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=EDUC 509A - Group 2 - April and Janette= = = =Inquiry Approach and Constructivism Model =

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This lesson incorporates an inquiry approach by allowing students to use manipulatives to visualize and verify that an infinite series equals 1. Instead of simply telling students that this is the case and they should just memorize this fact, students are allowed to explore and see for themselves in order to validate this fact. The lesson builds upon a constructivist model by allowing students to build on their prior knowledge, work together and reflect on their findings for deeper understanding. Students are encouraged to ask questions and work together to share thoughts and understandings. This lesson emphasizes a larger concept through the use of manipulatives and allows the teacher to interact with the students and inquire about their thoughts and ideas. =====

= = =__Stage 1 - Desired Results__ = =__Lesson Goals__ =
 * Students will be introduced to the concept of a series; both finite series and infinite series.
 * Concepts of convergent and divergent series will be introduced.
 * The symbol of summation will be reinforced and used with the series.
 * Students will see why the infinite series 1/2 + 1/4 + 1/8 + 1/16 + ...= 1
 * ===== Students will learn that the equation for this series is [[image:SeriesEquation.JPG width="305" height="33"]] =====

 a. Derive and apply the formulas for the sums of finite arithmetic series and finite and infinite geometric series
=__Lesson Understandings__=
 * Students will create a physical example of a partial infinite series, which will lead to the understanding of a basic series.
 * Students will understand that a series is the summation of a sequence
 * Students will explore the equation form of a series
 * Students will understand that some series are finite, while others are infinite
 * Students will understand that some infinite series converge, while others do not and are divergent (brief introduction only)
 * Students will be introduced to the vocabulary of sequence, series, finite series, infinite series, convergent series and divergent series

<span style="font-family: Tahoma,Geneva,sans-serif;">What happens with **divergent series**?
=__Stage 2 - Assessment Evidence__=

=__Lesson Assessment__=

<span style="font-family: Tahoma,Geneva,sans-serif;">Lesson Assessment will occur in two portions:

 * =====<span style="font-family: Tahoma,Geneva,sans-serif;">Primarily the assessments will be made during the Inquiry portion of the Lesson as the students are discovering the series and determining the equation and result of the series in their groups. =====
 * =====<span style="font-family: Tahoma,Geneva,sans-serif; line-height: 20px;">There will also be an assessment where the students write in their own words a summary of what they learned about series, how the equation was derived for the series in class, and why the result was one for that series. =====

=__Stage 3 - Learning Plan__=

__Overview of Lesson__

 * 1) =====<span style="font-family: Tahoma,Geneva,sans-serif;">The lesson will begin with a brief review of sequences, series and summation. =====
 * 2) =====<span style="font-family: Tahoma,Geneva,sans-serif;">Students will be placed in small work groups 3 to 4 students per group. Each student in the group will start with a single square piece of paper.=====
 * 3) =====The stude<span style="font-family: Tahoma,Geneva,sans-serif;">nts will cut the paper in half so that they now have two equal size rectangles =====
 * 4) =====<span style="font-family: Tahoma,Geneva,sans-serif;">Understanding that the original square is equal to 1, they now have 2 halves or 1/2 + 1/2 =1. The <span style="font-family: Tahoma,Geneva,sans-serif;">students make note of this on a separate sheet of paper as the teacher checks on each group and their findings. =====
 * 5) =====<span style="font-family: Tahoma,Geneva,sans-serif;">Students now take one of the rectangles and set it aside and cut the other rectangle (one half of the original square) and cut it in half. =====
 * 6) =====S<span style="font-family: Tahoma,Geneva,sans-serif;">tudents should understand that each of these new smaller rectangles equals 1/4 of the original whole square. =====
 * 7) =====<span style="font-family: Tahoma,Geneva,sans-serif;">Students verify that 1/2 + 1/4 + 1/4 = 1 because together these pieces still make one whole square. =====
 * 8) =====<span style="font-family: Tahoma,Geneva,sans-serif;">Students continue by taking one of the 1/4 pieces and cutting it in half again to create two more congruent pieces which now represent 1/8 each of the original square. =====
 * 9) =====<span style="font-family: Tahoma,Geneva,sans-serif;">Students again verify that 1/2 + 1/4 + 1/8 + 1/8 = 1. =====
 * 10) <span style="font-family: Tahoma,Geneva,sans-serif;">At this point the students have a //finite sequence// and this can be noted on their work papers.
 * 11) The teacher helps the students put these findings into notes and encourages them to continue until they are satisfied and see a pattern occurring. Students should reflect on this series and understand why it is that if they keep cutting the remaining pieces in half all the pieces together will still equal 1.
 * 12) <span style="font-family: Tahoma,Geneva,sans-serif;">The students would see that this pattern would always lead to 1 no matter how many times the paper was cut; i.e. if the series continues infinitely it will always equal 1 and never more.
 * 13) <span style="font-family: Tahoma,Geneva,sans-serif;">Teacher assists students and reviews their notes with them and encourages each group to develop their findings into an equation or a formula that represents this repeating series i.e 1/2n
 * 14) =====<span style="font-family: Tahoma,Geneva,sans-serif;">This result is discussed in groups and the teacher provides additional examples for the students to work with to further understand the nature of series, such as convergence and divergence.=====

Constructivist Learning Theory
Constructivism is largely influenced by the work of Jean Piaget and is based upon the notion that knowledge cannot be disseminated merely from one person to another, but rather it is built as a result of experiences, or constructed by the individual learner (Brahier, 2009). Piaget, a Swiss biologist, is well-known for his work in the 1920's on cognitive development in children and adolescents. Several of Piaget's basic assumptions about cognitive development include important components of modern constructivism, namely, that children are active and motivated learners and children //construct rather than absorb// knowledge (Ormrod, 2011). Piaget noted that children construct their understandings from their cumulative experiences; though he did not coin the term himself, his theory is known as constructivist theory or constructivism. While still heavily debated among educational psychologists as to its proper place in classrooms, constructivism offers important distinctions from traditional classroom settings.
 * Content is presented as a larger context, focusing on big concepts rather that basic skills.
 * Questioning ideas and "thinking outside the box" are encouraged, rather than sticking to a strict plan.
 * Activities focused around manipulatives and direct discovery through primary sources of data rather than textbooks only.
 * Students are acknowledges as independent thinkers with prior knowledge and experience, rather than "blank slates".
 * Teachers are interactive and mediate learning rather than dictate and lecturing information.
 * Teachers seek to understand each students unique point of view and logic, rather than looking for one specific answer.
 * Assessment is based largely on evidence other than traditional exams and is usually occurs during activities rather than after.
 * Students are encouraged to work as groups rather than individually.



Links
[|Additional Series Examples]

[|Curriculum Framework]

[|Overview of Constructivism]

[|Constructivism History]

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