Based on work by Pip Arnold, TEAM Solutions Takapuna Devonport Lead Teachers Workshop 3, 2010 Facilitators: 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 2

Release the Prisoners 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. Mathwire.com

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.

Engaging learners with mathematics Ministry Professional Development Modules – Jigsaw Activity Module 7 & 9 Engaging learners with mathematics http://nzcurriculum.tki.org.nz/National-Standards/Professional-development/Professional-learning-modules/Overview

Fitting It In The Sugar Cube Problem Using rich tasks to engage learners in mathematics To explore how numeracy underpins our ability to complete a measurement task… Fitting It In The Sugar Cube Problem Without a problem, there is no mathematics. Holton et al. (1999)

The Sugar Cube Problem The 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?

How rich was this task? 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

Using rich tasks to engage learners in mathematics 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 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 Communicating findings Gathering, sorting and displaying It doesn’t matter where within the cycle the investigation starts – other than the conclusion which may lead to further investigation. Needs a purpose – for example students may decide there aren’t enough rugby balls to play with at luchtime so may wish to investigate what games children in different year levels like playing in their leisure time. So the problem is identified and now needs planning. 11

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 12

What’s changed? Statistics in the “old” curriculum Statistics in the “new” curriculum Let’s look at the changes - wordle data Display / displays / graphs 13

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 Go through the PPDAC cycle - what does this mean? USE OF ACTIVE VERBS - the process is ongoing and variable ING Specific graphs (ie ‘create a stem & leaf graph L3’ not mentioned - NEW EMPHASIS on DATA DISPLAYS, NOT graphs. 14

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

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? To collect this data students will need to decide on variables (ie rugby, netball, handball, volleyball. What questions will they ask. How many students will need to be surveyed? How will we record the information when surveying? 16

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) 17

Data cards Year 1-3 teachers collect this data on yellow cards Year 4-6 on blue cards Non-classroom teachers can choose! Arm span Leisure activity No. of members in your family Category data,, whole no. data, Measurement data, multivariate data Height

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 set e.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?” 20

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? 23

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) Spread (difference between lowest & highest values) Middle group(s) 24

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 27

Collecting category data using post it notes Leisure activity = Reading 28

Collecting bivariate data using post it notes Leisure activity = Reading Leisure activity = Playing sport Yrs 1-3 teachers Yrs 5-8 teachers 29

Collecting multivariate data using post it notes What school subject do you most enjoy teaching? Birth month What time did you go to bed last night? What school subject did you most enjoy at school as a child? 30

Analysis: Key words for describing data display 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 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…

“What are typical birth months for people in this group?’ I notice… 32

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. 33

Greater Heights (FIO 2-3, pg.4) Dot plots are used to show number data that comes from counting or measuring. What is the same and/or different about the girls’ and boys’ data? How might Ahere’s idea of finding the ‘middle’ help answer Tim’s question “I wonder if the boys are taller than the girls?”. Do you agree or disagree with Ahere’s statement? Support your views with at least three statements based on the data.

Thinking Routines What data? How shall I collect it? What do I think might happen? I used to think… now I think…. I notice…. I conclude that…. Making Links to the Data Detective PPDAC White Hat = Information - the recipe - identifying the problem / planning the investigation Blue Hat = processing the investigation (data gathering etc) Black Hat = what limitations does this data have to my question or the problem Red Hat = analysing trends and relatiionships Yellow Hat = Concluding and justifying results of the investigation Blue Hat / Green Hat = commuicating findings What limitations does this data have for my question?. I wonder…. 35

Questioning to elicit open-ended investigation

SOLO TAXONOMY (after Biggs and Collis 1982) Evaluate Theorise Generalise Predict Create Imagine Hypothesise Reflect Compare/contrast Explain causes Sequence Classify Analyse Part/whole Relate Analogy Apply Formulate questions Define Describe List Do algorithm Combine Define Identify Do simple procedure Prestructural Unistructural Multistructural Relational Extended abstract

CensusAtSchool http:/// www.censusatschool.org.nz CensusAtSchool http:/// 38

Useful Websites: http://www.stats.govt.nz/ http://www.babynamewizard.com/

Resources: www.nzmaths.co.nz (Second tier material, statistics units) www.censusatschool.org.nz Figure It Out Statistics, Data Cards: Gender: male Age: 12 Height: 163 cm Arm span: 163 cm Travel: walk Time: less 10 Lunch: ran Gender: female Age: 12 Height: 155 cm Arm span: 155 cm Travel: walk Time: 10 - 20 Lunch: ran 41

Concluding thought… 98% of all statistics are made up. 42