ModellingTasks LessonsAssessment Reflecting Lessons

ModellingTasks LessonsAssessment Reflecting Lessons Competencies

Lessons Comptencies 3 ICTMethodsContent Session 1 Session 2 Image:

Lessons Comptencies 4 ICTMethodsContent Session 1

Lessons Comptencies 5 ICTMethodsContent You will consider: the sub-competencies required by students when modelling how to design lessons in order to support students with their development of particular modelling competencies how to support students‘ competencies in reasoning. Objectives

Lessons Comptencies 6 ICTMethodsContent You will develop some tasks to promote particular modelling competencies and plan how to use them Outcomes

Lessons Comptencies 7 ICTMethodsContent Reporting the work “Real world” “Mathematical world” Real-world problem Mathematical Problem Mathematical solution Real solution Validating & reflecting Setting up the model Working accurately Interpreting

Lessons Comptencies 8 ICTMethodsContent Reporting the work “Real world” “Mathematical world” Real-world problem Mathematical Problem Mathematical solution Real solution Validating & reflecting Setting up the model Working accurately Interpreting Modelling requires sub-competencies and competencies related to the overall process

Lessons Comptencies 9 ICTMethodsContent From the “problem in the real world” to the “mathematical problem” (1, 2, 3) (horizontal mathematization, De Lange, 1987) identifying the relevant mathematics with respect to a problem situated in reality; representing the problem in a different way, including organising it according to mathematical concepts and making appropriate assumptions; understanding the relations between the language of the problem, and symbolic and formal language needed to understand it mathematically; finding regularities, relations and patterns; recognising aspects that are isomorphic with known problems; translating the problem into mathematics; i.e., to a mathematical model Modelling (mathematization) process – PISA framework 2003 – p. 39 The modelling cycle (PISA, 2003) Setting up the model

Lessons Comptencies 10 ICTMethodsContent Working in the “mathematical world” (4) (vertical mathematization, De Lange, 1987) using and switching between different representations; using symbolic, formal and technical language and operations; refining and adjusting mathematical models; combining and interpreting models; argumentation; generalisation. Modelling (mathematization) process – PISA framework 2003 – p. 39 The modelling cycle (PISA, 2003) Working accurately

Lessons Comptencies 11 ICTMethodsContent Going back to the real world situation (5) (interpreting and validating both the solution and the model) understanding the extent and limits of mathematical concepts; reflecting on mathematical arguments, and explaining and justifying results; communicating the process and solution; critiquing the model and its limits. Back to presentation Modelling (mathematization) process – PISA framework 2003 – p. 39 The modelling cycle (PISA, 2003) Validating & reflecting Interpreting

Lessons Comptencies 12 ICTMethodsContent This remark came from teachers who had been dealing with modelling in their lessons: “In order to enhance the development of modelling competencies students need to practice the single steps of the modelling process first. Only when they are able to carry out the single steps, tasks requiring to carry out a whole modelling process can be given.” Discussion

Lessons Comptencies 13 ICTMethodsContent Imagine that you are using the “signing” task you met in “What is modelling?” with a group of students. You want to support them with “setting up the model”. Now imagine you are the students. You will be given a set of cards. Sort each card into one of three groups. Those that contain:  facts you need to use  facts you do not need  assumptions you need to make Activity 1

Lessons Comptencies 14 ICTMethodsContent Task 1: “Signing against a new law” Recently, the 25th of April of 2006, the Spanish party in the opposition presented in the congress signatures against a new law promoted by the government. All Spanish newspapers published pictures with the big boxes and the 10 vans needed to transport the sheets of paper to the congress. Do you think there was a political intention behind this staging or all these boxes and vans were really necessary to carry the signatures?

Lessons Comptencies 15 ICTMethodsContent Work with a partner on the following task. Journey Please look at a map. How far away is Madrid from Paris? If you drive by car? As the crow flies? If you know one distance can you predict the other? Activity 2

Lessons Comptencies 16 ICTMethodsContent Now consult with another pair who has been working with different data. Consider the validity of your solutions. Some questions to consider: Is one solution better for longer distances? Is one solution better for shorter distances? If you could only recommend one solution which would it be? Activity 2 continued

Lessons Comptencies 17 ICTMethodsContent Try to develop / adapt a task to focus on a particular sub- competency of the modelling process. Reporting the work “Real world” “Mathematical world” Real-world problem Mathematical Problem Mathematical solution Real solution Validating & reflecting Setting up the model Working accurately Interpreting Activity 3 (optional)

Lessons Comptencies 18 ICTMethodsContent Competencies to carry out the single steps of the modelling process Competencies to carry out a whole modelling process Competencies to reason / justify decisions in relation to the modelling process Metacognitive modelling competencies Modelling competencies

Lessons Comptencies 19 ICTMethodsContent At Europa Park you want to enter an attraction where there is a 70 metre queue. How long will you have to wait? Activity 4

Lessons Comptencies 20 ICTMethodsContent Make a poster of your solution. At each step give reasons why you have made the decisions you have. Justify your choices! Reporting the work “Real world” “Mathematical world” Real-world problem Mathematical Problem Mathematical solution Real solution Validating & reflecting Setting up the model Working accurately Interpreting Ask the question “why?” Activity 4

Lessons Comptencies 21 ICTMethodsContent Translation here

Lessons Comptencies 22 ICTMethodsContent How can students learn to reason in mathematics? „Reasoning is is a major issue in other subjects but not mathematics“ „Either students know how to reason or not. There is no way of supporting this competency in mathematics lessons“ Plenary discussion

Lessons Comptencies 23 ICTMethodsContent “I have tried to get students to write down their reasons for decisions they make, but they just can‘t do it. So I gave up.“ “Students may validate tasks when asked to do so in “Validation-tasks” but they always forget when carrying out the whole modelling process, so there is no use in these tasks.” “If students present their methods to their peers in class and the results are discussed and viewed critically, they learn to reason and justify their methods.” Plenary discussion

Lessons Comptencies 24 ICTMethodsContent You will consider: The sub-competencies required by students when modelling. How to design lessons in order to support students with their development of particular modelling competencies How to support students‘ competencies in reasoning. Objectives

Lessons Comptencies 25 ICTMethodsContent You will develop some tasks to promote particular modelling competencies and plan how to use them – do this in your Teacher Diary Outcomes

Lessons Comptencies 26 ICTMethodsContent Session 2 Metacognition

Lessons Comptencies 27 ICTMethodsContent Competencies to carry out single steps of the modelling process Competencies to carry out a whole modelling process Competencies to reason / argue in relation to the modelling process Metacognitive modelling competencies Modelling competencies

Lessons Comptencies 28 ICTMethodsContent In this module you will consider why having an overview of the modelling cycle is important for students how to support students in the development of metacognitive strategies. Objectives

Lessons Comptencies 29 ICTMethodsContent You will design / adapt schema for the modelling-cycle or information about modelling at a meta-level for students of different ages You will develop ideas (guidelines) about how to introduce meta-knowledge about modelling in your lessons Outcomes

Lessons Comptencies 30 ICTMethodsContent Competencies to carry out single steps of the modelling process Competencies to carry out a whole modelling process Competencies to reason / argue in relation to the modelling process Metacognitive modelling competencies Modelling competencies

Lessons Comptencies 31 ICTMethodsContent It involves diagnostic knowledge about one’s own thinking, critical thinking about tasks and strategic knowledge about ways to solve problems planning, surveying and judging, which includes monitoring of one’s owns actions motivation and willpower to use metacognition. (Sjuts 2003, S. 18) Metacognition Metacognition is the thinking about one’s own thinking and management of one’s own thinking.

Lessons Comptencies 32 ICTMethodsContent Work on the new task so that you reach a solution. At Europa Park you want to enter an attraction where there is a 70 metre queue. How long will you have to wait? Draw a quadrilateral. Find the mid-point of each side. Join the points to form a new quadrilateral. How are the areas related? Activity 1

Lessons Comptencies 33 ICTMethodsContent In groups: reflect on the processes that you used to solve the two problems (– you may wish to first of all confirm how the modelling cycle schema applies to the modelling task and then think how you would adapt this for the non-modelling task) what are the important and distinctive aspects of modelling? Make a mini- poster Activity 1 continued

Lessons Comptencies 34 ICTMethodsContent Develop a scema/diagram for a modelling cycle or information for students about modelling at a meta-level for students aged –7 / 8 –13 / 14 –17 / 18 Activity 2

Lessons Comptencies 35 ICTMethodsContent How important is it that students have knowledge about the modelling process at a meta-level? Give arguments for and against the introduction of knowledge about the modelling process. Discussion

Lessons Comptencies 36 ICTMethodsContent Students have guidelines about how to proceed when modelling Metacognition can reduce students anxiety when modelling Metacognition help students to see the world through mathematical lenses Metacognition: arguments “for“

Lessons Comptencies 37 ICTMethodsContent The whole process may be difficult for students to understand This is not “real mathematics” There is no time to work on metacognitive strategies (due to curriculum pressures) Metacognition: arguments “against“

Lessons Comptencies 38 ICTMethodsContent Can you think of ways in your lessons that you can encourage students to think about modelling on a meta-level? Discussion continued

Lessons Comptencies 39 ICTMethodsContent Competencies to carry out single steps of the modelling process Competencies to carry out a whole modelling process Competencies to reason / argue in relation to the modelling process Metacognitive modelling competencies Modelling competencies

Lessons Comptencies 40 ICTMethodsContent In this module you will consider why having an overview of the modelling cycle is important for students how to support students in the development of metacognitive strategies. Objectives

Lessons Comptencies 41 ICTMethodsContent You will develop some tasks to promote particular modelling competencies and plan how to use them – do this in your Teacher Diary Outcomes