jane watson university of tasmania

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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

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Issues

• Statistics curriculum - New Zealand.

• Statistical Literacy more generally.

• Grades K-10.

• Student understanding.

• Classroom experiences.

• Links across the curriculum.

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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

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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.

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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.

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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.

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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).

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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.

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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.

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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

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Performance Across Levels by Grade (Watson & Callingham, 2004)

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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

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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”

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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”

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Levels of ResponseSophisticated definitions appear with critical thinking

Level 2Informal

Sample - Code 1

Level 3 Inconsistent

Variation - Code 1

Level 4 Consistent non-critical

Sample - Code 2

Variation - Code 2

Level 5 Critical

Sample - Code 3

Variation - Code 3

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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

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Example 3: How Children Get to School One Day

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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?

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Responses to Pictograph QuestionSummary 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.

Not sure, not enough information.

Car, so he doesn’t get a cold.

The same way he does every day.

[2] Patterns or Anything can Happen

Boy, because there is a pattern (GGBGG…)

Girl, she is at the end.

Could be either.

Car, because there is a pattern (GGBGG…)

Anything, it’s chance!

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Responses to Pictograph QuestionSummary New student boy or girl? How does Tom get to

school?

[3] Balancing Boy, cause it’s the only boy that goes by car.

Boy, it could make 14 of both in the class.

Train, because there is no one on the train today.

[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.

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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.

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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].

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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].

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• 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.

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Example 4: Expectation and Variation

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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.”

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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.”

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• 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.”

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• 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)

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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.”

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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

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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%].

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• 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.

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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.

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Interviews with 6-year-olds• Creating a pictograph - cards representing books

and children who had read them.

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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…

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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.

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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.

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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.

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• 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?”

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Interviews on Expectation and Variation

• Drawing 10 lollies from a container with 50% red - predicting, experimenting, representing.

• How many red? The same every time?

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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.

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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.

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• 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.

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• 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.

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Pictures to show the results of 40 draws of 10 lollies

• Drawing lollies (Level 2)

• Variation without proportional reasoning (Level 3)

• Unconventional expectation and variation (Level 4)

• Conventional expectation and variation (Level 5)

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Interviews about the weatherSome 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 17C.

• What does this tell us about the temperature in Hobart?

• Do you think all the days had a maximum of 17C? Why or why not?

• What do you think the maximum temperature in Hobart might be for 6 different days in the year?

______, ______, ______, ______, ______, ______

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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.

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Drawing by a 6-year-old

• Describing variation in the weather with an average yearly temperature of 17oC.

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• 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.

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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.

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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.

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• 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

Desktop.scfea, maximum averages.

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• 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.

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• 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.

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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 non-random – generators, data collection from populations, ‘drawing’ outcomes).

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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.

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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.

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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?

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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]

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• 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.

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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.

jane watson university of tasmania

Documents

statistical information

statistical terminology

student understanding

summariesinterpret data

analysing data

multiple elements

various contexts

links assessment tasks