1. The Synergy of Adolescent Identity Formation, Mathematics Content and Reflective Abstraction Initiates Alan Zollman Northern Illinois University National Council of Teachers of Mathematics Annual Meeting April 24, 2009 Washington, DC

2. Question 1: Take the following proposal title of 22 words and edit it to 15 words or fewer: The Synergy of Adolescent Identity Formation, Mathematics Content, and Reflective Abstraction Initiates: Where the Combined Results Are Greater Than the Individual Effects Question 1: Take the following proposal title of 22 words and edit it to 15 words or fewer: The Synergy of Adolescent Identity Formation, Mathematics Content, and Reflective Abstraction Initiates: Where the Combined Results Are Greater Than the Individual Effects

3. Question 1: Take the following proposal title of 22 words and edit it to 15 words or fewer: The Synergy of Adolescent Identity Formation, Mathematics Content, and Reflective Abstraction Initiates: Where the Combined Results Are Greater Than the Individual Effects One possible answer: Adolescent Identity Formation, Mathematics Content, Reflective Abstraction Initiates Synergy: Combined Results Greater than Individual Effects Question 1: Take the following proposal title of 22 words and edit it to 15 words or fewer: The Synergy of Adolescent Identity Formation, Mathematics Content, and Reflective Abstraction Initiates: Where the Combined Results Are Greater Than the Individual Effects One possible answer: Adolescent Identity Formation, Mathematics Content, Reflective Abstraction Initiates Synergy: Combined Results Greater than Individual Effects

4. Question 2: As a lieutenant you need to get a 40-foot telephone pole raised. You have 5 privates and 1 sergeant. Specifically, how do you get the pole raised properly? Question 2: As a lieutenant you need to get a 40-foot telephone pole raised. You have 5 privates and 1 sergeant. Specifically, how do you get the pole raised properly?

5. Question 3: When you were in the first grade, what did you plan to be? Question 3: When you were in the first grade, what did you plan to be?

6. Question 3: When you were in the first grade, what did you plan to be? WHY? Question 3: When you were in the first grade, what did you plan to be? WHY?

7. Question 4: If you grow up, what do you plan to be? Question 4: If you grow up, what do you plan to be?

8. Question 4: If you grow up, what do you plan to be? WHY? Question 4: If you grow up, what do you plan to be? WHY?

9. Why, in one class, do we have some students mathematically achieve, and other students in the same class with similar aptitude and background do not? What is that one attribute we cannot put a finger on that is the difference between these students? Why, in one class, do we have some students mathematically achieve, and other students in the same class with similar aptitude and background do not? What is that one attribute we cannot put a finger on that is the difference between these students?

10. Why, in one class, do we have some students mathematically achieve, and other students in the same class with similar aptitude and background do not? What is that one attribute we cannot put a finger on that is the difference between these students? I say the major influence is self identity. Why, in one class, do we have some students mathematically achieve, and other students in the same class with similar aptitude and background do not? What is that one attribute we cannot put a finger on that is the difference between these students? I say the major influence is self identity.

11. Students ’ Development 1)Physical Development Students ’ Development 1)Physical Development

12. Students ’ Development 1)Physical Development 2)Social Development Students ’ Development 1)Physical Development 2)Social Development

13. Students ’ Development 1)Physical Development 2)Social Development 3)Cognitive Development Students ’ Development 1)Physical Development 2)Social Development 3)Cognitive Development

14. Students ’ Development 1)Physical Development 2)Social Development 3)Cognitive Development 4)Identity Development Students ’ Development 1)Physical Development 2)Social Development 3)Cognitive Development 4)Identity Development

15. Identity Formation Identity formation is the fundamental development task of psychological maturity Identity formation is a striving to achieve a unified, integrated sense of self. This requires the incorporation of past and present identifications with significant others, recognition of one ’ s aptitudes and skills, and occupational goals and aspirations. Identity Formation Identity formation is the fundamental development task of psychological maturity Identity formation is a striving to achieve a unified, integrated sense of self. This requires the incorporation of past and present identifications with significant others, recognition of one ’ s aptitudes and skills, and occupational goals and aspirations.

16. Identity is how we respond to the environmental, cognitive, and social affects in our lives. All affective domain topics — motivation, persistence, self-esteem, self-confidence, attitude, even behavior — are outcomes of our personal identity. Forming one ’ s identity is as important as developing one ’ s social skills or cognitive abilities. Identity is how we respond to the environmental, cognitive, and social affects in our lives. All affective domain topics — motivation, persistence, self-esteem, self-confidence, attitude, even behavior — are outcomes of our personal identity. Forming one ’ s identity is as important as developing one ’ s social skills or cognitive abilities. Identity Formation

17. Eight Psychosocial Stages 1)trust versus mistrust, in infancy; 2)autonomy versus shame and doubt, in toddlers; 3)initiative versus guilt, in early childhood; 4)industry versus inferiority, in middle childhood; 5)identity versus role diffusion, in adolescence; 6)intimacy versus isolation, in young adulthood; 7)generativity versus stagnation, in middle adult life; 8)integrity versus despair, in old age (Erikson, 1950). Eight Psychosocial Stages 1)trust versus mistrust, in infancy; 2)autonomy versus shame and doubt, in toddlers; 3)initiative versus guilt, in early childhood; 4)industry versus inferiority, in middle childhood; 5)identity versus role diffusion, in adolescence; 6)intimacy versus isolation, in young adulthood; 7)generativity versus stagnation, in middle adult life; 8)integrity versus despair, in old age (Erikson, 1950).

18. Identity Crisis Identity crisis occurs when the adolescent attempts to integrate childhood identification with ideas about what one wants to be and become as an adult. Identity Crisis Identity crisis occurs when the adolescent attempts to integrate childhood identification with ideas about what one wants to be and become as an adult.

19. Identity Crisis Identity crisis occurs when the adolescent attempts to integrate childhood identification with ideas about what one wants to be and become as an adult. Adolescents initiate identity work as they begin to think about their competencies and attributes, academic and occupational goals, and personal beliefs. Identity Crisis Identity crisis occurs when the adolescent attempts to integrate childhood identification with ideas about what one wants to be and become as an adult. Adolescents initiate identity work as they begin to think about their competencies and attributes, academic and occupational goals, and personal beliefs.

20. Identity Crisis Identity crisis occurs when the adolescent attempts to integrate childhood identification with ideas about what one wants to be and become as an adult. Adolescents initiate identity work as they begin to think about their competencies and attributes, academic and occupational goals, and personal beliefs. School and peers are important social contexts where much identity work occurs. Identity Crisis Identity crisis occurs when the adolescent attempts to integrate childhood identification with ideas about what one wants to be and become as an adult. Adolescents initiate identity work as they begin to think about their competencies and attributes, academic and occupational goals, and personal beliefs. School and peers are important social contexts where much identity work occurs.

21. Possible selves are influenced by social, cultural and historical contexts that surround the individual and function to generate feelings of a)competence (when a goal is attained), b)self-efficacy (beliefs about one ’ s personal competence in mathematics), and b)personal control (what one can do to achieve a hoped-for self). Possible selves are influenced by social, cultural and historical contexts that surround the individual and function to generate feelings of a)competence (when a goal is attained), b)self-efficacy (beliefs about one ’ s personal competence in mathematics), and b)personal control (what one can do to achieve a hoped-for self).

22. Possible Selves Theory is a theoretical foundation to promote teachers ’ understanding of identity formation — their students ’ and their own. Possible selves are one ’ s ideas about what one can become in the future. These perceptions of one ’ s future self can be highly motivating to students. When students have clear ideas about what they want to become, they are more willing to put forth the effort needed to attain their goals. Possible Selves Theory is a theoretical foundation to promote teachers ’ understanding of identity formation — their students ’ and their own. Possible selves are one ’ s ideas about what one can become in the future. These perceptions of one ’ s future self can be highly motivating to students. When students have clear ideas about what they want to become, they are more willing to put forth the effort needed to attain their goals.

23. Hoped-for possible selves, in particular, are strong predictors of mathematics achievement. A hoped-for self that is concrete, realistic, detailed, and invokes necessary strategies for achieving the goal that will guide student behavior and produce the intended results over time (Oyserman & Markus, 1990). Hoped-for possible selves, in particular, are strong predictors of mathematics achievement. A hoped-for self that is concrete, realistic, detailed, and invokes necessary strategies for achieving the goal that will guide student behavior and produce the intended results over time (Oyserman & Markus, 1990).

24. When students feel committed to, and invested in, working towards the attainment of hoped-for selves, and when they connect current behaviors to the accomplishment of future goals, their possible selves serve a self-regulatory role. Students with a self-regulatory focus are better able to make changes in behavior which can lead to goal achievement. When students feel committed to, and invested in, working towards the attainment of hoped-for selves, and when they connect current behaviors to the accomplishment of future goals, their possible selves serve a self-regulatory role. Students with a self-regulatory focus are better able to make changes in behavior which can lead to goal achievement.

25. It is important for students to become self-regulated learners who can - set learning goals - create action plans - then monitor their progress towards their goals - by assessing their efforts and making adaptations as necessary It is important for students to become self-regulated learners who can - set learning goals - create action plans - then monitor their progress towards their goals - by assessing their efforts and making adaptations as necessary

26. Reflective Abstraction Initiates Instructor

27. Reflective Abstraction Initiates Instructor Curriculum Instructor Curriculum

28. Reflective Abstraction Initiates Instructor Curriculum Peer Instructor Curriculum Peer

29. Reflective Abstraction Initiates Instructor Curriculum Peer Individual Instructor Curriculum Peer Individual

30. Instructor Initiates to Promote Identity Development

31. Show your genuine warmth (not fuzziness) for students Cultivate teacher-student professional relationships Model yourself as a life-long, inquisitive learner in front of your students Infuse passion for mathematics in your lessons Respect students as partners in the learning relationship Show your genuine warmth (not fuzziness) for students Cultivate teacher-student professional relationships Model yourself as a life-long, inquisitive learner in front of your students Infuse passion for mathematics in your lessons Respect students as partners in the learning relationship

32. Curriculum Initiates of Identity Development

33. Set up authentic problem-solving situations that students (not just teachers) value Build lessons with drama for “ inquiring minds who want to know ” -- the “ aha! ” vs. the “ oh no! ” problems Plan for students to discover patterns and relationships between and among the topics of mathematics Set up authentic problem-solving situations that students (not just teachers) value Build lessons with drama for “ inquiring minds who want to know ” -- the “ aha! ” vs. the “ oh no! ” problems Plan for students to discover patterns and relationships between and among the topics of mathematics

34. Peer Initiates to Promote Identity Development

35. Design cooperative learning activities where each student has a role; rotate roles Conduct role playing situations, where students view themselves and each other as mathematicians, engineers, investigators and scientists as they work on problems Design cooperative learning activities where each student has a role; rotate roles Conduct role playing situations, where students view themselves and each other as mathematicians, engineers, investigators and scientists as they work on problems

36. Individual Initiates to Promote Identity Development

37. Charge students to set goals, so they know where they want to be and what they have to do to get there; students feel they can take control of their own learning Ask students to regularly self-reflect on what and how they are learning mathematics Teach students to continually self-assess progress of learning Provide choices for students to demonstrate growth in self-determination, self-efficacy, self-regulation Charge students to set goals, so they know where they want to be and what they have to do to get there; students feel they can take control of their own learning Ask students to regularly self-reflect on what and how they are learning mathematics Teach students to continually self-assess progress of learning Provide choices for students to demonstrate growth in self-determination, self-efficacy, self-regulation

38. Multiple Methods - Multiple Choices TEAM Alice: Always works a problem using algebra TEAM Cheryl: Always works a problem making a chart TEAM Darrell: Always works a problem drawing a picture TEAM Thomas: Always works a problem guessing & testing TEAM Marvin: Always works a problem using manipulatives TEAM Gwen: Always works a problem graphing it Multiple Methods - Multiple Choices TEAM Alice: Always works a problem using algebra TEAM Cheryl: Always works a problem making a chart TEAM Darrell: Always works a problem drawing a picture TEAM Thomas: Always works a problem guessing & testing TEAM Marvin: Always works a problem using manipulatives TEAM Gwen: Always works a problem graphing it

39. Problem #5: from “ Are You Smarter than a Fifth Grader? ” There are 7 girls in a bus. Each girl has 7 backpacks. In each backpack, there are 7 big cats. For every big cat, there are 7 little cats. The bus driver is not in the bus at this time. Question: How many legs are there in the bus? Problem #5: from “ Are You Smarter than a Fifth Grader? ” There are 7 girls in a bus. Each girl has 7 backpacks. In each backpack, there are 7 big cats. For every big cat, there are 7 little cats. The bus driver is not in the bus at this time. Question: How many legs are there in the bus?

40. Problem #5: from “ Are You Smarter than a Fifth Grader? ” There are 7 girls in a bus. Each girl has 7 backpacks. In each backpack, there are 7 big cats. For every big cat, there are 7 little cats. The bus driver is not in the bus at this time. Question: How many legs are there in the bus? 10,990 legs Problem #5: from “ Are You Smarter than a Fifth Grader? ” There are 7 girls in a bus. Each girl has 7 backpacks. In each backpack, there are 7 big cats. For every big cat, there are 7 little cats. The bus driver is not in the bus at this time. Question: How many legs are there in the bus? 10,990 legs

41. Problem #6: The Driveway Problem Sarah can sweep the driveway in 40 minutes. And Robert can sweep the driveway in 50 minutes. If Sarah begins 4 minutes before Robert joins her, how long will it take them both to finish the whole driveway?

42. 20 minutes Problem #6: The Driveway Problem Sarah can sweep the driveway in 40 minutes. And Robert can sweep the driveway in 50 minutes. If Sarah begins 4 minutes before Robert joins her, how long will it take them both to finish the whole driveway? 20 minutes

43. Question 2: The correct response is (supposedly): Order the sergeant and the privates to get the job done, leave, and return later. Question 2: The correct response is (supposedly): Order the sergeant and the privates to get the job done, leave, and return later.

44. Question 2: The correct response is (supposedly): Order the sergeant and the privates to get the job done, leave, and return later. This solution displays an understanding and communication of the officer ’ s expectations of: trust, autonomy, initiative, persistence, role identification, team membership, respect and integrity. Question 2: The correct response is (supposedly): Order the sergeant and the privates to get the job done, leave, and return later. This solution displays an understanding and communication of the officer ’ s expectations of: trust, autonomy, initiative, persistence, role identification, team membership, respect and integrity.

45. Question 2: The correct response is (supposedly): Order the sergeant and the privates to get the job done, leave, and return later. This solution displays an understanding and communication of the officer ’ s expectations of: trust, autonomy, initiative, persistence, role identification, team membership, respect and integrity. It allows the privates and sergeant to become self- regulated problem solvers. Question 2: The correct response is (supposedly): Order the sergeant and the privates to get the job done, leave, and return later. This solution displays an understanding and communication of the officer ’ s expectations of: trust, autonomy, initiative, persistence, role identification, team membership, respect and integrity. It allows the privates and sergeant to become self- regulated problem solvers.

46. Question 4: The correct response is: Question 4: The correct response is:

47. Question 4: The correct response is: Be sitting in your chair Question 4: The correct response is: Be sitting in your chair

48. Dr. Alan Zollman Dept. of Mathematical Sciences Northern Illinois University DeKalb, IL 60115 815/753-6750 zollman@math.niu.edu http://www.math.niu.edu/~zollman zollman@math.niu.edu http://www.math.niu.edu/~zollman Dr. Alan Zollman Dept. of Mathematical Sciences Northern Illinois University DeKalb, IL 60115 815/753-6750 zollman@math.niu.edu http://www.math.niu.edu/~zollman zollman@math.niu.edu http://www.math.niu.edu/~zollman