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.
