Initial Publication Date: May 4, 2004

7-26 AM Posters

Rate of Change

Strategies that Work:

Use familiar concepts (e.g. speed, human growth) to start
Emphasize proper units/magnitude
Move from verbal to data table to plot to equation
Move from linear to non-linear examples
Involve students in active simulations (coin-flipping, dice rolling, etc.)
Compare plots of distance, speed, acceleration

Examples in a Geoscience context:

Determine average plate motion by measuring distance between two Hawaiian islands of known age. Determine average for several pairs of islands and discuss whether plate motion is a constant rate.
Determine the rate of retreat of St. Anthony Falls based on location in historical record and location today on topographic maps. Assuming retreat began at the confluence of the Minnesota and Mississippi Rivers and that the glacier was present at this location, and a constant rate of retreat, calculate when the glacier began to retreat from the Minneapolis area.
Illustrate half-lives and exponential decay by giving each student a coin. All flip coins, if a student has "heads" they are not "decayed" and can flip again. Count students with "heads," flip again, and repeat. Data can be plotted, etc.

Remote Sensing Examples

Matrices and Eigen vectors:

Question: Why? Is it worth the math?

Strategy that works: Principle components (also teaches covariance)

See diagram on poster

Challenge: Getting math poor/phobic/atrophied students to appreciate it

Systems of equations:

Strategy that works: Endmember analysis/classification

(Spectral unmixing)
See diagram/example on poster

Challenges: undersolving and oversolving

Simultaneous equations

Vectors

Jordan, Hutchings, Hutchings

Focus: Using vectors for navigation and mapping

Strategies that work:

Don't be abstract - Go outdoors and take measurements of distance between real objects and establish direction
Use a magnetic compass (quality will determine accuracy and precision) and your pace or tape measure.
Use protractor and scale to draw a representation of real situation. (Add points to map as a function of precision - users)
Triangulate, triangulate, triangulate OR trilaterate, trilaterate, trilaterate volcano, your house (or any other spatially distributed feature)

Graphical Literacy

Moe M., Carol D., Janet A., Cathy S.

Strategies that work:

Decide

FINISH

Approximation

Kohn, Kroeger, Macdonald, Shell-Gellasch

Reasonableness

Order of magnitude
Realistic (range of possible values)

Purpose

Level of accuracy needed
Level of accuracy possible

Implementation

Conditions
Audience
Methods

Evaluation and refinement

Example: Approximate rainfall within a drainage basin.

Purpose: determining water budget for a given area
Given: rain gauge data
Implementation

Contour maps
Statistical averaging
Weighted average
Histogram (1 and 2 dimensional)

Evaluation: consider "outliers"

Data Analysis/ Intro. Statistics

Carolyn Dobler, Chris Gellash, Patty Crews, Marvin Bennett and Rick Ford

Critical steps/strategies in teaching this concept:

What is the problem / research question?
Which data will be useful in answering the question?

Issues related to data collection:

Sampling strategies / constraints
Sample size
Representative samples
Samples over time (time series)
Data quality - accuracy and precision;
Quality assurance and control
Nature of the phenomena being studied (geoscience context)

Data description

Averages, mean, mode, median, variance, etc.
Weighted mean - important in many geoscience contexts
Correlations / regression
Graphs

Drawing conclusions from our data (statistical inference)

Estimation, hypothesis testing, prediction

Units and Significant Digits

Check errors in work using significant digits

Motivate significant digits from error analysis

Convert units

Dimensional analysis

Strategies that work

Order of magnitude reasoning to check work

Rivers on earth do not flow at 500mph
Hailstones do not grow to the size of basketballs

Example showing how errors propagate (significant digits)

Q=Av
A= 2.3 +/- 10^-2 m²
v =1.5 +/- 10^-2 m/s
(10^-2 represents uncertainty)
Q= (2.3 +/- 10^-2)(1.5 +/- 10^-2) m3 / s = 3.45 +/- .015 +/- .023 +/- 10^-4 m3 / s
=3.4 (must throw out the 5)

70 miles/hour * 1 hour/3600 seconds * 1.6 km/1 mile * 1000m/1 km = 31m/s

See diagram on chart
Identify important parameters (g, t, z)
Identify units ( L / T2, T, L)
Laws of the universe don't care about units:
Construct nondimensional group: z / (g * t2) = constant
z=cgt2

Example in a geoscience context:

(a) Use dimensional analysis to obtain the drag law for a spherical particle dropped through a viscous fluid (water or magma)
(b) Conduct physical experiments to determine the viscosity coefficient
Repeat the hydraulic example using order of magnitude reasoning:
Q= Av: A is order 1, v is order 1, so Q must be order 1
i.e. Q cannot be greater than 10 or less than 1

Optimization in Geology

Co, Bruce, Will

Math

Definition:: Min/max functions, typically misfits between data and a model
Issues:: Constraints
Global vs. local minima
Difficulties:: "Word Problem"
No such thing as perfect data

Geology

Definition:: Finding the best earth model to explain a given set of data
Issues:: Realism of results
Sufficiency of model
Difficulties:: Lack of perfect answer
Equivalence of solutions

Strategies that work:

Concrete examples
Diagrams, photos, road cuts, graphic solutions
Guidance and comfort

Example: Overdetermined 3-pt. Problem

Data: 5 points, (x_i, y_i, z_i)

Model: ax_i + by_i + gz_i = d_i
- Minimize: (from i = 1 to 5) ==> S [z_i - ((d_i - ax_i - by_i) / g)]z
NB: system is unsolvable (singular matriatic) if five points are Colinear

NB2: Measurements errors add uncertainty

Examples [to what?]

Problem: Are containments leaking from an underground storage tank?

Data:: Nature of phenomenon - soil and water
Variables to measure: hydrocarbon, lead
Data collection issues:
Data quality:: Data collection - field collection protocol; Certified lab quality
Data description:: Plot data on map
Inspect for variability; Draw contours
Data Conclusions:: If values are unusal, are they significant?
If values are not unusual, how confident are you that the site is clean?

Graphical Literacy

Strategies that work:

Decide upon what type of graphs you want students to work with:
(bar and scatterplot vs. [absolute?] lines - drawn on poster
Be clear about how students should work with graphs
- simply read information
- find slopes and calculate percent change
- fit a line to a "bunch of points" (i.e. data)
- interpret and predict (requires use of symbols)
- differentiate between different types of graphs and define behavior of system (is it linear or exponential? Power function made linear on log graph)
- understand scatter and error

Examples in Geoscience:

Exponential:: Number of aftershocks / time
Population Growth Energy consumption
Atmospheric carbon dioxide concentrations
Draw down in well
Land subsidence
Radiometric decay
Linear:: Travel-time of seismic waves (approximately)
Topographic slope (gradient)
Water-table slope
Miscellaneous: Bowen's reaction series
Sea-level curves
Climatic data/variation
Pressure-temperature curves

Helping Students Visualize Better in 3-D

Vince Schielack, Jim Sochacki, Denyse Lemaire

Our Goals:

Increase awareness of angle measurement;
Help students understand how arcs of longitude decrease in length as one moves toward the poles;
Have students ocmputer the chagnes in arcs of latitude as one nears the pole;
Have students use Geometer's Sketch Pad

Problem:

Students have a hard time visualizing volumes
When studying geographic coordinates, students often get confused about angles
Rotating meridians adds insult to injury

See diagram depicting earth with tropics, arctic circles, equator marked

3D Visualization of Earth

(Coordinates of Rotation)

Strategies: (Diagrams included for each step on original)

Axis of Earth
Meridian and coordinates on surface of Earth
If a point moves in miles along the equator, how far would another point at a degrees of latitude move at the same time?

x / r = sin (90-a)

x = r cos (a)

Degrees Latitude: x/r Proportion => 90:0, 80:0.1736, 70:0.3420, 60:0.5, 50:0.6428, 40:0.7660, 30:0.8660, 20:0.9397, 10:0.9848, 0:1.0

Cycles of changes

Specific topics: e.g. tides, cycles of sea-level change

Strategies: (use tide as example)

Display measured tidal curves (60-day record)
Question: What are the tidal ranges (maximums, minimums, "average")? What is the periodicity of the tides?

Let's take a step further than just visual observation
Let's try to describe tides with mathematical functions

The math tool - simple: y = sin x

Includes graph of y = sin x
(we got the cycle!!)

F(T) = A sin ((t * T) + f)
A = tide magnitude; t = tide frequency; T = time - the variable; f = a phase angle

Apparently, nature is more complicated than just one sine curve
What do we do?
Add several sine curves together.

Exercise II: Let's Predict Tides (with 6 components)

Given: T= (2*p / t) - period of the component
Create sine curves (in excel) with

F1(T) =A1 sin (t1 T + f1)
F2 (T) = A2 sin (t1 T + f1) -- Note: only A changed
F3 (T) = A3 sin (t2 T + f1) -- Note: both A and t changed
F4 (T) = A4 sin (t2 T + f1) -- Note: only A changed
F5 (T) = A5 sin (t1 T + f2) -- Note: A and changed
F6 (T) = A6 sin (t3 T + f3) -- Note: A, t, and f all changed
Hint: same Lt

Examples in a geoscience context: Oceanography (intro)

Sedimentary Geology (sedimentary processes)