An Investigation Into the van Hiele Level of Understanding Geometry of Preservice Elementary and Sec

1
An Investigation Into the van Hiele Level of
Understanding Geometry of Preservice Elementary
and Secondary Mathematics Instructors

• by Kathleen Chesley Knight
• Thesis Committee Eric Pandiscio, Advisor
• Robert Franzosa
• Tod Shockey

2
Introduction

• SMT 506 from the Teaching and Learning Geometry
  chapter of the NCTM Research Companion
• Most current curricular and teaching practices
  are simply, abominable. They promote little
  learning or conceptual change. They often do
  more harm than good. They leave students
  unprepared for further study of geometry and the
  many other mathematical topics and subjects that
  depend on geometric knowledge. -Clements (2000)

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Introduction, contd

• August 2005 Maine State Certification
  Requirements
• Requires 24 CH of Math Content but does not
  specify disciplines
• Historically, the curriculum for teacher
  education programs have been designed to meet the
  certification requirements
• The changing curriculum since the early 20th
  Century
• Formal geometry, now taught at the secondary
  level
• Informal geometry, needed to prepare students for
  formal geometry
• The common belief is that the more a teacher
  knows about a subject and the way students learn,
  the more effective that individual will be in
  nurturing mathematical understanding. Swafford,
  Jones Thornton (1997)

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Introduction, contd

• Vygotskys Theory
• Scaffolding
• Zone of Proximal Development
• van Hiele Theory
• Levels of Understanding
• Properties inherent to each level
• Phases of learning that lead to attainment of
  understanding
• NCTM Expectations
• Upon completion of grade 8 students are expected
  to exhibit behaviors that are similar to the van
  Hiele Theorys Level 3, identified as simple
  deduction
• Upon completion of grade 12 students are expected
  to exhibit behaviors that are similar to the van
  Hiele Theorys Level 4, identified as formal
  deduction

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Thesis Questions

• Are prospective elementary and secondary
  mathematics teachers at or above the van Hiele
  level of understanding geometry expected of their
  future students prior to completion of their
  program required geometry course?
• Are prospective elementary and secondary
  mathematics teachers at or above the van Hiele
  level of understanding geometry expected of their
  future students after completion of their program
  required geometry course?

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Literature

• TIMSS (Third International Mathematics and
  Science Study)
• geometry at the grade 8 level, 24 nations scored
  significantly higher than students in the United
  States. Only four nations scored significantly
  lower
• Range of application
• From advanced instruction
• to vocational careers to
• to daily life.

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Literature, contd

• Justification for Informal Geometry
• builds a foundation
• strong similarities in the natural world
• practical application in vocational careers and
  in the pursuit of advanced degrees

8
Methodology

• Sample Population
• Students enrolled in one section of MAT475 at the
  University of Maine during the Fall 2005
  Semester.
• Students enrolled in two sections of MAT107 at
  the University of Maine during the Fall 2005
  Semester.
• Test Tool
• The Cognitive Development and Achievement in
  Secondary School Geometry project (CDASSG) van
  Hiele Test.
• 25 questions, 5 blocks of 5 questions each.
• Procedure
• Pretest administered during the first class of
  the semester.
• Posttest administered upon completion of the
  formal instruction for each section.

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Presentation of Data

• Identification of the van Hiele Level
• Two cases
• Classical
• Highest consecutive level attained
• No skipped levels
• Modified
• Highest consecutive level attained
• Level 5 excluded from skipped levels

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Presentation of Data, contd

• Identification of the van Hiele Level, contd
• Two criteria
• 4 of 5 questions correctly answered
• Reduces Type I error, assigning a subject to a
  higher level than they understand.
• 3 of 5 questions correctly answered
• Reduces Type II error, failing to assign a
  subject to a level for which they have attained
  understanding

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Presentation of Data, contd

• Justification of case/criterion used
• Classical vs Modified
• Modified scenarios gave a higher percentage of
  subjects that could be analyzed (reduced number
  of no fit subjects).
• 3 of 4 Correct vs 4 of 5 Correct
• Reduces the likelihood that a student attained a
  level because of guessing correctly not because
  of understanding the content.

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Presentation of Data, Contd

• Frequency
• Total of 68 individuals were tested
• Breakdown as a result of Add/Drop, Withdrawal,
  and absenteeism,
• Pretest Only 17 (25)
• Posttest Only 9 (13.2)
• Both Pretest and Posttest 42 (61.8)
• Of the 42 individuals who took both pre- and
  posttest
• MAT107 30 (71.4)
• MAT475 12 (28.6)

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Distribution of Pretest M4 van Hiele Level by
Course of Instruction
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Distribution of Posttest M4 van Hiele Level by
Course of Instruction
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Null and Alternate Hypotheses of Pretest and
Posttest Level Thesis Questions

• Are prospective elementary and secondary
  mathematics teachers at or above the van Hiele
  level of understanding geometry expected of their
  future students prior to or after completion of
  their program required geometry course?
• MAT107 Null and Alternate Hypotheses
• H0 µ 3
• H1 µ lt 3
• MAT475 Null and Alternate Hypotheses
• H0 µ 4
• H1 µ lt 4

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Results of a One-Sample t Test of the Pretest van
Hiele Levels
Results of a One-Sample t Test of the Pretest van
Hiele Levels shows that the pretest van Hiele
level of prospective Elementary Teachers is
statistically significantly lower than the Level
3 expected of students completing grade 8 and
that the pretest van Hiele level of prospective
Secondary Mathematics teachers is statistically
significantly lower than the Level 4 expected of
students completing grade 12.
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Results of a One-Sample t Test of the Posttest
van Hiele Levels
Results of a One-Sample t Test of the Posttest
van Hiele Levels shows that the posttest van
Hiele level of prospective Elementary Teachers is
statistically significantly lower than the Level
3 expected of students completing grade 8 and
that the posttest van Hiele level of prospective
Secondary Mathematics teachers is statistically
significantly lower than the Level 4 expected of
students completing grade 12.
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Further investigation based on findings

• Are students showing a gain in understanding of
  at least one level as a result of completing
  their program required geometry course?
• Null and Alternate Hypotheses
• H0 µ 1, Upon completion of the program
  required geometry course students show a gain of
  at least one level of understanding
• H1 µ lt 1, Upon completion of the program
  required geometry course students show a gain of
  at least one level of understanding

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Distribution of the Mean Change in van Hiele
Level by Course of Instruction
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Results of a One-Sample t Test of the Mean Change
in the Pretest van Hiele Levels
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Results of a One-Sample t Test of the Mean Change
in the Pretest van Hiele Levels, contd

• Results of the One-Sample t Test of the mean
  change in the van Hiele Level shows that the
  students who complete the course in Higher
  Geometry and who are enrolled in the Secondary
  Mathematics Education program have a mean gain
  that is statistically significantly equal to or
  greater than one, meaning these students do
  attain the next higher van Hiele Level of
  Understanding Geometry
• Results of the One-Sample t Test of the mean
  change in the van Hiele Level also shows that the
  students who complete the course in Elementary
  Descriptive Geometry and who are enrolled in the
  Elementary Education program have a mean gain
  that is statistically significantly equal to or
  greater than one, meaning these students also
  attain the next higher van Hiele Level of
  Understanding Geometry

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The ?2 Test for Independence on the Change in van
Hiele Levels

• Is a students change in their van Hiele Level of
  Understanding Geometry independent of Pretest
  Level of Understanding?
• Null and Alternate Hypotheses
• H0 The change in van Hiele Level of
  Understanding is independent of the subjects
  pretest van Hiele level.
• H1 The change in van Hiele Level of
  Understanding is dependent on the subjects
  pretest van Hiele level.

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The ?2 Test for Independence on the Change van
Hiele Levels
Results of the ?2 Test of the mean change in the
van Hiele Level shows that the change in van
Hiele Level is independent of the Pretest M4
measured van Hiele Level of Understanding
Geometry for students enrolled in the Higher
Geometry course of instruction. However, the
results of the ?2 Test of the mean change in the
van Hiele Level also shows that the change in van
Hiele Level is dependent on the Pretest M4
measured van Hiele Level of Understanding
Geometry for students enrolled in the Elementary
Descriptive Geometry course of instruction.
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The t Test for Two Dependent Means of the Change
in Pretest van Hiele Level

• Does the M4 measured van Hiele Level of
  Understanding Geometry increase as a result of
  the geometry course they take, the education
  program that they are enrolled in or the M4
  measured Pretest van Hiele Level of Understanding
  Geometry?
• Null and Alternate Hypotheses
• H0 µPost µPre 0
• H1 µPost µPre gt 0

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The t Test forTwo Dependent Samples
For these five samples, the t Test for Two
Dependent Samples does indicate that there is
statistically significant evidence that students
who are enrolled in the Elementary Education
Program do have an increase their level of
understanding, not necessarily a full level, and
not necessarily to the level expected of their
future students
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The t Test forTwo Dependent Samples, contd
For these six samples, the t Test for Two
Dependent Samples does indicate that there is
statistically significant evidence that students
who have a van Hiele Level of 2 do have an
increase their level of understanding, not
necessarily a full level, and not necessarily to
the level expected of their future students No
Fit indicates that these subjects were unable to
be identified as having a level of understanding
that fit the van Hiele model. However, upon
completion of their required course all four
subjects were found to fit the van Hiele Model.
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Conclusions

• Study findings
• One Sample t Test of Pretest and Posttest van
  Hiele Levels
• One Sample t Test of the Change in Pretest van
  Hiele Level
• ?2 Test for independence of Gain vs. Pretest van
  Hiele Level
• t Test of Two Dependent Means on the Change in
  Pretest van Hiele Level

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Conclusions

• Study suggestions
• additional instruction is needed for the
  preservice Elementary teachers, who are not
  already at a Level 2
• additional instruction is needed for the
  preservice Secondary Mathematics teachers, who
  are not already at a Level 3
• develop a seminar course that is designed to
  provide students an opportunity to attain an
  understanding of at least a Level 2 and consider
  requiring these future educators to take a
  placement exam prior to registering for the
  courses currently being offered

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Conclusions

• Study limitations
• Traditionally only four sections of MAT107 are
  offered yearly, two in the Fall Semester and two
  in the Spring Semester
• Traditionally only one section of MAT475 is
  offered yearly, only in the Fall Semester
• Data should be collected for several years and a
  reinvestigation into the findings with a larger
  sample size should be considered

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Conclusions

• Future studies
• investigate the van Hiele Level of Understanding
  Geometry using subjects from all the University
  of Maine System campus, specifically Fort Kent,
  Presque Isle, Machias, Farmington, and the
  University of Southern Maine
• investigate the van Hiele Level of Understanding
  Geometry of inservice Elementary and Secondary
  Mathematics teachers, and compared these studies,
  to see if the number of years teaching this
  content area is a factor in the van Hiele Level
  of Understanding Geometry for this population

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Impact of this Study on Myself

• As a prospective secondary mathematics teacher
• Focus instruction more on understanding and less
  on memorization