Transcript Ratio and Proportion Strategies

Data Analysis and
Statistical Reasoning
Today’s Agenda

Geometric and Spatial Reasoning: Review &
Sharing

What is Statistical Reasoning?

Statistical Reasoning Assessment (SRA)

Statistics and Graphical Reasoning

Data Analysis Problems
Geometric and Spatial Reasoning Review

Pass around and look at student baseline assessments
within your group.

Assess and sort according to students’ abilities with
geometric and spatial reasoning
◦ 0: no spatial reasoning skills exhibited
◦ 1: some evidence of spatial reasoning
◦ 2: fluent spatial reasoner.


Was there a correlation between students’ abilities with
planar figures and the 3D isometric problems?
Are your good geometric reasoners also good at
algebraic reasoning?
Geometric and Spatial Reasoning Review
What did you learn most from the
baseline and summative assessments?
 Describe at least two ways that you
helped your students with geometric and
spatial reasoning.
 Describe one classroom situation where
you saw a student exhibit growth in
thinking about proof.

What is Statistical
Reasoning?
Definition
Statistical reasoning may be defined as
the way people reason with statistical
ideas and make sense of statistical
information. This involves making
interpretations based on sets of data,
representations of data, or statistical
summaries of data.
(Garfield & Chance, 2000)
Mathematics vs. Statistics
Many people think of mathematics and statistics as the
same thing, and therefore confuse statistical reasoning with
mathematical reasoning.
Today’s leading statistical educators see these disciplines
and types of reasoning as quite distinct.
(Garfield & Gal, 1999)
Distinguishing Characteristics I
1. In statistics, data are viewed as
numbers with a context, which can
affect procedures and interpretation.
2. Statistics has “messy” data compared
to the precision of mathematical
reasoning and proof.
Distinguishing Characteristics II
3. The fundamental nature of statistics
problems is that they do not have a single
mathematical solution. Rather, realistic
statistics problems begin with a question
and culminate with an opinion based on
certain assumptions. The “correctness” of
the answer is subjective, depending on the
communities standards for techniques and
reasoning.
Example of Context
An economics teacher asks five
students how much cash they are
carrying in their pockets. The results
are:
$6.50 $7.10 $15.80 $7.30 $6.80
The teacher wishes to find the
average amount of cash that each of
the five students has. What is the
answer?
Example of Context – Part II
A physics teacher gives five students
a stopwatch and asks them to time
how long it takes for a ball to drop
from the roof of a tall building to the
ground. In seconds, the times were:
6.5
7.1
15.8 7.3
6.8
How should she average her students’
times to come up with the “average”
observation?
SRA
Joan Garfield developed the Statistical
Reasoning Assessment (SRA) as part of
the ChancePlus Project.
You will take the assessment before
discussing what it measures, to avoid any
bias in your answers!
SRA Discussion
Did you notice the built-in ambiguity with some of
the questions and answers?
Did you find any problems more difficult than
others?
Did you feel any were unfair?
What does the SRA measure?
Correct Reasoning Skills:
1. Correctly interprets probabilities
2. Understands how to select an appropriate average
3. Correctly computes probability
a. Understands probabilities as ratios
b. Uses combinatorial reasoning
4. Understands independence
5. Understands sampling variability
6. Distinguishes between correlation and causation
7. Correctly interprets two-way tables
8. Understands importance of large samples
What does the SRA measure?
Misconceptions:
1. Misconceptions involving averages
a. Averages are the most common number
b. Fails to take outliers into consideration when
computing the mean
c. Compares groups based on their averages
d. Confuses mean with median
2. Outcome orientation misconception
3. Good samples have to represent a high percentage of
the population
4. Law of small numbers
5. Representativeness misconception
6. Correlation implies causation
7. Equiprobability bias
8. Groups can only be compared if they are the same size
Compute your SRA “score”
Statistics and Graphical
Reasoning
Fuel Economy
NYC Weather
Oil Prices
Data Analysis
Setup
One of your clever and precocious students, Harvey,
challenges you to a game: “I toss a coin 20 times. If 4060% of all my tosses turn out heads, I win. Otherwise you
win.”
You are tempted to play, but wonder about the choice of 20
flips. On one hand, if you asked for 40 flips you have more
chances to win. But you also think that the more times the
coin is flipped, the more likely it is that about half of them
will be heads, making you lose. Should you agree to play,
but with fewer than 20 flips?
Investigate this game on your handout!
Baseline
Assessment
Baseline Assessment
Baseline Assessment