PD: Thinking about Inquiry Tasks

James Calleja ©2015

2 Teaching and Learning Mathematics through Inquiry

Plan for the session Session 2 Thinking about Tasks for Mathematical Inquiry

Task Topic Time

Introduction What do you expect to get from today’s session? Aims of the session ¼ h

Working on a Mathematical

Task Teachers work on the ‘Web Pattern’ investigation. 1 h

Reflecting on the Task

Teacher views on their experience working on the ‘Web Pattern’ task ½ h

Video Discussion

Teachers reflect on student experiences working on the ‘Web Pattern’ 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

Planning for Tasks

Teachers work in groups to plan a task for inquiry. Teacher present and share their work with the whole group.

¾ h

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

Teaching and Learning Mathematics through Inquiry 3

WORKING ON A MATHEMATICAL TASK 60 min

Investigating Web Patterns

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 and to identify any mathematical aspects connected to this web pattern

7 minutes

• You will be asked to share your ideas with the group 5 minutes

• It is now time for you to work in groups of three to solve the problem. You need to select ideas that you wish to investigate

30 minutes

• You will present and share your findings to the whole group 15 minutes

On your graph paper, use 1 cm to represent 1 unit to draw the x-‐axis

from –8 to 10 and the y-‐axis from –12 to 12.

Then plot the points: (0, 1), (2, 0), (0, –3), (–4, 0) and (0, 5).

Continue this sequence of points as far as you can go.

Join lines through the points and identify the geometric pattern being

formed.

Investigate this web pattern.



SPACE FOR WORKING


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 (Keith) using the ‘Web Pattern’ task with his form two class of students.

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.

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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?


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 Worksheet 2


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

How can you justify each of (a), (b), (c) as the odd one out?


5. Analyzing reasoning and solutions – diagnose errors and comparing solutions

INQUIRY MATHS PROMPTS From http://www.inquirymaths.co.uk

On the website pages:

PROMPT 1: A NUMBER PROMPT Why is one statement correct when the other one is not?

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.

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.


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

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

⇒ What is different and the same about the rectangles?

⇒ How many rectangles are possible with the same area?

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

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

BOWLAND MATHS TASKS From http://www.bowlandmaths.org.uk

On the website pages:

𝒚− 𝒙 = 𝟒

Bowland Maths aims to make maths engaging and relevant to pupils aged 11-‐14, with a focus on developing thinking, reasoning and problem-‐solving skills. In these materials, the maths emerges naturally as pupils tackle problems set in a rich

mixture of real-‐life and fantasy situations.


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!


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


SESSION EVALUATION 10 min

Ø Briefly describe your experience during today’s session.

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Ø What did you feel un/comfortable doing during the session?

Comfortable: ___________________________________________________________________________________

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Uncomfortable: ________________________________________________________________________________

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Ø I used to think... but now I know…

I used to think __________________________________________________________________________________

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Now I know ____________________________________________________________________________________

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Ø What will you take with you and try to implement in your class?

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Ø Any other comments/suggestions that you would like to add.

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Thank you for your participation and reflections.

Documents

PD: Thinking about Inquiry Tasks