Presentation for the Delta 2013 Kiama, Australia.

1. Developing prospective
mathematics teachers in Mexico:
A lesson on the relationship between
integration and differentiation
Juan Carlos Ponce Campuzano
Faculty of Educational Sciences
University of Colima

2. Mexico

3. Mexico
• Population:
117 million
(World rank: 11th).
• Language:
Spanish (92%) &
Native languages
(8%).
Mexico
City

4. Mexico City

5. 20 million population ≈ Australian population
(Mexico city + Metropolitan area)

8. Developing prospective
mathematics teachers in Mexico:
A lesson on the relationship between
integration and differentiation

9. Teachers of mathematics
in Mexico

10. Where are they trained?
•
•
•
•
For basic education:
Normales (Teachers Colleges) and Universities
For High School and Higher Education:
Universities

11. General information about
Education in Mexico

12. Phases of Education in Mexico
Basic:
High School:
Higher:
Ages: 5 to 15 years
Ages: 15 to 18 years
Ages: 18 to 29 years
Mandatory
pre-K, grades 1-9
grades 10-12
23 years studying
Undergraduate
Postgraduate

13. Phases of Education in Mexico
Basic:
High School:
Higher:
Ages: 5 to 15 years
Ages: 15 to 18 years
Ages: 18 to 29 years
Mandatory
Average: Only 9 years of education

14. Phases of Education in Mexico
Basic:
Ages: 5 to 15 years
Mandatory
Average: Only 9 years of education

15. Phases of Education in Mexico
High School:
Higher:
Ages: 15 to 18 years
Ages: 18 to 29 years

16. Phases of Education in Mexico
Higher:
Ages: 18 to 29 years
Average: Only 13 in every 100 students
finish the university
Only 3 in every 100 students
will continue with a post grade

17. Issues in Mexico
•
•
•
•
•
•
Educational
Economical
Lack of jobs
Corruption
Political
Violence

18. There is hope!
• Mexican authorities and Universities have
been working together for improving
education at all levels.

20. University of Colima

22. * Colima

23. Faculty of Educational Sciences
• Programme for training mathematics teachers
for basic and high school level.

24. Mathematics curriculum
Basic
(1) number sense and
algebraic thinking
(2) shape, space and
measurement;
(3) data analysis;
(4) attitudes toward
the study of
mathematics.
High School
Algebra,
Analytic geometry,
Trigonometry,
Calculus,
Probability and Statistics

25. A lesson on the relationship between
integration and differentiation

26. The Fundamental Theorem of Calculus

27. Fundamental Theorem of Calculus

28. Fundamental Theorem of Calculus
b
x
d
f t dt
dx a
f t dt
f x
F b
a
Inverse relationship between
integration and differentiation
F a

29. Fundamental Theorem of Calculus
ò
x
x dx = + C
3
3
2
æ x3
ö
d
2
ç +C÷ = x
dx è 3
ø

30. Fundamental Theorem of Calculus
b
x
d
f t dt
dx a
f t dt
f x
F b
a
Inverse relationship between
integration and differentiation
F a

31. Fundamental Theorem of Calculus
• Applets
• Geogebra

32. Example 1

34. Time
Velocity

36. Example 2

41. F(x) = at + bt + c
2
x
ò f (t ) dt
a
x
d
f t dt
dx a
f x

42. What can you say about the car?

43. Another example

46. x
ò f (t ) dt
a

47. x
ò f (t ) dt
a
d
dx
x
ò f (t ) dt = f ( x) ?
a

48. f (t) = at + bt + c
2

49. • Intuitive idea of the Fundamental Theorem of
Calculus.

50. •
•
•
•
•
Use of technology
Dynamic interaction and visualization
Theory
History of mathematics
…

51. Thanks