6. For Shulman (1986, 1987) , teachers’ strategic judgment makes teaching, a profession. For Ball (2008, 2009) , teachers’ content knowledge makes mathematics teaching, a profession.
19. Allie’s Response to 1(a) • “ Think about what you could do to these numbers to make adding them together easier. Ask them if they could round the numbers up or down to make the addition easier. • For other learning styles, ask students what place you would add 1st (tens, ones, etc.) • Prepare students for carrying w/ base-10 blocks • You could draw boxes above the #’s to remind them to carry.”
20.
21. Preservice Teacher A I would start by discussing places (ones, ten, hundred, and so on). Then I would discuss how you add in columns and how to “carry over” to the next place showing that 9+8=17 which has 10 ones + 7 ones so we need to add 1 to the tens column and so on .”
22. Preservice Teacher B Before introducing this problem I would first review place value concepts . If it were for a younger classroom I would use some math manipulatives to demonstrate the concept of “carrying over,” or if it were an older class I would probably ask for volunteers to demonstrate “carrying over” on the board . After either task is completed I would do some similar examples on the board using class participation. Once I felt confident that everyone knew what they were doing , I would give them this problem to work out on their own.
23. Preservice Teacher C I would start by asking the students how many places there are in this problem and distinguishing between the ones, tens, and hundreds place. Then I am going to ask them to think about whether or not we are going to have to do some regrouping.
24.
25.
26. Defining MKT vs Examples of MKT Subject Matter Knowledge Pedagogical Content Knowledge
27. Defining MKT vs Examples of MKT Subject Matter Knowledge Pedagogical Content Knowledge
28. Defining MKT vs Examples of MKT Subject Matter Knowledge Pedagogical Content Knowledge
29. Defining MKT vs Examples of MKT Subject Matter Knowledge Pedagogical Content Knowledge
30. Defining MKT vs Examples of MKT Subject Matter Knowledge Pedagogical Content Knowledge
31.
32.
33.
34. Knowledgeable Practices of Allie • “ Think about what you could do to these numbers to make adding them together easier. Ask them if they could round the numbers up or down to make the addition easier. • For other learning styles, ask students what place you would add 1st (tens, ones, etc.) • Prepare students for carrying w/ base-10 blocks • You could draw boxes above the #’s to remind them to carry.”
35.
36. Studying Practices of Mathematics Teaching Sandra Crespo Michigan State University April 21, 2010
37. Welcome to the World of Cooking In concept the art of cooking attracts the bachelor, but in practice many are often at a loss as to how to bridge the gap between wanting to cook and actually knowing where to start . Perhaps he has prepared a meal or two. Perhaps the meals were not bad. But often the bachelor is robbed of the excitement of cooking through a lack of basic cooking know-how” (p. 7).
48. Sample Questions Can someone explain why Susie has 130 in the second line and not 13? Does someone have a different way to solve this problem? Why did this student put zeros in their work? Why didn’t this student carry the numbers? Will this work for any 3 digit + problems? For any + problem? For any problem? Can you see how this student was able to get to the answer? Why did they add 17, 130, and 500? How do you know that this answer is correct?
53. Ball, D. L., Hill, H. C., & Bass, H. (2005). Knowing mathematics for teaching: Who knows mathematics well enough to teach third grade, and how can we decide? American Educator, 29 (3), 14-17, 20-22, 43-46. Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education , 59 (5), 389-407. Bizzell, P., & Herzberg, B. (2001). The rhetorical tradition: Readings from classical times to the present (2 nd ed.). Boston: Bedford/St. Martin’s. Crowley, S., & Hawhee, D. (2009). Ancient rhetorics for contemporary students (4 th ed.) New York: Pearson/Longman . Delaney, S., Ball, D., Hill, H., Schilling, S., & Zopf, D. (2008). “Mathematical knowledge for teaching”: adapting U.S. measures for use in Ireland. Journal of Mathematics Teacher Education, 11 (3), 171-197. Franke, M. L. & Chan, A. G. (n.d.) High leverage practices . Retrieved April 16, 2010, from http://gallery.carnegiefoundation.org/insideteaching/quest/megan_loef_franke_ and_angela_grace_chan_high.html Hawhee, D. (2004). Bodily arts: Rhetoric and athletics in ancient Greece . Austin, TX: University of Texas Press. Hawhee, D. (2004). Bodily pedagogies: Rhetoric, athletics, and the Sophists’ three Rs. College English , 65 (2), 142- 162. Hill, H.C., & Ball, D. L. (2009). The curious—and crucial—case of mathematical knowledge for teaching. Phi Beta Kappan , 91 (2), 68-71. Hill, H. C., Ball, D. L., & Schilling, S. G. (2008). Unpacking Pedagogical Content Knowledge: Conceptualizing and Measuring Teachers' Topic-Specific Knowledge of Students. Journal for Research in Mathematics Education, 39 (4), 372-400. Hill, H. C., Blunk, M. L., Charalambous, C. Y., Lewis, J. M., Phelps, G. C., Sleep, L., & Ball, D. L. (2008). Mathematical Knowledge for Teaching and the Mathematical Quality of Instruction: An Exploratory Study. Cognition and Instruction, 26 (4), 430-511. Shulman, L. S. (1986/2004). Those who understand: Knowledge growth in teaching. In S. M. Wilson (Ed.), The wisdom of practice: Essays on teaching, learning, and learning to teach (pp. 187-216). San Francisco: Jossey-Bass. Shulman, L. S. (1987/2004). Knowledge and teaching: Foundations of the new reform. In S. M. Wilson (Ed.), The wisdom of practice: Essays on teaching, learning, and learning to teach (pp. 217-248). San Francisco: Jossey-Bass
