1. Based on work by Pip Arnold, TEAM Solutions
Takapuna Devonport Lead Teachers
Workshop 3, 2010
Facilitators: Heather Lewis
Christine Hardie
Based on work by Pip Arnold, TEAM Solutions

2. Overview 8.45-9.20 Discussion – National Standards implementation
Module 9 + Rich Task – Engaging Learners with Mathematics
Morning Tea
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

3. 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

4. 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.

5. Engaging learners with mathematics
Ministry Professional Development
Modules – Jigsaw Activity
Module 7 & 9
Engaging learners with mathematics

6. 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)

7. 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?

8. 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

9. 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)

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

11. 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

12. 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

13. 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

14. 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

15. 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?

16. 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

17. 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

18. 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

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

20. 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

21. Classifying
Sort / classify the questions according to the following categories:
Summary
Comparison
Relationship

22. Category Data
Numerical Data
Time-Series Data

23. 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

24. 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

25. Relationship Question
Are you a masterpiece?
What is the relationship between your height and arm span?

26. Statistics in the NZC and Standards
Highlight the difference in progression from Y1 to Y8
Circle any vocabulary that you are unsure of.

27. Problem Statistical investigation cycle
Has at its heart a starting point based on a problem.
Data driven or Question driven 27

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

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

30. 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

31. 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…

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

33. 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

34. 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.

35. 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

36. Questioning to elicit open-ended investigation

37. 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

38. CensusAtSchool http:///
CensusAtSchool 38

39. Useful Websites: http://www.stats.govt.nz/

41. Resources: www.nzmaths.co.nz (Second tier material, statistics units)
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:
Lunch: ran 41

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