Download Calculus Curriculum Questionnaire for Greece

CalGeo: Teaching Calculus using dynamic geometric tools
Outcome 1.1.1 : “Report on existing curricula. Questionnaire, Greece”
COMENIUS PROGRAM
CalGeo
CALCULUS CURRICULUM
QUESTIONNAIRE
**********************************
Greece
i
OCTOBER 2004
▄ National Curriculum
_____________________________________________________________________
IMPORTANT: Throughout this questionnaire, the term “mathematics national
curriculum” is intended to include any centrally-supported curriculum. This
curriculum may not be articulated in a formal document, or different aspects of the
curriculum may appear in different documents.
_____________________________________________________________________
1. Does your country have a mathematics national curriculum that includes
calculus at grades 10, 11 and 12?
Fill in one circle only_______________________________________
Yes
No

O
Note: If No, please complete the remainder of the questionnaire based on your best-informed
judgment of the intended mathematics curriculum for the majority of grades 10, 11 and 12
students in your country. If it is impossible to answer a particular question, just make a note
and move to the next question.
2. In what year was the current intended calculus curriculum for grades 10, 11
and 12 introduced?
The reform of national curriculum concerning mathematics in grades 10 to 12 started at 1989.
The application of this new curriculum started gradually during the period 1990-1993 and it is
still active, with some revisions, until nowadays.
3. What are the main differences of the current calculus curriculum from the
previous one? (if any)
According the previous curriculum:
o
The Sequences of Real numbers, their limits and properties were taught.
o
The exponential function was defined as an application of sequences of real numbers.
o
Emphasis to epsilonic definition of limit and continuity.
o
There were definitions and theorems about bounded functions.
o
Much more proofs of theorems.
o
No indefinite integration was included
o
Volume of a solid of revolution was included.
ii
4. Is the intended calculus curriculum that includes grades 10, 11 and 12
currently being revised?
Fill in one circle only _______________________________________
Yes
No

O
5. Which best describes how the calculus national curriculum at grades 10, 11
and 12 addresses the issue of students that take mathematics as a selective
course?
Fill in one circle for each column
Grade 10
The same curriculum is prescribed for all
students (i.e. There is not an option for basic or
advanced mathematics)
Different curriculum is prescribed for the
students that take mathematics as a selective
course
Grade 11
Grade 12


O
O
O

6. Does the national curriculum contain statements/policies about the use of
graphic calculators in grades 10, 11 and 12 calculus?
Fill in one circle only _______________________________________
Yes
No
O

If YES, what are the statement/policies?
____________________________________________________________
____________________________________________________________
7. Does the national curriculum contain statements/policies about the use of
computers/computer software in grades 10, 11 and 12 calculus?
Fill in one circle only _______________________________________
Yes
No
O

If YES, what are the statement/policies?
____________________________________________________________
____________________________________________________________
8. Which are the current requirements for being a mathematics teacher at highschool/lyceum?
Fill in one circle for each row
Yes
No
iii
a. Pre-practicum and supervised practicum in the field___ O

b. Passing an examination_________________________ 
O
c. Acquiring a University degree____________________ 
O
d. Completion of a probationary teaching period________ O

e. Completion of a mentoring or induction program______ O

f. Other________________________________________ O
O
(Please specify: _______________________)
9. Is there a process to license or certify high-school-lyceum mathematics
teachers?
Fill in one circle only _______________________________________
Yes
No
O

10. Do high-school/lyceum mathematics teachers receive specific compulsory
preparation in teaching the intended calculus curriculum at grades 10, 11 and
12?
Fill in one circle for each row
Yes
No
a. As part of pre-service education___________________ O
b. As part of in-service education____________________ O


If you answered YES to either (a) or (b), describe the nature of the
preparation.
____________________________________________________________
____________________________________________________________
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11. According to the national mathematics curriculum, at which grades the
following calculus topics are taught?



If at a particular grade there is a different curriculum for students that take
mathematics as a selective course, please complete the extra columns.
Please write other topics or subjects if necessary.
Please indicate with an asterisk (in the topic column) if a topic is not actually
taught (or taught in a different way) although it is included in the curriculum
of a specific grade. Explain in the “Notes” space if this is the case.
Curriculum for all
students
Topic
Curriculum for students
having mathematics as a
selective course
Grade
Primarily
Grade
Primarily
taught
taught
10 11 12
10 11 12
A. Real Numbers*
a)
b)
c)
d)
Natural numbers - Induction
Rational numbers (basic
properties)
Irrational numbers
Other topics or theorems: (Please
X
X
X
X
specify)
……………………………..
……………………………..
Notes:* There are introductory lessons about set of numbers (Natural, Rational, Irrational, Real
numbers) at grades less than 10. In these lessons is highlighted the differences between the different
meanings and symbolizations of these numbers and there is no emphasis in the characteristic properties
of each set. In Grade 10 there is a general representation of set of real numbers and its properties
concerning operations and ordering. In addition, the real number line is used as an external
representation for real numbers.
B. Real Functions
a)
b)
c)
d)
e)
f)
g)
h)
i)
Definition of a function
Operations between functions
Composition of functions
One-to-one functions
Inverse of a function
Other topics or theorems:
Graph of function
Definitions of odd and even
functions
Definitions of monotone
functions
Definitions of extrema
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
v
j)
Equal functions
Notes:
X
C. Sequences of real numbers
a)
b)
c)
d)
e)
f)
g)
h)
i)
Definition
Monotone sequences
Bounded sequences
lim n→∞ an=ℓ, ℓ real number
lim n→∞ an=±∞
Εpsilonic definitions of the
limits d) and e) above
Basic properties of the limits of
sequences, i.e., uniqueness,
algebra of limits
The Pinching Theorem
Other topics or theorems: (Please
X*
X**
X*
X**
X**
specify)
……………………………..
……………………………..
Notes: * Only for the needs of arithmetic and geometric progression and for the definition of number e.
** There is only a reference of the above definitions with no applications or exercises.
D. Limits of functions
lim x→c f(x)=ℓ, c,ℓ real numbers
X
lim x→±∞ f(x)=ℓ
lim x→c f(x)= ±∞
lim x→±∞ f(x)=±∞
Εpsilonic definitions of limits
a), b), c), d) above*
f)
Basic properties of the limits i.e.,
uniqueness, algebra of limits **
g) Operations between +∞, -∞ and
real numbers
Other topics or theorems:
h) Limit and ordering
i)
The Pinching Theorem
Notes: * Although this definition there is in book, it is not taught.
** Except uniqueness.
a)
b)
c)
d)
e)
E. Function continuity
a) Definition of continuity
b) Different types of discontinuity
c) ε, δ definition of function
d)
e)
f)
continuity
The algebra of continuous
functions
Composition of continuous
functions
Bolzano theorem (Intermediate-
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
vi
g)
h)
i)
value theorem)
The maximum-minimum
theorem
Continuity of the inverse of a
function
Other topics or theorems: (Please
X
specify)
……………………………..
……………………………..
Notes:
F. Differentiation and Applications
a)
b)
c)
d)
e)
Definition of the derivative
X
X
Derivative of basic functions
X
X
Differential of a function
The tangent line to a curve
X
X
The derivative as a rate of
X
X
change, velocity
f)
Differentiability and continuity
X
g) Differentiation rules, e.g., the
product rule, the quotient rule,
X
X
the chain rule
h) Derivatives of inverse functions
i)
Implicit differentiation
j)
Higher order derivatives
X
X
k) Rolle’s Theorem
X
l)
Mean-Value Theorem
X
m) L’ Hospital´s Law, (0/0), (∞/∞)
X
n) Derivative of a function given in
parametric form
o) Increasing and decreasing
functions, local and global
X
X
extrema, critical points, the first
and second derivative tests *
p) Concavity, points of inflection,
X
the second derivative test
q) Asymptotes, i.e., horizontal,
X
vertical, oblique
r)
Graphs of functions
X
s) Further applications of
X
X
differentiation, eg., acceleration
Other topics or theorems:
t)
Other consequences of MeanValue Theorem:
If f is defined on an interval and
X
f ( x)  0 for all x in the
interval, then f is constant on the
interval.
u) Study and sketching the graph of
X
a function
Notes: * Except the second derivative tests which although there are in book, they are not taught.
vii
G. Indefinite Integration
a)
b)
c)
Definition
Simple cases of integration
Properties of integrals, e.g.,
(summation, multiplication by
constants)
d) Integration by parts
e) Integration by change of
variables
f)
Integration of rational functions
Other topics or theorems:
g) Definition of antiderivative
function
Notes:
X
X
X
X
X
X
X
H. Definite Integration - Applications
a)
b)
c)
d)
e)
f)
g)
h)
i)
Definition by means of:
i. Rieman’s sums *
ii. indefinite integration
Fundamental theorem of calculus
Definite integration calculations
Interpretation of definite integral
as the area under a curve, area
problems
The length of a parameterized
curve
Area of a surface of revolution
Volume of a solid of revolution
Mean value theorem of
integration **
Other topics or theorems: (Please
X
X
X
X
X
specify)
……………………………..
……………………………..
Notes: * It is taught graphically but it not used in applications or exercises.
** Although there is in book, it is not taught.
I. Exponential and logarithmic functions
a) The definition of e
X
b) Exponential functions, ex , ax
X
c)
d)
e)
f)
g)
h)
Properties of exponential
functions
Logarithms to the base 10 and
natural logarithms
Change the base of a logarithm
Logarithmic functions
Properties of logarithmic
functions
Exponential equations
X
X
X
X
X
X
X
X
X
X
viii
i)
j)
k)
l)
m)
n)
Logarithmic equations
Limits of exponential and
logarithmic functions
Continuity of exponential and
logarithmic functions
Derivatives of exponential and
logarithmic functions
Inequalities of exponential and
logarithmic functions *
Other topics or theorems: (Please
X
X
X
X
X
specify)
……………………………..
……………………………..
Notes: * As applications and exercises.
J. Trigonometric functions
a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
k)
l)
Definitions of trigonometric
functions
Limits of trigonometric functions
Continuity of trigonometric
functions
Derivatives of trigonometric
functions
Graphs of trigonometric
functions
Definition of inverse
trigonometric functions
Limits of inverse trigonometric
functions
Continuity of inverse
trigonometric functions
Derivative of inverse
trigonometric functions
Graphs of inverse trigonometric
functions
Integration of trigonometric
functions
Other topics or theorems: (Please
X
X
X
X
X
X
X
X
X
specify)
……………………………..
……………………………..
Notes:
K. Differential equations *
a)
b)
c)
d)
e)
i)
Definition
Formation of differential
equations
First-order
Second-order
Applications
Other topics or theorems: (Please
X
X
X
X
ix
specify)
……………………………..
……………………………..
Notes: * Although there are in book, they are not taught.
L.
Notes:
M.
Notes:
12. The teaching of sequence limits takes place before the teaching of function
limits in curriculum.
Fill in one circle only _______________________________________
Yes
No
O

13. The teaching of indefinite integration takes place before the teaching of
definite integration in curriculum.
Fill in one circle only _______________________________________
Yes
No

O
x
14. Among the curriculum aims for Calculus are:
Yes
No
a. Intuitive understanding of the concept of limit

O
b. Graphical understanding of the concept of limit

O
c. Understanding the epsilonic definition of limit
O

d. Limit Calculations

O
e. Intuitive understanding of the concept of continuity 
O
f. Graphical understanding of the concept of

O
g. Understanding the ε δ of the concept of continuity O


O

O

O
Continuity
h. Test of the continuity of a function
i. Calculus theorems proofs understanding
j. Calculating skills development
1
1. Note : Significant decrease of the number of theorems proofs but still exist some.
xi