# IMPACT Live!

## IMPACT Plus - The case for active, student-centered, equity focused, inquiry-based learning in college mathematics

Posted on behalf of Stan Yoshinobu

Instructors at two-year colleges teach a large number of students across the nation. Successful math teaching at two-year colleges is essential in student success, participation in STEM careers, and more generally contributing to quantitative and scientific literacy. In addition to academic success and just as important is the lived experiences of students. Students from all backgrounds thrive when equitable teaching practices are used. In this post I provide some highlights of evidence in support of an inquiry-based learning (IBL) framework for teaching math courses. This is not a comprehensive review of the research, and is meant to provide a clear rationale for why we should let math problems and students do more of the talking.

What is IBL? IBL is a broad framework for teaching based on the four pillars (1) student engagement in rich mathematics, (2) regular opportunities for students to collaborate with each other and with their instructors, (3) instructor inquiry into student thinking, and (4) instructor focus on equity. These four pillars are common characteristics across the different variations of IBL teaching (Laursen, Rasmussen 2019).

IBL courses can look different from one another. In one class, students may primarily be working in groups, whereas in another class, students may be doing group work with some time spent on students presenting and evaluating mathematical arguments. These variations depend on the instructor, the needs of the students, class size, the course topic, and other institutional factors.

Across the different versions, an observer would notice the four pillars, and most strikingly that students are doing the sense making. Students are asking questions, sharing ideas, and helping one another with support from their instructor. Mini lectures are used by the instructor in a timely fashion to set the stage, frame a topic, pull together and make public a big idea, offer expert insights, and more.

Let’s now turn to the evidence from education research. What we know about teacher-centered instruction is that math classes are unfortunately not a level playing field. Ellis (Hagman), Fosdick, and Rasmussen (Ellis, Fosdick, Rasmussen 2016) found that women are 1.5 times as likely to leave the STEM pipeline compared to men, where women report significantly lower confidence in math compared to men, driving them out of the pipeline in calculus courses. The authors found that it wasn’t their math knowledge, and mathematically capable students *lost* confidence in math in calculus, suggesting that calculus courses may be pushing women out.

Further, Muis (Muis 2004) found across levels of education that students learn or acquire non-availing beliefs, which are beliefs that do not support or hinder learning. A partial list of non-availing beliefs are listed here.

- Only geniuses can understand math
- All math problems can be done in 5 minutes or less, otherwise there is something wrong with the problem (or the student)
- Math is about getting the answer the way the teacher does it only.
- Only the textbook or teachers can determine what is correct.
- Math is not a creative subject
- The point is to get to the answer quickly, not to understand
- Proof and understanding and doing mathematics are unrelated.

Students holding such beliefs will engage with math in ways that are not productive for their own intellectual development. For instance, a student believing that only geniuses can understand math will be far less likely to try and figure something out for themselves, hence limiting their learning.

In a study done with “good” calculus students, who had done well in calculus. They have the basic skills and knowledge needed to solve some non-routine (to them) problems. Despite having all the necessary ingredients, these students were not able to apply their calculus knowledge in problem-solving situations, relying primarily and unsuccessfully on algebra, which were not enough to solve these problems. (Selden, Selden, Mason 1994) Thus, even what would be considered a conventional success, where students earn an A or B, is not all that it’s cracked up to be. Math is a language and environment for quantitative problem solving, and students who are unable to use their knowledge effectively is a sign that we need to emphasize not only learning the core ideas but also provide students with opportunities to learn how to problem solve.

In other studies, researchers invested students thinking in more ways. They investigated whether students can unpack mathematical statements informally and found only a small percent of these attempts were successful (less than 10%) (Selden, Selden 1995). In a somewhat related line of work Moore found that students struggled with constructing basic proofs in an introduction to proof course, suggesting that more higher-level thinking material is needed in prior courses. What these authors found is that students, who were successful in the sense that they made it through the first two years of college math, still had incomplete skills and understanding in areas such as basic logic and mathematical argumentation in mathematics. One interpretation is that we have sacrificed higher order thinking for more time spent on computational skills.

But there is good news. In a comparison study between two differential equations courses, students in an inquiry-oriented course and traditionally taught course were given the same posttest right after and 1 year later. The students in the inquiry-oriented course retained the same level of procedural skill and further performed significantly better on conceptual understanding problems one year later. Students who developed knowledge with conceptual understanding and visual models had more durable learning outcomes (Kwon, Rasmussen, Allen 2005).

In a large, multi-institution study of the IBL centers by Laursen et al (Laursen, Hassi, Kogan, Weston 2014) found evidence that IBL courses can level the playing field for women, while still benefiting, and helping lower achieving students bridge gaps. The aggregate result is various student groups do better in IBL courses vs. non-IBL students, with women and men showing significant gains including affective gains in confidence in Mathematics. In particular, women gain enough to level the playing field. So all groups showed gains or similar levels of performance, and the gender gap was minimized. This is the way it should be, because there should be no advantage being one gender or group over another, and what matters most is what students and instructors do.

In two large meta-studies by Freeman et al (Freeman 2014) and Theobald at al (Theobald 2020), active learning outperforms lecture method courses statistically significantly, where the DFW rate is 50% higher in lecture-based courses. Further, in the 2020 study, the achievement gaps for minoritized groups in undergraduate sciences, technology, engineering and math are reduced in both examination scores and pass rates.

Thus, active learning is a critically necessary strategy not just in improving courses outcomes, but also in efforts related to social justice. Women and people of color have historically been excluded or pushed away from STEM careers, and one of the solutions we have is within our locus of control. Implementing active learning methods, such as IBL has simultaneous benefits on several levels, from achievement and learning, to reducing the gender gap, and supporting minoritized groups.

Beyond achievement is also the lived experiences of students and instructors. One of the four pillars of IBL teaching is that instructors focus on equity. Students from all backgrounds deserve a dignified experience, where students feel welcome, supported, and included in the classroom environment. Teaching is a cultural activity, and people bring with them all of the things that shape our society and who we are. In college courses, we have an opportunity to not only teach math well, but also to create communities of learners that are also inclusive and caring. We can include assignments and lessons about growth mindset, productive failure, and provide students with opportunities to learn how to think mathematically and problem solve.

Teaching matters and college math instructors have power in shaping the future. For faculty interested in learning more about IBL methods, please visit the Academy of Inquiry Based Learning or AIBL for short (www.inquirybasedlearning.org). On the AIBL website, we have a portfolio of support programs, including videos, a collection of blog posts on IBL and general teaching topics, student testimonials, instructor insights on video, self-paced workshops on equity, and more. Together with an amazing group of colleagues, we also developed and offered professional development workshops, funded by the National Science Foundation (NSF PRODUCT DUE-1525058). One of the levers for change is faculty professional development in teaching, where instructors come together to collaborate, learn from one another, and also support each other. While funding is currently not available for the present time, we do plan to offer workshops in the future for faculty at 2-year colleges on IBL methods.

Let’s end with a student quote.

*“… I learned that failing is necessary for success. This is applicable to my life in many ways, not only in calculus, and it was helpful for me to have the idea that failure is not a bad thing reinforced. I also learned strategies to think more effectively; understand deeply, make mistakes, raise questions, follow the flow of ideas and allow yourself to change. These strategies have encouraged me because they have helped me to see that I can change the way I think to become more effective in my daily life and tasks.” *

- Calculus 1 student* *

## References

Ellis J., Fosdick B.K., Rasmussen C. (2016) Women 1.5 Times More Likely to Leave STEM Pipeline after Calculus Compared to Men: Lack of Mathematical Confidence a Potential Culprit. PLoS ONE 11(7): e0157447. doi:10.1371/journal.pone.0157447

Freeman, S., Eddy, S., McDonough, M., Smith, M., Okoroafor, N., Jordt, H., Wenderoth, M., (2014). Active learning boosts performance in STEM courses.

Proceedings of the National Academy of Sciences Jun 2014, 111 (23) 8410-8415;

DOI: 10.1073/pnas.1319030111

Kwon, O. N., Rasmussen, C., & Allen, K. (2005). Students’ retention of knowledge and skills in differential equations. *School Science and Mathematics, 105*(5), 227-239.

Laursen, S., Hassi, M-L, Kogan, M., Weston, T., (2014). Benefits for Women and Men of Inquiry-Based Learning in College Mathematics: A Multi-Institution Study. Journal for Research in Mathematics Education, 45(4), 406-418. doi:10.5951/jresematheduc.45.4.0406

Laursen, S.L., Rasmussen, C. (2019) I on the Prize: Inquiry Approaches in Undergraduate Mathematics. Int. J. Res. Undergrad. Math. Ed. 5, 129–146. https://doi.org/10.1007/s40753-019-00085-6

Moore, R. (1994). Making the Transition to Formal Proof. Educational Studies in Mathematics, 27(3), 249-266. Retrieved from http://www.jstor.org/stable/3482952

Selden, J. Selden, A., Mason, A.. (1994). Even good calculus students can't solve non-routine problems. Research Issues in Undergraduate Mathematics Learning. 19-26.

Selden, J., Selden, A. (1995) Unpacking the logic of mathematical statements. Educ Stud Math 29, 123–151. https://doi.org/10.1007/BF01274210

Theobald, E., et al (2020). Active learning narrows achievement gaps for underrepresented students in undergraduate science, technology, engineering, and math. PNAS March 24, 2020 117 (12) 6476-6483; first published March 9, 2020; https://doi.org/10.1073/pnas.1916903117