James Calleja
©2015
OBJECTIVES OF PROFESSIONAL DEVELOPMENT
Ø To understand the role of tasks in planning to teach mathematics through inquiry
Ø To explore tasks that provide opportunities for students to engage in mathematical inquiry
Ø To learn about characteristics of tasks promoting inquiry
Ø To understand the importance of pedagogical issues in creating a classroom culture around inquiry
Ø To plan for inquiry tasks with a focus on classroom management and organisation
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Teaching and Learning Mathematics through Inquiry
LITERATURE ON MATHEMATICAL TASKS Mathematics classroom instruction is generally organised around and delivered through students’ activities on mathematical tasks. The tasks that teachers assign usually determine how students come to understand mathematics. In other words, mathematical tasks serve as a context for students’ thinking, during and after instruction. Choosing and setting appropriate tasks is key to the success of teaching mathematics (Doyle, 1988). Tasks can vary not only with respect to content but also with respect to the cognitive processes involved in working on them. Willis (2010) speak of tasks providing students with an ‘achievable challenge’. Achievable-‐challenge tasks require students to employ mental effort, performing a task that is just difficult enough to hold student interest but not so difficult that they give up in frustration. Tasks are worthwhile when they offer students the opportunity to extend what they know and stimulate their learning. Doyle (1988) argues that tasks with different cognitive demands are likely to provoke different kinds of learning. Tasks that require students to solve complex problems can be considered to be cognitively demanding tasks. In contrast, cognitively undemanding tasks are those that give less opportunity for students to engage in high-‐level cognitive processes. The role and the different types of cognitive demand that tasks assume, thus becomes key to students’ opportunities for learning. Using cognitively demanding tasks in the classroom is not a straightforward endeavor. Teachers need to be cautious not to minimize the cognitive demands of a task. For example, when teachers introduce tasks, they need to focus on stating the purpose of the task, how the lesson will be structured (specifying the different phases of the lesson), how students are expected to work (their role in working individually and/or collaboratively) and indicating the time allowed. It is crucial not to give away too much information – for example, about what students should be doing or providing examples that lead students to particular ways/methods of working on the problem assigned. The literature promotes the use of rich mathematical tasks. But what make a task rich? ‘Rich’ is understood in terms of the potential that a task may have in providing sufficient complexity as to admit a variety of approaches, and therefore to have the capacity to reveal differences in student conceptions of relevant mathematical concepts and procedures. Rich mathematical tasks usually have most of the following characteristics: ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒
Connect with what has been taught; Address a range of outcomes; Allow all students to undertake the activity; Provide multiple entry points; Can be successfully undertaken using a range of methods or approaches; Encourage students to reveal their understanding of what they have learned; Allow students to make connections between the concepts they have learned; Are worthwhile activities for students’ learning; Draw the attention of teachers to important aspects of mathematical activity; Help teachers to the kind of help that students may require.
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If you choose to use cognitively challenging mathematics tasks, the challenge would be to implement them so that their potential for student higher-‐level thinking is maintained. This is especially true if you are working with low-‐attaining students. When using cognitively demanding mathematical tasks, teachers need to be mindful not to simplify tasks. This is a common pitfall for teachers who tend to alleviate students’ struggles and speed up task completion. Teachers may also tend to model specific algorithms for their students and provide the procedures needed to solve the tasks. To avoid this, teachers need to be clear and adamant about their role as students engage with tasks. It is useful to communicate to the students that it is natural to get stuck when working on challenging problems. Encouraging students to take time to consider different approaches in tackling the problem is usually helpful. You may find it difficult to stand by and watch students struggle. However, try to avoid stepping in prematurely to relieve your students of their uncertainty and frustration at not being able to make progress. Try instead to encourage them to look for alternative possibilities. When students ask for help, provide help that supports their process of inquiry rather than answer getting. For example, try using the following prompts: ⇒
There is no need to rush things, take your time.
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Don’t give up on it too quickly. I believe you can do a lot if you think it out together. Give it your best shot first.
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What do you know about the problem?
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What are you trying to do?
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What do you need to find out?
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What is fixed? What can be changed?
Open-‐ended tasks provide a better window into student thinking and reasoning processes because students are required to show their solution processes and provide justifications for their answers. Open-‐ended tasks can promote students' conceptual understanding, foster their ability to reason and communicate mathematically. Tasks should involve a range of possibilities and offer students opportunities to discuss ideas and to make choices. In addition, tasks should be appropriately challenging and be chosen to enable learners to explore significant mathematical concepts, to connect mathematical ideas and thus make sense of mathematics. References: Doyle, W. (1988). Work in Mathematics Classes: The context of students’ thinking during instruction. Educational Psychologist, 23(2), 167-‐180. Willis, J. (2010). Learning to Love Math: Teaching strategies that change student attitudes and get results. Alexandria, Virginia: ASCD.
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Teaching and Learning Mathematics through Inquiry
TASKS FOR MATHEMATICAL INQUIRY Collaborative Learning Tasks
From http://www.educationaldesigner.org/ed/volume1/issue1/article3/
Malcolm Swan created a framework with five ‘types’ of activities that encourage distinct ways of thinking and foster conceptual understanding. These include: 1. Evaluating mathematical statements – ask students whether statements are always, sometimes or never true, and developing proofs
2. Classifying mathematical objects – ask students to devise/apply a classification
Worksheet 1
Worksheet 2
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How c an you justify each of (a), (b), (c) as the odd one out?
3. Interpreting multiple representations – draw links and develop mental images for concepts
4. Creating and solving problems – ask students to create problems for the class
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Teaching and Learning Mathematics through Inquiry
5. Analyzing reasoning and solutions – diagnose errors and comparing solutions Cut up the following cards. Rearrange them to form two proofs. The first should prove that: If n is an odd number, then n2 is an odd number The second should prove that: If n2 is an odd number, then n is an odd number. You may need to use all the cards.
The tasks described are designed as classroom activities, and according to Swan, the tasks that teachers use should be accessible, extendable, engage students in decision-‐making, promote discussion, stimulate creativity, and encourage ‘what if’ and ‘what if not?’ questions. Inquiry Prompts
From http://www.inquirymaths.co.uk
On the website pages: Inquiry maths is a model of teaching that encourages students to regulate their own activity while exploring a mathematical statement (called a prompt). Inquiries can involve a class on diverse paths of exploration or in listening to a teacher's exposition. In inquiry maths, students take responsibility for directing the lesson with the teacher acting as the arbiter of legitimate mathematical activity. Prompts are mathematical statements, equations or diagrams stripped back to the bare minimum, while simultaneously loaded with the potential for exploration. In short, a prompt should have “less to it and more in it”. Inquiry is not about discovering a pre-‐determined outcome; rather, it is a joint mathematical exploration initiated by the student and supported by knowledgeable others, be they peers or adults.
NUMBER PROMPTS – AN EXAMPLE
Why is one statement correct when the other one is not?
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ALGEBRA  PROMPTS  –  AN  EXAMPLE   Encourage  students  to  come  up  with  the  questions  on  the  following  prompt! Â
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GEOMETRY  PROMPTS  –  AN  EXAMPLE              Class  posing/answering  some  questions  in  response  to  the  prompt:  ⇒
What  is  different  and  the  same  about  the  rectangles?  Â
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How  many  rectangles  are  possible  with  the  same  area? Â
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Which  has  the  longest  perimeter?  ...  the  shortest? Â
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Is  there  a  rectangle  with  an  area  equal  to  the  length  of  its  perimeter? Â
 Bowland  Maths  Tasks  From  http://www.bowlandmaths.org.uk Â
On  the  website  pages:  Bowland  Maths  aims  to  make  maths  engaging  and  relevant  to  pupils  aged  11-Ââ€?14,  with  a  focus  on  developing  thinking,  reasoning  and  problem-Ââ€?solving  skills.  In  these  materials,  the  maths  emerges  naturally  as  pupils  tackle  problems  set  in  a  rich  mixture  of  real-Ââ€?life  and  fantasy  situations. Â
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Teaching  and  Learning  Mathematics  through  Inquiry Â
BOWLAND MATHS TASKS Three Unstructured Problems
PROBLEM 1:
ORGANISING A TABLE TENNIS TOURNAMENT
You have the job of organising a table tennis league. • 7 players will take part • All matches are singles. • Every player has to play each of the other players once. •
• •
There are four tables at the club.
Games will take up to half an hour. The first match will start at 1.00pm.
Plan how to organise the league, so that the tournament will take the shortest possible time. Put all the information on a poster so that the players can easily understand what to do.
PROBLEM 2:
DESIGNING A BOX FOR 18 SWEETS
You work for a design company and have been asked to design a box that will hold 18 sweets. Each sweet is 2 cm in diameter and 1 cm thick. The box must be made from a single sheet of A4 card with as little cutting as possible. Compare two possible designs for the box and say which is best and why. Make your box.
PROBLEM 3:
CALCULATING BODY MASS INDEX
This calculator shown is used on websites to help an adult decide if he or she is overweight. What values of the BMI indicate whether an adult is underweight, overweight, obese, or very obese? Investigate how the calculator works out the BMI from the height and weight. Note for pupils: If you put your own details into this calculator, don’t take the results too seriously! It is designed for adults who have stopped growing and will give misleading results for children or teenagers!
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