Lessons: Competencies
Introduction
This sub-module involves participants in considering teaching methods which support the development of students’ modelling competencies. It consists of two sessions.
In the first participants will reflect on the competencies students need to have when carrying out modelling (the module “Assessment” also deals with these sub-competencies). They will also develop guidelines about how to support students in developing some of these competencies. Special emphasis is given to students’ competencies in developing reasoned arguments.
In the second session participants will consider how to support students in developing knowledge about modelling at a meta-level (metacognition). They will also reflect on why this is necessary.
Materials for participants- Resource L.2.1 – Cards for use in developing “setting up the model” competencies
- Resource L.2.2a and L.2.2b – Tasks to provoke “validation” discussion
- Resource L.2.3 – Some ideas for teachers about how to focus on modelling competencies in lessons
- Resource L.2.4 – possible schema for the modelling cycle for use with students of different ages
- Resource L.2.5 – Some ideas for teachers about how to focus on modelling competencies at a meta – level in lessons
- Teacher diary pages for this sub-module
Materials you will need
- Power Point file: Lessons_competencies.ppt
- Poster paper
- Poster pens
- Scissors
Introducing the sub-module
/ Inform participants that this sub-module consists of two sessions.
In the first we will look at sub-competencies in the modelling process and how we might support these.
In the second session we will “zoom out” and consider the modelling cycle as a whole and the “meta-cognitive” competencies required to deal with modelling.
Session 1
Allow about 1 ½ to 2 hours
/ Emphasise that this session of the sub-module will allow participants to explore how best to support students with their mathematical modelling. In particular it will focus on how modelling lessons might best focus on sub-competencies of the modelling process.
/ Inform participants of the expected outcomes of Session 1.
Suggest that they should be thinking about how they might plan lessons to support their students with modelling competencies as they work through the session.
Main activities of the sub-module
/ Explain that participants will have met the PISA schema for the modelling cycle in the very first sub-module of the TTC “What is modelling?” They may have already met it again in the sub-module “Summative” of the Assessment module.
It is given in this slide with the sub-steps simplified (as you click to progress through the presentation).
This highlights and simplifies the sub-competencies into four key areas:
setting up the model
working accurately – this is the “working title” given to what PISA refers to as working in the mathematical world
interpreting
validating and reflecting
(Re-)Introduce this here.
You might ask participants to briefly think about where their students may have most problems when working on modelling tasks – to assist them they might like to think how they have coped when working on modelling tasks so far. It is likely that they will consider setting up the model and interpreting as difficult steps particularly as students meet modelling for the first time.
Slide 8 raises the issue that students need to develop sub-competencies and competencies related to the overall process of modelling. This will be discussed in greater detail in Session 2.
The slides that give details of the PISA sub-competencies are repeated here if you need them by clicking on the appropriate sub-competency coloured ovals in slide 8.
If you wish to skip straight to the discussion click on the arrow in the bottom right corner of the slide.
/ The detailed descriptions have already been looked at in the module “modelling”.
This slide explains in detail the competencies required to carry out the steps identified above; e.g. what it really means to proceed from the real world problem to the mathematical problem. The fact that you may have to consider so many different things when making this step is extremely important particularly for primary school pupils. Teachers need to be aware of the challenge this poses for students.
The complete list gives insight into the complexity of modelling. However, this does not mean that modelling is something students cannot learn, but it does show that we need to be aware that modelling requires considerably more from students than merely practising mathematical exercises
You will have to decide whether or not to use this and the next two slides or not.
/ Highlight, that making sense of the mathematical solution in terms of the real situation requires in fact two steps:
Interpreting the result (What does this mathematical result mean in the real situation?)
Validation of the result (Is the solution suitable for the problem?)
/ Now move the discussion on to start to consider how in teaching students can be assisted with sub-competencies whilst not neglecting others. We advocate that for problems to be meaningful the whole modelling process needs to be followed, but when doing this it is possible to emphasise particular sub-competencies.
That is, whilst working on a modelling task in full, students may be supported in developing particular sub-competencies.
This will be illustrated in the next activity.
/ This activity briefly involves participants in the type of activity they could use to focus students on a specific sub-competency needed when modelling.
It refers to the “signing” task that participants will have met in the first sub-module “What is modelling?” and may have considered in the Assessment module.
The next slide is a reminder of the task.
/ At this point distribute a set of cards to each group made from Resource L.2.1.
Allow participants to sort the cards into three groups:
· facts you need to use
· facts you do not need
· assumptions you need to make.
This will not take very long but allows participants to think of one method that supports learners in setting up a model.
You might suggest that the method of sorting all ideas that crop up when setting out to devise a model might be something teachers discuss with their students.
/ Please provide teachers with a map or ask them to bring a map from Europe with them. Ask participants to work on this modelling task. Arrange participants in groups of 4 but at first ask them to work in pairs. In each four give one pair resource L.2.2a and the other pair L.2.2.b. These may lead to different results which should lead to discussion later about the validity of models.
The Cities have been divided into two groups because the ratio of the two distances is different: those in Resource L.2.2a have a ratio of
distance by road :straight line distance
of about 1.2, whereas those in L.2.2b have a ratio of around 1.5. See below:
It is therefore likely that different models will be developed for the different groups of cities.
Emphasise that the questions asked here have been deliberately chosen to focus on the validity of the models that teachers developed, e.g. for cities in L2.2a distance by road = 1.2 × straight line distance, whereas for cities in L2.2b the multiplying factors is 1.5. If participants have difficulty in getting started perhaps drawing a graph of one valuable plotted against the other will help.
Suggest that it is important when developing / adapting tasks to consider objectives for teaching and learning: it is possible to focus learners on particular sub-competencies of the modelling cycle by judicious selection of tasks and questions.
The next activity is optional – if you wish to skip it click on the arrow in slide 16.
/ This activity is optional: allow participants some time to consider how they might adapt some tasks so that students are able to focus on particular sub-competencies of the modelling cycle.
/ This slide gives an overview. So far we have dealt with some sub-competencies in carrying out single steps of the modelling process (particularly the problematic steps of setting up the model, interpreting and validating). Participants also gained insight into the fact that students have to carry out the whole modelling process in order to learn modelling.
The next competencies we will deal with are those in reasoning / justifying. These are necessary in all steps of the modelling cycle.
In the next activity participants will have an opportunity to experience this for themselves.
/ Suggest that participants work on this task in pairs / groups. Tell them that it has been used with pupils with a range of ages. They will have an opportunity to see the solutions of some 9-10 year olds later. Make sure that you show participants the next slide which asks them to be involved in expressing their reasons for decisions they make at each stage of the modelling process.
/ Emphasise that at each step participants should ask the question “why?” in relation to the decisions they make.
Note the “bubble” suggesting this appears only when you click to advance the presentation.
This models the type of activity that participants can use with their students in an attempt to make explicit the reasons behind their decisions.
/ Participants may be interested in seeing this solution from a French pupil aged 9/10.
Plenary
/ It is suggested that you have a plenary discussion that focuses on the main aspects of this session. Perhaps you should start with asking participants about their thoughts relating to reasoning in mathematics lessons. This slide shows some quotes that may be useful in developing the debate.
/ This slide broadens the focus of the discussion to include some of the earlier issues dealt with in the session, ie considering how to develop sub-competencies important in modeling.
/ Return to the objectives and outcomes for the session. Ask participants to plan how they might incorporate the ideas of the session into a forthcoming lesson(s).
If possible they should try this before the next session.
At this point you may wish to distribute Resource L.2.3 which outlines some ideas for lessons that can be used to focus students on developing specific sub-competencies necessary for modelling.
Session 2
Allow about 1 ½ to 2 hours
/ This is the second session of the sub-module
/ Explain that in this session participants will have an opportunity to zoom out from considering sub-competencies and turn their attention to the modelling cycle as a whole and the competencies that are required to do this effectively.
Main activities of the sub-module
/ Refer to the list of modelling competencies in order to highlight that metacognition is an important modelling competency for students to develop.
The difficult thing when teaching is how to encourage students to think about this.
/ This slide gives an overview of meta-cognition. In many ways it is what mature learners do naturally: they stand back and take a more distant view of what they are doing and possibly adjust their approach according to their analysis of how they are proceeding. In order to do this they have to have a clear overview of what is expected. In terms of modelling the schema you have met/worked with so far are very helpful at allowing students to gain this overview.
The next activity will allow participants to experience one way of working they may use with students so that they may have an opportunity to gain such an overview.
/ Participants are asked to work on an open-ended task that is not a modelling task so that they can later reflect on competencies that are specific to mathematical modelling and how these are different to competencies that are used in solving other “open” mathematical tasks.
They are asked to do this in the next slide
If groups have trouble getting started you may advise them to perhaps work with simple situations (eg squares, or rectangles) first.
/ Participants should contrast how they tackled the new task with how they worked on the modelling task in Session 1 – the queuing problem.
Although there are some similarities (for example, simplifying the situation, being systematic and so on) there are distinctive differences. Of course, the main of these being that the modelling task is based in reality – this leads to some very important processes being necessary to work on the problem.
Ask groups to make a mini-poster of their reflections on problem-solving processes.
You can use these to prompt some discussion about metacognition but you may wish to leave this until after the next activity when some points to stimulate discussion are provided.
/ For possible solutions to this activity see Resource L.2.4.
Let groups work on this with each group working initially on the age range of their choice. If they have time they should repeat for a different age range.
Each group may present their solution on an overhead transparency and present it to the whole group.
Let teachers discuss the advantages and disadvantages of their proposals.
Emphasise that there is no ideal solution. Teachers should decide what they like to use with their particular classes.
/ One way of organising the discussion is to ask approximately half of the group (working in small groups or pairs) to develop arguments for and the other half to develop arguments against.
The next two slides give some arguments “for” and “against”. If you wish to skip these click on the arrow.
/ Some additional arguments “for”
· Metacognition about the modelling cycle has to be adapted to the age and to the level of competency
· Metacognitive strategies lead to reflection about how mathematics can contribute to solve problems within the real world
· Metacognitive strategies are an essential part of modeling (e.g., reflection)
· Metacognition makes modelling more effective, students learn a lot about problem solving/modelling which is necessary for doing mathematics.
/ You might first let participants work first individually, than in pairs. (Think – pair – share)
Take notes on a slide the participants cannot see; this will allow you to show an overview if you wish.
Resource L.2.5 gives some additional ideas that you may wish to distribute at the end of the session.
Emphasise that to enable students to reflect at a meta-level must form a continuous part of the mathematics lessons. During the discussion make sure it becomes clear, that it is not sufficient, to talk only once about the modelling process. It is an aim of modelling lessons that students learn how to proceed when they are given new modelling problems
Plenary
/ In a plenary session discuss with the whole group the session in relation to the objectives set at the start.
These relate to the final competency on slide 39 – those related to meta-knowledge in relation to modelling.
/ Prompt participants to use their Teacher Diary to reflect on how they can help their students develop meta-cognitive skills.
Suggest that it would be good to plan specific activities they might use and try these before the end of the course if they have an opportunity.
LESSONS ▪ Competencies ▪ page 3