Takapuna Devonport Lead TeachersWorkshop 3, 2010Facilitators: Heather Lewis hs.lewis@auckland.ac.nz

Christine Hardie c.hardie@auckland.ac.nz

Based on work by Pip Arnold, TEAM Solutions

Overview

8.45-9.20

• Discussion – National Standards implementation

9.20-10.30

• Module 9 + Rich Task – Engaging Learners with Mathematics

Morning Tea

10.45-11.45

• Become familiar with Statistics in the new Curriculum document and Standards

• Use the Statistical Enquiry Cycle

• Know the ‘Census at School’ website and other resources

Mathwire.com

Who will free their prisoners first? Students use subtraction facts to find the difference of two dice. Directions plus both the 6-sided dice and 12-sided dice gameboards are included so teachers can target subtraction practice while helping students develop an intuitive appreciation of probability.

Release the Prisoners

Discussion

What have you done since our last workshop to support teachers and/or leaders with the implementation of the National Standards?

Identify any key issues that you are finding challenging.

Ministry Professional DevelopmentModules – Jigsaw Activity

Module 7 & 9Engaging learners with mathematics

http://nzcurriculum.tki.org.nz/National-Standards/Professional-development/Professional-learning-modules/Overview

To explore how numeracy underpins our ability to complete a measurement task…

Fitting It InThe Sugar Cube Problem

Using rich tasks to engage learners in mathematics

Without a problem, there is no mathematics.Holton et al. (1999)

The Sugar Cube ProblemThe Sweet-tooth Company has hired you to design boxes to hold sixty-four sugar cubes. Each cube has edges of 2 cm, just like multilink cubes. The boxes have to be the shape of boxes (cuboids) as there should not be sugar cubes sticking out. What sizes of boxes could they have? Do not make the boxes, just sketch rough plans of them showing the length of the edges.How many different boxes could be made? How could this be worked out without having to build each shape with cubes?

The Managing Director now walks into the design room to say that market researchers say that 2cm cubes make the consumer’s tea too sweet – how many 1cm cubes could you fit into the box you have designed?

Would this be suitable? Or do you need to design a new box?

• It must be accessible to everyone at the start.• It needs to allow further challenges and be extendable.• It should invite learners to make decisions.• It should involve learners in speculating, hypothesis

making and testing, proving and explaining, reflecting, interpreting.

• It should not restrict learners from searching in other directions.

• It should promote discussion and communication.• It should encourage originality/invention.• It should encourage 'what if' and 'what if not' questions.• It should have an element of surprise.• It should be enjoyable.Ahmed (1987), page 20

How rich was this task?

While there is a place for practice and consolidation, “tasks that require students to engage in complex and non-algorithmic thinking promote exploration of connections across mathematical concepts” p.97

Using rich tasks to engage learners in mathematics

Without a problem, there is no mathematics.Holton et al. (1999)

Engaging learners with mathematics – how did these tasks engage,…….

Discuss.

Posing and answering questions

Gathering, sorting and displaying

Communicating findings

Mathematics Statistics

Exploration of and use of patterns and relationships in… quantities, space and time

Set answer

Exploration of and use of patterns and relationships in… data

No definitive answer

Statistics in the “old” curriculum

Statistics in the “new” curriculum

What’s changed?

How is Statistics different in the new curriculum?

• Data is still key• Enquiry cycle (PPDAC)• Verbs

– Posing, gathering, sorting, displaying, communicating, displaying, using

• Specific graph types not mentioned

Leonardo da Vinci (1452-1519) was a scientist and an artist. In 1492 he drew this picture. Can you see how the man is standingIn a circle and a square? Leonardo thought that The span of someone’s arms is equal to theirheight. Why do you think he was interested in working out body proportions?

Do you think Leonardo’s theories still work today?

Are you a Masterpiece?

Plan

– What variables do we need to collect?– How shall we pose the survey questions.– Who shall we ask / how many?– How will we know when we have asked everyone?– How are we going to record and collect the data?

Collecting data

What are these data types?

• Category data (Y1 onwards)

• Whole Number data (Y3 onwards)

• Multivariate category or whole number data (Y6 onwards)

• Time-series data (Y6 onwards)

• Measurement data (Y7 onwards)

Data cards

Leisure activity

Arm span

No. of members in your family

Height

Year 1-3 teachers collect this data on yellow cardsYear 4-6 on blue cards Non-classroom teachers can choose!

Data cards

Brainstorm all possible questions from the available information on the data cards.

Problem Question Types

• Summary (Years 1- 8)– A description of the data, usually a single data sete.g. “What is the most common birth month in our class”

• Comparison (Y5 onwards)– Comparing two (or more) sets of data across a common

variable, e.g. “Do females typically live longer than males?”

• Relationship (Y7 onwards)– Interrelationship between two paired variables,e.g.“Does watching a lot of TV increase your IQ?”

Classifying

Sort / classify the questions according to the following categories:

• Summary• Comparison • Relationship

Category Data

Numerical Data

Time-Series Data

Analysis

• Make a graph using your data cards that will help you to answer your question.

• Describe the graph identifying patterns and trends in context.

• Remember the context. If I cover any labels can I still tell what the graphs are showing?

Analysis• Use I notice… as a starter for statements.

• For category variables: (e.g. birth month etc)– Shape– The most common category, the least common category,

other categories of interest– Anything unusual, or of interest

• For measurement variables: (e.g. bed time)– Shape – Spread (difference between lowest & highest values)– Middle group(s)– Anything unusual, or of interest

Relationship Question

• Are you a masterpiece?

• What is the relationship between your height and arm span?

Statistics in the NZC and Standards

Highlight the difference in progression from Y1 to Y8

Circle any vocabulary that you are unsure of.

Problem

• Statistical investigation cycle• Has at its heart a starting point based on a

problem.• Data driven or Question driven

Collecting category data using post it notes

Leisure activity

= Reading

Collecting bivariate data using post it notes

Leisure activity

= Reading

Leisure activity

= Playing sport

Yrs 1-3 teachers

Yrs 5-8 teachers

Collecting multivariate data using post it notes

What school subject do you most enjoy teaching?

What time did you go to bed last night?

What school subject did you most enjoy at school as a child?

Birth month

Analysis: Key words for describing data display

Shape Middle Spread

Clump (s)

gap,

symmetrical, rectangular,

most of the data is, a few points are

Same/different

The middle of the data is …..

about..,

between,

higher/lower

Close together, spread out,

evenly spread, mostly between,

less/more spread out than…

Describing Categories

Most (N.B. “most” must be more than half), least, some, all, more than, less than, more than half, about half, roughly a quarter, a lot, not many, a few, most popular, least popular, most typical, least typical

“What are typical birth months for people in this group?’

I notice…

( )count

1

2

3

4

5

6

Birth_monthFebruary March April May June July August September October December

• I notice that the most common birth month is August with 5 people in the group.

• I notice the least common birth months are January and November with no one in the group born in these months.

• I notice that four months have four people born in them, they are May, June, October and December.

• I notice that the Winter months have the most people born in them, 12 people. Spring has the least number of people born with only 5 people born then.

( )count

1

2

3

4

5

6

Birth_monthFebruary March April May June July August September October December

Greater Heights (FIO 2-3, pg.4)

Dot plots are used to show number data that comes from counting or measuring.

1. What is the same and/or different about the girls’ and boys’ data?

2. How might Ahere’s idea of finding the ‘middle’ help answer Tim’s question “I wonder if the boys are taller than the girls?”.

3. Do you agree or disagree with Ahere’s statement? Support your views with at least three statements based on the data.

Thinking Routines

PROCESSJUSTIFICATIONS

IMPROVEMENTS

EMOTIONS

INFORMATION

I notice….

I used to think… now I think….

What limitations does this data havefor my question?.

I wonder….

I conclude that….

What data? How shall I collect it? What do I think might happen?

Questioning to elicit open-ended investigation

Unistructural Multistructural Relational Extended abstract

DefineIdentifyDo simple procedure

DefineDescribeListDo algorithmCombine

Compare/contrastExplain causesSequenceClassifyAnalysePart/wholeRelateAnalogyApplyFormulate questions

EvaluateTheoriseGeneralisePredictCreateImagineHypothesiseReflect

SOLO TAXONOMY(after Biggs and Collis 1982)

Prestructural

CensusAtSchool

http:///

www.censusatschool.org.nz

Useful Websites:

http://www.stats.govt.nz/

http://www.babynamewizard.com/

Gender: femaleAge: 12Height: 155 cmArm span: 155 cmTravel: walkTime: 10 - 20Lunch: ran

Gender: maleAge: 12Height: 163 cm Arm span: 163 cmTravel: walkTime: less 10Lunch: ran

Resources:

• www.nzmaths.co.nz (Second tier material, statistics units)

• www.censusatschool.org.nz

• Figure It Out Statistics,

• Data Cards:

Concluding thought…

98% of all statistics are made up.