Humanizing Teaching and Learning with an Expert Help
Eva Čoupková and Vojtěch Kejzlar, Czech Republic
Eva Čoupková has been a lecturer and assistant professor at Masaryk University Language Centre, Brno, Czech Republic, since 1997. She teaches Academic English and English for Specific Purposes for Mathematics students. She obtained her Ph.D. in 2003 from Palacký University in Olomouc for her dissertation on Gothic Novels and Drama as two related genres of English literature.
E-mail: coupkova@sci.muni.cz
Vojtěch Kejzlar is a Ph.D. student and graduate teaching assistant in the Department of Statistics and Probability at Michigan State University. He holds a Bachelor’s Degree in Mathematics with focus on Modelling and Calculations from Masaryk University in the Czech Republic.
E-mail: kejzlarv@msu.edu
Menu
Introduction
Background
Cooperation with experts (?)
Deciding what to do – Bachelor courses
Deciding what to do – Master courses
Implementing Task/Problem-Based Approach
Topic Selection
Language teacher-Expert-Students Cooperation
Problems
Conclusion
References
Appendix 1 – Task Based Course Topics
Appendix 2 – Task Based Course – Population Dynamics
As a university teacher of a foreign language (English) trying, sometimes desperately, to employ the subject-specific topics and vocabulary, along with the Task/Problem Based Approach in my lessons; and a doctoral student of mathematics interested in English and teaching English to the students of mathematics, we would like to share our positive experience obtained in the course of our very fruitful cooperation. The essay text is meant to be mainly practical, aiming at concrete materials and steps we used; however, a short theoretical perspective is also included.
Masaryk University Language Centre is a university-wide institute providing education in foreign languages and testing language proficiency of students of all individual faculties of Masaryk University. It consists of eight departments and a common coordination workplace. Students of all fields of study (i.e. Bachelor’s, Master’s and Doctoral study programmes in presentational and combined form), can learn five foreign languages at the academic or technical level – English, French German, Russian and Spanish. Furthermore, students in the faculties of Medicine and Law can take courses in Latin. The Language Centre also offers intensive courses in Czech for foreigners. (For information about Masaryk University Language Centre, see www.cjv.muni.cz/en/about-us/). At our department at the Faculty of Science we cater for the needs of a wide range of students of scientific subjects – biology, chemistry geography, geology, mathematics and physics. We teach both academic and subject-specific language as the main purpose of our courses is to prepare students for their professional life, which means to develop their use of English in a specialized field of science, and enable them to use English as the main means of communication and cooperation with partners in their expert fields. This aim and approach presents many challenges for the language teachers – I believe that most of us would agree with Tom Hutchinson and Allan Waters who claim that “ESP teachers are all too often reluctant dwellers in a strange and uncharted land” (Hutchinson, Waters, 1994, p.158).
The reasons for the feelings of alienation that many of my colleagues and I experience when trying to design and teach courses for Science students are well-known. The main obstacle is evidently the insufficient expertise in a specialist field. Even if some teachers are exceptions, having studied language and a scientific subject at college, the majority of them are still linguistics majors combined with a humanities discipline. It is true that as teachers prepare and adapt materials related to the specific scientific disciplines, they learn along the way and may become learned practitioners in the field. However, this knowledge is still quite limited and not sufficient to protect the teacher from being seen by students as unprepared and incompetent, and sometimes even ridiculous. So, where can language teachers look for help?
One obvious answer would be cooperation with a teacher of the subject (Helsvig, 2012, p. 4). Developing a project based on collaboration of the language teacher and the subject teacher has its merits, but there are also obstacles to overcome. The main one is, at least at our university, a heavy workload on the part of subject teachers, which may decrease their ability and willingness to cooperate with language teachers. Therefore I was exceptionally lucky when my former student, who has attended the course of English for Mathematicians, at that time completing his thesis and preparing for his further studies abroad, contacted me and offered his help. Fortunately our department management agreed to provide support and encouragement, so we were able to start.
The total available workload of the student assistant was just 2 hours a week for the period of six months; therefore we had to consider carefully what to concentrate on. After discussing the options, we agreed on two main areas of cooperation. With the Bachelor (B1 level) course students, Vojtěch was willing to share his experience he had obtained in the course of his previous studies abroad. As a former student of mathematics at Beloit College (Wisconsin, USA) in the ISEP study programme, he was able to share his learning experience and point out the differences between the Czech and American university environments. That was very interesting and relevant for our Bachelor students,’ since some of them are or will be in the future considering the completing or furthering their studies abroad. To further demonstrate the different methods employed at American universities, Vojtěch prepared an example expert topic with links and references, and then gave a short talk in the class followed by a plenary discussion. Our students liked his open and friendly approach, and appreciated his authentic, first-hand experience. Moreover, they realized that the study in the US would require a thorough homework preparation and independent thinking, as well as the active participation in the class discussions.
For the advanced (B2 level) Master courses we intended to work on a more complex task. Master students’ main needs (that) comprise improving their presentation skills and practising English in a scientific context. To address this problem we agreed on implementing the Task/Problem Based Approach consisting of team or pair work dealing with related mathematical topics, and then preparation for a class presentation followed by the discussion.
Why have we decided to base our language teacher-expert-students cooperation on this approach? We believe that some features characteristic of the ‘Task/Problem Based Approach’, as defined by David Boud and Grahame Feletti, correspond particularly well to the needs of our mathematics students, especially the requirement of being able to deliver a speech detailing the problem as a simulation of professional practice or a “real life” situation, because Scientists are often supposed to present their findings at professional conferences. The method also appropriately guides students’ critical thinking and provides an unlimited resource in helping them learn from defining and attempting to resolve the given problem. Most importantly, students are encouraged to work cooperatively as a group, exploring information in and out of class, with access to a tutor (in our case both a language teacher and a subject specialist) who knows the problem well and can facilitate the group’s learning process. Last but not least, students get used to identifying their own learning needs and using available resources, as well as evaluating their learning process. (Boud, Feletti, The Challenge, 1997, p. 2)
The first point is particularly relevant as the Problem-Based courses and curricula are generally designed to enable learning to understand and solve real-life problems (Ross, The Challenge, 1997, p. 28). This aspect was a key concern for us, since, in the Czech Republic, education at many schools is more theoretical than practical, providing students with a quantity of detailed information that they are not able to implement later in real life. Also solving problems in a foreign language (English) played an important role as students may find themselves working and studying in varied international environments in the course of their future careers.
Moreover, we were able to build on positive experience with this method obtained during the course of collaborative learning run by our department at the same time. In this course we successfully employed various teaching and learning methods including the Task/Problem Based Approach (see Collaborative teaching and learning in an interdisciplinary problem based language course. https://munispace.muni.cz/index.php/munispace/catalog/book/747 for more information about the course).
Defining and selecting a problem to deal with was a complex question that would be extremely difficult for a language teacher to solve without the help of the expert-student assistant. Firstly, he proposed some areas that could be of interest to students, comprising topics in pure mathematics, applied mathematics, statistics or modelling. (Appendix 1) Secondly, it was necessary to choose a topic that would provide enough stimuli in terms of discipline specific and more general questions, and could be appropriately challenging to students. We decided on the topic of Population Dynamics as there are various applications that can be relevant for the mathematicians of most branches. Vojtěch developed a set of more concrete tasks related to the main problem of Population Dynamics for the students of different branches to choose from: students’ working in the field of Financial Mathematics and Analysis could be interested in The Limits of Growth or Population Dynamics of Fisheries, prospective teachers of Maths and Biology could choose Epidemic Modelling or Age-Structured Population Models; many other possibilities were open for other branches (Appendix 2).
The students were given limited, but in our opinion, sufficient resources to work with. As Vojtěch provided a short description/introduction to the problem, possible tasks to solve, problems to discuss, and a list of useful links which the students were encouraged to use but were not limited to. The real scope of the problem and the range of questions students decided to address in their presentations were left for them to discuss and decide.
The “technical” support in the form of modelling tips, along with free and user-friendly software designed to model system dynamics, were given by Vojtěch. In fact, we used this as a “trigger” (Lovie-Kitchin, The Challenge, 1997, p.204) in the form of a short talk in which Vojtěch introduced the software and tried to motivate students to use it in preparation of their presentations.
The equally important aim was to create an environment in which students of different branches of mathematics (i.e. algebra, analysis, modelling, financial mathematics, prospective teachers of mathematics, etc.) could cooperate on a solution of one concrete problem. Again, despite the fact that multi-disciplinary strategies are quite common nowadays, and many fields of science join in addressing complex questions facing us in today’s world, the students of individual fields or branches at our faculty tend to be rather isolated, not having or creating enough opportunities to work with their colleagues.
Therefore our aim was to give the students opportunity to work in pairs or groups and solve problems in and outside the class. In the process of preparation students had access to both expert help (Vojtěch), dealing with mathematical questions, and language service (me), addressing problems related to the correct usage of English. The only requirement was specified beforehand – that both students in a pair or group should contribute in the topic presentation and their involvement in the preparation and presentation should be clearly indicated.
As we have discovered, students were well able to address questions related to their branch of science, develop a number of possible solutions, and sometimes even discover new materials relevant to the main topic. However, it was far more difficult for them to see the problem in its broader context and explain complex concepts in a simple, comprehensible way. The students of Science normally concentrate on one concrete question and try to discover or offer a specific final solution. As our main aim was cooperation in pairs or groups and looking at the problem from different perspectives, some students felt at the beginning that the questions were rather vague and not “scientific” enough for them to work with. Some declared simply that the solutions lied outside their fields of interest. Therefore we had to repeat many times that not only showing expertise in their disciplines, but also cooperation and communication play a crucial role in a foreign language learning community.
Evaluation
To sum up, the main advantages of the approach, as seen by a student assistant, language teacher and expressed by students, both during the discussion after the presentations and in their written comments and feedback on the course, are the following:
- Working with real-life problems and situations;
- Development of a number of alternative solutions to the main problem;
- Interesting and engaging classroom environment – working with an expert;
- More responsibility and independence of students;
- Thinking “outside the box” and not concentrating on one’s branch only;
- Cooperation and mutual support of students, an expert and a language teacher.
To conclude, we should admit that employing the Task/Problem-Based Approach in the mathematics courses required concentration and hard work. However, we think, and the reactions and feedback of students have proved it, that it can be stimulating and beneficial both for the teachers and learners. We can therefore agree with Dudley-Evans and St John who argue that “if we are to meet students’ needs we must deal with subject/specific matters” (Dudley-Evans, St John, 1998, p.51).
We believe that for language teachers who are preparing subject-specific courses an expert involvement can be extremely helpful and motivating, as they can easily avoid the aforementioned feelings of alienation and incompetence. Moreover, it is easier to persuade students’ to engage in activities designed and supported by a professionals in their field or related fields. As not many subject teachers are available or willing to help, we think that cooperating with an advanced interested student in the field, who is also language competent and friendly, is an immeasurable help. For the language teachers such a fruitful cooperation based on mutual respect can undoubtedly be one of the highlights of their language teaching careers.
Boud, D., Feleti, G. (1997). “Changing Problem-based Learning. Introduction to the Second Edition.” The Challenge of Problem-Based Learning. New York: Routledge. 1–14.
Cambridge ESOL. (2010). Teaching Maths through English - a CLIL approach. [online] Cambridge ESOL. [accessed 2013-09-16]. Available from WWW:
Čoupková, E. (2015). “ESP and Task/Problem-Based Approach in English for Science.“ Collaborative teaching and learning in an interdisciplinary problem based language course. [online] Brno: Munipress. 16–21. Available from WWW: https://munispace.muni.cz/index.php/munispace/catalog/book/747
Dudley-Evans,T., St. John, M. J. (1998). Developments in English for Specific Purposes. A multi-disciplinary approach. Cambridge University Press.
Helsvig, J. ESP – Challenges for learners and teachers in regard to subject-specific approach. [online] Vilnius: University of Vilnius. [accessed 2013-04-22]. Available from WWW:
Hutchinson, T., Waters, A. (1984). English for Specific Purposes. Cambridge University Press.
Lovie-Kitchin, J. (1997). “Problem-based Learning in Optometry.” The Challenge of Problem-Based Learning. New York: Routledge. 203–210.
Ross, B. (1997). “Towards a Framework for Problem-based Curricula.” The Challenge of Problem-Based Learning. New York: Routledge. 28–35.
Pure Mathematics
• Differential Equations – Differential equations are an integral part of mathematical analysis that characterize the relation between a function and its derivatives. The potential applications of differential equations are vast, and therefore, we shall focus a one specific equation: the heat equation. It is a parabolic partial differential equation that describes the distribution of heat in a given region over time. Among others, the equation is used to model particle diffusion, Brownian motion, intravenous injection, and options in financial mathematics.
Task: Choose a phenomenon which can be described through the heat equation, perform experiments, and interpret your observations.
• Graph Theory and Networks – From the point of view of mathematics and computer science, a graph is a pair of two sets G = (V, E). Where V is a set of vertices, and E is a set of edges. Graph theory has its beginnings in 1736, when arguably the most prominent mathematician in the history, Leonhard Euler, studied different paths through the system of bridges in Königsberg, Russia. In the 21st century, however, we use graph theory to model data structures in computer science, decision trees, and most importantly to study networks. Network theory is a part of graph theory which is concerned with the study of relations. It reaches into many fields such as electrical engineering, biology, and social sciences. For instance, one can represent a public electrical grid as a network and then study its resilience to black outs.
Task: Model a social, biological, or logistical network and analyze its resilience, robustness, and redundancy.
Applied Mathematics for Multi-Branches Study + Economics
• Time Series Forecasting and Its Applications in Economics – Time series is an ordered set (sequence) of data points that usually consist of measurements over certain period of time. Nevertheless, time does not necessarily need to be the independent variable here. This definition immediately gives a hint that time series are heavily used in weather forecasting, electrical engineering, earthquake forecasting, and mathematical finance. Time series analysis is, therefore, especially important in predicting the future values of a given variable based on the previous measurements, and it can be used to predict the development of a wide range of phenomena in economics.
Task: Use the means of time series analysis to forecast development of an economic phenomenon and interpret the implications of your model.
• Data Mining – Data mining is, generally speaking, the process of analyzing and summarizing large data sets from different perspectives or angles. In particular, it is used to discover correlations and patterns among many variables while working with significant amount of data. A typical task of data mining would be to analyze buying patterns of a customer to increase the revenue of a given food or grocery chain.
Task: Study the different tools used in data mining process and analyze one of the publicly available data sets.
Modelling and Calculations
• Population Models – We have all heard about the famous (or maybe infamous) model of Thomas Malthus which predicts exponential growth of a population. Even though the model provides oversimplified description of the changes in population size, it has revolutionized the way we look at the population dynamics. Currently, models that describe the difference between the number of deaths and the number of newly born are widely used to determine the maximum harvest for agriculturists, to understand dynamics of diseases, and have also many environmental conservation applications.
Task: Pick a population of your choice and use an epidemiological or a growth model to evaluate the dynamics of the population under specific conditions (disease, limits of growth, etc.).
• Analysis of Biological Signals – A signal is any phenomenon of a physical, chemical, or any other material based nature that carries information about a state of the system that generates the signal and about the dynamics of the system. If the system happens to be of a biological nature, we speak about biosignals. In the last few decades, the development of new imaging technologies, better sensors, and high-capacity hard drives caused biosignal analysis to be one of the most prominent areas of application for mathematical sciences. The methods used for analysis of the electrocardiogram (ECG), electroencephalogram (EEG), or magnetic resonance images range from statistics to graph theory.
Task: Use one of the publicly available data sets to analyze a biological phenomenon of your choice. Then, analyze the data set and interpret the results.
Statistics and Data Analysis
• Sports and Statistics – Sabermatrics is statistical analysis of baseball. This term was firstly coined by Bill James in 1980, who defined sabermatrics as “the search for objective knowledge about baseball.” Nevertheless, statistics and statistical analysis of data is not only the domain of baseball, but it is widely used across the spectrum of sports ranging from basketball to ice hockey. Such analysis can be later used by teams to form business plans, and to buy a new players that contributes best to the overall performance of the team.
Task: Attempt to predict the performance of a specific player in a given sport based on the data about his or her performance during the previous seasons.
• Survival Analysis – Imagine that you are a medicine doctor and you have a group of patients with a certain disease. A typical task of survival analysis would be to determine for how long will the patient survive or what is the risk that the patient dies. That is, survival analysis is a branch of statistics which analyzes the data where the outcome of a random variable is the time until the event of interest occurs. In general, the event does not have to be death, but it is often the occurrence of a disease, marriage, or divorce.
Task: Pick a disease of your choice and perform basic survival analysis, for example, the time to the occurrence of AIDS in HIV positive patients.
Introduction
We have all heard about the famous (or maybe infamous) model of Thomas Malthus which predicts exponential growth of a population. Even though the model provides oversimplified description of the changes in population size, it has revolutionized the way we look at the population dynamics. Currently, models with ideas rooted in population dynamics are used to describe interactions between different species, to determine the maximum harvest for agriculturists, to understand dynamics of diseases, to model demographic transition, and to predict the limits of population growth.
1. Tasks
Population Growth Models
Population growth refers to the change of a population over time. As we know from the introduction, the simplest model was introduced by Thomas Malthus, and therefore, it is called the Malthusian growth model. However, there are other more complex and accurate ways to describe population growth. The first possible extension is to introduce so called carrying capacity of a biological species in an environment. This extension ultimately leads to the well know logistic curve with the population size approaching the carrying capacity. Nevertheless, population growth models go beyond simple logistic curve, for instance, Richards growth model or Gompertz growth model.
Task: Compare and contrast Malthusian, Logistic (Verhulst), Richards, and Gompertz growth models. That is, pick a country of your choice and estimate population model parameters based on historical data of population growth. How well does each model correspond to the historical data? Choose the best model and make predictions about the population size in 5, 10, 20, 50, and 100 years from now. Interpret the results.
Useful links:
Population models:
www.iba.muni.cz/res/file/ucebnice/hrebicek-uvod-do-matematickeho-modelovani.pdf
Annual growth rates by country:
http://data.worldbank.org/indicator/SP.POP.GROW
Population size by country:
http://data.worldbank.org/indicator/SP.POP.TOTL
Predator-Prey Interaction
Population growth of species depends on many factors including the carrying capacity of an environment, potential presence of a disease in the population, or the predation. Since the introduction of population models, the predation has been an active subject of study for scientists in numerous fields. The main difference between a classic population growth model and a predator-prey model is that the model describes population dynamics of two species which interact. Probably the most pronounced model is a pair of first-order, nonlinear differential equations commonly known as the Lotka-Volterra system of equations.
Task: Pick a population of your choice that is predominantly hunted by a single predator, for instance the populations of fox and hare. Estimate the parameters of the Lotka-Volterra model and describe the population dynamics of the predator and the pray. How well does each model correspond to the historical data? Make predictions about the population dynamics in 5, 10, 20, 50, and 100 years from now. Will the sizes of both populations reach an equilibrium? Interpret the results.
Useful links:
Predator-prey models:
www.iba.muni.cz/res/file/ucebnice/hrebicek-uvod-do-matematickeho-modelovani.pdf
http://download.springer.com
Age-Structured Population Models
Let’s now take a one step back and think about possible ways to improve a model of a single population. The models that we often think about are those which describe a population as a single unit with homogenous birth rate and death rate. However, it is a great simplification to assume a homogenous population. A 45 years old woman is less likely to have a child than a 25 years old one. This is exactly the kind of refinement which we often consider, when we model a single population. The commonly used approach is to divide a population into cohorts by age, 0 - 15, 16 - 30, 31 - 45, and 45+ for instance. Sometimes, we even go beyond the age and distinguish between sexes as well. One of the most popular model in population ecology, which builds on the aforementioned premise, is the Leslie matrix model invented by Patrick H. Leslie.
Task: Use the Leslie matrix to model a population in country of your choice. Estimate the model parameters based on historical data. How well does the model correspond to the historical data? Make predictions about the population size and its age distribution in 5, 10, 20, 50, and 100 years from now. Will the demographic pyramid in your country of choice reach an equilibrium? Interpret the results and present them in form of demographic pyramids.
Useful links:
Leslie matrix:
http://web.stanford.edu/~jhj1/teachingdocs/Jones-Leslie1-050208.pdf
Population by age, sex, and urban/rural residence:
http://data.un.org/Data.aspx?d=POP&f=tableCode%3A22
Population Dynamics of Fisheries
Since the human population on the planet Earth has been rapidly growing over the past decades, sustainable fish yields are growing concern of fisheries scientist. Therefore, precise restrictions on the amount of fish one can harvest in given areas are needed. One of the ways to find the right gap for harvesting is to make a model. In other words, our goal will be to describe the way in which given population grows and shrinks over time with controlled birth and harvest rates and a given death rate. There are numerous approaches to tackle this problem, including but not limited to systems of differential equations and modified Leslie matrix model.
Task: Use a population model to describe population dynamics of a fishery of your choice. Estimate the model parameters based on historical data. How well does the model correspond to the historical data? Try to find a sustainable fish yield and make predictions about the population size in 5, 10, 20, 50, and 100 years from now. Interpret the results.
Useful links:
Population dynamics of fisheries overview:
http://en.wikipedia.org/wiki/Population_dynamics_of_fisheries
Harvesting in matrix population models (accessible through discovery.muni.cz):
www.jstor.org
Commercial fisheries statistics:
www.st.nmfs.noaa.gov/commercial-fisheries/
Epidemic Modeling
Even though epidemic models are not strictly population models, they describe dynamics of a population over time under certain conditions (disease). Also, both models approach their goals from a very similar angle. An epidemic model is a compartment model with population separated into different groups (compartments). The simplest epidemic models differentiate between susceptible and infectious individuals. More sophisticated models then extends this concept by another compartment, the group of recovered individuals.
Task: Use the SIR (or SIRS) epidemic model to simulate an epidemic outbreak in given population. For example, you can model the potential outbreak of the 2009 influenza H1N1 in Brno. Estimate the model parameters based on historical data. Given your analysis of the model, present a few policies that you think will decrease both the total number infected and the time that it takes for the disease to run its course through our population. Interpret the results.
Useful links:
Epidemic models overview:
http://en.wikipedia.org/wiki/Epidemic_model
Simple epidemic models:
http://mysite.science.uottawa.ca/rsmith43/MAT4996/Epidemic.pdf
WHO influenza statistics:
http://apps.who.int/globalatlas/dataQuery/default.asp
The Limits of Growth
The Limits of Growth refer to the now almost legendary study in the mathematical modeling circles. It is also the name of a book that presents the result of a mathematical model called World3 created by Donella Meadows, Denis Meadows, and Jorgen Randers. The authors of the model asked themselves as how the growing human population interacts with expanding global economy, and how they can both adapt to carrying capacity of the planet Earth. They created a complex model which is composed from five interrelated systems, the food system, the industrial system, the population system, the non-renewable resources system, and the pollution system.
Task: Use the ideas of World3 model to simulate the interaction between the economics and the limits of the planet Earth. Estimate the model parameters based on historical data. Given your analysis of the model, present several scenarios of the population growth and interpret the results. Especially, make predictions about the population size and its age distribution in 5, 10, 20, 50, and 100 years from now, and explain how the population is influenced by different parts of the model.
Useful links:
The Limits of Growth overview:
http://en.wikipedia.org/wiki/The_Limits_to_Growth
Simple World model:
www.mdpi.com/2071-1050/5/3/896/pdf
2. Modeling tips
STELLA Modeling & Simulation software is an easy to understand software designed to model system dynamics. It’s very simple to use and doesn’t require and prior programing knowledge. Practically all of the tasks above can be easily modeled using STELLA (the software should be installed in some of the universities’ computers).
Free trial version is available here:
www.iseesystems.com/softwares/Education/StellaSoftware.aspx
Please check the How to be a Teacher Trainer course at Pilgrims website.
Please check the Teaching Advanced Students course at Pilgrims website.
|
