Tennessee Technological University. AMS/MAA Special Session. Research in Undergraduate Mathematics. Education. SAN DIEGO, JANUARY 1997 ...
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Mathematicians' Views of Mathematics Preliminary Report (Partially supported by NSF award no. 9355841)
John Selden Mathematics Education Resources Company
Annie Selden Tennessee Technological University
AMS/MAA Special Session Research in Undergraduate Mathematics Education
SAN DIEGO, JANUARY 1997
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DIFFERING VIEWS OF MATHEMATICS
School teachers and their students Mathematics education researchers Mathematicians
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School Teachers' and their Students' Views of Mathematics
Change: Unchanging* -- Changing Kinds of Knowledge: Procedural* -- Conceptual Warrants: Authority*
--
Deductive Argument
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Mathematics Education Researchers' Views of Mathematics There is only one well-articulated view as a philosophy of mathematics: The Philosophy of Mathematics Education, by Paul Ernest, Falmer Press, London, 1991. Mathematics changes: It grows and is essentially fallible. Truth and objectivity are replaced by social acceptance.
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Mathematicians' Views of Mathematics Mathematicians do have views on the nature of mathematics. However, not in the form of a wellarticulated philosophy of mathematics. From (another) earlier study: Q: "How do you define mathematics?" A: "I don't."
Comment by professor, to a student: "We do mathematics, we don't talk about it.
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The Current Study SUBJECTS Junior college mathematics teachers (17) From two summer institutes (on calculus, cooperative learning)
Upper-division mathematics teachers (34) From two summer institutes (on calculus, cooperative learning)
Major research university professors (3) Individually interviewed
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7 Questions 1. How is mathematics done? A. Conceptually. B. Procedurally.
2. How is the truth of theorems established? A. Via deductive reasoning. B. It cannot be completely established -- social factors.
3. Who decides what's true? A. An individual. B. The community.
4. How is the correctness of solutions established? A. Same as proofs. B. Authority of the community.
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5. Does mathematics change/develop? A. It is fixed and unchanging. B. It changes/develops -- invention/discovery.
6. Is mathematics permanently reliable? A. It is independent of time, place, social considerations. B. Truth depends on time, community.
7. How does mathematics relate to reality? A. People discover pre-existent objects. B. People invent/construct objects. C. People invent/construct objects which influence the way reality is seen.
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Results 2. How is the truth of theorems established? A. Via deductive reasoning. B. It cannot be completely established -- social factors. Jr. College Teachers n=17 A.
76.5% (13)
A.&B. B.
76.5% (26) 14.5% (5)
11.7% (2)
Other Left Blank
Upper-division Research univ. Teachers Professors n=34 n=3
2.9% (1) 5.9% (2)
11.7% (2)
*1 person included some aspects of B.
100% (3)*
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3. Who decides what's true? A. Individual. B. Community. Jr. College Teachers n=17 A.
41.2% (7) 47.1% (16)
A.&B. B. Other
Upper-division Research univ. Teachers Professors n=34 n=3
5.9% (2) 47.1% (2)
44.1% (15)
5.9% (1)
2.9% (1)
66.7% (2) 33.3% (1)*
Left 5.9% (1) Blank *"Most people believe Wiles' proof -- there's no way I'd be able to work through it myself."
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6. Is mathematics permanently reliable? A. Independent of time, place, social considerations. B. Truth depends on time, community. Jr. College Teachers n=17 A.
Upper-division Research univ. Teachers Professors n=34 n=3
64.7% (11)
76.5% (26)
23.5% (4)
17.6% (6)
5.9% (1)
2.9% (1)
100% (3)*
A.&B. B. Other
Left 5.9% (1) Blank *"Standards of rigor have changed (over time). I don't agree with 'community of scholars'."
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1. How is mathematics done? A. Uses conceptual knowledge to produce original solutions, applications, proofs , or theorems B. Relies mainly on procedural knowledge – one carries out appropriate methods properly. Jr. College Teachers n=17
Upper-division Research univ. Teachers Professors n=34 n=3
A.
41.2% (7)
44.1% (15)
33.3% (1)*
A.&B.
17.6% (3)
35.3% (12)
66.6% (2)*
B.
17.6% (3)
14.7% (5)
Other
11.8% (2)
2.9% (1)
Left Blank
11.8% (2)
2.9% (1)
*The conceptual/procedural distinction is not one research mathematicians make. They say technical knowledge and insight are both important.
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4. How is the correctness of solutions established? A. Deductive reasoning, as with proofs. B. Depends on proper application of algorithms – authority of math community. Jr. College Teachers n=17 A.
Upper-division Research univ. Teachers Professors n=34 n=3
58.8% (10)
76.5% (26)
29.4% (5)
11.8% (4)
Other
5.9% (1)
5.9% (2)
Left Blank
5.9% (1)
5.9% (2)
A.&B. B.
100% (3)
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5. Does mathematics change/develop? A. It is fixed and unchanging. B. It changes through invention/discovery.
Jr. College Teachers n=17 A.
Upper-division Research univ. Teachers Professors n=34 n=3
11.8% (2)
5.9% (2)
A.&B.
5.9% (1)
5.9% (2)
B.
76.7% (13)
88.2% (30)
100% (3)*
Other Left Blank
5.9% (1)
*"It's growing. Things in the past remain true."
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7. How does mathematics relate to reality? A. People discover pre-existent objects. B. People invent/construct objects. C. People invent/construct objects which influence the way reality is seen. Jr. College Teachers n=17
Upper-division Research univ. Teachers Professors n=34 n=3
A.
17.6% (3)
8.9% (3)
33.3% (1)*
B.
29.4% (5)
32.4% (11)
33.3% (1)
C.
17.6% (3)
14.7% (5)
A&B&C
17.6% (3)
38.2% (13)
5.9% (1)
5.9% (2)
D. Left Blank
33.3% (1)
11.8% (2)
*"A question of what one means by reality. Math objects have an existence outside our minds, so we discover them – not a physical reality."
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1. How is mathematics done? A. Doing mathematics mainly requires using conceptual knowledge to produce original solutions, applications, proofs, or theorems. That is, often one solves problems, proves theorems, or develops applications without having explicit sample solutions, or even general methods, such as integration by parts, to follow. B. Doing mathematics relies mainly on procedural knowledge. That is, one usually follows or combines several sample solutions or general methods, such as integration by parts, to solve problems. The key to success is selecting and carrying out appropriate methods properly. C. ___________________________________________ ___________________________________________
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2. How is the truth of theorems established? A. The truth of theorems is determined by the validity of their proofs, which in turn, depends on deductive reasoning and previously accepted mathematical results. Proofs are often expanded by their readers until further details are seen as unnecessary. B. The truth of theorems cannot be completely established. A theorem's truth/acceptability depends largely on social factors such as the reputation of the prover, consistency with known results, lack of known counterexamples, and explanatory value. C. ___________________________________________ ___________________________________________
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3. Who decides what's true? A. The truth of a theorem can be checked by a single, or a few, knowledgeable individuals (mathematicians) examining its proof for themselves. When this is not done for oneself, the implicit assumption is that someone else has done so. B. A theorem's truth/acceptability cannot be fully established by a single, or a few, individuals, however knowledgeable they may be. It ultimately rests on the experience and authority of the mathematical community as a whole as judged by features such as the reputation of the prover, consistency with known results, lack of known counterexamples, and explanatory value. C. ___________________________________________ ___________________________________________
4. How is the correctness of solutions established?
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A. The correctness of the solutions to problems ultimately depends on the same kind of deductive reasoning as the validity of proofs. In addition, if the problem is applied, one might also check with the "real world" from which it came to see if the solution "makes sense." B. The correctness of problem solutions depends on the proper application of formulas and algorithms -- to decide this, one ultimately turns to the authority of the mathematical community. In addition, if the problem is applied, one might also check with the "real world" from which it came to see if the solution "makes sense." C. ___________________________________________ ___________________________________________
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5. Does mathematics change/develop? A. Mathematics is mostly fixed and unchanging, although any given individual's knowledge of it may change over time. B. Mathematics changes/develops through the invention/discovery of new mathematical objects and theorems. C. ___________________________________________ ___________________________________________
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6. Is mathematics permanently reliable? A. Mathematics is permanently reliable in the sense that, once a theorem's proof has been found free of errors, its truth is independent of time, place, applicability, or social considerations such as the reputation of the prover. B. Mathematics, being a human construct, is inherently impermanent and unreliable. What is true in one century or community of scholars may not be true in another. C. ___________________________________________ ___________________________________________
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7. How does mathematics relate to reality? A. People (often mathematicians) discover mathematical objects which pre-exist in reality. B. People (often mathematicians) invent/construct mathematical objects which can model, or be applied to, reality. C. People (often mathematicians) invent/construct mathematical objects which can structure reality, i.e., influence the way reality is seen, interpreted or experienced. D. ___________________________________________ ___________________________________________
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Contrasts School College/ Teachers/ Students
(Some) Ed. Fallibilist
Univ.
Change: Unchanging
Develops/ Fallible
Develops/ Infallible
Kinds of Knowledge: Procedural Conceptual Conceptual & Emphasis
Procedural
Social
Indiv. Reasoning
Warrants: Authority
Much Agreement
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Amongst Mathematicians (at all levels)
The correctness of solutions is decided in the same way as the correctness of proofs.
