Mathematics Teaching Practices and their Effects

1
Mathematics Teaching Practices and their Effects

• Mary Schatz Koehler
• San Diego State University
• Douglas A. Grauws
• University of Missouri-Columbia

2
History of Research on Math Teaching

• Rosenshine (1979)
• Cycle 1Teacher personality and characteristics
• Cycle 2Teacher-student interaction
• Cycle 3Student attention and the content
  students mastered.
• Medley (1979) identified effective teachers
• Characteristics
• Methods they used
• Behaviors and classroom climate
• Command of a repertoire of competencies.

3
Levels of Complexity in Teacher Research
4
Levels of Complexity in Teacher Research Level 1

• Teacher Effectiveness Model
• Specific component of teaching or teacher
  characteristic is studies in isolation.
• Little or no attention is given to other factors
  or to the quality of teaching.
• Early examples (most before1950) examine
• teacher characteristics
• Years of teaching experience
• Number of math courses taken, etc
• Personality traits
• enthusiasm

5
Basic Level 1 Research
Teacher Characteristics
Pupil Outcomes
Based on opinions of supervisors, principals, and
occasionally students!
6
Basic Level 1 Research

• Emphasized teachers rather than teaching.
• Identified important traits of effective
  teachers
• Good judgment
• Considerateness
• Enthusiasm
• Personal magnetism
• Personal appearance
• Loyalty
• Charters Waples (1929) and Barr Emans (1930)

7
Basic Level 1 Research

• Later examples examine only one component
• Time allocated within a math class period.
• Shipp Deer (1960)
• Shuster Pigge (1965)
• Zahn (1966)
• Sindelar, Garrland Wilson (1984)
• Teacher clarity
• Smith (1977)
• Smith and Cotton (1980)
• Hines, Cruickshank, and Kennedy (1985)

8
Elaborated Level 1 Research
Teacher Characteristics
Pupil Outcomes
Teacher Behavior
Clear acknowledgement of the influence teacher
behaviors have on pupil outcomes.
9
Levels of Complexity in Teacher Research Level 2

• Multiple classroom observations that provide
  extensive detail.
• Often referred to as process-product research.
• Classroom processes are observed.
• Frequency of particular student-teacher behaviors
  are noted.

10
Level 2 Research Model
Pupil Outcomes
Teacher Characteristics
Teacher Behavior
Pupil Behavior
Careful documentation of what teachers and
students do during math instruction.
11
Level 2 Research

• Coding schemes for recording classroom events
  during observations
• Types of questions asked
• Length of responses to questions
• Number and type of examples used
• Amount of instructional time devoted to practice
  activities
• Frequency of use of manipulative materials
• Amount of time allocated to developing new
  concepts
• Amount of time spent on review activities
• Student outcomes correlated with the frequency of
  observed behaviors.
• Again, not much attention was paid to quality of
  teaching.

12
Level 2 Research Examples

• Process-product studies
• Brophy Good (1986)
• Studies that contributed to the knowledge base of
  math teaching
• Texas Teacher Effectiveness Study
• Differences in math teaching and other subjects
• Evertson, Anderson, Anderson, Brophy, 1980
• Successful classroom management techniques
• Evertson, Emmer, Brophy, 1980
• Missouri Mathematics Program
• Active Mathematics Teaching
• Good, Grouws, Ebmeier, 1983

13
Levels of Complexity in Teacher Research Level 3

• Primary distinctions
• Inclusion of the category of pupil
  characteristics
• Gender
• Race
• Confidence level
• Broadening of the category of pupil outcomes to
  include attitudes as well as achievement.

14
Level 3 Research Model
Pupil Characteristics
Achievement
Pupil Outcomes
Teacher Characteristics
Teacher Behavior
Pupil Behavior
Attitudes
15
Level 3 Research Examples

• Autonomous Learning Behavior Study (Fennema
  Peterson, 1985-1986
• Large scale observational study that correlated
  classroom processes with student achievement
• Considered both high and low cognitive-level
  gains
• Considered different gains for females and males

16
Level 3 Research Examples

• Blending of methodologies in later studies
• Hart (1989) found gender differences in
  student-teacher interaction patters.
• Koehler (1986, 1990) found females performed
  better in classes where they were given less
  teacher help.
• Stanic Reyes (1986) found that differential
  outcomes can result from differential teacher
  treatment of students, but also from equal
  treatment of students.

17
Levels of Complexity in Teacher Research Level 4

• Teaching is very complex!
• Teachers have a wider range of ability levels to
  reach,
• A broader range of math topics to teach (NCTM,
  1989), and
• A greater assortment of teaching methodologies to
  choose from (ex. manipulatives, small groups)
• The most substantial change
• Need to pair research on teaching with research
  on learning.

18
Level 4 Research Model
Content
Math
Self
Pedagogy
Pupil Characteristics
Pupil Attitudes towards
Cognitive
Teacher Knowledge of
Student Learning
Gender
Race
Teacher Behavior
Pupil Behavior
Pupil Outcomes
Teacher Attitudes
Race
Gender
Classroom Processes
Teacher Beliefs about
Affective
Math
Teaching
19
Multiple Research Perspectives
20
Constructivist Approach

• Mathematical learning is not a process of
  internalizing carefully packaged knowledge but is
  instead a matter of reorganizing conceptual
  activity or thought. Cobb (1991)
• Students construct knowledge for themselves by
  restructuring their internal cognitive
  structures.
• The goal is no longer one of developing
  pedagogical strategies to help students receive
  or acquire knowledge, but rather to structure,
  monitor, and adjust activities for students to
  engage in.

21
Constructivist Approach

• Teaching behavior is examined from the viewpoint
  of how much it encourages or facilitates learner
  construction of knowledge.
• Teaching is viewed on a continuum between
  negotiation and imposition, and the teachers
  role is to find and adjust activities for
  students.

22
Constructivist Approach

• Much learning or construction of knowledge takes
  place through social interactions, with the
  teacher and peers as part of problem solving.
• When children are given the opportunity to
  interact with each other and the teacher, they
  can
• Verbalize their thinking
• Explain or justify their solutions
• Ask for clarifications
• Incorporate alternative solution methods
• (Yackel, Cobb, Wood, Wheatley, Merkel, 1990)

23
Constructivist Approach

• Research Techniques
• Teaching episode
• Teaching experiment

24
Cognitively Guided Instruction (CGI)

• Teachers make instructional decisions about their
  teaching based on their knowledge and beliefs
  about how students learn.
• Students learn by linking new knowledge to
  existing knowledge, so the role of the teacher is
  to provide instruction appropriate for each
  student

25
Cognitively Guided Instruction (CGI)

• Listening to students is critical.
• Teachers need to be aware of the knowledge their
  students have at various stages so they can
  provide appropriate instruction.
• --Fennema, Carpenter, and Peterson, 1989

26
Cognitively Guided Instruction (CGI)
Teachers Knowledge
Teachers Decisions
Classroom Instruction
Students Cognition
Students Learning
Teachers Beliefs
Students Behavior

• Major Beliefs of CGI
• Instruction must be based on what each learner
  knows,
• Instruction should take into consideration how
  childrens
• mathematical ideas develop naturally, and
• Children must be mentally active as they learn
  
• mathematics.

27
Cognitively Guided Instruction (CGI)

• Research involves
• Informing teachers about theories of how children
  learn and the research on how childrens
  mathematical ideas develop on particular topics.
• Monitoring how this new knowledge might change
  the teachers behavior in the classroom.
• Example
• Carpenter, Fennema, Peterson, Chiang, and Loef
  (1989) Study

28
Cognitively Guided Instruction (CGI)

• Carpenter, Fennema, Peterson, Chiang, and Loef
  (1989) Study Results indicated that
• Students in the experimental classes
• performed more favorably on measures of problem
  solving and also on recall of number facts.
• Experimental teachers
• spent more time on word problems,
• focused more often on the process students used
  to solve problems, and
• allowed students a wider variety of strategies to
  solve problems.

29
Expert-Novice Paradigm

• Consider teaching as one of the more interesting
  and complex cognitive processes in which human
  beings engage. (Leinhardt 1989)
• Two categories of teachers are observed, experts
  are contrasted with novices, with the intention
  of identifying the qualities or behaviors
  necessary for successful teaching.

30
Expert-Novice Paradigm

• The goal behind this research is more than just
  identifying the behaviors of expert teachers in
  their crafting of lessons.
• It is to identify the developmental process that
  teachers go through as they move from novice to
  experts.
• Teaching is viewed as both a complex cognitive
  skill and an improvisational performance.

31
Expert-Novice Paradigm

• Research methodology involves extensive
  observations and in-depth analysis of the
  implementation of lessons.
• Leinhart Study (1989)
• Four expert teachers two novice teachers

32
Expert-Novice Paradigm

• Leinhart Study (1989)
• Differences
• The ability to construct and teach
  lessonscrafting of lessons.
• Experts had rich agendastheir plans contained
• more detailed information,
• explicitly referenced student actions, and
• planned instructional actions.

33
Expert-Novice Paradigm

• Leinhart Study (1989)
• Experts spent less time in transitions, and more
  consistently distributed their time among other
  lesson components.
• Novices were much more variable in the amount of
  time spent in each lesson segment.
• Experts gave better explanations of new material
  and had fewer errors.

34
Expert-Novice Paradigm

• The student needs
• An action system, which enables him to act
  appropriately in school.
• A lesson parser, which enables the student to
  recognize and anticipate lesson components.
• An information gatherer, which absorbs
  information from a lesson and incorporates it
  into existing knowledge
• A knowledge generator, which seeks new knowledge
  and acts as a motivator
• An evaluator, which assesses the meaningfulness
  of the new material.
• --Lenhart Putnam (1987)

35
Expert-Novice Paradigm

• Livingston and Borko Study (1990)
• Complex Cognitive Skill
• Pedagogical reasoning
• Pedagogical content knowledge
• Schemata
• Improvisational Performance
• Overall plan or outline, but does not follow a
  script
• Relies on repertoire of routines and
  instructional moves
• Respond to needs or actions of students

36
Expert-Novice Paradigm Livingston and Borko
Study (1990)Results

• Novice
• Limited pedagogical content knowledge about
  student learning.
• Little knowledge about student misconceptions
• Schemata was adequate for their own
  understanding, but was insufficiently developed,
  interconnected, and accessible to enable them to
  be responsive, flexible teachers.
• Less skilled at improvisation.

37
Expert-Novice Paradigm Livingston and Borko
Study (1990)Results

• Experts became in some ways like novices when
  teaching new content.
• One cannot acquire pedagogical reasoning or
  pedagogical content knowledge without actually
  teaching the specific content.

38
Sociological Epistemological View

• New Mathematics is brought about through a
  process of conscious guessing about
  relationships among quantities and shapes.
• Proof follows a zigzag path starting from
  conjectures and moving to the examination of
  premise through the use of counter-examples.
• --Lamper, 1990

39
Sociological Epistemological View

• Doing mathematics means following the rules laid
  down by the teacher.
• Knowing mathematics means remembering and
  applying the correct rule when the teacher asks a
  question.
• Mathematical truth is determined when the answer
  is ratified by the teacher.
• The ultimate goal of teaching is to encourage
  conjecturing and arguing and to make the
  environment safe for students to express their
  thinking.
• --Lambert, 1990

40
Sociological Epistemological View

• Tasks for Teachers
• Choosing and posing problems, including raising
  questions and asking for clarification in order
  to engage students in math discourse.
• Familiar to student
• Potential to lead students into unfamiliar math
  territory.
• Finding language and symbols that students and
  teachers can use to enable them to talk about the
  same math content.

41
Mathematics Content View

• Teaching is regarded as an agent of cognitive
  change for the learner.
• The goal is to design instructional sequences and
  develop instructional techniques that would
  readily facilitate this cognitive growth and
  change.

42
Mathematics Content View

• Research methodology
• Well defined content domain is selected.
• The key cognitive processes for successful
  performance in that domain are identified.
• Find existing instruction or design special
  instruction that promotes the use of the key
  cognitive processes.
• Evaluate the instruction, both in terms of direct
  assessment of performance in the content domain
  and in transfer tasks.

43
Mathematics Content View

• The appropriate model for the development of
  understanding may be one of change and flux and
  reorganization rather than steady monotonic
  growthDisconnecting, connecting and reorganizing
  appear to be the rule rather than gradual
  addition to a stable structure.
• --Hiebert, Wearne, and Taber (1991)

44
Mathematics Content View

• The results add more evidence to the argument in
  favor of teaching concepts prior to procedures
  and suggest that students can construct
  meaningful algorithms by building upon informal
  knowledge.
• --Mack , 1990

45
Summary and Comparison

• All of the perspectives
• Accept the premise that students are not passive
  absorbers of information, but rather have an
  active part in the acquisition of knowledge and
  strategies.
• Basically view the teacher as an informed and
  reflective decision maker.

46
Summary and Comparison

• Students construct knowledge
• In much the same way that knowledge is
  constructed within the discipline of
  mathematicsLampert
• Through their interpretation of the
  lessonLeinhardt
• Through acquisition of key cognitive
  processesHiebert Wearne
• With informal knowledge as the basis for
  understandingMack
• IdiosyncraticallyCobb the CGI group

47
Summary and Comparison

• Putman, Lampert, and Peterson discuss
  understanding in terms of
• Representing
• Knowledge structures
• Connections among types of knowledge
• Active construction of knowledge
• Situated cognition

48
Summary Conclusion

• Different Views of Effective Teaching
• Hiebert and Wearne seem to provide more detail
  and structure in their attempt to identify a
  sequence of lessons that teachers can use to
  facilitate student learning in specific content
  domains.
• CGI seems most open and idealistic in suggesting
  that teachers know for each student precisely
  their stage of cognitive development with respect
  to a given content area and that they modify
  instruction continually to meet individual needs.

49
Summary Conclusion

• Different Views of Effective Teaching
• Cobb believes that teachers can work from lessons
  that have been developed with cognitive
  strategies in mind and can modify those lessons
  as necessary
• Lampert believes that teachers need to focus on
  selecting and posing appropriate problems.
• Leinhart focuses attention less on specific
  content and more on structure of lessons.
• Borko and Livingston explain that teachers
  improvise as they teach, using a rich repertoire
  of instructional moves.

50
Conclusions and Future Directions

• Although progress in the last twenty years has
  been remarkable, research on classroom teaching,
  including research on school mathematics
  instruction , is still in its infancy
• --Brophy, 1986