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Status and implementation of measures described in the last action plan:There is no prior action plan from previous years. However, there has been a goal to increase the perceived and actual relevance of the course for the BIT students by, e.g., including more programming. This was partially implemented this year with two of the problem sets featuring problems involving some use of python via Jupyterhub. Feedback on this seemed positive from the BIT-students and neutral or negative from the non-BIT-students.   Another measure that we continued to employ from last year was to highlight analogues between the mathematical notions we introduce and use in the course and notions in computer science and programming, emphasizing particularly data structures in programming languages. For instance, we formally defined strings mathematically, proved that concatenation of strings is associative via induction, and then compared this definition to lists, concatenation of lists and how the role of the empty string and ‘null’ were similar. Hence, this also indicated how one could use induction to prove correctness of algorithms, which was something that was suggested in the past by people in charge of the BIT-program.  Action plan. Measures with responsibilities to follow up and deadline for completion of such: As also suggested last year, one possible measure is to change the course description in such a way to allow the exam to include problems involving both discrete mathematics and some light programming. Without this, it is likely that the parts of the course including programming would be disregarded by the students.   Additionally, in the same vein one should consider including problems related to applications on the exam. Adapting some of the STACK-problems used in the other similar courses at NTNU could be considered.  One could consider making a pdf-document available to the students including concise information on how they can or should prepare for the lectures, what the specific learning goals are for each lecture. This could also include simple tips about standard mathematical problem solving techniques. This would be similar in form to existing documents already used in, e.g., NTNU’s course on algorithms and data structures. Another point to consider is to dedicate more time early in the course to explaining the notion of proof and why it is relevant for their study program. For instance, one could dedicate a lecture to the notion of proofs and we note that there are relevant and suitable chapters in the textbook for this.  Finally, one could consider re-arranging the course contents so that material relevant to courses taken at the same time as MA0301 would be introduced in time to be of benefit. There are tradeoffs to consider, though. For instance, much of the early material in the course seems to take some time for the students to absorb and this was highlighted in discussions with the learning assistants and the reference group.

Quality of the course, as evaluated by the course coordinator: Fair overall given the conditions and resources.  BackgroundThe lectures were given in person without recordings being made available to all students. Notes for the lectures were made available to the students the same week the lectures were given in. There were 12 weeks of problem sets with tutorials, of which students had to have 4 out of the first 6 and 4 out of the last 6 approved to be able to take the exam. T Lectures and tutorials  Attendance in the lectures began at around 50 students for the first lecture, increased to around 70-90 after the first lecture and continued at this level until the earliest point at which the students could have 4 out of the first 6 problems sets approved, following which attendance dropped to around 50-60 and then steadily declining to around 25-35 in the last two weeks of lectures.  The first meeting of the reference group included some important feedback on the lectures: namely that many students found the tempo too high when introducing new concepts. After some adjustments made following this, feedback on the lectures from the reference group was generally positive, e.g., it seems the tempo was better. Further adjustments were also made following the second meeting with positive feedback as well. The adjustments made included the following:   <ul><li>Generally lowering the tempo of the lectures.</li><li>More time spent on examples, and including more examples relevant for the problem sets.</li><li>When presenting proofs, more emphasis on explaining what we were going to do and summarizing what we had done.</li><li>Using more time on explaining the idea of a proof using an example whenever this could be done profitably.  </li></ul> Nevertheless, by survey, one sees that the results there indicate that the students found the teachers’ explanations very good at a lower rate than the IMF average. At the same time, the survey indicated that the students thought that the teachers tried to understand the challenges the students might have with the course and listened to their feedback at a slightly above average rate.  Note, though, that the survey response rate was somewhat low and it is thus difficult to say what weight one should assign this. Moreover, note that the survey was given at the end of the course and the students are giving an overall evaluation of the entire course. Finally, note also that the formulation of the relevant question means that these questions could include the learning assistants’ explanations as well, and that other questions indicated that the quality of the service provided by the learning assistants’ was rated as lower than the IMF average as well. This was also something we received feedback on from the reference group in the second meeting and made some efforts to fix.  Having said all that, it could be that this was not just an issue of tempo or quantity of explanation as suggested by reference groups feedback, but also one of quality. For instance, it could be that the explanations in the lectures were not sufficiently clear due to wordiness. ProofsThere are two more points raised by the survey results that we briefly discuss here: 1) lack of engaging learning activities and 2) a somewhat high level of difficulty. Both of these could be related to the emphasis in the course on proofs. On the one hand, from feedback from the reference group, students and past teachers of this course, it seems that the students both generally find proofs difficult and do not like or enjoy proofs. On the other hand, having some emphasis on proofs seems necessary if this course is to serve as adequate preparation for courses like TDT4120 - Algorithms and data structures. It could be that proofs were emphasized too much this year. Additionally or alternatively, it could be that we should have spent more time early on the course introducing the notion more carefully, using more time to explain why it is important and even relevant for future courses many of the students will take, e.g., that in TDT4120 one may need to understand how one proves correctness of algorithms or understand how one proves that a problem is NP-hard. Gap between lectures and problem setsAs was the case last year, one issue brought up by the reference group was that many students felt there was a gap in difficulty between the examples in the lectures and the problems on the problem sets. To help with this, as mentioned previously, we tried to include more examples of similar difficulty in the lectures, and we made some adjustments when making the later problem sets. This seemed to have helped based on feedback from students and the reference group, albeit with the result of the later problem sets becoming longer. Last year, the learning assistants noted that this gap claimed by some of the students was occasionally the result of them not recognizing that a problem was a generalization or a variation on an example in the lectures or the textbook. While we tried to account for this in the lectures and in the earlier problem sets this year, it seems this should perhaps have been emphasized more.  TexbookAs was the case last year, feedback from the reference group indicated that the current textbook worked well, highlighting that the chapters are short and concise, and possible “to read through all the way”.  The examWe included a Norwegian/English terms list on the exam. According to the report by the reference group, the level of the exam was appropriate given the contents of the course.

A survey was conducted near the end of teaching for the semester.

NTNU:TIA:REST

Institutt for matematiske fag

Department of Mathematical Sciences

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Dersom studenten også etter utsatt eksamen har sluttkarakteren F/ikke-bestått, må studenten gjenta emnet i sin helhet. Arbeider som teller med i sluttkarakteren må gjentas. For mer informasjon om vurdering, se «Læringsformer og aktiviteter».Ved utsatt eksamen (kontinuasjonseksamen) kan skriftlig eksamen bli endret til muntlig eksamen.

Anbefalte forkunnskaper

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Faglig innhold

Emnet er først og fremst ment som et tilbud til informatikkstudenter, og bygger ikke på forkunnskaper i matematikk utover det som undervises i videregående skole. Emnet vil også være av interesse for matematikkstudenter. Emnet gir en innføring i elementær kombinatorikk og mengdelære, setnings- og predikatslogikk, induksjon og rekursjon, relasjoner og funksjoner, grafteori, formelle språk og endelige automater.

Kursmateriell

Læringsutbytte

1. Kunnskap. Studenten har grunnleggende kunnskaper innen matematisk logikk, mengdelære og kombinatoriske tellemetoder. Studenten har kunnskaper om rekursjon og induksjon, samt om relasjoner, grafer og trær. Videre har studenten grunnleggende kunnskaper om formelle språk, grammatikker og endelige automater.2. Ferdigheter: Studenten kan gjøre praktisk bruk av elementær logikk og mengdelære, kan føre enkle induksjonsbevis, kan anvende kombinatoriske tellemetoder til å løse praktiske problemer og kan gjøre praktisk bruk av endelige automater.Studenten kan gjenkjenne, forstå og bruke begreper som relasjoner, grafer og trær i anvendelser, f.eks. innen informasjonsteknologi.

Læringsformer og aktiviteter

Forelesninger og obligatoriske øvinger. Karakter basert på avsluttende skriftlig eksamen.

Øvinger

Elementær diskret matematikk

Matematikk

Informatikk - bachelorstudium

Matematiske fag - bachelorstudium

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Status and implementation of measures described in the last action plan:There is no prior action plan from previous years. However, there has been a goal to increase the perceived and actual relevance of the course for the BIT students by e.g. including more programming.  One measure taken towards this is to highlight analogues between the mathematical notions we introduce and use in the course and notions in computer science and programming, emphasizing particularly data structures in programming languages. For instance, we formally defined strings mathematically, proved that concatenation of strings is associative via induction, and then compared this definition to lists, concatenation of lists and how the role of the empty string and ‘null’ were similar. Hence, this also indicated how one could use induction to prove correctness of algorithms. Another was that we removed the one and a half lectures on Boolean algebra to have more time for applications. This included in particular examples of using induction to verify correctness of algorithms and explicit applications of finite automata. For the latter, this included e.g. how finite automata could be used for auto complete, for AI in games, and for async/await. Action plan. Measures with responsibilities to follow up and deadline for completion of such: One possible measure is to change the course description in such a way to allow the exam to include problems involving both discrete mathematics and some light programming. Without this, it is likely that the parts of the course including programming would be disregarded by the students. Additionally, in the same vein one should consider including problems related to applications on the exam. One could consider making a pdf-document available to the students including concise information on how they can or should prepare for the lectures, what the specific learning goals are for each lecture, and some simple tips about standard mathematical problem solving techniques. This would be similar in form to similar documents already used in e.g. NTNU’s course on algorithms and data structures. Finally, one could consider re-arranging the course contents so that material relevant to courses taken at the same time as MA0301 would be introduced in time to be of benefit. There are tradeoffs to consider, though. For instance, much of the early material in the course seems to take some time for the students to absorb and this was highlighted in discussions with the learning assistants and the reference group.

Quality of the course, as evaluated by the course coordinator: Good overall assuming given the conditions and resources, namely that we had to give the early parts of the course digitally. In fact, because of demand from students, we ended up giving the lectures digitally throughout the semester, including making recordings available.  Digital lectures were given using an iPad. and complete lecture notes were made available before the lectures as an aid to remedy the lack of "board space" an iPad has compared to using a blackboard and projector.  Feedback on the lectures from the reference group was positive, the tempo was fine and having the lecture notes available was useful. The option of using a presentation at the same time as the iPad was raised. We evaluated this and decided that sticking with the iPad and lecture notes system was preferable.  Attendance digitally was generally reasonably good compared to other mathematics courses while attendance in person was generally low. More specifically, the attendance digitally remained reasonably good (greater than or equal to 20%) compared to other courses until halfway through the course, and only dropped to at or below 10% after most of the students had passed enough problem sets to take the exam. One should keep in mind that both weekly lectures had the 8-10 time slot. Viewership numbers for the recordings showed a similar pattern, although the number of unique viewers was between 30-55% before most students had passed enough problem sets. Finally, we note that both digital attendance and unique viewership started at below 60%.  One issue brought up by the reference group was that many students felt there was a gap in difficulty between the examples in the lectures and the problems on the problem sets. To help with this, we made available suggestions for additional problems to work on, tried to include more examples of similar difficulty in the lectures, and we made some adjustments when making the later problem sets. This seemed to have helped based on feedback from students and the reference group.  Later on, the learning assistants noted that this gap claimed by some of the students was occasionally the result of them not recognizing that a problem was a generalization or a variation on an example in the lectures or the textbook. Based on discussion with the reference group, in the future, we could try to include some hints in the early problem sets pointing to relevant examples to help with this.  One smaller issue raised by the reference group towards the end of the course, was that the course should’ve used announcements in Blackboard to a greater extent as this was common in the other courses they had or were taking. This might also have helped e.g. with letting students know about the discussion forum for the course. Feedback from the reference group indicated that the current textbook worked well, highlighting that the chapters are short and concise, and possible “to read through all the way”. Additionally, the reference group noted that it might be advantageous to advertise the Norwegian/English word list more and possibly to include relevant parts in problem sets.   As for the exam, the multiple choice problems ended up being too difficult and the grade threshold values used were adjusted to account for this.

Dersom studenten også etter utsatt eksamen har sluttkarakteren F/ikke-bestått, må studenten gjenta emnet i sin helhet. Arbeider som teller med i sluttkarakteren må gjentas. For mer informasjon om vurdering, se «Læringsformer og aktiviteter».

1. Kunnskap. Studenten har grunnleggende kunnskaper innen matematisk logikk, mengdelære og kombinatoriske tellemetoder. Studenten har kunnskaper om rekursjon og induksjon, samt om relasjoner, grafer og trær. Videre har studenten grunnleggende kunnskaper om formelle språk, grammatikker og endelige automater. 2. Ferdigheter: Studenten kan gjøre praktisk bruk av elementær logikk og mengdelære, kan føre enkle induksjonsbevis, kan anvende kombinatoriske tellemetoder til å løse praktiske problemer og kan gjøre praktisk bruk av endelige automater. Studenten kan gjenkjenne, forstå og bruke begreper som relasjoner, grafer og trær i anvendelser, f.eks. innen informasjonsteknologi. 

Forelesninger og obligatoriske øvinger. Karakter basert på avsluttende skriftlig eksamen. Ved utsatt eksamen (kontinuasjonseksamen) kan skriftlig eksamen bli endret til muntlig eksamen. 

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Overview. This introductory course is for computer scientists. The course provides an introduction to combinatorics, set theory, logic, induction and recursion, relations and functions, graph theory, formal languages and finite state automata.General evaluation. I believe we have achieved our goal, in particular: <ul><li>From the final exam, students showed that they understand the basic knowledge about discrete mathematics and can use the knowledge they learn from the course to solve some challenging problems. Moreover, most of them get used to the computer science mindset for their future study. For instance, they know how to use Kruskal’s algorithm to find a minimum spanning tree in a graph.</li><li>I handed out an overall survey. From it, many students are very satisfied with the contents and the way how it is explained. One of them even wrote, “… I've had almost all of the other math courses at NTNU thusfar, but you have been my favorite lecturer:-) …”</li></ul>Please note that our lecture was conducted during the challenging COVID time. It is only in the digital mode.

This course is based on two textbooks: Harry Lewis, Rachel Zax, &lt; Essential Discrete Mathematics for Computer Science&gt;; and Ralph P. Grimaldi, &lt; Discrete and Combinatorial Mathematics&gt;. I prefer the latter one, since it is better organized and easier to access the contents with examples. Those examples are very helpful for the students. I think its difficulty is suitable for a course like this, and the first book seems too easy. But one thing needs to be careful is the latter book covers more contents and more examples, and it would be stressful for the students to study them all. I think by making clear what is required and what is not and being slow, it would be fine. For instance, one student wrote “I think it is good that you explain most things slow and in detail, I feel like everyone could understand the topics in lecture, which is great.” I think in a long term we can consider having discrete mathematics for one year, which is the standard way of teaching discrete mathematics in many universities (at least for all the universities I have been to and worked at). The main reason is that it needs time for students to get used to the kind of thinking, and the contents could be challenging for students just coming from high schools.

Because of the COVID situation, lectures of this course are all digital and recorded. This course has compulsory exercises. It works very well, and I received many positive feedbacks. Many of them even want to continue this digital form of teaching. I think the COVID situation gives us a chance to re-think about what is the “right” form of lecturing in this new age. For instance, a student suggested to have timestamp for different topics in the lecture recordings, which has been done in many of nowadays’ YouTube videos. This is a great suggestion! Motivated by this, I think we can consider doing a flipped classroom: Students can watch the recorded lectures in advance, and then use the classroom time as a chance to interact with the lecturer by asking questions, etc. Of course, that will increase the students’ work, but they will become active learners and the outcome will be better.

Forelesninger og obligatoriske øvinger. Karakter basert på avsluttende skriftlig eksamen. Ved utsatt eksamen (kontinuasjonseksamen) kan skriftlig eksamen bli endret til muntlig eksamen.


1. Kunnskap. Studenten har grunnleggende kunnskaper innen matematisk logikk, mengdelære og kombinatoriske tellemetoder. Studenten har kunnskaper om rekursjon og induksjon, samt om relasjoner, grafer og trær. Videre har studenten grunnleggende kunnskaper om formelle språk, grammatikker og endelige automater.
2. Ferdigheter: Studenten kan gjøre praktisk bruk av elementær logikk og mengdelære, kan føre enkle induksjonsbevis, kan anvende kombinatoriske tellemetoder til å løse praktiske problemer og kan gjøre praktisk bruk av endelige automater.
Studenten kan gjenkjenne, forstå og bruke begreper som relasjoner, grafer og trær i anvendelser, 
f.eks. innen informasjonsteknologi. 

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Elementær diskret matematikk - MA0301