Algebra Curriculum and Instruction Mike Roach mroachdoe'in'gov 317 2340325

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Algebra Curriculum and InstructionMike
Roachmroach_at_doe.in.gov(317) 234-0325
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Indianas Core 40 Curriculum

• Established as Indianas college-prep curriculum
  in 1994
• Voluntary for students
• Required to be offered by schools
• Modified to reflect updated college- and
  workplace-readiness requirements beginning with
  the class of 2010
• Made the required high school curriculum for all
  students beginning with the class of 2011
• Indianas need-based financial aid policy awards
  low income students additional financial aid if
  they graduate with Core 40

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Core 40 and Academic Honors together represent
67 all Indiana high school diplomas after a
decade of voluntary participation

Source Indiana Department of Education
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Indiana Core 40 diplomas awarded

Source Indiana Department of Education
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Percent of Indiana students enrolling in Algebra
I by the end of grade 8.
Source Calculations based on unpublished data
provided by the Science and Math Indicator
Project team at the Council of Chief State School
Officers. Rolf K. Blank and Doreen Langesen.
State Indicators of Science and Mathematics
Education 2005 State-by-State Trends and New
Indicators from the 200304 School Year.
Washington, D.C. Council of Chief State School
Officers, 2005. (As reported by the National
Center for Public Policy and Higher Education in
Measuring Up 2006).
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ISTEP ResultsPercent Passing

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Comparison of grade 8 students enrolling in
Algebra I, 2003-04
Source Calculations based on unpublished data
provided by the Science and Math Indicator
Project team at the Council of Chief State School
Officers. Rolf K. Blank and Doreen Langesen.
State Indicators of Science and Mathematics
Education 2005 State-by-State Trends and New
Indicators from the 200304 School Year.
Washington, D.C. Council of Chief State School
Officers, 2005. (As reported by the National
Center for Public Policy and Higher Education in
Measuring Up 2006).
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Comparison of high school students enrolling in
at least one upper-level mathematics course,
2003-04
Source Rolf K. Blank and Doreen Langesen. State
Indicators of Science and Mathematics Education
2005 State-by-State Trends and New Indicators
from the 200304 School Year. Washington, D.C.
Council of Chief State School Officers, 2005. (As
reported by the National Center for Public Policy
and Higher Education in Measuring Up 2006).
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Remediation at college

• 26 of recent high school graduates take remedial
  courses (mathematics, English or both) in college
• 76 of remedial reading students and 63 of
  remedial mathematics student do not complete a
  college degree
• 35 of students at a public university receive
  low grades (D, F or withdrawal) in their first
  college-level mathematics course

Source Indiana Commission for Higher Education
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Effects of high school mathematics completion on
college success

• Completing one additional unit of intensive high
  school mathematics (i.e., Algebra II or higher)
  increases the odds of completing a bachelors
  degree by 73.

Source Trust, J. (2004, January). Effects of
students middle-school and high-school
experiences on completion of the bachelors
degree. Research Monograph 1. Center for
School Counseling Outcome Research, School of
Education, University of Massachusetts, Amherst.
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Algebraic Habits of Mind

• Doing/Undoing
• Building Rules to Represent Functions
• Abstracting from Computation

Adapted from Driscoll, M. (1999). Fostering
algebraic thinking A guide for teachers grades
6-10. Portsmouth, NH Heinemann.
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Algebraic Habits of Mind

• Doing/Undoing
• Input from output
• Working backwards

Adapted from Driscoll, M. (1999). Fostering
algebraic thinking A guide for teachers grades
6-10. Portsmouth, NH Heinemann.
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Algebraic Habits of MindDoing/Undoing

• Which process reverses the one Im using?
• How is this number in the sequence related to the
  one that came before it?
• What if I started at the end?
• Can I decompose this number of expression into
  helpful components?

Adapted from Driscoll, M. (1999). Fostering
algebraic thinking A guide for teachers grades
6-10. Portsmouth, NH Heinemann.
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Algebraic Habits of Mind

• Building Rules to Represent Functions
• Organizing information
• Predicting patterns
• Chunking the information
• Describing a rule
• Different representations
• Describing change
• Justifying a rule

Adapted from Driscoll, M. (1999). Fostering
algebraic thinking A guide for teachers grades
6-10. Portsmouth, NH Heinemann.
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Algebraic Habits of Mind Building Rules to
Represent Functions

• Is there a rule? How does the rule work and how
  is it helpful?
• How are things changing?
• Is there information here that lets me predict
  what is going to happen?
• What steps am I doing over and over?
• Now that I have an equation, how do the numbers
  in the equation relate to the problem context?

Adapted from Driscoll, M. (1999). Fostering
algebraic thinking A guide for teachers grades
6-10. Portsmouth, NH Heinemann.
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Algebraic Habits of Mind

• Abstracting from Computation
• Computational shortcuts
• Calculating without computing
• Generalizing beyond examples
• Equivalent expressions
• Symbolic expressions
• Justifying shortcuts

Adapted from Driscoll, M. (1999). Fostering
algebraic thinking A guide for teachers grades
6-10. Portsmouth, NH Heinemann.
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Algebraic Habits of Mind Abstracting from
Computation

• How is this situation like (or unlike) that one?
• How can I predict what is going to happen without
  doing all of the calculations?
• What are the operation shortcuts options?
• When I do the same thing with different numbers,
  what still holds true?
• What are other ways to write this expression that
  will bring out different information?

Adapted from Driscoll, M. (1999). Fostering
algebraic thinking A guide for teachers grades
6-10. Portsmouth, NH Heinemann.
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Cognitive Demand

• Blooms Taxonomy
• NAEP Mathematical Complexity
• Webbs Depth of Knowledge
• Porters Cognitive Levels

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TIMSS 1999 Video Study

• Although teachers from the United States
  presented problems of both types (practicing
  skills vs. making connections), they did
  something different than their international
  colleagues when working on the conceptual
  problems with students. For these problems, they
  almost always stepped in and did the work for the
  students or ignored the conceptual aspects of the
  problem when discussing it.
• Source Hiebert, J. Stigler, J. (2004). A
  world of difference Classrooms abroad provide
  lessons in teaching math and science. Journal of
  Staff Development, 25(4), pp. 10-15.

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TIMSS 1999 Video Study

• Teachers in high-achieving countries differed
  considerably from each other in how many problems
  of this kind they presented, but when such
  problems were presented, they implemented a
  similar percentage (about 50) in such a way that
  students studied the connections or relationships
  embedded in the problems.
• Source Hiebert, J. Stigler, J. (2004). A
  world of difference Classrooms abroad provide
  lessons in teaching math and science. Journal of
  Staff Development, 25(4), pp. 10-15.

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Cognitive Demand

• Marthas Carpeting Task
• Martha was recarpeting her bedroom, which was 15
  feet long and 10 feet wide. How many square feet
  of carpeting will she need to purchase?

Source Stein, M. K., Smith, M. S., Henningsen,
M. A., Silver, E. A. (2000). Implementing
standards-based mathematics instruction A
casebook for professional development. New York
Teachers College Press.
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Cognitive Demand

• The Fencing Task
• Ms. Browns students want their rabbits to have
  as much room as possible for their spring science
  fair. They have to keep 24 feet of fencing with
  which to build a rectangular rabbit pen to keep
  the rabbit.
• If Ms. Browns students want their rabbits to
  have as much room as possible, how long would
  each of the sides of the pen be?
• How long would each of the sides of the pen be if
  they had only 16 feet of fencing?
• How would you go about determining the pen with
  the most room for any amount of fencing? Organize
  your work so that someone else who reads it will
  understand it.

Source Stein, M. K., Smith, M. S., Henningsen,
M. A., Silver, E. A. (2000). Implementing
standards-based mathematics instruction A
casebook for professional development. New York
Teachers College Press.
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Formative Assessment

• Formative assessment is a planned process in
  which assessment-elicited evidence of students
  status is used by teachers to adjust their
  ongoing instructional procedures or by students
  to adjust their current learning tactics.
• Source Popham, W. J. (2008). Transformative
  assessment. Alexandria, VA Association for
  Supervision and Curriculum and Development.

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Assessments Within the System
Q U A R T E R L Y

A N N U A L
W E E K L Y
D A I L Y
U N I T
Student
Source Heritage, M. (2008, April 5). Formative
assessment. Presented at the Association of State
Supervisors of Mathematics Annual Meeting.
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Formative Assessment

• Typical effect sizes of the formative assessment
  experiments were between 0.4 and 0.7. These
  effect sizes are larger than most of those found
  for educational interventions. The following
  examples illustrate some practical consequences
  of such large gains.

Source Black, P. Wiliam, D. (1998). Inside the
black box Raising standards through classroom
assessment. Phi Delta Kappan, 80(2), 139-149.
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Formative Assessment

• An effect size of 0.4 would mean that the average
  pupil involved in an innovation would record the
  same achievement as a pupil in the top 35 of
  those not so involved.
• An effect size gain of 0.7 in the recent
  international comparative studies in mathematics5
  would have raised the score of a nation in the
  middle of the pack of 41 countries (e.g., the
  U.S.) to one of the top five.

Source Black, P. Wiliam, D. (1998). Inside the
black box Raising standards through classroom
assessment. Phi Delta Kappan, 80(2), 139-149.
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Five Key Strategies for Effective Formative
Assessment

• Clarifying, sharing and understanding what
  students are expected to know
• Creating effective classroom discussions,
  questions, activities and tasks that offer the
  right type of evidence of how students are
  progressing to learning goals

Source National Council of Teachers of
Mathematics. (2007). What is formative
assessment? Retrieved January 28, 2009 from
http//www.nctm.org/clipsandbriefs.aspx
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Five Key Strategies for Effective Formative
Assessment

• 3. Providing feedback that moves learning forward
• 4. Encouraging students to take ownership of
  their own learning
• 5. Using students as learning resources for one
  another

Source National Council of Teachers of
Mathematics. (2007). What is formative
assessment? Retrieved January 28, 2009 from
http//www.nctm.org/clipsandbriefs.aspx
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Recommended Books

• Driscoll, M. (1999). Fostering algebraic
  thinking A guide for teachers grades 6-10.
  Portsmouth, NH Heinemann.
• Popham, W. J. (2008). Transformative assessment.
  Alexandria, VA Association for Supervision and
  Curriculum Development.
• Stein, M. K., Smith, M. S., Henningsen, M. A.,
  Silver, E. A. (2000). Implementing
  standards-based mathematics instruction A
  casebook for professional development. New York
  Teachers College Press.

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