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1. Teaching Mathematics to Prospective Elementary School TeachersPresentation at the PMET Summer WorkshopHumboldt State UniversityArcata, CAJune 13, 2004Dr. Randy PhilippSan Diego State UniversityRphilipp@mail.sdsu.edu(619) 594-2361

2. Session Outline A Secondary Example from Trigonometry An Elementary Example: Place Value The Role of the Unit in Fractions Preparing for our Lesson on Fractions To ground our work, we’ll look to: 1) Examples from Preassessment and Prior Research 2) Video Clips of Children’s Reasoning

3. What do Mathematics Educators Think About? A Secondary Example 8th grade geometry student, Rick Introduction to trigonometry Rick can compute sin, cos, tan, and inverse functions. For example, he can determine the sin (38˚), or the angle whose cosine is .58. Rick is supposed to understand the fundamentals of right angle trigonometry.

4. My goal Trig functions depend upon the ratios of the lengths of the sides. Question: Rick, you have a right triangle, and one angle is 38˚. Can you find the ratio of the opposite side of the 38˚ angle to the hypotenuse of the triangle?

5. Rick draws a right triangle, using a protractor to make one acute angle approximately 38. He measures the two sides, and divides the opposite side by the hypotenuse. I ask him if he realizes that the opposite side divided by the hypotenuse is the same as the sin 38˚. He says he does. I ask, “Rick, can you do the same for another angle, say, 21˚?”

6. Pause. No, not without knowing the lengths of the sides. He draws another right triangle, and again uses a protractor to make one acute angle approximately 21˚. He measures the two sides. He divides. He sees it is the same as Sin 21˚. This goes on. And on.

7. What’s going on here? Rick knows that Sin (x) = Opp/Hyp Rick knows that the value he got by dividing Opp/Hyp is the same thing he got when he punched “sin 21˚” into his calculator. Why then, when I ask him to determine the ratio of the opposite side to the hypotenuse of a particular angle, must he draw the triangle? Sketchpad

8. Why then, when I ask him to determine the ratio of the opposite side to the hypotenuse of a particular angle, must he draw the triangle?

9. What is a ratio? Rick was conceptualizing a ratio as the result of dividing two quantities. He was not conceptualizing it as a single quantity. If he can not conceptualize a ratio as a single quantity, can he possibly understand triangle trigonometry?

10. A related example There are students who can determine the volume of the prism on the left, but not the prism on the right. Do you see why? 6 units 17 square units 3 units 3 units 8 units

11. What do Mathematics Educators do? Among other things, we try to go beyond determining whether a student is right or wrong, and try to understand how students are looking at something. Then we try to share this information with teachers, because we have found that, at least at the elementary school level, when teachers understand their students’ thinking, they are able to better support their students’ learning.

12. The Situation with Prospective Elementary School Teachers We know that developing deep understanding of the mathematics of elementary school is far more difficult than was once thought. -(Ball, 1990; Ma, 1999; Sowder, Philipp, Armstrong, & Schappelle, 1998) Even when PSTs attend a thoughtfully planned course designed to engage them in rich mathematical thinking, too many of them go through the course in a perfunctory manner.

13. Example: Place Value 527 -135 Can PSTs apply the standard subtraction algorithm?

14. Yes. 100% of the15 PSTs in Eva Thanheiser’s study could correctly apply the whole-number multidigit subtraction algorithm. 45127 -1 35 392

15. Eva then asked:Does Regrouping Change the Value of the Numbers? 45127 527 100% of the 15 PSTs stated that the value was the same. But 10 of the 15 could not explain why.

16. Related Example: Place Value Below is the work of Terry, a student who solved one addition and one subtraction problem. A) 1 B) 2 5 9 3412 9 + 3 8- 3 4 2 9 7 3 9 5 a) Does each 1 in these problems mean the same thing? b) Terry knows that when adding, as in Problem A, he adds 1 to the 5, but when subtracting, as in Problem B, he adds 10 to the 2, but he doesn’t know why. Can you explain the reasons to Terry?

17. Does each 1 mean the same thing?Humboldt Students: NO:11 Yes: 4 A) 1 B) 2 5 9 3412 9 + 3 8- 3 4 2 9 7 3 9 5 Comments for “NO” In A we add 1 to the tens column. In B we add 10 to the tens column. (3) The 1 in the addition problem is meant to be added (1 + 5 + 3), and the 1 in the subtraction problem is changing the 2 to a 12, it is “borrowed” from the 4 adding 10 to the 2.

18. 527 -135 • How would you solve it? • How would you explain it? • How would you like your child’s teacher to think about it?

19. 527 -135

20. 527 -135 2

21. 527 -135 2

22. 45127 -1 35 92

23. 45127 -1 35 392

24. 45127 -1 35 392

25. 45127 -1 35 392

26. How would you like children to be able to think about the red numbers? 45127 -1 35 392

27. Questions for PSTs: Look at the red numbers. Can you see 12 of something? If so, what? Can you see 120 of something? If so, what? 45127 -1 35 392

28. Eva’s Findings 2 of 15 students could only see ones. 45127 -1 35 392 Carmen: “I don’t get how the 1 can become a 10. One and 10 are two different numbers. How can you subtract 1 from here and then add 10 over here? Where did the other 9 come from?”

29. Eva’s Findings 2 of 15 students could only see ones. 45127 -1 35 392 Carmen: “I don’t get how the 1 can become a 10. One and 10 are two different numbers. How can you subtract 1 from here and then add 10 over here? Where did the other 9 come from?”

30. Eva’s Findings 2 of 15 students could only see ones. 45127 -1 35 392 Carmen: “I don’t get how the 1 can become a 10. One and 10 are two different numbers. How can you subtract 1 from here and then add 10 over here? Where did the other 9 come from?”

31. Unit is only “1” All groupings are only in ones. These two students knew that there were ten ones in one ten, but they did not connect that knowledge to the algorithm. How can a person “seeing” only ones correctly add multi-digit numbers? 36 + 57

32. The same way that most of us think when using algorithms. 36 + 57 “Six plus seven is Thirteen. Write down the 3 and carry the 1.” 1 6 + 7 3

33. 36 + 57 13 6 + 5+ 7 9 3

34. Eva’s Findings 8 of the 15 PSTs were aware that a digit in the hundred’s place represented 100s, but viewed the digits in the ten’s place as representing ones rather than tens. 45127 -1 35 392 Delia: “When you are borrowing it [1 in the hundred’s place] from over here … it’s a hundred. But once you put it [the borrowed 100] into the number, it becomes a ten.”

35. Eva’s Findings The last 5 provided valid explanations for why the value of the number stayed the same when it was regrouped. Two of those 5 reasoned in terms of groups of 100 ones, not ten tens.

36. Seeing “Groups of Ones” 45127 -1 35 392 Veronica: “So in reality when you are taking a 1 from the 5 you are not really just taking a 1 and putting it next to the 2, you are taking 100 from the 5 – so that is making it 400 and you are borrowing, you are adding the 100 to the 20, so in reality it is 120.”

37. 45127 • -1 35 • 392 Seeing “Groups of Ones,” But Not Seeing Tens When asked about the 12, Veronica did not show any evidence of thinking in terms of 12 tens. She referred to the 12 as “the 12 part of the 120.” Veronica: “Just out of reflex it’s 12, but if you like, look at it and you think about it, it’s actually 120 and 30. If you separate it into the separate components, but like out of just like reflex and human nature it’s a 12 and you are just subtracting 3 from it.”

38. 45127 • -1 35 • 392 Seeing “Groups of Ones,” But Not Seeing Tens When asked whether she could think of it as 12, she covered up the one’s column and stated “we can just like pretend that you’re taking 10 from the 5 and then just adding it to the 2 and then it would be 12.” When asked whether she could think of 12 without covering up the last digit she stated “no.”

39. Explaining the Students’ Conceptions Different Conceptual Structures for one Hundred

40. 5 2 7 How might one think about the 5?

41. 45 12 7

42. 45 12 7

43. 45 12 7