B eyond  M athematical  C ontent  K nowledge  Presenters Aaron Brakoniecki  Sandra Crespo Ann M. Lawrence MSU Discussant M...
ingredients   Beyond
Beyond   tools
Beyond   techniques
Beyond   Subject Matter  Knowledge Pedagogical Content Knowledge
For Shulman  (1986, 1987) , teachers’  strategic judgment  makes teaching, a profession. For Ball  (2008, 2009) , teachers...
knowledgeable practice mathematical topic pedagogical move strategic judgment &
P(osing), I(nterpreting), R(esponding) <ul><li>NSF Career Grant  Award Number 0546164 </li></ul><ul><li>Begin to unpack ho...
The Problem of Knowledge Categorizing Practice Aaron Brakoniecki Michigan State University April 21, 2010
The Chef & The Meal <ul><li>Interested in uncovering what good chefs need to know in order to make great meals. </li></ul>
Mathematical Knowledge for Teaching Subject Matter  Knowledge Pedagogical Content Knowledge
Common Content Knowledge <ul><li>“ ’ common’ knowledge of mathematics that any well-educated adult should have” (Ball, Hil...
Specialized Content Knowledge <ul><li>“ mathematical knowledge that is “specialized” to the work of teaching and that only...
Horizon Content Knowledge <ul><li>“ awareness of how mathematical topics are related over the span of mathematics included...
Knowledge of Content and Students <ul><li>“ knowledge that combines knowing about students and knowing about mathematics” ...
Knowledge of Content and Teaching <ul><li>“ knowing about teaching and knowing about mathematics” (Ball et al., 2008, p. 4...
Knowledge of Content and Curriculum <ul><li>Allusions to the MKT framework’s category being similar to Shulman’s original ...
Task 1(a) <ul><li>Preservice teachers are free to imagine their classroom exactly as they want it. </li></ul><ul><ul><li>S...
Allie’s Response to 1(a) • “ Think about what you could do to these numbers to make adding them together easier. Ask them ...
Intended Study <ul><li>Look at statements made by respondents and categorize them by their corresponding MKT </li></ul><ul...
Preservice Teacher A I would start by discussing places (ones, ten, hundred, and so on). Then I would discuss  how you add...
Preservice Teacher B Before introducing this problem I would first review place value concepts . If it were for a younger ...
Preservice Teacher C I would start by asking the students how many places there are in this problem and distinguishing bet...
The Problem of ‘Why’ <ul><li>Most all students described what they did, but not all students described why they were doing...
Analogy to Cooking <ul><li>How do you take a finished dish and figure out what went into it? </li></ul><ul><li>What does o...
Defining MKT vs Examples of MKT Subject Matter  Knowledge Pedagogical Content Knowledge
Defining MKT vs Examples of MKT Subject Matter  Knowledge Pedagogical Content Knowledge
Defining MKT vs Examples of MKT Subject Matter  Knowledge Pedagogical Content Knowledge
Defining MKT vs Examples of MKT Subject Matter  Knowledge Pedagogical Content Knowledge
Defining MKT vs Examples of MKT Subject Matter  Knowledge Pedagogical Content Knowledge
MKT to Describe Practice <ul><li>MKT is described as “practice-based theory of mathematical knowledge for teaching” (Ball ...
If not knowledge, then what? <ul><li>MKT was designed to describe teachers’ knowledge </li></ul><ul><li>Knowledge may be n...
An Example of a Knowledgeable Practice <ul><li>Using manipulatives to discuss rebundling/regrouping of addition </li></ul>...
Knowledgeable Practices of Allie • “ Think about what you could do to these numbers to make adding them together easier. A...
Knowledgeable Practice and Teacher Education <ul><li>MKT is often used to describe what teachers don’t have, a deficit mod...
Studying Practices of Mathematics Teaching Sandra Crespo Michigan State University April 21, 2010
Welcome to the World  of Cooking In concept the art of cooking attracts the bachelor, but in practice many are often at a ...
<ul><li>A.  Imagine you are going to ask your class to solve the following addition. What can you imagine  saying  and  do...
<ul><li>Start with the last digits of each number, 8 and 9. Add these together and then determine if the sum has two digit...
<ul><li>I would encourage them to use what they know about place value to solve this task.  'Look at the 1's place. What i...
<ul><li>Today we are going to add two numbers that hold 3 place values, ones, tens, and hundreds. What different methods c...
<ul><li>This is hard to imagine without creating a context! I think I would choose to give this problem because it require...
Repertoire
Contrasting  Repertoire
<ul><li>Is this a valid method – does it produce the correct answer? </li></ul><ul><li>How does the method work? Can we al...
Practice <ul><li>Noun </li></ul><ul><li>- a customary way of operation or behavior  </li></ul><ul><li>- recurrent socially...
Rehearsal of Practice
Sample Questions Can someone explain why Susie has 130 in the second line and not 13? Does someone have a different way to...
Teaching Knowledgeable Practices Ann Lawrence Michigan State University April 21, 2010
Presenting a Probability Experiment <ul><li>Giving students directions on how to approach the task </li></ul><ul><li>Elici...
Strategic  Rehearsal commonplaces   kairos   progymnasmata   (Bizzell & Herzberg, 2001; Crowley & Hawhee, 2009; Hawhee, 20...
knowledgeable practice practice high-leverage &
Ball, D. L., Hill, H. C., & Bass, H. (2005). Knowing mathematics for teaching: Who knows mathematics well enough to  teach...
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NCTM 2010

  1. 1. B eyond M athematical C ontent K nowledge Presenters Aaron Brakoniecki Sandra Crespo Ann M. Lawrence MSU Discussant Megan Franke UCLA
  2. 2. ingredients Beyond
  3. 3. Beyond tools
  4. 4. Beyond techniques
  5. 5. Beyond Subject Matter Knowledge Pedagogical Content Knowledge
  6. 6. For Shulman (1986, 1987) , teachers’ strategic judgment makes teaching, a profession. For Ball (2008, 2009) , teachers’ content knowledge makes mathematics teaching, a profession.
  7. 7. knowledgeable practice mathematical topic pedagogical move strategic judgment &
  8. 8. P(osing), I(nterpreting), R(esponding) <ul><li>NSF Career Grant Award Number 0546164 </li></ul><ul><li>Begin to unpack how teachers learn the practices of posing, interpreting, and responding in mathematics classrooms. </li></ul><ul><li>Study looking at students in our elementary education program </li></ul><ul><ul><li>Current students (preservice teachers) </li></ul></ul><ul><ul><li>Graduated students (in-service teachers) </li></ul></ul>
  9. 9. The Problem of Knowledge Categorizing Practice Aaron Brakoniecki Michigan State University April 21, 2010
  10. 10. The Chef & The Meal <ul><li>Interested in uncovering what good chefs need to know in order to make great meals. </li></ul>
  11. 11. Mathematical Knowledge for Teaching Subject Matter Knowledge Pedagogical Content Knowledge
  12. 12. Common Content Knowledge <ul><li>“ ’ common’ knowledge of mathematics that any well-educated adult should have” (Ball, Hill, & Bass, 2005, p. 22) </li></ul><ul><li>“ the mathematical knowledge known in common with others who know and use mathematics” (Ball, Thames, & Phelps, 2008, p. 403), </li></ul><ul><li>“ some of the mathematical resources that teaching requires are similar to the mathematical knowledge that other professionals use. We labeled this common content knowledge…” (Ball & Hill, 2009, p. 70). </li></ul>
  13. 13. Specialized Content Knowledge <ul><li>“ mathematical knowledge that is “specialized” to the work of teaching and that only teachers need know” (Ball et al., 2005, p. 22) </li></ul><ul><li>“ the mathematical knowledge and skill unique to teaching” (Ball et al., 2008, p. 400) </li></ul><ul><li>“ content knowledge that is tailored in particular for the specialized uses that come up in the work of teaching, and is thus not commonly used in those ways by most other professions or occupations” (Hill et al., 2008, p. 436). </li></ul>
  14. 14. Horizon Content Knowledge <ul><li>“ awareness of how mathematical topics are related over the span of mathematics included in the curriculum” (Ball et al., 2008, p. 403) </li></ul><ul><li>“ a kind of mathematical &quot;peripheral vision&quot; needed in teaching, that is, a view of the larger mathematical landscape that teaching requires (Ball and Bass 2009).” (Ball & Hill, 2009, p. 70) </li></ul>
  15. 15. Knowledge of Content and Students <ul><li>“ knowledge that combines knowing about students and knowing about mathematics” (Ball et al., 2008, p. 401) </li></ul><ul><li>“ knowing about both mathematics and students” (Delaney, Ball, Hill, Schilling, & Zopf, 2008, p. 175) </li></ul><ul><li>“ content knowledge intertwined with knowledge of how students think about, know, or learn this particular content” (Hill et al., 2008, p. 375) </li></ul>
  16. 16. Knowledge of Content and Teaching <ul><li>“ knowing about teaching and knowing about mathematics” (Ball et al., 2008, p. 401) </li></ul><ul><li>“ knowing about both mathematics and teaching” (Delaney et al., 2008, p. 175) </li></ul>
  17. 17. Knowledge of Content and Curriculum <ul><li>Allusions to the MKT framework’s category being similar to Shulman’s original conception of Curricular Knowledge. </li></ul>
  18. 18. Task 1(a) <ul><li>Preservice teachers are free to imagine their classroom exactly as they want it. </li></ul><ul><ul><li>Student </li></ul></ul><ul><ul><li>Resources </li></ul></ul><ul><ul><li>Autonomy </li></ul></ul><ul><li>Imagined practice scenarios can serve as a control next to actual practice </li></ul>
  19. 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. 20. Intended Study <ul><li>Look at statements made by respondents and categorize them by their corresponding MKT </li></ul><ul><li>Look at responses as a whole and categorize the depth of their MCK & PCK </li></ul><ul><li>Problems Occurred </li></ul>
  21. 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. 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. 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. 24. The Problem of ‘Why’ <ul><li>Most all students described what they did, but not all students described why they were doing those things. </li></ul><ul><li>The “why” is necessary to identify what knowledge may be being enacted. </li></ul><ul><li>Any attempt to investigate practice will not have access to all the reasons affecting their practice </li></ul>
  25. 25. Analogy to Cooking <ul><li>How do you take a finished dish and figure out what went into it? </li></ul><ul><li>What does one dish tell us about a chef’s repertoire? </li></ul>
  26. 26. Defining MKT vs Examples of MKT Subject Matter Knowledge Pedagogical Content Knowledge
  27. 27. Defining MKT vs Examples of MKT Subject Matter Knowledge Pedagogical Content Knowledge
  28. 28. Defining MKT vs Examples of MKT Subject Matter Knowledge Pedagogical Content Knowledge
  29. 29. Defining MKT vs Examples of MKT Subject Matter Knowledge Pedagogical Content Knowledge
  30. 30. Defining MKT vs Examples of MKT Subject Matter Knowledge Pedagogical Content Knowledge
  31. 31. MKT to Describe Practice <ul><li>MKT is described as “practice-based theory of mathematical knowledge for teaching” (Ball et al., 2008, p. 395) </li></ul><ul><li>The detail and nuance of the classroom has been stripped away in creating this framework </li></ul><ul><li>Using MKT to go back and describe the practice upon which the framework was based is challenging </li></ul>
  32. 32. If not knowledge, then what? <ul><li>MKT was designed to describe teachers’ knowledge </li></ul><ul><li>Knowledge may be necessary, but it is not sufficient </li></ul><ul><li>If our goal is to improve practice, we should try to use a frame that puts practice at the foreground </li></ul><ul><li>Knowledgeable Practices </li></ul>
  33. 33. An Example of a Knowledgeable Practice <ul><li>Using manipulatives to discuss rebundling/regrouping of addition </li></ul><ul><ul><li>Mathematical Topic: Adding numbers whose sums are larger than 10 </li></ul></ul><ul><ul><li>Pedagogical Move: Using beads or chips to illustrate addition </li></ul></ul><ul><ul><li>Knowledgeable Practice: Using beads or chips to model rebundling/regrouping in addition </li></ul></ul>
  34. 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. 35. Knowledgeable Practice and Teacher Education <ul><li>MKT is often used to describe what teachers don’t have, a deficit model. </li></ul><ul><li>Knowledgeable Practices are a way to think about and describe what teachers can do. </li></ul><ul><li>MKT is often used as a way to think about how we can bridge the “gap” between knowledge and practice. </li></ul><ul><li>Knowledgeable Practices are one bridge that we can start with. </li></ul>
  36. 36. Studying Practices of Mathematics Teaching Sandra Crespo Michigan State University April 21, 2010
  37. 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).
  38. 38. <ul><li>A. Imagine you are going to ask your class to solve the following addition. What can you imagine saying and doing to get them ready to work on this task? </li></ul><ul><li> 258 </li></ul><ul><li>+389 </li></ul><ul><li> </li></ul>Addition with Regrouping B. Imagine that a student shows on the board the following strategy for adding. i.) What would you want to make sure your class notices in this student’s work? ii.) What are three different questions you can imagine asking to the class about this student’s strategy, and say a bit about how these three are questions different? 258 +389 17 130 500 647
  39. 39. <ul><li>Start with the last digits of each number, 8 and 9. Add these together and then determine if the sum has two digits. Take the 10’s digit, if there is one, and add it to the next two digits, 5 and 8. Do this again, including the added digit, and carry over to the final pair of digits, 2 and 3. This time, if the sum is two digits, keep them at the bottom of the line with the others. For example: </li></ul><ul><li> </li></ul>A Recipe for Adding
  40. 40. <ul><li>I would encourage them to use what they know about place value to solve this task. 'Look at the 1's place. What is 8+9? (17). Will the answer, 17, fit in the 1's place? (NO) What can we do to the 17 to make it fit? (Break it up into 7 ones and 1 ten and put the 7 ones in the 1's place, and move the 1 ten to the 10's place.) Next, with the additional ten in the 10's place, what is 10+50+80? (140). Will 140 fit into the tens place? (No) What can we do to make 140 fit? (Break it up into 4 tens and 1 hundred. Keep the 4 tens (40) in the 10's place and move the 1 hundred (100) to the 100's place. With the additional hundred in the 100's place, what is 100 + 200 + 300? (600). Does the answer, 600, fit into the hundreds place? (YES, so we're done). </li></ul>Interactive Demonstration
  41. 41. <ul><li>Today we are going to add two numbers that hold 3 place values, ones, tens, and hundreds. What different methods could we use to solve this problem, besides just adding the columns? How could we add these numbers together, why? Could we first round our hundred column down and add them together, 200+300? What would we add next? </li></ul><ul><li> </li></ul>Interactive Demonstration 5 0 + 8 0 1 3 0 So what else would we have to add? 8 + 9 1 7 Then what do we do with these numbers? 500 + 130 17 647 How could you check this answer?
  42. 42. <ul><li>This is hard to imagine without creating a context! I think I would choose to give this problem because it requires “regrouping” twice. I think I would give it after students had worked on problems with smaller numbers. I find myself thinking about this problem from two different perspectives. I can imagine that students would have had multiple opportunities to work on place value issues and that perhaps the challenge in this problem is coordinating place value in the hundreds. I can also imagine giving this problem at the end of a unit on addition and subtraction (perhaps late 2 nd grade or early 3 rd grade) when I’m feeling comfortable with the students’ understanding of place value and I want them to explore the algorithm. </li></ul>A Different Approach
  43. 43. Repertoire
  44. 44. Contrasting Repertoire
  45. 45. <ul><li>Is this a valid method – does it produce the correct answer? </li></ul><ul><li>How does the method work? Can we all try explaining it and see if it makes sense to us? </li></ul><ul><li>Does it work for other sets of numbers? How generalizable is it? </li></ul><ul><li>How is this method related to other methods? </li></ul><ul><li>Is this method efficient? Would we want to use it in all, most, or some occasions? </li></ul>Heuristics of Practice
  46. 46. Practice <ul><li>Noun </li></ul><ul><li>- a customary way of operation or behavior </li></ul><ul><li>- recurrent socially organized activities that permeate daily life (Saxe, 1999) </li></ul><ul><li>I consider classroom practices to be the recurrent activities and norms that develop in classrooms over time, in which teachers and students engage (Boaler, 2002) </li></ul><ul><li>Verb </li></ul><ul><li>- is the act of rehearsing a behavior over and over, or engaging in an activity again and again, for the purpose of improving or mastering it. </li></ul>
  47. 47. Rehearsal of Practice
  48. 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?
  49. 49. Teaching Knowledgeable Practices Ann Lawrence Michigan State University April 21, 2010
  50. 50. Presenting a Probability Experiment <ul><li>Giving students directions on how to approach the task </li></ul><ul><li>Eliciting student recall of these directions </li></ul><ul><li>Engaging students in strategic rehearsal of the task— What would you do? </li></ul><ul><li>Performing a sample approach to the task </li></ul>
  51. 51. Strategic Rehearsal commonplaces kairos progymnasmata (Bizzell & Herzberg, 2001; Crowley & Hawhee, 2009; Hawhee, 2002, 2004)
  52. 52. knowledgeable practice practice high-leverage &
  53. 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

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