Teach the Earth > Quantitative Skills > Community > Workshop 2002 > July 26 AM Poster Session Notes

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:

  1. 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.
  2. 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.</span>

  3. 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:

  1. Don't be abstract - Go outdoors and take measurements of distance between real objects and establish direction
  2. Use a magnetic compass (quality will determine accuracy and precision) and your pace or tape measure.
  3. Use protractor and scale to draw a representation of real situation. (Add points to map as a function of precision - users)
  4. 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:

  1. 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:

  1. What is the problem / research question?
  2. 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)
  3. Data description
    • Averages, mean, mode, median, variance, etc.
    • Weighted mean - important in many geoscience contexts
    • Correlations / regression
    • Graphs
  4. 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)

    Multiply by a "convenient one" to convert units
    • 70 miles/hour * 1 hour/3600 seconds * 1.6 km/1 mile * 1000m/1 km = 31m/s

    Using dimensional analysis to solve simple problems
    • 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:

  1. (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
  2. 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, (xi, yi, zi)

  • Model: axi + byi + gzi = di

    • Minimize: (from i = 1 to 5) ==> S [zi - ((di - axi - byi) / 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:
  • Sampling strategy:
  • Sample size - number and placement of wells
  • Sample design - multi-stage to determine ground water flow direction
    • Place additional wells in down gradient direction
    • Background sample
Data quality:
Data collection - field collection protocol
  • Chain of custody
Certified lab quality
Data description:
Plot data on map
Inspect for variability
  • Compare concentrations with background value to define presence of a plume
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:

  1. Decide upon what type of graphs you want students to work with:
    (bar and scatterplot vs. [absolute?] lines - drawn on poster
  2. 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</br> 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)

  1. Axis of Earth
  2. Meridian and coordinates on surface of Earth
  3. 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?

  • Lead to: Tidal ranges vary with time. Tides vary with more than one period. Therefore, tidal cycles are quite complicated.

    We need to be more quantitative

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

    Introducing math concepts and Exercises I

    • The math tool - simple: y = sin x</span>

      • Includes graph of y = sin x

      • (we got the cycle!!)

      Let's be a bit more "mathematical"

      • F(T) = A sin ((t * T) + f)

      • A = tide magnitude; t = tide frequency; T = time - the variable; f = a phase angle

      Let's take a step back and compare the above sine curve with "real" tides

      • 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

    How do all your curves look?

    Question: Similarities? Differences?

    Now, the big moment: Add all the F(t)'s together.

    Hint: make sure they are scaled the same!

    You just developed a predictive tool for Tampa Bay tides!

Examples in a geoscience context: Oceanography (intro)

Sedimentary Geology (sedimentary processes)