1. Engaging Students and Teaching Mathematics for Understanding Carol E. Malloy, Ph.D. University of North Carolina at Chapel Hill cmalloy@email.unc.edu Milwaukee Mathematics Partnership August 27, 2007

2. Our Task To MOBILIZE our forces to educate students who are not succeeding in learning and understanding mathematics

3. Issues  Equity  The Achievement Gap  NCLB  Instruction

4. A Student’s Feelings

5. Equity NCTM Equity Principle Excellence in mathematics education Excellence in mathematics education requires equity—high expectations and strong support for all students.

6. Gap: Achievement or Opportunity What is the purpose of continually finding and reporting differences in racial and ethnic group achievement? A new way of normalizing self and stereotyping others.

7. NCLB No Child Left Behind of the Federal Government Leave No Child Behind of the Children’s Defense Fund

8. Breaking Down the Issues through Knowledge  Students  Advocates  Teachers  Learning

9. Students Student are Messy Functions in relationship to the clean and often predictable nature of mathematics functions.

10. Students Interest in mathematics Perception of mathematics Aptitude for reasoning with abstractions Perception of what is important Experiences Prior mathematical knowledge Communication skills Coursework in other academic areas Self-confidence Attitude toward learning Home and social life Special needs Cultural values Personal values Home resources Access to calculators, computers, and other tools Classroom citizenship

11. Happiness and Joy

12. REAL Issues of Opportunity  Recommendations for special education  Limited access to gifted education  Suspensions  Quality of curriculum and instruction  Beliefs of school administrators, teachers, parents, and students

13. Advocates Parents, Community members, and Society are dependent upon TEACHERS

14. Teachers Teachers are the key to students’ mathematics learning. More than any other single factor, teachers influence what mathematics students should learn and how well they learn it. (NCTM Teaching Principle) Teachers are the key to students’ mathematics learning. More than any other single factor, teachers influence what mathematics students should learn and how well they learn it. (NCTM Teaching Principle)

15. But sometimes

16. Advocates and Teachers Should focus on  Pedagogy  Mathematical Tasks with appropriate Content  Interaction within the classroom And actively and persistently  Convince students that they can learn  Believe that they can learn

17. Cognitive/Affective Opposition  Many communities value and encourage the acquisition of unique verbal expressiveness,  BUT schools place highest value on the written demonstration of verbal knowledge.  Many students’ view of the world is that of a unified environment, thus they use a mixture of holistic and analytical reasoning;  BUT many schools still concentrate on analytical reasoning.

18. For Example Introducing the Golden Ratio

19. Learning Preferences Analytic, field-independent, individual learners should be instructed in ways that encourage them to:  Focus on detail and use sequential/structured thinking,  Recall abstract ideas and irrelevant detail, engage in inanimate material, respond to intrinsic motivation, focus on the task,  Learn from formal lecture,  Achieve individually, and  Emphasize facts and principles.

20. Knowing the Golden Ratio The Golden Ratio can be seen in rectangles and is visually pleasing. The ratio of the width to length of the Golden Ratio is about 1 to 1.618. The Golden Ratio can be seen in rectangles and is visually pleasing. The ratio of the width to length of the Golden Ratio is about 1 to 1.618. 1.618 1.618

21. Learning Preferences Holistic, field-dependent, interdependent learners should be instructed in ways that encourage them to:  Focus on the whole, use improvisational intuitive thinking,  Recall relevant verbal ideas,  Engage in human and social content material,  Respond to extrinsic motivation,  Focus on interests,  Learn from informal class discussion,  Achieve interdependently, and  Narrate human concepts.

22. Understanding the Golden Ratio 1.Determine your ratio by measuring the length from the navel to chin, length of head, navel to ground, and navel to top-of- head. What ratios did you find for your navel height: total height and navel to top of head: navel height? 2. Use the chart below to find our class ratios. Make a conjecture about the ratios. Consider the reciprocals of your ratios. Class member Navel height Navel to head Total height Navel height Total height Navel – head Navel height

23. Engaging Extensions The Fibonacci Sequence begins 1,1,2,3,5,8,.... Expand the sequence to at least 12 terms. The Fibonacci Sequence begins 1,1,2,3,5,8,.... Expand the sequence to at least 12 terms. –Find the ratio of consecutive pairs of numbers in the sequence. –What conjecture can you make about the limit of these pairs of Fibonacci Numbers ? Try to find the solution to this system of equations first by guess and check: Try to find the solution to this system of equations first by guess and check: x – y = 1 and xy = 1. x – y = 1 and xy = 1. Now try another method of solution. Now try another method of solution.

24. We Must Use Varied Approaches

25. NCTM Equity Principle for Every Child  Equity requires high expectations and worthwhile opportunities for all.  Equity requires accommodating differences to help everyone learn mathematics.  Equity requires resources and support for all classrooms and all students.

26. Equity Equity does not mean that every student should receive identical instruction; instead, it demands that reasonable and appropriate accommodations be made as needed to promote access and attainment for all students. (PSSM, 2000)

27. Comparison Using Second Holistic Approach PreferencesAccommodationsRecommendations Students Teachers should Students should Focus on the whole Use improvisational, intuitive thinking Acknowledge and use individual student preferences in the acquisition of knowledge Use mathematical, reasoning, conjecturing, and problem solving Have teachers who value differences and who help everyone learn Learn from informal discussion Develop activities and questioning to promote mathematical discourse. Challenge, justification, and explanation Experience classrooms as mathematical communities Learn to communicate and reason mathematically Learn to communicate and reason mathematically

28. Comparison PreferencesAccommodation Recommendations Recommendations Students Teachers should Students should Recall relevant verbal ideas Narrate human concepts Value student discourse and verbal knowledge Listen and use student understanding Employ logic and mathematical evidence as verification for answers Connect mathematics, its ideas and its applications Respond to extrinsic motivation Encourage, support, and provide feedback to students as they learn Expect that students can and will achieve conceptual and procedural understanding of the mathematics content Have resources and support Become confident in their own ability Experience teachers who have high expectations and worthwhile opportunities for all

29. Comparison PreferencesAccommodationsRecommendations Students Teacher should Students should Focus on interests Engage in human and social content material Create and use mathematical tasks that require students to “do mathematics” Learn to value mathematics Become mathematical problem solvers Achieve interdependently Create interdependent learning communities within the classroom Experience classrooms as mathematical communities

30. Many of us have seen and heard the statements in the last column of the charts. They are the recommendations of the NCTM standards documents from the 1989 through 2000.

31. Working Toward Success These are exciting but challenging times in mathematics education. These are exciting but challenging times in mathematics education. We know more about teaching and learning than we did in the past. We know more about teaching and learning than we did in the past. We know students come to school with some understanding. We know students come to school with some understanding. We have the ability to have students learn more with understanding. We have the ability to have students learn more with understanding.

32. Learn, Reflect, Articulate Group A: Create geoboard squares with the given areas. Record your figures on geoboard dot paper. Group A: Create geoboard squares with the given areas. Record your figures on geoboard dot paper. 1.A square with an area of four square units. 1.A square with an area of four square units. 2.A square with an area of nine square units. 2.A square with an area of nine square units. 3.A square with an area of sixteen square 3.A square with an area of sixteen square units. units. 4.A SQUARE WITH THE AREA OF 5 SQUARE UNITS. 4.A SQUARE WITH THE AREA OF 5 SQUARE UNITS. 5.A square with an area of two square units. 5.A square with an area of two square units. 6.A square with an area of eight square units. 6.A square with an area of eight square units. 7.A square with an area of ten square units. 7.A square with an area of ten square units.

33. Finding Square with Area 10

34. Let’s think about Fractions and Triangles How would you teach middle or high school students the relationship between a fraction an its equivalent percent? How would you teach middle or high school students the relationship between a fraction an its equivalent percent? How would you teach a high school student when the segments from a vertex in a triangle are concurrent? How would you teach a high school student when the segments from a vertex in a triangle are concurrent?

35. Percentage Example Students can use trial and error to find that each section is 1/3 of the circle or about 30--33%. This totals 90--99% if the circle. They could become more exact by computing similar findings for the remaining 10—1%, resulting in 1/3 equals 33 1/3 %. Students can use trial and error to find that each section is 1/3 of the circle or about 30--33%. This totals 90--99% if the circle. They could become more exact by computing similar findings for the remaining 10—1%, resulting in 1/3 equals 33 1/3 %.

36. Exploration with Technology Compare the positions of segments in triangles from one vertex, including the altitudes, medians, perpendicular bisectors and angle bisectors. Use triangles that are isosceles, equilateral, and scalene. Include answers to the following in your comparison. Compare the positions of segments in triangles from one vertex, including the altitudes, medians, perpendicular bisectors and angle bisectors. Use triangles that are isosceles, equilateral, and scalene. Include answers to the following in your comparison. –In what types of triangles will all of the segments from one vertex coincide? –When will all of the sets of segments from each vertex coincide?

37. Results

38. Isosceles and Equilateral

39. What are these Tasks Really about? They are moving students from doing problems through memorization of procedures …. They are moving students from doing problems through memorization of procedures …. To doing and thinking mathematics. To doing and thinking mathematics.

40. What would you expect if you asked students to write a fraction that is larger than 2/7? What would you expect if you asked students to write a fraction that is larger than 2/7? Look at these replies. Look at these replies.

43. Talk with your Neighbor Which students demonstrate that they conceptually understand the size of fractions? Which students demonstrate that they conceptually understand the size of fractions? What could you do to help those who do not demonstrate their conceptual understanding? What could you do to help those who do not demonstrate their conceptual understanding?

44. The Task at Hand Is for US What can we do as teachers to help our students understand mathematics? What can we do as teachers to help our students understand mathematics? How can we ensure that they have understanding? How can we ensure that they have understanding? What can be done to help them apply their understanding to other situations? What can be done to help them apply their understanding to other situations?

45. First consider the process through which students gain conceptual understanding. Dimensions of Understanding [1]: [1] Students should Construct relationships Construct relationships Extend and apply mathematical knowledge Extend and apply mathematical knowledge Reflect about experiences Reflect about experiences Articulate what they know Articulate what they know Make the mathematical knowledge their own Make the mathematical knowledge their own [1][1] Carpenter, Thomas. P., and Lehrer, Richard. “Teaching and Learning Mathematics with Understanding.” In Mathematics Classrooms That Promote Understanding, edited by Elizabeth Fennema and Thomas. A. Romberg, pp. 19-32. Mahwah, NJ: LEA, 1999.. [1]

46. Second, consider the tasks you are asking students to complete. Levels of Tasks: Memorization Tasks: involve either reproducing previously learned facts, rules, formulae, or definitions OR committing facts, rules formulae or definitions to memory. Memorization Tasks: involve either reproducing previously learned facts, rules, formulae, or definitions OR committing facts, rules formulae or definitions to memory. Procedures without Connections Tasks: are algorithmic. The use of the procedure is either specifically called for is its use is evident based on prior instruction, experience, or placement of the task. Procedures without Connections Tasks: are algorithmic. The use of the procedure is either specifically called for is its use is evident based on prior instruction, experience, or placement of the task. Procedures with Connections Tasks: focus students' attention on the use of procedures for the purpose of developing deeper levels of understanding of mathematical concepts and ideas. Procedures with Connections Tasks: focus students' attention on the use of procedures for the purpose of developing deeper levels of understanding of mathematical concepts and ideas. Doing Mathematics Tasks: require complex and non- algorithmic thinking (i.e. there is not a predictable, well- rehearsed approach or pathway explicitly suggested by the teas, task instructions, or a worked out example.) Doing Mathematics Tasks: require complex and non- algorithmic thinking (i.e. there is not a predictable, well- rehearsed approach or pathway explicitly suggested by the teas, task instructions, or a worked out example.) Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. Implementing standards-based mathematics instruction: A casebook for professional development. New York, NY: Teachers College Press.

47. We Can Do It!

48. “Never doubt that a small group of thoughtful citizens can change the world. Indeed it is the only thing that ever has.” --Margaret Mead