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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

				
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