# Summary of Discussion surrounding "A Geoscience versus a Mathematical Approach to Problem Solving"

Small groups of geologists and mathematicians were given a map of Lake Erie with depths shown in different colors. Each group was asked to discuss ways to go about calculating the volume of the lake. Not surprisingly, geologists and mathematicians thought about this problem in different ways; nonetheless, there were similarities in the approaches. A summary of the varied approaches, similarities and differences and sentiments about working in a multidisciplinary group follows:

## Geoscience approaches

- Visualize surface of the lake as one rectangle and approximate average depth
- Visualize lake as one prism (trapezoidal cross-section)
- Approximate surface area as rectangle, assume equal depth
- Divide lake into three basins (use shape of basin to calculate volume of water in lake)
- Use statistical sampling with grid and representative depths
- Make grid and randomly place on map
- Cut map, weigh each piece and multiply by depth
- Use photometer
- Make clay model in bathtub, add water and scale
- Make nail model (depths represented by length of nails)

## Math approaches

- Reimann sums
- Statistical sampling
- Visualize surface of the lake as one rectangle and approximate average depth
- Make wooden model and fill with water

## Similarities between Math and Geoscience approaches

- divide into easy-to-manage pieces and sum
- Simplify to easier/known problem, in doing this we make assumptions and it is important to point this out to students (and ourselves)
- Many felt a need for context because that influences the method of determination
- Some made analogue models (more often geologists but both disciplines used this method)
- We all bring more than our discipline to the problem -- personal experience and personality play a role in our problem solving abilities
- Both disciplines used low-order modeling

## Differences in approaches

- Less variation from mathematicians -- tend to be more uniform in ways of approaching problem
- Geoscientists tend to use common words (i.e. grids), mathematicians tend to use mathematical vocabulary (i.e., statistical sampling)
- Many more geoscientists called upon physical models whereas mathematicians used equations (concrete vs. abstract)
- Mathematicians "lumped" approaches -- viewed many of the geoscience approaches as "the same" where as geologists thought they were "different"
- Geoscientists looked at the physical structure of the lake
- Way the problem is viewed -- "basin" (earth feature) vs. "lake" (problem/numbers)

## How did we benefit from group work?

- Came up with more ideas and were able to refine those ideas
- Exchange of disciplinary vocabulary
- More fun!
- Combining ideas led to a better approach because we could examine assumptions and determine the "goodness" of an answer
- Found out that what we viewed as different ideas are really the same
- Could hash out technical differences
- Could compare solutions and techniques