Embedding Physics into Mathematics

Introduction

This study aimed at identifying the effect of the embedding of physics into mathematics and whether it would improve the students’ learning, classroom environment and students’ engagement in the subjects by utilising an integrated cross curricular pedagogy. It is undoubtedly a critical time of human existence, in a world that is plagued by great issues, extending from climate change and its consequences to progressively adverse diseases and pandemics, innovation and rational skills across all disciplines that are of a greater necessity than ever. Haylock and Thangata (2007) argued that, it is mathematics that enables solving of real-world problems in various disciplines including science and social science.

Studies in Japan have shown that, despite the significantly higher competency of Japanese students in mathematics in relation to the world average, they had the poorest attitude towards the subject of mathematics with only approximately 70% of Japanese students, who think that “Mathematics is important to everyone’s life” (Saeki, Ujiie and Tsukihashi, 2001). Similarly, poor attitudes towards mathematics have also been found in the United Kingdom, where the students have positive attitudes towards mathematics and they are between the ages of eleven to sixteen (Biatchford, 1996), this suggests that this is not an isolated issue but rather a global one. These poor attitudes demonstrate the lack of intrinsic motivation of students which is well known to be generally superior to teachers providing extrinsic incentives for achievement (Middleton and Spanias 1999:81). In other words, the students do not perceive the mathematic instruction that they are receiving as useful which has impacts on the development of their skills.

This concept is appropriate for an action research, because a major paradigm shift is required to develop and enhance students’ critical thinking abilities and to meet the STEM requirement of a future that is increasingly in need of innovative solutions. Saeki, Ujiie and Tsukihashi (2001) continued their thesis by arguing the most effective approach that will improve these attitudes towards an integrated cross curricular learning experience. A classroom environment should be set that the students can work independently and collaboratively to explore scientific concepts in maths with a full context, this will aid them in realising the importance of both mathematics and the sciences.

Positionality

I am inclined to believe that, based on my own experiences and research of the positionality that an integrated cross curricular learning experience is required for the students to fully grasp the skills, interdisciplinary and otherwise, it required to face the challenges of a modern world (Jacob, 2015). I have become aware of the benefits of utilising the sciences which can positively influence and develop key skills in mathematics. It is my own experiences, as both a student and a teacher in the United Kingdom and also in Pakistan, which influence my teaching and the manner in which my students interact with me. I tend to ask more complex questions that require a logical thought process and that do not always have a very linear answer to nurture my students’ ability to reflect on their own approaches (Kolb, 1984) and rational thinking skills. I also endeavour to create a classroom environment in which students are able to freely ask questions about different topics and relate it back to a mathematical perspective. For making my positionality visible, I often incorporate key concepts in physics and sometimes other disciplines into mathematics not as a mere example but for synthesising knowledge and understanding from these concepts in physics with concepts in mathematics.

Context

Nordvedt and Siqveland (2018) and Pepper, Chasteen, Pollock, and Perkins (2012) amongst an abundant number of studies that have investigated the conceptual and mathematical difficulties of the students at introductory level STEM courses, most notably in Physics. Likewise, there is also significant emerging research about students in more advanced physics courses that are also having difficulties. These studies have demonstrated that, lack of mathematic competency is one of the foremost barriers in further education of STEM subjects and it shows that, it must be an issue that is resolved at younger ages to prevent its damaging impact on students entering into further education.

This study was conducted in a selective grammar school. As the school had a comprehensive selection process, the students were generally very competent and often very high achievers; this was indicated by the extremely impressive GCSE results in mathematics, where in excess of 92% of the students received grade 7 – 9, which was significantly greater than the national average of 20% (Adams, 2021). The following active enquiry was conducted with a year 9 class of males, this class was known to be some of the most mathematically able students in their year group and hence were designated as the ‘top set’.

I personally chose to explore this topic due to the fact that, I noticed a significant decline between the school’s GCSE Physics grades and A-level Physics grades of the same year. Likewise, there was a very similar discrepancy in the GCSE mathematics and A-level mathematics grades. Talented boys, who had achieved exceptional results at GCSE made very little progression into their A-level course. As approximately 30% of GCSE Physics and 40% of A-level Physics (AQA, 2021) respectively consisted of mathematics it seemed rational that they may be some mutual benefit for the pupils in integrating both subjects.

Action Research

According to Mills (2003), action research is a systematic enquiry, conducted by the teachers, administrators, school boards or other stakeholders with an endowed interest in the teaching and learning process, with the intention of collecting data about the operation of schools, how they teach and how the students learn. Action researchers engage in a “systematic, self-critical enquiry” (Stenhouse, 1985), by following a cycle of enquiry and reflection where the focus is on bringing about change in practice, improving student outcomes, and empowering the teachers (Mills 2017) by implementing reforms in accordance with the collected data. In this case, action research has been employed to explore an effective way of addressing the disinterest and lack of motivation of the students in relation to mathematics and physics.

Evidence drawn from my own teaching

Generally, studies have shown the poor attitudes of the students in the United Kingdom towards mathematics. This is also evident from my own teaching, which demonstrated that, despite the fact that my students were of a higher ability and competency, they still had a poor attitude towards mathematics. The students often appeared disinterested in the lessons and were eager to leave the classroom even though the majority of them completed their classwork with adroitness and without great exertion.

Closed questions can be described as a part of a thematic outline that is given by a teacher, where they are expecting a single probable answer, when multiple answers are reciprocated by the students and it is the responsibility of the teacher to identify and approve the correct response. Conversely, an open question does not always have a single correct answer rather it can take many forms of answers, where the result is often a subject of discussion.

When the students were presented closed questions that were purely mathematic and absent of any scientific or real-world context such as “What is the length of the third side of a right-angled triangle when the other two sides have a value of such and such?”, the majority of the students answered correctly with moderate ease. On the contrary, when asked an open question that required an application of their skills in the real world, the pupils were not as successful. For example, when the students were asked to discuss and calculate the distance a ball would travel when kicked in a straight line from the corner to the centre spot of a football pitch measuring 105 metres by 68 metres. The students debated a number of different ideas amongst themselves, but most did not consider the use of Pythagoras’ theorem with only approximately 40% of the class computing the correct answer.

This demonstrated the fact that, most of the students did not acknowledge the applicability of their knowledge and skills from the mathematics that they had learnt in the real world. This is despite the fact that almost all the students showed a strong competency in Pythagoras’ theorem calculation in a solely mathematical context of triangles and other shapes. This further shows that, students have a strong dissociation between the mathematics they have learnt in the classroom and its applicability to the real world which strongly suggests that they do not perceive the mathematic instruction they are learning to be as useful in the real world which will limit their intrinsic motivation and impact their development (Middleton and Spanias, 1999).

Theoretical Framework

The idea of cross curricular studies can be traced to Dewey and the pragmatist view of learning (Hammond, 2017) and in the United Kingdom to the Plowden report (1967). Although the National Curriculum (DfES 1999) has prescribed that, knowledge is to be learned in discrete subject areas, the Primary Strategy (DfES 2003) gave the teachers the flexibility regarding how the programmes of study are to be taught. As a direct result of the Primary Strategy, curriculum models like the International Primary Curriculum which allowed for cross curricular links, increased in popularity in English primary schools (Hammond, 2017).

According to Barnes (2018), a connecting curriculum does not merely bind together the curriculum subjects to the students’ lives but also attempts to positively link the different layers of experience within their lives; this broad connection making provides the underpinning of cross-curricular learning. In studies where the students and teachers were under observation, the researchers noted that, the teachers found the most effective, memorable and engaging sessions were those that employed personal stories, humour, cognitive challenge, gave multiple opportunities for reflection and illustrations among other factors (Grainger et al., 2004). These results may be explained from a neuroscientific perspective, where it is known that, “few if any perceptions of any object or event, actually present or recalled from memory, are ever neutral in emotional terms” (Damasio, 2003:93). This reinforces Barnes (2018) ideas of the necessity of creating a positive pedagogy within schools. In addition, this study found that, personal anecdotes played a distinctly important role in connecting the pupils with the subject manner on offer, similar results were found across a number of different countries including Malaysia, Rwanda, India and Tanzania (Barnes, 2018). Overall, a successful connecting curriculum does not solely produce effective and measurable progress in and between subjects but also connects the learners and teachers on many levels.

‘Token’ connectivity, as defined by Barnes (2018) is as an incorrect or false way of connecting the subjects of a curriculum. These are methodologies that appear to look like cross-curricular learning but actually only aids learning in one subject. One subject, often a ‘foundation’ subject is utilised to enhance the learning in another ‘core’ subject, real connections are not made between the two subjects and no growth is expected in the foundation subject. An example of this would be if a question from physics was presented in a maths lesson without any context from physics like a triangle consisting of two perpendicular forces and the students asked to calculate the third side of a triangle which would be the average force by utilising Pythagoras’ theorem. Although this would increase their learning in mathematics and understanding of the uses of Pythagoras’ theorem, this would not have significantly improved their learning in physics nor would they extensively understand why they are able to apply Pythagoras’ theorem to forces without a complete understanding about the physics background of scalar and vector quantities. This is of course not the most ideal way to teach as it fails to meet the aims of an integrated curriculum.

However, there is no uniform methodology to teaching a cross curricular approach, researchers of different schools of thoughts have implemented this approach in different ways. Beane (1996) argued that, curriculum integration requires four dimensions, the first of which is that, the curriculum is organised around problems and issues that personally and socially significant and relevant to the students’ lives; this is identified through collaborative planning of educators and students. Secondly, the curriculum is not organised by subject area lines where all the pertinent knowledge is integrated into the context of organising centres. Thirdly, that knowledge is utilised and developed to address each organising centre rather than merely to prepare for a test. Finally, emphasis is placed on activities and substantive projects that involve real application of knowledge. The methodology follows the idea that problems in the real world are not compartmentalised into distinct units, education likewise should not be presented as discrete subjects if children are to begin solving real world problems (Beane, 1996).

The most common approach is the multidisciplinary approach; where different courses are linked together but the separation of the disciplines remain. This arrangement mostly follows the same themes as Beane’s integrated curriculum including a focus of a more overarching question, which is solved with activities rather than the focus being on an assessment or grade. A study conducted by Rogers et al. (2015) further examined the effect of a multidisciplinary approach as opposed to the traditional discipline-specific learning with a specific focus on its effects on Stem subjects. It utilised a multi-method assessment research design, which incorporated pre-post course assessment, pre-post intervention assessment, technical reports, quizzes, exams and interviews to assess achievement of the students in a range of courses. The study further found that, the approach increased understanding about the characteristics of twenty-first-century problems, in particular those related to sustainability, and increased students' favourable perceptions of introductory calculus. The research, however, identified neither a loss nor gain in discipline specific learning and found it difficult to gauge if students understanding of how different disciplines can work together increased significantly due to the limitations of a project-developed assessment instrument.

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Furthermore, the studies have highlighted the true extent of variety in approaches to teaching a cross curricular programme of study. Fogarty (1995) identifies 10 levels of curricular integration which include:

Fragmented: separate and distinct disciplines.

Connected: topics within a discipline that are connected.

Nested: social, thinking and content skills are targeted within a subject area.

Sequenced: similar ideas are taught in concert, although subjects are separate.

Shared: team planning and/or teaching that involves two disciplines focuses on shared concepts, skills or attitudes.

Webbed: thematic teaching using a theme as a base for instruction in many disciplines.

Threaded: thinking skills, social skills, multiple intelligences and study skills are threaded throughout the disciplines.

Integrated: priorities that overlap multiple disciplines are examined for common skills, concepts and attitudes.

Immersed: learner integrates by viewing all learning through the perspective of one area of interest.

Networked: learner directs the integration process through selection of a network of experts and resources.

This demonstrates the fact that, a cross curricular approach is multi-faceted, where all of the above definitions are equally legitimate, this may be considered as an advantage because the concept is versatile and it can be implemented in different circumstances to achieve the desired results. Opponents to this style of teaching argue that, this style of teaching allows students liberty to avoid areas of learning that they may personally find difficult thereby cultivating poor work habits and attitudes as well as the notion that some subjects naturally bridge more easily than others such as mathematics and physics as opposed to geography and music (Hayes, 2010). These are all concerns a teacher should be observant with their students and must account for planning the lessons and projects.

Heywood, Parker and Jolley (2012) also studied a more general effect of integrating the curriculum. One of the major challenges the report was the ability of the teachers to implement such a curriculum, due to the difficulty of navigating between contributary subject-specific discourse and the particular demands of cross curricular activities. Indeed, interdisciplinary syntheses are ‘among the most epistemologically complex endeavours that humans can attempt’ due to the ‘deep differences of perspective that must be bridged in order to carry out interdisciplinary projects’ (Stein, Connell, & Gardner, 2008: 401). As cross-curricular integrated learning is a relatively new and distinct method of teaching, it is not as well understood as disciplinary research and gives rise to its own unique ‘quality control’ challenges (Boix-Mansilla, 2006). Venville et al. (2002) found that, although enthusiastic, the teachers themselves were often unable to explain their desired outcomes for their actions. These are influenced by a number of different factors including curriculum requirements and guidance, the school context, personal motivations and epistemological commitments,

The results of the study demonstrated that, some teachers with topic-based teaching experience believed that, it may not be the fact that, teacher’s subject knowledge is limiting but rather what is required for the best learning experience is a teacher to have an overarching knowledge through which they can establish a foundation and develop links and greater depth. This was found to be similar to the proposal put forward by Alexander (2012) who considered producing a curriculum with an essential ‘core’, which would be derived from a range of disciplines.

However, studies have shown that when implemented well a cross-curricular fully integrated learning experience can have a positive impact on numerous domains of education including collaborative skills and attitudes towards mathematics and the sciences, this is in spite of the aforementioned challenges of such an approach (Saeki, Ujiie and Tsukihashi, 2001). The study was conducted such that, all the students were divided into groups of three or four to plan, design, conduct and evaluate a set of experiments that was dependent on their year group. The students conducted a series of experiments mostly using familiar objects and situations such as: evaluating the motion of a walking person, the dropping of common objects and the cooling rate of water. The students were required to work both collaboratively as a working group and at other times, when in teams of four students, the students were required to work individually to their own goals. This independence however was paired by teacher support in the form of bi-weekly lessons.

The students were interviewed both before and after the experiments. The results of the study showed a positive impact in almost all the themes addressed in the interviews. Firstly, after completing their experiments, the researchers found that the students had largely replaced their naïve assumptions regarding the laws of physics with scientific concepts; for example, 45% of students prior to the study has the conceived notion that the relationship between the periodic motion of a pendulum and length of string could be modelled with a linear function, two weeks after the experiment only 10% of students utilised a linear function. Likewise, when plotting their results graphically the students not only utilised the correct trigonometric model but were also able to independently apply their knowledge of translation onto the trigonometric equations.

Furthermore, post-study interviews showed that, students’ interest in scientific phenomena has been increased by 26% and more significantly 48% of the students reported agreed or strongly agreed that they now felt that mathematics is more important than they had believed previously. The students even reported an increase in their positive attitudes towards cooperative work, with 54% of students saying that they agreed or strongly agreed that they now felt that cooperative work as more important they had believed than before. This further demonstrates that, when the students are given a greater level of independence as well as allowed opportunities to collaborate with one another in examining familiar, real-world situations through a full integrated cross curricular course they are able to engage in higher order thinking skills such as making predictions, analysing data, and modelling data with equations. This develops more positive attitudes towards mathematics and the sciences as well as further developing the understanding and skills required in the STEM subjects and team projects.

Action Steps/Interventions

Aim of the Study: To increase interest and motivation in mathematics and the other STEM subjects and attract more students towards STEM in further education and the workplace by showing the utility of mathematics via fully contextualised application in physics. In order to achieve the above aims, the following learning objectives have been identified:

Collecting and analysing data of students to understand the mindset and attitudes of students

To analyse the efficacy of interventions

To show how an integrated learning experience with real world context is essential for motivation

To Influence mindset and attitudes of students towards mathematics and STEM

1st stage:

The process of intervention was initiated by interviewing a sample of students from my class immediately after I had explained the mathematical solution to the question ‘How far can the eye see into the horizon?” I utilised a name generator programme that would randomly select the name of five students from my class for interview. I proceeded to then informally interviewed the five students of varying ability individually with questions based on three categories:

Interest in maths

Utility of mathematics in the real world

Motivation for Mathematics

Interestingly, the students’ responses to the questions were almost uniform even though they were asked separately. With regards to the first question, the students’ answers generally correlated with previous research conducted on British students’ attitudes to mathematics; none had a substantial interest in mathematics, and they all said, it was ‘quite’ or ‘very’ boring. Subsequently, when asked about the utility of mathematics and the extent of its applicability in the real world, almost all viewed mathematics as merely a qualification that is necessary for employment opportunities in future and the majority believed that the knowledge and skills that they had gained in mathematics would most likely to be unutilised after leaving education unless they pursued niche professions such as engineering.

Finally, when asked what motivated them to learn mathematics and continue their studies, they all mentioned that, their primary aim was to earn a good grade in their GCSE examinations so that they could meet parental and scholastic expectations.

2nd step:

After analysing the students’ attitudes and their roots, I sought to enact the next step to begin changing this mindset. As a group, I introduced the pupils to another open question that had a context based solely on physics without any mathematical element. The question included a fridge, an every-day object that all the children would be familiar of, to demonstrate the applicability of scientific education in school in relation to the real world. The pupils were asked if a fridge whose door had been left opened would cause the temperature of the surroundings to increase, decrease or remain constant. This question proved to be controversial as the pupils passionately debated, which of the three answers was the correct response. After they were allowed to discuss amongst themselves for some time, they were not able to reach a consensus.

I proceeded to explain a simplified, general modus operandi of fridges by using a compressor and utilised the analogy of pumping a ball with air, once more a familiar scenario for the students, to illustrate the principles of energy in physics, where work done on a gas causes a temperature increase. This example encouraged the students to think about a number of different ways and the scientific principles that they have learnt. It can be used to tackle problems in their daily lives. They were also motivated by the fact that a number of problems could be solved using just one simple idea or principle.

3rd Step:

The next and final stage of my intervention was to create fully synthesised lessons, where I would teach mathematics alongside a fully contextualised GCSE physics background. In the first lesson, the focus was on Pythagoras’ Theorem and its applicability in physics, primarily in the study of vector quantities. I began the lesson by explaining the formula of Pythagoras’ Theorem and applying it to its usual mathematical context: right-angles triangles. Once the students were confidently and independently able to answer questions on the topic, I proceeded to present the idea of scalar and vector quantities and gave a number of different examples of each; this provided a thorough physics background. The students were then able to apply their knowledge of Pythagoras’ Theorem to resolve vector quantities like velocity for identifying the average speed of familiar vehicles such as cars, boats and aircraft amongst other assorted questions involving perpendicular vectors.

The final question proved challenging, where the students were asked “How far are you able to see on the horizon if you are approximately 1.5 metres above sea level?” this immediately sparked some debate with two or three ideas, produced by the students being debated amongst themselves. After some deliberation, the students reached a general consensus that you will be able to see continuously as long as there are no obstructions to one’s sight. Despite reaching an erroneous answer, most of the students said they had considered the fact that the earth is spherical and some attempted to use Pythagoras’ theorem but were unsure of how to apply it to a round surface. I illustrated using a diagram how by using their knowledge from mathematics and physics such as the fact that the Earth is spherical and the value of the radius of the Earth, we could use Pythagoras’ Theorem to calculate the approximate distance we are able to see on the horizon. Once the students were able to visually see the diagram, it became apparent to the majority of them how they would proceed with the calculations.

As a continuation of the previous lesson, I utilised a similar methodology and format with my lesson on trigonometric functions: sin, cos and tan. I began the lesson by teaching and explaining the steps that are essential to solving a trigonometric problem, firstly in a purely mathematical context only using triangles. After revising the concepts of scalar and vector quantities in Physics, the students were presented a variety of familiar situations as real-world problems, such as a train moving up a steep hill and the length of string of a flying kite, these questions were more difficult as the vectors were no longer perpendicular. As a concluding assessment, the students were given a question with a basis in astronomy, the students were asked to determine the distance of a nearby star from Earth, utilising the assumption that, the star was vertical to the position of the Sun. Relative to the previous lesson, the pupils were able to resolve this question successfully by using trigonometry and the parallax effect with less guidance.

Evaluation

In order to evaluate the overall outcome of the interventions, I conducted one to one interviews with the students once more. The principal rationale for these interviews was to discern whether the intervention had been successful in changing the students’ negative attitudes towards mathematics and its utility in the real world. Using a random name generator, the class was divided into four quartiles on the basis of ability in accordance with their previous grades and then, using the generator, two students were selected from each quartile randomly to act as my sample. The students were asked individually to counteract the effects of conformity. When interviewed, more than half said they felt that their skills had improved in both maths and science and almost all the students said they found mathematics much more compelling post-intervention. More than half the students said that their attitudes towards the applicability of mathematics in the real world had also changed.

Furthermore, I found through my own observations of the class through the integrated lessons that the students were generally more engaged and were asking more questions than in previous lessons. They also worked more independently, as they pursued their own individual reasoning to address each question, and these positive differences were also noted by an impartial observer of the lesson.

The premise of this study could be expanded further to integrate not only mathematics and physics but other scientific disciplines such as biology and chemistry. This would help bridge students’ knowledge and may increase the scope for the application of students’ skills and competences. For example, the measurements and compound units’ modules in the mathematics correlates significantly with quantitative chemistry likewise the statistics module of mathematics relates well to the ecology component of biology. This would further demonstrate how students can use mathematic skills as a tool to analyse and explore the world around them, whilst also nurturing the mathematic cognizance and skills essential for the scientists.

Conclusion

Research has shown how poor attitudes towards mathematics can have impact on student’s development not only in the scope of mathematics but across disciplines, most acutely in the sciences and social sciences. Moreover, it is an issue that must be resolved with haste if the young students of today are to meet the STEM requirements of the future.

Teaching must adapt and change to tackle these issues directly, this current study has demonstrated the effect of an intervention that included the introduction of a fully integrated, cross-curricular module, which aided the students to understand the applicability of mathematics, as well as physics, to the world around them thus sparking interest and motivation.

In order to gain the greatest benefit from the integrated style of teaching for the students, they should have full context of both disciplines being studied not one where a subject is used merely as an example. This allows the students to gain the broadest comprehension of both disciplines. In addition, it is essential for the questions to have an element of familiarity and relatability, this allows the students to more easily understand how what they are studying can be useful to their day to day lives and will support them in building the intrinsic motivation that they would most benefit from.

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