Outline of Talk
- Background
- Models of Proof Understanding
- Students' Beliefs About Proofs
- Students' Ability to Construct Proofs
- Video Excerpts
- Concluding Remarks
Background-NCTM Standards
Standard 7: Reasoning and Proof
Mathematics instructional programs should focus on learning to reason and construct proofs as part of understanding mathematics so that all students --
recognize reasoning and proof as essential and powerful parts
of mathematics;
make and investigate mathematical conjectures;
develop and evaluate mathematical arguments and proofs;
select and use various types of reasoning and methods of proof
as appropriate.
Background-Proof Processes
Exploration Formulating
and Conjectures
Problem (bullet #2)
Posing
(bullet #2)
Need for Proof or
Verification
(bullet #1)
Proof and Justification
(bullets #3 and 4)
Background-Research Findings
"Throughout the history of American education, learning
to write proofs has been an important objective of the geometry
curriculum for college-bound students. At the same time, proof
writing has also been perceived as one of the most difficult
topics for students to learn."
(Senk, 1985)
"fewer than 15% of high school graduates in the United
States master proof writing."
(Senk, 1985)
"research on mathematics education needs to identify
cognitive and affective prerequisites ."
(Senk, 1985)
Background- TIMSS Results
Data from the Third International Mathematics and Science Study (TIMSS) indicate that students at the 8th grade and 12th grade level perform poorly in geometry.
In the geometry portion of the TIMSS, the United States scored the lowest of all countries.
TIMSS results show that, in general, students worldwide have particular difficulties organizing arguments.
(National Center
for Education Statistics, 1998)
Background TIMSS Results
Background Research Goals
Overall Goal:
To develop a theoretical model that relates pedagogy to student
understanding of geometric proofs.
Students' beliefs about what constitutes a proof.
-What are their beliefs?
-How are beliefs influenced by instruction and pedagogy?
Students' proof-construction ability
-How well do students construct proofs?
-How is ability linked to pedagogy?
Background-Research Projects
NSF-funded Project
Three-year study: Pedagogical Factors Influencing Student Understanding of Geometric Proof
Focus on the teachers
Focus on the students
University-funded Project
Pilot study: An Investigation of High School Students' Understanding of Geometric Proof
Focus on the assessment instruments
Focus on student understanding of proof
Types of Proof or Justification Schemes (Sowder & Harel,
1998)
Proof Scheme whatever constitutes ascertaining and persuading for that person
Externally-based Appeal to an external source
for convincing and persuading
Authoritarian rely on textbook, teacher, or more
knowledgeable classmate
Ritual correctness judged by the form of the argument
rather than the reasoning
Symbolic - rely on symbol manipulation, regardless of
correctness
Empirical Justifications made on the basis of
examples
Perceptual rely on how a figure looks (e.g., the
triangle looks equilateral)
Examples-based convinced by one or more examples
(e.g., seeing the pattern)
Analytic Mathematical proofs
Transformational based on general aspects of a situation,
perceiving underlying structure behind a pattern
Axiomatic an ability to work within an axiomatic
system
Balacheff's 4 Stages of Understanding Mathematical Proof
1) Naïve Empiricism
Inductive perspective
Conclusions based on small number of cases
2) Crucial Experiment
Question of generalization is considered
Examination of extreme cases
3) Generic Example
Arguments based on a class of objects
Highest level prior to deductive proof
4) Thought Experiment
Transition from practical to intellectual proofs
Development of deductive proofs
(Balacheff, 1987)
Six Principles of Geometric Proof
1) Implications of Truth Statements are true if and only if they are true for all cases.
a) A theorem has no exceptions.
b) A counterexample disproves a general statement.
2) Purposes of Proof - The dual role of proof is to convince and to explain.
a) Proofs are required to establish truth.
b) Proofs can explain.
3) Generality Requirements - A proof must be general.
a) Generality can be achieved by checking all cases.
b) Generality can be achieved by reasoning about general statements.
c) Generality is not achieved by reasoning inductively.
d) Generality is not achieved by checking special cases.
4) Internal Logic Requirements - The validity of
a proof depends on its internal logic.
a) Conditional statements contain distinct components.
b) The logical order of statements matters.
c) Ritualistic aspects of proof are irrelevant to its
validity.
5) Logical Equivalences - Statements are logically equivalent to their contrapositives, but not necessarily to their converses or inverses.
6) Role of Diagrams - Diagrams that illustrate statements have benefits and limitations.
a) Diagrams are limited by their specificity.
b) Diagrams may assist with visualization of relationships.
About Proofs
Questionnaire Results Summary
Principle Range of Item Averages
1. Implications of Truth 50 83%
2. Purposes of Proof 50 100%
3. Generality Requirements 22 78%
4. Internal Logic Requirements 17 53%
5. Logical Equivalences 67 83%
6. Role of Diagrams 42 75%
Scores: 1 for correct; 0 for incorrect; 0.5 for unsure
Questionnaire Best Performance
Prin. 2: Purposes of Proof - The dual role of proof is
to convince and to explain.
1. Consider the true statement, "When you add the measures of the interior angles of any triangle, your answer is always 180º."
d. A proof of this statement might show me why this statement is true.
Reply Sem. 1 Sem. 2 Overall
True 100% 100% 100%
False 0% 0% 0%
Unsure 0% 0% 0%
Questionnaire Strong Performance
Prin. 2: Purposes of Proof - The dual role of proof is to
convince and to explain.
13. If a statement makes sense and seems to be true, then it doesn't need to be proven.
Reply Sem. 1 Sem. 2 Overall
True 0% 25% 11%
False 100% 75% 89%
Unsure 0% 0% 0%
Questionnaire Strong Performance
Prin. 1: Implications of Truth Statements are true if
and only if they are true for all cases.
17. If you determine that a statement is true for 1,000,000 examples and false for one example, then you have proven that the statement is false.
Reply Sem. 1 Sem. 2 Overall
True 80% 88% 83%
False 20% 12% 17%
Unsure 0% 0% 0%
Questionnaire Strong Performance
Prin. 5: Logical Equivalences Statements are logically
equivalent to their contrapositives, but not necessarily to their
converses or inverses.
1. Consider the true statement, "When you add the measures of the interior angles of any triangle, your answer is always 180º."
c. Since this statement is true, I know that if the measures of the interior angles of a polygon do not add to 180º, then the polygon is not a triangle.
Reply Sem. 1 Sem. 2 Overall
True 80% 88% 83%
False 20% 12% 17%
Unsure 0% 0% 0%
Questionnaire Strong Performance
Prin. 3: Generality Requirements - A proof must be general.
1. Consider the true statement, "When you add the measures of the interior angles of any triangle, your answer is always 180º."
e. If someone could make a list of all possible triangles and confirm that the measures of the interior angles in each triangle summed to 180º, then they would have proven that the statement is true.
Reply Sem. 1 Sem. 2 Overall
True 60% 100% 78%
False 30% 0% 17%
Unsure 10% 0% 6%
Questionnaire Poor Performance
Prin. 4: Internal Logic Requirements - The validity of a proof
depends on its internal logic.
15. You are given the statement "The base angles of an isosceles triangle are congruent." You may use the fact that the base angles are congruent as given information in your proof.
Reply Sem. 1 Sem. 2 Overall
True 70% 88% 78%
False 20% 0% 11%
Unsure 10% 12% 11%
Questionnaire Poor Performance
Prin. 3: Generality Requirements - A proof must be general.
2. Dylan attempted to prove the statement, "When you add the measures of the interior angles of any triangle, your answer is always 180º." His work is shown below.
Dylan's Work
I measured the angles of all sorts of triangles accurately
and made a table.
A B C Total
34 110 36 180
95 43 42 180
35 72 73 180
10 27 143 180
They all added up to 180º, so the statement is true.
Questionnaire Poor Performance (continued)
2a. Since Dylan checked that the statement is true for both obtuse and acute triangles, his work shows that the statement is always true.
Reply Sem. 1 Sem. 2 Overall
True 60% 100% 78%
False 40% 0% 22%
Unsure 0% 0% 0%
Questionnaire Poor Performance
Prin. 4: Internal Logic Requirements - The validity of a proof
depends on its internal logic.
12. Geometric proofs must list statements and reasons in two separate columns.
Reply Sem. 1 Sem. 2 Overall
True 90% 50% 72%
False 0% 50% 22%
Unsure 10% 0% 6%
Questionnaire Poor Performance
Prin. 4: Internal Logic Requirements - The validity of a proof
depends on its internal logic.
8. Natalie attempted to prove the statement, "If C is any point on the perpendicular bisector of segment AB, then ABC is always isosceles."
Natalie's work shows that the statement is true.
Reply Sem. 1 Sem. 2 Overall
True 80% 50% 67%
False 20% 25% 22%
Unsure 0% 25% 11%
Questionnaire Poor Performance (continued)
Statements Reasons
1. CD is the | bisector of segment AB. 1. Given.
2. m ADC = 90° 2. Def. of perp. bisector.
3. mBDC = 90° 3. Def. of perp.bisector.
4. AD = BD 4. Def. of perp. bisector.
5. CAD @ CBD 5. Base angles of an isosceles triangle
are @.
6. CAD @ CBD 6. Two sides and included angle the same (ASA).
7. AC = BC 7. Corresponding parts of congruent triangles are
equal.
8. ABC is isosceles 8. Def. of isos. triangle.
Questionnaire Moderate Performance
Prin. 6: Role of Diagrams - Diagrams that illustrate statements
have benefits and limitations.
9. Consider the dialogue between Juan and Ling.
Juan: In both of our diagrams ABCD is a rectangle and
E is a point on segment AD. The height of BEC is the length of
segment CD. Since the area of a triangle is 1/2(baseoheight),
the area of BEC is half the area of rectangle ABCD. The same
is true in your diagram Ling.
Ling: In my diagram, E is in a different place on segment
AD. So, your argument doesn't apply to my diagram.
Juan's Diagram Ling's Diagram
Questionnaire Moderate Performance (continued)
b. Ling is correct. Since the diagrams are different, the conclusion that Juan made for his diagram doesn't apply to Ling's diagram.
Reply Sem. 1 Sem. 2 Overall
True 60% 37% 50%
False 30% 63% 44%
Unsure 10% 0% 6%
Open-ended Questionnaire Results
Prin. 3: Generality Requirements
Statement 1:
"In a triangle, a line connecting the midpoints of two of
its sides is parallel to the third side."
Argument 1:
I drew three different triangles. I labeled each triangle
ABC. In each triangle, D and E are the midpoints of the sides
AB and AC, respectively. I measured the angles in each of the
three different triangles and in each case was congruent to .
Since these angles are corresponding angles (relative to line
, line , and transversal ), is parallel to . Therefore, the statement
is always true.
1. Does argument 1 show that the statement is true for all
triangles? Why or why not?
Open-ended Questionnaire Results
Prin. 3: Generality Requirements
Question 1 responses:
Correct 4 (not all with valid reasons)
Incorrect 14
Open-ended Questionnaire Results
Prin. 4: Internal Logic Requirements
Statement 3:
"Supplements of congruent angles are congruent."
Or, equivalently...
"If A is supplementary to B, C is supplementary
to D, and B is congruent to D, then A and
C are congruent."
Argument 3:
Statements Reasons
1. A and B are supplementary angles 1. Given
2. C and D are supplementary angles 2. Given
3. mA + mB = 180° and mC + mD = 180°
3. Definition of suppl. angles
4. mA = 180° - mB and mC = 180° - mD
4. Subtraction prop. of equality
5. mA = 180° - mD 5. Substitution since B
@ D
6. B @ D 6. Given
7. A @ C 7. Substitution.
Open-ended Questionnaire Results
Prin. 4: Internal Logic Requirements
Can the ordering of the statements in a proof affect its validity? Explain.
Question 6 responses:
Correct 13 (most reasonable)
Incorrect 1
Open-ended Questionnaire Results
Prin. 5: Logical Equivalences
Statement 4:
"If a figure is a zapazoid, then it has six
vertices."
7. If we assume that statement 4 is true, say whether each of the following statements is TRUE or FALSE. Justify your answers.
If a figure has six vertices, then it is a zapazoid.
Question 7a responses:
Correct 6 (about half without valid reasons)
Incorrect 10
Construct Proofs
Proof Quiz
Summary of Results
Problem Range of Average Scores
1 29% 44%
2 6% - 31%
3 29% - 60%
4 0% - 22%
5 17% - 33%
6 22% - 32% (sem. I)
14% - 29% (sem. II)
Proof Quiz Results
1.
Fill in missing statements or reasons to form a valid proof.
Given: ABE @ DCE
Prove: EBC @ ECB
Statements Reasons
1. 1. Given.
2. mEBC+mABE=180° 2.
3. 3. Straight angle or linear pair.
4. mEBC + mABE =
mDCE + mECB 4.
5. mEBC + mABE =
mABE + mECB 5.
6. mEBC = mECB 6. Subtraction property of equality.
7. 7. Definition of congruent angles.
Results for Problem 1 (5 pts.):
Semester Mean Score
Version 1 Mean Score
Version 2
I 2.22 (44%) 2.22 (44%)
II 1.43 (29%) 2.14 (43%)
Combined 1.88 (38%) 2.19 (44%)
Proof Quiz Results
2. In the quadrilateral WXYZ below, diagonals and intersect at point P.
PROVE that point P is the midpoint of segments and . Show all your work. Be sure to provide reasons for your statements.
Results for Problem 2 (5 pts.):
Semester Mean Score
Version 1 Mean Score
Version 2
I 1.56 (31%) 1.11 (22%)
II 0.57 (11%) 0.29 (6%)
Combined 1.13 (23%) 0.75 (15%)
Proof Quiz Results
3. Conjecture: When I draw a line parallel to a side of a
triangle it creates a new triangle. I checked several examples
and noticed that this smaller triangle is always similar to the
original triangle.
Write the conjecture as a conditional statement (a statement
in "if-then" form).
Conditional Statement:
If you were asked to prove this statement you would first need to identify the "given" and the "prove." Write the "given" and "prove" information below, but DO NOT PROVE the statement.
Given:
Prove:
Results for Problem 3 (7 pts.):
Semester Mean Score
Version 1 Mean Score
Version 2
I 4.22 (60%) 3.67 (52%)
II 4 (57%) 2 (29%)
Combined 4.13 (59%) 2.94 (42%)
Proof Quiz Results
4. Consider the conditional statement and the accompanying
diagram.
"If two altitudes, and , in ABC intersect at point S
and are congruent, then ABC is isosceles."
Write a proof of the statement.
Give geometric reasons for the statements in your proof.
Results for Problem 4 (5 pts.):
Semester Mean Score
Version 1 Mean Score
Version 2
I 0 (0%) 1.11 (22%)
II 0.43 (9%) 0.57 (11%)
Combined 0.19 (4%) 0.87 (17%)
Proof Quiz Results
5.
Given: Quadrilateral KLMN is a parallelogram.
Segments and intersect at P.
N is on line
Prove:
KLP is similar to NQP
Several hints about how this proof may be constructed are provided below. Please use some of these hints to write a valid proof that KLP is similar to NQP.
Hints:
Recall that proving that triangles are similar requires the
identification of several pairs of congruent angles. Use the
quadrilateral to identify a pair of parallel lines. Use properties
of parallel lines and related angles to identify pairs of congruent
angles.
Results for Problem 5 (5 pts.):
Semester Mean Score
Version 1 Mean Score
Version 2
I 1.44 (29%) 1.67 (33%)
II 0.86 (17%) 1.29 (26%)
Combined 1.19 (24%) 1.5 (30%)
Proof Quiz Results
6.
For each part, write a logical conclusion that follows from the
given set of conditions. Also, record a reason that supports
each conclusion.
a. Given: Three distinct points A, B, and C lie on a line.
AB=BC.
Conclusion:__________________________________
Reason:_____________________________________
b. Given: intersects at point P. Point P is between X and
Y. Point P is between Z and W.
Conclusion:__________________________________
Reason:_____________________________________
c. Given: LMN and PQR. . . .
Conclusion:__________________________________
Reason:_____________________________________
d. Given: Line l and line m are both cut by
transversal line n. Line n is not perpendicular
to either line l or line m. The alternate exterior
angles are supplementary.
Conclusion:__________________________________
Reason:_____________________________________
Results for Problem 6 (12 pts.):
Semester Mean Score
Version 1 Mean Score
Version 2
I 3.67 (30%) 2.67 (22%)
Proof Quiz Results
6.
Given: Circle A with radius .
is the perpendicular bisector of .
Point C is on the circle.
Prove: ABC is equilateral.
Several hints about how this proof may be constructed are provided below. Please use some of these hints to write a valid proof that ABC is equilateral.
Hints:
Recall that proving a triangle to be equilateral requires
showing that several segments are congruent. Use the circle to
find some congruent segments. Using 's relationship to may help
you find a relationship between ADC and BDC.
Results for Problem 6 (5 pts.):
Semester Mean Score
Version 1 Mean Score
Version 2
II 1.43 (29%) 0.71 (14%)
Bibliography
Balacheff, N. (1987). Treatment of refutations: Aspects of the complexity of a constructivist approach to mathematics learning. Radical Constructivism in Mathematics Education, 89-110.
Elliot, L., & Knuth, E. (1998). Characterizing students' understandings of mathematical proof. Mathematics Teacher, 91, 714-717.
Harel, G., & Sowder, L. (1998). Types of students' justifications. Mathematics Teacher, 91, 670-675.
Senk, S. (1985). How well do students write geometry proofs? Mathematics Teacher, 78 (6), 448-456.
U.S. Department of Education. National Center for Education Statistics. (1998) Pursuing Excellence: A Study of U.S. Twelfth-Grade Mathematics and Science Achievement in International Context. Washington D.C.: U.S. Government Printing Office.