Developing prospective
mathematics teachers in Mexico:
A lesson on the relationship between
integration and differentiatio...
Mexico
Mexico
• Population:
117 million
(World rank: 11th).
• Language:
Spanish (92%) &
Native languages
(8%).

Mexico
City
Mexico City
20 million population ≈ Australian population
(Mexico city + Metropolitan area)
Developing prospective
mathematics teachers in Mexico:
A lesson on the relationship between
integration and differentiatio...
Teachers of mathematics
in Mexico
Where are they trained?
•
•
•
•

For basic education:
Normales (Teachers Colleges) and Universities
For High School and Hi...
General information about
Education in Mexico
Phases of Education in Mexico
Basic:

High School:

Higher:

Ages: 5 to 15 years

Ages: 15 to 18 years

Ages: 18 to 29 yea...
Phases of Education in Mexico
Basic:

High School:

Higher:

Ages: 5 to 15 years

Ages: 15 to 18 years

Ages: 18 to 29 yea...
Phases of Education in Mexico
Basic:
Ages: 5 to 15 years
Mandatory

Average: Only 9 years of education
Phases of Education in Mexico
High School:

Higher:

Ages: 15 to 18 years

Ages: 18 to 29 years
Phases of Education in Mexico
Higher:
Ages: 18 to 29 years

Average: Only 13 in every 100 students
finish the university
O...
Issues in Mexico
•
•
•
•
•
•

Educational
Economical
Lack of jobs
Corruption
Political
Violence
There is hope!
• Mexican authorities and Universities have
been working together for improving
education at all levels.
University of Colima
* Colima
Faculty of Educational Sciences
• Programme for training mathematics teachers
for basic and high school level.
Mathematics curriculum
Basic
(1) number sense and
algebraic thinking
(2) shape, space and
measurement;
(3) data analysis;
...
A lesson on the relationship between
integration and differentiation
The Fundamental Theorem of Calculus
Fundamental Theorem of Calculus
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 dif...
Fundamental Theorem of Calculus

ò

x
x dx = + C
3
3

2

æ x3
ö
d
2
ç +C÷ = x
dx è 3
ø
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 dif...
Fundamental Theorem of Calculus
• Applets
• Geogebra
Example 1
Time

Velocity
Example 2
F(x) = at + bt + c
2

x

ò f (t ) dt
a

x

d
f t dt
dx a

f x
What can you say about the car?
Another example
x

ò f (t ) dt
a
x

ò f (t ) dt
a

d
dx

x

ò f (t ) dt = f ( x) ?
a
f (t) = at + bt + c
2
• Intuitive idea of the Fundamental Theorem of
Calculus.
•
•
•
•
•

Use of technology
Dynamic interaction and visualization
Theory
History of mathematics
…
Thanks
Developing prospective mathematics teachers in Mexico
Developing prospective mathematics teachers in Mexico
Developing prospective mathematics teachers in Mexico
Developing prospective mathematics teachers in Mexico
Developing prospective mathematics teachers in Mexico
Developing prospective mathematics teachers in Mexico
Developing prospective mathematics teachers in Mexico
Developing prospective mathematics teachers in Mexico
Developing prospective mathematics teachers in Mexico
Developing prospective mathematics teachers in Mexico
Developing prospective mathematics teachers in Mexico
Developing prospective mathematics teachers in Mexico
Developing prospective mathematics teachers in Mexico
Developing prospective mathematics teachers in Mexico
Developing prospective mathematics teachers in Mexico
Developing prospective mathematics teachers in Mexico
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Developing prospective mathematics teachers in Mexico

  1. 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. 2. Mexico
  3. 3. Mexico • Population: 117 million (World rank: 11th). • Language: Spanish (92%) & Native languages (8%). Mexico City
  4. 4. Mexico City
  5. 5. 20 million population ≈ Australian population (Mexico city + Metropolitan area)
  6. 6. Developing prospective mathematics teachers in Mexico: A lesson on the relationship between integration and differentiation
  7. 7. Teachers of mathematics in Mexico
  8. 8. Where are they trained? • • • • For basic education: Normales (Teachers Colleges) and Universities For High School and Higher Education: Universities
  9. 9. General information about Education in Mexico
  10. 10. 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
  11. 11. 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
  12. 12. Phases of Education in Mexico Basic: Ages: 5 to 15 years Mandatory Average: Only 9 years of education
  13. 13. Phases of Education in Mexico High School: Higher: Ages: 15 to 18 years Ages: 18 to 29 years
  14. 14. 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
  15. 15. Issues in Mexico • • • • • • Educational Economical Lack of jobs Corruption Political Violence
  16. 16. There is hope! • Mexican authorities and Universities have been working together for improving education at all levels.
  17. 17. University of Colima
  18. 18. * Colima
  19. 19. Faculty of Educational Sciences • Programme for training mathematics teachers for basic and high school level.
  20. 20. 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
  21. 21. A lesson on the relationship between integration and differentiation
  22. 22. The Fundamental Theorem of Calculus
  23. 23. Fundamental Theorem of Calculus
  24. 24. 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
  25. 25. Fundamental Theorem of Calculus ò x x dx = + C 3 3 2 æ x3 ö d 2 ç +C÷ = x dx è 3 ø
  26. 26. 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
  27. 27. Fundamental Theorem of Calculus • Applets • Geogebra
  28. 28. Example 1
  29. 29. Time Velocity
  30. 30. Example 2
  31. 31. F(x) = at + bt + c 2 x ò f (t ) dt a x d f t dt dx a f x
  32. 32. What can you say about the car?
  33. 33. Another example
  34. 34. x ò f (t ) dt a
  35. 35. x ò f (t ) dt a d dx x ò f (t ) dt = f ( x) ? a
  36. 36. f (t) = at + bt + c 2
  37. 37. • Intuitive idea of the Fundamental Theorem of Calculus.
  38. 38. • • • • • Use of technology Dynamic interaction and visualization Theory History of mathematics …
  39. 39. Thanks

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