Rigor, Relevance, and Relationships by Design

Rigor Relevance, - Rigor Relevance and Relationships by Design in High School Mathematics by Eric Robinson Margaret Robinson NC Raising Achievement

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Unformatted text preview: Rigor, Relevance, and Relationships by Design in High School Mathematics by Eric Robinson, Margaret Robinson NC Raising Achievement and Closing Gaps Conference March 27, 2007 Session Purpose: Session To move forward the North Carolina Raising To Achievement and Closing Gaps Commission’s mission to assist “schools and school systems mission in identifying and developing programs and strategies to raise achievement and close gaps.” gaps.” Not just about doing things better, but doing better things! better, Session Overview: Session • Part I: Design components needed in Design curriculum programs to address rigor, rigor relevance, and relationships relevance and relationships • Part II: Evidence – Realization of design principles – Effectiveness Part I Part Design A look at the terms: look Rigor Exposing students to challenging class work with academic and social support Relevance Relevance Demonstrating how students will use their learning Relationships Relationships Building caring and supportive connections with students, parents, and communities Rigor Rigor Exposing students to challenging class work Rigor Rigor Exposing students to challenging class work Deep mathematical understanding Deep Rigor Rigor Exposing students to challenging class work Deep mathematical understanding Deep that 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. Rigor Rigor Exposing students to challenging class work Deep mathematical understanding Deep that 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 . Design elements Design • • • • Connections Mathematical thinking Problem-solving Flexible and fluent Rigor Rigor Exposing students to challenging class work Relevance Demonstrating how students will use their learning Deep mathematical understanding Deep that 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. Rigor, Relevance Rigor, Relationships Relationships … building…connections…with students… Deep mathematical understanding Deep that 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 Rigor, Relevance Rigor, Relationships Relationships … building…connections…with students… Deep mathematical understanding Deep that 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 Design elements Design • • • • Connections Mathematical thinking Problem-solving Flexible and fluent • Mathematically Model Mathematically • Communicate Communicate • Collaborate Collaborate Relationships Relevance Social Need Knowledge Knowledge of Teaching and Learning and Mathematics Mathematics Rigor What does mathematics as a discipline say? discipline Mathematics is a way of Mathematics thinking about, understanding, explaining, and expressing phenomena. phenomena. Mathematics is about inquiry and Mathematics insight. Computation is (usually) a means to an end. means Body of Knowledge Method of Thinking, Reasoning, and Explaining Explaining Collection of Skills Collection and Procedures and Language MATHEMATICAL PROCESSING MAKE & TEST CONJECTURES WONDER SEARCH FOR USE INDUCTIVE REASONING PATTERNS DRAW ANALOGIES FORMULATE ALGORITHMS GENERALIZE RESULTS/ ALGORITHMS CREATIVITY CHOOSE A STRATEGY/METHOD IMAGINATION CHOOSE REPRESENTATIONS MATHEMATICALLY MODEL DETERMINE WHAT’S KNOWN & WHAT’S LOGICALLY NEEDED DEDUCE RESULTS/ ALGORITHMS JUSTIFY SPECIAL CASES EXPLORE EXAMPLES ABSTRACT PROPERTIES POSE PROBLEMS Mathematical IMPLEMENT ALGORITHM/ PROCEDURE/ FORMULA Reasoning INTUITION Words such as conjecture, show, explain, justify, prove, abstract, and generalize are central components of a rigorous mathematics program of ─that students need to do. Relationships Relevance Social Need Knowledge Knowledge of Teaching and Learning and Mathematics Mathematics Rigor What does the research on learning suggest? suggest? We learn new knowledge by attaching it We to our current knowledge. We tend to learn by proceeding from the “concrete” to the “abstract.” “concrete” There are multiple learning styles. Contextualized development of content content Context: An environment in which Context: mathematics is developed or mathematical understanding is augmented. A context should be a familiar and engaging environment for the student. 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? ... . .. .. . . . Real World Mathematical Model Clearly identify situation Abstract Build math model Apply Real World Conclusions Mathematically Modeling Compute Process Deduce Revise Pose well-formed question Mathematical results Interpret Mathematical Conclusions Welcome to Lineville! 1.) Where would you build the fire 1.) station if there were only two houses? Explain. houses? 2.) Where would you build the fire 2.) station if there were only three houses? Explain. houses? . . . .. 1 1 5 4 5 3.) Where would you build the fire station of there were 4 houses? 5 3.) houses? Explain. houses? 4.) Make a conjecture about the location of the fire station if there were n 4.) houses. Can you justify your conjecture? houses. Background includes some linear modeling, Background some Euclidean and coordinate geometry, and the mean of a quantitative data distribution. the 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). in Integrated topics include algebra, geometry, and pre-calculus. pre-calculus. Contextualized development of content content Context: An environment in which Context: mathematics is developed or mathematical understanding is augmented. A context should be a familiar and engaging environment for the student. Contextual Development Contextual • 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 Making Connections: integrating 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 Relationships Relevance Social Need Knowledge Knowledge of Teaching and Learning and Mathematics Mathematics Rigor Relevance and relationships: Relevance What about all students? What all Curricular Curricular Objectives • Create mathematically literate citizens • Prepare students for the workplace • Prepare students for further study in Prepare disciplines that involve mathematics disciplines • Prepare students to be independent learners • Provide an appreciation of the beauty, power, Provide and significance of mathematics in our culture culture 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 Call for better things. Consider: Call • Updating, refocusing, and resequencing content within state sequencing guidelines-or change them guidelines-or • Incorporating concepts and methods Incorporating from statistics, probability, and discrete mathematics discrete Closing the Gap: Methods of Closing addressing equity in curriculum: addressing • Students feel at home in the Students curriculum curriculum • Students see a reason for doing Students problems problems • Students are actively involved in their Students learning learning • Students are respected and feel Students personally validated personally ..more on addressing equity: • Problems that allow multiple Problems approaches approaches • Problems that are open-ended – Students make (mathematical) choices Students choices • Problems that allow investigation and Problems response at multiple levels response • Different gradations of problems • Verbalization and varied representation • Reading Curriculum designed to raise achievement and close gaps with rigor relevance and relationships should include: relationships 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. Not addressed in depth in this presentation: presentation: • Topical content – But should include data analysis and But statistics statistics • Technology Part II Evidence 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 at http://www.ithaca.edu/compass Does this approach raise Achievement? Achievement? Achievement Goal: Achievement Deep understanding of mathematical Deep concepts and processes that includes the ability to use mathematics effectively in realistic problem-solving situations problem-solving A growing body of evaluation evidence suggests that it can* suggests Cumulatively, the summary of evidence below Cumulatively, 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. vary. On Evaluation of Curricular Effectiveness: Judging the Quality of K-12 Mathematics Evaluations Evaluations ─National Research Council (2004) On average, the evaluations in this On subgroup had reported stronger patterns of outcomes in favor of [these curricula and their K-8 counterparts]…than the evaluations of commercially-generated curricula. curricula. ─this result is not sufficient to establish the this curricular effectiveness of these programs as a whole with absolute certainty. whole A short list of summary references: short • Senk, S. L. and Thompson, D. R. (Eds.) StandardsSenk, Standardsbased school mathematics curricula? what are they? based what do students learn?; Lawrence Erlbaum Associates (2003) Associates • Harwell, M.R., Post T.R.,Yukiko M., Davis J.D., Cutler Harwell, A.L., Anderson E., Kahn J.A., “Standards-based A.L., mathematics curricula and secondary students’ performance on standardized achievement tests, Journal of Research in Mathematics Education (January, 2007) (January, • 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) America • On standardized tests that measure On quantitative thinking, reasoning and realistic problem-solving ability, students in all five curricula mentioned above most often do significantly better than their traditional counterparts. 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 • On tests that included measures of updated or On non-traditional mathematical science content (including statistics) students from several of these programs who were tested scored above their traditional counterparts their – 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 • Students from these programs generally Students 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 ACT, – 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/Assess CostIndex.htm ; Schoen and Hirsch; Senk and Thompson • Results on achievement with regard to Results symbol manipulation within first editions of these programs are mixed. these – Schoen, H.L. Hirsch, C.R. ; Huntley, et. al. ibid Do Programs with these design principles close the achievement gap? • There is growing evidence that when There changing to such a program, the lowest achievers will realize the largest gains. achievers – Merlino & Wolff; Harwell, et. al. ; Webb, N. L., & M. Dowling (1996), Impact of the Interactive Mathematics Program on the retention of underrepresented students: Cross-school analysis of transcripts for the class of 1993 for three high schools. Project Report 96-2. Madison: University of Wisconsin–Madison, Wisconsin Center for Education Research (WCER) • Data about CPMP, IMP and MMOW suggest that Data students at the high achievement levels are well served through programs with these design elements elements – Abeille and Hurley Final Evaluation Report of MMOW curriculum (2001) at http://www.comap.com/highschool/projects/mmow/FinalReport.pdf ; Harwell et. al., Merlino and Wolff. • Students in these programs take more Students mathematics courses (including AP courses). mathematics – Kramer, S. L. (2003). The joint impact of block scheduling and a standards-based curriculum on high school algebra achievement and mathematics and course taking. Ph. D. dissertation, University of Maryland; Webb and Dowling Harwell et. al., Senk and Thompson • Students in these programs tend to have a Students better attitude toward mathematics better – Clarke, D., et al. (1992). The other consequences of a problem-based mathematics curriculum, Research Report No. 3. Mathematics Teaching and Learning Centre, Australian Catholic University; Schoen and Prickett (1998) Students perceptions and attitudes in a standards-based high school mathematics curriculum, paper presented to the American Educational Research Association; Senk and Thompson Secondary Mathematics curriculum programs with these design elements : Evaluations: • Contemporary Mathematics in Context (Core-Plus Mathematics Project; CPMP) (Glenco/McGraw Hill, Publisher) http://www.wmich.edu/cpmp/evaluation.html [2:30-4:00 PM, Cedar B, Billie Bean] • Integrated Mathematics: A Modeling Approach Using Technology (SIMMS IM) (Kendal Hunt, Publisher) http://www.montana.edu/~wwwsimms/others.html [2:30-4:00 PM, Imperial A, Gary Bauer] • Interactive Mathematics Program (IMP) (Key Curriculum Press, Publisher) • Mathematics: Modeling Our World (ARISE) (COMAP, Publisher) http://www.comap.com/highschool/projects/mmow/introduction.htm • MATH Connections: A Secondary Mathematics Core Curriculum (MATH Connections) (IT’s About Time, Publisher) http://www.its-about-time.com/htmls/mc/mccasestudies.html http://www.mathimp.org/ Links to all at http://www.ithaca.edu/compass Relationships: Relationships: Building caring and supportive connections with Building students, parents, and communities students, (addressed to administrators, teachers, staff) *Success depends on relationships: • The development of a common belief system for all constituencies all •Support for and engagement of teachers in a Support strong, ongoing curriculum- centered professional development program professional • Support for programs from administrators Support • Recognition of the needs of administrators and parents parents •Implementation with fidelity •And atmosphere of communication and And cooperation ...
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This note was uploaded on 11/11/2011 for the course MATH 112 taught by Professor Jarvis during the Winter '08 term at BYU.

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