Tokyo Polytechnic University Repository / 東京工芸大学学術リポジトリ

9KJ00006190522論文Paperdepartmental bulletin pape

西村, 安弘

2893

ニシムラ, ヤスヒロ

NISHIMURA, Yasuhiro

2894

Alarcon, Camerini and Kinoshita : A Comparative Study on El sombrero de tres picos and Its Adaptataions


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編集後記

1KJ00006909639論文Articledepartmental bulletin pape

中根, 静男

ナカネ, シズオ

Nakane, Shizuo

1256

Postcritical sets and saddle basic sets for Axiom A
polynomial skew products
Shizuo Nakane 
In this note, the dynamics of Axiom A polynomial skew products on C2 is investigated. Espe-
cially, the postcritical sets are described in terms of the saddle basic sets.
1 Introduction
In this note, we consider Axiom A regular polynomial skew products on C2. It is
of the form : f(z; w) = (p(z); q(z; w)); where p(z) = zd +    and qz(w) = q(z; w) =
wd +    are polynomials of degree d  2. Then its k-th iterate is expressed by :
fk(z; w) = (pk(z); qpk 1(z)      qz(w)) =: (pk(z); Qkz(w)):
Hence it preserves the family of bers fzg  C and this makes it possible to study
its dynamics more precisely. Let K be the set of points with bounded orbits and
put Kz := fw 2 C; (z; w) 2 Kg and KJp := K \ (Jp C). The ber Julia set Jz is
the boundary of Kz.
Let 
 be the set of non-wandering points for f . Then f is said to be Axiom A if

 is compact, hyperbolic and periodic points are dense in 
. For polynomial skew
products, Jonsson [J2] has shown that f is Axiom A if and only if the following
three conditions are satised :
(a) p is hyperbolic,
(b) f is vertically expanding over Jp,
(c) f is vertically expanding over Ap := fattracting periodic points of pg.
Here f is vertically expanding over Z  C with p(Z)  Z if there exist  > 1 and
C > 0 such that j(Qkz)0(w)j  Ck holds for any z 2 Z;w 2 Jz and k  0.
Professor, General Education and Research Center, Tokyo Polytechnic University　　　　　　
　　 Received Sept. 15. 2010
1 
ACADEMIC REPORTS Fac. Eng. Tokyo Polytech. Univ. Vol. 33 No.1 (2010) 
We are interested in the dynamics of f on Jp C because the dynamics outside
Jp C is fairly simple. Consider the critical set
CJp = f(z; w) 2 Jp C; q0z(w) = 0g
over the base Julia set Jp. Let  be the ergodic measure of maximal entropy for
f (see Fornaess and Sibony [FS1]). Its support J2 is called the second Julia set of
f . Let PCJp := [n1fn(CJp) be the postcritical set of CJp . Jonsson [J2] has shown
that
(d) J2 = [z2Jpfzg  Jz (Corollary 4.4),
(e) the condition (b) () PCJp \ J2 = ; (Theorem 3.1),
(f) J2 is the closure of the set of repelling periodic points of f (Corollary 4.7).
By the Birkho ergodic theorem, -a.e. x has a dense orbit in J2. Especially,
J2 = supp is transitive. Hence J2 coincides with the basic set of unstable dimension
two. See also [FS2].
For any subset X in C2, its accumulation set is dened by
A(X) = \N0[nNfn(X):
DeMarco & Hruska [DH1] dened the pointwise accumulation set by
Apt(CJp) = [x2CJpA(x):
Let  be the closure of the set of saddle periodic points in JpC. It decomposes
into a disjoint union of saddle basic sets :  = tmi=1i. Put z = fw 2 C; (z; w) 2
g. The stable and unstable manifolds of  are respectively dened by
W s() = fy 2 C2; fk(y)! g;
W u() = fy 2 C2; 9 backward orbit y^ = (y k) tending to g:
Theorem A ([DH1]).
Apt(CJp) = ; A(CJp) =W
u() \ (Jp C):
We dene a relation  among saddle basic sets by i  j if (W u(i) n i) \
(W s(j)nj) 6= ;. A cycle is a chain of basic sets : i1  i2      in = i1 . For
Axiom A open endomorphisms, there is no trivial cycle because W u(i)\W s(i) =
i holds for any i. See [J2], Proposition A.4. Jonsson has also shown that, for
Axiom A polynomial skew products on C2, the non-wandering set 
 coincides with
the chain recurrent set R. This leads to the following lemma, which we use later.
ACADEMIC REPORTS Fac. Eng. Tokyo Polytech. Univ. Vol. 33 No.1 (2010) 2
Lemma 1.1. ([J2], Corollary 8.14) Axiom A polynomial skew products on C2 have
no cycles.
Put
C0 := CJp nKJp ; Ci := CJp \W s(i) (1  i  m):
Lemma 1.2. CJp = tmi=0Ci.
proof. By Proposition 3.3 in Jonsson [J1], 
^ has local product structure for
open Axiom A endomorphisms. Theorem A implies A(x)   for any x 2 CJp . If
A(x) = ;, then x 2 C0. Otherwise there exist an n and y 2  such that fn(x) 2
W sloc(y). Hence A(x)  i if y 2 i. Thus we conclude CJp = tmi=0Ci.
We put 0 = f[0 : 1 : 0]g, which is a superattracting xed point of f in P2. Hence
it is not in the saddle set . We also dene W s(0) = fx 2 Jp  C; fn(x) ! 0g.
For x 2 (JpC)nKJp , fn(x) tends to [0 : 1 : 0]. Then we have A(Ci)  Apt(Ci) = i
for any i  0.
Theorem B (Nakane [N]). For any i  0, we have
A(Ci) = i () Ci is closed:
The main purpose of this note is to consider the case where Ci is not necessarily
closed. Put M = f0; 1; 2;    ;mg and Ii = fj 2M ;Ci \ Cj 6= ;g.
Theorem 1.1. A(Ci) W u([j2Ii j) \W s([j2Ii j) (0  i  m):
If Ci is closed, then Ii = fig and Theorem 1.1 says
A(Ci) W u(i) \W s(i) = i:
Since A(Ci)  i, Theorem 1.1 generalizes Theorem B.
2 Proof of Theorem
First we will give preliminary lemmas. Take a small open neighborhood Uk of k
for 1  k  m so that f(Uk) \ U` = ; for k 6= `. The argument in the proof of the
following lemma will be frequently used later.
Lemma 2.1. Suppose i 6= j and a sequence fxng in Ci converges to a point x0 in
Cj. Then A(fxn;n  1g) contains a point y in W u(j) n .
Postcritical sets and saddle basic sets for Axiom A polynomial skew products 3 
proof. Since x0 2 Cj , there exists a K > 1 such that fk(x0) 2 Uj for k  K.
Then fK(xn) 2 Uj for large n. Since xn 2 Ci, the orbit of xn eventually leaves Uj .
Hence put kn := minfk  K; fk(xn) =2 Ujg. We will show kn ! 1. Otherwise,
taking a subsequence, we may assume fkng is bounded. Then there is a subsequence
n` such that kn` = L for all `. That is, f
L(xn`) =2 Uj . Taking ` ! 1, we have
fL(x0) =2 Uj , which contradicts L  K. Now let y be an accumulation point of the
sequence ffkn(xn)g. From the denition of Uk, we have y =2 [Uk, hence y =2 . Thus
the set A(fxn;n  1g) contains a point y outside .
Next we show y 2 W u(j). In fact, by taking subsequences if necessary, put
y ` = limn!1 fkn `(xn). Then fy `; `  0g forms a backward orbit of y in Uj . By
the local product structure of 
^, we conclude y ` ! j , hence y 2 W u(j). This
completes the proof.
Lemma 2.2. If C is a closed set contained in Ci, then A(C)  i.
proof. If C  C0, then A(C) = ; since C is compact. Suppose C  Ci and there
exists x 2 A(C)ni for i  1. By taking Ui small, there exists a neighborhood V of x
such that V \Ui = ;. Since x 2 [kNfk(C) for any N  0, there exist mn %1 and
xn 2 C such that fmn(xn) 2 V , i.e. fmn(xn) =2 Ui for any n. Since C is closed, we
may assume xn tends to some x0 2 C  Ci. By the same argument as above, if we
put kn := minfk  K; fk(xn) =2 Uig, we have an accumulation point y of ffkn(xn)g
inW u(i)n. We have y =2W s(i) becauseW u(i)\W s(i) = i. Since y 2 A(C),
y 2 KJp nJ2 =W s(). Thus y must belong to W s(i1) for some i1 6= i. That is, we
have a sequence ffkn(xn)g inW s(i) tending to y 2 (W u(i)ni)\(W s(i1)ni1),
hence we have an order i  i1 .
By successively applying this argument, we have a chain of saddle basic sets :
i  i1  i2     ; i 6= i1 6= i2 6=    :
Since there exist only nitely many basic sets, we must have a cycle of them, which
contradicts Lemma 1.1.
Proposition 2.1. For any i; j 2M , the following four conditions are equivalent to
each other.
(1) Ci \ Cj 6= ;;
(2) A(Ci) \W s(j) 6= ;;
(3) W s(i) \W s(j) 6= ;;
(4) W s(i) \W u(j) 6= ;:
ACADEMIC REPORTS Fac. Eng. Tokyo Polytech. Univ. Vol. 33 No.1 (2010) 4
proof. Since, for any i  1, all the sets Ci\C0; A(Ci)\W s(0);W s(i)\W s(0)
and W s(i) \W u(0) are empty, we may assume j  1.
((1) ) (2)) Suppose Ci \ Cj 6= ;. We will show A(Ci) \ j 6= ;. Let xn 2 Ci
tending to p 2 Cj . Then there exist y 2 j and an k such that fk(p) 2 W sloc(y).
Hence, for any n > 0, there exists an Ln such that d(f
`(p); f ` k(y)) < 1=n for
`  Ln. For each xed n, take kn so that d(fLn(xkn); fLn(p)) < 1=n. Then it
follows d(fLn(xkn); f
Ln k(y)) < 2=n. Since fLn k(y) 2 j , we conclude that A(Ci)
intersects j .
((2)) (3)) It is evident since A(Ci) W s(i).
((3) ) (1)) Suppose W s(i) 3 xn ! p 2 W s(j). Then there exist k and y =
(z0; w0) 2 j such that fk(p) 2 W sloc(y). Let Uy be the connected component
of intKz0 containing w0. Then Uy  W sloc(y) and there exists L > 0 such that
fL(Uy) = UfL(y) contains a critical point c in Cj . If we put (zn; wn) = f
k(xn), then
zn 2 Jp. By the continuity of the map z 7! Jz in Jp, fL(Ufk(xn)) ! fL(Ufk(p)) =
fL(Uy). Thus, for large n, there exist critical points cn 2 fL(Ufk(xn)) tending to c.
Since cn 2 Ci, we conclude that Ci \ Cj 6= ;:
((3)) (4)) Suppose W s(i) 3 xn ! p 2W s(j). By the above argument, we have
Ufk(xn) ! Ufk(p) = Uy. Since, for any prehistory y^ 2 ^j of y, W uloc(y^) is transversal
to the ber, it follows that Ufk(xn) \W uloc(y^) 6= ; for large n. Thus we conclude
W s(i) \W u(j) 6= ;:
((4)) (3)) We need a lemma.
Lemma 2.3. Suppose W s(i) \ W s(j) = ;: Then there exists  > 0 such that
W s(i) \W u (y^) = ; for any y^ 2 ^j.
proof. Suppose, for any n  1, there exist y^n 2 ^j and xn 2W s(i)\W u1=n(y^n).
Since fyng  j , it follows that fxng and fyng are bounded. Hence there exist their
subsequences fxnkg and fynkg respectively tending to x0 and y0: Since d(xn; yn) <
1=n, we have x0 = y0 2 j . Thus we conclude W s(i) \W s(j) 6= ;:
Now supposeW s(i)\W u(j) 6= ;: If we take p 2W s(i)\W u(j), there exists
a prehistory p^ = (p n) of p tending to j . Assume W s(i)\W s(j) = ;: Let  > 0
be the constant in Lemma 2.3. Then there exists L > 0 such that d(p n;j) < 
for any n  L, hence, p L 2 W u (y^) for some y^ 2 ^j . Since p L 2 W s(i), we have
W s(i) \W u (y^) 6= ;; which contradicts Lemma 2.3. This completes the proof of
Proposition 2.1.
Postcritical sets and saddle basic sets for Axiom A polynomial skew products 5 
Corollary 2.1. A(Ci)  [j2IiW s(j) (0  i  m):
proof. By Lemma 3.5 in [DH2], we haveW u()\KJp =W u()\W s(). Then,
for i  1 it follows A(Ci) W u()\KJp W s(). Now the assertion follows from
Proposition 2.1.
Now we prove Theorem 1.1. We give a proof mainly for the case i  1. The case
i = 0 can be done with a minor change and will be given at the end.
Suppose p 2 A(Ci). Then there exists xn 2 Ci and mn % 1 such that
fmn(xn) ! p. If p 2 j for some j, then A(Ci) \ W s(j) 6= ;. Hence we have
j 2 Ii by Proposition 2.1. Thus, if p 2 , then p 2 [j2Iij . This holds also for
i = 0.
In the sequel, we assume p =2 . We may assume p =2 [mj=1Uj . We also assume
xn ! x0. Suppose x0 2 Ci. Then by Lemma 2.2, we have p 2 i, which contradicts
p =2 . Hence x0 2 Ci1 for some i1 6= i and it follows i1 2 Ii. As in the proof of
Lemma 2.1, we take K1 so that f
k(x0) 2 Ui1 for k  K1 and put k(1)n = minfk 
K1; f
k(xn) =2 Ui1g. If mn < k(1)n for innitely many n, then p = lim fmn(xn) 2 Ui1 , a
contradiction. Thus mn  k(1)n for large n. We may assume fk
(1)
n (xn) tends to some
y(1) 2W u(i1) n . Suppose y(1) 2W s(i2). Since y(1) 2 A(Ci), we have i2 2 Ii.
We repeat the argument in the proof of Lemma 2.1. PutK2 so that f
k(y(1)) 2 Ui2
for k  K2 and k(2)n = minfk  k(1)n +K2; fk(xn) =2 Ui2g. If k(1)n  mn  k(1)n +K2
for innitely many n, there exists j  K2 so that mn = k(1)n + j for innitely many
n. Then we have
p = lim fmn(xn) = lim f
k
(1)
n +j(xn) = f
j(y(1)) 2W u(i1):
Otherwise, we have mn  k(1)n +K2 for large n. We may assume fk
(2)
n (xn)! y(2) 2
W u(i2) n . If mn < k(2)n for innitely many n, then p = lim fmn(xn) 2 Ui2 , a
contradiction. Thus mn  k(2)n for large n.
Repeating this argument, we eventually meet i. That is, there exist ` and
ij 2 Ii; 1  j  ` such that
i1  i2      i` = i:
Suppose k
(`)
n <1 for innitely many n. Then, further repeating this argument, we
must meet i again. That is, there exists a sequence :
i1      i` = i      i:
ACADEMIC REPORTS Fac. Eng. Tokyo Polytech. Univ. Vol. 33 No.1 (2010) 6
This contradicts Lemma 1.1. Thus we conclude k
(`)
n = 1 and fk(xn) 2 Ui` (k 
k
(` 1)
n +K`) for large n. Since p =2 Ui` , we may conclude k(` 1)n  mn  k(` 1)n +K`
for large n. Then, there exists j  K` such that mn = k(` 1)n + j for innitely many
n and
p = lim fmn(xn) = lim f
k
(` 1)
n +j = f j(y(` 1)) 2W u(i` 1):
Thus we conclude A(Ci)  [j2Ii (W u(j) \ (Jp C)). Combining this with Corol-
lary 2.1, we have
A(Ci) W u([j2Ii j) \W s([j2Ii j):
Let us consider the case i = 0. Let p 2 A(C0). The argument as above works
as long as y(`) 2 W s(). Suppose y(`) =2 W s(). Then it belongs to W s(0). If
mn  k(`)n , then p 2 Ui` , a contradiction. Now suppose 0 < mn   k(`)n < K`+1 for
some K`+1 > 1. Then there exists j  1 such that mn = k(`)n + j for innitely many
n and p = lim fmn(xn) = f
j(y(`)) 2W u(i`). Next suppose the sequence mn   k(`)n
is unbounded. By taking a subsequence tending to 1, we have
p = lim fmn k
(`)
n  fk(`)n (xn) = [0 : 1 : 0]:
which implies p 2 W u(0) = f[0 : 1 : 0]g. This completes the proof of Theorem
1.1.
Note that Cj is open in CJp if and only if Ci \ Cj = ; for any i 6= j. After
Proposition 2.1, it is equivalent to W s(i) \W s(j) = ; for any i 6= j.
Theorem 2.1. For any xed j  1, we have
Cj is open in CJp () the map z 7! j;z is continuous in Jp;
which is also equivalent to W u(j) \ (Jp C) = j :
proof. ()) Suppose the map z 7! j;z is discontinuous at z0 2 Jp. Then there
exist y = (z0; w0) 2 j ; a sequence fzng in Jp tending to z0 and  > 0 such that
D(w0; ) \ j;zn = ; for any n  1. By Lemma 2.3, there exists  > 0 such that, for
any prehistory y^ 2 ^j of y,
W u (y^) \ (Jp C) nKJp =W u (y^) \ (Jp C) \W s(0) = ;;
W u (y^) \ (Jp C) \KJp =W u (y^) \W s() =W u (y^) \W s(j)  j :
Postcritical sets and saddle basic sets for Axiom A polynomial skew products 7 
Hence we haveW u (y^)\(JpC)  j . This contradicts the fact D(w0; )\j;zn = ;
for large n since W u (y^) is transversal to the vertical ber.
(() Suppose Cj is not open in CJp . Then there exist i 6= j and a sequence xn
in W s(i) tending to p 2 W s(j). Take k  1 and y = (z0; w0) 2 j so that
fk(p) 2 W sloc(y). Note that (zn; wn) := fk(xn) 2 W s(i) tends to fk(p). By the
continuity of the map z 7! Jz, we conclude that D(w0; )\j;zn = ; for large n and
for small  > 0. This contradicts the continuity of the map z 7! j;z.
References
[DH1] L. DeMarco & S. Hruska: Axiom A polynomial skew products of C2 and
their postcritical sets. Ergod. Th. & Dynam. Sys. 28 (2008), pp. 1749{1779.
[DH2] L. DeMarco & S. Hruska: Corrections to \Axiom A polynomial skew products
of C2 and their postcritical sets". Preprint 2009.
[FS1] J.E. Fornaess & N. Sibony: Complex Dynamics in higher dimension II. Ann.
Math. Studies 137 (1995), pp. 134{182.
[FS2] J.E. Fornaess & N. Sibony: Hyperbolic maps on P2. Math. Ann. 311 (1998),
pp. 305{333.
[J1] M. Jonsson: Dynamical studies in several complex variables, I. Hyperbolic dy-
namics of endomorphisms. PhD thesis, Royal Institute of Technology, 1997.
[J2] M. Jonsson: Dynamics of polynomial skew products on C2. Math. Ann. 314
(1999), pp. 403{447.
[N] S. Nakane: Component-wise accumulation sets for Axiom A polynomial skew
products. Preprint.
ACADEMIC REPORTS Fac. Eng. Tokyo Polytech. Univ. Vol. 33 No.1 (2010) 8


Postcritical sets and saddle basic sets for Axiom A polynomial skew products

8KJ00006910959論文Articledepartmental bulletin pape

植野, 義明

1274

ウエノ, ヨシアキ

Ueno, Yoshiaki

1275

数学のメタファー
植野義明
概要
レイコフとヌーニェスが [1]で展開している数学的概念の分析学は, 言語学に起源をもつメタファー理論を
数学的概念に応用したものである。人間がなぜ数学を理解できるのかという問題を考えるとき, 認知科学, 脳
神経科学の最近の成果から学べることが多い。実証的なデータに基づくこれらの科学が示すのは, 人間が実際
に数学的概念やその他の概念を理解するやり方により近いメカニズムである。生徒の発達についてより深く,
現実に即して理解することは, 教材研究や授業実践に役立つ。数学的な考え方のよさを人類共通の財産として
活用するためにも, 数学を理解する脳のメカニズムに注意を払うことは有用である。
検索語：数学教育, 認知科学, 数学の概念分析学, メタファー
1 はじめに
言語学者 George Lako と心理学者 Rafael E.
Nu~nez は共著 [1] の前書きで, 認知的視点から捉え
た数学的概念の分析学|数学の認知科学|を立ち
上げたいと書いている。
認知科学では, 人間の概念をそれを理解する脳の
メカニズムに基づいて分析することを目指してい
る。それが概念分析学である。脳のメカニズムは完
全に解明されてはいないが, 1960～70年代の認知科
学で言われていた「人間の脳はコンピュータのよう
なものである」というドグマは大幅に変更されてき
ている。例えば, 脳にはさまざまなモジュールが集
まっているが, 数学の理解に特化されたモジュール
は存在しない。数学に関係するのは, 言語を操るモ
ジュール, 図形的イメージを操るモジュール, 指の
運動・感覚モジュール, 3～4個くらいまでのものの
個数を瞬時に把握し, 計算を行なうモジュールなど
である。量の大小感覚を伴って数を理解するのは自
然なやり方であり, おそらくはそのためのモジュー
ルが存在するのではないかと予想されているが, 文
字式を規則に従って変形するような「計算」は別の
 Associate professor, General Education and Research Center, Tokyo Polytechnic University,
Received Sept.17, 2010.
部分が司っているらしいということが脳疾患の症例
から言われている。
本稿では, 数学教育の立場から, 実証的データに
基づく新しい認知科学の結果を, 数学の現場教師の
実践にどのように生かすべきであるかを論じたい。
現在, 日本では脳科学の目覚ましい発展がよく話
題になっており, そのことは数学教育とも無縁では
ない。実際, 算数の問題を解くときに, 脳の中でど
のような変化が見られるかといった実験が各地の
大学で計画, 実行されている。このような脳科学の
進展によって, 数学教育の立ち位置が微妙に変化し
てきていると感じる。今まで, 数学科教員は生徒の
数学理解に関して疑問を感じることがあると, 数学
者に質問するしか方法がなかった。しかし, 教育現
場で起こってくる問題の中には, 数学者ではなかな
か答えにくいものも多い。脳科学が身近な学問にな
るにつれて, 人間の脳の中で実際にどんなことが起
こっているのかを実証的なデータを通して理解でき
るようになってきた。より生徒の心理プロセスの実
態に即した教育を行える可能性が見えてきたといえ
るだろう。
2 数学の基礎と認知科学
数学の認知科学とは何だろうか。それは数学の基
礎たりうる学問であろうか。数学の認知科学を学
94
東京工芸大学工学部紀要 Vol. 33 No.1（2010） 
ぶことは特にどのような人たちにとって有効だろ
うか。
2.1 数学史における数学の基礎付け運動
数学史では, 19世紀後半から 20世紀初頭におい
て, 数学を厳密化する動きがあった。ニュートン以
来, 微積分は科学の道具として非常に発展してきた
が, 厳密性がそれに追いついていない部分があった。
関数の概念や収束の定義に対する見直しをする中
で, デデキント, クロネッカー, カントールなどに
より, 実数論, 集合論が起こった。また, 数学から矛
盾を排するために, 確かな土台の上に数学の理論体
系全体を基礎づける必要が叫ばれ, 公理的集合論に
よって数学を「基礎づける」運動があった。
クロネッカーは「物理学までも含めた数学の算術
化をしなければならない」と書いている*1。算術は
人間にとって最も自然で, 生まれて最初にであう整
合的な体系であり, 単純であるだけに矛盾はないと
考えられている。算術化 (arithmetization) はまた
離散化 (discretization) とも呼ばれるが, それは数
学の発達のひとつの段階として自然な段階であっ
た。カントルが集合論を建設したとき, 数学界から
さまざまな批判があったが, 集合のような単純な概
念を無限を扱う数学の基礎に据えようとしたからで
あった。
20 世紀に入り, ブルバキが集合論を基礎とした
全数学の書き換えを実行しようと試みた。ブルバキ
流の数学は世界の数学界を席巻したかのように見え
る。空集合の記号 ;などが世界の数学者の共通語に
なり, 新しい分野を学ぶときもいつも入口は同じで
あるという安心感に寄与した。
しかし,数学が牽引したといわれた 20世紀構造主
義の思想が他の分野に波及し, とくに哲学や心理学
にも強い影響を及ぼしたことによって, 西洋思想は
行き過ぎた公理主義の影響を受けることとなった。
2.2 数学教育の現代化運動
スプートニクショックを受けて, 1960 年代から
70 年代にかけて世界中の小学校から高校までの教
*1 現代的な言い方では, 代数化という方が相応しい。
室を巻き込んだ数学教育の現代化運動は, 現在では
歴史的な誤謬ということになっている。現代化運動
の熱がまだ冷めていなかった当時は, 現代化運動そ
れ自体は間違っておらず, 現場の教師の理解が十分
でなかったことが敗因であるとみなす空気があっ
たが, 21世紀に入った現在では, 誰もがはっきりと
数学教育の現代化は間違っていたと言っている。だ
が, 何が間違っていたのか。そして, なぜ間違って
しまったのだろうか。
数学教育が数学史とある程度連動するように進ん
できたことは, 自然な成り行きであったとはいえ,
一部には行き過ぎもあったといえる。公理的集合論
や数学基礎論が目指している「数学の基礎付け」の
動機は, 数学の中からなるべく矛盾を排したり, 数
学自体を分りやすいモデルの上に構築しなおすこと
によって, 数学的命題間の関係をより深く理解する
ことである。
一方, 数学教育にとって重要な「数学理解の基礎」
とは, 児童・生徒がなぜ数学を理解できるのか (ある
いは, できないのか)ということが根本問題であり,
それに付随して, 数学の理解には発達段階があるの
かどうか, 数学的概念は日常生活のどのような概念
と関係しているのか, どのような状態になったとき
に人間はある概念を理解したと感じるのか, などの
問題が重要である。
集合論はその上に数学を建設するためのひとつの
モデルであるが, それが唯一ではなく, また, 生徒の
数学理解を必然的に基礎づけるものでもない。数学
的概念にはさまざまな起源があり, 人間による数学
的概念の理解にもさまざまな自由度がある。集合論
はその中で比較的うまくいった数学理解の方法論で
あり, 現代の数学者に受け入れられることによって,
数学者の間での世界的標準になっている。これから
の数学教育では, 数学を基礎付けるモデルと数学理
解の人間的な基礎をはっきり区別して扱わなければ
ならない。
要約していうと, たとえば具体的に高校でベクト
ルを教えたときに, それを生徒が理解できるのは,
集合論に即してすべての定理の証明を形式的に遂行
することを通してではない。数学者が数学を理解す
東京工芸大学工学部紀要 Vol. 33 No.1（2010） 95 
るときとも共通するが, ベクトルという概念に人間
的に自然な意味があり, それぞれの定理にも人間の
日常体験に即した意味があるから理解できるので
ある。そして, 一度そのようにして理解した概念は,
体験していない場面にも適用できる。
3 数学理解の基礎とメタファーの使用
小平邦彦は [5]の中で,「数学を理解するとは，数
学的現象を「数覚」という感覚で「見る」ことであ
る．「数覚」は感覚なので頭の良し悪しとは関係が
ない」と書いている。
スタニスラス・ドゥアンヌ [2]では数覚について,
進化生物学的に説明されている。数覚は動物にもあ
り, ヒトの新生児でも実験で確かめることができる。
脳の中に, 数を知覚することに特化されたモジュー
ルが存在し, おそらく, 生物にとって生存に必要な
機能なのであろう。だが, その範囲は非常に限られ
ている。足し算, 引き算ができるのは高々 3までの
自然数であり, それ以上の数になると大雑把な量の
感覚しかない。耳で聞いて分かる数の範囲はもう少
し広いが, それは記憶できるという意味であって,
計算はできない。
このように, [2]では「数覚」が偶然キーワードと
して登場しているが, 数学者がその存在を予想し,
あるいは感覚的に感じている「数覚」と, 生物学的
に検証できる数覚との間には大きな隔たりがある。
数学者が主張する数覚は, 脳の一つのモジュールで
はなく, 複合的な概念形成の結果生じている脳内の
現象である。[1] によると, この概念形成にはメタ
ファーというメカニズムが重要な役割を果たして
いる。
メタファーの仕組みは脳生理学的に解明されてい
るわけではないが, 実験によると, 簡単な算数の課
題を与えられた被験者の脳の中では, 次々にいろい
ろな部分が「発火」, すなわち, 活性化することが観
察されている。そして, 数学ではさまざまな複合的
な概念が使われる。このような状況証拠から, レイ
コフ [1]はメファターという認知メカニズムが概念
と概念を繋いで複合的な概念を構築するのに使われ
ているという仮説を立てた。
言語学において, 数学のメタファーより以前にメ
タファーが発見されていた。レイコフは, 言語現象
におけるメタファーの使用は離れた文化の間でも共
通のものがあり, それは, 人類であれば生理学的に
共有している脳の構造に原因があるだろうと推測し
ている。数学についても同じことが言える。全く異
なる文化的背景をもっていても, 教育を受けた人間
ならば標準的な数学の内容を理解できるのは, 人類
として同じ脳の構造を共有しているからであろう。
数学のメタファーが使用されている例として, [1]
では単位円が扱われている。単位円は, 図形とし
ての円の概念と平面上の座標の概念からなるメタ
ファーブレンドである。
4 メタファーの存在の論拠
レイコフとヌーニェス [1] では, 集合論における
無限概念, 微積分における無限小概念, 群論におけ
る同型概念, 複素数の幾何学的理解とオイラーの関
係式 ei  = cos  + i sin  など, 高校から大学初年
級程度のやや高度な数学に現れる数学的概念の概念
分析に相当のページ数が割かれている。
認知科学, 心理学, 神経生理学などの分野にわた
る論拠については, [1] では初めの方の数章で簡単
に触れられているだけであるが, それらについては
[2] でより詳しく解説されている。[2] では, たとえ
ば, 2 桁以上の掛け算を含む算数の範囲の計算能力
について, 脳のメカニズムの解明がなされている。
ヒトを含む動物の数概念はあいまいな概数を把握す
る量感覚であって, コンピュータでの計算のような
記号を用いたものとは性質が違っている。これを説
明するのに, 脳内に計器のようなアナログ的な機能
をもったモジュールがあるとするモデルで説明して
いる。この説明は現実の実験結果とうまく合う。数
直線に当たるものは小学校の課程を経て徐々に形成
されるとしている。その他, 脳疾患の患者によって
得られる貴重なデータについて触れられている。
数学のメタファー 96
5 結論
数学の認知科学的研究, とくに, 数学的概念のメ
タファーによる研究は, 数学教育を重要な応用分野
としている。そこには, 数学教育が, 言語学, 心理学
などの臨床的な分野の研究と連携する可能性が秘め
られている。現在, 脳内の活動がリアルタイムで測
定できるようになり, さまざまな実験が可能となっ
ている。さまざまな環境の中で展開される教育実践
の経験に合致し, 現場のニーズに応える数学教育の
理論展開が俟たれる。本論では述べることができな
かったが, 数理哲学上のさまざまな立場についても
認知科学の立場から批判, ないしは検討することが
できる。
参考文献
[1] George Lako, Rafael E. Nu~nez, Where Math-
ematics Comes From, Basic Books, 2000.
[2] スタニスラス・ドゥアンヌ著, 長谷川眞理子, 小
林哲生訳, 数覚とは何か|心が数を創り, 操る仕
組み, 早川書房 (2010)
[3] 茂木健一郎,脳の中の小さな神々,柏書房 (2004)
[4] 茂木健一郎,脳と仮想 (新潮文庫),新潮社 (2007)
[5] 小平邦彦, 怠け数学者の記 (岩波現代文庫), 岩波
書店 (2000)
[6] 国岡高宏, 数学教育におけるアナロジーの
研究 (2) |概念メタファーによる数学学習
の分析|, 全国数学教育学会誌「数学教育
学研究」第 15 巻, 第 2 号 (2009), pp.17{27.
http://hdl.handle.net/10132/2663
東京工芸大学工学部紀要 Vol. 33 No.1（2010） 97 


数学のメタファー

18KJ00006190879作品Workdepartmental bulletin pape

渡辺, 仙一郎

2911

ワタナベ, センイチロウ

WATANABE, Senichiro

2912

Harmony of Furniture and the Organic

1KJ00006911179論文Articledepartmental bulletin pape

田邉, 寛

3200

タナベ, ヒロシ

Tanabe, Hiroshi

1825

* Associate Professor. General Education and Research Center, Tokyo Polytechinic University 
 Received Sept. 1, 2010 
 
 
Hiroshi TANABE* 
 
This paper introduced the total appearance of the program for the preparation for the TOEIC test of 
Tokyo Polytechnic University since the introduction of the TOEIC test in 2002 to the present situation 
where 155 students took the TOEIC test and 3 examinees scored over 800 points, 3 over 700, 7 over 600 
and 3 over 500 in 2009 and suggested the points of improvement. For the improvement, the program could 
be extended as a one year course with 30 lecture hours while the students are required to study 1 hour at 
home, where 300 hours of training for the TOEIC test recommended by Chida, 2010 would be realized 
ignoring the condition that should be a conversation class with native speakers.  
 
Introduction  
Since 2002, Tokyo Polytechnic University has provided the opportunity for the students to take the 
College TOEIC test, which was a kind of the Institutional Program (IP) test*1 for college students and the 
score had the same quality and value of the public version of the TOEIC test, once or twice in a year on the 
Atsugi campus. Table 1 shows the results of the TOEIC test and the highest scores and the average scores. 
 
Table 1. Numbers of Examinees and Scores 
 
Schedule  Examinee  Highest Score Lowest Score Average 
2002  202  905  115  333 
2003  114  525  145  302 
2004  129  770  120  326 
2005    55  665  155  337 
2006  125  685  135  323 
2007  135  910  180  335 
2008  253  915  165  341 
2009  155  855  160  364 
 
    The TOEIC test was held twice in 2009 and 155 students took the examination with the highest score 
of 855 and the average score of 364. According to ETS (ETS, 2010)*2, the average score of the year 2008 
was 447 and the average score of science technology and agriculture major students was 399*3. 
 
Administration of the TOEIC test  
    IP test of the TOEIC examination had been held twice in a year. All the students of Tokyo Polytechnic 
University could take the test for free. The test itself was held by the co-op of Tokyo Polytechnic 
University and the handbills were prepared by the co-op and they were distributed to the students in the 
required classes of English. For the announcement for the higher graders who had already taken the 
required credits of English, the school public-address system was used during the lunch time. Posters 
announcing the schedule of the TOEIC test were put on the walls of the co-op and the school cafeteria. All 
1 
ACADEMIC REPORTS Fac. Eng. Tokyo Polytech. Univ. Vol. 33 No.2 (2010) 
Improving TOEIC Training Program of Tokyo Polytechnic University 
the applicants were accepted by registering by seven p.m. for the test held in the next day if there was 
room.  
 
TOEIC seminar 
(1) Size of the class  
    TOEIC seminar had been a selective course for all the students of the faculty of engineering since 
2006. It was a three month course for one credit with 15 times of classes and each class had 90 minutes. 45 
students registered the course and 10 to 15 students attended the lecture as auditors in the first semester of 
the year 2010, for example.  
 
(2) Text 
Shin TOEIC Tesuto Kanzen Kouryaku Gaido (Complete Guide for the New TOEIC Test) (Helki and 
Maeda, 2005 ) was used in the class. The listening part of the textbook was used for the limitation of the 
schedule. 
 
(3) Philosophy of the class 
The TOEIC test (The Test of English for International Communication) had been accepted widely as a 
valid measure of evaluating the level of the practicality of English of its examinees. The vocabulary and the 
expressions tested by the TOEIC test had been general and practical and studying for the TOEIC test will 
guide the students to plan their study and improve the level of their English for general purposes.*4 The 
learning contents in the textbook would contain useful vocabulary for the general and practical English as 
well as for the TOEIC test itself and the students were strongly recommended to completely master the 
whole textbook contents as the first step to score the higher points in the TOEIC test.  
 
(4) Syllabus 
1) Guidance 
2) Practice test 
3) Answering questions and explanation (1) 
4) Answering questions and explanation (2) 
5) Answering questions and explanation (3) 
6) Answering questions and explanation (4) 
7) Midterm examination and summary of the points 
8) Answering questions and explanation (5) 
9) Answering questions and explanation (6) 
10) Answering questions and explanation (7) 
11) Answering questions and explanation (8) 
12) Answering questions and explanation (9) 
13) Answering questions and explanation (10) 
14) Final examination (1) tested with the same questions and summary of the points  
15) Final examination (2) tested with the new questions and summary of the points  
 
In the 14th class the same 100 listening questions that were discussed in the class were asked and the 
new questions for the students that were not dealt with in the class were asked in the 15th class for the 
Improving TOEIC Training Program of Tokyo Polytechnic University 2
comparison of the mastery of the learning contents of the texts and the mastery of the skills for answering 
for the TOEIC type of the English test, the acquisition of the general knowledge of English and the 
improvement in the listening skill.  
 
 
(5) Rules of the class 
Students were required to attend at the first class and follow the guidance, where they were told to 
keep the higher level of motivation to keep their study to improve the score of the TOEIC test. More than 
four times of the absences of the class would lead to the failure of taking the credit of the class. The review 
of the contents were needed by listening to the same part of the CD as they learned in the class every time. 
The same part of the questions were tested each time to evaluate their mastery of the contents.  
  
(6) Teaching contents 
Strategies and skills to deal with the TOEIC type of the English tests were introduced before the  
practice of answering listening questions. Following are the strategies and skills of each section*5: 
 
Techniques 
a) Part 1 (Photographs) 
Hilke (2005) pointed out the techniques for Part 1: 
i) Deleting misleading pronunciation 
ii) Checking general information such as the situation, the subject, the act of the people in the  
picture 
iii) Comparing the people in the picture 
iv) Ignoring non-existent 
v) Avoiding guessing 
In addition to the descriptions above, the following points were introduced in the lecture. 
vi) Guessing points of questions according to the picture given 
vii) Ignoring the less obvious points in the picture 
viii) Ignoring the misleading words and expressions  
 
b) Part 2 (Question-Response) 
Hilke (ibid.) pointed out the techniques for Part 2: 
i) Listening to the first word carefully 
ii) Avoiding keywords 
iii) Listening to WH-questions  
iv) Finding correct answers to the patterns such as suggestions, orders and requests 
v) Not answering in Yes or No forms for yes/no questions 
 
c) Part 3 (Short Conversations) 
Hilke (ibid.) pointed out the techniques for Part 3: 
i) Reading the choices before listening to the questions 
ii) Reading the choices as the clues to the questions 
iii) Understanding the patterns of the questions for Part 3 
ACADEMIC REPORTS Fac. Eng. Tokyo Polytech. Univ. Vol. 33 No.2 (2010) 3 
 
d) Part 4 (Short Talks) 
Hilke (ibid.) pointed out the techniques for Part 4: 
i) Reading the choices before listening to the questions 
ii) Ranking the questions according to the difficulty order 
iii) Understanding the patterns of questions according to the topic types 
 
Contents 
As was already stated Shin TOEIC Tesuto Kanzen Kouryaku Gaido (Complete Guide for the New 
TOEIC Test) (Helki and Maeda, 2005 ) was the only text used in the class. The coverage of the vocabulary 
and expressions assumed to be enough for the beginners as had not been examined. The acquisition of the 
basic vocabulary and expressions for the TOEIC test was expected by mastering the whole contents of the 
textbook. 
 
Procedure 
To realize the mastery of the learning contents, the students answered the questions, checked their 
answers, listening to the explanations and repeating answering questions at home. The achievement of their 
study was checked in each class. 
     
Method 
There were three times of practice examinations that were given exactly in the same way and with the 
same type, number and level of questions. The first test was given in the second and third class and the 
second test was given in the fourteenth class and the third test was in the fifteenth class. The tests given in 
the first and the fourteenth class were exactly the same one to check the mastery of the contents. The test 
contents of the fifteenth’s*6 were new and unknown for the examinees . 
 
Results and Disscussion 
 
Table 2. Scores  
  Listening Reading  Total Converted Score* 
1) Initial (N=38 31.1 27.8 51.9 352.4
2) Same (N=33 74.9  N/A  N/A  N/A
3) New (N=12 32.7 27.6 60.3 347.9
*Converted score was calculated according to the table attached in the practice tests. Hilek&Maeda (2005) for the first test and Ishi 
(2007) for the third test.  
 
In the comparison between the first and the third test, The average number of the correct answers of 
the listening part improved from 31.1 to 32.7 (+1.6) while that of the average number of the correct 
answers of the reading part was from 27.8 to 27.6 (-0.2). The average of the total number of the correct 
answers of the listening and reading parts improved from 51.9 to 60.3 (+8.4). 
The comparison of converted scores, though, showed the different view about the result. The result of 
the third test was lower than the initial test by 4.8 points.  
Improving TOEIC Training Program of Tokyo Polytechnic University 4
As only the contents of the listening part of the TOEIC test was taught in the class, the reading part of 
the third test might be assumed to be the transfer of the practice in the listening part, but as far as the result 
of the comparison between the first test and the third test showed there did not seem to be any significant 
improvement in the score in the third test. That might tell that in the class in that condition the basic skills 
and knowledge to improve the score of the TOEIC test were not directly related to the improvement of the 
score. 
The comparison of the average number of correct answers between the first and the second showed 
the significant difference, though. In the comparison between the first and the second test, The average 
number of the correct answers of the listening part improved from 31.1 to 74.9 (+43.8). 
As was stated before, the most important goal of the class was to master the fundamentals of 
vocabulary and the expressions for the practical use of English, which might be reflected in the 
improvement of the score of the TOEIC test in the future. As shown in Table 1, the total average score of 
the students began to improve from 2007 one year after the TOEIC seminar class began, even though there 
was no evidence that the class affected the score. 
Chida (2010) said to raise 100 points in the TOEIC test 300 hundred hours of learning in the English 
conversation course by native lectures of English was necessary*7. The class in this paper offers 15 lecture 
hours in less than 3 months. Calculated according to the criterion the raise of the score in 15 hours would 
be one twenties of 100 points, which is 5 and as has been already discussed. The average of the total 
number of the correct answers of the listening and reading parts improved from 51.9 to 60.3 (+8.4) in the 
seminar, which might be in a margin of error, though if the number of the correct answer would be 
calculated according to the conversion table. 
 
Improvement of the class 
To cover the gap of the length of the time, the self-training by the student themselves would be 
necessary. Assume the goal of the class is to raise the points by 100 points, it would be impractical for the 
science major students to study English for 100 hours a month that will be more than three hours a day. If 
students are required to study 1 hour everyday for 1 year, 300 hundred hours of study will be realized only 
if the length of the time for study is in focus ignoring the quality of the training. The problem would be to 
what extent the difference in the condition between the conversation class with native instructors and the 
self-learning by the students based on studying textbooks for the TOEIC test. 
The seminar could be a one year course with 30 lecture hours while the students are required to study 
1 hour everyday at home, where 300 hours of training for the TOEIC test would be realized ignoring the 
condition that should be a conversation class with native speakers. 
The effects of the 300 hours of self-training by students with 30 hours of TOEIC preparation course 
will be the next theme of inquiry. 
 
 
ACADEMIC REPORTS Fac. Eng. Tokyo Polytech. Univ. Vol. 33 No.2 (2010) 5 
References  
[1]  Chida Junichi. 2010. Level Up Dojo (Level Up Class) in the Official TOEIC Homepage  
 
[2]  Hilke Robert, Maeda Horo. 2005. Shin TOEIC Tesuto Kanzen Kouryaku Gaido (Complete Guide  
for the New Version of the TOEIC Test) , ALC,  
 
[3]  Ishi Tatsuya. 2007. Shin TOEIC Test Zen Pato Kanzen Kouryaku (Perfect Capture for the New  
Version of the TOEIC Test). ALC 
 
Appendix  
 
*1 Following are the explanation of IP test provided by Educational Testing Service (ETS, 2009): TOEIC 
programs are administrated through two programs: the Secure Program (SP) test and the Institutional 
Program (IP) test. SP test is conducted on a date and at location designed by the Institute for International 
Business Communication while the IP test is conducted on the date and at location determined by each 
school, corporation or organization. 
 
*2 According to TOEIC test data & analysis 2008 in the official homepage of ETS, 2010. 
 
*3 Number of examinees and average scores by types of major in Toeic test data & analysis 2008. (ETS, 
2009) According to ETS, the data had been obtained from examinees who filled in the section on “Major” 
in the TOEIC IP tests conducted by universities, junior colleges, vocational schools and technical colleges 
in the official homepage of ETS, 2010. 
 
*4 This is from the syllabus of TOEIC seminar of Tokyo Polytechnic University for the year 2010. 
 
*5 Hilke presented the skills and strategies for scoring points in the TOEIC test in Robert Hilke, Horo 
Maeda. 2005. Shin TOEIC Tesuto Kanzen Kouryaku Gaido (Complete Guide for the New TOEIC Test) , ALC.  
 
*6 Tatsuya Ishi ( 2007) showed a conversion table from the raw points of the total to the imaginable score 
of the TOEIC test in Shin TOEIC Test Zen Part Kanzen Kouryaku (Perfect Capture for the New Version of 
the TOEIC Test). ALC. 
 
*7 Chida (2010) in the Reberu Appu Dojo (Level Up Class) in the Official TOEIC Homepage said to raise 
100 point in the TOEIC test 300 hundred hours of learning in the English conversation course by native 
lectures of English was necessary.  
 
Improving TOEIC Training Program of Tokyo Polytechnic University 6


Improving TOEIC Training Program of Tokyo Polytechnic University

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