Diana Piccolo, Alpaslan Sahin, Heather Louder, Amanda Ross, Mary Margaret Capraro, Robert Capraro

1
What Teaching Strategies Can Best Improve
Students Understanding of Algebraic Concepts?

• Diana Piccolo, Alpaslan Sahin, Heather Louder,
  Amanda Ross, Mary Margaret Capraro, Robert
  Capraro
• Texas A M University

2
Objectives of our Symposium

• a)Demonstrate how to increase middle grades
  students understanding of variables and
  equations
• b) Examine various pedagogical strategies that
  produce positive student outcomes and performance
  in middle grades algebra

3
Our session will

• emphasize the connections between research and
  practice
• present four different research-based teaching
  strategies for improving students understanding
  of algebraic variables and equations
• a) effective questioning,
• b) teacher responses,
• c) use of manipulatives,
• d) conceptually-based teaching.

4
Algebra in Middle School

• One major goal of mathematics education is to
  improve the teaching of algebra to ALL students
  (NCTM, 2000).

5
NCTM P S says

• Using symbolic algebra to represent and solve
  linear equations is one of the expectations
  within the Algebra content standard for grades
  (NCTM P S, 2000).

6
Algebra is essential

• Understanding linear equations and algebraic
  relationships is fundamental to preparing
  students for advanced algebraic concepts.
• Middle grades students need to develop
  representational techniques for a profound
  understanding of, and fluency with linear
  equations (Silver, 2000).

7
Algebra is critical

• Algebra is widely regarded as a gatekeeper.
  Students in the US fail math more frequently than
  any other subject (Jacobson, 2000).
• Higher-level mathematics and opportunities that
  come with it are closed to students who do not
  succeed in high school algebra.
• Preparation for algebra in the middle grades is
  critical to student success in high school
  mathematics (Silver, 2000).

8
Algebra begins early

• Teachers should build upon algebra in the early
  grades. This development of reasoning is closely
  related to students language development and is
  dependent on their abilities to explain their
  reasoning rather than just give the answer
  (NCTM, 2000)

9
Manipulatives Research by Amanda Ross

• Statement of the Problem
• The purpose of this study was to determine if
  access to virtual manipulatives, or use of
  kinesthetic manipulatives have a statistically
  significant effect on middle school students
  understanding of equations.

10
Review of Relevant Literature

• Included literature that investigated kinesthetic
  manipulatives, as well as virtual manipulatives.
• Theoretical framework was based upon importance
  of discovery learning (Bruner, 1966) and need for
  constructivist teaching strategies (Piaget,
  1970b).

11
Research Questions

• Does access to virtual manipulatives or use of
  kinesthetic manipulatives have a statistically
  significant impact on students understanding of
  equations on sixth-grade level questions, or
  questions covering material from the fifth grade
  through eighth grade?

12
Research Questions

• Does access to virtual manipulatives or use of
  kinesthetic manipulatives have a statistically
  significant impact on students beliefs
  concerning the usefulness of computers in heeding
  the process of understanding ideas?

13
Research Questions

• Does access to virtual manipulatives or use of
  kinesthetic manipulatives have a statistically
  significant impact on students attitudes
  concerning confidence and usage with computers,
  as well as attitudes concerning dedication and
  appearance of assignments with their use?

14
Research Questions

• Does access to virtual manipulatives or use of
  kinesthetic manipulatives have a statistically
  significant impact on students attitudes
  concerning hands-on learning?
• Does the use of visualization have a
  statistically significant impact on students
  understanding of equations on sixth-grade level
  questions?

15
Methodology

• Participants included 25 sixth grade students
  enrolled in two different schools within the same
  rural school district.
• Instruments included ten 2004 TAKS questions,
  interview questions related to processes involved
  in solving problems, and attitude survey
  questions.

16
Methodology

• Quasi-experimental design
• Results from test questions and attitude survey
  were used in quantitative analysis.
• Interview questions were included in the
  qualitative analysis to provide information
  concerning students thought processes,
  strategies, and beliefs.

17
Methodology

• These questions were specifically chosen to
  assess students understanding, not simply
  knowledge-base of algebraic equations.
• Questions pertaining to sixth-grade level
  material were examined separately from the
  material for grades fifth through eighth.

18
Analysis

• Test data and survey data
• Separate independent samples t-tests were used
  to examine students understanding of equations,
  beliefs and attitudes towards computers, and
  attitudes towards hands-on learning.
• Interview data Constant comparison was used to
  determine students level of visualization, as
  well as steps taken to solve problems.

19
Results and Discussion

• For the sixth grade questions, the students with
  access to virtual manipulatives performed higher,
  p gt .05.
• For the fifth-eighth grade questions, the
  students with access to virtual manipulatives
  performed higher again, p gt .05.
• Students who used kinesthetic manipulatives had a
  higher attitude towards use of computers in
  relation to their understanding of mathematics, p
  gt .05.

20
Results and Discussion

• Students who had access to virtual manipulatives
  had a higher attitude concerning confidence with
  the use of computers, p gt .05.
• Students who used kinesthetic manipulatives had
  higher scores for perseverance or dedication with
  the use of computers, p gt .05.
• Students with access to virtual manipulatives had
  higher scores for feelings about the increase in
  appearance with computer use, p gt .05.

21
Results and Discussion

• There was a statistically significant difference
  in preference towards hands-on learning with the
  students using kinesthetic manipulatives having a
  higher mean, p lt .05.
• On the sixth grade questions, those students who
  did not visualize anything performed higher on
  the test, p gt .05.
• Both groups of students used much guess-and-check
  in their steps.
• Most of the processes provided from both groups
  aligned with the status of the correct or
  incorrect response they were explaining, although
  several had difficulty explaining steps taken to
  solve 4finding equation for basic fee plus
  additional amount per hour.

22
Concept-Based InstructionHeather Louder

• Definition of terms
• Conceptual knowledge
• understanding of ideas and generalizations that
  connect mathematical constructs (Ashlock, 2002)
• rich in relationships and connections (Hiebert
  Lefevre, 1986)
• Procedural knowledge
• understanding that is focused on skills and
  step-by-step procedures without explicit
  reference to mathematical ideas (Ashlock, 2002,
  p. 8).

23
Research Question

• How does the degree to which a teacher emphasizes
  conceptual and procedural knowledge affect
  students ability to write equations that
  represent problem situations?

24
Method

• Participants 2 Seventh grade teachers and their
  targeted classes
• Teacher A Works in suburban district
• Teacher B Works in a rural district
• Both general seventh grade math classes
• 33 students completed pretests and posttests (20
  for A and 13 for B)

25
Method

• Video Analysis
• One video for each teacher was split into
  10-second intervals
• Each interval coded for type of understanding
  emphasized (Conceptual, Procedural, or Neither)
• Reliability checked by another graduate student
• Two other available videos were analyzed for
  their potential effect on student responses to
  test items
• Item Analysis
• Types of conceptual or procedural knowledge
  needed was noted
• Correct and incorrect responses were analyzed in
  light of each teachers instruction

26
Summary of Teachers Instructional Delivery

• Teacher A
• Connections emphasized
• Mathematical communication
• Teacher B
• Teaching more procedural
• Stand-alone concepts and skills
• Video clip

27
Results of Video Analysis
28
Results Student Achievement
Average total scores on Algebra test
Statistically significant difference resulted
only between posttest scores
29
Results Item Analysis

• Item 2 Modeling equations from verbal
  representations

30
Results Item Analysis

• Item 3 Modeling equations from verbal
  representations

31
Results Item Analysis

• Item 8 Modeling equations from verbal
  representations

32
Discussion Item Analysis

• Items 2, 3, 8
• Teacher A Students relative success stems from
  mathematical communication in classroom
  activities (Schoenfeld Arcavi, 1988 Kieran
  Chalouh, 1993)
• Teacher B Entirely procedural teaching of this
  topic did not appear to help students succeed
  (Lodholz, 1990)

33
General Conclusions

• Two teachers in the study represented two ends of
  a continuum
• Conceptual teaching produces students who are
  more capable problem-solvers and more flexible
  when encountering unfamiliar problems
• Connections explicitly made help students develop
  a network of knowledge

34
Sixth Grade Mathematics Teachers Use of
Probing, Guiding, and Factual Questions

• Alpaslan Sahin

35
Questioning

• Research indicates that questioning is second in
  popularity as a teaching method and classroom
  teachers spend anywhere from 35 to 50 percent of
  their instructional time conducting questioning
  sessions (Cotton, 1998).
• On the average, approximately 60 of the
  questions asked were lower order, 20 higher
  order, and 20 were procedural (Cotton, 1998).
• The issue of questioning has also received
  attention in contemporary studies of education
  (e.g., Harrop Swinson, 2003 Ilaria, 2002
  Kawanaka Stigler, 1999 Martino Maher, 1994
  Sahin, Bullock, Stables, 2002).

36
Research on Questioning

• There is a body of studies on question typologies
  asked by teachers.
• Some of them categorized teacher questioning
  (Cotton, 1989 Cunningham, 1987 Ilaria, 2002
  Kawanaka Stigler, 1999 Styless
  Cavanagh,1980).
• Some of them specifically tried to define certain
  types of teachers questions such as
• Higher order (Bloom et al., 1956 Brualdi,
  1998Gall, 1984 Newmann, 1988 ).
• Factual (Brualdi, 1998Gall, 1984Vacc, 1993).
• Open-ended (Hancock, 1995 Vacc, 1993),
• Probing (Kawanaka Stigler, 1999 Moyer
  Milewicz, 2002 Newmann, 1988).

37
Probing, Guiding, and Factual Questions

• Student-centered instruction is an essential
  component of reform-based teaching and a key
  characteristic of student-centered instruction is
  the use of questions by teachers to probe student
  understanding and to guide students as they
  construct knowledge.
• A focus of the current study was to investigate
  characteristics of probing and guiding questions
  and thus develop clear definitions of them.
• Factual questions to this study because we
  believed that it was important to understand the
  link between factual questions and guiding
  questions.

38
Theoretical Framework

• Questions are a way that teachers use to bring
  students around to the correct mathematical
  concepts and procedures through the negation of
  meaning for necessary condition of learning
  (Voigt, 1992, p. 43).
• the course of negotiation, the teacher and the
  students (or the students among themselves)
  accomplish relationships of mathematical meanings
  taken to be shared (p. 35).
• We tried to develop some indicators through
  literature and conducted teachers interviews to
  see how much teachers probing, guiding, and
  factual questions carry out this negatotation of
  meaning.

39
Research Questions

• What types and frequencies of questions are asked
  by two middle grades mathematics teachers?
• How do the types and frequencies of questions
  vary within major parts of a lesson and across
  lessons?
• What were the teachers intentions or purposes
  in asking the questions?

40
Methodology

• Data Source
• Collected through systematic observations of
  videotaped lessons from public schools in Texas
  as part of a five-year longitudinal study.
• Videotapes of seventh-grade lessons were
  transcribed.
• The teachers used different textbooks but the
  lessons addressed the same mathematical content
  dealing with solving and graphing linear
  equations.

41
Data Source cont.

• Textbooks are intended to support teachers in
  reform-oriented approaches, including the use of
  student-centered learning and inquiry strategies.
  
• Participants ranged from first year to
  experienced teachers.
• The students in project schools were demography
  diverse.

42
Procedure

• The videotapes from the classroom sessions were
  transcribed and checked for accuracy.
• All teacher questions were identified and coded
  into one of three question types probing,
  guiding, and factual questions.
• The coding of question types was done by two
  others independently of each other.

43
Procedure

• The criteria we chose to adopt for identifying
  probing questions were
• Asks students to explain or elaborate their
  thinking
• Asks students to use prior knowledge and apply it
  to a current problem or idea
• Asks students to justify or prove their ideas.

44
Procedure

• We have adopted the following criteria for
  guiding questions
• Provides students a specific suggestion or hint
  about the next step of solution
• Provides students with a general heuristic
  (Polya, 1947)
• Provides a sequence of ideas or hints that
  scaffolds or leads toward convergent thinking.

45
Procedure

• We adopted the following as defining
  characteristics of factual questions
• Asks student for a specific fact or definition
  (Vacc, 1993).
• Asks a student for an answer to an exercise
• Asks students to provide the next step in a
  procedure.

46
Results

• On the average, 15 percent of teachers questions
  was probing.
• The frequency of asking guiding questions was low
  for all teachers.
• Some teachers asked few guiding questions some
  did not ask any.

47
Results for Factual Questions

• Approximately 80 percent of all questions asked
  by teachers were factual.
• Teachers used factual questions consistently
  during all parts of the lessons.
• Guiding questions were mainly found as a series
  of factual questions.

48
Conclusion

• Teachers still ask more factual questions than
  other types of questions as reported before by
  Stevens (1912) and Myhill and Dunkin (2002).
• Results of this study were consistent with the
  results of Watson and Young (1986) in which they
  found that 80 percent of questions were memory
  related.

49
Assessing the Type of Teacher Responses to
Students Questions byDiana Piccolo

• Research Questions
• What type of responses do teachers give to
  students answers?
• How are those responses different in whole group
  versus individual instructional settings?

50
Literature Review

• Rather than simply telling and validating answers
  for students, teachers should work towards
  promoting an environment where students share,
  discuss, and debate the reasonableness and
  validity of their ideas (Crespo, 2000).

51
Literature Review

• Feedback is important to both motivate students
  and to let them know how they are doing (Good
  Brophy, 2003).
• The quality of teacher responses to those
  questions is an important contributor to student
  understanding and learning. Classroom discourse
  features a complex interaction of questions and
  responses (Cazden, 1988)

52
Methodology

• Participants-
• Teacher A- 7th grade math teacher who works in
  a rural district.
• Teacher B- 6th grade math teacher who works in
  an urban district.
• Both classrooms consisted of 20-25 students
  each. Math lessons for both classes were
  approximately 50 minutes.

53
Methodology

• Instrumentation
• Video observation instrument was used to record
  types of responses from teacher.
• Video clip

1. Teacher asks a question

This is the focus of my study!
3. How does the teacher respond to that students
answer?
2. Student gives an answer.
54
Methodology

• Teacher responses in the
• instrument were divided into 6
• areas
• 1. N (no reaction to the students question
  and/or response)
• 2. S (short answer response, such as, yes/no)
• 3. L (limited response, such as, yes, that is
  correct.)
• 4. R (recitation of students question or answer,
  such as, correct, the answer is 16)
• 5. I (involvement of other students, such as,
  Does anyone else agree?)
• 6. P (probing question to elicit further
  responses from the student, such as, if we know
  this, then why do you think we get that?)

55
Methodology

• A tally of the number of teacher responses in
  both whole class and individual student
  situations was recorded.
• The percentage of whole class instructional time
  and individual student/teacher instructional time
  was also recorded.

56
Results Percentage of instructional time during
the 50 minute lesson
57
ResultsVideo Analysis for both whole group
individual instruction
58
ResultsWhole class compared to individual
instruction
59
Discussion

• For both whole class and individual instruction,
  the highest percentage of teacher responses were
• (P) probing for further information from the
  student (teacher A-37, teacher B-51),
• (R) recitation of the students answer (teacher
  A-39, teacher B- 36), and
• (L) limited response to students answer
  (teacher A-22, teacher B- 12)

60
Discussion

• When comparing the type of teacher response to a
  students answer in either a whole group setting
  or in an individual student setting
• teachers spent more time probing to elicit
  further responses from the student during
  individual instruction time (Teacher A- 75,
  Teacher B- 67).
• However, the least amount of individual
  instruction time was allocated during the 50
  minute lesson in comparison with whole group
  instruction time (Teacher A- 1942, Teacher B-
  2103).

61
Discussion

• Teachers must realize that not only are the type
  of questions they ask students important, but
  their responses to those questions are just as
  important.
• Continued learning and development should result
  in response to students answers.

62
Discussion

• Lack of time during a lesson and teacher
  knowledge of the subject matter are a few reasons
  why teachers typically do not respond or spend
  the time needed to effectively respond to a
  students answer.

63
Conclusion

• As Schwartz (1996) suggested, teachers should
  respond with questions that encourage students to
  revisit their work and/or own questions, thus
  extending the experience that invites further
  exploration.

64
Assessing the Type of Teacher Responses to
Students Questions byDiana Piccolo

• Research Questions
• What type of responses do teachers give to
  students answers?
• How are those responses different in whole group
  versus individual instructional settings?

65
Literature Review

• Rather than simply telling and validating answers
  for students, teachers should work towards
  promoting an environment where students share,
  discuss, and debate the reasonableness and
  validity of their ideas (Crespo, 2000).

66
Literature Review

• Feedback is important to both motivate students
  and to let them know how they are doing (Good
  Brophy, 2003).
• The quality of teacher responses to those
  questions is an important contributor to student
  understanding and learning. Classroom discourse
  features a complex interaction of questions and
  responses (Cazden, 1988)

67
Methodology

• Participants-
• Teacher A- 7th grade math teacher who works in
  a rural district.
• Teacher B- 6th grade math teacher who works in
  an urban district.
• Both classrooms consisted of 20-25 students
  each. Math lessons for both classes were
  approximately 50 minutes.

68
Methodology

• Instrumentation
• A video observation instrument was used to record
  types of teacher responses.
• Video clip

1. Teacher asks a question

This is the focus of my study!
3. How does the teacher respond to that students
answer?
2. Student gives an answer.
69
Methodology

• Teacher responses in the
• instrument were divided into 6
• areas
• 1. N (no reaction to the students question
  and/or response)
• 2. S (short answer response, such as, yes/no)
• 3. L (limited response, such as, yes, that is
  correct.)
• 4. R (recitation of students question or answer,
  such as, correct, the answer is 16)
• 5. I (involvement of other students, such as,
  Does anyone else agree?)
• 6. P (probing question to elicit further
  responses from the student, such as, if we know
  this, then why do you think we get that?)

70
Methodology

• A tally of the number of teacher responses in
  both whole class and individual student
  situations was recorded.
• The percentage of whole class instructional time
  and individual student/teacher instructional time
  was also recorded.

71
Results Percentage of instructional time during
the 50 minute lesson
72
ResultsVideo Analysis for both whole group
individual instruction
Total 45
Total 58
73
Discussion

• For Teacher A, both whole class and individual
  instruction, the highest percentage of teacher
  responses were
• (L) Limited response to students answer or
  question - 40
• (R) recitation of the students answer 22,
• (P) probing for further information from the
  student 18

74
Discussion

• For Teacher B, both whole class and individual
  instruction, the highest percentage of teacher
  responses were
• (R) recitation of the students answer 31
• (L) limited response to students answer 24
• (P) probing for further information from the
  student- 19

75
ResultsWhole class compared to individual
instruction
76
Discussion

• When comparing the type of teacher response to a
  students answer in either a whole group setting
  or in an individual student setting
• teachers spent more time probing to elicit
  further responses from the student during
  individual instruction time (Teacher A- 11,
  Teacher B- 14).
• However, the least amount of individual
  instruction time was allocated during the 50
  minute lesson in comparison with whole group
  instruction time (Teacher A- 1942, Teacher B-
  2103).

77
Discussion

• Teachers must realize that not only are the type
  of questions they ask students important, but
  their responses to those questions are just as
  important.
• Continued learning and development should result
  in response to students answers.

78
Discussion

• Lack of time during a lesson and teacher
  knowledge of the subject matter are a few reasons
  why teachers typically do not respond or spend
  the time needed to effectively respond to a
  students answer.

79
Conclusion

• As Schwartz (1996) suggested, teachers should
  respond with questions that encourage students to
  revisit their work and/or own questions, thus
  extending the experience that invites further
  exploration.