In the "Whole Lot of 'Shakin' Going On" challenge, students have opportunities to employ all five process standards. They will:

• problem solve as they decide on a strategy for solving a problem where the answer is not immediately obvious.

• reason as they think about and justify their solution as the only possible correct answer.

• communicate about their thinking by discussing it with classmates, parents, and teachers, as well as by posting their written solution on the message board, and communicating via the message board with other students from around the world.

• make connections between mathematics and a real world event – ice cream combinations.

• use representations (drawings, charts, words, equations, manipulatives, etc.) to provide a record of their efforts to understand the mathematics of this challenge and make their understanding available to others.

About the mathematical content in this challenge:
The "Whole Lot of 'Shakin' Going On" challenge is an example of discrete mathematics – a combinatorics problem where systematic listing and counting are used to determine how many solutions exist for a situation. Discrete mathematics falls into the content standards of Patterns, Functions and Algebra, because, while children may initially approach such problems randomly, they will eventually notice patterns that lead to more efficient solutions. It also falls into the content standard Data Analysis, Statistics and Probability, because, in order to solve the problem satisfactorily, data is generated and recorded.

About the challenge:
Why in the world would teachers or parents want to present discrete mathematics activities with young children when there are so many other mathematical topics of importance in today's world? Isn't this just a "frill" or unnecessary "extra"? Absolutely not! Even though we probably did not learn such mathematical topics in school, there are important reasons why children should be exposed to the kind of mathematical thinking that these kinds of problems call forth. According to Joe Rosenstein, in Discrete Mathematics in the Schools: An Opportunity to Revitalize School Mathematics (NCTM Yearbook), discrete mathematics problems:

• lend themselves to mathematical exploration, enabling all students of all ages, grade levels and ability levels, to experience the excitement of doing mathematics.

• are easily presented in terms of the real- world problems that they are used to solve.

• require few mathematical prerequisites so that all students can use discrete mathematics as an entry point into mathematics.

• they have a strong visual component so that they are particularly accessible to learners who have difficulty with the more symbolic aspects of school mathematics.

• they are ideal for group learning situations, for stimulating mathematical discussion, and for alternative assessment strategies.

It is important to encourage children not to stop when they have found a few combinations, but to figure out a way to prove that they have found them all. Younger children may want to have paper cutouts or drawings of the problem posted in the room so they are free to return to them and add other combinations when they think of them.

After an appropriate amount of time has been spent of seeking a solution, a whole class session should be scheduled to prove that all combinations have been found. (It will be hard for some children to understand that a shake that is half vanilla and half chocolate is the same combination as a shake that is half chocolate and half vanilla – in combinatory problems the order doesn't matter.) Recently a first grade class used the following diagram to prove they found all flavor combinations:

Extensions of the Challenge:
Math/Science/Health Connections:
With the increase in childhood obesity in this country, Aunty thinks that a milkshake should be just an occasional treat. A good mathematics/ science/health/connection activity for older children would be to compare the nutritional information regarding milkshakes at local fast food restaurants. For instance, a small Burger King chocolate milkshake contains 410 calories and has 13 grams of fat. A small Dairy Queen chocolate milkshake contains 560 calories and has 15 grams of fat. How many calories could you save by ordering a Burger King milkshake? How many could you save if you split a small milkshake with a friend or sibling?

Using nutritional guides, figure out which of the children's favorite recipes for summer cool treats are most nutritious and contain less calories. (An easy way to reduce calories and fat is to use 2% or skim milk and ice milk instead of ice cream in recipes.)