# «PROCEEDINGS OF THE 15TH ANNUAL CONFERENCE ON RESEARCH IN UNDERGRADUATE MATHEMATICS EDUCATION EDITORS STACY BROWN SEAN LARSEN KAREN MARRONGELLE ...»

Interestingly enough, nearly 75% (75 out of 101) of students surveyed are deciding that they are going into a mathematics-based STEM field before or during their freshman year at the university. This follows other research findings about the significance of the first year of college to students’ selection of mathematics-based majors (Muller & Pavone, 1998). This tells us that the window of opportunity to recruit majors into mathematics-based STEM fields begins even before they enter the university system. This also means that institutions need to develop ways to understand how women and other underrepresented groups experience the mathematics classroom in order to develop a receptive climate that encourages their success in the classroom.

What do you feel has contributed most to your success?

Overall, the majority of students—both men and women—attributed their success to their personal drive and ability (males (42): 49.4%; females (6): 37.5%) and to their classmates (males (20): 23.5%; females (4): 25%). Parental support (males (1):.03%; females (0): 0%) and enjoyment of material (males (0): 0%; females (1):.06%) were the lowest contributing factors selected by the students. More details, including comments from students, will be included in the final version of this paper.

Educational significance Our goal in this paper was to provide insights on how professors can better serve underrepresented groups in the mathematics-based STEM disciplines to be successful and have a pleasant experience in their mathematics courses. The results from our research once again illustrate the pronounced lack of representation of particular groups (e.g., females and non-white students) in disciplines that require a strong mathematics background. However, these data tell 460 15TH Annual Conference on Research in Undergraduate Mathematics Education us that the students who are successful in mathematics are selecting their majors either very early on in their college careers or before they enter college. This should be a call to educators to communicate with high school teachers and people who are teaching algebra courses at the university level.

Full analysis of the data will also help professors be cognizant of how they can help to develop a community and culture that supports women and underrepresented groups. This study provides a starting point for discussion and a call for additional research on building an institutional environment that fosters women’s success and values their presence in STEM disciplines. We will pose the following questions to the audience to push the research beyond the preliminary stage.

1. What ways do you know of that institutions understand how women and other underrepresented groups experience the mathematics classroom?

2. What ways do you know of that institutions develop a receptive climate that encourages their success in the classroom?

3. What would be interesting to come out of the analysis of the open-ended questions?

With this discussion, we hope to gain thoughtful insights to strengthen our research, data analysis, and further data collection so that we can disseminate quality research to the mathematics/mathematics education community.

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15TH Annual Conference on Research in Undergraduate Mathematics Education 461 Pattatucci, A. (1998). Women in science: Meeting career challenges. Thousand Oaks: Sage Publications.

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## Abstract

Students face an array of difficulties when they learn to understand and write proofs by mathematical induction (MI). This paper describes the responses of students in an inquiry-based (IBL) number theory course when presented with a false proof by MI asserting that all humans are the same height. The proof schemes of Harel, Sowder and others provided a lens through which to analyze student responses. Some were consistent with the misconceptions already in the literature on MI, while others may be especially revealing of IBL students’ ways of understanding mathematical proof in general and MI in particular.** Keywords: Inquiry-Based Learning, Mathematical Induction, Proof Schemes**

1 Introduction Undergraduate mathematics students experience a variety of challenges learning to understand and construct proofs by Mathematical Induction (MI). Ernest (1984) reported that many view the basis step as unnecessary. Avital and Libeskind (1978) and other authors have noted students’ serious difficulties with the logical complexity of the inductive step due to the universally-quantified implication ∀k ≥ 1 [P (k) → P (k + 1)], where one wishes to prove the statement P (n) for all natural numbers n. Baker (1996) and others have described how many students focus on form over substance when writing and reading proofs by MI, and Harel (2002) has described how students accept the MI procedure as a rule handed down by an authority – a textbook author or an instructor – without developing an understanding of why MI constitutes valid reasoning and when it should be used.

An excellent way to assess a student’s beliefs about proof is to ask him or her to critique a proof attempt. Some textbooks offer students an opportunity to critique a purported proof by MI of a silly and clearly false statement. Pólya (1954) offered an early example in Mathematics and Plausible Reasoning, suggesting a “proof” that all ladies have the same color eyes. Brumfiel (1974) reported that in a university honors calculus class, none of the students readily identified the error in a similar argument asserting that all billiard balls are the same color.

What can be learned from undergraduate students’ written reactions to a Pólya-style argument? We propose to analyze the work of students in an Inquiry-Based (IBL) Number Theory course critiquing a false proof that all humans are the same height. Please see Figure 1.

2 Theoretical Framework and Methodology The proof analysis task in Figure 1 was given to 27 students in a Number Theory class at a commuter university in a working class, urban/suburban area in California. The majority of undergraduates at this university are first-generation college students, and the main ethnic subgroups in 2009 (when the task was assigned) were Hispanic 40%, White 28% and African American 11%. At this university, the Number Theory course functions as a transition to the upper division mathematics curriculum.

The instructor used IBL (also known as the Modified Moore Method) along with the text by Marshall et al. (2007).

Students had spent the first class meeting exploring MI problems from their text. At subsequent class meetings students

The group on the left contains k humans, therefore all in that group have the in thatheight. have the the k humans in the right The group on the left contains k humans, therefore all same group Similarly, group also must have the same height. All of them have the samethe right group− 1 humans who belong to both groups. Therefore, same height. Similarly, the k humans in height as the k also must all have the same height. All of them have the same height as the k-1 humans who belong in any group of k + 1 humans, groups. Therefore, in any group of k+1 humans, all must have the have the same height.

to both all must have the same height. By induction, this proves that all humans same height. By induction, this proves that all humans are the same height.

You know that something must be wrong here. After all, the conclusion is false! Write a brief letter to the editor of Amazing Induction!!! explainingYou know that something must be wrongspecific as possible. conclusion is false!to look at examples to see what is wrong with this argument. Be as here. After all, the [Hint: it may help Write a brief letter to the editor of Amazing Induction!!! explaining what is how this argument works for specific numbers of people.] wrong with this argument. Be as specific as possible.

2. (Personal reflection) Write one or two paragraphs containing your thoughts about the mathematics being1: Proofin Math 345. Is there something that Figure learned Analysis Task you learned that was a big breakthrough? Are you having difficulty with a concept? Is there something that interests you about the mathematics we solutions andlearned up in whole-class discussion of you concern? Explain. instructor.

have engaged till now? Is something causing MI facilitated by the presented their All students had Junior or Senior standing and just over half were mathematics majors.

Students were required to write their own critique of the Pólya-style argument. This assignment was graded for completion (as were similar short writing assignments given throughout the course). Written responses were collected and coded for common responses.

The work of Harel and Sowder (1998) on proof schemes provided a theoretical framework for understanding student responses. A student’s proof scheme describes what that student tends to find convincing in mathematical argumentation. Students with faulty conceptions of proof may have an external proof scheme – taking the instructor’s authority or surface features of an argument (such as the two-column format or the use of algebraic symbolism) as sources of validity. Studies by Harel with Sowder and others describe students moving through empirical schemes – in which a general statement is “proved” if it is seen to be true for a particular figure or for several example cases – towards deductive schemes as students gain mathematical maturity. Within the category of deductive schemes, students may possess a transformational scheme – where one freely manipulates a generic expression or figure to explain why a statement holds in all cases – or an axiomatic scheme approaching a modern mathematician’s concept of proof.

Student responses shed light on their ways of understanding mathematical proof in general and MI in particular. A few responses may be especially revealing of IBL students’ ways of understanding.

3 Preliminary Results and Discussion Fifteen students consented to have their work considered for this study, and this sample appeared to give a faithful representation of the entire class. Their responses are summarized in Table 1. Some students incorporated several coded responses in their work, so the total in the Frequency column exceeds 15.

The flaw in Figure 1 (and all similar arguments) lies in the implicit assumption that k is a generic large number (at least 2) in the inductive step, whereas the basis step can only establish the trivial case k = 1. The picture shows k greater than or equal to 3, with some unspecified number of people to the right of the 2nd human and to the left of the

k th. But when k = 1, we have k − 1 = 0 and the two sets of equal-height humans have an empty intersection. This argument cannot show that any two humans are the same height. We discuss some particularly interesting responses below.

Response B: Basis Step and Inductive Step Don’t Work Together 3.1 A remarkably sophisticated critique that showed a mature understanding of the relationship between the two steps in a proof by MI was offered by two students who appeared to possess a modern axiomatic proof scheme. The student

**response excerpted below is notable for its reliance on the properties of equivalence relations:**

As I was trying to understand how the proof worked, I realized that the base case did not give enough ground to work with. Proving the base case works for a group of one only shows a reflexive property: that a person is identical to his or herself. What is needed and what is being used in the rest of the proof is a relation between two or more people: that they have the same height. If we were to show that any two people have the same height, we could see how a larger group of people would have to have the same height. [ …] We could follow the pattern forever and prove that it would work for groups of k + 1. Of course, proving that any two people have the same height would not be possible because plenty of counterexamples could be given.

Response C: The Basis Step is Trivial 3.2 C responses suggested that because the case n = 1 is vacuous it is illegitimate, or even that a comparison can only be made between two distinct objects.

The problem with this proof is the base case. You can’t say for k = 1 compare your height to your own.

k = 1 should have been a comparison of at least two humans because this is a comparison proof.