Mathematica を活用する数学教材とその検証 (数式処理と教育)

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1 $\bullet$ $\bullet$ Mathematica (Shuichi Yamamoto) College of Science and Technology, Nihon University 1 21 ( ) 1 3 (1) ( ) (2 ) ( ) 10 Mathematica ( ) Mathematica $?$ 10 Mathematica ( ) Mathematica 1

2 $\bullet$ 116 (i) (ii) (iii) (2007 ) Mathematica Web (1) (2) ( ) (3) Mathematica (4) Mathematica (5) Mathematica (6) (7) (8) (9) (10) 2 (11) $n$ (12) (13) (14) (15) a $b$ $c$ $d$ $e$ Mathematica (i) Mathematica NBP (ii) ( ) Mathematica ( ) ( ) 2

3 $\bullet$ $\bullet$ $\bullet$ 117 ( ) (1) Mathematica :Mathematica Mathematica Mathematica Player (2) (NBP ) : Conceptual understanding (3) Mathematica : ( ) NBP 4.1 ( ) $II$ $III$ (NBP ) (a) :Manupulate $\sin(x+t),$ $\sin x\cos t,$ $\cos x\sin t$ ( 1) $1\sin(x+t)\pm_{\ovalbox{\tt\small REJECT}_{\square }^{B\wedge-}}$ 3

4 118 (b) $\sin(x-t)$ $\sin(x+t)$ ( ) : $\sin(x-t)$ $\sin(x+t)$ ( ) ( ) (c) : $\sin x\cos t,$ $\cos x\sin t$ $(\sin x\cos t+\cos x\sin t$ $)$ 1.? 2. $\sin(x+t)$ $0$? 3. $\sin x\cos t$ $0$? 4. $\sin x\sin t$ $0$? 5. $\sin(x+t)=\sin x\cos t+\cos t\sin x$? $a$ : $b$ : $c$ : $d$ : $n^{e}$ : 4.2 Conceptual understanding 4

5 119 conceptual understanding (a) 1 procedural understanding conceptual understanding $1$ (Roddick [3]) 2 (b) 2 conceptual understanding 2 3 2( 1) 3( 2) (c) $NB$ $(-1,2)$ 3 1. $y=x^{3}+x^{2}-2x$ 1 $y=a(x+1)+b$ $a,b$ 2 (I) ( 4) 2 $a,b$ 5

6 $ \hat$ $ _{\vee}\vee E$ $ \wedge\hat \check\check $ $\emptyset$ $\rangle + $ $-\beta$ $\rangle + $ $\otimes-=$ $-\overline{ - \prime} + $ $-\beta\beta$ $ $ $ $ 4 5 (II) $f(x)=x^{3}+x^{2}-2x$ $(-1,2)$ $y=f(x)$ 2 $(-1,2),$ $(-1+ \frac{1}{n}, f(-1+\frac{1}{n}))$ $y= \frac{f(-1+\frac{1}{n})-f(-1)}{\frac{1}{n}}(x-(-1))+2$ $n$ $(-1+ \frac{1}{n}, f(-1+\frac{1}{n}))$ $(-1,2)$ ( 5) 2. $\frac{f(-1+\frac{1}{n})-f(-1)}{\frac{1}{n}}=-1+\frac{1}{n^{2}}-\frac{2}{n}arrow-2(narrow\infty)$ $f (-1)$ $(-1, f(-1))$ $y=f (-1)(x-(-1))+f(-1)$ 3. $f (x)$ ( ) $y=f(x)$ ( ) $f(x)=x(x-a)(x-2)$ $f(x)=x^{3}+ax$ $a$ $y=f(x)$ $y=f (x)$ 4.2 (I) 1 2 6

7 (II)3. abcdefabcdef $f (x)$ $f(x)$ abcdefabcdef 4.3 Mathematica $n$ Mathematica (i) PDF Sum $f(x)=x^{2}$ : (1) $[0,1]$ Clear $[f]$ ; $f[x_{-}]$ $:=x^{-}2;f[x]$ 7

8 122 (2) Clear [S] ; $S[n_{-}]$ $:=$Sum $[f[(i-1)/n]*(1/n), \{i, 1,n\}]$ ; $S[2]$ $S[3]$ $,$ $S[10]$ Mathematica $\{S[3\}, S[10]\}$ $n$ $S[n]$ $n$ $=$ 1 (ii) $NBP$ $f(x)=\cos x$ $[0,2\pi]$ $x_{k}$ $x_{i}= \frac{2i\pi}{n}(i=0,1, \ldots,n)$ $n$ $y_{k}$ $\sum_{i=0}^{k-1}f(x_{i})\cross\frac{a}{n}$ $y_{0}=0$ $y_{0},y_{1},$ $\ldots,y_{n}$ $(x_{i},y_{i})(i=0,1, \ldots, n)$, $n$ $\cos x$ $\sin x$ ([7]) 43 (i) ( ) (ii) abcdefabcdef 5 8

9 $\square$ $\square ^{b}$ : $\square ^{d}$ : abcdefabcdef 1 6 ( 21 : 12 ) a: C: 2 $O$ 7 ( 22 : 16 ) 7 a: 9

10 124 3 $a$. $b$. (1) (2) (3) (4) (5) (1) (2) ( ) (3) ( ) (4)IT (5) (6) 1 40 (7) (8) 4 2( ) Mathematica ([4], [5], [6]) 21 ( ) ( ) (i) O

11 $\square$ 125 (ii) $O$ 9 $\square ^{b}$ : 9 $\circ$ a:. $\circ$. 7 9 (16 13 ) 5 Mathematica 11

12 [1] 21 7 [2] [3] Cheryl D Roddick, Differences in learning outcomes: Calculus & mathematica vs. traditional calculus, Primus XI(2), 2001, $\sim$ [4] : $\sim$, ( ), Vol. 16, No. 1, pp , 2007 [5] :Mathematica - -, 82 pp , 2007 [6] S. Yamamoto and N. Ishii : A way of computer use in mathematics teaching-the effectiveness that visualization brings -, Proc. of the 10th International conference of The Mathematics Education into the 21st Century Project, Sep , University of Applied Sciences, Dresden, pp , 2009 [7] 34, ,

1 (1) ( i ) 60 (ii) 75 (iii) 315 (2) π ( i ) (ii) π (iii) 7 12 π ( (3) r, AOB = θ 0 < θ < π ) OAB A 2 OB P ( AB ) < ( AP ) (4) 0 < θ < π 2 sin θ

1 (1) ( i ) 60 (ii) 75 (iii) 315 (2) π ( i ) (ii) π (iii) 7 12 π ( (3) r, AOB = θ 0 < θ < π ) OAB A 2 OB P ( AB ) < ( AP ) (4) 0 < θ < π 2 sin θ 1 (1) ( i ) 60 (ii) 75 (iii) 15 () ( i ) (ii) 4 (iii) 7 1 ( () r, AOB = θ 0 < θ < ) OAB A OB P ( AB ) < ( AP ) (4) 0 < θ < sin θ < θ < tan θ 0 x, 0 y (1) sin x = sin y (x, y) () cos x cos y (x, y) 1 c