Drawing from Traditional Andragogy
Wisdom is knowledge of general principles distilled in time. Thus, wisdom is delivered by means of traditions. One of the traditions, among others, that I have most benefited from, is that of the Protestant-Reformed. Below I list four objectives of education that is (or should be) pursued in the Protestant-Reformed tradition. Even if one is not aware of the Protestant-Reformed worldview, I think one can still see the value of them:
Love. The beauty sought out in mathematics is a rational beauty and, thus, for rational beings. Rationality belongs to persons, so mathematics that does not lead to compassion and love is an ugly one.
Artistry. Engaging in mathematics, just like in any other arts, cultivates one’s artistry in understanding the world and expressing ideas.
Liberty. The liberty sought after in mathematics begins with self-directed learning and ultimately takes form in creativity.
Harmony. Neither the individual nor the society takes priority when it comes to harmony. Indeed, a human being is a social being, and a society presupposes the existence of individuals. So living in harmony and pursuing teamwork is not just a skill but the very manner by which we exist.
Drawing from Scientific Andragogy
We are now in a privileged era where experimental results regarding educational practices are rushing in. The experiments have brought a challenge to conventional ideas and practices—such as lecture-based classroom, isolated pencil-and-paper exam, and so on—at least in the so-called STEM fields.
Indeed, the evidence is now overwhelming [1] [2]; conventional methods are effective, but nowhere close to active learning that involves graded preparatory assignments, extensive student in-class engagement, and graded review assignments. Moreover, it has been demonstrated that, regardless of the particular discipline in the so-called STEM area, active-learning practices:
- benefit all students, but especially women, minorities, and first-generation students (in other words, the conventional classroom is biased against students that are not white, male, and affluent)
- increases the communal atmosphere in class and sense of belonging, which is crucial for retention
It is no wonder that the executive summary of a joint report issued in 2015 by the five [3] major mathematical societies is that “the status quo is unacceptable” [4]. In July 2016, the Conference Board of the Mathematical Sciences (CBMS) comprising seventeen [5] societies also issued a statement [6] on active learning, calling on “institutions of higher education, mathematics departments and the mathematics faculty, public policy-makers, and funding agencies to invest time and resources to ensure that effective active learning is incorporated into post-secondary mathematics classrooms.” And most recently, in August 2016, the White House Office of Science and Technology Policy issued a national Call to Action [7] to “educators in K-12 and higher education, professional development providers, non-profit organizations, Federal agencies, private industry, and members of the public to participate in a nationwide effort to meet the goals of STEM for All through the use of active learning at all grade levels and in higher education.”
It may be worthwhile to mention that active-learning is nothing but new. In the words of a prominent mathematician and educator P.R. Halmos [8], “[t]he best way to learn is to do; the worst way to teach is to talk.” The only difference between now and then is that we now have a strong scientific case for the age-old wisdom.
Computers Were Built by Mathematicians for Mathematics
The impetus towards active-learning is also brought by the mighty waves of change brought forth by the unprecedented advancements in electronic technology. Anyone connected to the internet has access to books and articles that cannot be contained in any private library that has existed. Finding a primitive of $f\colon x\mapsto (\ln (x)+1)x^x$ is a breeze. In short, tasks consisting of information retrieval and procedural problem solving has been and will continue to be replaced by computers and robots. As a consequence, many jobs we see today will not be available to our students.
Really this is good news for mathematicians because, after all, mathematics is about creative ideas interwoven by strands of logic, attempting to solve problems whose method of solution is unknown. The advancement of technology provides us more room to engage in such higher cognitive skills.
Practice
“New wine is for fresh wineskins” (Mk 2:22). I believe that the converse of this dictum is also true, that is, new wineskins (andragogical practices) are best used to hold new wine (active-learning paradigm). Put in another way, we must start with the “wine” first, that is, take on active-learning, and then seek “wineskins,” that is, employ reproducible evidence-based andragogical practices.
Below are some of the practices that I bring into the classroom to provide an environment in which students can grow towards the goals I mentioned in the very beginning.
Reading with Annotations Assignments. I ask the students to annotate their reading. That encourages them to reinterpret the stated facts in their own words and formulate questions that isolate sources of confusion. It also helps students grow in self-directed learning.
I also ask the students to share their annotations and engage in discussions, helping them make reading a social experience. (To that end, I use the online service provided by Perusall.)
Moreover, research shows that seasoned mathematicians read mathematical literature differently than undergraduate students—both groups spend more-or-less equal amount of time on formulas, but mathematicians spend more time on words and written arguments. The annotated-reading helps student focus more on the text than formulas, which ultimately helps them grow in mathematical maturity.
Peer Instruction. The more I teach a certain subject, the more likely I forget what it is like to approach the subject as a novice. But the students still have a fresh memory of the difficulties they encountered, and it stands to reason that the students are in a prime position to help each other’s learning. One way to encourage that is to bring conceptual questions that need a solid understanding of the material in order to answer; then ask students to choose an answer with commitment; have them discuss with peers; and finally debrief with the instructor. This method of Peer Instruction pioneered by Eric Mazur has shown to bring higher learning gains and better retention of information and discipline.
Liberal Use of the Plain Language. I emphasize and ask the students to write solutions or proofs using plain language liberally. Writing is not just a tool for transmitting ideas but a means to produce ideas and elucidate them. Carefully laying out one’s ideas or solutions helps the student make the following transition of mindset when approached with a problem:
\boxed{\text{What do I do here?}\quad \xrightarrow{\mathrm{transition}} \quad \text{Why would I do that?}}Making Room for Errors and Mistakes. I make room for productive errors and mistakes in the courses that I design so that the students’ learning is maximized—for “an expert is a man who has made all the mistakes which can be made, in a narrow field” (Niels Bohr).
As an example, for evaluating assignments I use the so-called Standards Based Grading system; it is essentially a rating system with values [9] from 0 to 3 that has the following interpretations: 3 = exceptional (rarely given), 2 = meets expectations, 1 = needs improvement, 0 = fragmentary. The criteria (rubric) for these ratings are clearly laid out to students via the syllabus. In this way, the students get a better idea of the meaning of the results of the assessment, rather than having them figure out what it means to get, say, 87% instead of 83% on homework. Consistent ratings of 2 (or higher) are enough to guarantee an A-grade, so students have a clearer idea of the quality of work that goes beyond what is asked of them (rating of 3).
Taking Calculated Risks. The willingness to take calculated risks goes hand in hand with innovation and creativity. I try to encourage such an attitude in my classroom.
Time to time I intentionally assign unfamiliar problems. Some students, unfortunately, take this unacceptable, perhaps due to the schooling they received; indeed, there is something comforting about being told what you need to know or to do to reach a certain result and avoiding failures as much as possible. But discontent is the first necessity of progress as Thomas Edison said. I remind the students that the ability to deal with unfamiliar problems is one of the key objectives of the courses that I teach and the skill that distinguishes us from computers.
Assessment that Mimics Real Life. Rarely one is faced with problems in life under severe conditions, cut off from information and coworkers. On the contrary, one is expected to use all the resources available to solve any problem. Likewise, I am fond of using open-book, open-notes, and open-internet assessment activities. Often, discussion with peers is also open. Another benefit of such practice is that I could, therefore, give problems that demand higher thinking skills.
End Notes
- Freeman, S., Eddy, S. L., McDonough, M., Smith, M. K., Okoroafor, N., Jordt, H., & Wenderoth, M. P. (2014). Active learning increases student performance in science, engineering, and mathematics. Proceedings of the National Academy of Sciences, 111(23), 8410–8415. http://doi.org/10.1073/pnas.1319030111 ↩
- Wieman, C., & Gilbert, S. (2015). Taking a Scientific Approach to Science Education, Part I–Research. Microbe Magazine, 10(4), 152–156. http://doi.org/10.1128/microbe.10.152.1 ↩
- AMATYC, AMS, ASA, MAA, and SIAM ↩
- K. Saxe and L. Braddy, A Common Vision for Undergraduate Mathematical Sciences Programs in 2025, 2015 ↩
- AMATYC, AMS, AMTE, ASA, ASL, AWM, ASSM, BBA, IMS, INFORMS, MAA, NAM, NCSM, NCTM, SIAM, SOA, and TODOS ↩
- Active Learning in Post-Secondary Mathematics Education, Conference Board of the Mathematical Sciences, 15 Jul 2016, ↩
- J. Handelsman and Q. Brown, A Call to Action: Incorporating Active STEM Learning Strategies into K-12 and Higher Education, The White House Office of Science and Technology Policy, 17 Aug 2016 ↩
- P. R. Halmos, E. E. Moise, and G. Piranian, The Problem of Learning to Teach, The American Mathematical Monthly 82 (1975), 466–476. Sec. I: The teaching of problem solving. ↩
- The original system uses the letters E-M-N-F instead of numbers because students can misinterpret the meaning of the rating of 2 as 66% achievement, which is certainly not the case. ↩