CalGeo: Teaching Calculus using dynamic geometric tools
Outcome 1.1.3 : “Report on existing curricula. Questionnaire, Bulgaria”
COMENIUS PROGRAM
CalGeo
CALCULUS CURRICULUM
QUESTIONNAIRE
**********************************
Bulgaria
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?
Yes
Fill in one circle only_______________________________________
No
X
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?
2003________________
3. What are the main differences of the current calculus curriculum from the
previous one? (if any)
____________________________________________________________
____________________________________________________________
4. Is the intended calculus curriculum that includes grades 10, 11 and 12
currently being revised?
Yes
Fill in one circle only _______________________________________
No
X
ii
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
X
X
X
6. Does the national curriculum contain statements/policies about the use of
graphic calculators in grades 10, 11 and 12 calculus?
Yes
No
X
Fill in one circle only _______________________________________
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?
Yes
Fill in one circle only _______________________________________
No
X
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
a. Pre-practicum and supervised practicum in the field___ X
O
b. Passing an examination_________________________ X
O
c. Acquiring a University degree____________________ X
O
iii
d. Completion of a probationary teaching period________ O
X
e. Completion of a mentoring or induction program______ O
X
f. Other________________________________________ O
X
(Please specify: _______________________)
9. Is there a process to license or certify high-school-lyceum mathematics
teachers?
Yes
Fill in one circle only _______________________________________
O
No
X
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
a. As part of pre-service education___________________ X
b. As part of in-service education____________________ O
No
O
X
If you answered YES to either (a) or (b), describe the nature of the
preparation.
There is a Bachelor program “Mathematics & Informatics for high-school
teachers” contains obligatory courses devoted to “High-school Mathematics” , in
particular High-school calculus.
iv
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
X
X
X
X
specify)
X
Complex numbers………..
……………………………..
Notes:
B. Real Functions
a)
b)
c)
d)
e)
f)
Definition of a function
Operations between functions
Composition of functions
One-to-one functions
Inverse of a function
Other topics or theorems: (Please
X
X
X
X
X
X
specify)
……………………………..
……………………………..
Notes:
C. Sequences of real numbers
a)
b)
c)
d)
Definition
Monotone sequences
Bounded sequences
lim n→∞ an=ℓ, ℓ real number
X
X
X
X
X
X
v
e)
f)
g)
h)
i)
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
specify)
Arithmetic and geometric
progression
Geometric series
X
X
X
X
Notes:
D. Limits of functions
a)
b)
c)
d)
e)
f)
g)
h)
lim x→c f(x)=ℓ, c,ℓ real numbers
lim x→±∞ f(x)=ℓ
lim x→c f(x)= ±∞
lim x→±∞ f(x)=±∞
Εpsilonic definitions of limits a),
b), c), d) above
Basic properties of the limits i.e.,
uniqueness, algebra of limits
Operations between +∞, -∞ and real
numbers
Other topics or theorems: (Please
X
X
X
X
X
X
X
specify)
lim
x 0
sin x
x
X
Notes:
E. Function continuity
a) Definition of continuity
b) Different types of discontinuity
c) ε, δ definition of function continuity
d) The algebra of continuous functions
e) Composition of continuous
f)
g)
h)
i)
functions
Bolzano theorem (Intermediatevalue theorem)
The maximum-minimum theorem
Continuity of the inverse of a
function
Other topics or theorems: (Please
X
X
X
X
X
X
specify)
……………………………..
vi
……………………………..
Notes:
F. Differentiation and Applications
a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
k)
l)
m)
n)
o)
p)
q)
r)
s)
t)
Definition of the derivative
Derivative of basic functions
Differential of a function
The tangent line to a curve
The derivative as a rate of change,
velocity
Differentiability and continuity
Differentiation rules, e.g., the
product rule, the quotient rule, the
chain rule
Derivatives of inverse functions
Implicit differentiation
Higher order derivatives
Rolle’s Theorem
Mean-Value Theorem
L’ Hospital´s Law, (0/0), (∞/∞)
Derivative of a function given in
parametric form
Increasing and decreasing functions,
local and global extrema, critical
points, the first and second
derivative tests
Concavity, points of inflection, the
second derivative test
Asymptotes, i.e., horizontal,
vertical, oblique
Graphs of functions
Further applications of
differentiation, eg., acceleration
Other topics or theorems: (Please
X
X
X
X
X
X
X
X
X
X
X
X
X
X
specify)
……………………………..
……………………………..
Notes:
G. Indefinite Integration
a)
b)
c)
d)
Definition
Simple cases of integration
Properties of integrals, e.g.,
(summation, multiplication by
constants)
Integration by parts
vii
e)
f)
g)
Integration by change of variables
Integration of rational functions
Other topics or theorems: (Please
specify)
……………………………..
……………………………..
Notes:
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
specify)
……………………………..
……………………………..
Notes:
I. Exponential and logarithmic functions
a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
k)
l)
The definition of e
Exponential functions, ex , ax
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
Logarithmic equations
Limits of exponential and
logarithmic functions
Continuity of exponential and
logarithmic functions
Derivatives of exponential and
logarithmic functions
X
X
X
X
X
X
X
X
X
X
X
X
viii
m)
n)
Inequalities of exponential and
logarithmic functions
Other topics or theorems: (Please
X
specify)
Graphic solutions of logaritmic
equations and inequalities
Notes:
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
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
specify)
……………………………..
……………………………..
ix
Notes:
L.
Notes:
M.
Notes:
12. The teaching of sequence limits takes place before the teaching of function
limits in curriculum.
Yes
Fill in one circle only _______________________________________
X
No
O
13. The teaching of indefinite integration takes place before the teaching of
definite integration in curriculum.
Yes
Fill in one circle only _____________N.A.______________________
O
No
O
x
14. Among the curriculum aims for Calculus are:
Yes
No
a. Intuitive understanding of the concept of limit
X
O
b. Graphical understanding of the concept of limit
X
O
c. Understanding the epsilonic definition of limit
O
X
d. Limit Calculations
X
O
e. Intuitive understanding of the concept of continuity O
X
f. Graphical understanding of the concept of
Continuity
X
O
g. Understanding the ε δ of the concept of continuity O
X
h. Test of the continuity of a function
O
X
i. Calculus theorems proofs understanding
O
X
j. Calculating skills development
X
O
xi