Transcript University of Auckland 10 October, 2002 Tasmanian Research

New Zealand Numeracy Facilitators
Conference
10 February 2005
Statistics Education: Towards 2010
What do we need and how do we do it?
Jane Watson
University of Tasmania
1
Issues
• Statistics curriculum - New Zealand.
• Statistical Literacy more generally.
• Grades K-10.
• Student understanding.
• Classroom experiences.
• Links across the curriculum.
2
What is the context within which
we are working?
• Curriculum change
• The statistical literacy needs of ALL
students and the foundation for those going
on at the highest level in the final years
• Teachers’ needs
• Students’ starting points
• Ways of assessing change
• Desired end points
3
New Zealand - Statistics
• Recognise appropriate statistical data for
collection, and develop the skills of
collecting, organising, and analysing data,
and presenting reports and summaries;
• Interpret data presented in charts, tables,
and graphs of various kinds;
• Develop the ability to estimate probabilities
and to use probabilities for prediction.
4
Statistical Literacy
- Gal & Garfield
• Comprehend and deal with uncertainty,
variability, and statistical information in the
world around them, and participate
effectively in an information-laden society.
• Contribute to or take part in the production,
interpretation, and communication of data
pertaining to problems they encounter in
their professional life.
5
Student Understanding and
Development - Structure
[Biggs & Collis 1982; Pegg 2002]
• Prestructural: no facet of the task.
• Unistructural: single elements relevant to the task — if a
contradiction occurs, it is not recognized.
• Multistructural: multiple elements in a sequential fashion
— if conflict likely to be recognized but not resolved.
• Relational: multiple elements integrated into a whole —
conflict resolved.
• Extended Abstract: beyond the expectations of the task,
bring in unexpected more sophisticated insights.
6
Student Understanding and Goals
- Appropriateness [Watson, 1997]
• Understanding of the statistical terminology to be
used.
• Understanding of the terminology when it appears
in various contexts, including social, scientific and
technical contexts appearing in media or other
reports.
• Ability (and motivation) to question claims that are
made without proper statistical justification (and
even to explore and assess those made with
adequate justification).
7
Added dimension: The Dilemma of
Expectation versus Variation
David Moore (1990) stresses… The
omnipresence of variation in processes.
Individuals are variable; repeated measurements
on the same individual are variable.
AND…Phenomena having uncertain individual
outcomes but a regular pattern of outcomes in
many repetitions are called random.
In the curriculum we have stressed pattern/
expectation/probability at the expense of
variation.
8
Levels of Statistical Literacy
1. Idiosyncratic
Tautologies, one-to-one counting, read cells
2. Informal
Intuitive non-statistical beliefs (3 is lucky),
one-step calculations
3. Inconsistent
Limited appreciation of content and context
without justification; qualitative ideas.
4. Consistent
Non-critical
Straight-forward engagement with context;
means, simple probabilities and graphs.
5. Critical
Questioning engagement; appreciation of
variation; qualitative interpretation of chance.
6. Critical
Mathematical
Questioning critical engagement with context,
proportional reasoning, subtle language.
9
Goals and Levels
of Statistical Literacy
Tier 1
Terminology
Tier 2
Context
Tier 3
Questions
1. Idiosyncratic
2. Informal
3. Inconsistent
4. Consistent
Non critical
5. Critical
6. Critical
Mathematical
10
Performance Across Levels
by Grade (Watson & Callingham, 2004)
11
Links
• Assessment tasks help determine students’
current levels of understanding
• Assessment tasks can inform classroom
activity (can be the basis for it!)
• Teachers need an understanding of what to
expect from students (as well of course as
the understanding of the statistics involved!)
• Teaches also need appreciation of possible
progressions in understanding
12
Example 1:
What Does Sample Mean?
• Code 0 – Idiosyncratic or tautological
responses, “put on a letter”
• Code 1 – Example of an isolated idea, such
as “try” or “piece”
• Code 2 – Partial definition based on several
aspects, such as “part of something”
• Code 3 – Related aspects of definition, such as
“small part of the whole to test or taste”
13
Example 2:
What Does Variation Mean?
• Code 0 – Idiosyncratic or tautological
responses
• Code 1 – Example of an isolated idea, such
as “lots of choices”
• Code 2 – Simple definition based on
difference between things
• Code 3 – Subtle change, such as “slight
change or difference”
14
Levels of Response
Sophisticated definitions appear with critical thinking
Level 2
Sample - Code 1
Informal
Level 3
Variation - Code 1
Inconsistent
Level 4
Consistent
non-critical
Level 5
Critical
Sample - Code 2
Variation - Code 2
Sample - Code 3
Variation - Code 3
15
Teaching Implications
• Time is required to absorb structure and
sophistication of definitions
• Can’t wait to introduce investigations until
the definitions are mastered
• Investigations are likely to help the
development of understanding of definitions
• Never stop discussing, reinforcing
terminology
16
Example 3: How Children Get to
School One Day
17
How Children Get to
School One Day
• How many children walk to school?
• How many more children come by bus than by car?
• Would the graph look the same everyday? Why or why
not?
• A new student came to school by car. Is the new student a
boy or a girl? How do you know?
• What does the row with the Train tell about how the
children get to school?
• Tom is not at school today. How do you think he will get to
school tomorrow? Why?
18
Responses to Pictograph Question
Summary
New student boy or girl?
How does Tom get to
school?
[0] Inappropriate
There were more kids.
He will feel better after a
day off.
[1] No interaction with
the graph
Boy, I just guessed.
Car, so he doesn’t get a
cold.
Not sure, not enough
information.
[2] Patterns or Anything Boy, because there is a
can Happen
pattern (GGBGG…)
Girl, she is at the end.
The same way he does
every day.
Car, because there is a
pattern (GGBGG…)
Anything, it’s chance!
Could be either.
19
Responses to Pictograph Question
Summary
New student boy or girl?
How does Tom get to
school?
[3] Balancing
Boy, cause it’s the only boy Train, because there is no
that goes by car.
one on the train today.
Boy, it could make 14 of
both in the class.
[4] Statistical reasons
but no uncertainty
Girl because the majority
who come by car are girls.
Bike, the majority of boys
ride to school.
Bus, more people catch
the bus.
[5] Statistical reasons
acknowledging
uncertainty
You don’t know but it is
just more likely to be a girl
‘cause more come by car.
Probably by bus because
1/3 of the children caught
it today.
20
Responses to Pictograph Question
• Large numbers of students in the middle two categories,
especially for the new student question.
• Teachers are not surprised.
• Interference from the pattern work that is done to prepare
students for work with algebra.
• In statistics we are interested in different sorts of patterns
than those related to algebra.
• Very few middle school students acknowledge uncertainty
and potential variation.
• Teachers may be over-emphasizing the deterministic power
of information obtained from graphs.
21
Relating the Hierarchical Goals to
the Pictograph Task
• Pre Tier 1: Students don’t interact with the
graph at all [0], or they don’t know what to
do with the information in it [1].
• Tier 1: Students read the information from
the graph but their interpretations are based
on information that is not relevant to the
context of the question: patterns [2] or
suggestions considered “fair” [3].
22
Applying the Hierarchical Model to
the Pictograph Task
• Tier 2: Students are able to read the graph in the
intended context and use it to make appropriate
interpretations for the data [4]. These statements,
however, are deterministic in nature.
• Tier 3: Students go beyond the basic
interpretation of the information in the graph to
include an element of uncertainty in their
predictions, acknowledging that variation is
possible [5].
23
Teaching Implications
• Always ask for the reasons behind answers.
• Stress sharing of views in the class,
consideration of contextual knowledge,
different kinds of patterns in mathematics.
• Possibly a task for group work.
• Have high expectations for discussion.
• Continue to use pictographs in the middle
years.
24
Example 4: Expectation and
Variation
25
Levels of Response to the
“60 tosses of a die” Task - Code 0
• Prestructural 30.76%
• Description: Do not add to 60 or have
unrealistic value.
• Example: “6, 3, 2, 1, 4, 5 - Because the one
might have a bigger chance of coming up
more than the other numbers.”
26
Levels of Response to the
“60 tosses of a die” Task -Code 1
• Unistructural 28.05%.
• Description: Add to 60 without appropriate
variation and explanation or do not add to
60 with aspect of variation.
• Examples: “10, 10, 10, 10, 10, 10 - It was a
guess”
“10, 20, 10, 5, 5, 10 - Because it adds to 60”
“19, 18, 5, 7, 23, 10 - Because any number
can come up.”
27
Levels of Response to the
“60 tosses of a die” Task -Code 2
• Transitional 22.76%
• Description: Strict probability or too little
variation.
• Examples: “10, 10, 10, 10, 10, 10 - They all
have the same chance of coming up.”
“10, 10, 9, 11, 10, 10 - These numbers are
reasonable because there is a chance in six.”
28
Levels of Response to the
“60 tosses of a die” Task -Code 3
• Multistructural 13.96%.
• Description: Conflict of probability and variation,
variation with no explanation, or explanation but
too much variation.
• Examples: “10, 10, 10, 10, 10, 10 - In theory all
numbers should come up equally. They probably
will not.” (Realised Conflict of probability and
variation)
“9, 12, 10, 7, 6, 16 - I used these numbers based
on what usually happens to me.” (No explanation)
“15, 8, 10, 2, 19, 6 - Because there is one of each
so it could be any number.” (Too much variation) 29
Levels of Response to the
“60 tosses of a die” Task - Code 4
• Relational 4.47%
• Description: Appropriate variation and
explanation.
• Example: “12, 9, 11, 10, 10, 8 - Because
they’re all around the same but you can’t
know if they will come up that number of
times.”
30
Outcomes: Codes Across Levels
Code 1: Level 1 Idiosyncratic
Code 2: Level 2 Informal
Code 3: Level 4 (low) Consistent non-critical
Code 4: Level 4 (high) Consistent non-critical
31
Outcomes: Codes Across Grades
• Average Code per grade:
3
5
7
9
0.79
1.43
1.50
1.58
• Improvement then levelling.
• Grade 9 highest Code 2 [39%].
32
Teaching Implications
• This is a tricky context for expectation
(theory from probability) and variation (that
surely occurs from the theoretical values).
• Need lots of classroom practice over the
middle years (once is not enough).
• Teachers need to be flexible in class
discussion - aware of interference of ideas.
• Opportunity for group work and report
writing.
33
What do students tell us across the years?
• A few examples from student interviews
• Don’t underestimate 6-year-olds
• Start them developing good habits of
statistical thinking
• Be aware that many students take a long
time to develop appropriate intuitions,
especially about expectation and variation.
34
Interviews with 6-year-olds
• Creating a pictograph - cards representing books
and children who had read them.
35
Prediction for 6-year olds
•
•
•
•
•
I: Suppose Paul came along. How many books do you think Paul’s read?
S: I don’t know.
I: Don’t know? Don’t want to make a guess?
S: No. My sister always makes me do guessing things. I always have to put up
with it.
I: Okay you don’t have to put up with it from me…
36
Prediction for 6-year-olds
•
•
•
•
Who do you think is most likely to want a book for Christmas?
Terry.
Why Terry?
… Just pretend, like he’s got […] book, and a dinosaur one, and a
skeleton one, and a giraffe one, and he wants one about plants, like …
to see how they grow.
37
Prediction for 6-year-olds
• Let’s suppose that Paul came along. How many books do you think
Paul has read?
• Three.
• Why do you think three?
• Because one of my sisters is three.
38
Prediction for 6-year-olds
•
•
•
•
From the picture can you tell who likes reading the most?
Umm … think … Anne or Jane… no Lisa.
Why Lisa?
Because she started off with 6 and then she got 7… and she’s got one
more, so that’s 7.
39
Teaching Implications
• Don’t avoid tasks that require representation
and prediction in early childhood
• Just be prepared for anything!
• Don’t just “correctness” but discuss
alternatives
• “What might we say if we only had your
display to look at?”
40
Interviews on Expectation and Variation
• Drawing 10 lollies from a container with 50% red
- predicting, experimenting, representing.
• How many red? The same every time?
41
6 years: Expectation and Variation (Level 3)
•
•
•
•
•
•
•
•
How many red ones do you think you might get?
I think I would get … about 5.
Why do you think you might get 5?
…Because there’s 50 so I think I might get 5, because there’s 5 pl.. 10, so…
…Would you get 5 again? [shakes head] Might get something different? [Nods
head] Why?
Because every time you do something it’s a different way.
How many would surprise you?
Umm I think 6, … because 6 is my favourtist number.
42
Grade 7: Expectation and Variation (Level 5)
•
•
•
•
•
•
… and pull out a handful, how many do you think you might get?
Five.
Why do you think you might get 5?
Because half of the contents of the container is red and so you should expect to
get half the amount in what you pull out.
Suppose you did this a few times… Would you expect to get the same number
of reds every time?
No. Because it’s just the luck of the draw most of the time. You’ll get around
the same amount but not exactly the same amount.
43
Grade 7: Expectation and Variation (Level 5)
•
•
•
•
How many reds would surprise you?
I reckon about 8 or 9.
So why do you think 8 or 9?
…cause again there’s only half the container filled. So you’d still
expect to get some yellow and green in there, so you wouldn’t expect
just to pull out this huge handful of red ones, cause they’d all be mixed
up.
44
Grade 7: Expectation and Variation (Level 5)
•
Suppose 6 of you’ve come along and done this experiment… Can you write
down for each of the people the number of red that would be likely?
• 5, 3, 6, 4, 5, 4
•
•
So why have you chosen these numbers?
I’ve chosen them because they’re around the middle number that I chose of 5
and so there’s a bit of give and take for different mixtures… cause obviously
they’d mix them up after each go and you never know they might bring all of
the other ones up to the top.
45
Pictures to show the results of 40 draws
of 10 lollies
• Drawing lollies (Level 2)
• Unconventional
expectation and variation
(Level 4)
• Variation without
proportional reasoning
(Level 3)
• Conventional expectation
and variation (Level 5)
46
Interviews about the weather
Some students watched the news every night for a
year, and recorded the daily maximum
temperature in Hobart. They found that the
average maximum temperature in Hobart was
17C.
• What does this tell us about the temperature in
Hobart?
• Do you think all the days had a maximum of
17C? Why or why not?
• What do you think the maximum temperature in
Hobart might be for 6 different days in the year?
______, ______, ______, ______, ______, ______
47
Interviews with 6-year-olds: Weather
•
•
•
•
•
•
What does this tell us about the temperature?
That is was quite hot if it was 17.
Do you think all of the days of the year had a temperature of 17˚?
No, because you get summer, winter - summer, spring, winter, autumn, then
summer again.
What does that mean?
You get, it’s like hot… mild or cool, cold, mild or cool, and then hot again.
48
Drawing by a 6-year-old
• Describing variation in the weather with an
average yearly temperature of 17oC.
49
Grade 3: Expectation and Variation (Level 1)
•
•
•
•
What does that tell you about the temperature in Hobart?
Well sometimes you can’t always rely on the weather… because I can
remember one day when I was down in Hobart, that it was freezing cold and it
was supposed to be 17˚ … and well sometimes it’s hard when you’re sort of
thinking about what the weather’s going to be, knowing what to put on, when
it can change later in the day.
Do you think that all of the days of the year had a maximum of 17˚?
No, because you can’t always be the same temperature… because you have
different seasons… well like you’ve got spring, summer, autumn, and winter,
and winter is one of the coldest seasons, and sometimes it can still be cold in
summer.
50
Grade 7 - Expectation and Variation (Level 4)
•
•
•
•
What does this tell us about the temperature in Hobart?
It’s not really high, like up in Darwin but it’s not absolutely freezing like in
Antarctica or somewhere.
So do you think all days might have a maximum of 17˚C?
No. Because some days you would get like a day that might go to 30˚C, if it is
really hot, and a lot of days could get much colder.
51
Grade 7 - Expectation and Variation
for 6 days of the year (Level 4)
• Temperatures: 12, 23, 17, 19, 14, 20
•
•
Why?
Because like, it could be anything basically - it depends, but the average is 17,
so it would be more likely to be within a certain range, but up like 40 or down
to zero.
52
• So what have you done there?
• It’s the highest in January, February, and December cause that’s the
middle of summer... The coldest would be around here in winter. In
around these sections, it’s around middling.
• It’s interesting you’ve got May a little bit higher here…
• Yea, it could change. There’d be a lucky day sometimes. It could just
go up over.
• So are these temperatures, are they what, maximums, or averages
or…?
• Y..\Application Data\Microsoft\Internet Explorer\Quick Launch\Show
53
Desktop.scfea, maximum averages.
Grade 7 - Expectation and Variation (Level 5)
•
•
•
It is cold. Either you get - in Hobart obviously it means that there’s a lot of
cold days but then there’s a few hot days in there but the number of cold days
is outnumbering the number of hot days bringing the total down.
So do you think all days have a maximum of 17?
No, I think they could be hotter - some of them - most of them - a fair few of
them might have been higher but then you have got all these ones that are
really low. Like dismal.
54
Grade 7 - Expectation and Variation (Level 5)
•
•
Temperatures - 19, 29, 15, 11, 35, 31
I just, I made the choices because to give a wide range of the possibilities
because quite often you have a very cold day but then of course you have very
hot days and so the rest are just spread out through the middle to show that
they are through the middle and all different, you can get all different
temperatures no matter what.
55
Summary
• Almost all students appreciate variation and
uncertainty.
• The ability to make predictions appropriately,
especially based on proportional thinking, grows
(idea not reached for all by grade 10).
• Sampling and representation are key issues in both
of these areas (e.g., trials of random – or nonrandom – generators, data collection from
populations, ‘drawing’ outcomes).
56
Issues across the statistics curriculum
•
•
•
•
Variation.
Pattern.
Manipulation of single & multiple ideas.
Considering different points of view in
decision-making.
• Judgments based on ‘fairness’.
• Determinism versus chance.
• Importance of context.
57
Questions to consider
• How do these ideas fit with the traditional ordered
view of teaching statistics?
–
–
–
–
–
Gather data
Represent the data
Calculate a statistic (the mean!)
Think about the chances
Draw a conclusion
• Critical thinking is required to relate the issues
throughout all aspects of statistical investigations.
58
Critical Statistical Literacy in the Media
• ABOUT six in 10 United States high school students
say they could get a handgun if they wanted one, a
third of them within an hour, a survey shows. The poll
of 2508 junior and senior high school students in
Chicago also found 15 per cent had actually carried a
handgun within the past 30 days, with 4 percent taking
one to school.
• Q1: Would you make any criticisms of the claims in
this article?
• Q2: If you were a high school teacher, would this
report make you refuse a job offer somewhere else in
the United States, say Colorado or Arizona? Why or
why not?
59
Responses to media article
• “Students shouldn’t have guns.” [Level 1]
• “No, because the whole of the US would be
exactly the same.” [Level 1]
• “How do you know they are not lying?” [Level 4]
• “If they wanted to get their facts right they should
survey every school in America.” [Level 4]
• Q2: “No because this poll is in Chicago. Results
may be different in Colorado.” [Level 5]
• Q1: “Yes. It is generalizing the whole of the USA
– when they only surveyed in Chicago.” [Level 6]
60
Teaching Implications
• This is an amazingly subtle task.
• Sample and population aren’t mentioned in the
article or the questions.
• How do students learn what to focus on?
• Having beliefs about guns shouldn’t colour
analysis of the text.
• Questions of the reliability of the data are also
important but shouldn’t overshadow the sampling
issue.
61
Conclusion
• Statistical literacy is relevant across the
entire statistics curriculum: thinking
critically at every point and being able to
express concerns in appropriate language.
• Expectation and variation permeate every
aspect of investigations.
• Integration is the key to high level
outcomes.
62