2013 - (Wheeler+Champion) Students’ proofs of 1-to-1 N onto properties in introd abstract algebra

International Journal of Mathematical Education in Science and Technology, 2013

Vol. 44, No. 8, 1107–1116, http://dx.doi.org/10.1080/0020739X.2012.756546

Students’ proofs of one-to-one and onto propertiesin introductory abstract algebra

Ann Wheelera∗ and Joe Championb

aDepartment of Mathematics and Computer Science, Texas Woman’s University, Denton, TX;bDepartment of Mathematics and Statistics, Texas A & M University - Corpus Christi, Corpus

Christi, TX

(Received 15 December 2011)

Learning to write formal mathematical proofs presents a major challenge to undergrad-uates. Students who have succeeded in algorithm-intensive courses such as calculusoften find the abstract logic and nonprocedural nature of proof writing to be technicallydifficult, ambiguous and filled with potential errors and misconceptions. This mixed-methods study examines 23 undergraduate students’ attempts to write one-to-one andonto proofs in an introductory abstract algebra course. Data collected consisted of sixrounds of assessments on one-to-one and onto proofs, including homework, quizzes andexams. Using an existing framework of undergraduate proof writing, the researchersfound that students’ misconceptions and errors varied substantially by student and task,with one-to-one proofs presenting unique challenges. Implications for teaching and re-search include emphasis on the logic of proof approaches and providing structured proofframeworks to assist undergraduates with the procedural and conceptual challenges inlearning to write proofs.

Keywords: abstract algebra; proof; one-to-one; onto; undergraduate mathematics;teacher education

1. Introduction

‘Proof provides a means for explaining why mathematics works and for conveying knowl-edge from one person (or generation) to the next’ [1,p.440]. Promoting the instructionof mathematical proof and its historical functions of explaining and justifying claims isa core value of the international mathematics community [2]. For example, the NationalCouncil of Teachers of Mathematics [3] considers mathematical reasoning and proof to beone of the five essential mathematical processes in school mathematics, and recommendsthat teachers should stress proof concepts in all mathematics classes. Consequently, thefoundations of informal and formal mathematical proof ‘permeate the whole mathemat-ics curriculum, from kindergarten on as well throughout the historical development ofmathematics’ [4,p.3]. Nonetheless, many students at the elementary, secondary and under-graduate levels consistently struggle with reading, interpreting and constructing formal andinformal proofs [5–7], and even many mathematics teachers have limited experience withmathematical proof [1].

In principle, effective proof instruction could help move students beyond traditionaltabular proofs in geometry towards an expanded view of proof as a generative and creative

∗Corresponding author. Email: [email protected]

C© 2013 Taylor & Francis

http://dx.doi.org/10.1080/0020739X.2012.756546

1108 A. Wheeler and J. Champion

process. This learning trajectory is marked by many challenges, however, and researchershave investigated the understanding of mathematical proof in a variety of courses andcontexts, including students’ (a) perceptions of what constitutes proof in mathematics[8,9], (b) performance in writing proofs [7] and (c) textual validation of mathematicalproof [10,11]. Within the area of abstract algebra, Weber [12] contrasted the proof-writingskills of undergraduate and doctoral students, finding doctoral students held a deeperunderstanding and specialized ‘knowledge of the domain’s proof techniques, knowledge ofwhich theorems are important and when they will be useful, and knowledge of when andwhen not to use “syntactic” strategies’ [12,p.111].

Importantly, Selden and Selden’s [13] study of introductory abstract algebra studentsled them to develop a list of misconceptions and errors demonstrated by undergraduatesin abstract algebra. The list catalogued students’ proof errors and misconceptions intoseveral types, including beginning with the conclusion, overextending symbols, holes, andsubstitution with abandon [13]. However, the Seldens’ study lacked data regarding thefrequency or distribution of these errors and misconceptions across abstract algebra topics,limiting the rationale for instructors to take specific interventions to address commonerrors and misconceptions. We addressed this limitation by implementing a modifiedversion of Selden and Selden’s rubric in the specific context of students’ proofs involvingone-to-one (injective) and onto (surjective) mappings in a first-semester abstract algebracourse.

Why focus on one-to-one and onto proofs in an abstract algebra class? Besides beingone of the first topics in the course, one-to-one and onto are important for undergraduatesbecause developing comprehensive knowledge of functions is essential to the study ofadvanced mathematics [14]. Understanding functions as mappings between sets and asobjects with properties rather than as equations or actions has consistently been identifiedas foundational to success in subjects ranging from analysis, linear algebra and grouptheory, to high school algebra and calculus [15]. Once students learn to view functions asobjects with properties, they can begin to understand abstract structures, including sets offunctions, groups of functions, functions which operate on functions and so on. In abstractalgebra, the standard definitions of one-to-one (f : A → B is one-to-one provided ‘for allx �= y in A, f (x) �= f (y)’) and onto (f : A → B is onto provided ‘for all b in B, there issome a in A with f (a) = b’) both establish general properties that can be tested for alltypes of functions, including homomorphisms.

In a first-semester abstract algebra class, proof writing about one-to-one and ontoproperties provides opportunities to gain insight into students’ schemas for functions,direct and indirect proof techniques, sets, logical implications and equations. In thisstudy, as in many introductory abstract algebra and introductory proof courses, proofsof one-to-one and onto properties are among the first and most procedural proof-writingexercises, and errors and misconceptions found in students’ novice attempts at constructingthese proofs can provide early indications of instructional interventions. Moreover,for undergraduate mathematics majors who may also become secondary mathematicsteachers, learning to recognize and justify the properties of one-to-one functions andonto functions through multiple representations (i.e. equations, tables, graphs anddiagrams) is a practical skill for the classroom [3]. With this in mind, we approachedthe study with a research question focused on description of both one-to-one and ontoproofs:

What characterizes students’ attempts to write formal proofs of (a) one-to-one functions and(b) onto functions in a first-semester undergraduate abstract algebra course?

International Journal of Mathematical Education in Science and Technology 1109

2. Methods

2.1. Sample

Following informed consent procedures, 23 undergraduates (16 female, 7 male) enrolled ina three-semester credit-hour introductory undergraduate abstract algebra class volunteeredto participate in the study. Of the 23 participants, 16 were prospective mathematics teachers(12 secondary, 4 middle school), six were non-teaching mathematics majors and one wasa kinesiology major. The class under investigation was the first proof-writing mathematicscourse for 13 of the participants, while six of the remaining 10 students had previouslyfailed or withdrawn from abstract algebra. Twelve of the students were 19–22 years old,8 were 23–30 years old and 3 were over 30 years old. Table 1 lists the participants bypseudonym, age, undergraduate major and previous proof-writing course attempts.

2.2. Instruction in one-to-one and onto proofs

As noted in the description of the sample, abstract algebra is typically the first undergraduateproof-writing course for mathematics students at the research site. The instructor wasaware that many of the students had limited experience in formal proof writing and chosean abstract algebra text [16] that included an introductory chapter with topics that helpstudents ease into proof writing through ideas which are foundational to groups and rings.

Table 1. Background summary of study participants.

LevelPseudonym Age Major Prior proof courses of prior courses

Adrian 19 Math (8-12)Akira 22 Math (8-12) aAbstract Algebra JuniorAriana 25 Math (8-12)Brooke 35 Chemistry/Math Symbolic Logic I and II, Linear

AlgebraJunior

Brenda 23 Math (8-12)Caleb 20 Math (8-12) Abstract Algebra JuniorCole 23 Math (8-12) Discrete Math JuniorEmma 20 Math (8-12)Haley 46 Math (4-8)Hayden 19 Math (8-12)Jasmine 22 Math (4-8) Abstract Algebra JuniorJuan 29 Math Calculus I, Discrete Math Sophomore, JuniorLainey 24 Math Abstract Algebra JuniorMakayla 28 Math/Account. Symbolic Logic, Discrete

Math, Abstract AlgebraJunior

Mallory 36 Math (8-12) Geometry, Matrix Math,Statistics, Abstract Algebra

Junior, Senior

Mia 20 Math (8-12)Michelle 20 Math (8-12)Naomi 21 MathPaul 28 Math (8-12)Roberto 24 Kinesiology Discrete Math JuniorTensia 20 Math (4-8)Tina 19 MathVanessa 20 Math (8-12)

Note: a‘Abstract Algebra’ refers to students who are retaking the class under study.


,

( )

Figure 1. One-to-one and onto problem on quiz.

Through exercises and examples in the initial chapter, class participants read examples andattempted to write formal proofs of basic statements relating to topics such as one-to-one,onto, closure and equivalence relations. Our study tracks the participants’ proof attemptsregarding the first two mathematical concepts through three separate assessments in thecourse.

In the United States, students are often introduced to one-to-one and onto as propertiesof functions during secondary and undergraduate algebra classes, such as Algebra II andCollege Algebra. Instructors in these settings may draw diagrams or graphs to illustrate theproperties, and may state formal definitions, but rarely prove functions are one-to-one oronto using general arguments. Besides College Algebra, students at the research site wereunlikely to have had formal instruction on one-to-one and onto functions prior to enrollingin the introductory abstract algebra course.

During class in the initial weeks of the abstract algebra course, the instructor drewmapping diagrams with action gestures [15] as visual representations of one-to-one andonto. For one-to-one functions, the instructor also drew graphs of functions and discussedthe horizontal-line test (a real-value function on the reals is one-to-one if and only if everyhorizontal line intersects the graph of the function in at most one point) as a way to helpstudents develop insight into whether a function might possess the one-to-one property. Theinstructor modelled several example proofs of one-to-one and onto properties, and suggestedthat students use the contrapositive of the one-to-one criterion (f (x) = f (y) ⇒ x = y)when proving functions are one-to-one. During the initial class discussion, the instructornoted that some students confused the processes of trying to understand the behaviour ofa function and the process of trying to prove the function possessed the one-to-one and/oronto properties.

2.3. Data collection

The data sources included copies of students’ individual work on each of 24 one-to-one andonto proof tasks spread across three assessments: a homework assignment, a quiz and anexam. The homework assignment included 10 one-to-one proofs and 10 onto proofs fromthe course text. The subsequent in-class quiz and exam each included 1 one-to-one and 1onto proof, which were similar in nature to the homework tasks (see Figures 1 and 2).

( )

Figure 2. One-to-one and onto problem on exam.


Table 2. Defined codes for misconceptions and errors in one-to-one and onto proofs.

Error/misconception Example

E1: Misuse of symbols Use = in place of ⇒.E4: Misuse of given information Prove f is not one-to-one, then give a counterexample.eE6: Unintelligible proof No part of the proof is understandable.E7: Unjustifiable substitution Declare a & b even, and use undeclared x & y in the proof.E8: Misuse of quantifiers Conclude for all integers when only true for a subset.E9: Logical holes Omit multiple consecutive steps in a logical argument.E11: Computational error Incorrectly simplify an expression.E12: Undefined variables Use one or more variables without definition.E13: Set membership Fail to verify an inverse element is in the domain.M1: Begin with the conclusion Assume a1 = a2 at the beginning of a one-to-one proof.M5: Confused real number laws Conclude b/2 is not in the integers for b in the integers.M8: Interfering knowledge Model an onto proof after a proof of one-to-one.M9: Proof by example Conclude a function is one-to-one by graphing.No response No visible proof attempt.

Note: Only codes identified in the data are listed; see [13] for additional examples of most codes.

All assessments occurred during the first 6 weeks of the class. We initially also includedstudents’ work on 8 total one-to-one and onto proofs in the context of verifying the propertiesof homomorphisms, but subsequently restricted the sample to the initial 24 proof tasks toallow for better comparisons across students.

2.4. Data analysis

After making copies of student work, a research assistant and the first author coded the datausing an adapted version of Selden and Selden’s [13] rubric of undergraduate students’ proofmisconceptions and errors. Prior to the initial structured coding of students’ proofs usingthe rubric, the researchers discussed the meanings of the codes and made modifications toalign with the topics of the proofs, such as renaming some codes and including new onessuch as M9 (proof by example), E11 (computational error), E12 (undefined variables) andE13 (set membership). Table 2 gives the altered rubric with just those codes that were foundin the student work for one-to-one and onto proofs.

We defined misconceptions to be beliefs evidenced in a student’s proof writing that con-flict with formal mathematics, including context-dependent operational definitions of theSeldens’ errors and misconceptions. These included assuming the conclusion to be logicallytrue at the beginning of the proof (M1), misunderstanding the properties of real numbers(M5), using prior knowledge of similar but interfering content as proof (M8) and presentingone or more examples (verified instances of the general statement) as a proof (M9). Besidesmisconceptions, students’ one-to-one and onto proofs included some mathematical errors,coded as: incorrect mathematical symbols (E1), misunderstanding what must be shown inthe proof (E4), writing a grammatically nonsensical argument (E6), substituting quantitiesfor one another without justification (E7), misuse of quantifiers (E8), skipping one or morelogical steps in a sequence of statements (E9), using mathematical ideas that are not relevantto the argument (E10), making computational errors with symbols or numbers (E11), notdefining variables used in a proof (E12) and failing to verify membership in a set (E13).

Analysis consisted of three rounds of coding for each proof exercise. The researchassistant first coded each student’s proof attempt for the problem. Then, the first authorchecked the codes and suggested any changes to the coded misconceptions or errors.


Finally, the two coders reached consensus and the overall codes were aggregated with anemphasis on both describing and exemplifying the apparent challenges that students facedwhen writing proofs of one-to-one and onto properties.

3. Findings

Students left about one in nine (11.1%) of the 552 total one-to-one and onto proof tasksblank (no response). In addition, the most common error or misconception identified in theproof attempts was using one or more undefined variables in a proof (E12), which occurredin 9.4% of the proof attempts. Other less common errors included E4: misuse of giveninformation (4.2%), E13: failing to verify set membership (3.1%) and E9: logical holes(2.4%). The percentages of errors and misconceptions identified in the students’ proofattempts for one-to-one (Table 3) and onto (Table 4) tasks suggests somewhat increasedprevalence of the error and misconception patterns on exams and quizzes as compared withhomework.

3.1. Qualitative nature of errors and misconceptions in one-to-one proofs

Through the homework, quiz and exam, students had 12 opportunities to write a formalproof of whether or not a given function was one-to-one. Based on the analysis of theseproofs using the modified misconception and errors rubric, students struggled the most with(a) using one or more variables without definition, (b) understanding how to utilize giveninformation to write a correct mathematical proof and (c) distinguishing among differentproof formats.

An interesting example of a proof misconception included Jasmine’s work. Jasminetried plotting points both as a way to determine if a function was one-to-one and as a way ofproving her choice. One proof submitted on Jasmine’s first homework assignment, whichincluded one-to-one proofs, involved proving whether the function f : Z → Z, given byf(x) = 3x, was one-to-one. Jasmine’s proof attempt included only a table of values for f

Table 3. Percentages of errors and misconceptions in students’ proofs of one-to-one.

Error/ Homework Quiz Exam Combinedmisconception (10 tasks) (1 task) (1 task) (12 tasks)

E1: Misuse of symbols 0.9 0.0 8.7 1.4E4: Misuse of given information 0.4 8.7 39.1 4.3E6: Unintelligible proof 1.7 4.3 0.0 1.8E7: Unjustifiable substitution 0.0 4.3 0.0 0.4E8: Misuse of quantifiers 0.0 0.0 21.7 1.8E9: Logical holes 0.0 4.3 4.3 0.7E11: Computational error 0.4 0.0 0.0 0.4E12: Undefined variables 7.0 21.7 43.5 11.2E13: Set membership 0.0 0.0 0.0 0.0M1: Begin with the conclusion 0.0 0.0 8.7 0.7M8: Interfering knowledge 1.7 8.7 4.3 2.5M9: Proof by example 2.2 4.3 0.0 2.2No response 9.6 0.0 13.0 9.1

Note: Percentages of errors and misconceptions = 100∗(number of errors)/[(sample size)∗(number of tasks)].Example: the number of E1 errors on the homework was 2, so the E1 error rate on the homework was100∗(2)/(23∗10) = 0.9, meaning error E1 was present in less than 1% of the homework proofs.


Table 4. Percentages of errors and misconceptions in students’ proofs of onto.

Error/ Homework Quiz Exam Combinedmisconception (10 tasks) (1 task) (1 task) (12 tasks)

E1: Misuse of symbols 0.9 4.3 13.0 2.2E4: Misuse of given information 1.3 0.0 34.8 4.0E6: Unintelligible proof 0.9 4.3 8.7 1.8E7: Unjustifiable substitution 0.0 8.7 0.0 0.7E8: Misuse of quantifiers 1.7 0.0 4.3 1.8E9: Logical holes 2.2 4.3 21.7 4.0E12: Undefined variables 6.5 13.0 13.0 7.6E13: Set membership 2.6 4.3 43.5 6.2M1: Begin with the conclusion 0.0 0.0 4.3 0.4M5: Confused real number laws 0.4 0.0 0.0 0.4M8: Interfering knowledge 0.9 0.0 8.7 1.4M9: Proof by example 0.4 0.0 0.0 0.4No response 13.5 0.0 21.7 13.0

Note: Percentages of errors and misconceptions = 100∗(number of errors)/[(sample size)∗(number of tasks)].Example: the number of E1 errors on the homework was 2, so the E1 error rate on the homework was100∗(2)/(23∗10) = 0.9, meaning error E1 was present in less than 1% of the homework proofs.

at integer values of x between –2 and 2, along with the phrase ‘one-to-one mapping’ (seeFigure 3). Jasmine’s work consistently suggested a belief that a table of values for a functionover a finite subset of the domain was sufficient to prove that the function is one-to-oneover its entire domain.

While Jasmine consistently struggled in early work on writing one-to-one proofs usingexamples, other students seemed to be able to mostly write correct one-to-one argumentsbut lacked true understanding of their work. For example, the first part of Mallory’s proofon a quiz that f (x) = 10x − 7 is one-to-one on the integers (see Figure 4) demonstrates acorrect symbolic proof that a function was one-to-one. The remaining portion of her work,however, shows that Mallory had not developed a strong conceptual understanding of theprocedural steps in her proof. Having written a correct direct argument that f (a1) = f (a2)implies a1 = a2, Mallory chose values a1 = 3 and a1 = 7 and rewrote the general argumentwith these specific values. This led to an apparent contradiction (23 = 63), leading Malloryto erroneously conclude that f was in fact not one-to-one.

3.2. Qualitative nature of errors and misconceptions in onto proofs

Students made more mistakes with proofs involving the onto property of functions thanwith one-to-one proof writing. The most common error was students’ failures to define

Figure 3. One-to-one proof for f : Z → Z, f (x) = 3x, by Jasmine on homework. Error: M9.


Figure 4. One-to-one proof for f : Z → Z, f (x) = 10x − 7, by Mallory on quiz. Errors: E4, E12.

variables. For example, Akira often forgot to include parts of proofs, such as defining newvariables and checking to see whether or not an element found by reversing a mappingbelonged to the domain of the function. In a proof from an early quiz (Figure 5), Akiraintroduced a variable b ∈ B to presumably be in the co-domain of the function f : Z → Z,given by f (x) = 10x − 7. She assumed that the value was of the form b = 10a − 7 andfound a = b+7

10 . Akira then noted that this element was not an integer and concluded thatthe function was not onto. However, this statement is sometimes false (e.g. for b = 3,a = 1 ∈ Z), and Akira neglected to make the necessary qualification. However, Akira’sstatement need only be true for one element of the co-domain to prove the function isnot onto, so Akira’s proof can be considered a ‘partially correct’ proof, with a gap thatunderscores the difficulty she was having in learning the logic of proving the negative of auniversally quantified statement.

Jasmine’s logic was typical of a student who struggled with proofs of both one-to-oneand onto properties throughout the course. For example, in her attempted proof that thefunction f : Q → Q, given by f (x) = x+5

7 , is onto (left panel of Figure 6), Jasmine actuallysubmitted a mostly correct and complete proof that f is one-to-one. She left the onto exerciseof an isomorphism proof on the final exam completely blank.

4. Discussion

In the light of the many complex settings in which students learn abstract algebra at post-secondary institutions, it is important to interpret the study findings in the context of therelatively small sample size, limited duration, and specific instructional and curricularapproach. The study findings, though adding substantially to the work of Selden and Selden[13], by providing quantitative and qualitative descriptions of errors and misconceptionsevidenced in one-to-one and onto proofs, should be considered preliminary. Nonetheless,the findings suggest that through extensive practice, most undergraduate abstract algebrastudents can write one-to-one and onto proofs during the first 6 weeks of the course with few

Figure 5. Onto proof for f : Z → Z, f (x) = 10x − 7, by Akira on quiz. Error: E12, E13.


Figure 6. Onto proof by Jasmine for f : Q → Q, f (x) = x+57 on exam. Misconceptions: M8.

errors. Moreover, the qualitative nature of the students’ errors and misconceptions in theproof attempts provided several insights into their understanding of functions, computationand deductive reasoning.

As proof-based course instructors know, some proof errors are more harmful for stu-dents’ later progress in the course than others. Several students, including Jasmine andMallory, struggled especially with writing logically consistent one-to-one proofs. Sinceone-to-one proofs at this level often include proving a contrapositive statement, we notedthat these students seemed to have difficulty understanding that the contrapositive is logi-cally equivalent to the original conditional statement, a difficulty that is supported by thefindings of Stylianides et al. [17]. Moreover, based on our qualitative analysis of students’submitted proofs, proofs involving the onto property of functions appeared to be less chal-lenging for students, who sometimes omitted certain elements of their proofs but typicallyunderstood what they were being asked to prove, started the proof correctly and includedother key features of a mathematically consistent argument.

Finally, despite the relatively low numbers of proof errors during the initial instructionalperiod on one-to-one and onto, the students in the study had substantial difficulty inimplementing the basic one-to-one and onto proof frameworks when later asked to verifythe one-to-one and onto properties of a homomorphism. This suggests that many of thestudents took the approach of memorizing a set of procedures for writing one-to-one andonto proofs, which may have led to the difficulty in adapting the procedures to new andmore abstract contexts later in the course.

4.1. Implications for teaching

Based on our findings, we believe abstract algebra instructors can implement teachingtechniques to head off potentially detrimental student errors in proof writing by highlightingproofs involving one-to-one and onto properties of functions in the early stages of the course.For example, some students initially believed that proof by example and proof by graphconstituted convincing mathematical arguments that a function was one-to-one or onto.Instructors can choose examples to help students see the need for formal symbolic proofs,and choose homework and quiz items that specifically ask students to prove one-to-oneand onto properties of functions in a variety of contexts and representations. In addition,we recommend paying special attention to outlining a framework for one-to-one and ontoproofs (e.g. ‘Suppose x, y in A with f (x) = f (y), then show that x must equal y usingthe definition of f and properties of the set A’), while highlighting that the particulars ofindividual proofs may vary greatly. Last, several of the error patterns suggest that abstractalgebra students could benefit from carefully editing their proofs. Providing students with


opportunities to practice reading and editing their own proofs could help students developimportant metacognition and proof validation skills [7]. One strategy that might serve thispurpose would be to assign students the task of reading and editing some of their ownrecently submitted proofs using a list of errors provided by the instructor.

4.2. Implications for future research

In follow-up studies, we plan to investigate the role that a student’s career plans (e.g.secondary teaching, graduate school, etc.) plays in the student’s development of proof-writing skills. We also plan to test for an effect on proof writing as a result of requiringstudents to understand and use a list of common misconceptions and errors as a self-analysistool for their own proofs.

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2013 - (Wheeler+Champion) Students’ proofs of 1-to-1 N onto properties in introd abstract algebra