Transcript Improving Teacher Knowledge in Geometry and Measurement

Journey in Developing
Statistical Reasoning
in Elementary and Middle School
Teacher Leaders
This material is based upon work
supported by the National
Science Foundation under Grant
No. 0314898. Any opinions,
findings and conclusions or
recommendations expressed in
this material are those of the
author and do not necessarily
reflect the views of the National
Science Foundation (NSF).
DeAnn Huinker & Janis Freckmann
Milwaukee Mathematics Partnership
University of Wisconsin–Milwaukee
[email protected], [email protected]
Association of Teachers of Mathematics (AMTE) Annual Meeting
February 5, 2009, Orlando, Florida
www.mmp.uwm.edu
Session Goals



Summarize the GAISE report Framework Model
and key concepts (ASA, 2005).
Review a year-long sequence of activities
used to strengthen statistical reasoning of
teacher leaders.
Report impact on teachers’ statistics and
probability content knowledge and knowledge
for teaching (University of Louisville).
Ponder and Discuss…
As you reflect on your work with preservice and
inservice teachers, what are some concepts and
ideas in statistical reasoning that challenge
their understanding?
Turn to the person next to you,
share and explain your responses.
Statements from the MET Report
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
Statistics is the science of data, and the daily display of
data by the media notwithstanding, most elementary
teachers have little or no experience in this vitally
important field.” (p. 23)
“Of all the mathematical topics now appearing in the
middle grades curricula, teachers are least prepared to
teach statistics and probability.” (p. 114)
“Statistics is now widely acknowledged to be an
extremely valuable set of tools for problem solving and
decision making. But, despite the production of
interesting statistics materials for the schools, it has been
hard to find room for the subject in curricula dominated
by preparation for calculus.” (p. 137)
www.cbmsweb.org/MET_Document/
Statements from Teacher Leaders


It was good to revisit some content that I had not used
since high school or college. I enjoyed analyzing the data
and deepening my understanding of probability. It is an
area that scares people a bit (myself included) and
therefore teachers may not teach it well because of the
lack of understanding.
Statistics, probability, and data analysis has always been
an area of weakness for me, but the content sessions have
really allowed me to expand my knowledge. They have
also given me the opportunity to see a continuum of
thinking across grade levels that I didn't have before.
GAISE
Guidelines for Assessment and
Instruction in Statistics Education
A Pre-K–12 Curriculum Framework
American Statistical
Association (2005)
www.amstat.org/education/gaise/
The Nature of Variability



A major objective of statistics education is to
help students develop statistical thinking.
Statistical thinking must deal with the
omnipresence of variability.
Statistical problem solving and decision
making depend on understanding, explaining,
and quantifying the variability in the data.
It is this focus on variability in data that sets
apart statistics from mathematics.
(GAISE report, p. 6)
The Role of Context
Statistics requires a different kind of thinking,
because data are not just numbers, they are
numbers with a context.
 In mathematics, context obscures structure.
In data analysis, context provides meaning.

(GAISE report, p. 7)
The GAISE Framework
I. Formulate Questions
II. Collect Data
III. Analyze Data
IV. Interpret Results
-----------------------------------Nature of Variability
Focus on Variability
Three Developmental Levels
Level A  Level B  Level C

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Articulates development in an orderly way.
Builds on previous concepts.
Increases in complexity.
Progression is based on experience, not age.
Collect Data
Level A
Level B
Level C
• Do not yet design • Beginning awareness • Students make
for differences
of design for
design for
differences
differences
• Census of
classroom
• Sample surveys;
• Sampling designs
begin to use random
with random
• Simple
selection
selection
experiment
• Comparative
• Experimental
experiment begin to
designs with
use random
randomization
allocation
District & Participants


Milwaukee Public Schools, 90,000 students
 127 Elementary Schools (K-5, K-8)
 17 Middle Schools
 58 High Schools, 11 Combined M/H Schools
Math Teacher Leaders (MTL)
 Grades K–7
 Grades 8–9
PD Sequence
Aug
Sept
Oct
Dec
Jan
Feb
Mar
Apr
June
Likely and Unlikely Events; Simple Experiments
Experimental and Theoretical Probability
Role of Questions in Statistical Investigations
Collecting Data: Sampling, Bias, Randomness
Analyzing Variability in Data
Interpreting Data: Measures of Central Tendency
Interpreting Data: Stem and Leaf Plots, Box Plots
Interpreting Data: Revisiting Box Plots
Fair and Unfair Games
www4.uwm.edu/Org/mmp/_resources/math_content.htm
Green Fields Golf
th
18
Hole
The math group went golfing as a way of
celebrating the 5th year of the grant. Which
team did better on this hole?
Team A Scores
4, 5, 5, 18
Team B Scores
6, 7, 8
Ritzy & Normal Counties
In Ritzy County, the average annual household income is
$100,000. In neighboring Normal County, the average
annual household income is $30,000. Sally thinks the
average income of the two counties is $65,000.
Do you agree with Sally or not?
If so, explain why; if not, explain why not.
---------------------------------------------------------------
What other information would you need in order
to calculate the average annual household income
in the two-county area?
Are You Typical?
In MPS there are: 4,793 teachers with 58,414
combined years of teaching experience
Mean
12.19 years
years
Median 10 years
Mode 5
Same Median, Different Mean
Nine teachers reported their years of experience as
follows: 7, 5, 5, 4, 6, 8, 7, 6, 6
Draw cubes to represent this data set.
What is the median? the mean?
Rearrange the cubes in your drawing to represent
possible data sets for the following. You may add or
remove cubes. Record each data set and its mean.
 Sample of 9 teachers with a median of 6 years
experience, but a mean less than 6.
 Sample of 9 teachers with a median of 6 years
experience, but a mean greater than 6.
Consolidate ideas
What changes occurred in the data set that allowed
you to keep the median but change the mean?

To lower the mean…

To increase the mean…
Big Ideas: Analyzing Data
When analyzing data, key features of the
data are measures of center, spread, and
shape.
 The question affects the choice of measure of
central tendency.
 The median is a more robust measure of
central tendency.
 The mean is more influenced by outliers.

Clarity of Concepts: “Mean”
Level A: Mean as an idea of fair share or
redistribution and leveling.
Level B: Mean as a balancing point.
Level C: Mean as an estimate from a sample
to make an inference about a population.
Procedures
Pretest: September 2007
20 items on Statistics and Probability
School Year: Monthly PD sessions
16 hours Grades K–7 Math Teacher Leaders
12 hours Grades 8–9 Math Teacher Leaders
About 200 Teacher Leaders
Posttest: May/June 2008
20 items on Statistics and Probability
Diagnostic Mathematics Assessments
for Middle School Teachers

Level 1. Declarative Knowledge
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Level 2. Conceptual Knowledge
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Level 3. Problem Solving and Reasoning

Level 4. Mathematical Knowledge for Teaching
University of Louisville, Center for Research in
Mathematics and Science Teacher Development
URL: louisville.edu/education/research/centers/crmstd
Instruments

Statistics and Probability Pretest & Posttest

Form A (v2.3), Form B (v5.3) (Reliability 0.90)
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20 items: 10 multiple choice, 10 open response
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Open response items score up to 3 points

Each Level I–IV has a possible score of 10

Statistics sub-score (20 points)
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Probability sub-score (20 points)
Test Results by Strand
30
25
20
15
Pretest
Posttest
10
5
0
K-7 MTL
8-9 MTL
Statistics
K-7 MTL
8-9 MTL
Probability
K-7 MTL
8-9 MTL
Total
K-7: n=62; Gr 8-9: n=32
Types of Knowledge
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Level 1. Declarative Knowledge
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Level 2. Conceptual Knowledge
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Level 3. Problem Solving and Reasoning
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Level 4. Mathematical Knowledge for Teaching
Test Results by Type of Knowledge
Type II. Conceptual Knowledge
Type I. Declarative Knowledge
9
9
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
1
0
0
Pretest
Posttest
K-7 MTL
8-9 MTL
Pretest
Posttest
7
Type IV. Mathematics Knowledge for
Teaching
Type III. Problem Solving and
Reasoning
7
6
6
5
5
4
4
3
3
2
2
1
1
0
0
Pretest
Posttest
K-7 MTL
8-9 MTL
Pretest
Posttest
Using the GAISE Report
How Can I Use the GAISE Report?
 As a quick reference for the components of the
framework.
 As a resource for explanations of each component in
practice—and as a series of examples by which we can
teach each component.
Focus Questions
 Considering the three levels, where would you place the
majority of the lessons that are taught in your school?

How might you use the GAISE Report for professional
development at your school?
MTL comment: Variability
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I have a better understanding of variability. I
realize that statistics is centered more on the main
term "variability" than it is on mean, median, and
mode. Usually when statistics is mentioned,
someone automatically talks about mean, median,
and mode. Now I understand that these "m” terms
are used to help describe variability.
MTL comments: Formulate Questions
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We had grade level discussions about formulating
good questions and the implications of teachers
always giving students the questions rather than
having students develop them, too.
I have seen some of the teachers applying this idea
to questioning in several different subject areas. For
example, the Kindergarten teachers were having
students ask questions that they thought they might be
able to answer from data they were planning to
graph.
MTL comment: Sampling
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I took back to my students the idea of collecting
data. We designed surveys about favorite foods. I
think that the learning was deeper because I
better understood the importance of formulating
a good question and identifying the sample
population.
Several students asked for my answer, so we had
to discuss whether or not I was included in “the
class.” We were able to talk about how this
impacts the results and accuracy of our data.
MTL comment: Role of Context
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The GAISE Report clearly outlined the major parts of
statistics. Many teachers create graphing projects
without knowing how to make them appropriate for
their grade levels.
I found that I could push my students to make some
statements that were more precise than saying "pink
has more than yellow.” They were able to connect
their representations back to their questions and the
context, and say "more children in our class like pink
than yellow.”
It's just a small step, but it's the kind of tweaking that
could make our students more successful with statistics.
MTL comments: Development

Because some of the content presented in these meetings
was challenging for me at the K–5 level, it helped me
reach beyond what I might have done in the past with my
students, giving me the enrichment I needed as a teacher
to thoroughly teach concepts at deeper levels.
Thank You!
MMP website
www.mmp.uwm.edu
PD Resources
www4.uwm.edu/Org/mmp/_resources/math_content.htm
DeAnn Huinker, [email protected]
Janis Freckmann, [email protected]