### On short-video productions of a linear algebra topic for engineering students

#### Ragnhild Johanne Rensaa, UiT The Arctic University of Norway, Campus Narvik

*(This research was supported by a MatRIC Small Research Grant, project number 140605)*

**Abstract**

The research project reported here focuses on making short videos in mathematics and interpretations of the content of such videos. Two series of short videos about a topic in linear algebra have been produced with a particular aim of being relevant for engineering students. Additionally, feedback from a mathematics interested audience about one particular video have been analysed in order to gain knowledge about how mathematical content in videos is being apprehended. The paper reports on experiences gained both from designing and producing videos and from getting feedback about mathematical content from an audience. Based on this, some concluding remarks are given about how the results may prove useful for future video production.

*Introduction*

The present research project originally had a two-fold aim. One was to design and record two series of short videos in linear algebra. The other was to get feedback about the two series of videos from a selected group of colleague experts. However, as the project developed, it turned out to be valuable to adjust the part about responses from colleagues and instead include results from a pilot study done ahead of making the sets of videos. This pilot study was done by gathering feedback on a YouTube video from a larger group of mathematics interested people. Thus, one part of the project was done differently from that initially intended, where selected colleague experts giving feedback to two sets of recorded videos rather became a larger group of mathematics interested people that gave responses to one short video. There were reasons for making the adjustment: The feedback became more trustworthy when giving response to a video made by an unknown teacher on the web and the larger number of responses represented a greater variety from institutions and thereby a clearer picture of judgements.

The design and making of two sets of mathematical videos, though, was done according to plan. The theme of these sets was basis of general vector spaces. Relevant concepts were presented with slightly different approaches to the mathematics in the two sets, but as far as possible this was the only difference between the designs of the videos in the sets. This was done to enable a comparison between the different approaches.

This report is presented in line with the progress of the research project. First the method is presented, followed by a description of how the content, preparing and production of videos were designed. Next, the response from a pilot expert is discussed. This feedback led attention back to a pilot research project which had been run in an initial stage of the project but had not been properly analysed earlier. An analysis, however, became relevant at this stage, and the results are summarized. Finally, some concluding remarks are given about how the experiences and results may provide useful for future video productions.

*Method*

The design of videos was done with a comparative aim, in which two contrasting cases are in focus (Bryman, 2004, p. 53). In the present project the cases were the two divergent sets of videos dealing with basis of general vector spaces. The videos were intended for students that need to take a course in linear algebra as part of their studies to become engineers. All videos were approximately of the same length, had the same lecturer to present the content and had similar lecturing formats. Thus, as far as possible, the only difference between the two sets of videos was the slightly different didactical approaches to how the concepts were explained. This made it possible to compare the sets of videos.

The planning of the videos was done with instances of design science involved (Wittmann, 1995). This was since the planning developed in stages: In the pilot study the idea of searching for conceptual and procedural approaches in videos was introduced to a conference audience. Feedback from the audience was consulted when preparing the videos. The next phase was to make some video recordings based on the planning. These first recordings soon revealed the need for tighter planning of video content in order to satisfy more of the media principles set out by Mayer (2005). Thus, a refined planning of content was done before most of the videos where made. Finally, feedback from a pilot expert implied some adjustments and one new video made, before the sets of videos were decided in their final form.

As for the collection of data about content of a particular linear algebra video, this was done somewhat out of the ordinary way. The setting was the MatRIC November conference in 2014, and we were giving a presentation at this conference (http://www.matric.no/events/4). In this presentation we introduced and showed a short video from the web, followed by a discussion. Among the wealth of linear algebra teaching videos that can be found on the web, we selected one to use in the session. This was one of the better videos found, produced by MIT, on basis and dimension of a vector space (https://www.youtube.com/watch?v=AqXOYgpbMBM). The video deals with dimension and basis of a vector space spanned by four given vectors. The teacher solves this by letting vectors become rows in a matrix and do Gaussian operations on the obtained matrix. Next, an alternative method is shortly presented, here the vectors are put as columns in a matrix instead of rows, but with a rather similar procedure to solve the case. We analysed this video by splitting in sections, describing steps in the progress of the video. These headlines were put on two sheets of paper with 5 cm space between in order to provide room for comments. Each person in the audience of our session at the MatRIC conference was given copies of these two pages of headings and asked the following question: "Can you locate which parts of the video that you interpret as being *procedural* and which parts being *conceptual*? Please explain you answers shortly". We introduced web-videos about linear algebra in general and the MIT-video in particular, and then presented the MIT-video. Afterwards the audience was given some additional time to write their comments on each video section on the distributed papers. Then we initiated a discussion about the video in plenary before collecting the responses at the end of the session. All together 18 replies were submitted.

*The preparing for and production of videos*

Two sets of videos were made, each contained nine recordings. Four of the videos were common in both sets. For instance, the introduction and the summing up of the theme 'bases for general vector spaces' were identical in both sets. Thus, five videos were diverging, engaging with the content in different ways.

All the videos were produced with the same setting in order to provide cases being as similar as possible – except for the mathematical content. The making of videos was implemented in stages as described in the Method section. The initial plan was to produce shorter recordings with talking head, hand-writing there and then and with a somewhat faster talking than in a normal lecture setting. These requirements are in line with Mayer's principles for designing videos (2005) together with the empirical study by Guo, Kim and Rubin (2014) about how video designs affected student engagement in online lectures. Thus, the main writing format was hand-writing on sheets of paper that was published by using a document camera. However, when starting the recording it was soon experienced that the use of time was a major problem. If wanting to introduce a topic by relating to previous concepts and motivate engagement because of its importance, one could easily use three minutes before actually having started discussions of the topic itself. When the research-based guidelines suggest that videos should be three to seven minutes, there was not much time left for the topic. Thus, a refinement of the design of content of the videos was needed. For instance, carrying out an example with vectors in (R-four) gives a lot of numbers and writing, and even if writing and talking fast, it takes time. Thus, some formulations and calculations had to be premade. Thorough introductions also had to be avoided and discussions of the theme in each video had to start immediately. Based on these experiences, the final sets of videos were made.

The set of videos to be made were intended to have two different mathematical approaches, one set of a more procedural type and one set being more conceptually oriented. This was based on my original idea about procedural and conceptual knowledge. In this idea, a procedural approach meant calculations and following recipes in order to obtain answers to problems. Thus, in the procedural set of videos no proofs of statements and few arguments were offered. Instead, theorems were illustrated by examples showing how the theorems 'worked' in tasks; sometimes offering more than one method to solve a task, sometimes illustrating by making a picture. In the set of conceptual videos, more arguments were given. Lines of reasoning were offered but also formal proofs of statements were presented. This set also included examples, but with more emphasis on arguments about why we solved a task in a particular way. An example to illustrate the different approaches in the two sets of videos, was the recordings presenting properties of a basis. In the conceptual set, this was taken care of by discussing what happens if at least one of the constants is zero in a linear combination of vectors like

how this produces linear dependency of the vectors and not a basis. The corresponding recording in the procedural set of videos presented an example with concrete vectors. In this example, an approach to decide about linear independency was given by calculating the determinant of the matrix that was made by the vectors.

In order not to influence the viewers, the two sets of videos were not given any labels to connect them to procedural or conceptual ideas. They were just labelled 'the S-set' and 'the E-set' before being sent to the pilot expert for feedback.

**Response from a pilot expert**

The pilot expert's feedback was thorough and detailed, both about particular videos and general sets. Two comments were of particular interest. One was that two parallel videos, one in each of the sets, were too similar. This puzzled me since I meant to have given more arguments about 'why' in one than in the other. However, the expert suggested a more illustrative way of doing the task which I found illuminating, and a new video based on this idea was made. The pilot expert's arguments about the content of this new video convinced me to replace the video in the intended conceptual set by this new one. The other expert comment was about two parallel videos following after the unique representation theorem. Both videos used the theorem to represent a concrete 2*x*2-matrix as a vector in (R-four). In one video this was followed by a mathematical argument about linear independency and spanning of vectors, thus confirming a basis. This video was part of the intended conceptual videos set. The other video argued for basis properties by pointing at two types of procedures that could be utilized to state linear independency and spanning. It did not go into details, thus was originally meant to be of a more procedural type involving mainly methods for calculations. However, the pilot expert argued that the reference to two alternative solution methods demonstrated a more conceptual teaching style. The feedback resulted in a switch of these two videos, letting the intended conceptual one become the procedural and vice versa.

*Going back to the pilot study*

The feedback from the pilot expert with some diverging interpretations of the mathematics content compared to my intentions awoke my interest: What is emphasized when we watch a video? What is actually meant by conceptual and procedural approaches in a video? Didacticians seem to have some sort of agreement when it comes to the notions conceptual and procedural, originating in Hiebert and Lefevre's description (1986). Reconceptualizations and further developments have taken place, but still there is a common core of understanding of the concept. A recent working definition by Bergsten and colleagues is as follows:

A *conceptual approach* includes translations between verbal, visual (graphical), numerical and formal/algebraic mathematical expressions (representations); linking relationships; and interpretations and applications of concepts (for example by diagrams) to mathematical situations.

A *procedural approach* includes (symbolic and numerical) calculations, employing (given) rules, algorithms, formulae and symbols (Bergsten, Engelbrecht, & Kågesten, 2015)

The definition is interesting since the authors report a study involving engineering students, but is this opinion about what is meant by conceptual and procedural shared across the mathematical community? In order to contribute to an answer to this question, the pilot study carried out ahead of the making of videos received renewed interest. Up to now the feedbacks about the MIT-video had been used mainly as input when planning for our set of videos. However, since the question we asked the audience before showing the video was "Which parts do you interpret as procedural and which parts are of a conceptual type (about the concept)?", the answers were relevant to our interest in content of videos. A detailed analysis of the 18 submitted responses at the MatRIC conference will be dealt with in (Rensaa & Vos, in preparation). However, the results showed that the mathematical content of the MIT-video was interpreted in a variety of ways, ranging from pure procedural or pure conceptual to a combination of procedural and conceptual. A fourth group did not classify in conceptual or procedural at all, signaling unfamiliarity either with the concepts or the mathematics dealt with in the MIT-video.

**Results**

From the planning, design and implementation of the making of videos in the present project a number of experiences were gained. For instance, it is difficult to manage to introduce and provide satisfying arguments within a 7 minutes long video, it is challenging to relate concepts to each other when presenting these in separate videos, and comparative settings make videos rather similar and somewhat 'boring'. However, many aspects may be improved by practise. In order to manage within 7 minutes, one needs to plan carefully so that the structure for each video is ready. To connect topics in different videos together, it may be useful to show readymade arguments and results which have been explained in previous videos and relate to these when introducing a new topic. New arguments, however, should preferably be written there and then in accordance with Mayer's principles (2005). It is also possible to talk and write rather fast in videos. Students use the stop button if arguments are presented too fast, thus the original speed may very well be faster than in a traditional lecture in class (Rensaa, 2015). However, a careful and ready structure of a video together with speaking and writing fast require careful planning. Often repeated recordings are needed before a final result is obtained. Thus, a main experience from the making of videos is that it calls for detailed planning and repeated recordings which takes time, often more time than initially planned for.

The results from the pilot study indicate that in the community of mathematics interested people, either being mathematicians, didacticians, teachers or students, there is no overall apprehension of what is meant by a mathematical approach in a video being conceptual or procedural. The initial aim of the present project; to produce one set of videos with a conceptual approach and one set with a more procedural approach, is therefore not straight forward. First it is necessary to decide what is meant by these concepts and whether this meaning is what we want to emphasize in the design of videos. It may sometimes be more useful just to list requirements for a set of videos rather than being busy with having conceptual or procedural approaches.

A next step for the present project will be to analyse feedbacks from a larger expert group about the set of videos in order to make the descriptions of the sets more detailed. Then, based on this knowledge of how the sets are interpreted by experts, it will be valuable to investigate engineering students' reactions to the different sets of videos.

**References**

Bergsten, C., Engelbrecht, J., & Kågesten, O. (2015). Conceptual or procedural mathematics for engineering students - views of two qualified engineers from two countries. *International Journal of Mathematical Education in Science and Technology*.

Bryman, A. (2004). *Social research methods*. Oxford: Oxford University Press.

Guo, P. J., Kim, J., & Rubin, R. (2014). How video production affects student engagement: An empirical study of MOOC videos. Association for Computing Machinery. from http://pgbovine.net/publications/edX-MOOC-video-production-and-engagement_LAS-2014.pdf

Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.), *Conceptual and procedural knowledge: The case of mathematics* (pp. 1-27). Hillesdale, NJ: Erlbaum.

Lithner, J. (2003). Students' mathematical reasoning in university textbook exercises. *Educational Studies in Mathematics, 52*(1), 29-55.

Mayer, R. E. (2005). *The Camebridge Handbook of Multimedia Learning*. New York: Camebridge University Press.

Rensaa, R. J. (2015). Ingeniørstudenters bruk av læringsverktøy i et lineær algebra-emne - hvilken rolle spiller nettbaserte forelesninger? [Engineering students' use of learning tools in linear algebra – which role does web based lectures play?] *UniPed, 38*(4), 345-352.

Wittmann, E. C. (1995). Mathematics education as a 'design science'. . In A. Sierpinska & J. Kilpatrick (Eds.), *Mathematics education as a research domain: A search for identity* (Vol. 4, pp. 87-103): Kluwer Academic Publisher.