What Our Students Really Know About Proof and Reasoning in Geometry: A Look at Classroom-based Research Data

 

Tami S. Martin
tsmartin@math.ilstu.edu
Sharon M. Soucy McCrone
smccrone@math.ilstu.edu
Cynthia A. Pulley
capope@ilstu.edu

Illinois State University

 

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

Models of Proof Understanding


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.

Students' Beliefs
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

Students' Ability to
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%)

 

Video Excerpts


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.


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Last updated April 10, 2000