1 ) = 1 + ∑ n = 2 N ( 1 n − 1 − 1 n ) = 1 + 1 − 1 N ⟶ N → ∞ 2. {\displaystyle {\begin{aligned}\sum _{n=1}^{N}{\frac {1}{n^{2}}}&<1+\sum _{n=2}^{N}{\frac {1}{n(n-1)}}\\&=1+\sum _{n=2}^{N}\left({\frac {1}{n-1}}-{\frac {1}{n}}\right)\\&=1+1-{\frac {1}{N}}\;{\stackrel {N\to \infty }{\longrightarrow }}\;2.\end{aligned}}} This gives us

5060-439: The limit as t {\displaystyle t} approaches zero and use L'Hôpital's rule thrice. By Tannery's theorem applied to lim t → ∞ ∑ n = 1 ∞ 1 / ( n 2 + 1 / t 2 ) {\textstyle \lim _{t\to \infty }\sum _{n=1}^{\infty }1/(n^{2}+1/t^{2})} , we can interchange

5170-697: The logarithm of a power identity and the substitution u = x / 2 {\displaystyle u=x/2} . This makes it possible to determine c = π 2 6 {\displaystyle c={\frac {\pi ^{2}}{6}}} , and it follows that I ( − 2 ) = 2 ∫ 0 ∞ ln ⁡ ( 1 − e − x ) d x = − π 2 3 . {\displaystyle I(-2)=2\int _{0}^{\infty }\ln(1-e^{-x})dx=-{\frac {\pi ^{2}}{3}}.} This final integral can be evaluated by expanding

5280-1898: The orthonormal basis on this space defined by e k ≡ e k ( ϑ ) := exp ⁡ ( 2 π ı k ϑ ) {\displaystyle e_{k}\equiv e_{k}(\vartheta ):=\exp(2\pi \imath k\vartheta )} such that ⟨ e k , e j ⟩ = ∫ 0 1 e 2 π ı ( k − j ) ϑ d ϑ = δ k , j {\displaystyle \langle e_{k},e_{j}\rangle =\int _{0}^{1}e^{2\pi \imath (k-j)\vartheta }\,d\vartheta =\delta _{k,j}} . Then if we take f ( ϑ ) := ϑ {\displaystyle f(\vartheta ):=\vartheta } , we can compute both that ‖ f ‖ 2 = ∫ 0 1 ϑ 2 d ϑ = 1 3 ⟨ f , e k ⟩ = ∫ 0 1 ϑ e − 2 π ı k ϑ d ϑ = { 1 2 , k = 0 − 1 2 π ı k k ≠ 0 , {\displaystyle {\begin{aligned}\|f\|^{2}&=\int _{0}^{1}\vartheta ^{2}\,d\vartheta ={\frac {1}{3}}\\\langle f,e_{k}\rangle &=\int _{0}^{1}\vartheta e^{-2\pi \imath k\vartheta }\,d\vartheta ={\Biggl \{}{\begin{array}{ll}{\frac {1}{2}},&k=0\\-{\frac {1}{2\pi \imath k}}&k\neq 0,\end{array}}\end{aligned}}} by elementary calculus and integration by parts , respectively. Finally, by Parseval's identity stated in

5390-752: The primitive function of the integrand cannot be expressed in terms of elementary functions, by differentiating with respect to α {\displaystyle \alpha } we arrive at d I d α = ∫ 0 ∞ e − x 1 + α e − x + e − 2 x d x , {\displaystyle {\frac {dI}{d\alpha }}=\int _{0}^{\infty }{\frac {e^{-x}}{1+\alpha e^{-x}+e^{-2x}}}dx,} which can be integrated by substituting u = e − x {\displaystyle u=e^{-x}} and decomposing into partial fractions . In

5500-1257: The recurrence relation that I j , k = { 1 j + 1 , k = 0 ; − 1 2 π ı ⋅ k + j 2 π ı ⋅ k I j − 1 , k , k ≠ 0 = { 1 j + 1 , k = 0 ; − ∑ m = 1 j j ! ( j + 1 − m ) ! ⋅ 1 ( 2 π ı ⋅ k ) m , k ≠ 0. {\displaystyle {\begin{aligned}I_{j,k}&={\begin{cases}{\frac {1}{j+1}},&k=0;\\[4pt]-{\frac {1}{2\pi \imath \cdot k}}+{\frac {j}{2\pi \imath \cdot k}}I_{j-1,k},&k\neq 0\end{cases}}\\[6pt]&={\begin{cases}{\frac {1}{j+1}},&k=0;\\[4pt]-\sum \limits _{m=1}^{j}{\frac {j!}{(j+1-m)!}}\cdot {\frac {1}{(2\pi \imath \cdot k)^{m}}},&k\neq 0.\end{cases}}\end{aligned}}} Then by applying Parseval's identity as we did for

5610-818: The sine function sin ⁡ x = x − x 3 3 ! + x 5 5 ! − x 7 7 ! + ⋯ {\displaystyle \sin x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots } Dividing through by x {\displaystyle x} gives sin ⁡ x x = 1 − x 2 3 ! + x 4 5 ! − x 6 7 ! + ⋯ . {\displaystyle {\frac {\sin x}{x}}=1-{\frac {x^{2}}{3!}}+{\frac {x^{4}}{5!}}-{\frac {x^{6}}{7!}}+\cdots .} The Weierstrass factorization theorem shows that

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5720-898: The upper bound 2, and because the infinite sum contains no negative terms, it must converge to a value strictly between 0 and 2. It can be shown that ζ ( s ) has a simple expression in terms of the Bernoulli numbers whenever s is a positive even integer. With s = 2 n : ζ ( 2 n ) = ( 2 π ) 2 n ( − 1 ) n + 1 B 2 n 2 ⋅ ( 2 n ) ! . {\displaystyle \zeta (2n)={\frac {(2\pi )^{2n}(-1)^{n+1}B_{2n}}{2\cdot (2n)!}}.} The normalized sinc function sinc ( x ) = sin ⁡ ( π x ) π x {\displaystyle {\text{sinc}}(x)={\frac {\sin(\pi x)}{\pi x}}} has

5830-734: The (finite) power sums H n ( 2 k ) {\displaystyle H_{n}^{(2k)}} in terms of the elementary symmetric polynomials , e i ≡ e i ( − π 2 1 2 , − π 2 2 2 , − π 2 3 2 , − π 2 4 2 , … ) , {\displaystyle e_{i}\equiv e_{i}\left(-{\frac {\pi ^{2}}{1^{2}}},-{\frac {\pi ^{2}}{2^{2}}},-{\frac {\pi ^{2}}{3^{2}}},-{\frac {\pi ^{2}}{4^{2}}},\ldots \right),} but we can go

5940-422: The 20 countries examined in this study, 13-year-old boys tended to score higher than girls. Moreover, mathematics is often labeled as a masculine ability; as a result, girls often have low confidence in their math capabilities. These gender stereotypes can reinforce low confidence in girls and can cause math anxiety as research has shown that performance on standardized math tests is affected by one's confidence. As

6050-630: The Internet that help students learn about many things from probability distributions to linear regression. These applets are commonly used in introductory statistics classes, as many students benefit from using them. Active learners ask critical questions, such as: Why do we do it this way, and not that way ? Some teachers may find these questions annoying or difficult to answer, and indeed may have been trained to respond to such questions with hostility and contempt, designed to instill fear. Better teachers respond eagerly to these questions, and use them to help

6160-489: The MARS does measure mathematics anxiety". This test was intended for use in diagnosing math anxiety, testing the efficacy of different math anxiety treatment approaches and possibly designing an anxiety hierarchy to be used in desensitization treatments. The MARS test is of interest to those in counseling psychology and the test is used profusely in math anxiety research. It is available in several versions of varying lengths and

6270-982: The Math Girls Talk About... ( 数学ガールの秘密ノート , Sūgaku Gāru no himitsu nōto , lit., 'Secret Notebooks of the Math Girls') series of mathematics primers. These books take the form of characters from the Math Girls series discussing various topics from mathematics, but can be considered nonfiction in that they are intended to be strictly instructional, and do not advance the storyline of the Math Girls series. The following titles from this series are available in English translation from Bento Books : Other books in this series currently available only in Japanese cover sequences and series , calculus , vectors , probability , and statistics . Basel problem The Basel problem

6380-725: The construct of this test by sharing previous results from three other studies that were very similar to the results achieved in this study. They also administered the Differential Aptitude Test, a 10-minute math test including simple to complex problems. Calculation of the Pearson product-moment correlation coefficient between the MARS test and Differential Aptitude Test scores was −0.64 (p < .01), indicating that higher MARS scores relate to lower math test scores and "since high anxiety interferes with performance, and poor performance produces anxiety, this result provides evidence that

6490-1265: The corresponding hyperbolic function : π cot ⁡ ( − π i t ) 2 i t = π 2 t i cot ⁡ ( π i t ) = π 2 t coth ⁡ ( π t ) . {\displaystyle {\frac {\pi \cot(-\pi it)}{2it}}={\frac {\pi }{2t}}{i\cot(\pi it)}={\frac {\pi }{2t}}\coth(\pi t).} Then ∑ n = 1 ∞ 1 n 2 + t 2 = π ( t e 2 π t + t ) − e 2 π t + 1 2 ( t 2 e 2 π t − t 2 ) = − 1 2 t 2 + π 2 t coth ⁡ ( π t ) . {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{2}+t^{2}}}={\frac {\pi \left(te^{2\pi t}+t\right)-e^{2\pi t}+1}{2\left(t^{2}e^{2\pi t}-t^{2}\right)}}=-{\frac {1}{2t^{2}}}+{\frac {\pi }{2t}}\coth(\pi t).} Now we take

6600-690: The courses. In the same culture, there is little difference in anxiety scale that is associated with gender, while the anxiety is more related with its type. Samples show greater degree of anxiety at subscale. MEA (Mathematical Evaluation Anxiety) compared with LMA (Learning Mathematical Anxiety). Another difference in mathematic abilities often explored in research concerns gender disparities. There has been research examining gender differences in performance on standardized tests across various countries. Beller and Gafni's have shown that children at approximately nine years of age do not show consistent gender differences in relation to math skills. However, in 17 out of

6710-426: The fact that he wears glasses, his physical characteristics are not described. He has a quiet personality, but is somewhat self-conscious of his mathematical ability as compared to Miruka and Tetra, and when he has difficulty solving a math problem he tends to become depressed. He is very conscious of Miruka and Tetra as potential love interests, but is too shy to initiate a relationship with either one. He acts as both

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6820-1270: The first case above along with the linearity of the inner product yields that ‖ f j ‖ 2 = 1 2 j + 1 = 2 ∑ k ≥ 1 I j , k I ¯ j , k + 1 ( j + 1 ) 2 = 2 ∑ m = 1 j ∑ r = 1 j j ! 2 ( j + 1 − m ) ! ( j + 1 − r ) ! ( − 1 ) r ı m + r ζ ( m + r ) ( 2 π ) m + r + 1 ( j + 1 ) 2 . {\displaystyle {\begin{aligned}\|f_{j}\|^{2}={\frac {1}{2j+1}}&=2\sum _{k\geq 1}I_{j,k}{\bar {I}}_{j,k}+{\frac {1}{(j+1)^{2}}}\\[6pt]&=2\sum _{m=1}^{j}\sum _{r=1}^{j}{\frac {j!^{2}}{(j+1-m)!(j+1-r)!}}{\frac {(-1)^{r}}{\imath ^{m+r}}}{\frac {\zeta (m+r)}{(2\pi )^{m+r}}}+{\frac {1}{(j+1)^{2}}}.\end{aligned}}} It's possible to prove

6930-447: The following activities can help students in reducing and mitigating mathematical anxiety: B R Alimin and D B Widjajanti recommend teachers: Several studies have shown that relaxation techniques, including controlled breathing, can be used to help alleviate anxiety related to mathematics. In her workbook Conquering Math Anxiety , Cynthia Arem offers specific strategies to reduce math avoidance and anxiety. One strategy she advocates for

7040-946: The following formula: ζ ( s ) = ∑ n = 1 ∞ 1 n s . {\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}.} Taking s = 2 , we see that ζ (2) is equal to the sum of the reciprocals of the squares of all positive integers: ζ ( 2 ) = ∑ n = 1 ∞ 1 n 2 = 1 1 2 + 1 2 2 + 1 3 2 + 1 4 2 + ⋯ = π 2 6 ≈ 1.644934. {\displaystyle \zeta (2)=\sum _{n=1}^{\infty }{\frac {1}{n^{2}}}={\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{4^{2}}}+\cdots ={\frac {\pi ^{2}}{6}}\approx 1.644934.} Convergence can be proven by

7150-416: The following known formula expanded by the Bernoulli numbers : ζ ( 2 n ) = ( − 1 ) n − 1 ( 2 π ) 2 n 2 ⋅ ( 2 n ) ! B 2 n . {\displaystyle \zeta (2n)={\frac {(-1)^{n-1}(2\pi )^{2n}}{2\cdot (2n)!}}B_{2n}.} For example, let

7260-1950: The form above, we obtain that ‖ f ‖ 2 = 1 3 = ∑ k ≠ 0 k = − ∞ ∞ 1 ( 2 π k ) 2 + 1 4 = 2 ∑ k = 1 ∞ 1 ( 2 π k ) 2 + 1 4 ⟹ π 2 6 = 2 π 2 3 − π 2 2 = ζ ( 2 ) . {\displaystyle {\begin{aligned}\|f\|^{2}={\frac {1}{3}}&=\sum _{\stackrel {k=-\infty }{k\neq 0}}^{\infty }{\frac {1}{(2\pi k)^{2}}}+{\frac {1}{4}}=2\sum _{k=1}^{\infty }{\frac {1}{(2\pi k)^{2}}}+{\frac {1}{4}}\\&\implies {\frac {\pi ^{2}}{6}}={\frac {2\pi ^{2}}{3}}-{\frac {\pi ^{2}}{2}}=\zeta (2).\end{aligned}}} Note that by considering higher-order powers of f j ( ϑ ) := ϑ j ∈ L per 2 ( 0 , 1 ) {\displaystyle f_{j}(\vartheta ):=\vartheta ^{j}\in L_{\operatorname {per} }^{2}(0,1)} we can use integration by parts to extend this method to enumerating formulas for ζ ( 2 j ) {\displaystyle \zeta (2j)} when j > 1 {\displaystyle j>1} . In particular, suppose we let I j , k := ∫ 0 1 ϑ j e − 2 π ı k ϑ d ϑ , {\displaystyle I_{j,k}:=\int _{0}^{1}\vartheta ^{j}e^{-2\pi \imath k\vartheta }\,d\vartheta ,} so that integration by parts yields

7370-2672: The function f ( x ) = x ) to obtain ∑ n = − ∞ ∞ | c n | 2 = 1 2 π ∫ − π π x 2 d x , {\displaystyle \sum _{n=-\infty }^{\infty }|c_{n}|^{2}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }x^{2}\,dx,} where c n = 1 2 π ∫ − π π x e − i n x d x = n π cos ⁡ ( n π ) − sin ⁡ ( n π ) π n 2 i = cos ⁡ ( n π ) n i = ( − 1 ) n n i {\displaystyle {\begin{aligned}c_{n}&={\frac {1}{2\pi }}\int _{-\pi }^{\pi }xe^{-inx}\,dx\\[4pt]&={\frac {n\pi \cos(n\pi )-\sin(n\pi )}{\pi n^{2}}}i\\[4pt]&={\frac {\cos(n\pi )}{n}}i\\[4pt]&={\frac {(-1)^{n}}{n}}i\end{aligned}}} for n ≠ 0 , and c 0 = 0 . Thus, | c n | 2 = { 1 n 2 , for  n ≠ 0 , 0 , for  n = 0 , {\displaystyle |c_{n}|^{2}={\begin{cases}{\dfrac {1}{n^{2}}},&{\text{for }}n\neq 0,\\0,&{\text{for }}n=0,\end{cases}}} and ∑ n = − ∞ ∞ | c n | 2 = 2 ∑ n = 1 ∞ 1 n 2 = 1 2 π ∫ − π π x 2 d x . {\displaystyle \sum _{n=-\infty }^{\infty }|c_{n}|^{2}=2\sum _{n=1}^{\infty }{\frac {1}{n^{2}}}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }x^{2}\,dx.} Therefore, ∑ n = 1 ∞ 1 n 2 = 1 4 π ∫ − π π x 2 d x = π 2 6 {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{2}}}={\frac {1}{4\pi }}\int _{-\pi }^{\pi }x^{2}\,dx={\frac {\pi ^{2}}{6}}} as required. Given

7480-477: The gender gap has almost been reversed, showing an increase in female presence. This is being caused by women's steadily increasing performance on math and science testing and enrollment, but also by males' losing ground at the same time. This role reversal can largely be associated with the gender normative stereotypes that are found in the Science, technology, engineering, and mathematics (STEM) field, deeming "who math

7590-461: The influence of mathematical anxiety on math-related performance increases over time due to the accumulation of passive experience in the subject or other factors like more requirements on mathematics as children grow up. These findings demonstrate the clear link between math anxiety and reduced levels of achievement, suggesting that alleviating math anxiety may lead to a marked improvement in student achievement. A rating scale for mathematics anxiety

7700-476: The influence of parents' and teachers' attitudes on " 'the child's expectations in that area of learning.'...   It is less the actual teaching and more the attitude and expectations of the teacher or parents that count". This is further supported by a survey of Montgomery County, Maryland students who "pointed to their parents as the primary force behind the interest in mathematics". Claudia Zaslavsky contends that math has two components. The first component

7810-591: The interchange of the derivative and infinite series is permissible). After dividing the equation by 2 x {\displaystyle 2x} and regrouping one gets 1 2 x 2 − π cot ⁡ ( π x ) 2 x = ∑ n = 1 ∞ 1 n 2 − x 2 . {\displaystyle {\frac {1}{2x^{2}}}-{\frac {\pi \cot(\pi x)}{2x}}=\sum _{n=1}^{\infty }{\frac {1}{n^{2}-x^{2}}}.} We make

7920-769: The introduction of newer instrumentation, the use of the MARS test appears to be the educational standard for measuring math anxiety due to its specificity and prolific use. While there are overarching similarities concerning the acquisition of math skills, researchers have shown that children's mathematical abilities differ across countries. In Canada, students score substantially lower in math problem-solving and operations than students in Korea, India and Singapore. Researchers have conducted thorough comparisons between countries and determined that in some areas, such as Taiwan and Japan, parents place more emphasis on effort rather than one's innate intellectual ability in school success. By placing

8030-438: The lack of competency in math because of math avoidance. Ashcraft determined that by administering a test that becomes increasingly more mathematically challenging, he noticed that even highly math-anxious individuals do well on the first portion of the test measuring performance. However, on the latter and more difficult portion of the test, there was a stronger negative relationship between accuracy and math anxiety. According to

8140-474: The late 19th century to create an environment which is ideal for fostering fear and anxiety, and for preventing or delaying learning. Many who are sympathetic to Gatto's thesis regard his position as unnecessarily extreme. Diane Ravitch , former assistant secretary of education during the George H. W. Bush administration, agrees with Gatto up to a point, conceding that there is an element of social engineering (i.e.

8250-1641: The left-hand side is the product of linear factors given by its roots, just as for finite polynomials. Euler assumed this as a heuristic for expanding an infinite degree polynomial in terms of its roots, but in fact it is not always true for general P ( x ) {\displaystyle P(x)} . This factorization expands the equation into: sin ⁡ x x = ( 1 − x π ) ( 1 + x π ) ( 1 − x 2 π ) ( 1 + x 2 π ) ( 1 − x 3 π ) ( 1 + x 3 π ) ⋯ = ( 1 − x 2 π 2 ) ( 1 − x 2 4 π 2 ) ( 1 − x 2 9 π 2 ) ⋯ {\displaystyle {\begin{aligned}{\frac {\sin x}{x}}&=\left(1-{\frac {x}{\pi }}\right)\left(1+{\frac {x}{\pi }}\right)\left(1-{\frac {x}{2\pi }}\right)\left(1+{\frac {x}{2\pi }}\right)\left(1-{\frac {x}{3\pi }}\right)\left(1+{\frac {x}{3\pi }}\right)\cdots \\&=\left(1-{\frac {x^{2}}{\pi ^{2}}}\right)\left(1-{\frac {x^{2}}{4\pi ^{2}}}\right)\left(1-{\frac {x^{2}}{9\pi ^{2}}}\right)\cdots \end{aligned}}} If we formally multiply out this product and collect all

8360-578: The level of mathematical anxiety and increase students' self-esteem. Visualization has also been used effectively to help reduce math anxiety. Arem has a chapter that deals with reducing test anxiety and advocates the use of visualization. In her chapter titled Conquer Test Anxiety (Chapter 9) she has specific exercises devoted to visualization techniques to help the student feel calm and confident during testing. Studies have shown students learn best when they are active rather than passive learners. The theory of multiple intelligences suggests that there

8470-2008: The limit and infinite series so that lim t → 0 ∑ n = 1 ∞ 1 / ( n 2 + t 2 ) = ∑ n = 1 ∞ 1 / n 2 {\textstyle \lim _{t\to 0}\sum _{n=1}^{\infty }1/(n^{2}+t^{2})=\sum _{n=1}^{\infty }1/n^{2}} and by L'Hôpital's rule ∑ n = 1 ∞ 1 n 2 = lim t → 0 π 4 2 π t e 2 π t − e 2 π t + 1 π t 2 e 2 π t + t e 2 π t − t = lim t → 0 π 3 t e 2 π t 2 π ( π t 2 e 2 π t + 2 t e 2 π t ) + e 2 π t − 1 = lim t → 0 π 2 ( 2 π t + 1 ) 4 π 2 t 2 + 12 π t + 6 = π 2 6 . {\displaystyle {\begin{aligned}\sum _{n=1}^{\infty }{\frac {1}{n^{2}}}&=\lim _{t\to 0}{\frac {\pi }{4}}{\frac {2\pi te^{2\pi t}-e^{2\pi t}+1}{\pi t^{2}e^{2\pi t}+te^{2\pi t}-t}}\\[6pt]&=\lim _{t\to 0}{\frac {\pi ^{3}te^{2\pi t}}{2\pi \left(\pi t^{2}e^{2\pi t}+2te^{2\pi t}\right)+e^{2\pi t}-1}}\\[6pt]&=\lim _{t\to 0}{\frac {\pi ^{2}(2\pi t+1)}{4\pi ^{2}t^{2}+12\pi t+6}}\\[6pt]&={\frac {\pi ^{2}}{6}}.\end{aligned}}} Use Parseval's identity (applied to

8580-752: The limiting recurrence relation (or generating function convolution, or product ) expanded as π 2 k 2 ⋅ ( 2 k ) ⋅ ( − 1 ) k ( 2 k + 1 ) ! = − [ x 2 k ] sin ⁡ ( π x ) π x × ∑ i ≥ 1 ζ ( 2 i ) x i . {\displaystyle {\frac {\pi ^{2k}}{2}}\cdot {\frac {(2k)\cdot (-1)^{k}}{(2k+1)!}}=-[x^{2k}]{\frac {\sin(\pi x)}{\pi x}}\times \sum _{i\geq 1}\zeta (2i)x^{i}.} Then by differentiation and rearrangement of

8690-458: The manufacture of the compliant citizenry) in the construction of the American education system, which prioritizes conformance over learning. The role of attachment has been suggested as having an impact in the development of the anxiety. Children with an insecure attachment style were more likely to demonstrate the anxiety. Math used to be taught as a right and wrong subject and as if getting

8800-399: The narrator teaching math to Tetra (who she has never met) in the library, and responds by kicking her chair out from under her. She also stomps on the narrator's foot and walks off alone when she goes to meet him, but finds him talking to Tetra. She stops such behavior after becoming friends with Tetra. There are scenes in the book that indicate that she thinks of the narrator as more than just

8910-400: The narrator to ask him to tutor her in math. She is petite with short hair and large eyes, and is a sweet and energetic girl. The narrator comments that she reminds him of a squirrel eating a nut. Tetra's expression clearly shows what she is thinking, which the narrator appreciates because it lets him know if she is following his explanations. She uses exaggerated gestures, frequently making

9020-407: The narrator when they meet with a series of math problems. She tends to like anyone who shows an interest in math. She is at first ambivalent, even antagonistic towards Tetra. She later warms up to Tetra, though, when she sees that she is earnestly pursuing mathematical studies with the narrator, and after she hears that Tetra has independently come up with an idea that leads to an elegant solution to

9130-480: The narrator wish that she would calm down. She is romantically attracted to the narrator, who she also greatly respects because he takes the time to listen to what she has to say. She also thinks very highly of Miruka, and is also romantically attracted to her, what with her being bisexual She first begins studying mathematics seriously because she thinks that she would like to go on to some kind of computer-related work, or some other career that might involve math. She

9240-455: The natural logarithm into its Taylor series : Mathematical anxiety Mathematical anxiety , also known as math phobia , is a feeling of tension and anxiety that interferes with the manipulation of numbers and the solving of mathematical problems in daily life and academic situations. Mark H. Ashcraft defines math anxiety as "a feeling of tension, apprehension, or fear that interferes with math performance" (2002, p. 1). It

9350-470: The normative data, including a mean score of 215.38 with a standard deviation of 65.29, collected from 397 students that replied to an advertisement for behavior therapy treatment for math anxiety. For test-retest reliability, the Pearson product-moment coefficient was used and a score of 0.85 was calculated, which was favorable and comparable to scores found on other anxiety tests. Richardson and Suinn validated

9460-570: The original infinite series expansion of ⁠ sin x / x ⁠ , the coefficient of x is − ⁠ 1 / 3! ⁠ = − ⁠ 1 / 6 ⁠ . These two coefficients must be equal; thus, − 1 6 = − 1 π 2 ∑ n = 1 ∞ 1 n 2 . {\displaystyle -{\frac {1}{6}}=-{\frac {1}{\pi ^{2}}}\sum _{n=1}^{\infty }{\frac {1}{n^{2}}}.} Multiplying both sides of this equation by − π gives

9570-3672: The partial product for sin ⁡ ( x ) {\displaystyle \sin(x)} expanded as above be defined by S n ( x ) x := ∏ k = 1 n ( 1 − x 2 k 2 ⋅ π 2 ) {\displaystyle {\frac {S_{n}(x)}{x}}:=\prod \limits _{k=1}^{n}\left(1-{\frac {x^{2}}{k^{2}\cdot \pi ^{2}}}\right)} . Then using known formulas for elementary symmetric polynomials (a.k.a., Newton's formulas expanded in terms of power sum identities), we can see (for example) that [ x 4 ] S n ( x ) x = 1 2 π 4 ( ( H n ( 2 ) ) 2 − H n ( 4 ) ) → n → ∞ 1 2 π 4 ( ζ ( 2 ) 2 − ζ ( 4 ) ) ⟹ ζ ( 4 ) = π 4 90 = − 2 π 4 ⋅ [ x 4 ] sin ⁡ ( x ) x + π 4 36 [ x 6 ] S n ( x ) x = − 1 6 π 6 ( ( H n ( 2 ) ) 3 − 2 H n ( 2 ) H n ( 4 ) + 2 H n ( 6 ) ) → n → ∞ 1 6 π 6 ( ζ ( 2 ) 3 − 3 ζ ( 2 ) ζ ( 4 ) + 2 ζ ( 6 ) ) ⟹ ζ ( 6 ) = π 6 945 = − 3 ⋅ π 6 [ x 6 ] sin ⁡ ( x ) x − 2 3 π 2 6 π 4 90 + π 6 216 , {\displaystyle {\begin{aligned}\left[x^{4}\right]{\frac {S_{n}(x)}{x}}&={\frac {1}{2\pi ^{4}}}\left(\left(H_{n}^{(2)}\right)^{2}-H_{n}^{(4)}\right)\qquad \xrightarrow {n\rightarrow \infty } \qquad {\frac {1}{2\pi ^{4}}}\left(\zeta (2)^{2}-\zeta (4)\right)\\[4pt]&\qquad \implies \zeta (4)={\frac {\pi ^{4}}{90}}=-2\pi ^{4}\cdot [x^{4}]{\frac {\sin(x)}{x}}+{\frac {\pi ^{4}}{36}}\\[8pt]\left[x^{6}\right]{\frac {S_{n}(x)}{x}}&=-{\frac {1}{6\pi ^{6}}}\left(\left(H_{n}^{(2)}\right)^{3}-2H_{n}^{(2)}H_{n}^{(4)}+2H_{n}^{(6)}\right)\qquad \xrightarrow {n\rightarrow \infty } \qquad {\frac {1}{6\pi ^{6}}}\left(\zeta (2)^{3}-3\zeta (2)\zeta (4)+2\zeta (6)\right)\\[4pt]&\qquad \implies \zeta (6)={\frac {\pi ^{6}}{945}}=-3\cdot \pi ^{6}[x^{6}]{\frac {\sin(x)}{x}}-{\frac {2}{3}}{\frac {\pi ^{2}}{6}}{\frac {\pi ^{4}}{90}}+{\frac {\pi ^{6}}{216}},\end{aligned}}} and so on for subsequent coefficients of [ x 2 k ] S n ( x ) x {\displaystyle [x^{2k}]{\frac {S_{n}(x)}{x}}} . There are other forms of Newton's identities expressing

9680-548: The problem. The Basel problem asks for the precise summation of the reciprocals of the squares of the natural numbers , i.e. the precise sum of the infinite series : ∑ n = 1 ∞ 1 n 2 = 1 1 2 + 1 2 2 + 1 3 2 + ⋯ . {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{2}}}={\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+\cdots .} The sum of

9790-468: The problem. A growth mindset can benefit everyone, not just people trying to solve math computations. Moreover, parents in these countries tend to set higher expectations and standards for their children. In turn, students spend more time on homework and value homework more than American children. In addition, researchers Jennifer L. Brown et al. shows that difference in level of mathematical anxiety among different countries may result from varying degrees of

9900-516: The profession depending on coursework preparation. A study conducted by Kawakami, Steele, Cifa, Phills, and Dovidio examined attitudes towards math and behavior during math examinations. The study examined the effect of extensive training in teaching women how to approach math. The results showed that women who were trained to approach rather than avoid math showed a positive implicit attitude towards math. These findings were only consistent with women low in initial identification with math. This study

10010-451: The properties of the odd-indexed zeta constants , including Apéry's constant ζ ( 3 ) {\displaystyle \zeta (3)} , are almost completely unknown. The Riemann zeta function ζ ( s ) is one of the most significant functions in mathematics because of its relationship to the distribution of the prime numbers . The zeta function is defined for any complex number s with real part greater than 1 by

10120-815: The range − 2 ≤ α ≤ 2 {\displaystyle -2\leq \alpha \leq 2} the definite integral reduces to d I d α = 2 4 − α 2 [ arctan ⁡ ( α + 2 4 − α 2 ) − arctan ⁡ ( α 4 − α 2 ) ] . {\displaystyle {\frac {dI}{d\alpha }}={\frac {2}{\sqrt {4-\alpha ^{2}}}}\left[\arctan \left({\frac {\alpha +2}{\sqrt {4-\alpha ^{2}}}}\right)-\arctan \left({\frac {\alpha }{\sqrt {4-\alpha ^{2}}}}\right)\right].} The expression can be simplified using

10230-464: The research found at the University of Chicago by Sian Beilock and her group, math anxiety is not simply about being bad at math. After using brain scans, scholars confirmed that the anticipation or the thought of solving math actually causes math anxiety. The brain scans showed that the area of the brain that is triggered when someone has math anxiety overlaps the same area of the brain where bodily harm

10340-479: The result using elementary calculus by applying the differentiation under the integral sign technique to an integral due to Freitas: I ( α ) = ∫ 0 ∞ ln ⁡ ( 1 + α e − x + e − 2 x ) d x . {\displaystyle I(\alpha )=\int _{0}^{\infty }\ln \left(1+\alpha e^{-x}+e^{-2x}\right)dx.} While

10450-444: The right answer were paramount. In contrast to most subjects, mathematics problems almost always have a right answer but there are many ways to obtain the answer. Previously, the subject was often taught as if there were a right way to solve the problem and any other approaches would be wrong, even if students got the right answer. Thankfully, mathematics has evolved and so has teaching it. Students used to have higher anxiety because of

10560-401: The series is approximately equal to 1.644934. The Basel problem asks for the exact sum of this series (in closed form ), as well as a proof that this sum is correct. Euler found the exact sum to be π 2 / 6 {\displaystyle \pi ^{2}/6} and announced this discovery in 1735. His arguments were based on manipulations that were not justified at

10670-521: The sine function as an infinite product is valid, by the Weierstrass factorization theorem ), but even without justification, by simply obtaining the correct value, he was able to verify it numerically against partial sums of the series. The agreement he observed gave him sufficient confidence to announce his result to the mathematical community. To follow Euler's argument, recall the Taylor series expansion of

10780-408: The students deepen their understanding by examining alternative methods so the students can choose for themselves which method they prefer. This process can result in meaningful class discussions. Talking is the way in which students increase their understanding and command of math. Teachers can give students insight as to why they learn certain content by asking students questions such as "what purpose

10890-399: The subject. In fact, Ashcraft found that the correlation between math anxiety and variables such as self- confidence and motivation in math is strongly negative . According to Schar, because math anxiety can cause math avoidance, an empirical dilemma arises. For instance, when a highly math-anxious student performs disappointingly on a math question, it could be due to math anxiety or

11000-391: The sum of the reciprocals of the positive square integers. ∑ n = 1 ∞ 1 n 2 = π 2 6 . {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{2}}}={\frac {\pi ^{2}}{6}}.} This method of calculating ζ ( 2 ) {\displaystyle \zeta (2)}

11110-399: The term mathemaphobia to describe the phobia-like feelings of many towards mathematics. The first math anxiety measurement scale was developed by Richardson and Suinn in 1972. Since this development, several researchers have examined math anxiety in empirical studies . Hembree (1990) conducted a meta-analysis of 151 studies concerning math anxiety. The study determined that math anxiety

11220-470: The terms in the previous equation, we obtain that ζ ( 2 k ) = [ x 2 k ] 1 2 ( 1 − π x cot ⁡ ( π x ) ) . {\displaystyle \zeta (2k)=[x^{2k}]{\frac {1}{2}}\left(1-\pi x\cot(\pi x)\right).} By the above results, we can conclude that ζ ( 2 k ) {\displaystyle \zeta (2k)}

11330-400: The time, although he was later proven correct. He produced an accepted proof in 1741. The solution to this problem can be used to estimate the probability that two large random numbers are coprime . Two random integers in the range from 1 to n {\displaystyle n} , in the limit as n {\displaystyle n} goes to infinity, are relatively prime with

11440-579: The times, as she has explored in her book Women Becoming Mathematicians: Creating a Professional Identity in Post-World War II America , "Since 1980, women have earned over 17 percent of the mathematics doctorates.... [In The United States]". The trends in gender are by no means clear, but perhaps parity is still a way to go. Since 1995, studies have shown that the gender gap favored males in most mathematical standardized testing as boys outperformed girls in 15 out of 28 countries. However, as of 2015

11550-841: The translations of the other books in the series "Math Girls." A Math Girls manga, illustrated by Mika Hisaka, serialized 14 chapters between April 2008 and June 2009 in Comic Flapper (except for the November 2008 issue). The chapters were subsequently published in two tankōbon volumes. This was followed by subsequent manga and tankōbon versions of Math Girls 2: Fermat's Last Theorem (illustrated by Kasuga Shun) and Math Girls 3: Gödel's Incompleteness Theorems (illustrated by Matsuzaki Miyuki). Math Girls Manga appeared in English translation from Bento Books in 2013 ( ISBN   978-0983951346 ), followed by Math Girls Manga 2 in 2016 ( ISBN   978-0983951353 ). Hiroshi Yuki also authors

11660-476: The type of anxiety. Tests triggers greater anxiety in girls compared with boys, but they feel same level of anxiety learning math. The principles of mathematics are generally understood at an early age; preschoolers can comprehend the majority of principles underlying counting. By kindergarten, it is common for children to use counting in a more sophisticated manner by adding and subtracting numbers. While kindergarteners tend to use their fingers to count, this habit

11770-414: The way math was taught. "Teachers benefit children most when they encourage them to share their thinking process and justify their answers out loud or in writing as they perform math operations. ... With less of an emphasis on right or wrong and more of an emphasis on process, teachers can help alleviate students' anxiety about math". There have been many studies that show parent involvement in developing

11880-410: Was developed in 1972 by Richardson and Suinn. Richardson and Suinn defined mathematical anxiety as "feelings of apprehension and tension concerning manipulation of numbers and completion of mathematical problems in various contexts". Richardson and Suinn introduced the MARS (Mathematics Anxiety Rating Scale) in 1972. Elevated scores on the MARS test translate to high math anxiety. The authors presented

11990-421: Was replicated with women who were either encouraged to approach math or who received neutral training. Results were consistent and demonstrated that women taught to approach math had an implicit positive attitude and completed more math problems than women taught to approach math in a neutral manner. Johns, Schmader, and Martens conducted a study in which they examined the effect of teaching stereotype threat as

12100-461: Was twenty-eight. Euler generalised the problem considerably, and his ideas were taken up more than a century later by Bernhard Riemann in his seminal 1859 paper " On the Number of Primes Less Than a Given Magnitude ", in which he defined his zeta function and proved its basic properties. The problem is named after Basel , hometown of Euler as well as of the Bernoulli family who unsuccessfully attacked

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