Initial Planning Reflection for - An Activity Sequence to identify a procedure to systematically identify elements or subsets in a finite set

Generalization

A procedure can be made to systematically identify elements or subsets in a finite set.

Possible Related Concepts

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What instructional theory and learning theory should be used to begin to facilitate student learning for these ideas?

I would want to begin with exploration so the students will learn through their own actions and reactions in a new situation.  If they explore new materials and new ideas with minimal guidance or expectation of specific accomplishments, the new experience should raise questions they cannot answer with their present ideas and patterns of reasoning.  Having made an effort that most likely will not be completely successful, they should be motivated to ask questions and begin to look for ideas that will lead them to self-regulation. Meanwhile, I will have collected enough assessment data to begin to know what they understand and where I might begin to facilitate their learning in the second phase: invention.

What activities could be used for the first activity with students for identification of a procedure to systematically identify elements or subsets in a finite set?

Review a resource file or a list of possible activities that would fit the concepts and generalization.

List of possible activities

1.  Give the class several easy combinatorial problems which use all subsets of three elements.  Examples:

1.  How many ice cream sundaes can one make with apple, banana, and cherry topping?

2.  Give each student or have each student make the 8 line code:

000
001
010
011
100
101
110
111

3.  In small groups have students compare the combinatorial problems 1. and 2. and the pattern given to them in 2.

4.  Ask the class to share their discoveries.  After sufficient discussion if students haven’t mentioned a relationship with A-apple, B-banana, and C-cherry and the code in the columns, then write apple above the first column, banana above the second, and cherry above the third.  Then have students, in their groups compare combinations of a,b,c and the three combinations of 0 and 1.  The goal is to have them match them 1-1.

5.  Have students solve the number combination for $1, $.01, $.05.  Repeat 1-4 as feel necessary.

6.  Suggest some problems or have students create problems to solve with the algorithm.  Have them list all subsets of two or four elements. 

7.  If appropriate, have students state the correspondence between n-digit binary numbers and subsets of sets on n elements.  If appropriate interpret as characteristic functions.

One could consider one through four as an exploration phase.  The students are working with some fairly concrete ideas and they gather information when they share their answers.  They may or may not have in their minds a pattern for both and if they do, they may not be aware they are mathematically the same.  They may be a bit puzzled by the apparent lack of order or by the indefinite order of some of the answers.

The purpose of the next task is to invite them to notice the one to one correspondence which is the basis for the algorithm.  Many students will notice this correspondence, but it can be explicitly stated for all to apply in other cases.  This stating of the correspondence, the “invention” stage.  Here the instructor is suggesting a new way of viewing the experiences in the exploration phase.  In essence the instructor is proposing a new structure which will account for the experiences of exploration. 

The third stage has been called applications, discovery, and expansion of new ways of using the invention.  This stage has been dissolved as a time to reinforce, refine apply, extend, and generalize the content of the invention.  It’s also good chance to assess student understanding.

Making a decision

Review what resources are needed for each activity and the preparation of students needed for each activity. Eliminate activities that wouldn't fit the availability of resources and the readiness of students. Then think about how each would or would not be good to use as the first activity choice.  When you have done that, compare the ideas below with yours, and if possible, with those of others.

Reflect on the positives and negatives for each and make a decision as to what you believe might be the best before continuing.

 

Dr. Robert Sweetland's Notes ©