建議

本研究使用的實驗活動教材是以學生學習為主要考量，這樣的教材設計方式，不 但有別於以往傳統以形式數學知識為主、以專家的觀點來進行教學。在教學實驗活 動中，教師是教學內容和任務的決策者，必須暸解學生的思維模式，以及思索如何幫 助學生發展預期的數學概念。

從文獻探討可知，若能夠利用「多重表徵」來表達同一個概念並在表徵之間順利 的做轉換，甚至懂得如何選擇適合的表徵來協助解題，都表示擁有更穩固的概念理解 (Davis, 1984; Even, 1998; Putman, Lampert & Peterson, 1990; Lesh et al., 1987)。從活動 錄影片段可知，學生會拉動捲動軸進行資料的交錯比對，即進行表徵之間連結，但仍 存在著以單一表徵思索問題之學生。研究者認為：學習者的學習態度會影響表徵之間 的連結，因此將於活動中增加表徵連結相關試題，進一步探究學生表徵連結之表現情 形，以提升極限概念之廣度。

Tall (1991) 指出：教師教學時常只呈現最後的結果，而沒有讓學生參與整個生產 的過程。學習極限概念不應只為了求極限值，而是了解極限程序與意涵。當學習者 主動參與各種學習活動時，給予更多探索的空間活動，將使學習者的學習更加具有 效果 (Duffy, Lowyck & Jonassen, 1993)。也就是 Dewey 所言「做中學」(learning by doing) 之理念。CAS 整合教學環境配合教師的引導，既可學到應有的知識，亦拓展 學生多面向思維；實驗活動之學習環境真正實踐了「做中學」的精神。但使用 CAS 時需教導學生如何解讀 CAS 技術限制下之結果，亦不可迴避圖形產生之詭異處，應 當適時地讓學生思考這些類似的現象，也必須注意到學生使用之態度。

針對極限概念而言，強化「量詞」與操作行動密切的結合，有助於使用「量詞化」

基模完整地表示形式定義。因此，在活動設計上必須考慮「量詞」問題，才能確保 學生將操作行動內化為正確的過程。此外，教學時須提供反思抽象化的情境，幫助學 生連結非形式定義與形式定義。有鑒於此，將進一步思考如何設計「量詞」融入極限 概念之實驗活動，進而幫助學生學習形式概念。

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(1996). Understanding the Limit Concept: Beginning with a Coordinated Process Scheme, Journal of Mathematical Behaviour, 15, 167-192.

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Heid, M. K. (1988). Resequencing Skills and Concepts in Applied Calculus Using the Computer as a Tool. Journal for Research in Mathematics Education, 19 (1), 3-25.

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Claude Janvier (ed.): Lawrence Erlbaum, Hillsdale, NJ. 67-71.

Kaput, J. (1987). Representation Systems and Mathematics. Problems of Representation in

the Teaching and Learning of Mathematics. Claude Janvier (ed.): Lawrence Erlbaum,

Hillsdale, NJ. 19-26.

Li, L. & Tall, D. (1993). Constructing Different Concept Images of Sequences and Limits by Programming, Proceedings of PME 17, Japan, 2, 41-48.

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Multiple research perspectives. In A. Silver (ed.), Hallsadale, N.J: Lawrence Erbaun

Associates.

Monaghan, J. (1991). Problems with the Language of Limit, For the Learning of

Mathematics, 11(3).

Monaghan, J., Sun, S. & Tall, D. (1994). Construction of the Limit Concept with a Computer Algebra System, Proceedings of PME 18, Lisbon, III, 279-286.

NCTM (2000). Principles and Standards for School Mathematics: Discussion Draft, Standards 2000.

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Mathematics, 14, 1-18.

Pierce, R. & Stacey, K. (2001). Observations on Students' Responses to Learning in a CAS Environment. Journal for Research in Mathematics Education, 13(1), 28-46.

Rochowicz, JR J.A. (1996). The Impact of Using Computers and Calculators on Calculus Instruction: Various Perceptions. Journal of Computers in Mathematics and Science

Teaching, 15(4), 423-435.

Stewart, J. (2003) Calculus: Early Transcendentals. Pacific Grove: Brooks

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Tall, D. (1980). Mathematical intuition, with special reference to limiting processes,

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Mathematics Education, Berkeley, 170-176.

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Research in Mathematics Education, 22(3), 219-236.

附錄一、電腦實驗活動

L-1.a

Execute the following commands to define the function sin( ) ( )

x

> f:=x->sin(x)/x;

> N:=10:

M:=matrix(N+1,4,(Row,Col)->0):

M[1,1]:='x': M[1,2]:='f(x)': M[1,3]:='x': M[1,4]:='f(x)':

for i from 1 to N do

x1:=1/2^i: x2:=-1/2^i:

M[i+1,1]:=evalf(x1): M[i+1,2]:=evalf(f(x1)):

M[i+1,3]:=evalf(x2): M[i+1,4]:=evalf(f(x2)):

od:

eval(M);

L-1.b

Notice that the column that gives the values of

f x

( ) for 1 2ⁿ

x

= , 1 2

n

=

, ,...,

10 is identical with the column that gives the values of

f x

( ) for 1

2ⁿ limit ? Give a reason to your answer.

Fill in the limit you get above to get a graph of sin( ) ( )

x f x

=

x

.

> L:=???:

> plot(f(x),x=-1..1,view=L-0.5..L+0.5,axes=boxed);

L-1.d

According to the graph above, can you determine how close to 0 (from either side of 0) x has to be to ensure that sin( )

x

(x

≠ 0

) is close to the limit you get in (c)

within 0.5. (Note that f is not defined at ^x

^{= 0}

.) If so, find a distance that measures the closeness of x to 0 and confirm your answer by modifying the graph above with a new horizontal range (x-range).

L-1.e

By zooming in the graph of sin( ) ( )

x

f x

=

x

on the limit point, can you determining how close to 0 (from either side of 0) does x have to be to ensure that

sin( )

x

x

(x

≠ 0

) is close to the limit you get in (c) within 0.1 ? If so, find a distance that measures the closeness of x to 0 and confirm your answer by a graph above with proper horizontal range (x-range).

> plot(f(x),x=???..???,thickness=2,view=???..???,axes=boxed);

L-1.f

By zooming in the graph of sin( ) ( )

x

f x

=

x

on the limit point, can you determining how close to 0 (from either side of 0) does x have to be to ensure that

sin( )

x

x

(x

≠ 0

) is close to the limit you get in (c) within 0.01 ? If so, find a distance that measures the closeness of x to 0 and confirm your answer by a graph above with proper horizontal range (x-range).

> plot(f(x),x=???..???,thickness=2,view=???..???,axes=boxed);

L-1.g

Do the graphical experiments in (d), (e) and (f) confirm your answer in (c) ? Give a reason to your answer.

2.

L-2.a

Execute the following commands to define the function 1 ( ) sin

> g:=x->x*sin(1/x);

N:=15:

M:=matrix(N+1,2,(Row,Col)->0):

M[1,1]:='x': M[1,2]:='g(x)':

for i from 1 to N do

x1:=1/2^i:

M[i+1,1]:=evalf(x1): M[i+1,2]:=evalf(g(x1)):

od: the limit. Give a reason to your answer.

Fill in the limit you get above to get a graph of 1

L-2.c

According to the graph above, can you determine how close to 0 (from either side of 0) x has to be to ensure that 1 the closeness of x to 0 and confirm your answer by modifying the graph above with a new horizontal range (x-range).

L-2.d

By zooming in the graph of 1 ( ) sin

g x x

x

= ⎛ ⎞⎜ ⎟⎝ ⎠ on the limit point, can you

determining how close to 0 (from either side of 0) does x have to be to ensure that sin 1

x x

⎛ ⎞⎜ ⎟

⎝ ⎠ (x

≠ 0

) is close to the limit you get in (b) within 0.1 ? If so, find a distance

proper horizontal range (x-range).

> plot(g(x),x=???..???,thickness=2,view=???..???,axes=boxed);

L-2.e

By zooming in the graph of 1 ( ) sin

g x x

x

= ⎛ ⎞⎜ ⎟⎝ ⎠ on the limit point further, can you

determining how close to 0 ( from either side of 0) does x have to be to ensure that sin 1

x x

⎛ ⎞⎜ ⎟

⎝ ⎠ (x

≠ 0

) is close to the limit you get in (b) within 0.01 ? If so, find a distance that measures the closeness of x to 0 and confirm your answer by a graph above with proper horizontal range (x-range).

> plot(g(x),x=???..???,thickness=2,view=???..???,axes=boxed);

L-2.f

Do the graphical experiments in (c), (d) and (e) confirm your answer in (b) ? Give reasons to your answer.

3. ₁

L-3.a

Execute the following commands to define the function

( ) ¹

₁

1 3

^x

> h:=x->1/(1-3^(1/x));

> N:=10:

M:=matrix(N+1,2,(Row,Col)->0):

M[1,1]:='x': M[1,2]:='h(x)':

for i from 1 to N do

x1:=1/2^i:

M[i+1,1]:=evalf(x1): M[i+1,2]:=evalf(h(x1)):

od: the limit ? Give a reason to your answer.

Fill in the limit you get above to get another graph of

( ) ¹

₁

> plot(h(x),x=-1..1,view=L-0.5..L+0.5,axes=boxed);

L-3.c

According to the graph above, can you determining how close to 0 (from either side of 0) does x have to be to ensures that

¹

₁

L-3.e

On the basis of the data in (d) , do you think ₁

0

lim 1 1 3

x

x

→

−

⎛ ⎞⎜ ⎟⎝ ⎠

exists ? If so, what is the limit ? Give a reason to your answer.

Fill in the limit you get above to get another graph of

( ) ¹

₁

1 3

^x

h x _{⎛ ⎞}

⎜ ⎟⎝ ⎠

=

−

.

> L:=???:

plot(h(x),x=-1..1,view=L-0.5..L+0.5,axes=boxed);

L-3.f

According to the graph above, can you determining how close to 0 (from either side of 0) does x have to be to ensures that

¹

₁

1 3

^x

⎛ ⎞⎜ ⎟

−

⎝ ⎠

(x

≠ 0

) is close to the limit you get

in (e) within 0.5. (Note that g is not defined at x

= 0

.) Does your answer support your answer in (e) ? Why ?

L-3.g

With all the investigation above, what can you conclude about the existence of

0 1

lim 1 1 3

x

x

→

−

⎛ ⎞⎜ ⎟⎝ ⎠

? Why ?

4.

⎝ ⎠ and create a table that gives the numerical values of

k x

( ) for 1 your answers.

Here is a graph of

k x

( ) sin

L-4.c

By zooming in the graph above properly if necessary, can you confirm your answer in (b) ? If so, explain why ? If not, do more calculations and plot more graphs to investigate

0

lim sin

x^→

x

⎛ ⎞π

⎜ ⎟⎝ ⎠ until you can confirm your answer.

附錄二、成效評量試題

P-1.a

Here is a Maple plot of sin x

y

=

x

with viewing rectangle [ 1,1] [0,1]− × . Does the graph of sin

( )

x

f x

=

x

actually passing through the point (0,1) ? Explain briefly.

P-1.b

The Maple plot of sin x

y

=

x

with viewing rectangle [ 0.1, 0.1] [0.99,1.01]− × is shown below. What can you say from the graph ?

P-1.c

The Maple plot of sin x

y

=

x

with viewing rectangle [ 0.1, 0.1] [0,1]− × , is shown below. Explain why the graph looks so much like a straight line and what does this say about

0

limsin

x

x

→

x

. Give a reason to your answer.

P-2.a

Do you think the statement “If reason to your answer.

P-2.b

Here is a Maple plot of 1 ₁

exists? Give a reason to your answe

Here are values of ( )

f x

sin

P-3.a

Does the table contradict to the graph ? Give a reason to your answer.

P-3.b

What can you say about the behavior of

f x

( ) sin

附錄三、成就測驗試題

Determining where each of the following statement is true. If it is true, give a brief explanation. If not, give a counterexample.

A-1.a

If lim ( )

Evaluate the following limits.

A-2.a

→ = , find the largest possible

δ

that works for any given

> 0

ε .

A-7.b

Given 1

=2

ε

, find the largest possible

δ

for showing that

1

lim 1

x

x

→ = .

附錄四、期中考試題

. Which of the following statement CAN show

0

Determining where each of the following statement is true. Given reasons for your answers.

M-II2.b

If lim ( )

附錄五、問卷 (極限部分)

1. 給一個函數

f ，你是從哪方面來猜測它的極限值

lim

f

(

x

)

x→a ？

□ 只由數值資料。

例如：在 worksheet 裡「Given a function f , the value of

f

(x) when

x

n

2

= 1

and 1 2ⁿ

x

= − , where

n

∈Ν.」所做的數值資料。

□ 只由函數圖形。

□ 數值和圖形做比對。

□ 其它。_______________________________________________________

2. 在這個活動中，對於「給定一個誤差，決定x，使得你所猜測的極限值與函數值的 差小於所給的誤差」這個問題，你了解" plot 指令中"

"view "

這個 option 所代表的意 義嗎？

□ 否

□ 是，請說明____________________________________________________

3. 在這個活動中，給定一個誤差，你是否能決定x，使得你所猜測的極限值與函數值 的差小於所給的誤差？

□ 否

□ 是，你會如何決定x？

□ 觀察圖形，亂槍打鳥式地去試。

□ 先限定" plot 指令中"

"view "

的範圍，從圖形中觀察，並適時地調整" plot 中" x 的範圍。

□ 其它。_______________________________________________________

4. 若你猜測

0

lim ( ) 1

x

f x

→ = ，而圖形顯示

這告訴你

□

0

lim ( )

x

f x

→ 不存在 □

0

lim ( ) 1

x

f x

→ ≠ □ Maple 把圖形畫錯了

□ 其它。___________________________________________________

請說明原因__________________________________________________

5. 這個活動中，重覆了許多次「給一個誤差，決定x，使得所猜測的極限值與函數 值的差小於所給的誤差」，這樣的動作是否有助於你了解

f x L

a

x =

→ ( )

lim 的非形式 定義：「

f

(x)要有多靠近就可以多靠近 L，當我們適當地選取 x (從 a 的兩邊，

但不等於 a ) 夠靠近 a」？

□ 否 □ 一點點 □ 是

6. 這個活動中，重覆了許多次「給一個誤差，決定x，使得所猜測的極限值與函數 值的差小於所給的誤差」，這樣的動作對於你了解 「極限的ε

−

δ定義」中ε 和δ 所 代表的意義有幫助嗎？

□ 沒有

□ 一點點

□ 有，請寫出你了解的ε 和δ 所代表的意義___________________________

7. 這個活動中，「給一個誤差，決定x，使得所猜測的極限值與函數值的差小於所給 的誤差」，這樣的動作對於你了解「極限的ε

−

δ定義」有幫助嗎？

□ 有很大的幫助。

□ 有，但幫助不大。不過在活動之後，經老師的講解才使我更了解。

□ 有，但幫助不大。即使在活動之後，老師的講解也是無法幫助我了解。

□ 沒有幫助，但老師課後的講解對我比較有幫助。

□ 沒有幫助，但老師課後的講解對我比較有幫助。

在文檔中 CAS 實驗活動的學習環境下極限概念之建構歷程 (頁 108-140)