Rigor, Relevance, and Relationships by Design in High School Mathematics

1. Rigor, Relevance, and Relationships by Design in High School Mathematics Eric Robinson, Margaret Robinson NC Raising Achievement and Closing Gaps Conference March 27, 2007

2. Session Purpose: To move forward the North Carolina Raising Achievement and Closing Gaps Commission’s mission to assist “schoolsand school systems in identifying and developing programs and strategies to raise achievement and close gaps.”

3. Not just about doing things better, but doing better things!

4. Session Overview: • Part I:Design components needed in curriculum programs to address rigor, relevance, and relationships • Part II:Evidence • Realization of design principles • Effectiveness

5. Part IDesign

6. A look at the terms: Relevance Demonstrating how students will use their learning Rigor Exposing students to challenging class work with academic and social support Relationships Building caring and supportive connections with students, parents, and communities

7. RigorExposing students to challenging class work

8. RigorExposing students to challenging class work Deep mathematical understanding

9. RigorExposing students to challenging class work Deep mathematical understandingthat allows students to: 1.) see the connections between bits of mathematical knowledge; 2.) apply mathematical thinking to formulate and execute problem-solving strategies; 3.) apply mathematics in novel situations.

10. RigorExposing students to challenging class work Deep mathematical understandingthat allows students to: 1.) see the connections between bits of mathematical knowledge; 2.) apply mathematical thinking to formulate and execute problem-solving strategies; 3.) apply mathematics in novel situations .

11. Design elements • Connections • Mathematical thinking • Problem-solving • Flexible and fluent

12. RigorExposing students to challenging class work Relevance Demonstrating how students will use their learning Deep mathematical understandingthat allows students to: 1.) see the connections between bits of mathematical knowledge; 2.) apply mathematical thinking to formulate and execute problem-solving strategies; 3.) apply mathematics in novel situations; 4.) see and use mathematics in real world situations.

13. Rigor, Relevance Relationships … building…connections…with students… Deep mathematical understandingthat allows students to: 1.) see the connections between bits of mathematical knowledge; 2.) apply mathematical thinking to formulate and execute problem-solving strategies; 3.) apply mathematics in novel situations; 4.) see and use mathematics in real world situations; 5.) communicate and collaborate mathematically.

14. Rigor, Relevance Relationships … building…connections…with students… Deep mathematical understandingthat allows students to: 1.) see the connections between bits of mathematical knowledge; 2.) apply mathematical thinking to formulate and execute problem-solving strategies; 3.) apply mathematics in novel situations; 4.) see and use mathematics in real world situations; 5.) communicate and collaborate mathematically.

15. Design elements • Connections • Mathematical thinking • Problem-solving • Flexible and fluent • Mathematically Model • Communicate • Collaborate

16. Relationships Relevance Knowledge of Teaching and Learning Social Need Mathematics Rigor

17. What does mathematics as a discipline say? Mathematics is a way of thinking about, understanding, explaining, and expressing phenomena. Mathematics is about inquiry and insight. Computation is (usually) a means to an end.

18. Body of Knowledge Method of Thinking, Reasoning, and Explaining Collection of Skills and Procedures Language

19. MATHEMATICAL PROCESSING MAKE & TEST CONJECTURES WONDER SEARCH FOR PATTERNS IMAGINATION USE INDUCTIVE REASONING CHOOSE A STRATEGY/METHOD CHOOSE REPRESENTATIONS DRAW ANALOGIES MATHEMATICALLY MODEL FORMULATE ALGORITHMS DETERMINE WHAT’S KNOWN & WHAT’S NEEDED GENERALIZE RESULTS/ ALGORITHMS JUSTIFY SPECIAL CASES EXPLORE EXAMPLES INTUITION CREATIVITY ABSTRACT PROPERTIES POSE PROBLEMS LOGICALLY DEDUCE RESULTS/ ALGORITHMS IMPLEMENT ALGORITHM/ PROCEDURE/ FORMULA Mathematical Reasoning

20. Words such as conjecture, show, explain, justify, prove, abstract, and generalize are central components of a rigorous mathematics program ─that students need to do.

21. Relationships Relevance Knowledge of Teaching and Learning Social Need Mathematics Rigor

22. What does the research on learning suggest? We learn new knowledge by attaching it to our current knowledge. We tend to learn by proceeding from the “concrete” to the “abstract.” There are multiple learning styles.

23. Contextualized development of content Context:An environment in which mathematics is developed or mathematical understanding is augmented. A context should be a familiar and engaging environment for the student.

24. From: Mathematics: Modeling Our World (COMAP) Unit 1; Course 2 Welcome to Gridville! This small village has grown in the past year. The people of Gridville have agreed they now need to build a fire station. What is the best location for the fire station? . . . . . . . . . . .

25. Real World Mathematical Model Abstract Build math model Clearly identify situation Pose well-formed question Mathematically Modeling Compute ProcessDeduce Revise Mathematical results Apply Interpret Mathematical Conclusions Real World Conclusions

26. Welcome to Lineville! . . 1.) Where would you build the fire station if there were only two houses? Explain. 1 5 . . . 2.) Where would you build the fire station if there were only three houses? Explain. 1 4 5 3.) Where would you build the fire station of there were 4 houses? 5 houses? Explain. 4.) Make a conjecture about the location of the fire station if there were n houses. Can you justify your conjecture?

27. Background includes some linear modeling, some Euclidean and coordinate geometry, and the mean of a quantitative data distribution. The mathematical content for this unit includes geometry (using a non-Euclidean metric in the plane), absolute value, functions and algebra involving the weighted sum of absolute value functions, piecewise linear functions, and minimax solutions (choosing the minimum value in a set of several maximum values). Integrated topics include algebra, geometry, and pre-calculus.

28. Contextualized development of content Context:An environment in which mathematics is developed or mathematical understanding is augmented. A context should be a familiar and engaging environment for the student.

29. Contextual Development • Provides cognitive “glue” for ideas and thought processes • Provides rationale for doing mathematical activities, such as finding patterns, making conjectures, studying quadratics, etc. • Allows development “from the concrete to the abstract” or the “extension” of ideas and structure • Real-world contexts add value to mathematical content

30. Making Connections: integrating mathematical topics • Permits synergistic development and multiple ways of connecting old and new content • Provides genuine opportunity to revisit topics in more depth • Addresses various student strengths • Presents mathematics as a unified discipline • Provides access to a broader collection of problems and solutions

31. Relationships Relevance Knowledge of Teaching and Learning Social Need Mathematics Rigor

32. Relevance and relationships: What about all students?

33. CurricularObjectives • Create mathematically literate citizens • Prepare students for the workplace • Prepare students for further study in disciplines that involve mathematics • Prepare students to be independent learners • Provide an appreciation of the beauty, power, and significance of mathematics in our culture

34. Mathematical needs of the workforce beyond computational skills • Understand the underlying mathematical features of a problem • Have the ability to see applicability of mathematical ideas in common and complex problems • Be prepared to handle open-ended situations and problems that are not well-formulated • Be able to work with others ─ Henry Pollack

35. Call for better things. Consider: • Updating, refocusing, and re-sequencing content within state guidelines-or change them • Incorporating concepts and methods from statistics, probability, and discrete mathematics

36. Closing the Gap: Methods of addressing equity in curriculum: • Students feel at home in the curriculum • Students see a reason for doing problems • Students are actively involved in their learning • Students are respected and feel personally validated

37. ..more on addressing equity: • Problems that allow multiple approaches • Problems that are open-ended • Students make (mathematical) choices • Problems that allow investigation and response at multiple levels • Different gradations of problems • Verbalization and varied representation • Reading

38. Curriculum designed to raise achievement and close gaps with rigor relevance and relationships should include: Mathematical connections, thinking and reasoning, problem-solving, modeling, and communication. It needs to address multiple learning styles, issues of equity and access, and multiple objectives. Methods suggested in this session include the contextual development of concepts; integration of topics, and placing mathematical methods of thinking and reasoning at the center of the curriculum.

39. Not addressed in depth in this presentation: • Topical content • But should include data analysis and statistics • Technology

40. Part II Evidence

41. Secondary Mathematics curriculum programs with these design elements : • Contemporary Mathematics in Context(Core-Plus Mathematics Project; CPMP)(Glencoe/McGraw Hill, Publisher) [2:30-4:00 PM, Cedar B, Billie Bean] • Integrated Mathematics: A Modeling Approach Using Technology(SIMMS IM) (Kendall Hunt, Publisher) [2:30-4:00 PM, Imperial A, Gary Bauer] • Mathematics: Modeling Our World(ARISE) (COMAP, Publisher) • Interactive Mathematics Program(IMP) (Key Curriculum Press, Publisher) • MATH Connections: A Secondary Mathematics Core Curriculum(MATH Connections) (IT’s About Time, Publisher) Links to all athttp://www.ithaca.edu/compass

42. Does this approach raise Achievement?

43. Achievement Goal: Deep understanding of mathematical concepts and processes that includes the ability to use mathematics effectively in realistic problem-solving situations

44. A growing body of evaluation evidence suggests that it can* Cumulatively, the summary of evidence below stretches from field test results from the early 1990’s to district adoptions in the 2000’s. It cuts across urban, suburban, and rural districts and ethnically and culturally diverse populations. Measurement instruments and research designs vary.

45. On Evaluation of Curricular Effectiveness: Judging the Quality of K-12 Mathematics Evaluations─National Research Council (2004) On average, the evaluations in this subgroup had reported stronger patterns of outcomes in favor of [these curricula and their K-8 counterparts]…than the evaluations of commercially-generated curricula. ─this result is not sufficient to establish the curricular effectiveness of these programs as a whole with absolute certainty.

46. A short list of summary references: • Senk, S. L. and Thompson, D. R. (Eds.) Standards-based school mathematics curricula? what are they? what do students learn?; Lawrence Erlbaum Associates (2003) • Harwell, M.R., Post T.R.,Yukiko M., Davis J.D., Cutler A.L., Anderson E., Kahn J.A., “Standards-based mathematics curricula and secondary students’ performance on standardized achievement tests, Journal of Research in Mathematics Education (January, 2007) • Schoen, H.L. Hirsch, C.R. “Responding to calls for change in high school mathematics: implications for collegiate mathematics” Mathematical Association of America Monthly, vol. 110, (February, 2003)

47. On standardized tests that measure quantitative thinking, reasoning and realistic problem-solving ability, students in all five curricula mentioned above most often do significantly better than their traditional counterparts. • Instruments included subtests from NAEP,ITED-Q • Senk and Thompson; Mary Ann Huntley,Chris L. Rasmussen, Roberto S. Villarubi, “Effects of Standards-Based Mathematics Education: A Study of the Core-Plus Mathematics Project Algebra and Functions Strand;” Journal of Research in Mathematics Education, May 2000, Vol.31

48. On tests that included measures of updated or non-traditional mathematical science content (including statistics) students from several of these programs who were tested scored above their traditional counterparts • Webb, N. and Maritza D., "Comparison of IMP Students with Students Enrolled in Traditional Courses on Probability, Statistics, Problem Solving, and Reasoning," Wisconsin Center for Education Research, University of Wisconsin-Madison, April, 1997; Senk and Thompson

49. Students from these programs generally received cumulative scores as high as and often higher than their traditional counterparts on traditional items on standardized tests such as the PSAT, SAT, ACT, SAT-9 • Merlino, J. & Wolf, E. (2001).Assessing the Costs/Benefits of an NSF“ Standards-Based" Secondary Mathematics Curriculum on Student Achievement. Philadelphia, PA: The Greater PhiladelphiaSecondary Mathematics Project : http://www.gphillymath.org/StudentAchievement/Reports/AssessCostIndex.htm ; Schoen and Hirsch; Senk and Thompson

50. Results on achievement with regard to symbol manipulation within first editions of these programs are mixed. • Schoen, H.L. Hirsch, C.R. ; Huntley, et. al. ibid