Process Standard: Connections

1
Process StandardConnections

• Instructional programs from prekindergarten
  through grade 12 should enable all students to
• recognize and use connections among mathematical
  ideas
• understand how mathematical ideas interconnect
  and build on one another to produce a coherent
  whole
• recognize and apply mathematics in contexts
  outside of mathematics.

2
Why are connections important?

• When students can see the connections across
  different mathematical content areas, they
  develop a view of mathematics as an integrated
  whole.

3
I have a dilemma. As you may know, I have a
faithful dog and a yard shaped like a right
triangle. When I go away for short periods of
time, I want Fido to guard the yard. Because I
dont want him to get loose, I want to put him on
a leash and secure the leash somewhere on the
lot. I want to use the shortest leash possible,
but wherever I secure the leash, I need to make
sure the dog can reach every corner of the lot.
Where should I secure the leash? The dog in
the yard problem(pp. 354-358)
4
The dog in the yard problem

• Now you have read the story carefully. Answer the
  following questions.
• What is the mathematical concept under
  development?
• What does the teacher do?
• What does the students do?
• What would be your own method of solving the
  problem?

5
Let us analyze the problem that we just did.

• The problem unites several mathematical concepts
  within a single investigation.
• It emphasizes the mathematical connections
  between a variety of topics.
• In the classroom, students should be encouraged
  to think of mathematics as the connected whole
  that is rather than see a course as a chapter of
  this and a chapter of that.
• As such, even the idea that one can take an
  algebra class one year and a geometry class the
  following year as if they are not inherently
  connected can be very misleading to the young
  learner.

6
What should be the teachers role in developing
connections?

• Problem selection is especially important because
  students are unlikely to learn to make
  connections unless they are working on problems
  or situations that have the potential for
  suggesting such linkages.
• Teachers need to take special initiatives to find
  such integrative problems when instructional
  materials focus largely on content areas and when
  curricular arrangements separate the study of
  content areas such as geometry, algebra, and
  statistics.
• One essential aspect of helping students make
  connections is establishing a classroom climate
  that encourages students to pursue mathematical
  ideas in addition to solving the problem at hand.

7
Teachers can foster independent thinking by
giving explanations and modeling why a procedure
works, not just how it works.Teachers can also
include assessment questions that ask students to
explain their thinking. For many students, if the
question why? is not on the test, it is not
important. As part of homework assignments, the
teacher can ask children to write about something
that they learned in class, for example, why 1 is
neither prime nor composite.(Flores, 2002)
8
Consider the following problem.(Phillips,
Gardella, Kelly, Stewart, 1991)

• Counting Paths from Oz The city of Oz is located
  at point A and that a person wants to travel to
  point B, moving only right along horizontal lines
  or up along vertical lines. How many paths are
  there to move from point A to point B?

B
A
9
Fibonacci Sequence 1, 1, 2, 3, 5, 8, 13, 21,
34, 55, 89, What is the process used in
generating successive terms?
10
The golden ratio, f

• It is also known as the divine proportion, golden
  mean, or golden section.
• Euclid ca. 300 BC defined the "extreme and mean
  ratios" on a line segment as following

11
Digital Cameras

• Water Fountain Activity
• Day of Week Formula
• EIU Sign Problems
• Other ideas

12
Mathematics Media

• Books
• Songs
• Video