PD: Thinking about Inquiry Tasks

James Calleja

©2015

Plan for the Session

Thinking about Tasks for Mathematical Inquiry

Task

Introduction

Topic Introducing the aims of this series of the four summer workshops What do you expect to get from today’s session?

Time

½ h

Aims of the session Working on a Mathematical Teachers work on the ‘Four Fours’ task.

Task

¾ h

Reflecting on the Task

Teacher views on their experience working on the ‘Four Fours’ task

½ h

Video Discussion

Teachers reflect and discuss the roles of the teacher and that of students during the ‘Four Fours’ task.

½ h

Looking at Tasks for Inquiry

Teachers are presented with a range of tasks.

Which tasks are more likely to promote inquiry?

What characteristics should they have?

½ h

Teachers work in groups to plan a task for inquiry.

Teacher present and share their work with the whole group.

¾ h

Planning for Tasks

Aims of the session For today’s session we will have the following aims: o To understand the role of tasks in planning to teach mathematics through inquiry o To experience mathematical inquiry by working on a task o To explore tasks that provide opportunities for students to engage in mathematical inquiry o To reflect critically on an ‘inquiry’ lesson o To experience features and aspects of mathematical inquiry

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Teaching and Learning Mathematics through Inquiry

WORKING ON A MATHEMATICAL TASK

45 min

You are provided with two sets of numbers:

Four 4’s

&

4 4 4 4

four basic operations

+ −

× ÷

Part 1 – On your own

10 min

Use all four 4’s and any of the four basic operations provided to write equations that give answers from 0 to 10.

You may want to use other operations than the ones given!

Part 2 – In Pairs

15 min

Work with a partner and investigate whether it is possible to come up with more than one equation for each answer 0 to 10.

Part 3 – In Groups of three

15 min

Use all four 4’s and any operation to write equations that give answers from 11 to 20. Explore possibilities for other ways of writing equations for each number.

This is how you will work on this task: • The task is presented to you

3 minutes

• You have some time to work individually on the problem

7 minutes

• You will be asked to work with a partner

10 minutes

• It is now time for you to work in a small-­‐group 10 minutes • You will present and share your findings to the whole group

15 minutes

Teaching and Learning Mathematics through Inquiry

3

SPACE FOR WORKING

4

Teaching and Learning Mathematics through Inquiry

DISCUSSION POINTS

30 min

As a whole group you are asked to reflect on the following questions: 1. What opportunities does the task provide for students to struggle with mathematical ideas?

2. How do you see students engaging with important mathematical ideas?

3. What could the mathematical goals for a lesson using this task be? How would you plan a lesson using this task?

4. How do you see this task integrated within a unit of study?

WATCHING A LESSON VIDEO

10 min

You will now watch a video of a teacher (David) using the ‘Four Fours’ task with his form two (year 8) levels 7-­‐8 class.

Note how the teacher structures the lesson, the mathematical ideas valued, the difficulties that students encounter and how the teacher deals with these issues.

For the follow-­‐up discussion, you are encouraged to write down some notes/points you might see as important. ___________________________________________________________________________________________________ ___________________________________________________________________________________________________ ___________________________________________________________________________________________________ ___________________________________________________________________________________________________

SOME POINTS TO THINK ABOUT

20 min

What are your comments about the lesson?

Would you structure the lesson as the teacher did or would you do it differently?

Who generates the mathematical ideas that get discussed? Who evaluates and/or responds to these ideas?

How deeply do students get to explain their ideas?

How does the teacher respond to students’ struggles?

To what extent, do you think, students engaged in inquiry?

Teaching and Learning Mathematics through Inquiry

5

LOOKING AT MATHEMATICAL TASKS FOR INQUIRY

30 min

You are presented with a set of tasks – also available on the teacher booklet.

These tasks are taken from the work of Malcolm Swan and two websites – Inquiry Maths and Bowland Maths.

COLLABORATIVE LEARNING TASKS Malcolm Swan created a framework with five ‘types’ of activities that encourage distinct ways of thinking and learning. These are: 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 or apply a classification

Worksheet 1

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Teaching and Learning Mathematics through Inquiry

Worksheet 2

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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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.

INQUIRY MATHS 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.

PROMPT 1: A NUMBER PROMPT

Why is one statement correct when the other one is not?

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Teaching and Learning Mathematics through Inquiry

PROMPT  2:  AN  ALGEBRA  PROMPT   Encourage  students  to  come  up  with  the  questions  on  the  following  prompt! Â

           Â

đ?’š − đ?’™ = đ?&#x;’ Â

 PROMPT  3:  A  GEOMETRY  PROMPT              Class  posing/answering  some  questions  in  response  to  the  prompt:  ⇒

What  is  different  and  the  same  about  the  rectangles?  Â

⇒

How  many  rectangles  are  possible  with  the  same  area? Â

⇒

Which  has  the  longest  perimeter?  ...  the  shortest? Â

⇒

Is  there  a  rectangle  with  an  area  equal  to  the  length  of  its  perimeter? Â

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  BOWLAND  MATHS  TASKS   From  http://www.bowlandmaths.org.uk  On  the  website  pages: Â

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Bowland  Maths  aims  to  make  maths  engaging  and  relevant  to  pupils  aged  11-­â€?14,  with  a  focus  on  developing  thinking,  reasoning  and  problem-­â€?solving  skills.  In  these  materials,  the  maths  emerges  naturally  as  pupils  tackle  problems  set  in  a  rich  mixture  of  real-­â€?life  and  fantasy  situations. Â

Teaching  and  Learning  Mathematics  through  Inquiry Â

Â

9 Â

BOWLAND MATHS TASKS

PROBLEM 1:

Three Unstructured Problems

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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Teaching and Learning Mathematics through Inquiry

REFLECTING ON TASKS FOR INQUIRY

30 min

What are the essential differences between these tasks and those commonly found in textbooks?

Why are these tasks more likely to promote inquiry?

What characterizes tasks that promote inquiry?

What pedagogical issues do you believe will arise when teachers use these tasks?

PLANNING MATHEMATICAL TASKS FOR INQUIRY

45 min

Choose a task (from the ones provided above) that you feel would be appropriate to use with one of your classes.

In groups, discuss how you will: ⇒

Organise the classroom and the resources needed

⇒

Introduce the problem to your students

⇒

Explain to students how you want them to work together

⇒

Challenge/assist student that find the problem straightforward/difficult

⇒

Help students share and learn from alternative problem-­‐solving strategies

⇒

Conclude the lesson

Teaching and Learning Mathematics through Inquiry

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SESSION EVALUATION

10 min

Ø Briefly describe your experience during today’s session. ___________________________________________________________________________________________________ ___________________________________________________________________________________________________ ___________________________________________________________________________________________________ ___________________________________________________________________________________________________ Ø What did you feel un/comfortable doing during the session?

Comfortable: ___________________________________________________________________________________ ___________________________________________________________________________________________________ ___________________________________________________________________________________________________ Uncomfortable: ________________________________________________________________________________ ___________________________________________________________________________________________________ ___________________________________________________________________________________________________ Ø I used to think... but now I know…

I used to think __________________________________________________________________________________ ___________________________________________________________________________________________________ Now I know ____________________________________________________________________________________ ___________________________________________________________________________________________________ Ø What will you take with you and try to implement in your class? ___________________________________________________________________________________________________ ___________________________________________________________________________________________________ ___________________________________________________________________________________________________ Ø Any other comments/suggestions that you would like to add. ___________________________________________________________________________________________________ ___________________________________________________________________________________________________ Thank you for your participation and reflections.

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Teaching and Learning Mathematics through Inquiry