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HOW TO “DOMESTICATE” LOGARITHMS IN SCHOOL?

HOW TO “DOMESTICATE” LOGARITHMS IN SCHOOL?. Presented at 5th International Colloquium  on the Didactics of Mathematics University of Crete, Greece April 2008 Dr. Marina Rugelj, SLOVENIA. St. Stanislav Institution. Ljubljana, Slovenia. Private high school

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HOW TO “DOMESTICATE” LOGARITHMS IN SCHOOL?

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  1. HOW TO “DOMESTICATE” LOGARITHMS IN SCHOOL? Presented at 5th International Colloquium on the Didactics of Mathematics University of Crete, Greece April 2008 Dr. Marina Rugelj, SLOVENIA

  2. St. Stanislav Institution Ljubljana, Slovenia Private high school Students from 15 to 19 years old

  3. Combined lessons • Dr. Marina Rugelj - maths teacher • Tine Golež, MSc. - physics teacher European project: ScienceMath Example: Logaritmic function

  4. The logarithmic function is introduced as the inverse function of the exponential one: logax = y x = ay a > 0 and x > 0

  5. The graph of f(x) = logax

  6. Rules of calculations with logs • loga 1 = 0 • loga a = 1 • loga (x y) = logax + logay • loga (x/y) = logax - logay • logaxr = r logax

  7. Students can: • Repeat the definition • Put down the rules of calculation with logs • Draw the graph of function: f(x) = log3 (x +3) - 1 • Solve the equation: log3 (x – 2) – log3 (x + 1) = 3

  8. Students have: • Repeat the definition KNOWLEDGE • Put down the rules of calculation with logs • Draw the graph of function: f(x) = log3 (x +3) - 1 SKILLS • Solve the equation: log3 (x – 2) – log3 (x + 1) = 3

  9. Knowledge and skills in themselves do not guarantee understanding. Rote knowledge generally defies active use, and routine skills often serve poorly because students do not understand when to use them. Teaching for understanding (Perkins, 1993)

  10. The spreading activation model Earthquake, Richtar scale … Exponential function Sound, dBel… Logarithmic function pH, acidity … Distances in Universe… Spreading Activation Model of Semantic Memory (Collins & Loftus, 1975)

  11. Applicability of logarithms Physic teacher: intensity of sound, loudness Chemistry teacher: definition of pH Geography teacher: earthquakes

  12. Example 1Loudness of sound

  13. Human is equiped with very sensitive ears: threshold of hearing: 10-12 W/m2 threshold of pain: 10W/m2 10-12 W/m2…1 W/m2 … 200 km/h 1.8 mm/year

  14. Coversation is 1 000 000 times more intenstive than TOH … 10-11 10-12 W/m2 10-10 10-6 1km Conversation TOH Rustling leaves Whisper

  15. Loudness: I …. Intensity of sound Io …. Intensity of threshold Alexander Graham Bell 1847 - 1922

  16. Sound level meter La, la La, la La, la La, la La, la La, la La, la 50 La, la La, la La, la La, la 60 If we like the loudness of 70 dB there should be 100 students (in an ideal condition).

  17. Example 2Map of the Universe

  18. Distance from the Earth to 286 000 km MOON 149 000 000 km aprox. 1 light second SUN 8,278 light minute

  19. The Sun is 520 times more far away from the Erath than the Moon.

  20. Moon 1 cm Sun 5 m Saturn 19 m Proxima Centauri 1422 km Center of Milky Way 8 600 727 km Adromeda 727 753 846 km

  21. Logaritmic scale The Moon 1.0 cm the Sun 2.7 cm, Saturn 3.3 cm, Proxima Centauri 8.2 cm, the center of the Milky Way 11.9 cm, Andromeda 13.9 cm.

  22. Example 3pH measurement

  23. pH = -log[0H3+] Soren Peter Lauritz Sorensen 1868 - 1939

  24. 10 ml add water 100 ml 1000 ml HCl pH: 1 2 3

  25. 1 liter of acid solution of pH 4 how much pure water do we need to get the solution of pH 9?

  26. Example 4Earthquakes

  27. Richter magnitude scale is Logaritmic scale: Charles F. Richter 1900 - 1985 Earthquake of magnitude 6 is 100 times stronger than Earthquake of magnitude 4.

  28. Find the strongest earthquakes in Slovenia! • How many earthquakes were last three days? • Find earthquakes where many people died! • How much stronger was the earthquake in Slovenia in 1998 (5.6) than in 2004 (4.2)?

  29. Observations

  30. At first students were not so enthusiastic about my new approach of teaching. • It was difficult for them to transfer concepts, ideas and procedures learned in mathematics to real life, to science. • They had to use knowledge from other subjects, such as physics, chemistry, biology ... • Active participation brought them a feeling of success. • They became motivated. • They better understood the concept of logs.

  31. Which part do you like the most? N=30

  32. How often would you like such practical lesson? N=30

  33. Have these lessons helped you to understand logarithms better? N=30

  34. Do you like these type of tasks would appear in exams? N=30

  35. Next year we will revise all the mathematical concepts that the students have learned in secondary school for the final exams. At that time I will check if my students have truly constructed a knowledge network and if they know better the concept of the logarithm.

  36. Thank you!

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