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  1. This virtual journey will include a survey of perils that must be overcome with resourcefulness in making mandatory algebraic connections. Recent research in teaching and learning that includes effective habits and strategies supporting Mathland survival will be discussed. Use of current and rising technologies will also be explored. This is a 21st century research-based exploration of k-14 math ed issues. Mathland’s Best Route via Arithmetic Canyon into Algebra Valley:A Virtual Hike Guide: Dr. June Gastón Borough of Manhattan Community College, CUNY AMATYC Annual Conference, November 12, 2009 flickr.com/photos/63615182@N00/2932586073

  2. Why do this exploration? article.apcs.vn/showdetail.php?id=6

  3. RESEARCH: American Mathematical Association of Two-Year Colleges Professionals should continue to search for strategies to address various mathematics education issues such as: • choice of appropriate mathematics content and effective instructional strategies • the use of technology in instruction • teacher preparation • professional development for full-time and adjunct faculty, and instructional support staff. Source: American Mathematical Association of Two-Year Colleges (2006). Beyond Crossroads: Implementing Mathematics Standards in the First Two Years of College, p.16-17.

  4. What mathematics must 21st century k-12 students know to be prepared for college-level mathematics? http://www.wizardnet.com/musgrave/color_range6.jpg

  5. RESEARCH: National Mathematics Advisory Panel • To prepare students for Algebra, the curriculum must simultaneously develop conceptual understanding, computational fluency, and problem solving skills. • Computational proficiency with whole number operations is dependent on practice to develop automatic recall of addition and related subtraction facts, and of multiplication and related division facts. It also requires fluency with the standard algorithms for addition, subtraction, multiplication, and division – and a solid understanding of core concepts, such as the commutative, associative and distributive properties. • The most important foundational skill not presently developed appears to be proficiency with fractions (including decimals, percent, and negative fractions). The teaching of fractions must be acknowledged as critically important and improved before an increase in student achievement in algebra can be expected. Source: National Mathematics Advisory Panel. (2008). Foundations for Success: The Final Report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education, p. xix, p.18-19.

  6. Adding It Up: Helping Children Learn Mathematics, Executive Summary, p. 5 http://books.nap.edu/catalog/9822.html RESEARCH: Commission on Mathematics and Science Education Instruction should emphasize inquiry, relevance, and a multilayered vision of proficiency as indicated by the National Research Council in Adding It Up and the National Mathematics Advisory Panel in Foundations for Success. Proficiencies are: • Conceptual understanding (comprehension of mathematical concepts, operations, and relations) • Procedural fluency (skills in carrying out procedures flexibly, fluently, and appropriately) • Strategic competence (ability to formulate, represent, and solve mathematical problems) • Adaptive reasoning (capacity for logical thought, reflection, explanation, and justification) • Productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy) Source: Carnegie Corporation of New York, Institute for Advanced Study, Commission of Mathematics and Science Education (2009).The Opportunity Equation, p.24-25.

  7. How should 21st century k-12 mathematics be taught? According to research, what approaches and strategies work? What are the best practices? What strategies can teachers use to help elementary school students achieve computational fluency? What instructional strategies can be used to keep adolescents motivated to learn abstract pre-algebraic and algebraic concepts? Are k-12 teachers prepared to teach these concepts? http://www.wizardnet.com/musgrave/ranges.jpg

  8. RESEARCH: How the Brain Learns K-12 Math Title: How the Brain Learns Mathematics (Paperback) Author: David A. Sousa Publisher: Corwin Press, CA Year: 2008 ISBN-10: 1412953065 ISBN-13: 978-1412953061 Product Description Discusses cognitive mechanisms for learning mathematics and factors that contribute to mathematics difficulties, examines how the brain develops an understanding of number relationships, and connects to NCTM curriculum focal points. http://www.corwinpress.com/booksProdDesc.nav?prodId=Book230967

  9. RESEARCH: How the Brain Learns K-12 Math Criteria for Long-Term Storage of Information In everyday life, working memory quickly and/or permanently stores information that has: • survival value and/or • an emotional connection In the classroom, working memory saves information if it: • makes sense when connected to past experiences and prior knowledge • has meaning or relevance for the learner Sousa, David A. How the Brain Learns Mathematics, Thousand Oaks, CA: Corwin Press, 2008, p. 54-56.

  10. RESEARCH: National Mathematics Advisory Panel Instructional Practices • Team Assisted Individualization (TAI) improves students’ computation skills, but not conceptual understanding and problem solving skills. This pedagogical strategy involves heterogeneous groups of students helping each other, individualized problems based on student diagnostic test results, specific teacher guidance, and rewards based on both group and individual performance. • Teachers’ regular use of formative assessment improves their students’ learning, especially if teachers use the assessment to design and to individualize instruction. • Mathematically gifted and motivated students appear to be able to learn mathematics faster than students proceeding through the curriculum at a normal pace, with no harm to their learning, and should be allowed to do so. Source: National Mathematics Advisory Panel. (2008). Foundations for Success: The Final Report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education, p. p xxii-xxv .

  11. RESEARCH: National Mathematics Advisory Panel Instructional Practices involving Computation and Calculators • Explicit instruction with students who have mathematical difficulties has shown consistently positive effects on performance with word problems and computation. Results are consistent for students with learning disabilities and students who perform in the lowest third of a typical class. • The Panel cautions that to the degree that calculators impede the development of automaticity, fluency in computation will be adversely affected. The Panel recommends that high-quality research on particular uses of calculators be pursued, including both their short- and long-term effects on computation, problem solving, and conceptual understanding. • Calculators should not be used on test items designed to assess computational facility. Source: National Mathematics Advisory Panel. (2008). Foundations for Success: The Final Report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education, p. p xxii-xxv .

  12. RESEARCH: National Mathematics Advisory Panel Instructional Practices using Computer Assisted Instruction (CAI) • CAI drill and practice, is a useful tool in developing students’ automaticity (i.e., fast, accurate, and effortless performance on computation), freeing working memory so that attention can be directed to more complex tasks. • CAI tutorials are useful in introducing and teaching specific subject-matter content to specific populations. Additional research is needed to identify which goals and which populations are served well by tutorials, and features of effective tutorials and of their classroom implementation. Source: National Mathematics Advisory Panel. (2008). Foundations for Success: The Final Report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education, p.51 .

  13. RESEARCH: National Mathematics Advisory Panel Teacher education programs and licensure tests for • early childhood and early childhood special education teachers should fully address the topics in the Critical Foundations of Algebra, as well as the concepts and skills leading to them • elementary and elementary special education teachers should fully address all topics in the Critical Foundations of Algebra and topics typically covered in introductory Algebra • middle school and middle school special education teachers should fully address all of the topics in the Critical Foundations of Algebra and all of the Major Topics of School Algebra. ...teachers must know in detail and from a more advanced perspective the mathematical content they are responsible for teaching and the connections of that content to other important mathematics, both prior to and beyond the level they are assigned to teach. Source: National Mathematics Advisory Panel. (2008). Foundations for Success: The Final Report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education, p. 19, p.38.

  14. What must 21st century developmental math students know to be prepared for college-level mathematics? http://www.idiom.com/~zilla/Work/Gsd/img90.gif

  15. RESEARCH: American Mathematical Association of Two-Year Colleges • The desired student outcomes for developmental mathematics courses should be developed in cooperation with the partner disciplines. The content for these courses also should address mathematics anxiety, develop study and workplace skills, promote basic quantitative literacy, and create active problem solvers. • Topics in algebra, geometry, statistics, problem solving and experience using technology should be integrated throughout developmental courses. However, students should still be expected to perform single digit arithmetic, without the use of a calculator. Source: American Mathematical Association of Two-Year Colleges (2006). Beyond Crossroads: Implementing Mathematics Standards in the First Two Years of College, p.41-42.

  16. What strategies can instructors use to help developmental students achieve computational proficiency? How should developmental mathematics be taught? According to research, what approaches and strategies work? What are the best practices? What causes the high failure rates? What strategies can help curb them? http://www.meta-synthesis.com/webbook/24_complexity/landscape.jpg

  17. RESEARCH: 100% Math Initiative Instructors should: • vary their classroom methodology to actively engage students in the learning process. (Most developmental education students have an attention span of approximately 15 minutes. Adjust the lesson appropriately. For example, switch from lecture to group discussion or student demonstration of a concept or procedure.) • employ a broad range of pedagogical approaches that both match the range of material and actively engage students. Use the “rule of four” and present information four ways: graphically, numerically, symbolically, and verbally. • orient their presentation to the real world application of the material. • be aware of different learning styles among their students and adjust their instructional approach accordingly. • identify and implement strategies for assisting students with study skills and integrate these skills directly into course and classroom activities. Source: “Building a Foundation for Student Success in Developmental Math,” Massachusetts Community College Executive Office. (2006). 100% Math Initiative: Building a foundation for student success in developmental math. Boston: Author, p. 31-33

  18. RESEARCH: 100% Math Initiative Instructors should also: • clarify specific competency-based expectations that a developmental math student must meet. These can include student portfolios and/or departmental final exams, which measure student proficiency. • convey the value of homework (emphasize it, offer incentives for completed homework, check it regularly, and provide feedback to students). • attend professional development workshops that address strategies for accommodating different learning styles, integrating study skills into instruction, using technology, creatively engaging students, and advising students. • regularly collaborate with specialists and support staff who work with learning-disabled students. • have access to a handbook for developmental mathematics instructors that includes materials concerning administrative and logistical issues, curriculum and syllabus information, recommended instructional approaches, and an inventory of academic support resources. • collaboratively select textbooks (in print or online) that include varied instructional approaches, are contextually rich, incorporate numerous applications, are activity-based, hands-on and most effectively meet student needs. Source: “Building a Foundation for Student Success in Developmental Math,” Massachusetts Community College Executive Office. (2006). 100% Math Initiative: Building a foundation for student success in developmental math. Boston: Author, p. 31-33

  19. RESEARCH: Technology Solutions for Developmental Math: An Overview of Current and Emerging Practices Instructors should use technology: • to facilitate acceleration of coursework • to transform developmental courses by offering traditional content in an intensified time frame • to target specific skills gaps Ex: Community College of Denver’s FastStart@CCD allows students to complete two levels of remedial math within one semester. Though acceleration is a key component, the holistic program draws effectiveness from a mix that includes student support, a learning community format, interactive teaching, and career exploration delivered in a first-year experience class. Ex: Ivy Tech Community College, Evansville, replicated the CCD accelerated model in fall 2007, with similar initial outcomes…efforts have increased to identify and remediate students who may test close to the “cut-off score” as a way to reduce unnecessary time and money spent in developmental coursework. Self-paced, competency-based, and modular courses have proven successful for some students, as have “refresher courses” for returning students. • for instruction Ex: Instructional software (e.g., tutorial) programs have been in use for many years in community college classrooms, yet less than 40% of two-year colleges reported use of a Learning Management System or Course Management System in 2007. • to supplement rather than replace traditional delivery methods because studies show no clear consensus on the effectiveness of technology-based delivery methods. Source:Technology Solutions for Developmental Math: An Overview of Current and Emerging Practices, Rhonda M. Epper and Elaine DeLott Baker, January 2009, p.8-10, http://www.gatesfoundation.org/learning/Documents/technology-solutions-for-developmental-math-jan-2009.pdf

  20. What current technology trends are helping to bridge the gap between arithmetic and algebra? http://farm1.static.flickr.com/48/143182831_1ad0a65010.jpg?v=1147241085

  21. RESEARCH: Technology Solutions for Developmental Math: An Overview of Current and Emerging Practices General Trends: Open Education Resources, Digital Game-Based Learning, Social Networking, Virtual Worlds • Use of Web 2.0 technologies in developmental math Advocates of digital game-based learning believe math drill-and-practice games could be effective for higher math levels if appropriately designed. “Net Generation” attraction to games and research on effectiveness of Digital Game-Based Learning (DGBL) may prove such math games helpful to developmental math students. • Reconciling “Net Generation” learner profiles with profiles of the developmental student population Penn State University researchers suggest that the game generation prefers doing many things simultaneously by using various paths toward the same goal, is less likely to become frustrated when facing a new situation, prefers being active, learning by trial and error, and figuring things out by themselves rather than by reading or listening. • Use of open courseware The National Repository of Courseware (NROC) maintains and expands a courseware library. The content is distributed free-of-charge to students and teachers at public websites including www.HippoCampus.org. • Web-base learning initiatives The Open Learning Initiative (OLI) of Carnegie Mellon University uses intelligent tutoring systems, virtual laboratories, simulations, and frequent opportunities for assessment and feedback. OLI builds college level web-based courses that are intended to provide effective instruction that promotes learning. Source: Technology Solutions for Developmental Math: An Overview of Current and Emerging Practices, Rhonda M. Epper and Elaine DeLott Baker, January 2009, p.15-16, http://www.gatesfoundation.org/learning/Documents/technology-solutions-for-developmental-math-jan-2009.pdf

  22. What other issues need to be explored? http://upload.wikimedia.org/wikipedia/commons/8/8b/Fractal_terrain_texture.jpg

  23. Ongoing explorations • How can high school coursework and testing that permits calculator use, and college developmental coursework and testing that prohibits calculator use, be brought into clearer alignment? Many students in high school experience mathematics in context, using technology. A majority of high school exit examinations allow the use of graphing calculators. In contrast, some higher education mathematics placement exams test only basic arithmetic and algebraic computation without technology. These differences in the use of technology need to be addressed. Collaborative efforts to implement standards-based mathematics can be an initial step in minimizing the need for remediation in postsecondary mathematics education, address the critical need for students to complete algebra, and help to ease the transition from high school to college. American Mathematical Association of Two-Year Colleges (2006). Beyond Crossroads: Implementing Mathematics Standards in the First Two Years of College, p.73-74.

  24. Ongoing explorations • How can high school testing be brought into better alignment with college testing so that high school students will be better prepared for college entry-level assessment exams in math? 70% of students entering CUNY’s community colleges failed their placement exams in reading, writing, or math and were required to take remedial courses. Grading of NYS high school mathematics Regents examinations …about the new Integrated Algebra Exam administered for the first time in June 2008…the NYSED announced today that a raw score of 30 points out of 87 (just 34.5 percent) was all that students were required to earn to achieve a passing grade of 65. In the State’s headlong race to lead American students to the bottom rung of the industrialized world’s academic ladder, we’ve proudly declared a 35 to be our 65. . . . What now passes in New York State for high school level competency, represented by the new Integrated Algebra I Regents Exam, is by any measure an international laughingstock, an exam that a typical sixth grader in China could ace with hardly a second thought. (“NYS Algebra Regents” 2008) Annenberg Institute for School Reform at Brown University (with John Garvey), Education Policy for Action Series, Education Challenges Facing New York City, (2009). Are New York City’s Public Schools Preparing Students for Success in College? p. vii, p.20.

  25. Ongoing explorations • What can be done about poor k-14 attendance in New York City? How does poor attendance impact math achievement? Are socioeconomic factors an issue? Chronic Absence Starts Young and Grows With Age Citywide, more than 20% of elementary school pupils missed more than a month of school in 2007-08; nearly 40% of high school students missed that much. Moreover, the rates of severe chronic absence increase with the grade level. While 4.5% of elementary pupils missed 38 days (nearly two months), 24% of high school students missed that much. Why Attendance Matters – early chronic absenteeism sets the stage for failure. ...recent research by the National Center for Children in Poverty at Columbia University shows that children who have poor attendance in kindergarten tend to do poorly in first grade, and that children with a history of poor attendance in the early elementary grades have lower levels of academic achievement throughout their school years...Schools with high levels of absenteeism tend to have slower-paced instruction overall, harming the achievement levels of strong students as well as those who struggle, a report by the Open Society Institute suggests...good attendance is a prerequisite for academic achievement. However, under No ChildLeft Behind, schools are primarily judged on their students’ performance on standardized tests in math and reading—not their attendance rates. Nauer, Kim, White, Andrew and Yerneni, Rajeev. Strengthening Schools by Strengthening Families, Community Strategies to Reverse Chronic Absenteeism in the Early Grades and Improve Supports for Children and Families, Center for New York City Affairs, The New School, Milano the new school for Management and Urban Policy, October 2008, p.14, p.20.

  26. Next explorations • What gets college students to math class and gets them there on time? Rewards for attendance and punctuality? Penalties for absence and lateness? Quiz at the beginning of each session? Accelerated coursework? Team or group activities? Individualized instructional approaches? An effective college online early intervention system? The Community College of Allegheny County (CCAC) has Online Early Intervention enabling faculty members to refer struggling students for assistance via a secure website. The student support staff members who receive these referrals attempt to contact the students in order to connect them to college resources. The goals of the CCAC Online Early Intervention System are to: - Reach struggling students as early as possible - Connect students to college resources to resolve their issues - Help students to maintain or improve their GPA - Improve student retention rates (CCAC defines struggling students as those who have poor attendance, are continually late for class or leave early, don’t take notes, are inattentive, don’t participate, or are unlikely to be successful in a given course.) CCAC Online Early Intervention, http://www.ccac.edu/default.aspx?id=151239

  27. Questions or interest in collaborating on k-14 mathematics education research? Contact: Dr. June Gastón BMCC-CUNY Mathematics Department Room N520 jgaston@bmcc.cuny.edu tel: (212) 220-1342 For information concerning the fractal landscapes in this presentation, use the link at the right corner under each image  article.apcs.vn/showdetail.php?id=6

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