### by Amelie G. Schinck-Mikel, Cal Poly, San Luis Obispo

**Abstract**

*This literature summary begins by discussing the various incarnations of Inquiry-Based Learning (IBL), elaborating on various modifications as well as providing links to resources. The often discussed coverage issue and the costs of choosing to teach in a lecture-intensive approach are also explained. Inquiry-Based Learning is offered as an alternative to the traditional lecture-based approach. IBL allows students to not only cover, but engage with the material in a more stimulating way. Strong evidence from the literature is presented in favor of using Inquiry-Based Learning in the teaching of proof. Furthermore, this summary highlights the positive effect between an inquiry approach to teaching and overall better acquisition and retention of conceptual understanding as well as improved enjoyment of the subject for the students and the teacher.*

Inquiry-Based Learning (IBL) is not a recent or passing movement in mathematics education. IBL is based on a wide body of research and has a long track record of success. The following is an outline of IBL’s theoretical background and empirical grounding.

At the university level, IBL is also known as the Modified Moore Method (MMM), named after professor R. L. Moore of the University of Texas. In the majority of undergraduate mathematics classrooms across the nation, “doing mathematics means following the rules laid down by the teacher; knowing mathematics means remembering and applying the correct rules when the teacher asks a question; and mathematical truth is determined when the answer is ratified by the teacher” (Lampert, 1988, p.437). Moore aimed to challenge students’ assumptions about what it is to do, know, and understand mathematics. Beginning in the 1920’s, and continuing for half a century, Moore taught collegiate mathematics through inquiry, challenging his students to think like mathematicians. Moore believed: “That student is taught the best who is told the least” (Parker, 2005, p.vii). Through a sequence of carefully crafted problems and theorems, Moore would have students pose conjectures, construct their own proofs, justify their reasoning to their peers at the board, and assess the validity of proposed solutions and proofs. Textbooks were generally not used. Lectures were kept to a minimum. Collaboration between classmates was strictly prohibited. For a biography of R. L. Moore, and an account of the origin and impact of the Moore Method, see Zitarelli (2004) and Whyburn (1970). For more information on R. L. Moore, also see http://legacyrlmoore.org.

The Modified Moore Method is a less strict version of Moore’s approach to the teaching and learning of mathematics. For instance, MMM courses may make use of an IBL inspired textbook. Varying degrees of importance can be placed on formal examinations. Student collaboration is sometimes encouraged, with solutions to problems shared during small-group and/or whole-group discussions. For descriptions of different modifications and their rationale, see Chalice (1995), Mahavier (1999), Padraig & McLoughlin, (2008), and Yoshinobu & Jones, 2012). For some examples of IBL textbooks, see Burger & Starbird (2005), Hale (2003), Schumacher (1995) and Starbird, Marshall & Odell (2007). For refereed, IBL classroom tested course notes for university level mathematics classes, visit the website for the Journal of Inquiry-Based Learning in Mathematics (www.jiblm.org).

Students are thus engaged in the creation of mathematics, allowing them to see mathematics as a part of human activity, not apart from it. MMM courses are in direct contrast to the traditional lecture-based approach to the teaching of mathematics. Reporting on his use of MMM, Chalice (1995) stated:

*While using this method, I have been able to cover as much material (and in few cases more material) as in the usual lecture-style course. More importantly, with the Modified Moore Method, the students and I have covered that material in a far more enlivening, enjoyable, and intellectually stimulating way (p.317).*

Yoshinobu & Jones (2012) tackle the coverage issue in undergraduate mathematics classes, that is “the set of difficulties that arise in attempting to cover a lengthy list of topics” (p.303), connecting the coverage issue with how teachers choose to teach. The perceived need to cover all the material in the syllabus encourages the use of a traditional, lecture-intensive approach to teaching. This choice, the authors argue, comes at a price: the development of beliefs by students that do not support the learning of mathematics, the inability of students to prove simple statements, the powerlessness of students in the face of non-routine problems, and the eventual disillusionment and departure of a significant number of students from the major. The article offers IBL as an alternative way of teaching that positively effects what Kilpatrick, Swafford, & Findell (2001,p.16) called the five components of mathematical proficiency: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition. With IBL, coverage is measured differently, not by the sheer amount of topics discussed, but by the time students are actually doing mathematics.

An inquiry-based approach was recommended by the National Science Foundation in their 1996 report of a year-long review of the state of undergraduate Science, Mathematics, Engineering and Technology (SME&T) education in the United States entitled Shaping the Future (NSF, 1996). In this report, the researchers stated that it is imperative that:

*All students have access to supportive, excellent undergraduate education in science, mathematics, engineering, and technology, and all students learn these subjects by direct experience with the methods and processes of inquiry (NSF, 1996, p.6).*

The IBL movement found in undergraduate mathematics, and supported by the Academy of Inquiry-Based Learning (AIBL), is in line with, and a natural extension of, the reform efforts in grades K-12. Recommendations by the National Council of Teachers of Mathematics (NCTM) for the past three decades (NCTM, 1980, 1989, 2000) and now the Common Core State Standards for Mathematics have consistently included a call for a focus on problem solving by teachers, positioning problem solving ability as the overarching goal of mathematics education. These recommendations are founded on the notion that the learning of mathematics is an active, social process in which students construct new ideas or concepts based on their current knowledge. Student understanding is connected to open-ended questions and an inductive teaching style. Principles and Standards for School Mathematics (NCTM, 2000) emphasizes the need for teachers to create a culture of learning in their classroom in which students learn with understanding and construct conceptual mathematical meaning through a problem-solving approach:

*Problem solving means engaging in a task for which the solution is not known in advance. In order to find a solution, students must draw on their knowledge, and through this process, they will often develop new mathematical understanding. Solving problems is not only a goal of learning mathematics but also a major means of doing so. (NCTM, 2000, p.51)*

Discussing the importance of fostering Reasoning and Proof in Grades 9-12, Principles and Standards for School Mathematics (NCTM, 2000) states:

*As in other grades, teachers of mathematics in high school should strive to create a climate of discussing, questioning, and listening in their classes. Teachers should expect their students to seek, formulate, and critique explanations so that classes become communities of inquiry (p.346).*

To sustain and support the recommended focus on problem solving, active learning and inquiry in grades K-12, undergraduate mathematics education must also change, especially in the area of teacher preparation (NSF, 1996).

As mathematics education researchers turn their attention to IBL, evidence is mounting that this approach to the teaching of mathematics is ideal for the teaching of proof (e.g. Smith, 2005; Dhaler, 2008). Despite the emphasis on proof in higher level undergraduate mathematics courses, research on students’ conception of proof consistently shows that most struggle with appreciating, understanding and producing mathematical proof (Dreyfus, 1999; Harel & Sowder, 1998; Jones, 2000; Selden & Selden, 1987, 2003; Weber, 2001). Many mathematics educators argue that students’ (mis)conceptions about proof are the inevitable result of the traditional, lecture-based approach to the teaching of proof (Dreyfus, 1999; Harel & Sowder, 1998; Jones, 2000; NCRTL, 1993; Shoenfeld, 1988; Silver, 1994; Smith, 2005; Weber, 2003). In his article Why Johnny can’t prove, Dreyfus (1999) noted that “the ability to prove depends on forms of knowledge to which students are rarely if ever exposed” (p.85). Dreyfus (1999) concluded that a classroom environment in which students are required to explain and justify their reasoning is key to helping students transition from a computational view of mathematics to a view that conceives of mathematics as a field of intricately related structures.

Smith (2005) reports on the results of an exploratory study of the perceptions of mathematical proof and strategies for constructing proof of undergraduate students enrolled in lecture-based and problem-based (MMM) “transition to proof/number theory” course. Smith (2005) found evidence that the problem-based approach provided students with more opportunities to make sense of the proof construction process in a personally meaningful way than the lecture-based approach. Smith (2005) noted marked differences between the two groups of students. For instance, students in the lecture-based course focused on the form of the proof rather than on its meaning, and were reluctant to work concrete examples. On the other hand, students in the MMM course emphasized meaning over surface features, introduced notation in the sense-making process, conjured up previous proof strategies on the basis of the concept under study, and made use of concrete examples to gain insight into the main idea. Based on these results and the preliminary analysis of other collected data, Smith (2005) hypothesized that classroom communities of inquiry (such as MMM) encourage students to produce proofs by making global or intuitive observations about the mathematical concepts and transform these observations into formal, deductive reasoning.

In a dissertation study on the effects of the Modified Moore Method on college students’ concept of proof, Dhaler (2008) found that MMM had a positive effect on student’s conceptualization of mathematical proof, as well as on self-confidence in their abilities, their appreciation of the relevance of proof, and their ability to be independent thinkers.

An inquiry approach to teaching has also been shown to have a positive effect on students’ acquisition and retention of conceptual understanding. At the K-12 level, Boaler (1998), for instance, showed that students who learned mathematics in an open, project-based approach developed superior conceptual understanding to their counterparts who had learned the same subject matter through a traditional, textbook approach. A central part of Boaler’s study was to compare students’ capacity to use their mathematical knowledge in new and unusual situations. Boaler (1998) found that students who had been taught in the traditional way “did not think it was appropriate to try to think about what to do; they thought they had to remember a rule or method they had used in a situation that was similar” (p.47). On the other hand, students who had been taught in an open approach could use mathematics in novel situations as they had developed the belief that mathematics required active, flexible thought. Furthermore, they had gained the capability to adapt strategies and methods depending on the situation. Though the project-based approach described in Boaler (1998) is not identical to the inquiry-based learning approach sponsored by AIBL, the implications of Boaler’s study remain ; conceptual understanding is improved when learn mathematics by engaging in inquiry.

At the undergraduate level, Rasmussen & Kwon (2007) provides a summary of two quantitative studies that assessed the effectiveness on student learning of an inquiry-based approach to the teaching of differential equations (as part of the Inquiry Oriented Differential Equations (IO-DE) project.) Rasmussen, Kwon, Allen, Marrongelle, and Burtch (2006) compared students that had taken inquiry-oriented differential equations (IO-DE) classes versus students that had been taught using a traditional approach. Rasmussen et al. (2006) found that although the two groups did not show a significant difference in procedural fluency (i.e. routine problems), the IO-DE group scored significantly higher on conceptual problems.

In a follow-up study one year later, Kwon, Rasmussen, and Allen (2005) compared the retention effect on procedural and conceptual understanding between the traditional and IO-DE group. The data showed no significant difference between the two groups in procedural fluency. However, the IO-DE group showed a significant positive difference compared to their traditional counterpart on conceptual understanding.

The IO-DE project described above uses an adaptation of Realistic Mathematics Education (RME), an inquiry approach to the teaching of K-12 mathematics in the Netherlands. RME is based on curriculum developed at the Freudenthal Institute. Through the posing of true problematic situations (not simply “word problems”), RME encourages student investigation and inquiry. Students’ construction and representation of mathematical concepts such as number sense is valued. The book series Young Mathematicians at Work by Fosnot and Dolk outline the translation of the Dutch approach to the teaching of mathematics to numerous American urban classrooms.

Lauren, Hassi, Kogan, Hunter and Weston (2011) performed a comprehensive, mixed-methods study of IBL in four university mathematics departments that host IBL Mathematics Centers supported by the Educational Advancement Foundation. Courses included in the study were varied, ranging from courses for pre-service K-12 teachers to more advanced courses for mathematics majors such as number theory, analysis, discrete math and geometry. The analysis of hundreds of hours of classroom observations, approximately 1000 surveys and 110 interviews with students, faculty and teaching assistants yielded many interesting results, some of which will be discussed here. For the full report, visit:

http://www.colorado.edu/eer/research/documents/IBLmathReportALL_050211.pdf

On surveys and interviews, math majors who took IBL classes reported greater cognitive gains, improved thinking and problem solving capabilities and persistence, enhanced ability to write and speak about proofs, and better confidence and attitude than their peers who took non-IBL sections of the same courses. Pre-service teachers reported gains in areas that are significant for their future careers as teachers such as more comfort in teaching mathematical ideas to others and collaboration. In general, students emphasized “the deep and lasting learning that came from working through ideas for themselves” (p.v) and the increased enjoyment and interest in mathematics gained from an IBL math class. Focusing on academic performance, the researchers stated that:

Overall, it appeared that non-IBL courses tended to reinforce prior achievement patterns, helpingthe “rich” to get “richer.” In contrast, IBL courses seemed to offer an extra boost to lower achieving students, especially among pre-service teachers. Yet there was no evidence of harm done to the strongest students. Indeed, high-achieving students may be encouraged by an IBL experience to take more mathematics courses, especially more IBL courses (p.vii).

In a recent paper in the Proceedings of the National Academy of Sciences, Freeman, Eddy, McDonough, Smith, Okoroafor, Jordt, & Wenderoth (2014) reported the findings of a meaanalysis of 225 studies which compared student performance in undergraduate STEM courses taught through traditional lecturing versus active learning on two outcome variables: (1) scores on identical or equivalent exams, concept inventories or other assessments; and (2) failure rates (DFW rates). For the purposes of their analysis, the researchers included studies with varying degrees of active learning interventions, from occasional problem solving in groups to studio or workshop settings. The results indicate that active learning improves examination performance by approximately 6%. In addition, lecturing was found to increase failure rates by 55% (approximately 22% for active learning vs. 34 % for traditional sections). The authors found that these results held even when comparing courses for majors versus non-majors or introductory versus upper-division courses. A press release in May 2014 by The National Science Foundation entitled Enough with the Lecturing applies the failure rates for active learning and traditional sections to the 7 million U.S. undergraduates who have stated that they would like to pursue studies in STEM fields. The conclusion is that 840,000 additional students fail under lecturing and that changing to an active learning approach would go a long way to answering the call by the President’s Council of Advisors on Science and Technology for a million more STEM majors in the future.

The famous Swiss psychologist Jean Piaget stated: “to understand is to invent”, highlighting the active nature of the learner. The above discussion provides an outline of the theoretical foundation on which Inquiry-Based Learning rests. Furthermore, it provides a summary of the mounting evidence that students who are given the opportunity to learn mathematics through inquiry develop better strategic competence and deeper conceptual understanding of mathematics.

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