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23 Months and Counting Ramping Up Students’ Algebra Skills Bonnie Goonen bv73008@aol.com Susan Pittman-Shetler skptvs@aol.com Webinar Overview ▫ Define the need for high-order algebraic thinking skills ▫ Identify strategies to integrate algebraic thinking and problem solving in the classroom ▫ Look at resources for you and your students Where Am I? Associated Press Poll • People have a “love- hate” relationship with mathematics ▫ Twice as many people hated it as any other school subject ▫ It was also voted the most popular subject Why Is Math so Important? The Building Blocks of Success Higher-Level Math for All Students http://www.achieve.org/files/BuildingBlocksof Success.pdf All Students Need Advanced Math • Algebra is widely regarded as a “gatekeeper.” • Higher-level mathematics and opportunities that come with it are closed to students who do not succeed in high school algebra (Silver, 2000). • Advanced math is needed: ▫ To boost college grades ▫ For career opportunities ▫ To improve earnings Advanced Math Boosts College Grades • “Years of mathematics instruction was a significant predictor of performance across all college science subjects • “Only high-school mathematics carries significant cross-subject benefit.” Source: Sadler, P. M. & Tai, R. H. (2007). The Two High-School Pillars Supporting College Science. Science, 317, 457-8. Advanced Math Improves Job Opportunities Professional and related occupations – fastest growth rate • Health care practitioners • Technical occupations • Computer and mathematical occupations Non-Degreed Jobs that Pay Well Require Strong Math Skills • Electricians, pipe fitters, sheet metal workers, and draftsman need courses like algebra, geometry, and trigonometry to be successful on the job. Source: Dohm, A. & Shniper, L. (2007, November). Employment Outlook: 2006–16. Washington, DC: U.S. Department of Labor, Bureau of Labor Statistics. (p. 895) Sources: ACT, Inc. (2006). Ready for College or Ready for Work: Same or Different? Iowa City, IA: Author. Association of General Contractors of New Hampshire website, www.agcnh.org/public/workforce_dev/employment_labor/job_opportunities.asp Advanced Math Improves Earnings Earnings Boost From Taking Advanced Math Courses 15% 12.0% 10% 8.8% 7.8% 6.1% 5% 0% Algebra/ geometry Algebra II Trig/ Pre-cal Calculus Source: Rose, H. & Betts, J. R. (2004, May). The Effect of High School Courses on Earnings. The Review of Economics and Statistics, 86(2), 497- 513. Based on data in Table 2 on p. 501. Change Is Coming! Integration of Common Core State Standards (college and career ready) Assess higher-level math skills Integration of more authentic problem solving Use of multiple item types to assess higher-level skills Use of computer-based testing Math A new approach to: • Quantitative Skills and Problem Solving – 45% • Algebraic Problem Solving – 55% • Descriptive Statistics and Basic Inference ▫ Embedded primarily on Science and Social Studies tests Math Quantitative Skills and Problem Solving • Computation • One-step and multi-step word problems • Rate, ratio, and percent word problems • Geometric measurement • Geometric thinking skills Math Algebraic Problem Solving • Transforming expressions (linear, polynomial, rational) • Solving equations (linear equalities, linear inequalities, quadratic equations) • Lines in the coordinate plane (graphing equations, equation of a line, slope criteria) • Function concepts (evaluating a function, comparing functions, identifying features from graphs or tables) Question Types for the Next Generation Math Test • Technology-Enhanced Items ▫ Drag-and-drop ▫ Hot spot ▫ Cloze ▫ Fill-in-the-blank ▫ Multiple choice ▫ Multiple select Students will need to be able to Mathematical Practice Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Model with mathematics. Attend to precision. Use appropriate tools strategically. Look for and express regularity in repeated reasoning. Look for and make use of structure. Construct viable arguments and critique the reasoning of others. Time Out for a Word on Calculators A virtual scientific calculator will be embedded in the computer-based delivery platform A limited number of calculator-free items will be included to test computational standards More Authentic Problems Examining the Components Let’s investigate the basics of algebra and see what we can include in our programs. Put Your Thinking Cap On - What’s Your Sign? In the equation below, replace each question mark with one of the four mathematical signs: +,-, ×, or ÷. Each sign can be used only once. Fill in the blanks to solve the equation. (Hint: the first sign is +.) 7 ? 5 ? 4 ? 7 ? 6 = 15 Put Your Thinking Cap On - What’s Your Sign? The Answer Is . . . (7 + 5) ÷ 4 × 7 - 6 = 15 If the first sign is +, there are only 6 possible combinations. You can get the answer by trying each one of them out. There is only one correct answer. Algebraic thinking is . . . “The ability to think about functions and how they work and to think about the impact that a system’s structure has on calculations.” Mark Driscoll Algebraic ideas or building blocks include: • patterns • variables • expressions • equations • functions Algebraic thinking . . . Involves the connection between all learning levels. • Concrete • Representational (semi-concrete) • Abstract Getting Started . . . My Math Journal • One secret I have about math is . .. • My best experience with math was when . . . • My worst experience with math was when . . . Some Big Ideas in Algebra • Variable • Symbolic Notation • Equality • Ratio and Proportion • Pattern Generalization • Equations and Inequalities • Multiple Representations of Functions Variable Some students believe that letters represent particular objects or abbreviated words Symbolic Notation A FewSign Examples Arithmetic Algebra = (equal) . . . And the Equivalence answer is between two quantities + Addition Positive number operation - Subtraction Negative operation number Which Is Larger? 23 or 32 34 or 43 62 or 26 89 or 98 Effective Questions • Ask challenging questions. • Ask well-crafted, uncluttered, open-ended questions, such as: ▫ What would happen if ... ? ▫ What would have to happen for ... ? ▫ What happens when ... ? ▫ How could you ... ? ▫ Can you explain why you decided to begin with ... ? Teacher Responses • Phrases to Avoid ▫ Let me show you how to do this. ▫ That’s not correct. ▫ I’m not sure you want to do that. • Phrases to Use ▫ I’m not sure I understand, could you show me an example of ... ? ▫ What do you think the next step should be? ▫ Where would you use ... ? ▫ Could ____ be an answer? ▫ How do you know you are correct? It’s All About Patterns! Banquet Tables Arrangement 1 Arrangement 2 Arrangement 3 You need to determine how many people can be seated at a 100 tables in the banquet hall. You are provided with the above chart of the arrangement of tables and the fact that Arrangement 1 seats four people. How many people can be seated at Arrangement 100? Multiple Representations • Represent problems using symbols, expressions, and equations, tables, and graphs • Model real-world situations • Complete problems different ways (flexibility in problem solving) Manipulatives for Algebra (the “C” of CRA) • Students with access to virtual manipulatives achieved higher gains than those students taught without manipulatives. • Students using hands-on and manipulatives were able to explain the how and why of algebraic problem solving. Recommendations • Integrate elements of algebraic thinking into all mathematic instruction • Use symbols, expressions, and equations, tables and graphs • Model real-world situations • Provide activities that require flexibility in problem solving (you can do it more than one way) • Use symbolic language ▫ letters can mean “what number” (e.g. 8 – x = 5) ▫ Letters can indicate “which values” can change (e.g., in the formula for area where the rule doesn’t change, but the numbers can Remember . . . • Arithmetic is doing something to numbers to get an answer. • Algebra is exploring the relationships between numbers. Research and Development Bonnie Goonen bv73008@aol.com Susan Pittman-Shetler skptvs@aol.com