MATHEMATICAL MODELLING COMPETENCIES MEANING TEACHING AND ASSESSMENT Gabriele

MATHEMATICAL MODELLING COMPETENCIES: MEANING, TEACHING AND ASSESSMENT Gabriele Kaiser University of Hamburg Australian Catholic University Gabriele Kaiser – University of Hamburg 1

Introduction The development of students’ competencies to use their mathematics for the solution of problems of daily life and from sciences is currently accepted aim worldwide for mathematics education. For example the PISA study formulates: Mathematical literacy also implies the ability to pose and solve mathematical problems in a variety of situations, as well as the inclination to do so, which often relies on personal traits such as self-confidence and curiosity But: Modelling still of low importance in ordinary mathematics teaching at school and university level. Still strong deficits of students in their competencies to use and apply mathematics in modelling problems or in their daily life. What can we do and are we successful in our activities? Gabriele Kaiser – University of Hamburg 2

Structure • Framework for modelling competencies • Description of two examples • Description of modelling activities in German schools and teacher education • Results of empirical evaluation of modelling competencies • Closing remarks Gabriele Kaiser – University of Hamburg 3

Theoretical framework for modelling competencies Gabriele Kaiser – University of Hamburg 4

Milestone for recent international initiatives for implementing modelling approaches in mathematics education: Hans Freudenthal’s Symposium “How to teach mathematics so as to be useful” in 1968, ICME-1, in Exeter, proceedings published as first volume of Educational Studies in Mathematics in 1969. In his welcome speech, “Why to teach mathematics so as to be useful”, Freudenthal made a strong plea to change mathematics education, to include real world examples and modelling into mathematics education in order to make mathematics more meaningful for students. The connection to real life, mathematizing the world, was important for him. He criticises usual teaching approaches neglecting relations to the real word, unfortunately still common in many classrooms until today. Gabriele Kaiser – University of Hamburg 5

Theoretical framework: How do we model? Several approaches to describe modelling processes Widely accepted are modelling cycles with different stages departing from the real world problem and coming back to its solution via mathematical models Gabriele Kaiser – University of Hamburg 6

Real world model Mathematisation Mathematical model Understanding Idealisation Validation Mathematical work Real situation Mathematical results Validation Real results Interpretation Real world Gabriele Kaiser – University of Hamburg Mathematics

Speed of a Tsunami wave We itend to develop an elementary model for calculating the speed of a Tsunami wave. Needs only knowledge about root functions and basic EXCEL-skills. From Henning & Freise (Eds. ) (2011), Realität und Modell (pp. 110 -125). Münster: WTM. Gabriele Kaiser – University of Hamburg 8

Real situation: In order to create a model for a single Tsunami wave, one needs to understand the underlying physical principles first, i. e. that the speed of a Tsunami wave is connected to its depth. Water depth (m) Speed (kph) 6000 800 2000 500 250 The spreadsheet on the right-hand side has been submitted by a geographic research institute (GFZ). 200 150 100 110 http: //www. gitews. org/fileadmin/documents/cont ent/press/GITEWS_Broschuere_DE_08. pdf 20 50 10 36 In a classroom situation students could use the internet to conduct the necessary information. Gabriele Kaiser – University of Hamburg 9

Understanding and simplifying… We need to make a few assumptions to be able to start working. Assumption 1: We do not take the magnitude of the wave into account as it is relatively small compared to its depth. Assumption 2: We do not care about friction. Assumption 3: As the speed of the wave is decreasing the nearer it gets to the surface, we assume that its speed is zero at the surface. Gabriele Kaiser – University of Hamburg 10

Mathematising… Gabriele Kaiser – University of Hamburg 11

Mathematical working… In order to plot the graph of the function, we have used the former spreadsheet again but calculated the speed in meters per second. This is the result: 250 200 150 100 50 0 0 1000 2000 Gabriele Kaiser – University of Hamburg 3000 4000 5000 6000 7000 12

Mathematical working… Gabriele Kaiser – University of Hamburg 13

Interpreting… Gabriele Kaiser – University of Hamburg 14

Validating… Gabriele Kaiser – University of Hamburg 15

Current discussion: focus on modelling competencies Globally: modelling competencies include ability and willingness to work out real world problems, which contain mathematical aspects through mathematical modelling. In detail that comprises variety of competencies such as: Competence to solve real world problem through mathematical description (model) developed by one’s own; Competence to reflect about the modelling process by activating metaknowledge; Insight into the connections between mathematics and reality, esp. its subjectivity; Social competencies such as working in groups, communicating Origin: KOM-project by Mogens Niss (and Thomas Jensen). Gabriele Kaiser – University of Hamburg

Modelling competencies in the KOM-project Proposal by Niss (1999) for 8 mathematical competencies KOM flower • Modelling competence as one component • Definition of competence as someone‘s insightful readiness to act in response to the challenges of a given situation • Specification of this construct as competency by Blomhøj & Jensen (2003, 2007) - analytical view in contrast to holistic view Gabriele Kaiser – University of Hamburg 17

Modelling cycle (Kaiser & Stender, 2013) mathematise Real model Mathematical model understand simplify Real situation mathematical work validate Real results Gabriele Kaiser – University of Hamburg interpret Mathematical results 18

Description of modelling competencies along the modelling process with subcompetencies (Blum & Kaiser, 1997 and continuation by Maaß, 2004) Competence to § understand the real problem § create a real model by simplifying the problem and concentrating on questions that can be worked out; recognising central variables and their relations; formulating assumptions and descriptions of the problem § § transfer the real model into a mathematical model work mathematically within the mathematical model, i. e. construct or select adequate mathematical models of the problem and develop solutions from within the model; § § interpret mathematical results in a real situation evaluate the solution’s adequacy and validate it Gabriele Kaiser – University of Hamburg Sub-process 1: Understanding – Simplifying – Mathematising Sub-process 2: Working mathematically Sub-process 3: Interpreting – Validating 19

Promotion of modelling competencies Gabriele Kaiser – University of Hamburg 20

How can we promote modelling competencies in mathematics education? Many important issues, in the following only a few key aspects selected. Two different approaches of fostering mathematical modelling competencies: • holistic approach: the development of modelling competencies should be fostered by performing complete processes of mathematical modelling, whereby the complexity and difficulty of the problems should be matched to the competencies of the learners (Haines, Crouch & Fitzharris, 2003). • atomistic approach: propagates the individual fostering of subcompetencies of mathematical modelling, because implementation of complete modelling problems would be too time-consuming and be not effective (Blomhøj & Jensen, 2003). Gabriele Kaiser – University of Hamburg 21

Consequences for integration the inclusion of modelling examples in teaching at all levels – two extremes: - “mixing approach”, i. e. “in the teaching of mathematics, elements of applications and modelling are invoked to assist the introduction of mathematical concepts etc. Conversely, newly developed mathematical concepts, methods and results are activated towards applicational and modelling situations whenever possible” (Blum & Niss, 1991, p. 61). - “separation approach”, i. e. instead “of including modelling and applications work in the ordinary mathematics courses, such activities are cultivated in separate courses specially devoted to them” or variations like the “twocompartment approach” or the “islands approach” (Blum & Niss, 1991, p. 60) Gabriele Kaiser – University of Hamburg 22

David Burghes Proceedings ICTMA 1, 1983, Exeter „The basic philosophy behind the approach … of the modelling workshop for higher education is that to become proficient in modelling, you must fully experience it – it is no good just watching somebody else do it, or repeat what somebody else has done – you must experience it yourself. I would liken it to the activity of swimming. You can watch others swim, you can practice exercises, but to swim, you must be in the water doing it yourself. “ Similar positions: Helmut Neunzert at ICTMA-14, 2009 in Hamburg: Modelling is not a spectator sport, in order to learn modelling, you have to model. Gabriele Kaiser – University of Hamburg 23

Authentic modelling problems in the context of modelling days/weeks in Hamburg Project at Hamburg University since 2000 in cooperation by applied mathematics and mathematics education: Modelling courses for students from upper secondary level (age 16 -18) and modelling days lasting three days with students from lower secondary level (grade 9, age 14) with whole age cohort of about 5 -6 schools supervised by prospective teachers as part of their master study. Core: groups of students working on modelling problems Gabriele Kaiser – University of Hamburg 24

Characteristics of modelling activities Teacher independent work on modelling examples by students necessary No fast intervention by the teacher, only support in case mathematical means are missing or if the students are in a cul-de-sac regarding the content (empirical studies by Leiß 2010 and Stender 2016); Experience of helplessness and insecurity a central aspect and a necessary phase. Gabriele Kaiser – University of Hamburg 25

Aims and characteristics of activities Usage of authentic examples: v authentic examples (in the sense of Niss, 1998) are recognised by practitioner working in this field as examples they could meet; v only little simplified, often no solution known, neither to us nor to the problem poser; v often only problematic situation described, students have to determine or develop a question, which can be solved (Pollak, 1969: “here is a problem, think about it”); v various problem definitions and solutions possible dependent of norms of the modellers. Aim is not: comprehensive overview on modelling approaches. Gabriele Kaiser – University of Hamburg

Characteristics of the examples and activities No high level of mathematical knowledge expected, complexity from the context, making assumptions, simplifications, interpretations Independent work is important Challenge does not come from algorithms, but lies in the independent realisation of the modelling process Gabriele Kaiser – University of Hamburg

Problems of recent modelling activities How can fishing be controlled sustainably by quotas? When does an epidemic break out and how many people fall ill? How can the arrangement of irrigation systems be planned in an optimal way? How can we design an optimal wind park, off-shore or on-shore? How can bush fires be hold off by cutting forest aisles? In the following an advanced solution for this problem, can be done more elementarily by students at school or at university. Gabriele Kaiser – University of Hamburg 28

How can bush fires be hold off by cutting lanes? Bush fires are a serious problem for large connected wooded areas. Therefore. it is important to raise questions about prevention and containment of bush fires. Methods like forest aisles or the controlled burning off of single forest areas can constitute effective measures. Construct a preventive model for bush fires while cutting down as many trees as necessary but as few as possible! Gabriele Kaiser – University of Hamburg 29

Simplify the situation, idealise and structure - Real model Assumptions: • quadratic forest area (commercial forest) • parallel forest aisles with a fixed width (e. g. 10 m) • Forest aisles are equidistant • the fire cannot skip over an aisle • no wind • the fire breaks out only in one separate forest area at one time • if no additional forest aisles will be cut in the area, where the fire breaks out, this whole forest area will burn down Gabriele Kaiser – University of Hamburg 30

Mathematise the real model - Mathematical model – First mathematical model Gabriele Kaiser – University of Hamburg Further mathematical model 31

Mathematical work and results Gabriele Kaiser – University of Hamburg 32

Mathematical work and mathematical results Gabriele Kaiser – University of Hamburg 33

Interpretation of the mathematical results - Real results – The lost forest area in the further mathematical model is lower (about 2. 97 km 2 instead of 6. 23 km 2) and only 20 aisles are needed instead of 31 in the first mathematical model Validate the results and improve the model: • Consider wind and its direction • Dynamic of a bush fire Ø cellular automata Gabriele Kaiser – University of Hamburg 34

Advanced mathematical model cellular automata: programming the dynamic of a bush fire via simulation with fixed parameter for wind direction, wind strength …… Gabriele Kaiser – University of Hamburg 35

Characteristical activities within our modelling activities with students Peer-explanations by students Gabriele Kaiser – University of Hamburg Work in groups with minimal help by tutors

Characteristical activities within our modelling activities with students Presentation of results by students in lecture hall Gabriele Kaiser – University of Hamburg Presentation of results via poster exhibition

Selected results of empirical studies Evaluation of different modelling activities with different instruments taken from the literature Evaluation of modelling competencies with the test instrument developed by Haines et al. and various further developments. For example In more recent activities by Brand (2014): comparison of efficiency of holistic and atomistic approach on the development of modelling competencies with an own modelling test developed. Gabriele Kaiser – University of Hamburg

How to measure modelling competencies? Historical development described by Houston (2007): - first phase of measuring mathematical modelling: holistic assessment with list of sub-competencies related to the different phases of modelling (Hall, 1984; Berry and Le. Masurier, 1984); already problem of compensating good grades with bad grades; proposal of multiplication of grades, not feasible - second phase: developing robust assessment criteria for projects by the UK Assessment Research Group (ARG): still strong connection to modelling phases, in addition high importance of communication skills (Haines et al. , 1996) - third phase of micro-assessment: development of multiple-choiceitems on eight phases of the modelling process Gabriele Kaiser – University of Hamburg

Micro-assessment of modelling competencies Modelling test developed by Haines & Crouch, Davis (2000), work extended by Haines & Crouch, Davis (2001), Haines & Crouch, Fitzharris (2003), Houston & Neill (2003) , Izard, Haines, Crouch, Houston and Neill (2003), strongly discussed in the context of the International Conferences on the Teaching and Learning of Mathematical Modelling (ICTMA). Items developed for mathematics undergraduate students and/or engineering students in their mathematics courses; 22 modelling questions, which allow pre- and post-test-design; multiple -choice test with partial credit system. Still problem of compensation Gabriele Kaiser – University of Hamburg 40

Example I - Assigning central variables Gabriele Kaiser – University of Hamburg

Example II - Interpreting the solution Gabriele Kaiser – University of Hamburg

Example III – Validating the solution Gabriele Kaiser – University of Hamburg

Results of the evaluation of modelling days (Kaiser, 2013) Modelling days at Gymnasium Grootmoor, school with students of higher ability track, grade 9 (age 14 -15), about 160 students. Aim: fostering modelling competencies of students Four authentic problems in small groups: • Optimal positioning of rescue helicopters; • Development of new dress sizes; • Optimal automatic garden irrigation • Optimal tariffs for communication networks (phone, mobile phone, internet). • Gabriele. Kaiser – University of Hamburg

Summary of the results Significant development of the sub-competencies within the modelling project; No significant differences between the improvements differentiated by gender or grade; Biggest changes of the scores in items which test competencies that are not fostered by usual mathematical tasks, namely description of the problem, identification of central variables; but problems with validation; Only little changes of the meta-cognitive modelling competencies measurable. Gabriele Kaiser – University of Hamburg

Results of studies with modelling integrated into whole school year (Kaiser, 2007) Usage of the original test with students from modelling courses in 20042005, modelling activities integrated into whole school year: 57 students, 11 th and 12 th year (age 16 -17), 39% students from advanced mathematics courses, year 12; 61% students from basic courses year 11; 67% boys, 33% girls, Three test times: beginning, middle, end of course, 8 items, 16 points maximum. Gabriele Kaiser – University of Hamburg 46

Results High results (about 50% of possible points), if compared to results of mathematics undergraduate students in original studies; Positive development already after 4 months, but small decrease from middle to end test Small gender differences in accordance with other studies, e. g. Houston & Neill (2003). Conclusions: long-term processes necessary. Gabriele Kaiser – University of Hamburg

Summarising Modelling competencies of students can be promoted by adequate courses, either within ordinary classroom setting or special modelling weeks; , strong increase in competencies But: somehow instable progress, decrease after some time, if no further activities are following; apparently longitudinal processes within whole curriculum change necessary. Promotion of meta-cognitive competencies necessary, i. e. competencies to plan and monitor the modelling process and to evaluate about it. In accordance with results from other studies (such as Haines et al. Galbraith/Stillman), new large scale study by Vorhölter (2017, 2018). Gabriele Kaiser – University of Hamburg

Possible consequences Necessity of explicit modelling activities such as modelling days or modelling weeks Necessity of meta-cognitive activities such as discussion of metacognitive strategies (monitoring, planning, usage modelling cycle). Integrated approach preferable, i. e. mixing of atomistic and holistic phases as both approaches foster different sub-competencies of modelling (Brand 2014). But: fostering of mathematical foundation necessary as well. And: long-term processes needed. Gabriele Kaiser – University of Hamburg 49

To summarise: Modelling as an established didactical approach can offer many examples for meaningful mathematics teaching at all levels from primary to tertiary; But not only examples are important, but also ways of fostering it. Holistic and atomistic approach necessary in an integrated way Focus on students‘ activities, students • need to carry out modelling processes by themselves (not a spectator sport) • need to take the examples seriously, authentic examples are necessary, which contain a serious problem to be posed or question to be anwered. Gabriele Kaiser – University of Hamburg 50

Outlook Theme: Empirical studies on the teaching and learning of mathematical modelling ZDM Mathematics Education, 50, issue 1 -2, 2018 Only 3% of papers in high-ranking journals in mathematics education on the teaching and learning of mathematical modelling. Hardly any paper from tertiary education, especially not empirically oriented; restriction to conference papers (mainly ICTMA). That must be changed. Gabriele Kaiser – University of Hamburg 51

Thank you very much for your attention. Gabriele Kaiser – University of Hamburg 52