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All measures proposed last autumn were implemented. We increased the number of student assistants and distributed them into fewer exercise sessions. The general feedback received from the students regarding the exercise sessions was positive. The plenary sessions were also made physical this year, at two alternative time slots every week. We further increased the number of mandatory exercises involving code interpretation, which was well met by the students. Also following lessons learned from previous semesters, within certain topics I decided to skip or replace some “subtopics” that traditionally had been taught before — still observing the official course description, naturally. This allowed us to spend more time within each content block, which I believe had a positive impact on the learning outcome. An example of what was skipped this semester is adaptive time-stepping, which is perhaps a bit too advanced for a 2nd/3rd year course that is otherwise rather superficial/broad. We also skipped solving PDEs on infinitely long domains, which is quite complex and of little relevance for our target group. That freed some time that was spent on the discrete Fourier transform (DFT), a topic which the students have historically struggled with — and one of the most important in the curriculum.

The course went generally well throughout the semester, according to continuous feedback by the reference group and other students. In particular, several students independently expressed their appreciation of practical examples and contextualisation. Attendance was moderate, with between 30% and 40% of enrolled students regularly present. However, the complete iPad annotations produced during the lectures were uploaded to our webpage, which is a resource many students regularly relied on.  The exam style and structure was similar to previous years, with 9 questions covering all the course’s major topics. The failure rate was under 5%, roughly a third of last autumn’s percentage. A possible reason is that the exam was, as a whole, perhaps a bit shorter (fewer lengthy hand calculations). Another factor is that, this time around, the exam question on the DFT was designed to fit the corresponding learning goal, instead of focusing on formulas and mathematical definitions. The fact that many students scored high marks on this particular question indicates that they have learned enough to understand and use the DFT in their careers, if needed. Another change to the exam was the formula sheet, which was extended to include more formulas deemed essential, such as Fourier coefficients.

A general opinion shared by lecturer and students is that the curriculum is too large for one regular course. Numerical methods, for example, are usually a course on its own, especially in engineering curricula. The course could use a major reformulation, which should be done in combination with other math courses. For example, the Laplace transform could be moved to an earlier math course: it (1) has basically no connection to any other topic taught in Mathematics 4D and (2) is usually needed by engineering students earlier in their studies. In the short run, though, there are a few simple measures that could be taken to sensibly shorten the curriculum, so that each topic can be taught (and learned) better. Some ideas are: 1) Remove the continuous (!) Fourier transform from the curriculum. It has little to no relevance to engineering, economics or computer science students, and takes a considerable amount of time and effort to be explained and understood. 2) Reduce the amount of time spent on interpolation. The students do not need to know, for example, multiple different methods to find the same polynomial, or how to compute them by hand. That is no longer modern.3) Spend less time on analytical solution of PDEs. That is of little use to the audience, whether they end up in industry, business, academia… Something that can also improve learning is investing some time on visualisation of results. On the internet there is a great variety of videos and interactive tools that can help students better understand the meaning behind, or beyond, the techniques they learn in Math 4D. Another easy idea to implement is conceiving exercises that mix different topics, such as Fourier series + numerical integration, or time-stepping + Newton’s method (see questions 2 and 3 from our exam, for instance). Math 4D could only benefit from actions that try to better connect the different topics taught.

NTNU:TIA:REST

Institutt for matematiske fag

Department of Mathematical Sciences

Spesifiserte trykte og håndskrevne hjelpemidler tillatt. Bestemt, enkel kalkulator tillatt.

Engelsk

Norsk

Mer om vurdering

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

Anbefalte forkunnskaper

Emnene TMA4100/10/15 Matematikk 1/3 eller tilsvarende.

Link til informasjon om emnet

Faglig innhold

Partiellderiverte. Laplacetransformasjonen og løsning av ordinære differensial- og integralligninger. Fourierrekker, Fouriertransformasjonen og løsning av lineære partielle differensialligninger. Diskret Fouriertransformasjon. Numeriske metoder: Interpolasjon, derivasjon og integrasjon. Teknikker for løsning av lineære og ikkelineære ligninger. Runge-Kutta-metoder for løsning av system av ordinære differensialligninger. Differensmetoder for løsning av partiell differensialligninger. Innføring i beregningsverktøy med eksempler.

Kursmateriell

Læringsutbytte

1. Kunnskap. Studenten kan gjenkjenne, forstå og anvende grunnleggende begreper og metoder fra teorien for Fourierrekker, Fouriertransformasjonen, Laplacetransformasjonen, ordinære og partielle differensialligninger og numerisk løsning av ligningssystem og differensialligninger. 2. Ferdigheter. Studenten kan anvende sin kunnskap om Fourierteori, ordinære og partielle differensialligninger og numerikk til å formulere og løse problemer i matematikk og naturvitenskap/teknologi, om nødvendig supplert med bruk av matematisk programvare.

Læringsformer og aktiviteter

Forelesninger og obligatoriske øvinger. Emnet kan bli forelest på engelsk.

Øvinger

Matematikk 4D

Teknologiske fag

Datateknologi - masterstudium (5-årig)

Kommunikasjonsteknologi og digital sikkerhet - masterstudium (5-årig)

Industriell økonomi og teknologiledelse - masterstudium (5-årig)

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- Make sure that enough student assistants are hired from the start. 

- Organise a smaller number of larger exercise sessions, with 2-3 student assistants present in each session. 

- Try to organise the “plenumsregning” as physical lectures (+ recording). 

Tiltak

Oppf

A key point in this semester (Autumn 2022) was that Mathematics 4N and 4D were run and coordinated closely together. In 4N we had two parallels (one in English, one in Norwegian), which also ran in parallel with 4D (in Norwegian). We tried to synchronise the content of all parallel lectures as best as possible, and we got positive feedback from students about that. The cooperation between 4N and 4D worked very well, but it does require more time and resources. In fact, the scale of these courses (we had around 1200 students combined) has made it such that we were very dependent on the availability of student assistants, which was a challenge throughout the semester.  As additional teaching resources, we offered exercise classes, plenary sessions with teaching assistants (plenumsregninger), a Mattelab forum for mathematical and organisational questions, new/revised lecture notes in the form of PDFs and Jupyter notebooks, and additional Jupyter notebooks and python snippets with focus on coding. In particular, the students have used the Mattelab forum a lot, and it seems to have been helpful – but we should have a teaching assistant to monitor and manage that (it was to a large degree the lecturers who dealt with it this time). A general (and obvious) feedback we got is that there should have been more student assistants. We tried to get more, but it is not easy to find so many. In order to improve the exercise classes is to organise fewer but larger parallel sessions, each led by 2-3 student assistants, ideally one experienced student assistant being paired with a new one; it seems to have happened often that students went to exercise sessions and left without having their questions answered, due to the limited number of assistants present.

Maths 4N/D is a challenging course to hold, especially due to the many different study programmes involved. It is hard to find applications and teaching ideas that fit well all these programmes, and the previous knowledge in programming also varies a lot between them. However, our assessment is that the courses went in general well, and we got mostly positive feedback from the students, especially in their individual evaluations. They were generally pleased with the lectures, but shrinking the pensum a bit (see below) should really improve the effective learning outcome of Math 4N/4D. Attendance in the lectures is a bit hard to measure, but we estimate that the ratio of students that attended the lecture to those who did the exam is of the order of 1/2. A large additional challenge in this course is the question how we can ensure that the students meet the learning requirements related to usage of mathematical software (i.e., python), as this is difficult to measure in a written exam. In this year, we included some programming problems on the exercise sheets whenever it made sense, but a large number of students simply skipped these problems, as it was possible to get the exercise approved without these problems. One possible suggestion for future iterations of this course is to have 1-2 mandatory exercise sheets wholly dedicated to programming instead.

An opinion shared by the lecturers, students and reference groups is that the pensum is way too fragmented. We have too many topics that do not relate well to each other, so it is hard to build a consistent learning curve. When the students are starting to “get the hang” of some topic, we must move to a completely different one. In particular, the Laplace transform is something that could be shifted to a different math course, especially since many engineering students need that tool much earlier in their curriculum. Some topics in numerics could also be seen in earlier math courses, such as numerical linear algebra and rounding errors. This time we changed the order of the topics a bit, also in connection with a suggestion by last year’s reference group, and we got positive feedback about having started with numerics for two-point boundary value problems (finite differences). As a negative point, there was much confusion among students about the pensum. We decided to omit/change some topics this semester, which created some misunderstanding because the students tend to rely heavily on previous exams to determine what to learn. They should of course be able to determine what is pensum from our wiki-page, course description, the lecture plan, etc., but in practice that does not always work so well. We noticed that the students expect a course structure like they had in Math 1, 2, 3, but that is not the case here. In Math 4 we do not have the same resources; besides, we expect students in their third and fifth semesters to be more independent in their studies.   Oppfølgingsplan for emnet - Make sure that enough student assistants are hired from the start. - Organise a smaller number of larger exercise sessions, with 2-3 student assistants present in each session. - Try to organise the “plenumsregning” as physical lectures (+ recording).

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<h1>TMA4135 Matematikk 4D</h1><h2> </h2><h2>Forelesninger.</h2> Emnet ble forelest fysisk på tavle med live stream over Panopto. Pga én-metersregelen måtte studentene, i store deler av semesteret, melde seg opp på forhånd via BB for å delta på de fysiske forelesningene. Vi hadde plass til ca. halvparten av antall registrerte studenter. Likevel, det var alltid plass til alle som ønsket å delta. Det ble endel prøving å feiling med Panopto-kameraet i starten av semesteret. Jeg endte opp med å la kameraet være fiksert på 2/3 midten av tavlen. Forelesningsnotatene ble publisert på wiki-siden i forkant av hver forelesning. Vi hadde felles fremdriftsplan med 4N bortsett fra den første uken da vi studerte partiellderivasjon.Dette opplegget fungerte godt. Undervisningsmaterialet kom fra Kreyszig, samt et antall andre kilder og tidligere års forelesningsnotater av varierende kvalitet. Jeg har inntrykk av at mange studenter ikke skaffet seg boken, og jeg synes også selv at den ikke er spesielt god eller godt egnet til å dekke pensum. Jeg tror det hadde vært en fordel å redusere antall læringskilder til én matematikkbok (ikke Kreyszig. Norsk?) og én programmeringsbok. Dette ville bedret studentenes oversikt over pensum og det er også en fordel å ha, og bli kjent med, kilder som kan være robuste referanser i fremtiden. <h2>Øvinger.</h2> Kurset hadde ukentlige øvinger, øvingstimer og plenumsregning, administrert av øvingslærer i samarbeid med de to øvingslærerne i 4N. Det var opprinnelig krav om 8 godkjente øvinger for å få gå opp til eksamen, men dette ble redusert pga at eksamen var allerede påfølgende uke etter siste forelesning. Jeg forstod at det ble mye jobb med å administrere forespørsler om utsettelse av innleveringer fra studentene. Det er uklart for meg om det finnes en overordnet politikk for disse spørsmålene. En ulempe med å gi utsettelser er at det også forsinker publiseringen av løsningsforslagene. Altså, øvingsopplegg der det er rom for utsettelser av frister fungerer ikke særlig godt. I fremtiden kan det være lurt at øvingslærer deltar på referansegruppemøtene. <h2>Eksamen.</h2> Vi hadde ordinær skriftlig, 4-timers eksamen uten hjelpemidler. Jeg besluttet å ikke videreføre tidligere års praksis med å tillate selvskrevne notater (gult A5). Derimot var en formelsamling inkludert i eksamen. Denne formelsamlingen bør revideres før eventuell videre bruk. Den bør utvides og gjøres mer generell. Jeg anbefaler å publisere en slik samling tidlig i semesteret. Jeg og ekstern sensor var enige i at eksamen var både lang og noe vanskeligere enn vanlig. Karaktergrensene ble derfor justert. Strykgrensen ble f.eks. satt til 35% score, fra standard 40%.Det ble klart under rettingen at det var store hull i forkunnskapene hos en stor del av studentene. To konkrete eksempler på temaer var kjerneregelen ved derivasjon og substitusjon i integraler. Jeg anbefaler at man tidlig i kurset gjør studentene oppmerksom på hva man forventer av ferdigheter innen grunnleggende kalkulus. Ettersom det kan være vanskelig å lage gode programmeringsrelaterte eksamensoppgaver, var vår (4D og 4N) tanke at læringsmålene innen programmering skulle oppnås gjennom øvingsopplegget. Det mener jeg er en fornuftig strategi. Jeg har inntrykk av at dette fungerte tilfredsstillende.

Jeg er usikker på verdien av forumet, IMF mattelab. Hvis man skal ha et forum bør det avklares på forhånd hvem som skal svare på hvilke kategorier innen hvor lang tid. Likevel så vi tendenser av at gode spørsmål drukner i repeterende og upresise henvendelser. Det er en fare for at et forum bare blir frustrerende, både for studenter og fagstaben.

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

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1. (general feedback on the final exam) The passing threshold is a little bit high. It was a quite hard exam. There are too many time consuming tasks, 24 minutes for each question is not enough. Before the exam, could be better to explain whether the students can get some points if they write down related formulas (even they could not solve the problem).2. (on the course) In general, it is fine. But would be better if we can make 4D to be a more independent course, so that both the teacher/students can decide/get information earlier.

<ol><li>Yes</li><li>In the first lecture, also the use of coursepage.</li><li>Yes</li></ol>

1. Kunnskap. Studenten kan gjenkjenne, forstå og anvende grunnleggende begreper og metoder fra teorien for Fourierrekker, Fouriertransformasjonen, Laplacetransformasjonen, ordinære og partielle differensialligninger og numerisk løsning av ligningssystem og differensialligninger.

2. Ferdigheter. Studenten kan anvende sin kunnskap om Fourierteori, ordinære og partielle differensialligninger og numerikk til å formulere og løse problemer i matematikk og naturvitenskap/teknologi, om nødvendig supplert med bruk av matematisk programvare.

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Matematikk 4D - TMA4135