Despite the modern consensus that traditional-style lectures do not promote deep and longlasting learning outcomes, many mathematics courses are delivered precisely in this manner.1 How does one incorporate elements of active learning in a pre-established course with traditional-style lectures? Are these additional activities introduced at the expense of covering lessmaterial and do more advanced students suffer from this style of teaching? If so, do the advantages outweigh the negative effects in the context of a master’s level mathematics course

Mancinska, Laura

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14Introducing elements of active learning in asmall courseLaura MančinskaDepartment of Mathematical SciencesUniversity of CopenhagenSetting the sceneDespite the modern consensus that traditional-style lectures do not promotedeep and longlasting learning outcomes, many mathematics courses are de-livered precisely in this manner.1 How does one incorporate elements ofactive learning in a pre-established course with traditional-style lectures?Are these additional activities introduced at the expense of covering lessmaterial and do more advanced students suffer from this style of teaching?If so, do the advantages outweigh the negative effects in the context of amaster’s level mathematics course?Context of the studyThis study will be conducted in the context of the 7.5 ECTS Master’s levelcourse “Introduction to Modern Cryptography” at the Department of Math-ematical Sciences (see Table 14.1 for a quick overview of the course). Theteaching format adopted in the previous years was traditional-style lecturesplus exercise classes which are normally used for going over assignment so-lutions or additional course material. This was the first time the course wasoffered as a graded rather than a pass/fail course. Therefore, to allow for1 A typical mathematics course consists of lectures and exercise classes. It is thelecture component that is often delivered in the traditional “instructor-at-the-blackboard” manner. Most of this article concerns the format of the lecture ratherthan exercise-class component.164 Laura Mančinskamore accurate individual assessment, I introduced a final exam. This wasdone in view of the fact that the students are encouraged to work on assign-ment problems collectively. Another change in comparison to the previousyears was that the deliverable component of the project was changed fromwritten to oral presentation. Oral presentations allow students to gain ex-perience and improve their presentation skills, as well as allowing them tolearn from each other.Table 14.1. The course "Introduction to Modern Cryptography" at a glance.Audience Study program Mathematics (majority), Computer Science, Statistics Level Master’s (majority), Bachelor’s Size 9 students Course Weekly contact hours 4 hours of lectures + 3 hours of exercise classes Assessment • 4 assignments• Individual project presentation (15 min)• Written final exam (3 hours)Credit 7.5 ECTS To evaluate and reflect on the planned intervention we must considerthem in the context of intended learning outcomes (ILOs). To comply withthe Danish Qualifications Framework for Higher Education (Danish Min-istry of Higher Education and Scince, 2008), ILOs are specified in terms ofknowledge, skills, and competencies (see Table 14.2).Table 14.2. Intended learning outcomes of the course "Introduction to ModernCryptography".Knowledge The students will have an understanding of the theoretical and mathematical basis of modern cryptographic systems. Skills The students will be able to give rigorous security proofs of basic cryptographic systems and connect various cryptographic primitives with rigorous reductions. Competencies Understanding theorems about theoretical cryptography; proving security reductions; reasoning about the limits of computationally-bounded adversaries. 14 Introducing elements of active learning in a small course 165The planned intervention and its theoretical backingIn this section we describe the planned intervention which can be summa-rized as incorporating elements of active learning in traditional-style lec-tures of a pre-existing course. We include a description of the elements tobe incorporated, give specific examples, explain the desired outcomes ofeach of the elements. We also provide theoretical backing of the interven-tion as a whole as well as its constituent elements.Theoretical backing and motivation behind the interventionRoediger et al. define learning as “Acquiring knowledge and skills and hav-ing them readily available from memory so you can make sense of futureproblems and opportunities.” (Roediger et al., 2014). According to the sem-inal work (Piaget, 1978), the learner is viewed as a goal-oriented agent whoactively seeks information. In contrast, the traditional didactic lectures areone-way exchanges, where the knowledge is supposed to flow from the in-structor to the students. In addition, the modern consensus is that the acquir-ing of usable knowledge and skills occurs when the learner examines differ-ent facets of the topic in question and connects it to already existing know-ledge and skills (see e.g. Bransford et al., 2000)). During a traditional-stylelecture it is easy for a student to take the role of a passive listener and leavethe auditorium without having connected the discussed topic with their pre-vious knowledge and experiences. Active and student-centered learning isan approach that seeks to correct this by putting the focus on the student andplacing them into situations where they are “forced” to take a more activerole and form the desired links. Of course, not all the students require suchforcing as they independently examine different aspects of the discussedtopic and contrast it with their pre-existing knowledge via an un-coercedinner reflection. However, it is not this group of students2 that an instructorshould focus on as they are bound to learn almost irrespective of the circum-stances. Instead one should encourage the more passively-inclined studentsand design activities that would encourage them to take up a more activerole therefore promoting the acquisition of deep, usable and long-lastingknowledge.2 For the purpose of this article, let us briefly refer to this group as “advancedstudents” or “advanced learners”.166 Laura MančinskaDescription of the interventionWe now proceed to describe the elements of active learning introduced inthe lecture-component of the course “Introduction to Modern Cryptogra-phy”. We also describe what aspects of active learning each of these ele-ments is meant to address.a) Short self- or peer-graded quizzes at the beginning of the lecture.Description: Start a lecture by asking the students to recall the pre-viously introduced concepts necessary for the upcoming lecture. Thestudents would be given a few minutes to write down their answerand then provided with a model solution along with a sample grad-ing scheme. Sometimes the students would grade their own solutionsand sometimes they would be asked to switch with a peer. The stu-dents’ performance in these quizzes had no effect on their final grade.However, similar style questions were asked in the first part of the fi-nal exam and I would remind the students of this in order to encouragemaximum effort.Intended outcome: Activate the students at the beginning of the lectureand refresh the concepts needed to place the forth-coming material inthe context of the previously covered topics. The purpose of the grad-ing scheme is to help the students identify the essential ingredients ofa correct answer. The intention behind peer-grading is to help the stu-dents recognize a correct and complete answer as we are often quickerto notice shortcomings of others’ work in comparison to our own.Table 14.3. An example quiz question.Question Define a private-key encryption scheme Answer A (simple) PK-encryption scheme  over  consists of three probabilistic algorithms Key generation algorithm Gen producing key Encryption algorithm EncEnc  Decryption algorithm Dec .Scheme must satisfy correctness requirement Dec Encfor any  and any key output by GenGrading scheme [1pt] mention the three sets  [1pt] for a correct description of each of the 3 algorithms [1pt] Correctness requirement [1pt bonus] For “probabilistic” and quantifiers on m, k. 14 Introducing elements of active learning in a small course 167b) Guided interactive investigation of a topic.Description: Rather than presenting the students with a ready-madetheory and concepts, sometimes I would attempt to develop the materialtogether with them via a series of leading questions. Precisely what ismeant here is best understood via an example (see Table 14.4).Intended outcome: Engage the students and allow them to actively dis-cover the material themselves, promote deeper understanding, developresearch skills.Further considerations: When employing this tactic, one should bearin mind that it is rather time-consuming. It is a very engaging way topresent new material and it undoubtedly activates students and leadsto a deeper understanding of the topic. However, due to the associatedtime cost and the fact that not every topic lends itself easily to thisapproach, I did not use it in every single lecture.Table 14.4. An example of guided investigation.Concept to be discovered Message authentication codes (MACs) for long messages. Question and answer sequence Initial question (instructor): Now that we have seen fixed-length MACs, how could we use them to authenticate longer messages? Answer (student): Split message into blocks and obtain the tag by concatenating tags for each of the blocks. Instructor (writes on the board): Alright, so we consider a block-message  and then obtain its tag as , where  Mac  Does anybody see a possible attack? (Give some time if no one raises their hand.) Student: We can request tags for two different messages and then produce a new valid message-tag pair.  Instructor: Formalizes the suggested answer on the board. Does anyone see how we could modify the previous idea to render this attack ineffective? (Give some time.) Students: … Instructor: Recall that we know from the last week that deterministic encryption schemes are insecure… Student: Offers an idea that is later formalized by the instructor as a complete construction for MACs that can be used to authenticate long messages. 168 Laura Mančinskac) Short activation questions.Description: While explaining the material pose quick questions to theaudience. For example, the question could be drawing students’ atten-tion to crucial or subtle aspects of the notion currently being introducedor asking them to compare it to a related previously introduced con-cept. Other questions of this type would be to encourage the studentsto suggest a relatively easy-to-anticipate next step in a proof or per-form a simple calculation. In order to avoid interrupting the flow of theexposition these questions are designed to be rather straightforward.Intended outcome: Activate the students that have slipped into passivelistening, encourage the students to link the new material with the pre-viously introduced one, give them a chance to gain further familiaritywith the concept in question.Evaluation and outcomes of the interventionThe intervention was evaluated via the following four methods:E1. A standard course evaluation form (Anonymous, Response rate: 5/9).E2. Additional free-form question specifically regarding the small exer-cises and quizzes during the lectures (Anonymous, Response rate: 5/9).E3. An individual, informal follow-up interview with select students.E4. Feedback from the peer-supervision group who observed two lectures.The course evaluation E1 was generally very positive. For instance, allof the students agreed that “they have acquired the competencies describedin the course objectives”. The students evaluated this year’s course a bithigher than the previous year’s one in practically all the categories. Also,based on evaluation E1, the Teaching Committee at MATH categorized thiscourse as belonging to (the highest) category A. This category is describedas “Courses where the teaching has worked particularly well and can in-spire others.”Of course, without a further follow-up E1 does not allow us to con-clude that the positive evaluation was due to the added elements of activelearning. In E2 the students explicitly mention that they found the smallexercises helpful and that they allowed them to discern the most importantaspects of definitions. However, most of the students did not like that someof the quizzes were peer-graded. Some mentioned feeling uncomfortablewhile others felt they could do just as well by grading their own work.14 Introducing elements of active learning in a small course 169According to the feedback from the peer-supervision group who ob-served two of the lectures (E4), the students were generally engaged duringthe lectures and the interactive student-activation elements were perceivedas aiding active learning. However, according to E4, student engagementlowered while a longer proof was being presented on the board.Finally, according to evaluation E3, students generally enjoyed theadded interactive elements. However, some of the more advanced studentsmentioned that the they found the pace of the course a bit slow. This wasalso echoed in one of the anonymous comments from E1.Discussion of the outcomes and further improvementsIn general, the introduced elements of active learning seem to have in-creased the student engagement and also their satisfaction with the course.However, the small number of involved students does not allow to concludewith certainty that the observed improvement is due to the intervention. Infact, to me the most convincing piece of evidence is my own empirical ob-servation of the student engagement during the lectures that suggests thatthe intervention worked well. However, the small exercises did take up timewhich slowed the pace of the lectures and reduced the amount of material Iwas able to cover.I believe that the effects of the intervention varied between differentgroups of students. For instance, if the definition from the last week is freshin your memory, you are not gaining much by being asked to repeat it. Also,the time used for the small exercises is generally tailored to the average paceof the students. Therefore, I believe that the intervention might have causedsome negative effects for the learning outcomes of the more advanced stu-dents. In general, I find that it is challenging to design activities that areequally beneficial to all groups of students.The evaluation shows that peer-grading was generally not perceivedwell. However, evaluations E1 and E3 show that this can be attributed tomiscommunication. The students seemed to believe that peer-grading wasused to promote fair assessment. However, this was not the intention be-hind it (see (a) Intended outcomes in Section 2.1). I believe the perceptionof peer-grading can be altered by better communicating the reasons behindit.In the future I would like to experiment with incorporating elementsof active learning into the presentation of mathematical proofs. Proofs are170 Laura Mančinskaimportant elements of virtually any mathematics course, as they form thefoundation of mathematical thinking. They are most often presented in themost elegant available form that highlights the crucial steps. This is usu-ally not the form the proof is first conceived. So often times the instructorpresents the proof without much interaction with the students. This canlower the student engagement. In fact, this was my experience during the“Introduction to Modern Cryptography” course and it was also pointed outby my peer-supervision group (evaluation E4).ReferencesDanish Ministry of Higher Education and Scince. (2008). "Danish Qualifi-cations Framework for Higher Education." Uddannelses og Forskningsmin-isteriet. ufm.dk.Bransford, J. D., Brown, A. L., & Cocking, R. R. (2000). How people learn(Vol. 11). Washington, DC: National academy press.Roediger, H. L., McDaniel, M. A., & Brown, P. C. (2014). Make it stick:The science of successful learning. Harvard University Press.Piaget, Jean. (1978). Success and Understanding. Harvard University Press.

Introducing elements of active learning in a small course

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Introducing elements of active learning in a small course

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