Russell Gersten

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Needed Future Research on Instructional Practice
in Mathematics

• Russell Gersten
• Professor Emeritus, University of Oregon
• Director, Instructional Research Group

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What does not change isthe will to change

• Charles Olson, The Kingfishers, 1949

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There Have Been Changes in Math Instruction
Knowledge Base

• Small but growing body of empirical research
• Active sustained engagement by research
  mathematicians in the process
• Insights gained from international comparisons
  .. Though correlational findings hard to
  interpret accurately
• Scores on NAEP grade 4 rising

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Response to Intervention (RtI) Increased Interest
in Interventionsfor Struggling Students

• However, problems include
• Lack of valid screening and progress monitoring
  measures
• Lack of reliable and valid formative assessment
  measures
• Need for Interventions at the Secondary Level
• Limited connections/misunderstandings between
  mathematics education and special education
  communities

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Objectives for the Session

• Suggest needed future directions for
  instructional research in mathematics
• Describe troubling conceptual issues that require
  advances, if not resolution
• Modus Operandi Build from pockets of strength in
  the research base.

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Sources of Inspiration

• National Mathematics Advisory Panel Report
• Assisting Students Struggling with Mathematics
  Response to Intervention (RtI) for Elementary and
  Middle Schools
• http//www.centeroninstruction.org/

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Sources of Inspiration

• NCTM Curriculum Focal Points
• Meta-analysis of research on teaching students
  with LD (Gersten, Chard, Jayanthi, Baker, Morphy
  Flojo, in press Review of Educational
  Research).

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Sources of Inspiration

• Research Mathematicians H. S. Wu, Sybilla
  Beckman, Jim Lewis, Dick Askey, Jim Milgram, Hy
  Bass
• Mathematics education Deborah Ball, Heather
  Hill, Skip Fennell, Jon Star, Joan Ferrini-Mundy,
  Jeremy Kilpatrick, Karen Fuson, Bill Schmidt
• Special education/ At risk learners Anne Foegen,
  Lynn Fuchs, Brad Witzel, Diane Bryant, Ben
  Clarke, Asha Jitendra
• Psychology Dave Geary, Bob Siegler, Bethany
  Rittle-Johnson

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Heavy Lifting Major Issues to Broach in Future
Research

• Instruction
• Teaching fractions and proportions (operations,
  concepts, word problems, linkage of number
  concepts to geometry concepts) so that students
  understand the mathematics
• Whole numbers algorithms and their link to
  number properties, number lines and number paths,
  number sense
• Effective instructional sequences that
  simultaneously build proficiency in multi-digit
  multiplication and distributive property of
  numbers
• Sequences that integrate word problems with work
  on procedural fluency

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Heavy Lifting

• Teacher Knowledge
• Content knowledge necessary to teach grades 3-5,
  6,
• Measures
• Valid screening measures and assessments for
  grades 4-8
• Predictors of success in algebra
• Evidence that facility with fractions really does
  predict success in algebra

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Other Interesting Areas

• Role of effort vs. talent in learning mathematics
• Use of this knowledge in interventions for
  struggling students
• Making middle school interventions come alive for
  students
• What should content be? Integrate with core grade
  level instruction
• Engagement (Bottge, Woodward, National Research
  Council)
• Ultimate goal of middle school double dose
  intervention

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Other Interesting Areas

• Professional development that really makes a
  difference
• Exploration of extant data bases (NAEP, state
  data bases)
• Is mastery of fractions the key?

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Where to Start?

• Findings with some empirical evidence that can
  serve as a basis for future research
• Clear areas of need that should serve as a focus
  for intervention research
• Areas of fairly broad consensus that require
  empirical validation or further study

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Starting Point For At-risk Learners/Intervention
Methods

• Explicit instruction (teachers model easy and
  hard problems, and think aloud steps in how to
  solve)
• Systematic instruction
• Students justify decisions they make
• Judicious use of concrete representations and
  consistent use of visual representations
• (Source Gersten, Chard, Jayanthi et al., in
  press)

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A Few Starting Points From Experimental
Research

• Use of contrasting examples (Star)
• Focus on fluency with combinations/facts
  (critical for understanding mathematics)
• a mix of practice and work with number families
• estimation is more potent
• work on fact retrieval is critical

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Starting Point Word Problems

• Use of common underlying structures to help
  students figure out how to solve word problems
  e.g., change problems (for time), compare
  problems (for quantity). (Cognitively guided
  instruction, Jitendra et al, Fuchs et al).
• Design curricular sequences so that students
  consistently focus on underlying structure and
  learn to ignore irrelevant information and
  translate information from different formats
  (pictorial, graphs, currency etc.) (Fuchs et al.)
• Link word problems with procedural examples
  (Singapore)

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A Stab at Putting our Needs and Starting Points
Together

• Longitudinal research Is proficiency with
  fractions and proportions the key to future
  success in algebra? (extant data bases, new
  longitudinal research)
• Word problems and visual representations
  extensive use of underlying structure approach
  and study of comparative impacts of various
  visual representations?
• Embedded in existing curricula? Asian curricula?
• Clarity as to measures of precisely what is
  taught versus broader measures of problem solving
  

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A Stab at Putting our Needs and Starting Points
Together

• Valid screening measures for grades 3-8 for RtI
• Are state assessments valid for this purpose?
• What is the validity of commonly used benchmark
  tests? Reliability?
• Effective interventions for middle school
• Build on what is available and what is known
• Continue to use quasi-experiments and descriptive
  research to understand impacts of double dose
  interventions
• Should second mathematics class be integrated
  with the core class? If so,how?

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Professional Development, Teaching and Policy

• Teaching content knowledge, especially to
  teachers in grades 3-8
• Evaluate use of departmentalized mathematics
  teachers in upper elementary schools
• Evaluate impact of standards that are challenging
  but focused and precise. (e.g. Massachusetts,
  Minnesota, California)

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Areas of emerging consensus(National Mathematics
Panel)

• Reciprocal relationship between proficiency with
  procedures and understanding of mathematical
  ideas
• Neither teacher directed nor student-centered
  learning should be sole instructional means of
  teaching
• Topics need much more in depth coverage.
• At risk learners need some explicit instruction

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But We Lack Precision

• What do these terms really mean?
• Conceptual understanding of distributive
  property a (bc) ab ac
• Explicit instruction
• Systematic instruction
• Guided inquiry
• Use of multiple representations
• Rich, deep mathematical problems

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And

• Precision is at the heart of mathematics
• (Wu, Milgram, and others)

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Final Thoughts

• Remember the importance of precise definitions
• Remember that there are serious mathematics ideas
  that need to be highlighted in texts, state
  standards, intervention programs
• Equivalence, number line, number properties,
  linear functions
• Use more operational language to increase
  precision of our research and our students
  understanding of mathematics
• e.g., minimal use of buzzwords such as
  conceptual knowledge, real world problems, rich
  mathematical problems

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Final Thought

• We can be precise
• Charles Olson, The Kingfishers, 1949