Mathematics Designed for Engagement: from tasks promoting situational interest to sustained positive mathematics experiences

doi:10.21203/rs.3.rs-3214644/v1

Download PDF

Research Article

Mathematics Designed for Engagement: from tasks promoting situational interest to sustained positive mathematics experiences

https://doi.org/10.21203/rs.3.rs-3214644/v1

This work is licensed under a CC BY 4.0 License

Version 1

posted

You are reading this latest preprint version

Efforts to engage students with mathematics should better take the knowledge of interest development into account. This is a comprehensive study combining personal, philosophical and evidence based considerations aiming for productive contributions to the understanding of the role of interest in long term mathematics engagement. The three parts of the study consist of 1) theoretical review of the interplay of interest development and long term engagement, following with a proposal of interest developing features in mathematics tasks, 2) case study of mathematics tasks following with a model for interest promoting task design and 3) pilot study of teachers’ role in applying interest developing tasks. I will explain how situational interest can be triggered and maintained in ways that result in long term engagement. I will show that common mathematics tasks may lack interest-promoting features. I will provide principles to interest-developing mathematics task design. Finally, I will argue that by unraveling the confusion of the interplay between interest and engagement in research, in practice, in policy documents and in teachers’ professional development, we can improve the quality of students' personal mathematics experiences. The study bridges important gaps in the knowledge of how to promote long term mathematical engagement.

Engagement in mathematics

Interest development

Mathematics tasks

Task design

Mathematics educators’ collective effort of motivating, engaging, attracting and entertaining students to study mathematics is a paramount endeavour that has been ongoing for decades. Interestingly, the issue of student dis/interest in mathematics continues to challenge educators globally despite all those efforts. I have personally been subject to this venture throughout my journey with mathematics, starting from being a child interested in science in the 80’s, a school student in the 90’s, a maths student around the Millenium, and finally in my current position as a mathematics education researcher.

Longitudinal studies can reveal students developing maths aversion during their schooling, as well as gradually worsened development over the years [1]. However, the intervention methodologies typically have fairly short term follow ups [2]. This way the engagement processes that develop slowly or non-linearly may have remained overlooked. In this study, I will argue that mathematics tasks are a natural starting point to mathematics engagement. As evidenced in the study of Middleton, Jansen & Goldin in 2016 [3], student engagement in mathematics can be reinforced through interest, namely, by using activities that trigger and hold student interest both momentarily and in the long run. Acknowledging the different stages of interest development, as well as understanding how to trigger them, may assist in promoting productive long term engagement.

The study addresses long term engagement through three sections: 1) Interest-promoting experiences, 2) Principles for task design, 3) Teachers’ role (Figure 1).

The motive of this study comes from the realisation of mathematics education being in vain if students end up actively discarding what they have studied. There is also a personal motive. In my first research of (dis)engagement about 10 years ago, the data of students’ drawings of their mathematics classrooms showed that by the end of 5th year, as many as 70% of the students in the cohort had ended up seeing their mathematics classes as an entirely negative learning environment [4]. These findings addressed Finnish students, who by the time were excelling in large scale studies, such as Programme for International Student Assessment (PISA) [5].

Later, when analysing longitudinal, nationally representative data of (again) Finnish students, I noticed that even the highest achievers often develop aversion towards mathematics: despite the good grades many students did not feel confident about their skills, and did not find mathematics enjoyable [1]. Girls and women in particular have widely reported to suffer from low self-efficacy and anxiety worldwide [6,7]. These issues likely have an impact on long term engagement: it has been noted in multiple studies internationally that students ‘opt out’ or ‘switch off’ mathematics when it is not obligatory anymore [8,9]. In the referred studies, this has happened with both high and low performing students, suggesting the reason for detrimental long term consequence, letting go of mathematics, might be due to affective issues in particular.

The findings about poor engagement with mathematics are consistent and global, and expands to avoiding other STEM subjects [10]. Such a predicament is against the explicit aims of mathematics education. In most curricula engagement, enjoyment, and interest in mathematics is called for explicitly. For example, in Australia, where I currently work, the recently updated local syllabus [11] states that ‘When students enjoy learning mathematics, they develop a positive self-concept and become self-motivated learners through active participation in appropriately challenging tasks. This can enhance their resilience in solving mathematical problems relevant to further education and their everyday lives.’. Similar statements are made in earlier Australian syllabi, so it is interesting to notice that this goal has not been met for the last decades in Australia [12,13]. In the syllabi, what is meant by interest, enjoyment and confidence, or how to achieve such outcomes, is left for the reader to interpret. In this study, the discussion of the three themes will shed light to why such obscurity might be the very reason for the poor results.

The method design for each theme is elaborated below, building a mixed method approach of a literature review, model proposals, and a small preliminary pilot study with a few case studies informing the themes. Using this mixed approach allows for an evaluative framework to investigate, improve, refine and suggest [14,15], and creates a logical methodology to the proposed models, as also seen in previous mathematics education articles such as Mcintosh, Reys and Reys [15]. The study is a dialogue between practical and theoretical. Drawing from individual cases, plus subjective anecdotal experiences, I will inform the use and applicability of theories. And vice versa: general theories and frameworks are applied to understand individual practices, cases and experiences. The nature of this paper is an initial exploratory study, and part of an ongoing Love Maths research project. This framework was created to assist in identifying key variables influencing interest promotion [16] as well as assist in broadening the research in this area [17].

Interest promotion experiences. Literature suggests a pipeline from emerging situational interest to long term engagement. Such progress, as well as tasks’ role in it, will be elaborated. I will respond to the first theme through discussing how engagement can be promoted through interest development. This section is a literature review, focusing on what is the role and potential of the different stages of interest in the process of engagement. I will proceed by introducing what makes a mathematics task appealing, as well as what are general features known to appeal to school aged children. Building on these findings in the literature, I will propose a theoretical model of interest promoting features in mathematics tasks, entitled ProInterest Features. The proposed model may act as a guide, based on professional experiences and is a framework or template model for further studies to come. The potential to promote interest in two sample tasks will be evaluated using this proposed model, shedding light on how the model looks in practice. This is an opportunity for the reader to have their own individual experience and assess the validity of the model.

Principles for task design. Experts have varying findings about what makes a mathematics task engaging, suggesting the knowledge of task design incomplete. In this section, I will start by discussing findings of a study where three different task types did not make a difference in how students experienced interest. By contrasting the three tasks with the ProInterst Features, I will be able to point out some gaps in task design. Conducting an in depth analysis by implementing a case study within the overall study I am making an investigation utilising qualitative research methods [18], therefore this allowed for a targeted and robust in depth analysis. Based on this ‘case study within a study’ I will discuss principles for task design acknowledging the ProInterest Features. These principles will construct the key model proposed in this study, Maths Designed for Engagement (MaDE). The model suggests a blueprint to design and execute mathematics tasks which both stimulate and maintain learner interests, the term task referring to a digital, textbook or alike visualised prompt intended to support the student’s learning.

Teachers’ role. What teachers view as appealing to students might differ from students’ view, but it is nevertheless the teacher who determines what activities to bring in the mathematics classroom [19]. I will briefly discuss what we know about teachers’ knowledge of engagement. I will then introduce a pilot data about teachers’ views when it comes to interest development, and discuss resources that are easily accessible to teachers using keywords such as' fun’, ‘interesting’ or ‘engaging '. From the survey data, I identified what teachers proposed to trigger and maintain student interest. I also wanted to see if additional themes would emerge, so I reviewed the data phenomenologically. An additional theme was available in the data: it was about the functions teachers give to maths activities. This resulted in the final case study within a study, where I researched what is readily available, ‘on the market’ to teachers and educators, and examined what ProInterest Features are present in a task that is advertised to be engaging for students.

The survey was short and semi-structured, the questions inviting the teachers to describe their initial “gut beliefs" about what their students would find catching or maintaining their interest in the tasks with ProInterest Features [20]. To collect data amid the global pandemic of Covid-19 (beginning of year 2021), when people spent time at home and online [21], I sent a short research participation invitation as a direct message to over 1000 Instagram accounts that clearly indicated a teacher account (teacher of mathematics or primary teaching). Instagram is a highly popular platform with a high engagement amongst teachers and educators [22,23,24] as well being a potential mathematics educational tool for students [25]. The data collection was approved by my university’s ethics panel. It was unlikely for the recipient to encounter the direct message I sent due to the platform’s high spam. As expected, the resulting number of voluntary participants in the survey that responded and did want to participate was 10 out of 1000. The teachers were located in Australia, India, The UK, The USA and Italy. The age range was 23-52, the years of experience ranged from one month to 27 years. I was unable to confirm the real identities of the participants, however all the respondents indicated in the consent form that they do teach mathematics, and they also publicly shared and discussed resources and insights regarding mathematics teaching.

During my four decades long relationship with mathematics, I have become well aware of the disheartening findings showing how difficult it is to attain students’ engagement [8,9,13,26]. The literature suggests students’ engagement connects with better learning quality; it can make teachers work easier; it plays a role in students choosing STEM-related careers and is a preferred learning outcome per se [27,28,29]. This means that the concept of engagement is of interest to students, teachers, researchers and stakeholders. The term itself can be defined through several key concepts, such as the type of behaviour, the depth+duration of commitment, or as a certain kind of cognition or emotion [30,31,32].

To unravel the problem of disengagement, I will start from another concept, interest, proposing it to be one of the key components to tackle in the journey towards long term engagement.As Middleton and others [3] state:

in the here and now, we can see students developing interest, and engaging with gusto, while over the long term, they may try to avoid further mathematics coursework. Or, students may be bored with a particular algebra task, but tend to enjoy and seek out algebraic puzzles in their free time. What we see is the cycle of self-regulation that serves to manage engagement behaviours, but may or may not result in long-term valuing of mathematics, positive affect, and improved mathematics performance. (p.19)

The statement underlines the importance of all aspects of engagement to develop. It is not enough to see students engaged (participating), if deep understanding and emotional fulfillment are not accompanied.

Middleton and others (ibid. p.19) shed more light to the issue by arguing that ‘When experiences tend to be consistent and coherent over time with regard to motivational affordances, it becomes more probable that the person will develop a (positive or negative) long-term disposition and identity toward mathematics.’. With motivational affordances, I understand the authors referring to the emotional aspect of engagement as described in the seminal research by Fredricks, Blumenfeld, and Paris in 2004 [30] where engagement is defined as a multidimensional construct including behavioural, affective and cognitive components. The three components are also manifested in the concept of interest: the psychological state of attention and affect toward a particular object or topic, and a motivation to engage, and reengage [33]. The process described by Middleton and others (ibid.) was confirmed empirically by Rotgans and Schmidt [34], who noticed that individual interest can be increased by a repeated arousal of situational interest.

To land on this positive long-term engagement through consistent motivational experiences, interest triggering activities are an important factor. This consequential function is one of the two perspectives of interest, focusing on ‘What would create interest that would result in…?’[35], and could range from what is asked from students to what the process looks like; what resources are given and what is the significance of the work [19]. Another way to define the concept is seeing interest as a reaction to something that has been present. This perspective leads to examining the emergence of interest, circling back to interest triggering activities. Both approaches see the concept as something that happens between an individual and an object, environment or situation and can be seen as representing the trait/state aspects of interest. A dual function of interest promoting activities appears: fueling interest to emerge, and being present to contribute to a motivational experience short and long term.

The two functions are woven into a four-phase model of interest development by Hidi and Renninger [36]. Frequent and successful situational interest leading in long term personal interest proposes a flow chart displayed in Figure 2 below.

Situational interest was elaborated by Mitchell [37] a few decades ago, however there is still more to investigate. Rotgans and Schmidt [34] called for studies uncovering the ‘underlying mechanisms' for the growth trajectory for individual interest modelled by Hidi and Renninger [36] (Figure 2). My contribution is: what features in mathematics tasks can promote the first two stages of interest development, triggering interest and then have it maintained? Researchers have often suggested specific means, such as modelling [38,39], to promote interest towards mathematics. Also Guberman and Leikin [40] state that mathematical problem solving is ‘at the heart of mathematics teaching and learning’ (p.33), seeing a challenge as ‘an interesting difficulty’ (p.36). Another commonly suggested approach is to contextualise problems [41]. In their study, Sullivan and others (ibid.) discuss the use of real life contexts in particular, but they end up challenging the consensus of using real life contexts as a straightforward solution that serves all students. Sullivan and others (ibid.) remind about some contexts being alienating to some students, also that adult’s real life contexts differ from those of students, and summarise other studies showing that imaginary contexts (fantasies) can have stronger responses than real life contexts. Ineffective use of context or forced modelling challenges may explain why only 34% of surveyed students found themselves interested in tasks they had themselves co-designed to enable reasoning, communicating, cognitive challenge and real life feel [42].

Harackiewicz and others [33] point out that interest is essential to academic success. The authors (ibid.) recommend interest interventions including situational interest, problem-based learning, and utility value, but the practical implementations are yet to be fully explored. In Nyman’s [43] study, students’ interest was said to be prompted by 1) having mathematics in the foreground, 2) giving challenge, 3) allowing student influence and 4) allowing students to present their results. Nyman’s list aligns with the suggestions discussed above, as well as with a list by Attard [44] (2021), where Nyman’s (ibid.) four aspects are supplemented by 5) relevance 6) positivity and 7) mixed format, such as adding technological affordances. My concern is: how is this done? The listed aspects seem subjective–what do these features mean for a student?

To make things more practical, I will bring into the discussion what appeals to young people in general, for example, what features are used to entertain or sell products to school aged children. According to market research [45], the triggers for students from Year 2 to Year 9 are: Sleek, neat designs; Stands out; Uniqueness; Humour; Does new stuff all the time; Creative; Gives choices; Positive attitude.

Psychology research knows a lot about how to arouse attention. If, for example, a cue is given with laughter, the human brain’s attention is better caught than when given in a neutral tone [46]. Whenever something unusual or unexpected occurs, the human brain easily pays attention. Maintaining the situational interest may be the tricky part: as soon as the initial surprise has passed students’ attention can begin to dissipate. On the other hand, maintaining interest might happen through very traditional means. For example, including aspects of drilling into activities may be one way to keep students engaged, and also promote further creativity, as noted by Csikszentmihalyi [47, p.123], who suggests that ‘It is a mistake to assume that creativity and rote learning are incompatible.’.

Combining what is suggested above by education research, psychology research and market research, I categorised the different ideas into ProInterest Features that trigger and maintain interest (Figure 3). It is important to note that I am only hypothesising how each feature might contribute to triggering or maintaining interest.

Next, I will introduce two tasks that manifest most or of the listed features. The tasks seem very different at the first glance, but as the reader will learn, they have many similarities when it comes to their potential to trigger and maintain situational interest. The first one is done by my own team and shared in social media as part of the many experiments I have run within the different stages of the Love Maths research project. In Figure 4, I discuss how the task meets the features promoting emerging and maintained situational interest.

The second task comes from a study of Stylianides and Stylianides [48]. I first encountered this task (see Figure 5) in a conference, and was much influenced by the ideas the authors introduced in their presentation. The task is confusing and intriguing, and in my experience surprisingly easy to get off to a good start if the facilitator directs the problem solver to start finding potential combinations of the three factors of 36. Most importantly, the problem solvers do not need any special skills, it is enough to do some basic calculations, and realise the logic behind what prompts Pythagoras to ask for one more hint. Stylianides and Stylianides (ibid.) findings show important positive changes in student’s beliefs about mathematical problem solving after having experienced just this one task. I have since applied the task on multiple occasions, and my experiences have confirmed the findings. My students have, albeit puzzled, eagerly taken on the challenge, persevered, and expressed immense content after having solved the problem. Every one of them has solved the problem, and many of them have expressed being pleased to experience mathematics as fascinating, some of them for the first time in their lives. Similar account was described in the findings of Stylianides and Stylianides (ibid.). It is interesting to see that the task meets most ProInterest Features (Figure 5).

Five out of seven features are common in the two tasks (Figures 4 and 5) despite their seemingly differing appearances. This suggests a variety of mathematical activities to meet the interest promoting features, resulting in long term engagement over time. Based on my anecdotal knowledge I argue that many mathematical resources are yet to manifest these features.

The learning of mathematics likely tolerates a few indifferent experiences, however when promoting interest and long term engagement is explicitly aimed, applying the interest promoting features might be critical. For example, in a study of Schukajlow, Leiss, Pekrun, Blum, Müller, & Messner [49], the authors were not able to find differences in how students assessed their interest between three different task types. The finding may be due to the fact that there were no essential differences between the task types when it comes to either triggering and maintaining student interest. In the next section, I will first take a closer look at the tasks used in Schukajlow and others (ibid.) study to see how they meet the interest promoting features (Figure 3). After that, I propose how to include all features in task design.

Schukajlow and others [49] call for references to reality to attain more student interest, increased motivation and positive emotions. In their empirical study (ibid.), three different task types were introduced, asking students to assess their interest towards the tasks. No differences were found. The three types described by Schukajlow and others (ibid.) were 1) ‘intra-mathematical problems’ (mathematics tasks without context), 2) ‘dressed up mathematical word problems’ (for example, asking students to apply a mathematical operation in a fabricated situation) and 3) ‘modelling problems’ (for example, using a real life context where mathematical operation needed to be applied). As discussed earlier, modelling and real life relevance have often been suggested to promote interest, which may explain the assumption of the degree of real life context accounting for interest. Since this was not the case, it is worth remembering that real-life contexts were problematised by Sullivan and others [41], and not included as one of the features suggested by Nyman [43] and Cuddeford-Jones [45] to promote interest (see Figure 3). Attard [44] calls for the relevance and explicit links and value to students’ lives, but is not suggesting manufactured contexts.

In Figures 6, 7 and 8, I will show how the three sample tasks representing the three task types in the study of Schukajlow and others [49] meet the ProInterest Features. The justifications are given after the images.

All three tasks are clearly mathematical. The first one is very straightforward, making it simple. None of the tasks stand out: in mathematics textbooks the use of images or pseudo context with characters is very common and do not make a task to stand out. On the contrary, the way images and context stories are used in the tasks provides a usual format that assists in completing the task (once details are collected, one can apply the appropriate operation and forget about the characters). None of the tasks communicate humour, and they are not creative.

The essential difference lies in challenge, in how much details are given to the students versus how much they need to figure out (i.e. model) themselves. Challenge is a feature to maintain interest, which becomes more apparent after having started with the task, but in the study of Schukajlow and others [49] the examinees were instructed not to solve them, but based on the reading to determine whether they would be interested in working on the problem. At one glance, one would likely pay attention to features that trigger interest, in which there were no essential differences. This might explain why the researchers did not find any statistically significant differences in students’ perceptions of these tasks: similar findings were introduced by Parhizgar and Liljedahl [39], who also noticed that students could not differentiate three types of mathematical problems. However, after having solved the given tasks the students reported more flow while solving word problems, which were mathematical and simple to trigger some interest, and also challenging and in a usual format to maintain some interest.

In Figure 9, I propose the model Maths Designed for Engagement (MaDE) that ensures that the features to trigger and maintain interest are present in mathematics tasks. The MaDE model is introduced through a version of the mathematical problem (using Pythagoras Theorem) that is common in the three tasks contrasted by Schukajlow and others [49], with a blueprint for task design that acknowledges all interest promoting features.

In the sample task (Figure 9), the problem solver must find out the length of the triangle’s sides, compare their added length to the shortcut, and use the length difference to estimate the number of their individual step length to cover this additional stretch. They need to remember the return trip, and multiply by the number of school days. In each step, similar mathematics is used even if the number of school days isn’t totally accurate, or the estimation of the step length does not hold. When getting to the end of the problem, one cannot check the answer key for the right answer, instead one has to rely on their own reasoning and assess how realistic their answer is. The task can evolve to make connections outside the mathematics classroom (why having green areas?). Such debates can turn meaningful for students: for example, when I have facilitated The Blonde Hair Problem (Figure 5), students have sometimes had opinionated conversations about gender differences while on the task, Pythagoras being known by everybody and female mathematician Hypatia by hardly anyone.

In the sample task in Figure 9, all features that trigger and maintain interest are present. In step 2, it is advised to introduce a recognizable situation. This can happen in multiple ways: it could be through a real life context that really matters to students, but it could also be something abstract, as long as it creates a clear entrance to the task. Adding a background story, such as Trainer Manfred and their team (Figure 7) does not necessarily make the situation more familiar, as pointed out by Sullivan and others [41]. We still do not know who Trainer Manfred is, the mere mention of them does not make us interested in them. If instead of Trainer Manfred it was somebody the student would recognise as someone who would ask their team to engage in silly training habits, the situation would make more sense; the context would truly contribute to the feel of the task instead of seeming tokenistic. For step 4, the sample task uses a meme. This results in copyright considerations; the suggested blueprint for interest developing task design does not take into account further challenges, it simply serves as a starting point to know what to look for. On the other hand, obstacles such as copyright issues do not need to be solved by sticking to what can be printed in a task, as a part of the task design could be a provision of original media to be shown and discussed. In steps six and seven, interactivity and assistance could be added by doing just that: providing teachers with technological tools, such as GeoGebra, to dynamically operate with variables like step length. These considerations point to multimedia format tasks, consisting of what is shown to students to ensure simplicity, and what additional resources are available; such greater digital affordances suggests a documentational approach [50].

Attard’s studies [26,44] confirm the principles the teacher could follow should they wish to engage their students: rich interaction, constructive feedback, allowing challenge and success for all, allowing making choices, mathematical, relevant, and served with enthusiasm. This was exemplified in Attard’s earlier study [26], where she showcases both poor and rich engagement as a result of teacher-student interaction. In both cases, the teacher has the right intentions and may very well use resources aligning to what should make it engaging for students. The difference is made in the nature of assisting. When a teacher promptly helps students out or encourages students to take on trivial challenges, the reward gets lowered and interest dissipates. On the contrary, Attard (ibid.) introduces a teacher who seems to allow risky tasks (in the sense of not being certain if the task gets solved), and then enters into an examination as a collaborator with their students. In both cases, a teacher helps and assists their students, but there is a greater fulfillment in the latter approach, likely to explain the higher engagement that was documented in Attard’s study (ibid.) for the adventurous teacher.

I saw this happening first hand in a project I ran five years ago, where the participating teachers, all passionate about making mathematics engaging, were given interest promoting activities. One of the teachers, after having introduced each task to their students, let the working phase evolve naturally, regenerating the activity together with their students. This led to engagement. Another teacher, although being very professional and liked by their students, followed the activity instructions to the T, held the reins and made sure the anticipated outcome was achieved. There was also a lot of housekeeping. One could observe active participation and even reflective thinking (cognitive and participatory aspects of engagement), but after the lesson, the students did not express specific excitement and value when casually interviewed. Even a great activity may fall short if the facilitation is very poor: in a study of Lee, Coomes, and Yim [51], it was noted that a similar task could be facilitated to prompt new knowledge, or as a mere exercise keeping students on their current level (by suppressing some of the task affordances).

Teachers use multiple criteria when they select their teaching resources. In a study of Siedel and Stylianides [52], how teachers select tasks were categorised in 1) considering students’ needs (student-driven selection), 2) finding resources that align one’s teaching preferences and style (teacher-driven), 3) focusing on content (mathematics-driven), 4) using what is accessible (constraints driven), 5) being inspired by encountered resources (resource-driven), and 6) adapting to sharing of resources, collaboration or other cultural aspects (culture-driven). Pepin, Gueudet, and Trouche [53] discuss reasons for teachers to apply/reject resources, showing evidence that resources that align with teachers’ beliefs or views of what to teach plays an important role.

Next, I will report on the findings of a sample data collected online about teachers’ view on tasks that have ProInterest Features. To select such tasks, a teacher might use a student-driven selection, or become resource-driven.

5.1 Findings of the online survey

Teachers were asked to pick a task that they felt would a) trigger, b) maintain their students’ interest [20]. They could pick the same task for both, or choose one that would trigger, another that would maintain interest. All tasks did get mentioned.

When describing what in the selected task would trigger students’ interest, teachers listed the following features: looking simple or being simple (e.g. little writing), having a nice layout (colourful, clear, visual), being unexpected, using storytelling (or real-life connections), looking fun or being fun, giving a challenge, being rich, allowing physical activity and/or creativity or roleplay. These features cover A2: Simple, A3: Stands out and A4: Humorous, indicating the teachers’ view on what triggers student interest well aligns with the proposed model. However, the features teachers mentioned also cover B1: Challenging and B3: Creative, which according to the proposed model likely maintain interest.

Interestingly, when describing what in the selected task what would maintain students’ interest, teachers listed mainly the same features: simple, unexpected (or novel), doesn’t feel like maths, outside the classroom/book, being short, fun, challenging, rich, physical, allowing creativity, discussion, roleplay, having social aspects, or being competitive. These features cover A2: Simple, A3: Stands out, A4: Humorous, B1: Challenging and B3: Creative. The exact same features were covered in both cases, showing evidence of the teachers being ignorant to the differences between what triggers, what maintains interest. Based on this evidence alone, we cannot say whether it is the teachers that know better: maybe in practice the features have double functions. However, in our data, teachers did not make any remarks about the different stages, namely, emerging and maintained situational interest [36], keeping it an open possibility that the stages are foreign to teachers.

I identified the teachers associating appeal in 1) the nature of the task and 2) what you get to do with it. This well aligns with the idea of triggering (the nature of the task) and maintaining (what you get to do) interest. Teachers seem to know what in general connects with interest-development.

Some teachers’ saw tasks being open as problematic (Feature B2 to maintain interest). The activity being short was seen as plausible:

If they were allowed a discussion over the task it could maintain their interest, but that kind of discussion would delve more into philosophy than math and take longer than a "starter" should (response #8)

This can be seen as one example of the multiple goals teachers have to entertain, causing tensions between the many aims and what gets actualised. The purpose of interest-developing activities is indeed getting students’ activated for a bit longer and even start exploring new grounds, but the teacher might feel pressured to keep the lesson focused and under control to reach cognitive learning goals and only allow minor flexibility, as also noted by Kaur and Chin [19].

One more topic was identified in teachers’ accounts. The role of ‘fun maths’ was acknowledged, not just as a feature to trigger interest (A4: Humorous: it could make students ‘excited’ and bring ‘joy’)but also as a segway to content teaching:

I could imagine using a task like this throughout primary, by getting students to use different mathematics symbols and increasing the difficulty. (response #9)

Evidence of similar endeavours, teachers looking for entertaining activities as a pathway to content teaching was recorded by Grave and Pepin [54]. In their study (ibid.), the authors identified teachers' usage of resources to ”inspire” teaching, being directly linked to the teaching objectives. The study also showed that the main sources teachers were using to search for resources to inspire teaching were textbooks and the Internet, the latter being mainly used to this purpose only by all four teachers who participated in their study (ibid.). The last bit of this study is to examine what the Internet has to offer.

5.2 Interest promoting features in “fun maths”

When using web-based search engines, such as Google, it is possible to find several million search hits using the key term ‘fun maths task’ or similar keywords (Fun Maths 952,000,000 results, Inspiring maths 52,900,000 results, Motivational maths 40,900,000 results, Enjoyable maths 29,100,000 results, Entertaining maths 33,500,000 results, tested 31st of January 2023 using Google). Looking at the results one gets when using ‘Fun maths’, it is not hard to notice certain common features in the provided activities. In Figure 10, you can see the first page of results in Google Images search using the key term ‘fun maths task’): there are some noticeable common features of the results. They predominantly use a bright colour palette, characters (such as imaginary creatures, plants and animals), hands-on interactive materials and the use of different fonts in one task. Visual richness may be prevalent in these tasks but cannot alone describe what ‘fun maths’ means. What one can do, however, is to examine how these tasks abundant for teachers manifest the ProInterest Features.

I have selected a representative example task ‘Silly Monster Number Towers’ (Figure 11) that is considered ‘fun maths’ based on the commonalities we have seen and described above. The task is advertised to be fun, and is offered by a platform entitled “Fun Learning for Kids” [55]. The prompt is to use two colours of blocks to build a tower that matches the numbers in dominoes the bright colored monsters have on their bellies. In addition, students are asked to write a number sentence for each tower. For example, if there is a monster with dominoes 2 and 5, students can choose 2 yellow blocks and 5 red blocks, pile them to represent a ‘monster tower’ and write 2+5=7 in a designated recording sheet.

Let us see what appealing features are present in ‘Silly Monster Number Towers’. It has mathematics in the foreground (has feature A1), and there is humour in the title of the task, the characters are indeed labelled as silly (has feature A4). The format is typical to a mathematics task; the use of blocks assists a student with the manipulation of the quantities, and the designated recording sheet gives a clear structure for documentation (has feature B3). However, there are several missing features when it comes to both triggering and maintaining interest. The task is not simple: it requires reading of a prompt written in a small font, plus grasping the different steps of what is asked of a student, not in terms of making mathematical discoveries, but following orders (does not have feature A2). The use of colours, characters and fonts likely aims for a neat look, however many textbooks and online resources nowadays use similar designs (does not have feature A3). The intellectual challenge is minor: it is a closed addition task where two integers are visually displayed and manipulated (does not have feature B1). The student influence is tokenistic at best, you only get to choose the colours; the task is not evolving or creative (does not have feature B3).

When I first encountered the ‘Silly Monster Number Towers’, my first impression was that the task would likely promote student interest. This was albeit having read literature on the topic and having engaged myself in the design of interest developing activities. The analysis here suggests otherwise: not many interest promoting features are available. If I am myself prone to being mistaken, I doubt the average teacher would be critical with a task’s potential to promote interest. It is not to say that the task wouldn’t be very applicable in mathematics class: my speculation is that procedural fluency can be developed through it. Interest could also be promoted if it was used surprisingly as an alternative to very traditional, dull mathematics exercises, simply by offering something different. In regular use, however, one can well argue that over time tasks with accessibility issues (not being simple), unimaginative design (not being particularly exciting), and omitted features to maintain interest (no challenge, minimally interactive, not creative) make a repetitive, meaningless experience.

Based on this pilot analysis, I cannot make generalisations over all tasks that are aimed to promote interest. I cannot confirm how students experience these tasks; much depends on teachers’ skills to use tasks as affordances, not in a tokenistic manner. These examinations will be my focus in the future. However, I have discussed a task that is easily available for teachers all around the globe through the engine search using the key term ‘fun maths task’ and shown that in such a task, the features that would trigger or maintain interest can be largely missing.

The findings of this study can inform why students are ‘switching off’ maths, as worded by Martin and others [8]. If situational interest is not being triggered or maintained, such detrimental experiences can feed disengagement over time. In the task design view, I have provided four features to trigger interest (Mathematical, Simple, Stands out, Humorous) and three features to maintain interest (Challenging, Format, Creative), and pointed out that typical mathematics tasks can lack many of these features. Mathematical tasks may thus be an important factor in students losing opportunities for interest development. Teachers’ role in the issue of student engagement lies in the unclarity of the stages of interest development, as well as being misguided by tasks that are easily accessible with keywords to engage students, such as ‘fun maths’, potentially lacking interest promoting features and doing a disservice to the aim to engage students. With the findings of this study, we can add to our knowledge about what we need engagement for (Figure 12).

Theoretical implementations. I have clarified the role of interest development in long term engagement, consisting of behavioral, cognitive and affective aspects [30]. In future studies, the importance of interest development as a separate process, in addition to being a factor in the learning process, should be better addressed. Warshauer [56] reminds how sometimes studying mathematics can feel momentarily difficult, exhausting and tedious, yet the student might experience this as a productive struggle and learn effectively, discovering the joy of knowing mathematics in the long run. Research shows some activities positively impacting students’ agency and examination skills and yet being detrimental to explicit learning outcomes; for example, inquiry-based approach was shown less effective than teacher-led approach in PISA 2015 data in the study of Oliver, McConney, & Woods-McConney [57]. The latter example shows that sometimes affective processes develop independently from the learning process, requiring explicit focus.

Practical implementations. The role of interest development as part of the learning process, but also as an independent process, must be acknowledged in policy documents. In task design, we must include features that trigger interest in particular, as well as features that maintain students’ interest. In teachers’ professional learning, there is a need to elaborate on interest development, and the subtle differences in teacher-student interaction that can either promote long term engagement or contribute to slow regression in student engagement. This study is a mere starting point; I challenge the mathematics education community working towards a well established framework for Maths Designed for Engagement. The model proposed here must be extensively tested and the features better defined. There is also a need to further examine the different manifestations of interest promoting activities, as well as students’ responses to them.

The preliminary pilot social media survey showed promising results that teachers are willing and wanting to engage their students further. In the teachers’ entries, seeing interest promoting activities as a segway to content teaching was addressed. What the interest promotion stimulates in many ways aligns with what is aimed by rich tasks [58,59], mathematical discourse [60], number talks [61] and also mathematical modeling [62]. Features promoting a comprehensive learning process may overlap with features promoting interest development, and yet another factor is how the affordances of the features are leveraged. The role of a teacher in selecting and applying tasks is of utter importance. In the next stages of the Love Maths project, we have collected more data on teachers’ views and application of interest developing tasks. We are looking forward to continuing the discussion with those data.

Some proportion of maths activities must appeal to the feelings aspect. Middleton and others [3] suggest that the in-the-moment engagement serves as a combination of intrinsic, extrinsic, social, and individual factors that are interrelating, that are drawn from personal, social, cultural and macro cultural constraints, and are on the other hand contributing to the development of a longer-term disposition towards the subject. We should not accept bringing students to the culture that mathematics is boring, hard and not relevant. Sullivan and others [41] proposed fantasies: I want to finish this study by mentioning what I once heard from a mathematics teacher student: people are interested in superheroes, or antiheroes like Donald Duck and other irrelevant stuff. In the student’s opinion, we just need more space for imagination: fantasies, curiosity, stories, humour. One can argue that mathematics inherently embodies features that appeal to the human mind. People are fascinated by imaginary realities, logical ponderings, and infinite ways to develop mathematical structures. It is time we bring that fascination to where new mathematics users are generated: mathematics classrooms.

Funding

This study was funded by the Rapid-Fire Research Grant Scheme Application, UNSW, Sydney, Australia.

Competing Interests

No competing interests to disclose

Data availability

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

Compliance with Ethical Standards

The Love Maths study is classified as low risk research under University of New South Wales’ (UNSW) criteria for classifying negligible risk for research (UNSW, 2022). The research ethics for the entirety of the Love Maths Research Project was approved by the UNSW Australia Human Research Ethics Advisory Panel prior to contacting schools for involvement.

Acknowledgements

I want to thank Scarlett Li-Williams and Samantha Gooch for their assistance in this paper.

Metsämuuronen, J., & Tuohilampi, L. (2014). Changes in achievement in and attitude toward mathematics of the Finnish children from grade 0 to 9—A longitudinal study. Journal of Educational and Developmental Psychology.
Woolcott, G., Le Tran, T., Mulligan, J., Davis, B., & Mitchelmore, M. (2020). Towards a framework for spatial reasoning and primary mathematics learning: an analytical synthesis of intervention studies. Mathematics Education Research Journal, 1-31.
Middleton, Jansen, & Goldin. (2016). Motivation. In G. A. Goldin, M. S. Hannula, E. Heyd-Metzuyanim, A. Jansen, R. Kaasila, S. Lutovac, P. D. Martino, F. Morselli, J. A. Middleton, M. Pantziara, & Q. Zhang (Eds.), Attitudes, beliefs, motivation and identity in mathematics education: An overview of the field and future directions.
Tuohilampi, L., Laine, A., Hannula, M. S., & Varas, L. (2016). A comparative study of Finland and Chile: the culture-dependent significance of the individual and interindividual levels of the mathematics-related affect. International Journal of Science and Mathematics Education, 14(6), 1093-1111.
OECD. (2010). PISA 2009 Results: What Students Know and Can Do – Student Performance in Reading, Mathematics and Science (Volume I). ISBN 978-92-64-09144-3. . https://doi.org/http://dx.doi.org/10.1787/9789264091450-en
Morán-Soto, G., & González-Peña, O. I. (2022). Mathematics Anxiety and Self-Efficacy of Mexican Engineering Students: Is There Gender Gap? Education Sciences, 12(6), 391.
OECD. (2013). PISA 2012 Results: Ready to Learn: Students’ Engagement, Drive and Self-Beliefs (Volume III).
Martin, A. J., Anderson, J., Bobis, J., Way, J., & Vellar, R. (2012). Switching on and switching off in mathematics: An ecological study of future intent and disengagement among middle school students. Journal of Educational Psychology, 104(1), 1.
Nyman, M. (2020). What do students’ feel about mathematics?: Compulsory school students’ emotions and motivation towards mathematics Stockholms universitets förlag].
Berger, N., Mackenzie, E., & Holmes, K. (2020). Positive attitudes towards mathematics and science are mutually beneficial for student achievement: a latent profile analysis of TIMSS 2015. The Australian Educational Researcher, 47(3), 409-444.
NSW Education Standards Authority (2023). The aim and objectives of the Mathematics K-10 Stage 1-5 Syllabus. Accessed 17 February 2023.
Watson, J., Beswick, K., & Brown, N. (2006). Teachers’ knowledge of their students as learners and how to intervene. Identities, cultures and learning spaces 29th annual conference of the Mathematics Education Research Group of Australasia, Canberra.
Beswick, K., Faragher, R., & Sullivan, P. (2021). The knowledge and skills gap in Australian secondary mathematics classrooms. Knowledge and Skills Gap Report 2021.
Brinkerhoff, J. M. (2002). Assessing and improving partnership relationships and outcomes: a proposed framework. Evaluation and program planning, 25(3), 215-231.
Mcintosh, A., Reys, B. J., & Reys, R. E. (1992). A Proposed Framework for Examining Basic Number Sense. For the Learning of Mathematics, 12(3), 2–44. http://www.jstor.org/stable/40248053
Sacred Heart University Library. (2020). Organizing Academic Research Papers: Theoretical Framework. Importance of Theory. Retrieved 6 Feburary fromhttps://library.sacredheart.edu/c.php?g=29803&p=185919#:~:text=Having%20a%20theory%20helps%20you,differ%20and%20under%20what%20circumstances.
Kamil Kazan, M. (1997). Culture and conflict management: A theoretical framework. International Journal of Conflict Management, 8(4), 338-360.
Feagin, J. R., Orum, A. M., & Sjoberg, G. (1991). A case for the case study. UNC Press Books.
Kaur, B., & Chin, S. L. (2022). Nature of mathematics tasks and what teachers do. Current Opinion in Behavioral Sciences, 46, 101169.
Love Maths. (2021). Love Maths Survey tasks for the pilot study 2021. https://docs.google.com/document/d/1xNjvKP63_0mQThUHn5NVXu9tYmn4bKjuDFvSjh-RFTY/edit?usp=sharing
Pandya, A., & Lodha, P. (2021). Social connectedness, excessive screen time during COVID-19 and mental health: a review of current evidence. Frontiers in Human Dynamics, 3.
Carpenter, J. P., Morrison, S. A., Craft, M., & Lee, M. (2020). How and why are educators using Instagram? Teaching and teacher education, 96, 103149.
Carpenter, J. P., Shelton, C. C., & Schroeder, S. E. (2022). The education influencer: A new player in the educator professional landscape. Journal of Research on Technology in Education, 1-16.
Shelton, C., Schroeder, S., & Curcio, R. (2020). Instagramming their hearts out: What do edu-influencers share on Instagram? Contemporary Issues in Technology and Teacher Education, 20(3), 529-554.
Mulenga, E. M., & Marbán, J. M. (2020). Is COVID-19 the gateway for digital learning in mathematics education? Contemporary Educational Technology, 12(2), ep269.
Attard, C. (2013). “If I had to pick any subject, it wouldn’t be maths”: foundations for engagement with mathematics during the middle years. Mathematics Education Research Journal, 25(4), 569-587.
McPhan, G., Morony, W., Pegg, J., Cooksey, R., & Lynch, T. (2008). Maths? Why Not? Final Report prepared for the Department of Education, Employment and Workplace Relations (DEEWR).
Nakakojia, Y., & Wilsona, R. (2014). MATHS IS A STRONG PREDICTOR OF STEM ATTAINMENT IN FIRST YEAR UNIVERSITY. Proceedings of The Australian Conference on Science and Mathematics Education.
Wang, M.-T., Degol, J., & Ye, F. (2015). Math achievement is important, but task values are critical, too: examining the intellectual and motivational factors leading to gender disparities in STEM careers. Frontiers In Psychology, 6, 36.
Fredricks, J. A., Blumenfeld, P. C., & Paris, A. H. (2004). School engagement: Potential of the concept, state of the evidence. Review of educational research, 74(1), 59-109.
Helme, S., & Clarke, D. (2001). Identifying cognitive engagement in the mathematics classroom. Mathematics Education Research Journal, 13(2), 133-153.
Stipek, D. J. (1996). Motivation and instruction. Handbook of educational psychology, 1, 85-113.
Harackiewicz, J. M., Smith, J. L., & Priniski, S. J. (2016). Interest matters: The importance of promoting interest in education. Policy insights from the behavioral and brain sciences, 3(2), 220-227.
Rotgans, J. I., & Schmidt, H. G. (2017). Interest development: Arousing situational interest affects the growth trajectory of individual interest. Contemporary Educational Psychology, 49, 175-184.
Renninger, K. A., Hidi, S., Krapp, A., & Renninger, A. (2014). The role of interest in learning and development. Psychology Press.
Hidi, S., & Renninger, K. A. (2006). The four-phase model of interest development. Educational psychologist, 41(2), 111-127.
Mitchell, M. (1993). Situational interest: Its multifaceted structure in the secondary school mathematics classroom. Journal of educational psychology, 85(3), 424.
Maiorca, C., & Stohlmann, M. S. (2016). Inspiring students in integrated STEM education through modeling activities.
Parhizgar, Z., & Liljedahl, P. (2019). Chapter 10: Teaching modelling problems and its effects on students’ engagement and attitude toward mathematics. Affect in mathematical modeling, 235-256.
Guberman, R., & Leikin, R. (2013). Interesting and difficult mathematical problems: changing teachers’ views by employing multiple-solution tasks. Journal of Mathematics Teacher Education, 16(1), 33-56.
Sullivan, P., Zevenbergen, R., & Mousley, J. (2003). The contexts of mathematics tasks and the context of the classroom: Are we including all students? Mathematics Education Research Journal, 15(2), 107-121.
Kohar, A. W., Zulkardi, Z., & Darmawijoyo, D. (2014). Developing Pisa-Like Mathematics Tasks To Promote Students? Mathematical Literacy.
Nyman, R. (2016). What Makes a Mathematical Task Interesting? Educational Research and Reviews, 11(16), 1509-1520.
Attard, C. (2021). Applying a framework for engagement with mathematics in the primary school. Australian Primary Mathematics Classroom, 26(4), 4-9.
Cuddeford-Jones, M. (2012). Cool for kids. https://www.marketingweek.com/cool-for-kids/
Pinheiro, A. P., Barros, C., Dias, M., & Kotz, S. A. (2017). Laughter catches attention!. Biological psychology, 130, 11-21.
Csikszentmihalyi, M. (2002). Flow: The classic work on how to achieve happiness. Random House.
Stylianides, G. J., & Stylianides, A. J. (2011). An intervention on students' problemsolving beliefs. In Proceedings of the 7th congress of the European Society for Research in Mathematics Education (pp. 1209-1218). Poland: Rzeszów.
Schukajlow, S., Leiss, D., Pekrun, R., Blum, W., Müller, M., & Messner, R. (2012). Teaching methods for modelling problems and students’ task-specific enjoyment, value, interest and self-efficacy expectations. Educational Studies in Mathematics, 79(2), 215-237.
Gueudet, G., & Trouche, L. (2009). Towards new documentation systems for mathematics teachers?. Educational studies in mathematics, 71, 199-218.
Lee, H. S., Coomes, J., & Yim, J. (2019). Teachers’ conceptions of prior knowledge and the potential of a task in teaching practice. Journal of Mathematics Teacher Education, 22(2), 129-151.
Siedel, H., & Stylianides, A. J. (2018). Teachers’ selection of resources in an era of plenty: An interview study with secondary mathematics teachers in England. In Research on mathematics textbooks and teachers’ resources (pp. 119-144). Springer, Cham.
Pepin, B., Gueudet, G., & Trouche, L. (2013). Re-sourcing teachers’ work and interactions: A collective perspective on resources, their use and transformation. ZDM, 45(7), 929-943.
Grave, I., & Pepin, B. (2015). Teachers’ use of resources in and for mathematics teaching. Nordic Studies in Mathematics Education, 20(3-4), 199-222.
Fun Learning for Kids. (2021). Silly Monsters Number Towers. https://funlearningforkids.com/monster-math-number-towers-activity/
Warshauer, H. K. (2015). Productive struggle in middle school mathematics classrooms. Journal of Mathematics Teacher Education, 18(4), 375-400.
Oliver, M., McConney, A., & Woods-McConney, A. (2021). The efficacy of inquiry-based instruction in science: A comparative analysis of six countries using PISA 2015. Research in Science Education, 51(2), 595-616.
Foster, C. (2013). Mathematical études: Embedding opportunities for developing procedural fluency within rich mathematical contexts. International Journal of Mathematical Education in Science and Technology, 44(5), 765-774.
NRICH (2023). About NRICH. University of Cambridge, Faculty of Mathematics. Available at: https://nrich.maths.org/about
Ryve, A. (2006). Approaching mathematical discourse: two analytical frameworks and their relation to problem solving interactions (Doctoral dissertation, Mälardalens högskola).
Parrish, S. D. (2011). Number talks build numerical reasoning. teaching children mathematics, 18(3), 198-206.
Boaler, J. (2001). Mathematical modelling and new theories of learning. Teaching Mathematics and Its Applications: International Journal of the IMA, 20(3), 121-128.

No competing interests reported.

Download PDF

Version 1

posted

You are reading this latest preprint version

Mathematics Designed for Engagement: from tasks promoting situational interest to sustained positive mathematics experiences

Status:

Version 1

Abstract

Figures

1. INTRODUCTION

2. METHOD

3. MODELLING INTEREST DEVELOPMENT AND LONG TERM ENGAGEMENT

4. PRINCIPLES FOR TASK DESIGN: A CASE STUDY

5. TEACHERS’ ROLE: THE PILOT SURVEY

6. DISCUSSION

Declarations

References

Additional Declarations

Status:

Version 1