**Special Note: The AMS Blog On Teaching and Learning Mathematics has started a six-part series on active learning.**

Over the past decade, the Educational Advancement Foundation has supported programs to promote Inquiry-Based Learning (IBL) in mathematics at four major universities. IBL is not a curriculum. Rather, it is a guiding philosophy for instruction that takes a structured approach to active learning, directing student activities and projects toward building a fluent and comprehensive understanding of the central concepts of the course. Ethnography & Evaluation Research (E&ER) at the University of Colorado, Boulder has studied the effectiveness of these implementations. Several research papers have resulted, of which the paper by Kogan and Laursen (2014), discussed in my column Evidence of Improved Teaching (October 2013), presented very clear evidence that IBL prepares students for subsequent courses better than standard instruction and that IBL can result in students taking more mathematics courses, especially when offered early enough in the curriculum. Two recent papers document the benefits of IBL in preparing future teachers and in building personal empowerment.

In

*Implementation and outcomes of inquiry-based learning in mathematics content courses for pre-service teachers*(Laursen, Hassi, and Hough, 2015), the authors focused on the development of Mathematical Knowledge for Teaching (MKT), a term coined by Deborah Ball to describe the kind of knowledge that teachers must draw upon to teach mathematics well and that reflects understanding of how ideas and concepts relate to one another as well as the common difficulties and misunderstandings that students are likely to encounter. Being prepared for teaching requires more than being able to find solutions to particular problems. A good teacher must have at her or his disposal a variety of approaches to a solution and the ability to take a student’s incorrect attempt at an answer, recognize where the misunderstanding lies, and build on what the student does understand.

In theory, IBL should help develop MKT because it focuses on precisely those characteristics of practicing mathematicians that teachers most need, the habits of mind than include sense-making, conjecture, experimentation, creation, and communication.

E&ER studied students in thirteen sections of seven courses for pre-service teachers at two of the four universities, courses that collectively spanned preparation for primary, middle school, and secondary teaching. They used an instrument developed by Ball and colleagues, Learning Mathematics for Teaching (LMT) that has been validated as an effective measure of MKT for practicing teachers. The results were impressive. The students had begun the term with LMT scores that averaged at the mean for in-service teachers across the country. Each of the IBL classes saw mean LMT scores rise by 0.67 to 0.90 standard deviations. In line with the results of the 2014 report, all students experienced gains from IBL, but the weakest students saw the greatest gains.

The second recent article is

*Transforming learning: Personal empowerment in learning mathematics*(Hassi and Laursen, 2015). In

*Adding It Up*(NRC 2001), mathematical proficiency is recognized as consisting of five strands: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition. They are each critically important. This paper investigates the effect of IBL on both strategic competence, what the authors term

*cognitive empowerment*, and productive disposition, which they separate into

*self-empowerment*and

*social empowerment*, the last of which also incorporates effective communication.

The study was conducted through interviews with students who had taken a class at one of the four universities using IBL. An overwhelming majority of students reported gains in each of the three areas of personal empowerment. Among women 77% and among men 69% reported an increase in self-esteem, sense of self-efficacy, and confidence from their IBL experience. For general thinking skills, deep thinking and learning, flexibility, and creativity, 77% of the women and 90% of the men described improvements. For ability to explain and discuss mathematics as well as skills in writing and presenting mathematics, 79% of the women and 76% of the men saw gains.

When pressed for what made the IBL experience special, students identified their own role in influencing the course pace and direction, the importance of combining both individual and collaborative work, and the fact that they were faced with problems that were both challenging and meaningful. They appreciated that they were given responsibility to think on their own. Such experiences were especially important for women and for first-year students.

In the very discouraging reports on the effects of Calculus I instruction in most US universities (Sonnert and Sadler 2015), we see courses that accomplish exactly the opposite of personal empowerment, courses that sharply decrease student confidence and sense of self-efficacy. It does not have to be this way.

References

Hassi, M.-L., and Laursen, S.L. 2015. Transformative learning: Personal empowerment in learning mathematics.

*Journal of Transformative Education*. Published online before print May 24, 2015, doi: 10.1177/1541344615587111.

M. Kogan and S. Laursen. 2014. Assessing long-term effects of inquiry-based learning: A case study from college mathematics.

*Innovative Higher Education*

**39**(3), 183–199. http://link.springer.com/article/10.1007/s10755-013-9269-9

Laursen, S.L., Hassi, M.-L., and Hough, S. 2015. Implementation and outcomes of inquiry-based learning in mathematics content courses for pre-service teachers.

*International Journal of Mathematical Education in Science and Technology*. Published online before print July 25, 2015, doi: 0.1080/0020739X.2015.1068390

National Research Council (NRC). 2001

*Adding it up: Helping children learn mathematics*. J. Kilpatrick, J. Swafford, and B. Findell (Eds.). Washington, DC: National Academy Press.

Sonnert, G. and Sadler, P. 2015. The impact of instructor and institutional factors on students’ attitudes. Pages 17–29 in

*Insights and Recommendations from the MAA National Study of College Calculus*, D. Bressoud, V. Mesa, and C. Rasmussen (Eds.). Washington, DC: Mathematical Association of America Press.