Welcome to the March 18, 2015 JSE webinar:
The Median has a Balance Representation, too!
presented by
Larry Lesser & Amy Wagler
Mathematical Sciences Dept. faculty at The University of Texas at El Paso
and lead authors of the Nov. 2014 JSE paper
http://www.amstat.org/publications/jse/v21n1/lesser.pdf
“Finding a Happy Median: Another Balance Representation for Measures of Center”
Outline
Nov. 2014 JSE paper
Today’s webinar
1. Introduction
2. Balance Representations:
Mean & Median
3. Pilot Assessment of Intervention
4. Classroom Exploration of Model
5. Results of Assessment
6. Discussion
Appendix: In-class Protocol
Appendix: Pre-post Test Questions
References
• Brief motivation and
illustration of Model
• Some pedagogical
considerations
• Assessment of Model
• Q&A
Students’ procedural understanding does
not imply conceptual understanding
Groth and Bergner (2006) report that:
• on a NAEP item, only 4% of Grade 12 students
correctly chose (with satisfactory explanation)
the median as the measure of center to
summarize a particular data set
• only 3 of 45 pre-service ES teachers gave
descriptions of the median that addressed
both ordering as well as when the median is
not a value in the dataset
balance model of the median
(from Figure 1 of our paper)
• 4” diameter clothesline pulley wheel
• 6.5’ piece of twine tied into a loop,
marked at 2” intervals
• several 2” office (colored) binder clips
(Lynch used spring-loaded clothespin-type mini-clamps instead)
mean & median
balance models are similar
• Can be built inexpensively (< $10)
• Support prediction questions
• Show that particular measure of center
need not be a value in the dataset,
but must be between minimum and maximum
balance models differ
MEAN
• Find balance point by trial and
error
• Board weight usually matters:
our 48” stick will balance at 24
for {12,30,30}, but won’t balance
at 18 for {12,12,30}; solution:
have weights much heavier than
the board or use a virtual applet
MEDIAN
• Find (a) median
automatically
• Weight of loop won’t
matter
1-min demo video of median loop model
(storyboarded by Lesser; the below URLs are on p. 7 of our paper)
•
•
amstat.org/publications/jse/v22n3/pulley_loop_physical_model_of_median.html
youtube.com/watch?v=-dDbWTPL0vE
• Questions posed so video can be paused
while your students make predictions.
Q1: When I lift my index finger
so the string is free to move,
what will be at the bottom of the loop
for this set of 3 unequally-spaced weights?
1-min demo video
Q2: Now, what happens
when I lift my finger
so the string is free to move
for an even number of values?
Inspired by the O’Dell (2012) task sequence for the mean,
Tables 2 & 3 use a more specific
dataset sequence (price in $ of new-release DVDs)
DATASET
COMMENT
{20,22,24,26,28}
odd number of equally-spaced values
{10,22,24,26,28}
one value changed to being an outlier
{12,24,26,28,30}
add 2 to each value in previous dataset
{12,24,24,30,30}
dataset includes some repetition
{10,11,12,24,24,30,30}
dataset includes some repetition that affects median
{12,24,24,30}
even number of values with middle two equal
{12,24,30,35}
even number of values with middle two not equal,
so median not unique
Assessment (Fall 2013)
• Population Description
• Method: In-class Assessment and Pre-post test
• Instrument: multiple choice and open ended
questions
• Results
• Discussion and Extensions
Fall 2013 Research
• Setting: Southwestern US research institution
with predominantly Latino/a student
population
• Population: Pre-service middle school teachers
• Sample Size: 10 completed pre and post tests
Method
• Pretest about median given in prior class with
100% informed consent
• Students participated in median model activity for
approximately 45 minutes in subsequent class
period
• Median activity included in-class voting questions
• Completed posttest directly after activity
• Data deidentified and analyzed by statistics
graduate student researcher
Excerpt of In-Class Assessment Protocol
Dataset
#1
Assessment
Instructor Prompts
Based on the sample size of the
data set, how many clips will we
need?
[Place clips as described in the
above the first row of Table 3]
Student Engagement
Ask students questions to verify
that they understand how to
“read” the number loop.
Targeted Observations
* There is an odd
number of values,
equally spaced.
Ask a student to place or guide
your placement of the clips to
correspond to the observations
listed.
Students predict using
choices:
A) 24 (9 votes)
B) something else (1 vote)
* Because the loop is formed
from a number line, the
values are already sorted.
Why do you think it was 24?
Is 24 a particular measure of
center?
Discussion
Now, where should I put
the fulcrum to make this
“platform model”
Discussion
The value 24 is actually
the value of several
measures of center in
this case (mean, median,
and midrange) and so
more exploration is
needed to make a
conjecture.
The midpoint value 24 is
the only point on the stick where
it balances.
Are the values already sorted?
Why?
When I let go of the string so the
loop is free to move, what value
will come to rest at the bottom of
the loop?
The number 24 quickly
moves to the bottom of the
loop.
10-question pre/post test
• 1 created by the authors from scratch;
• 9 adapted from “Measures of Center” ARTIST
database (Garfield & Gal, 1999)
• 5 multiple choice
• 2 multiple choice followed by “Explain”
• 3 open-ended
Pretest Item Example
1) A test was given to 60 individuals. Half answered 80% of the
questions correctly, the other half of the class answered 90%
correctly. With regard to the mean and median of the class scores,
which of the following statements is correct?
a. mean > median b. mean = median c. mean < median.
Explain.
Pretest Item Example
10) NCAA wrestler weights are summarized in the table below.
However, when determining these weights it was discovered that the
scale topped out at 250 lbs. so that the exact weight of any player 250
lbs. or more could not be determined. The following is a sample of
weights from a particular team:
Which can we say about the median?
a. It cannot be determined
b. The median is 175
c. The median is 200
d. The median is 225
e. The median is 250
f. The median can be determined, but is
not one of the choices
Weight
175
200
225
250+
Frequency
4
8
15
9
Results
• Overall test of proportion correct was
conducted for all 7 multiple-choice questions
(Ha: µpre-µpost < 0; Z = -1.71; p = 0.0438,
permutation test p = 0.034)
• Follow-up analysis of individual items using
permutation tests
Multiple-choice Item Results
Item: Concept
Proportion
correct
Pre Post
Permut.
p-value
Adjusted
p-value
Proportion
correct for
in-class polls
2: Compares mean
and median for
symmetric data
3: Def. of median
0.1
0.1
0.234
1.000
0.900 (#1)
0.5
0.7
0.077
0.539
0.678 (#1-7)
4: Compares
mean/median for
skewed data
5: Find median
0.2
0.9
0.000
0.000
0.900 (#6)
0.6
0.7
0.290
1.000
0.800 (#5)
6: Find median
0.7
0.7
0.323
1.000
0.100 (#7)
7: Interpretation
0.3
0.6
0.031
0.217
1.000 (#2)
10: Find median
from table
0.3
0.5
0.074
0.518
0.678 (#1-7)
Open-ended Items
3 statistics graduate students independently
graded open-ended responses using this rubric:
1 = virtually no relevance
2 = some relevance
3 = substantively relevant
4 = completely relevant
Responses to Open-ended Items
Description of Question
Pre-test
mean Likert rating
(intraclass
correlation
coefficient)
Post-test
mean Likert rating
(intraclass
correlation
coefficient)
1: Provides an example of skewed count data 2.13 (0.101)
and asks for reason why the mean would be
inappropriate as a measure of center.
2.60 (0.194)
2: Asks for comparison of mean and median 1.70 (0.291)
for symmetric data, then requests
explanation
1.73 (0.253)
4: Asks for comparison of mean and median 1.47 (0.594)
for skewed data, then requests explanation
8: Asks for approximate value of median from 1.87 (0.147)
a dotplot
9: Asks for comparison of mean and median 1.60 (0.290)
for a dotplot
2.37 (0.408)
1.63 (0.607)
1.93 (0.187)
Interpretation of Results
• Biggest insight gain: how mean and median
are affected differently by outliers
• Median need not be unique
• May not transfer well without physical model
• Students gain a concrete basis to visualize the
data as ordered when finding the median
• Reinforces idea that mean and median are not
always an observed value
(De)limitations
• Preservice middle school teachers provide
“upper bound” benchmark
– If they struggle with median concepts,
middle school students will even more
• Sample size small due to only one section of
class available
• Results are preliminary, not definitive